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FUNDAMENTALS OF HEAT EXCHANGER DESIGN

Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

FUNDAMENTALS OF HEAT EXCHANGER DESIGN

Ramesh K. Shah Rochester Institute of Technology, Rochester, New York Formerly at Delphi Harrison Thermal Systems, Lockport, New York

Dusˇan P. Sekulic´ University of Kentucky, Lexington, Kentucky

JOHN WILEY & SONS, INC.

1 This book is printed on acid-free paper. *

Copyright # 2003 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, e-mail: [email protected]. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best eﬀorts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication for x ¼ 0

0.0016 0.0027 0.0041 0.0051 0.0058 0.0065 0.0130 0.0169

0.0015 0.0026 0.0040 0.0050 0.0057 0.0064 0.0130 0.0169

0.0014 0.0024 0.0039 0.0049 0.0057 0.0063 0.0130 0.0168

0.005

0.0032 0.0053 0.0080 0.0099 0.0114 0.0127 0.0246 0.0311

0.0029 0.0050 0.0078 0.0097 0.0113 0.0125 0.0245 0.0310

0.0027 0.0048 0.0076 0.0095 0.0111 0.0124 0.0245 0.0310

0.010

"=" for x ¼ :

0.0047 0.0078 0.0119 0.0146 0.0168 0.0186 0.0350 0.0434

0.0042 0.0074 0.0115 0.0144 0.0166 0.0184 0.0350 0.0434

0.0039 0.0070 0.0112 0.0141 0.0163 0.0182 0.0349 0.0433

0.015

Source: 6¼0

" ¼ 1 "¼0 ¼ ð1 0:0445Þ0:7729 ¼ 0:7377 "

NTU = 6 C* = 1 λc = λh = 0.04

360°C

(ηohA)h/(ηohA)c = 1

25°C FIGURE E4.1

244

ADDITIONAL CONSIDERATIONS FOR THERMAL DESIGN OF RECUPERATORS

Thus, from the deﬁnition of eﬀectiveness, the outlet temperatures are Th;o ¼ Th;i "ðTh;i Tc;i Þ ¼ 3608C 0:7377ð360 25Þ8C ¼ 112:98C

Ans:

Tc;o ¼ Tc;i þ "ðTh;i Tc;i Þ ¼ 258C þ 0:7377ð360 25Þ8C ¼ 272:18C

Ans:

Discussion and Comments: For this particular problem, the reduction in exchanger eﬀectiveness is quite serious, 4.6%, with corresponding diﬀerences in the outlet temperatures. This results in a 4.6% reduction in heat transfer or a 4.6% increase in fuel consumption to make up for the eﬀect of longitudinal wall heat conduction. This example implies that for some high-eﬀectiveness crossﬂow exchangers, longitudinal wall heat conduction may be important and cannot be ignored. So always make a practice of considering the eﬀect of longitudinal conduction in heat exchangers having " 75% for single-pass units or for individual passes of a multiple-pass exchanger.

4.2

NONUNIFORM OVERALL HEAT TRANSFER COEFFICIENTS

In "-NTU, P-NTU, and MTD methods of exchanger heat transfer analysis, it is idealized that the overall heat transfer coeﬃcient U is constant and uniform throughout the exchanger and invariant with time. As discussed for Eq. (3.24), this U is dependent on the number of thermal resistances in series, and in particular, on heat transfer coeﬃcients on ﬂuid 1 and 2 sides. These individual heat transfer coeﬃcients may vary with ﬂow Reynolds number, heat transfer surface geometry, ﬂuid thermophysical properties, entrance length eﬀect due to developing thermal boundary layers, and other factors. In a viscous liquid exchanger, a tenfold variation in h is possible when the ﬂow pattern encompasses laminar, transition, and turbulent regions on one side. Thus, if the individual h values vary across the exchanger surface area, it is highly likely that U will not remain constant and uniform in the exchanger. Now we focus on the variation in U and how to take into account its eﬀect on exchanger performance considering local heat transfer coeﬃcients on each ﬂuid side varying slightly or signiﬁcantly due to two eﬀects: (1) changes in the ﬂuid properties or radiation as a result of a rise in or drop of ﬂuid temperatures, and (2) developing thermal boundary layers (referred to as the length eﬀect). In short, we relax assumption 8 postulated in Section 3.2.1 that the individual and overall heat transfer coeﬃcients are constant. The ﬁrst eﬀect, due to ﬂuid property variations (or radiation), consists of two components: (1) distortion of velocity and temperature proﬁles at a given free ﬂow cross section due to ﬂuid property variations (this eﬀect is usually taken into account by the property ratio method, discussed in Section 7.6); and (2) variations in the ﬂuid temperature along axial and transverse directions in the exchanger, depending on the exchanger ﬂow arrangement; this eﬀect is referred to as the temperature eﬀect. The resulting axial changes in the overall mean heat transfer coeﬃcient can be signiﬁcant; the variations in Ulocal could be nonlinear, depending on the type of ﬂuid. While both the temperature and thermal entry length eﬀects could be signiﬁcant in laminar ﬂows, the latter eﬀect is generally not signiﬁcant in turbulent ﬂow except for low-Prandtl-number ﬂuids. It should be mentioned that in general the local heat transfer coeﬃcient in a heat exchanger is also dependent on variables other than the temperature and length eﬀects, such as ﬂow maldistribution, fouling, and manufacturing imperfections. Similarly, the

NONUNIFORM OVERALL HEAT TRANSFER COEFFICIENTS

245

overall heat transfer coeﬃcient is dependent on heat transfer surface geometry, individual Nu (as a function of relevant parameters), thermal properties, fouling eﬀects, temperature variations, temperature diﬀerence variations, and so on. No information is available on the eﬀect of some of these parameters, and it is beyond the scope of this book to discuss the eﬀect of other parameters. In this section we concentrate on nonuniformities in U due to temperature and length eﬀects. To outline how to take temperature and length eﬀects into account, let us introduce speciﬁc deﬁnitions of local and mean overall heat transfer coeﬃcients. The local overall heat transfer coeﬃcient U ðx*1 ; x*2 ; T Þ, is deﬁned as follows in an exchanger at a local position ½x* ¼ x=ðDh Re PrÞ, subscripts 1 and 2 for ﬂuids 1 and 2] having surface area dA and local temperature diﬀerence ðTh Tc Þ ¼ T: U¼

dq dA T

ð4:24Þ

Traditionally, the mean overall heat transfer coeﬃcient Um ðTÞ is deﬁned as 1 1 1 þ Rw þ ¼ ðo hm AÞc Um A ðo hm AÞh

ð4:25Þ

Here fouling resistances and other resistances are not included for simplifying the discussion but can easily be included if desired in the same way as in Eq. (3.24). In Eq. (4.25), the hm ’s are the mean heat transfer coeﬃcients obtained from the experimental/empirical correlations, and hence represent the surface area average values. The experimental/ empirical correlations are generally constant ﬂuid property correlations, as explained in Section 7.5. If the temperature variations and subsequent ﬂuid property variations are not signiﬁcant in the exchanger, the reference temperature T in Um ðTÞ for ﬂuid properties is usually the arithmetic mean of inlet and outlet ﬂuid temperatures on each ﬂuid side for determining individual hm ’s; and in some cases, this reference temperature T is the logmean average temperature on one ﬂuid side, as discussed in Section 9.1, or an integralmean average temperature. If the ﬂuid property variations are signiﬁcant on one or both ﬂuid sides, the foregoing approach is not adequate. ^ A more rigorous approach is the area average U used in the deﬁnition of NTU [see the ﬁrst equality in Eq. (3.59)], deﬁned as follows: ^

U¼

1 A

ð Uðx; yÞ dA A

ð4:26Þ

This deﬁnition takes into account exactly both the temperature and length eﬀects for counterﬂow and parallelﬂow exchangers, regardless of the size of the eﬀects. However, there may not be possible to have a closed-form expression for Uðx; yÞ for integration. Also, no rigorous proof is available that Eq. (4.26) is exact for other exchanger ﬂow arrangements. When both the temperature and length eﬀects are not negligible, Eq. (4.24) needs to be integrated to obtain an overall U (which takes into account the temperature and length eﬀects) that can be used in conventional heat exchanger design. The most accurate approach is to integrate Eq. (4.24) numerically for a given problem. However, if we can come up with some reasonably accurate value of the overall U after approximately

246

ADDITIONAL CONSIDERATIONS FOR THERMAL DESIGN OF RECUPERATORS

integrating Eq. (4.24), it will allow us to use the conventional heat exchanger design methods with U replaced by U. Therefore, when either one or both of the temperature and length eﬀects are not negligible, we need to integrate Eq. (4.24) approximately as follows. Idealize local Uðx; y; TÞ ¼ Um ðTÞ f ðx; yÞ and U ðx*1 ; x*2 ; T Þ ¼ Um ðTÞ f ðx*1 ; x*2 Þ; here Um ðTÞ is a pure temperature function and f ðx; yÞ ¼ f ðx*1 ; x*2 Þ is a pure position function. Hence, Eq. (4.24) reduces to dq dA T

ð4:27Þ

f ðx*1 ; x*2 Þ dA

ð4:28Þ

Um ðTÞ f ðx*1 ; x*2 Þ ¼ and integrate it as follows: ð

dq ¼ Um ðTÞ T

ð

An overall heat transfer coeﬃcient U~ that takes the temperature eﬀect into account exactly for a counterﬂow exchanger is given by the ﬁrst equality of the following equation, obtained by Roetzel as reported by Shah and Sekulic´ (1998): 1 1 ¼ U~ ln TII ln TI

ð ln TII ln TI

dðln TÞ 1 UðTÞ ln TII ln TI

ð ln TII ln TI

dðln TÞ Um ðTÞ

ð4:29Þ

Note that UðTÞ ¼ Um ðTÞ in Eq. (4.29) depends only on local temperatures on each ﬂuid side and is evaluated using Eq. (4.25) locally. The approximate equality sign in Eq. (4.29) indicates that the counterﬂow temperature eﬀect is valid for any other exchanger ﬂow arrangement considering it as hypothetical counterﬂow, so that TI and TII are evaluated using Eq. (3.173). The overall heat transfer coeﬃcient Um ðTÞ on the left-hand side of Eq. (4.28) depends on the temperature only. Let us write the left-hand side of Eq. (4.28) by deﬁnition in the following form: ð

dq 1 ¼ Um ðTÞ T U~

ð

dq T

ð4:30Þ

Thus, this equation deﬁnes U~ which takes into account the temperature eﬀect only. The integral on the right-hand side of Eq. (4.30) is replaced by the deﬁnition of true mean temperature diﬀerence (MTD) as follows: ð

dq q ¼ T Tm

ð4:31Þ

Integration of the right-hand side of Eq. (4.28) yields the deﬁnition of a correction factor that takes into account the length eﬀect on the overall heat transfer coeﬃcient. ¼

1 A

ð A

f ðx*1 ; x*2 Þ dA

ð4:32Þ

where x*1 and x*2 are the dimensionless axial lengths for ﬂuids 1 and 2, as noted earlier.

NONUNIFORM OVERALL HEAT TRANSFER COEFFICIENTS

247

Finally, substituting the results from Eqs. (4.30)–(4.32) into Eq. (4.28) and rearranging yields q ¼ U~ A Tm ¼UA Tm

ð4:33Þ

Thus, an overall heat transfer coeﬃcient U which takes into account both the temperature ðU~ Þ and length eﬀects () is given by U¼

ð 1 Um ðTÞ f ðx; yÞ dA ¼ U~ A

ð4:34Þ

As noted before, f ðx; yÞ ¼ f ðx*1 ; x*2 Þ is a pure position function. We provide appropriate formulas for U~ and in the following sections. Note that since 1 (see Fig. 4.5), U U~ . Also from Eq. (4.30) we ﬁnd that U~ ¼ Um if the temperature eﬀect is not signiﬁcant (i.e., Um ðTÞ does not vary signiﬁcantly with T). It can be shown that for a counterﬂow exchanger, ^

U ¼ U~ ¼ U

ð4:35Þ

^

of UðTÞ or with U deﬁned by Eq. (4.26) and U~ deﬁned by Eq. (4.29) with Uðx; TÞ instead ^ Um ðTÞ. Hence, for evaluating U for a counterﬂow exchanger, one can use U , which is a function of only the area (ﬂow length), such as for laminar gas ﬂows, or U~ , which is a function of the temperature only such as for turbulent liquid ﬂows. These diﬀerent deﬁnitions of overall heat transfer coeﬃcients are summarized in Table 4.2. TABLE 4.2 Deﬁnitions of Local and Mean Overall Heat Transfer Coeﬃcients Symbol

Deﬁnition

Comments

U

dq U¼ dA T

Basic deﬁnition of the local overall heat transfer coeﬃcient.

Um

1 1 1 ¼ þ Rw þ Um A ðo hm AÞh ðo hm AÞc

Overall heat transfer coeﬃcient deﬁned using area average heat transfer coeﬃcients on both sides. Individual heat transfer coeﬃcients should be evaluated at respective reference temperatures (usually arithmetic mean of inlet and outlet ﬂuid temperatures on each ﬂuid side).

1 U˘ ¼ A

U˘

ð

Mean overall heat transfer coeﬃcient averaged over heat transfer surface area.

UðAÞ dA A

U~

U~ ¼ ðln T2 ln T1 Þ

U

U ¼ U~

ð ln T

2

ln T1

dðln TÞ UðTÞ

1

Mean overall heat transfer coeﬃcient that takes into account the temperature eﬀect only. Mean overall heat transfer coeﬃcient that takes into account the temperature and length eﬀects. The correction factor takes into account the entry length eﬀect.

248

ADDITIONAL CONSIDERATIONS FOR THERMAL DESIGN OF RECUPERATORS

Now we discuss methods that take temperature and length eﬀects into account, to arrive at U for the exchanger analysis.

4.2.1

Temperature Eﬀect

4.2.1.1 Counterﬂow Exchanger. Consider a single-pass counterﬂow exchanger in which U varies linearly with the temperature of either ﬂuid stream, U ¼ að1 þ bTÞ, where a and b are constants. In this case, the mean value of U T (where T is the temperature diﬀerence between the hot and cold ﬂuids) is as given by Colburn (1933): q U TII UII TI ¼ ðU TÞm ¼ I lnðUI TII =UII TI Þ A

ð4:36Þ

where UI and UII are overall heat transfer coeﬃcients determined at the exchanger hot and cold terminals, and TI and TII are given by Eq. (3.173). Note that ðU TÞm cannot be equal to U~ Tm . In this case, from Eq. (4.36), we get the exchanger heat transfer rate as q ¼ ðU TÞm A. Equation (4.36) represents a good approximation for very viscous liquids and partial condensation, and further discussion will be provided later with an example. An alternative approach to take into account the temperature eﬀect on U is to use the approximate method of integration by evaluating local U at speciﬁc points in the exchanger or perform a numerical analysis. Since such methods are more general, they are discussed next for all other exchanger ﬂow arrangements. 4.2.1.2 Other Exchangers. We ﬁrst illustrate the concept of how to include the eﬀect of variable UA for a counterﬂow exchanger and then extend it to all other ﬂow arrangements. To ﬁnd out whether or not variations in UA are signiﬁcant with temperature variations, ﬁrst evaluate UA at the two ends of a counterﬂow exchanger or a hypothetical counterﬂow for all other exchanger ﬂow arrangements. If it is determined that variations in UA are signiﬁcant, such as shown in Fig. 4.4, the average value U~ A can be determined by approximate integration of the variations in UA [i.e., Eq. (4.29) with the ﬁrst equality sign], by the three-point Simpson method as follows (Roetzel and Spang, 1993): 1 1 1 2 1 1 1 þ ¼ þ * A 6 UII A U~ A 6 UI A 3 U1=2

ð4:37Þ

where U *1=2 A ¼ U1=2 A

T1=2 T *1=2

ð4:38Þ

In Eq. (4.38), T1=2 and T *1=2 are deﬁned as T1=2 ¼ Th;1=2 Tc;1=2

and

* ¼ ðTI TII Þ1=2 T 1=2

ð4:39Þ

NONUNIFORM OVERALL HEAT TRANSFER COEFFICIENTS

249

FIGURE 4.4 Variable UA in a counterﬂow exchanger for the Simpson method.

where the subscripts I and II correspond to terminal points at the end sections, and the subscript 12 corresponds to a point in between deﬁned by the second equation, respectively. Here Th;1=2 and Tc;1=2 are computed through the procedure of Eqs. (4.43)–(4.45). Usually, uncertainty in the individual heat transfer coeﬃcient is high, so that the three-point approximation may be suﬃcient in most cases. Note that for simplicity, we have selected the third point as the middle point in the example above. The middle point is deﬁned in terms of TI and TII [deﬁned by Eq. (3.173)] to take the temperature eﬀect properly into account; it is not a physical middle point along the length of the exchanger. The step-by-step procedure that involves this approach is presented in Section 4.2.3.1. 4.2.2

Length Eﬀect

The heat transfer coeﬃcient can vary signiﬁcantly in the entrance region of laminar ﬂow. This eﬀect is negligible for turbulent ﬂows. Hence, we associate the length eﬀect to laminar ﬂow. For hydrodynamically developed and thermally developing ﬂow, the local and mean heat transfer coeﬃcients hx and hm for a circular tube or parallel plates are related as follows (Shah and London, 1978): hx ¼ 23 hm ðx*Þ1=3

ð4:40Þ

where x* ¼ x=ðDh Re PrÞ. Using this variation in h on one or both ﬂuid sides, counterﬂow and crossﬂow exchangers have been analyzed, and the correction factors are presented in Fig. 4.5 and Table 4.3 as a function of ’1 or ’2 , where ’1 ¼ o;2 hm;2 A2

1 þ 2Rw o;1 hm;1 A1

’2 ¼ Rw

1 1 þ ðo hm AÞ1 ðo hm AÞ2

1

ð4:41Þ

The value of is 0.89 for ’1 ¼ 1, (i.e., when the exchanger has the hot- and cold-side thermal resistances approximately balanced and Rw ¼ 0). Thus, when a variation in the heat transfer coeﬃcient due to the thermal entry length eﬀect is considered, U U~ or Um since U ¼ U~ from Eq. (4.34). This can be explained easily if one considers the thermal resistances connected in series for the problem. For example, consider a very simpliﬁed problem with the heat transfer coeﬃcient on each ﬂuid side of a counterﬂow exchanger

250

ADDITIONAL CONSIDERATIONS FOR THERMAL DESIGN OF RECUPERATORS

FIGURE 4.5 Length eﬀect correction factor for one and both laminar streams based on equations in Table 4.3 (From Roetzel, 1974).

varying from 80 to 40 W=m2 K from entrance to exit and A1 ¼ A2 , Rw ¼ 0, o;1 ¼ o;2 ¼ 1, and there is no temperature eﬀect. In this case, the arithmetic average hm;1 ¼ hm;2 ¼ 60 W=m2 K and Um ¼ 30 W=m2 K. However, at each end of this counterﬂow exchanger, U1 ¼ U2 ¼ 26:67 W=m2 K (since 1=U ¼ 1=80 þ 1=40). Hence U ¼ ðU1 þ U2 Þ=2 ¼ 26:67 W=m2 K. Thus U=Um ¼ 26:67=30 ¼ 0:89.

TABLE 4.3 Length Eﬀect Correction Factor When One or Both Streams Are in Laminar Flow for Various Exchanger Flow Arrangements One stream laminar counterﬂow, parallelﬂow, crossﬂow, 1–2n TEMA E Both streams laminar Counterﬂow

1 þ Rw a1

0:65 þ 0:23Rw ða1 þ a2 Þ 4:1 þ a1 =a2 þ a2 =a1 þ 3Rw ða1 þ a2 Þ þ 2R2w a1 a2

0:44 þ 0:23Rw ða1 þ a2 Þ 4:1 þ a1 =a2 þ a2 =a1 þ 3Rw ða1 þ a2 Þ þ 2R2w a1 a2 4 8 3 ¼ ð1 þ ’2 Þ 1 ’2 þ ’22 ln 1 þ 3 9 2’2

Parallelﬂow

¼1 ¼1

Crossﬂow

’1 ¼ a2

4 8 3 ¼ ð1 þ ’1 Þ 1 ’1 þ ’21 ln 1 þ 3 9 2’1

’2 ¼ 1

Rw 1=a1 þ 1=a2

a1 ¼ ðo hm AÞ1

a2 ¼ ðo hm AÞ2

NONUNIFORM OVERALL HEAT TRANSFER COEFFICIENTS

4.2.3

251

Combined Eﬀect

A speciﬁc step-by-step procedure is presented below to take into account the combined temperature and length eﬀects on U; the reader may refer to Shah and Sekulic´ (1998) for further details. First, we need to determine heat transfer coeﬃcients on each ﬂuid side by taking into account ﬂuid property variations due to two eﬀects: (1) distortion of velocity and temperature proﬁles at a given ﬂow cross section due to ﬂuid property variations, and (2) variations in ﬂuid temperature along the axial and transverse directions in the exchanger. In general, most correlations for the heat transfer coeﬃcient are derived experimentally at almost constant ﬂuid properties (because generally, small temperature diﬀerences are maintained during experiments) or are theoretically/numerically obtained for constant ﬂuid properties. When temperature diﬀerences between the ﬂuid and wall (heat transfer surface) are large, the ﬂuid properties will vary considerably across a given cross section (at a local x) and will distort both velocity and temperature proﬁles. In that case, the dilemma is whether to use the ﬂuid bulk temperature, wall temperature, or something in between for ﬂuid properties to determine h’s for constant property correlations. Unless a speciﬁc heat transfer correlation includes this eﬀect, it is commonly taken into account by a property ratio method using both ﬂuid bulk temperatures and wall temperature, as discussed in Section 7.6. Hence, it must be emphasized that the local heat transfer coeﬃcients at speciﬁc points needed in the Simpson method of integration must ﬁrst be corrected for the local velocity and temperature proﬁle distortions by the property ratio method and then used as local h values for the integration. The net eﬀect on U~ due to these two temperature eﬀects can be signiﬁcant, and U~ can be considerably higher or lower than Um at constant properties. The individual heat transfer coeﬃcients in the thermal entrance region could be generally high. However, in general it will have less impact on the overall heat transfer coeﬃcient. This is because when computing Ulocal by Eq. (4.25), with Um and hm ’s replaced by corresponding local values [see also Eq. (3.20) and the subsequent discussion], its impact will be diminished, due to the presence of the other thermal resistances in the series that are controlling (i.e., having a low hA value). It can also be seen from Fig. 4.5 that the reduction in Um due to the entry length eﬀect is at the most 11% (i.e., the minimum value of ¼ 0:89Þ. Usually, the thermal entry length eﬀect is signiﬁcant for laminar gas ﬂow in a heat exchanger. 4.2.3.1 Step-by-Step Procedure to Determine U. A step-by-step method to determine U~ A for an exchanger is presented below based on the original work of Roetzel and Spang (1993), later slightly modiﬁed by Shah and Sekulic´ (1998). In this method, not only the variations in individual h’s due to the temperature eﬀect are taken into account, but the speciﬁc heat cp is considered temperature dependent. 1. Hypothesize the given exchanger as a counterﬂow exchanger (if it is different from a counterﬂow exchanger), and determine individual heat transfer coefﬁcients and enthalpies at the inlet and outlet of the exchanger. Subsequently, compute the overall conductances UI A and UII A at inlet and outlet of the exchanger by using Eq. (3.24). 2. To consider the temperature-dependent speciﬁc heats, compute the speciﬁc enthalpies h of the Cmax ﬂuid (with a subscript j) at the third point (designated by 1/2 as a subscript, see Section 4.2.1.2) within the exchanger from the following equation

252

ADDITIONAL CONSIDERATIONS FOR THERMAL DESIGN OF RECUPERATORS

using the known values at each end: hj;1=2 ¼ hj;II þ hj;I hj;II

* TII T 1=2 TI TII

ð4:42Þ

* is given by where T 1=2 * ¼ ðTI TII Þ1=2 T 1=2

ð4:43Þ

Here TI ¼ ðTh Tc ÞI and TII ¼ ðTh Tc ÞII . If TI ¼ TII , (i.e., C* ¼ R1 ¼ 1), the rightmost bracketed term in Eq. (4.42) becomes 1/2. If the speciﬁc heat is constant, the enthalpies can be replaced by temperatures in Eq. (4.42). If the speciﬁc heat does not vary signiﬁcantly, Eq. (4.42) could also be used for the Cmin ﬂuid. However, when it varies signiﬁcantly as in a cryogenic heat exchanger, the third point calculated for the Cmax and Cmin ﬂuid by Eq. (4.42) will not be close enough in the exchanger (Shah and Sekulic´, 1998). In that case, compute the third point for the Cmin ﬂuid by the energy balance as follows: ½m_ ðhi h1=2 ÞCmax ¼ ½m_ ðh1=2 ho ÞCmin

ð4:44Þ

Subsequently, using the equation of state or tabular/graphical results, determine the temperature Th;1=2 and Tc;1=2 corresponding to hh;1=2 and hc;1=2 . Then T1=2 ¼ Th;1=2 Tc;1=2

ð4:45Þ

3. For a counterﬂow exchanger, the heat transfer coefﬁcient hj;1=2 on each ﬂuid side at the third point is calculated based on the temperatures Tj;1=2 determined in the preceding step. For other exchangers, compute hj;1=2 at the following corrected reference (Roetzel and Spang, 1993): 3 1F Th;1=2;corr ¼ Th;1=2 ðTh;1=2 Tc;1=2 Þ 2=3 2 1 þ Rh

ð4:46Þ

3 1F Tc;1=2;corr ¼ Tc;1=2 þ ðTh;1=2 Tc;1=2 Þ 2=3 2 1 þ Rc

ð4:47Þ

In Eqs. (4.46) and (4.47), F is the log-mean temperature diﬀerence correction factor and Rh ¼ Ch =Cc or Rc ¼ Cc =Ch . The temperatures Th;1=2;corr and Tc;1=2;corr are used only for the evaluation of ﬂuid properties to compute hh;1=2 and hc;1=2 . The foregoing correction to the reference temperature Tj;1=2 ð j ¼ h or cÞ results in the cold temperature being increased and the hot temperature being decreased. Calculate the overall conductance at the third point by 1 1 1 þ Rw þ ¼ U1=2 A o;h hh;1=2 Ah o;c hc;1=2 Ac Note that f and o can be determined accurately at local temperatures.

ð4:48Þ

253

NONUNIFORM OVERALL HEAT TRANSFER COEFFICIENTS

4. Calculate the apparent overall heat transfer coefﬁcient at this third point using Eq. (4.38): T1=2 * A ¼ U1=2 A U1=2 ð4:49Þ * T 1=2 5. Find the mean overall conductance for the exchanger (taking into account the temperature dependency of the heat transfer coefﬁcient and heat capacities) from the equation 1 1 1 2 1 1 1 þ ð4:50Þ ¼ þ * A 6 UII A U~ A 6 UI A 3 U1=2 6. Finally, the true mean heat transfer coefﬁcient U that also takes into account the laminar ﬂow entry length effect is given by UA ¼ U~ A

ð4:51Þ

where the entry length eﬀect factor 1 is given in Fig. 4.5 and Table 4.3. Example 4.2 In a liquid-to-steam two-ﬂuid heat exchanger, the controlling thermal resistance ﬂuid side is the liquid side. Let’s assume that the temperature of the steam stays almost constant throughout the exchanger (Tsteam ¼ 1088C) while the liquid changes its temperature from 26.78C to 93.38C. The heat transfer coeﬃcient on the steam side is uniform and constant over the heat transfer surface (12,200 W/m2 K), while on the liquid side, its magnitude changes linearly between 122 W/m2 K (at the cold end) and 415 W/m2 K (at the hot end). Determine the heat transfer surface area if the following additional ="; at large values of NTUo ( 9), increasing C*r from 1 to 10 decreases "=". Bahnke and Howard’s results show that Kays and London’s approximation (1998) " ¼ "

ð5:124Þ

is a very good engineering approximation for NTUo > 10 and < 0:1. Longitudinal heat conduction can have a serious impact on the regenerator eﬀectiveness or NTU for an ultra high-eﬀectiveness regenerator. For example, a Stirling engine regenerator may require 350 ideal NTU to get 200 usable NTU due to longitudinal heat conduction. As a result, such regenerators may require a stack of high thermal conductivity (copper or aluminum) perforated plates (Venkatarathnam, 1996) or wire screens, alternating with low thermal conductivity spacers made up of plastic, stainless steel,

352

THERMAL DESIGN THEORY FOR REGENERATORS

and so on. Such a design would signiﬁcantly reduce longitudinal heat conduction or the stack conduction. No detailed temperature distributions for ﬂuids and wall were obtained by either Lambertson (1958) or Bahnke and Howard (1964). Mondt (1964) obtained these temperature distributions by solving the diﬀerential equations numerically for some values of the associated dimensionless groups. Illustrative results are shown in Figs. 5.9–5.11. In Fig. 5.9, the matrix wall temperatures Tw* at x ¼ 0, x ¼ L=2, and x ¼ L are shown as functions of a dimensionless time for the ¼ 0 case. Also shown are the hot- and coldgas inlet and outlet temperatures. Experimental points shown for T *c;o are in good agreement with the theoretical predictions. The wall temperatures are linear with time except for the sections of hot- and cold-ﬂuid inlets. In Fig. 5.10, hot-gas and matrix temperatures are shown as a function of the ﬂow length X* for h ¼ 0; Ph =2, and Ph . Here, again, these temperature distributions are linear except for the regenerator ends. In Fig. 5.11, the matrix wall temperatures are shown at switching time ¼ Ph and Pc . The reduction in matrix wall temperature gradients at X * ¼ 0 due to longitudinal heat conduction is evident. Note that the time average T *c;o is reduced, which in turn indicates that the exchanger eﬀectiveness " is reduced due to longitudinal wall heat conduction, as expected. Skiepko (1988) presented three-dimensional temperature charts demonstrating how the longitudinal matrix heat conduction aﬀects the matrix as well as the gas temperature distributions, shown as dependent on coordinate x and time . Example 5.5 Determine the reduction in the regenerator eﬀectiveness of Example 5.3 due to longitudinal wall heat conduction given that the thermal conductivity of the

T *h,i

T *w,i

1.0 at x = 0 T *c,o T *w,m T* 0.5

at x = L/2 Ph

Pc

T *h,o T *w,o at x = L 0.0 0.0

0.5 _ hAτ/Cr

1.0

T *c,j

FIGURE 5.9 Cyclic temperature ﬂuctuations in the matrix at the entrance, midway, and exit of a regenerator. (From Mondt, 1964.)

INFLUENCE OF LONGITUDINAL WALL HEAT CONDUCTION

353

1.0

T *h T *w Ph T* 0.5

Ph/2

Hot fluid Matrix 0.0 0.0

τh = 0

0.5 X*

1.0

FIGURE 5.10 Fluid and matrix wall temperature excursion during hot-gas ﬂow period. (From Mondt, 1964.)

1.0

1/C *r T *h T* 0.5

Solid

T *w,m

Fluid T *c 0.0 0.0

0.5 X*

1.0

FIGURE 5.11 Balanced regenerator temperature distributions at switching instants. (From Mondt, 1964.)

ceramic matrix is 0.69 W/m K and its porosity is 0.7. Use all other pertinent information from Example 5.3. SOLUTION Problem 2 1

Long (Inﬁnite) Concentric Cylinders

r1

A1 r1 ¼ A2 r2

r2

q12 ¼

A1 ðT 41 T 42 Þ ð1="1 Þ þ ½ð1 "2 Þ="2 ðr1 =r2 Þ

F12 ¼ 1

Concentric Spheres A1 r21 ¼ A2 r22

r1

q12 ¼

A1 ðT 41 T 42 Þ ð1="1 Þ þ ½ð1 "2 Þ="2 ðr1 =r2 Þ2

F12 ¼ 1

r2

Small Convex Object in a Large Cavity A1 0 A2

A1, T1, ε1 •

q12 ¼ A1 "1 ðT 41 T 42 Þ

F12 ¼ 1 •

A2, T2, ε2

Source: w 1 g

ð7:173Þ

Since the convection heat transfer equation is in terms of the temperature diﬀerence ðTw Tg Þ, the radiation heat transfer coeﬃcient hrad can be presented similarly using Eq. (7.172): hrad ¼

"g Tg4 g Tw4 qrad "w

¼ AðTg Tw Þ 1 ð1 "w Þð1 g Þ Tg Tw

ð7:174Þ

In this equation, Tw and Tg must be given in K or 8R. Note that hconv for forced convection is generally not a strong function of the temperatures Tw and Tg , but hrad is a strong function of Tw and Tg , as shown in Eq. (7.174). In heat exchangers, the combined convection and radiation eﬀects are then taken care of approximately by considering the convection and radiation heat transfer phenomena in parallel. Hence, hcombined ¼ hconv þ hrad

ð7:175Þ

The rest of the heat exchanger analysis for a combined convection and radiation problem is performed the same as before by replacing h (or hconv ) with hcombined , assuming that hcombined is deﬁned using jTw Tg j ¼ jTw Tm j. Now we consider the determination of "g and g of Eq. (7.174) from the experimental results/correlations of Hottel and Saroﬁm (1967), and some of them reproduced by Incropera and DeWitt (2002), among others. The gas emissivity "g has been correlated in terms of the temperature Tg , the total pressure p of the gas, the partial pressure pg of the gas species (such as water vapor, carbon dioxide, etc.), and the radius L of an equivalent hemispherical gas mass. More precisely, Le , the mean beam length, is the required radius of a gas hemisphere such that it radiates a heat ﬂux to the center of its base equal to the average ﬂux radiated to the area of interest by the actual volume of gas (Siegel and Howell, 2002). In most exhaust gases, water vapor and carbon dioxide are the most important components from the radiation eﬀect viewpoint, and hence we only consider them here. For other gas components, refer to Hottel and Saroﬁm (1967). The water vapor emissivity "H2 O is presented in Fig. 7.35 as a function of the gas temperature Tg and pH2 O Le . Here pH2 O is the partial pressure of water vapor in the gas mixture at a total pressure of 1 atm; and Le is the mean beam length to take into account

INFLUENCE OF SUPERIMPOSED RADIATION

541

0.8 0.6 20 10 5 3 2

0.4 0.3 0.2 1 Emissivity, ε H O 2

0.6 0.4

0.1 0.08

0.2

0.06 0.1

0.04 0.06 0.04

0.03 0.02

0.02 0.015 0.01 0.007

0.01 0.008

pH OL e = 0.005 ft • atm 2 0.006 300 600 900 1200 1500 Gas temperature, Tg (K)

2100

1800

FIGURE 7.35 Emissivity of water vapor in a mixture with nonradiating gases at 1 atm total pressure and of hemispherical shape. (From Hottel, 1954.) (1 ft atm ¼ 0.305 m atm).

pH OLe = 2

1.8

0 – 0.05 ft • atm 0.25 0.55 1.0 2.5 5.0 10.0

Pressure correction, CH

2O

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

0

0.2

0.4 (pH

+ 2O

0.6 0.8 p)/ 2 (atm)

1.0

1.2

FIGURE 7.36 Correction factor for obtaining water vapor emissivity at pressures other than 1 atm; here, "H2 O; p6¼1 atm ¼ CH2 O "H2 O; p¼1 atm . (From Hottel, 1954.) (1 ft atm ¼ 0.305 m atm).

542

SURFACE BASIC HEAT TRANSFER AND FLOW FRICTION CHARACTERISTICS

TABLE 7.15 Mean Beam Lengths Le for Various Gas Geometries Geometry

Characteristic Length

Le

Sphere (radiation to surface) Inﬁnite circular cylinder (radiation to curved surface) Semi-inﬁnite circular cylinder (radiation to base) Circular cylinder of equal height and diameter (radiation to entire surface) Inﬁnite parallel planes (radiation to planes) Cube (radiation to any surface) Arbitrary shape of volume V (radiation to surface of area A)

Diameter D Diameter D Diameter D Diameter D

0:65D 0:95D 0:65D 0:60D

Spacing between planes L 1:8L Side L 0:66L Volume to area ratio V=A 3:6V=A

Source: ¼ 0:0017, and actual exchanger eﬀectiveness is "actual ¼ 0:8328 0:0017 ¼ 0:8311 The heat transfer rate q is then q ¼ "ðTg;i Ta;i ÞCmin ¼ 0:8311 ð900 200Þ8C 1863 W=K ¼ 1083:8 103 W The outlet temperatures are then Tg;o ¼ Tg;i

q 1083:8 103 W ¼ 9008C ¼ 318:38C ¼ 591:5 K Cg 1863 W=K

Ta;o ¼ Ta;i þ

q 1083:8 103 W ¼ 2008C þ ¼ 705:08C ¼ 978:2 K Ca 2146 W=K

Since these outlet temperatures are diﬀerent from those assumed for the initial determination of the ﬂuid properties, two more iterations were carried out with ﬂuid properties evaluated at the new average temperatures. The values of C*, NTU, ", Tg;o and Ta;o were 0.857, 7.082, 0.8382, 314.48C and 701.98C, respectively, and after the third iteration were 0.857, 7.079, 0.8381, 314.58C and 701.88C respectively. Pressure Drops. We use Eq. (9.20) to compute the pressure drop on each ﬂuid side. The densities are evaluated using the perfect gas equation of state:

Gas Air

Ti ðKÞ

To (K)

i (kg/m3 )

o (kg/m3 )

m (kg/m3 )

1173 473

591.5 978.2

0.4751 1.4726

0.9424 0.7123

0.6318 0.9602

Note that we have also considered the outlet pressures as 160 kPa and 200 kPa for gas and air, respectively, since the pressure drop across the core is usually small and hence is neglected in the ﬁrst iteration. The mean density in the last column is the harmonic mean value from Eq. (9.18). Now let us determine Kc and Ke . Oﬀset strip ﬁns are used on gas and air sides. In such ﬁn geometries, because of the frequent boundary layer interruptions, the ﬂow is well mixed and is treated as having the Reynolds number very large (Re ! 1). The aspect ratio of the rectangular passage, height/width ¼ 2:49=ð1=0:615 0:15Þ ¼ 1:15. Since Re ¼ 1 curves for parallel plate and square passage geometries of Fig. 6.3 are identical, we could determine Kc and Ke from either geometry for a ¼ 0:36 as Kc ¼ 0:36

Ke ¼ 0:42

616

HEAT EXCHANGER DESIGN PROCEDURES

Before we compute the pressure drop, we need to correct the values of the isothermal friction factors by the method of Section 7.6.1 to take into account the temperaturedependent properties. A review of Eq. (9.11) indicates that we need to calculate the ﬂuid bulk mean temperatures and the wall temperature. The mean temperatures on the gas and air sides, based on the latest outlet temperatures, are ð900 þ 318:3Þ8C ¼ 609:28C ¼ 882:4 K 2 ð200 þ 705:0Þ8C ¼ ¼ 452:58C ¼ 725:7 K 2

Tg;m ¼ Ta;m

The thermal resistances on the gas and air sides are Rg ¼

1 ¼ 0:3698 104 K=W ðo hAÞh

Ra ¼

1 ¼ 0:3911 104 K=W ðo hAÞa

Rg ¼ 0:946 Ra If we neglect the wall resistance, q¼

Tg;m Tw Tw Ta;m ¼ Rg Ra

so that Tw ¼

Tg;m þ ðRg =Ra ÞTa;m 609:28C þ 0:946 452:58C ¼ ¼ 533:08C ¼ 806:2 K 1 þ ðRg =Ra Þ 1 þ 0:946

Since the gas is being cooled, using Eq. (9.11) and the exponent m ¼ 0:81 from Table 7.12, m Tw 806:2 K 0:81 ¼ 0:0669 ¼ 0:0622 f ¼ fcp Tm 882:4 K Since the air is being heated, using Eq. (9.11) and the exponent m ¼ 1:00 from Table 7.12, m Tw 806:2 K 1:00 ¼ 0:0534 ¼ 0:0593 f ¼ fcp Tm 725:7 K Now let us calculate the pressure drops using Eq. (9.20). ð15:342 kg=m2 sÞ2 0:4751 kg=m3 2 pg ¼ 1 ð1 0:361 þ 0:36Þ þ 2 2 1 0:4751 kg=m3 0:9424 kg=m3 0:0622 0:3 m 0:4751 kg=m3 0:4751 kg=m3 2 þ ð1 0:361 0:42Þ ð0:00154=4Þ m 0:6318 kg=m3 0:9424 kg=m3 2 2 ð7:0484 kg=m sÞ 1:3827 kg=m3 2 pa ¼ ð1 0:437 þ 0:33Þ þ 2 1 2 1 1:3827 kg=m3 0:8194 kg=m3 0:0683 0:6 m 1:3827 kg=m3 1:3827 kg=m3 2 ð1 0:437 þ 0:31Þ þ ð0:002383=4Þ m 1:0290 kg=m3 0:9424 kg=m3

PLATE-FIN HEAT EXCHANGERS

617

¼ 247:71 Pa ð1:2297 0:9917 þ 36:4465 0:2267Þ ¼ 247:71 Pa 36:4578 ¼ 9031 Pa ¼ 9:031 kPa pa ¼

18:365 kg=m2 sÞ2 1:4726 kg=m3 2 ð1 0:363 þ 0:36Þ þ 2 1 2 1 1:4726 kg=m3 0:7123 kg=m3 0:0593 0:3 m 1:4726 kg=m3 1:4726 kg=m3 2 ð1 0:363 0:42Þ þ ð0:00154=4Þ m 0:9602 kg=m3 0:7123 kg=m3

¼ 114:51 Pa ð1:2282 þ 2:1348 þ 70:8660 0:9267Þ ¼ 114:51 Pa 73:3023 ¼ 8394 Pa ¼ 8:394 kPa With these values of the pressure drops, the outlet pressures were recomputed, and with the corresponding values of the new outlet densities, the pressure drops were recalculated. The gas and airside pressure drops after the second iterations were 9.571 and 8.776 kPa, and after the third iterations 9.050 and 8.757 kPa, respectively. Discussion and Comments: As shown above, the calculation procedure for rating of a crossﬂow exchanger is straightforward and demonstrates how all information presented in preceding chapters is integrated to obtain the performance of the exchanger. Note that this calculation assumes the validity of a number of assumptions. For example, the ﬁn eﬃciency calculations were performed idealizing perfect brazing between the plates and ﬁns. A manufactured heat exchanger would inevitably be characterized with a performance more or less diﬀerent than predicted, depending on a designer’s ability to incorporate relaxation of some of the assumptions into this procedure.

9.2.2

Sizing Problem

The sizing problem is more diﬃcult. Many early decisions to choose the construction type and basic geometries on each ﬂuid side are based on experience (including rules of thumb and engineering judgments), operating conditions, maintenance, manufacturing capability, and the expected life of the exchanger. Some of these issues were mentioned in Chapter 2. We discuss the selection of exchanger construction type, ﬂow arrangement, surface geometries, and so on, in Chapter 10. With those inputs, the sizing problem then reduces to the determination of the core or exchanger dimensions for the speciﬁed heat transfer and pressure drop performance. One could, of course, reduce this problem to the rating problem by tentatively specifying the dimensions, then calculate the performance for comparison with the speciﬁed performance. This type of search for a solution is usually performed in the case of shell-and-tube exchangers and regenerators where one needs to take care of leakage and bypass ﬂows in a very complex manner. However, leakage and bypass ﬂows are not signiﬁcant for plate-ﬁn and tube-ﬁn exchangers. The solution method can be made more straightforward, with fast convergence for these exchangers by reforming the surface characteristics input to include j=f vs. Re for surfaces on each ﬂuid side, in addition to the separate j and f versus Re characteristics. This coupling of heat transfer and ﬂow friction is now made in the derivation of the core mass velocity equation that has been proposed by Kays and London (1998). Once the

618

HEAT EXCHANGER DESIGN PROCEDURES

core mass velocity is determined, the solution to the sizing problem is carried out iteratively in a manner similar to the rating problem discussed in Section 9.2.1. 9.2.2.1 Core Mass Velocity Equation. The coupling of heat duty and pressure drops is done by the core mass velocity equation as follows. From the required heat duty (and hence the exchanger eﬀectiveness ") and known heat capacity rates on each ﬂuid side (known C*), the overall NTU is determined for the selected exchanger ﬂow arrangement. The overall conductance as a ﬁrst approximation is given by 1 1 1 þ UA ðo hAÞh ðo hAÞc

ð9:21Þ

Here we have neglected the wall and fouling thermal resistances. The overall NTU is related to individual side ntuh and ntuc as follows [see Eq. (3.67)]: 1 1 1 C* 1 þ þ ¼ ¼ NTU ntuh ðCh =Cmin Þ ntuc ðCc =Cmin Þ ntug ntua

ð9:22Þ

where the second equality is for Cc ¼ Cmin . From the known NTU, we need to determine ntuh and ntuc from this equation either from past experience or by guessing. If both ﬂuids are gases or both ﬂuids are liquid, one could consider that the design is ‘‘balanced’’ (i.e., the thermal resistances are distributed approximately equally on the hot and cold sides). In that case, Ch Cc , and ntuh ntuc 2NTU

ð9:23Þ

Alternatively, if we have liquid on one side and gas on the other side, consider 10% thermal resistance on the liquid side: 1 1 ¼ 0:10 UA ðo hAÞliq

ð9:24Þ

Substituting Eq. (9.24) into Eq. (9.21) with Cc ¼ Cgas ¼ Cmin and Ch ¼ Cliq , we can determine ntugas and with its subsequent substitution in Eq. (9.22), we get ntugas ¼ 1:11 NTU

ntuliq ¼ 10C* NTU

ð9:25Þ

The ntu on each ﬂuid side is related to the Colburn factor j as follows by using Eq. (7.31) and (7.33): ntu ¼

o hA h A A L ¼ o ¼ o j Pr2=3 ¼ o j Pr2=3 m_ cp Gcp Ao Ao rh

ð9:26Þ

The pressure drop on each ﬂuid side is given by Eq. (9.20). Substituting L=rh from Eq. (9.26) into Eq. (9.20) and simplifying, G reduces to

PLATE-FIN HEAT EXCHANGERS

f ntu 2=3 1 1 1 1 þ ð1 2 þ Kc Þ Pr þ2 j o m o i i 1 1=2 ð1 2 Ke Þ o

619

G ¼ ð2gc pÞ1=2

ð9:27Þ

Equation (9.27) is a more generalized core mass velocity equation than that provided by Kays and London (1998), who considered only the ﬁrst term in brackets in Eq. (9.27). Since the contribution of the last three terms in the brackets in Eq. (9.27) is generally very small, they can be neglected in light of other approximations already mentioned. In this case, Eq. (9.27) reduces to G¼

2gc o p ð1=Þm Pr2=3 ntu

1=2 j f

ð9:28Þ

Equation (9.28) is referred to as the core mass velocity equation. The feature that makes this equation so useful is that the ratio j=f is a relatively weak function of the Reynolds number for most extended surfaces (see Fig. 2-41 of Kays and London, 1998, and Fig. E9.2). Thus, one can readily estimate a fairly accurate value of j/f in the operating range of Re. Also, for a ‘‘good design,’’ the ﬁn geometry is chosen such that o is in the range 70 to 90% and higher. Hence, o 80% may be assumed for the ﬁrst approximation in Eq (9.28), unless a better value is known from the past experience. All other information in Eq. (9.28) is known or evaluated from the problem speciﬁcation. Thus, the ﬁrst approximate value of G can be computed from Eq. (9.28). As a result, the iterative solution to the sizing problem converges relatively fast with this estimated value of G. We will use Eq. (9.28) for the ﬁrst iteration of a sizing problem, as described next for single-pass counterﬂow and crossﬂow exchangers. 9.2.2.2 Sizing of a Single-Pass Counterﬂow Exchanger. Now we outline a detailed procedure for arriving at core dimensions for a counterﬂow exchanger for speciﬁed heat transfer and pressure drop. In a single-pass counterﬂow heat exchanger of any construction, if the core dimensions on one side are ﬁxed, the core dimensions for the other side (except for the passage height) are also ﬁxed. Therefore, the design problem for this case is solved for the side that has more stringent pressure drop speciﬁcation. This method is also applicable to the C* 0 exchanger, such as a gas-to-liquid or phase-changing ﬂuid exchanger. In this case, the thermal resistance is primarily on the gas side and the pressure drop is also more critical on the gas side. As a result, the core dimensions obtained are based on the gas-side p and ntugas . The dimensions on the other side are then chosen such that the calculated pressure drop is within the speciﬁed p (i.e., pcalculated pspecified ). Thus either for counterﬂow or for a C* ¼ 0 exchanger, the core dimensions are calculated for the side having the most stringent p. Following is a step-by-step procedure for the solution. 1. To compute the ﬂuid mean temperature and the ﬂuid thermophysical properties on each ﬂuid side, determine the ﬂuid outlet temperatures, for the speciﬁed heat duty, from the following equation considering ﬂuid 1 as the hot ﬂuid. q ¼ ðm_ cp Þ1 ðT1;i T1;o Þ ¼ ðm_ cp Þ2 ðT2;o T2;i Þ

ð9:29Þ

620

HEAT EXCHANGER DESIGN PROCEDURES

2.

3.

4.

5. 6.

7.

If the exchanger eﬀectiveness is speciﬁed, use Eqs. (9.9) and (9.10) to compute outlet temperatures. For the ﬁrst iteration, estimate the cp values to determine the outlet temperatures from Eq. (9.29). Subsequently, determine the mean temperatures on both ﬂuid sides using the procedure discussed in Section 9.1.2 or from Table 9.1. With these mean temperatures, determine the cp ’s, and iterate one more time for the outlet temperatures if warranted. Subsequently, determine , cp , k, Pr, and on each ﬂuid side. Calculate C* and " (if q is given), and determine NTU from the "-NTU expression, with tabular or graphical results for the ﬂow arrangement selected [in this case, use Eq. (3.86) for counterﬂow]. The inﬂuence of longitudinal heat conduction, if any, is ignored in the ﬁrst iteration since we don’t yet know the exchanger size. Determine ntu on each ﬂuid side by the approximations discussed with Eqs. (9.23) and (9.25) unless it can be estimated differently (i.e., instead of 50 : 50% or 90 : 10% thermal resistance distribution) from past experience. For the surfaces selected on each ﬂuid side, plot j=f vs. Re curve from the given surface characteristics, and obtain an approximate mean value of j=f over the complete Reynolds number range; an accurate mean value of j=f is not necessary since we are making a number of approximations to get the ﬁrst estimate of G. If ﬁns are employed, assume that o ¼ 0:80 unless a better value can be estimated based on the experience. Evaluate G from Eq. (9.28) on each ﬂuid side using the information from steps 1 through 4 and speciﬁed values of p. Calculate Reynolds number Re, and determine j and f on each ﬂuid side from the given design data for each surface. The design data may be in the form of graphs, curve ﬁt to tabulated data or an empirical generalized equation. Compute h, f , and o using Eqs. (9.13)–(9.15). For the ﬁrst iteration, determine U1 on the ﬂuid 1 side from the following equation derived from Eq. (3.24): 1 1 1 = = ¼ þ þ 1 2 þ 1 2 U1 ðo hÞ1 ðo hf Þ1 ðo hf Þ2 ðo hÞ2

ð9:30Þ

where 1 =2 ¼ A1 =A2 , ¼ A=V, and V is the exchanger total volume, and subscripts 1 and 2 denote the ﬂuid 1 and 2 sides. For a plate-ﬁn exchanger, ’s are related to ’s by Eqs. (8.96) and (8.97) with usually provided with the surface basic characteristics (see, e.g., Fig. E9.1B) 1 ¼

b1 1 b1 þ b2 þ 2w

2 ¼

b2 2 b1 þ b2 þ 2w

ð9:31Þ

Note that the wall thermal resistance in Eq. (9.30) is ignored in the ﬁrst iteration since we do not yet know the size of the exchanger (i.e., Aw =A1 is unknown). In second and subsequent iterations, compute U1 from 1 1 1 A A =A A =A ¼ þ þ w 1þ 1 2þ 1 2 U1 ðo hÞ1 ðo hf Þ1 kw Aw ðo hf Þ2 ðo hÞ2

ð9:32Þ

where the necessary area ratios A1 =A2 and A1 =Aw are determined from the geometry calculated in the preceding iteration.

PLATE-FIN HEAT EXCHANGERS

621

8. Now calculate the core dimensions. In the ﬁrst iteration, use NTU as computed in step 2. For subsequent iterations, calculate the longitudinal conduction parameter . With known ", C*, and , determine the correct value of NTU using Eq. (4.15) or (4.20) . Determine A1 from NTU using U1 from step 7 and known Cmin : NTU Cmin U1

ð9:33Þ

A2 A ¼ 2A A1 1 1 1

ð9:34Þ

A1 ¼ and hence A2 ¼

The free-ﬂow area Ao from known m_ and G is given by Ao;1 ¼

m_ G 1

Ao;2 ¼

m_ G 2

ð9:35Þ

Ao;1 1

Afr;2 ¼

Ao;2 2

ð9:36Þ

so that Afr;1 ¼

where 1 and 2 are generally speciﬁed for the surface or can be computed from Eqs. (8.94) and (8.95) as follows using given geometrical properties: 1 ¼

b1 1 Dh;1 =4 1 Dh;1 ¼ b1 þ b2 þ 2w 4

2 ¼

b2 2 Dh;2 =4 2 Dh;2 ¼ b1 þ b2 þ 2w 4

ð9:37Þ

where the term after the second equality sign comes from the deﬁnition of ’s from Eq. (9.31). In a single-pass counterﬂow exchanger, Afr;1 and Afr;2 must be identical, and those computed in Eq. (9.36) may not be identical. In this case, use the greater of Afr;1 and Afr;2 . Finally, the core length L in the ﬂow direction is determined from the deﬁnition of the hydraulic diameter of the surface employed on each ﬂuid side. L¼

Dh A Dh A ¼ 4Ao 1 4Ao 2

ð9:38Þ

The value of L calculated from either of the equalities will be the same, as can be shown using Eqs. (9.34), (9.36), and (9.37), and for Afr;1 ¼ Afr;2 . Once the frontal area is determined, any choice of exchanger width and height (product of which should be equal to the frontal area) will theoretically be a correct solution. If there are any constraints imposed on the exchanger dimension, select the frontal area dimensions accordingly. Also, from the header design viewpoint as well as from the ﬂow distribution viewpoint, select the frontal area dimensions to make it the least prone to maldistribution; see Chapter 12 for a discussion of related issues.

622

HEAT EXCHANGER DESIGN PROCEDURES

9. Now compute the pressure drop on each ﬂuid side, after correcting f factors for variable property effects, in a manner similar to step 8 of the rating problem (section 9.2.1). 10. If the values calculated for p are within the input speciﬁcations and close to them, the solution to the sizing problem is completed; ﬁner reﬁnements in the core dimensions, such as integer numbers of ﬂow passages, may be carried out at this time. Otherwise, compute the new value of G on both ﬂuid sides using Eq. (9.27), in which p is the input speciﬁed value, and f, Kc , Ke , and geometrical dimensions are from the previous iteration. 11. Iterate steps 6 through 10 until both heat transfer and pressure drops are met as speciﬁed. Probably, only one of the two pressure drops (whichever is the most critical) will be matched, the other will be lower than speciﬁed for a gas-to-gas exchanger. Only two or three iterations may be necessary to converge to the ﬁnal size of the exchanger within 1% or the accuracy desired. 12. If the inﬂuence of longitudinal heat conduction is important, the longitudinal conduction parameter is computed from Eq. (4.13), and subsequently, NTU is computed iteratively from the " formula/results of Section 4.1.2. This new value of NTU is then used in step 8 in the second and subsequent iterations. 9.2.2.3 Sizing of a Single-Pass Crossﬂow Exchanger. For a crossﬂow exchanger, determining the core dimensions on one ﬂuid side (Afr and L) does not ﬁx the dimensions on the other ﬂuid side. In such a case, the design problem is solved simultaneously on both ﬂuid sides. The solution procedure follows closely that of Section 9.2.2.2 and is outlined next through detailed steps. 1. Determine G on each ﬂuid side by following steps 1 through 5 of Section 9.2.2.2. 2. Follow steps 6 through 8 of Section 9.2.2.2 and compute A1 , A2 , Ao;1 ; Ao;2 , Afr;1 ; and Afr;2 . 3. Now compute the ﬂuid ﬂow lengths on each ﬂuid side (see Fig. E9.1A) from the deﬁnition of the hydraulic diameter of the surface employed on each ﬂuid side as follows: Dh A Dh A L2 ¼ ð9:39Þ L1 ¼ 4Ao 1 4Ao 2 Since Afr;1 ¼ L2 L3 and Afr;2 ¼ L1 L3 ; we obtain L3 ¼

Afr;1 L2

L3 ¼

Afr;2 L1

ð9:40Þ

Thus the noﬂow (or stack) height L3 can be determined from the deﬁnition of either Afr;1 or Afr;2 and known L2 or L1 and should be identical. In reality, they may be slightly diﬀerent because of the round-oﬀ error in calculations. In that case, consider an average value for L3 . 4. Now follow steps 9 and 10 of Section 9.2.2.2 to compute p on each ﬂuid side. If p on one of two ﬂuid sides does not match (i.e., too high compared to the speciﬁcation), calculate new values of G on both ﬂuid sides as mentioned in step 10 of Section 9.2.2.2.

PLATE-FIN HEAT EXCHANGERS

623

5. Iterate steps 1 through 4 until both heat transfer and pressure drops are met as speciﬁed within the accuracy desired. 6. If the inﬂuence of longitudinal heat conduction is important, the longitudinal conduction parameter h , c , and other appropriate dimensionless groups are calculated based on the core geometry from the preceding iteration and input operating conditions with the procedure outlined in Section 4.1.4. Subsequently, NTU is computed iteratively from the " results of Section 4.1.4. This new value of NTU is then used in step 8 of Section 9.2.2.2. It should be emphasized that since we have not imposed any constraints on the exchanger dimensions, the procedure above will yield unique values of L1 , L2 , and L3 for the surface selected such that theoretically the design will exactly meet the heat duty and pressure drops on both ﬂuid sides. Example 9.2 Consider the heat exchanger of the rating problem in Example 9.1. Design a gas-to-air single-pass crossﬂow heat exchanger operating at " ¼ 0:8381 having gas and air inlet temperatures as 9008C and 2008C respectively, and gas and air mass ﬂow rates as 1.66 kg/s and 2.00 kg/s, respectively. The gas side and air side pressure drops are limited to 9.05 and 8.79 kPa, respectively. The gas and air inlet pressures are 160 kPa and 200 kPa absolute. The oﬀset strip ﬁn surface on the gas and air sides has the surface characteristics as shown in Fig. E9.1B. Both ﬁns and plates (parting sheets) are made from Inconel 625 alloy (its thermal conductivity as 18 W/m K). The plate thickness is 0.5 mm. Determine the core dimensions of this exchanger. SOLUTION Problem Data and Schematic: The following information is provided for the sizing of a crossﬂow exchanger. " ¼ 0:8381

pg ¼ 9:05 kPa

pa ¼ 8:79 kPa

Basic surface geometry parameters on the gas and air sides along with their j and f data are given in Fig. E9.1B. Geometry and operating parameters are: Af ¼ 0:785 bg ¼ 2:49 mm Dh;g ¼ 0:00154 m g ¼ 2254 m2 =m3 g ¼ 0:102 mm A g Af ¼ 0:785 ba ¼ 2:49 mm Dh;a ¼ 0:00154 m a ¼ 2254 m2 =m3 a ¼ 0:102 mm A a m_ g ¼ 1:66 kg=s Tg;i ¼ 9008C

pg;i ¼ 160 kPa

m_ a ¼ 2:0 kg=s

pa;i ¼ 200 kPa

Ta;i ¼ 1008C

kf ¼ kw ¼ 18 W=m K

Determine: The length, width and height of this exchanger to meet speciﬁed exchanger eﬀectiveness (heat duty) and pressure drops. Assumptions: The assumptions listed in Section 3.2.1 applicable to a plate-ﬁn exchanger are invoked. Neglect the eﬀect of longitudinal heat conduction and treat the gas as air for ﬂuid property evaluation. Analysis: We will follow the steps outlined in Section 9.2.2.3 for the solution.

624

HEAT EXCHANGER DESIGN PROCEDURES

Outlet Temperatures. To determine outlet temperatures from the known ", we ﬁrst need to know which ﬂuid side is the Cmin side. Since m_ a > m_ g , the Cmin side will be the gas side. Assuming that the speciﬁc heats of air and gas in the ﬁrst iteration are the same ðcp;a cp;g Þ, using Eqs. (9.9) and (9.10), we get Tg;o ¼ Tg;i "ðTg;i Ta;i Þ ¼ 9008C 0:8381 ð900 200Þ8C ¼ 313:38C Ta;o Ta;i þ "

m_ g ðT Ta;i Þ ¼ 2008C þ 0:8381 0:83 ð900 200Þ8C ¼ 686:98C m_ a g;i

This value of Ta;o will be reﬁned after we determine the ﬂuid properties. Fluid Properties. Since C* m_ g =m_ a ¼ 0:83, we will evaluate the ﬂuid properties at the arithmetic mean temperatures. 9008C þ 313:38C ¼ 606:78C ¼ 879:8 K 2 2008C þ 686:98C ¼ ¼ 443:58C ¼ 716:6 K 2

Tg;m ¼ Ta;m

The cp values of gas and air at these temperatures are 1.117 and 1.079 J/kg K, respectively (Raznjevic´, 1976). Hence, using Eq. (9.9), the correct Ta;o will be

m_ g cp;g ðTg;i Ta;i Þ m_ a cp;a 1:66 kg=s 1:117 kJ=kg K ð9008C 200Þ8C ¼ 704:18C ¼ 2008C þ 0:8381 2:00 kg=s 1:079 kJ=kg K

Ta;o ¼ Ta;i þ "

Thus the reﬁned value of Ta;m is Ta;m ¼

704:18C þ 2008C ¼ 452:08C ¼ 725:2 K 2

The speciﬁc heat of air at 725.2 K is 1.081 kJ/kg K, which has negligible diﬀerence from the previous value of 1.079 kJ/kg K and hence there is no further need of iterations. The air properties at Tg;m ¼ 879:8 K and Ta;m ¼ 725:2 K from Appendix 1 are as follows.

Gas at 879.8 K Air at 725.2 K

ðPa sÞ

cp ðkJ=kg KÞ

Pr

Pr2=3

39:3 106 34:7 106

1.117 1.081

0.721 0.692

0.804 0.782

The inlet and outlet gas densities are evaluated at 160 kPa and 150.95 (¼ 160 9:05) kPa, respectively. The inlet and outlet air densities are evaluated at 200 kPa and 191.21 ð¼ 200 8:79Þ kPa, respectively. The mean densities are evaluated using Eq. (9.18):

PLATE-FIN HEAT EXCHANGERS

Gas Air

625

Ti (K)

To (K)

i (kg/m3 )

o (kg/m3 )

m (kg/m3 )

1173.2 473.2

586.5 977.2

0.4751 1.4726

0.8966 0.6817

0.6212 0.9319

C* and NTU. From the foregoing values of cp and given ﬂow rates, we evaluate Cg ¼ ðm_ cp Þg ¼ 1:66 kg=s ð1:117 103 Þ J=kg K ¼ 1854 W=K Ca ¼ ðm_ cp Þa ¼ 2:00 kg=s ð1:081 103 Þ J=kg K ¼ 2162 W=K C* ¼

Cmin 1854 W=K ¼ ¼ 0:858 Cmax 2162 W=K

Neglecting longitudinal heat conduction, NTU for a crossﬂow exchanger with both ﬂuids unmixed for " ¼ 0:8381 and C* ¼ 0:858, from the expression of Table 3.3, is NTU ¼ 7:079 Now we need to estimate ntug and ntua from the overall NTU. The better the initial estimate, the closer will be the value of G as a ﬁrst estimate. For a gas-to-gas heat exchanger, a good estimate would be equal resistances on each ﬂuid side, considering a thermally balanced design. This would correspond to Eq. (9.23). Hence, ntua ¼ 2 NTU ¼ 2 7:079 ¼ 14:16 Then neglecting the wall thermal resistance (since we do not know Aw yet), we get, from Eq. (9.22), ntua ¼ 2C* NTU ¼ 2 0:858 7:079 ¼ 12:15 While ntua ¼ 12:15 is a somewhat reﬁned value, we could have taken ntua ¼ 14:16 for the ﬁrst iteration. Core Mass Velocities. To determine G from Eq. (9.28), we need to estimate the values of j=f and o . Since j and f versus Re characteristics are speciﬁed for the surfaces on the gas and air sides, j=f versus Re curves are constructed as shown in Fig. E9.2. Since we do not know Re yet, an approximate average (‘‘ballpark’’) value of j=f over the complete range of Re is taken for each surface from this ﬁgure as ð j=f Þg 0:25

ð j=f Þa 0:25

Again, a more precise value of j=f is not essential since we are getting a ﬁrst approximate value of G with a number of other approximations. In absence of any speciﬁc values of o , we will assume o on both the gas the air sides to be 0.80. Now substituting all values on the right-hand side of Eq. (9.28), we get

626

HEAT EXCHANGER DESIGN PROCEDURES

0.40

j/f

Strip-fin plate-fin surface 1/8-19.86 0.30

0.20 200

300

500

1000 Re

2000

5000

FIGURE E9.2 j=f vs. Re characteristics of surfaces of Fig. E9.1B. (From Shah, 1981).

Gg ¼

2gc o p j ð1=Þm Pr2=3 ntu f

1=2 ¼ g

" #1=2 2 1 0:8 ð9:05 103 PaÞ 0:25 ð1=0:6212 kg=m3 Þ 0:8039 14:16

2

¼ 14:06 kg=m s

2gc o p j Ga ¼ ð1=Þm Pr2=3 ntu f

1=2 a

"

#1=2 2 1 0:8 ð8:79 103 PaÞ 0:25 ¼ ð1=0:9319 kg=m3 Þ 0:7823 12:15

¼ 18:56 kg=m s 2

Reynolds Numbers and j and f Factors. Compute the Reynolds number on each ﬂuid side from its deﬁnition as GDh 14:06 kg=m2 s 0:00154 m ¼ Reg ¼ ¼ 551 g 0:0000393 Pa s Rea ¼

GDh 18:56 kg=m2 s 0:00154 m ¼ ¼ 824 a 0:0000347 Pa s

From Fig. E9.1B, (or the curve ﬁt of j and f data), determine the j and f factors for these Reynolds numbers as follows:

Gas Air

Re

j

f

551 824

0.0174 0.0135

0.0695 0.0539

Since Reynolds numbers indicate the ﬂow as laminar on both gas and air sides, the correction to the j factor for the temperature-dependent property eﬀects is unity because n ¼ 0 from Table 7.12.

PLATE-FIN HEAT EXCHANGERS

627

Heat Transfer Coeﬃcients, Fin Eﬀectivenesses, and Overall Heat Transfer Coeﬃcient. We compute the heat transfer coeﬃcient from the deﬁnition of the j factor as follows: hg ¼ ha ¼

jGcp

Pr2=3 g jGcp Pr2=3

¼

0:0174 14:06 kg=m2 s ð1:117 103 Þ J=kg K ¼ 339:88 W=m2 K 0:804

¼

0:0135 18:57 kg=m2 s ð1:081 103 Þ J=kg K ¼ 346:36 W=m2 K 0:782

a

Let us calculate mg and ma in order to calculate the ﬁn eﬃciency on each ﬂuid side. Since the oﬀset ﬁns are used on both gas and air sides, we use Eq. (4.147) with Lf replaced by ‘s to take into account strip edge exposed area. mg ¼

2h 1=2 2 339:88 W=m2 K 0:102 mm 1=2 1 þ ¼ 1þ kf ‘s g 3:175 mm 18 W=m K 0:102 103 m

¼ 618:17 m1 2h 1=2 2 346:36 W=m2 K 0:102 mm 1=2 1 þ 1þ ma ¼ ¼ kf ‘s a 3:175 mm 18 W=m K 0:102 103 m ¼ 624:04 m1 ‘a ¼ ‘g

b ¼ ð2:49 mm=2 0:102 mmÞ ¼ 1:143 mm ¼ 0:00114 m 2

Thus f ;g ¼

tanhðm‘Þg tanhð618:17 m1 0:00114 mÞ ¼ 0:8609 ¼ ðm‘Þg 618:17 m1 0:00114 m

f ;a ¼

tanhðm‘Þa tanhð624:04 m1 0:00114 mÞ ¼ 0:8592 ¼ ðm‘Þa 624:04 m1 0:00114 m

The overall surface eﬃciencies with Af =A values from Fig. E9.1B or input are Af ¼ 1 ð1 0:8609Þ 0:785 ¼ 0:8908 A Af ¼ 1 ð1 f Þ ¼ 1 ð1 0:8592Þ 0:785 ¼ 0:8895 A

o;g ¼ 1 ð1 f Þ o;a

To calculate Ua from Eq. (9.30), we need to ﬁrst calculate a and g using Eq. (9.31): a ¼

ðbÞa 2:54 mm 2254 m2 =m3 ¼ 941:6 m2 =m3 ¼ ba þ bg þ 2w 2:54 mm þ 2:54 mm þ 2 0:5 mm

g ¼

ðbÞg 2:54 mm 2254 m2 =m3 ¼ 941:6 m2 =m3 ¼ ba þ bg þ 2w 2:54 mm þ 2:54 mm þ 2 0:5 mm

628

HEAT EXCHANGER DESIGN PROCEDURES

Hence, Aa a 941:6 m2 =m3 ¼ ¼ ¼ 1:0 Ag g 941:6 m2 =m3 Thus Ug from Eq. (9.30), with no fouling, is g =a 1 1 1 1:0 þ ¼ þ ¼ Ug ðo hÞg ðo hÞa 0:8908 339:88 W=m2 K 0:8895 346:36 W=m2 K ¼ 6:549 103 m2 K=W Ug ¼ 152:70 W=m2 K Surface Area, Free Flow Area, and Core Dimensions. Since NTU ¼ 7:079 and Cmin ¼ Cg ¼ 1854 W=K; Cg 1854 W=K ¼ 85:95 m2 Ag ¼ NTU ¼ 7:079 Ug 152:70 W=m2 K From the speciﬁed m_ and computed G, the minimum free-ﬂow area on the gas side is Ao;g ¼

1:66 kg=s m_ ¼ 0:1181 m2 ¼ G a 14:06 kg=m2 s

The air ﬂow length is then computed from the deﬁnition of the hydraulic diameter: Lg ¼

Dh A 0:00154 m 85:95 m2 ¼ ¼ 0:280 m 4Ao g 4 0:1181 m2

Since Ag =Aa ¼ 1 and Ag ¼ 85:95 m2 , we get Aa ¼ Ag ¼ 85:95 m2 Also, Ao;a

2:00 kg=s m_ ¼ 0:1078 m2 ¼ ¼ G a 18:56 kg=m2 s

and La ¼

Dh A 0:00154 m 85:95 m2 ¼ ¼ 0:307 m 4Ao a 4 0:1078 m2

To calculate the core frontal area on each ﬂuid side, we ﬁrst need to determine ¼ Dh =4 as a ¼

a Dh;a ð941:6 m2 =m3 Þ 0:00154 m ¼ ¼ 0:363 4 4

g ¼

g Dh;g ð941:6 m2 =m3 Þ 0:00154 m ¼ ¼ 0:363 4 4

PLATE-FIN HEAT EXCHANGERS

629

Hence, Afr;g ¼

Ao;g 0:118 m2 ¼ ¼ 0:3253 m2 g 0:363

Afr;a ¼

Ao;a 0:1078 m2 ¼ ¼ 0:2970 m2 a 0:363

Since Afr;a ¼ Lg L3 or Afr;g ¼ La L3 , we get L3 ¼

Afr;a 0:2970 m2 Afr;g 0:3253 m2 ¼ 1:061 m or L3 ¼ ¼ 1:060 m ¼ ¼ Lg 0:280 m La 0:307 m

The diﬀerence in two values of L3 is due strictly to the round-oﬀ error. Note that we obtained Lg , La , and L3 as 0.280, 0.307, and 1.060. Thus, even with the very ﬁrst iteration with very approximate value of G yielded the core dimensions that are within 6%. Pressure Drops. We will now use Eq. (9.20) to determine the pressure drop on each ﬂuid side. The entrance and exit loss coeﬃcients will be the same as those determined during the rating problem. Gas side:

Kc ¼ 0:36

Ke ¼ 0:42

Air side:

Kc ¼ 0:36

Ke ¼ 0:42

To correct f factors for the temperature-dependent property eﬀects, let us ﬁrst calculate Tw . The thermal resistances on the hot and cold ﬂuid sides are Rg ¼

1 1 ¼ ¼ 3:843 105 K=W ðo hAÞg 0:8908 339:88 W=m2 K 85:95 m2

Ra ¼

1 1 ¼ ¼ 3:776 105 K=W ðo hAÞa 0:8895 346:36 W=m2 K 85:95 m2

Therefore, Rg 3:843 105 K=W ¼ ¼ 1:018 Ra 3:776 105 K=W Now Tw ¼

Tg;m þ ðRg =Ra ÞTa;m 606:78C þ 1:018 452:08C ¼ ¼ 528:78C ¼ 801:8 K 1 þ ðRg =Ra Þ 1 þ 1:018

Since the gas is being cooled, using Eq. (9.11) and the exponent m ¼ 0:81 from Table 7.12, the corrected f factor is m T 801:8 K 0:81 ¼ 0:0695 ¼ 0:0645 fg ¼ fcp w Tm 879:8 K g

630

HEAT EXCHANGER DESIGN PROCEDURES

Since the air is being heated, using Eq. (9.11) and the exponent m ¼ 1:00 from Table 7.12, we have m T 801:8 K 1:00 fa ¼ fcp w ¼ 0:0539 ¼ 0:0596 Tm 725:2 K a The pressure drops, using Eq. (9.20), are " ! ð14:06 kg=m2 sÞ2 0:4751 kg=m3 2 ð1 0:363 þ 0:36Þ þ 2 1 pg ¼ 2 1 0:4751 kg=m3 0:8966 kg=m3 # 0:0645 0:280 m 0:4751 kg=m3 0:4751 kg=m3 2 ð1 0:363 0:42Þ þ ð0:00154=4Þ m 0:6212 kg=m3 1:8966 kg=m3 ¼ 208:04 Pa ð1:2282 0:9402 þ 35:8765 0:1123Þ ¼ 208:04 Pa 36:0522 ¼ 7500 Pa ¼ 7:50 kPa " ! ð18:56 kg=m2 sÞ2 1:4726 kg=m3 2 pa ¼ 1 0:363 þ 0:36 þ 2 1 2 1 1:4726 kg=m3 0:6817 kg=m3 þ

0:0596 0:307 m 1:4726 kg=m3 1:4726 kg=m3 2 ð1 0:363 þ 0:42Þ ð0:00154=4Þ m 0:9319 kg=m3 0:6817 kg=m3

¼ 116:96 Pa ð1:2282 þ 2:3204 þ 75:0999 0:9682Þ ¼ 116:96 Pa 77:6803 ¼ 9085 Pa ¼ 9:09 kPa Since the air-side p is higher than speciﬁed, new values of G on both gas and air sides are determined again from Eq. (9.20), considering G as unknown. 9:05 103 Pa ¼

G2g ½36:0522 ! Gg ¼ 15:44 kg=m2 s 2 1 0:4751 kg=m3

8:79 103 Pa ¼

G2a ½77:6803 ! Ga ¼ 18:26 kg=m2 s 2 1 1:4726 kg=m3

Knowing the rating problem solution, we can see that this new value of Gg and Ga have almost converged to the true values. Repeating Steps 2 through 4 of section 9.2.2.3 with the new values of G yields the following results: Iterations

NTU

Lg

La

L3

Original First Second Third

7.079 7.079 7.079 7.079

0.280 0.300 0.300 0.300

0.307 0.295 0.297 0.299

1.061 1.009 1.003 1.003

TUBE-FIN HEAT EXCHANGERS

631

Thus with the iterations, the solution can be converged to the actual core dimensions within any desired accuracy. Discussion and Comments: The foregoing method clearly indicates how fast a solution to the sizing problem will converge to the core dimensions that will meet the heat transfer and pressure drops on both sides for a crossﬂow exchanger when no constraints are imposed on the dimensions. However, with the imposed constraints on the core dimensions, the design will not meet the heat transfer and pressure drops speciﬁed on both ﬂuid sides, and a constraint on the geometric parameters or operating condition variables must be relaxed. Also, in a sizing problem, one would like to ﬁnd the optimum set of core/surface geometries, and/or operating conditions for the problem speciﬁcation. In that case, it becomes an optimization problem. We discuss it in Section 9.6.

9.3

TUBE-FIN HEAT EXCHANGERS

Tube-ﬁn exchangers are mostly used as single-pass crossﬂow or multipass cross-counterﬂow exchangers. A heat pipe heat exchanger is eﬀectively two tube-ﬁn single-pass crossﬂow exchangers placed side by side separated by a splitter plate and connected to each other by the same tubes, which are heat pipes. Fluids (usually air and exhaust gas) ﬂow in counterﬂow (opposite) directions (crossﬂow to ﬁnned tubes). The design theory for heat pipe heat exchangers has been presented by Shah and Giovannelli (1988), and will not be discussed here. The solution procedures for the rating and sizing problems for tube-ﬁn exchangers with individually ﬁnned tubes or ﬂat ﬁns (see Fig. 1.31), either in single-pass crossﬂow or two-pass cross-counterﬂow, are identical to those for plate-ﬁn exchangers described in detail in preceding sections. Hence, rather than repeating the same steps, only the diﬀerences are highlighted. 9.3.1

Surface Geometries

In this case, the surface area density , a ratio of total transfer surface area A on one side of the exchanger to total volume V of the exchanger is used for heat transfer surfaces used in tube-ﬁn exchangers. Hence, , , and Dh are computed from Eqs. (8.51)–(8.53). Note that Eq. (9.31) and the ﬁrst equality of Eq. (9.37) have no physical meaning for tube-ﬁn exchangers. The heat transfer surface area density does not have signiﬁcance for the tube-ﬁn exchangers. 9.3.2

Heat Transfer Calculations

All heat transfer equations, except for Rw and f , remain the same as those for plate-ﬁn exchangers. The overall thermal resistance Eq. (9.16) should include a term for contact resistance if ﬁns are wrapped tension wound or mechanically expanded onto the tubes. Also, the wall thermal resistance term should be for a tube. For a circular tube, it is as given in Eq. (3.26). The ﬁn eﬃciency for circular ﬁns of Fig. 1.31a or ﬂat ﬁns of Fig. 1.31b is diﬀerent from that for the straight ﬁns [Eq. (9.14)]. For circular ﬁns of Fig. 1.31a, the ﬁn eﬃciency is given by Eq. (4.151), and an approximate formula, which does not involve Bessel functions, is given in Table 4.5. The ﬁn eﬃciency of ﬂat ﬁns is obtained by an approximate

632

HEAT EXCHANGER DESIGN PROCEDURES

method referred to as the sector method, discussed in Section 4.3.2.3. The details of how to evaluate the ﬁn eﬃciency for this case are also presented there and hence are not repeated here. 9.3.3

Pressure Drop Calculations

The core pressure drop Eq. (9.20) for plate-ﬁn exchangers needs to be modiﬁed for tubeﬁn exchangers for the tube outside, as discussed in Section 6.2.2.2. For individually ﬁnned tube exchangers, the entrance and exit pressure losses cannot readily be measured and evaluated separately. Hence, they are lumped into experimentally determined friction factors. In this case, the pressure drop is computed from Eq. (6.32) if f is the Fanning friction factor based on the hydraulic diameter. If, instead, the Euler number or Hagen number per tube row for the tube bank is used for the pressure drop evaluation, Eq. (6.33) should be used for the pressure drop calculation. For continuous ﬂat ﬁns, the pressure drop components of Eq. (9.20) are all valid. However, while the entrance and exit pressure losses are evaluated based on the ﬂow area at the leading and trailing edges of the ﬁns, the core friction and momentum eﬀect terms are based on G computed from the minimum free-ﬂow area within the core. Thus, Eq. (6.36) should be used, instead of Eq. (9.20), for pressure drop evaluation of continuous ﬂat ﬁns on tubes. 9.3.4

Core Mass Velocity Equation

For a tube-ﬁn exchanger with ﬂat ﬁns, if the ﬂow friction and heat transfer correlations are based on the hydraulic diameter on the tube outside, the core mass velocity equation of Eq. (9.28) is also valid for the tube outside. For ﬁnned tubes, there are a number of diﬀerent ways of correlating heat transfer and ﬂow friction characteristics, such as that Nu is based on the tube outside diameter and the pressure drop is based on the Euler number or Hagen number. In that case, the core mass velocity equation should be derived for the speciﬁc cases depending on the nature of the correlations, such as using the Nu/Eu ratio instead of the j=f ratio.

9.4

PLATE HEAT EXCHANGERS

Plate heat exchangers can be designed with m passes on the ﬂuid 1 side and n passes on the ﬂuid 2 side, depending on ﬂow rate imbalance, available pressure drop, and other design criteria. One of the most common ﬂow arrangements is 1-pass 1-pass counterﬂow design, selected for reasonably balanced ﬂow rates on hot and cold ﬂuid sides. If the ﬂow maldistribution within the PHE is ignored and if all plates have the same geometry, the rating of this exchanger is identical to that for the counterﬂow plate-ﬁn exchanger described in Section 9.2.1 except that o and f are unity since there are no ﬁns in a PHE. Unlike plate-ﬁn exchangers, sometimes it is not possible to meet the pressure drop and heat transfer speciﬁed even on one side of a PHE. So let us ﬁrst discuss this condition and the limiting cases for the design of a PHE; these limiting cases involve limitations imposed on the speciﬁed heat transfer and/or pressure drops. For a PHE with mixed plate design (i.e., having two diﬀerent plate geometries), the channel-to-channel ﬂow maldistribution must be taken into account for rating and sizing. Detailed analysis of

PLATE HEAT EXCHANGERS

633

ﬂow maldistribution is given in Chapter 12. Here we present only a step-by-step method for rating a PHE with mixed plate design. We then very brieﬂy discuss sizing a PHE. 9.4.1

Limiting Cases for the Design

Let us discuss how the design of a 1 pass–1 pass counterﬂow PHE diﬀers from that of a pure counterﬂow plate-ﬁn exchanger. For a plate-ﬁn exchanger, the minimum free-ﬂow area Ao and the surface area A on each ﬂuid side are independent of each other. For example, for a speciﬁed (or selected) Ao , the surface area A could be varied by changing the ﬁn density (this change has a minor eﬀect on Ao ). As a result, both heat transfer and pressure drop on one of the two ﬂuid sides can be matched exactly. The design method for plate-ﬁn exchangers then involves the coupling of speciﬁed NTU and p through a core mass velocity equation as presented by Eq. (9.28). This approach cannot be used for plate heat exchangers since Ao and A are not independent. Once Ao is ﬁxed (i.e., the number of plates is selected), A is ﬁxed automatically (A ¼ 4Ao L=De ) for a speciﬁed plate geometryy (the plate pattern, De , and L speciﬁed) because there is no secondary (or extended) surface. Hence, in most cases, it is not possible (unless mixed channels are used as discussed later) to match speciﬁed pressure drop and heat transfer identically even on one ﬂuid side. The plate exchanger design in general is either pressure drop– or heat transfer–limited. In the pressure drop–limited design, the free-ﬂow area is determined that satisﬁes the pressure drop limit; however, the corresponding surface area will be higher than that required to meet the heat duty. In the heat transfer–limited design, the surface area speciﬁed transfers the required heat duty; however, the corresponding freeﬂow area will be higher than that required to take advantage of the available pressure drop on either ﬂuid side. Hence, the resulting pressure drops on both ﬂuid sides will be lower than speciﬁed. This is explained further through Fig. 9.2 next. A channel (a ﬂow passage) in a PHE is made of two plates. Two given plates (with two diﬀerent chevron angles) can be used to obtain three diﬀerent channel types. For example, a plate type has two chevron angles low and high ; three channel (plate) combinations are possible: low and low , low and high , and high and high . For discussion purposes, let us refer to them as channel types 1, 2, and 3, respectively. It is worth noting that there exists an apparent chevron angle app for a mixed-plate channel, which is approximately equal to ðlow þ high Þ=2. This approximate relationship can be unsatisfactory when a 908 plate is ‘‘mixed’’ with another plate having ( < 908Þ to form a channel. A more precise value of app can be obtained experimentally by testing the mixed-plate channels, determining their j (or Nu) and f vs. Re characteristics, comparing them with the data for one type of chevron plate of various , and ﬁnding the value of the closest match to the data. However, no such app values are available in the open literature. For a given design, there exists an eﬀective (or ideal) chevron angle eff that will meet the design criteria: the heat transfer speciﬁed, the pressure drop on one ﬂuid side matched, and the pressure drop on the other ﬂuid side lower than the value speciﬁed. In general, it is not possible to match the pressure drop on both sides because of the limited number of plate geometries available in a given size. In reality, the value of eff (consider it less than app for discussion purposes) indicates whether the heat exchanger should have channel types 1 and 2 or 2 and 3: In other words, if eff < app , channel types 1 and 2 should be used; and if eff > app , channel types 2 and 3 should be used in the y There are a limited number of plate patterns available, due to the very high cost of tooling and of the press required to stamp the plates.

634

HEAT EXCHANGER DESIGN PROCEDURES

q′2 ∆p′2

q

qr 2 2

∆pr

i

q1′

i 1

∆p

∆p′1

1 A′1 A′2

Aid

A1

A2

A FIGURE 9.2 Heat-transfer-limited (channel type 1) and pressure-drop-limited (channel type 2) designs. Curves labeled 1 have low low plates; curves labeled 2 have high high plates. (From Shah and Focke, 1988.)

PHE. In both cases, a speciﬁc combination of one type of plates and mixed plates are chosen to meet the required qr or pr on one ﬂuid side, depending on whether the design is heat transfer– or pressure drop–limited. We now explain the concept of heat transfer– and pressure drop–limited designs and its relation to the concept of mixing the plates using Fig. 9.2. Consider two diﬀerent channel types as possible candidates to meet the required heat duty qr and pressure drops pr for the two ﬂuid streams. Generally, in a heat exchanger, one ﬂuid stream has a more severe pressure drop restriction than the other stream. We consider only that ﬂuid stream having the more severe pressure drop constraint,y designated as pr (required or speciﬁed p). The heat transfer rate and pressure drop (on the more constrained side) as a function of the surface area (or the number of thermal plates) for these two channels are shown in Fig. 9.2 with solid line curves labeled as 1 and 2, with heat transfer rate and pressure drop scales on the left- and right-hand y axes, respectively. Also the speciﬁed qr and pr are shown in the same ﬁgure by horizontal long dashed lines. The following discussion assumes that eff lies between those given by channel types 1 and 2.

y Note that the ﬂuid side having a severe pressure drop constraint does not necessarily have a lower pressure drop speciﬁed than that on the other ﬂuid side; in fact, it could have higher pressure drop. All it means is that we have to ensure that the design pressure drop is equal or lower than the pressure drop speciﬁed.

PLATE HEAT EXCHANGERS

635

Channel Type 1. As shown in Fig. 9.2, heat transfer of channel type 1 matches the required heat transfer qr (see the intersection of dashed line qr and solid line q for surface 1) with surface area A1 and has pressure drop p10 ; thus, it does not utilize the available pressure drop since p10 < pr . If the entire speciﬁed pressure drop would have been utilized (consumed), the surface area required would be A10 , but the heat transfer rate of the exchanger would be only q10 , which is signiﬁcantly lower than qr . Hence, the exchanger design with channel type 1 is limited by the surface area A1 to transfer the required qr . Hence, it is designated as a heat transfer–limited design. Channel Type 2. Pressure drop of channel type 2 matches the pressure drop requirement pr with surface area A2 , but in doing so, it utilizes much more free-ﬂow area (and hence surface area), which yields the heat transfer rate q20 . This is higher than the value required. If the heat transfer would have been matched, the surface area required would have been A20 , but the resulting pressure drop would have been p20 , which is signiﬁcantly higher than pr . Hence, the exchanger design with channel type 2 is limited by the surface area A2 to meet the speciﬁed pr constraint. Hence, it is designated as a pressure drop–limited design. Mixed Channels. From Fig 9.2, it is clear that the ideal channel, designated as i, which meets the heat transfer and pressure drop speciﬁcations simultaneously, will require the ideal amount of the surface area Aid . Thus, a proper mixing (combination) of channel types 1 and 2 will yield the q and p curves as those indicated by i in Fig. 9.2. That design would enable the designer to satisfy the design heat duty and pressure drop on one of the two streams considered. The use of two channel types in a given pass creates channel-to-channel ﬂow maldistribution discussed in Section 9.4.2 and thereby a reduction in heat duty. Marriot (1977) reports that the eﬀect of maldistribution of this type on q is typically less than 7%. In practical situations, a design based on mixed channels should be rated to quantify the eﬀects of ﬂow maldistribution (refer to Sections 12.1.2 and 12.1.3 for detailed discussion of ﬂow maldistribution). If these eﬀects are too severe, a pressure drop–limited design (using uniform channels giving a chevron angle higher than ideal) or a heat transfer– limited design (using uniform channels, giving a lower chevron angle) may be preferable to a mixed-channel design. 9.4.2

Uniqueness of a PHE for Rating and Sizing

Since 1 pass–1 pass counterﬂow PHE is the most common in application, its rating and sizing can be accomplished by using the methods described in Sections 9.2.1 and 9.2.2 for plate-ﬁn exchangers, if the ﬂow distribution is assumed uniform through all ﬂow channels. However, due to the nature of exchanger construction, it leads to several ﬂow maldistributions: within the channel, channel to channel, and manifold induced. These maldistributions are described further in some detail in Sections 12.1.2 and 12.1.3. To explain the rating procedure for a PHE with a mixed-plate design, we must consider, as a minimum, channel-to-channel ﬂow maldistribution. This type of ﬂow maldistribution occurs due to the presence of two diﬀerent plate groups in a PHE. For example, consider two types of plates used in a PHE: part of an exchanger made up with all 308 chevron plates and the rest with alternating 308 and 608 chevron plates (i.e., having mixed-plate channels). In such a heat exchanger, in addition to having manifold-induced maldistribution in any plate group, the ﬂow will be maldistributed among diﬀerent plate groups, due to

636

HEAT EXCHANGER DESIGN PROCEDURES

their diﬀerent ﬂow resistance (such as f vs. Re) characteristics. This can be quantiﬁed readily as shown below on a given ﬂuid side if we imagine that the pressure drop across all channels (all plate groups) on a given ﬂuid side is the same. Hence, we ﬁrst summarize the theory as to how to determine diﬀerent ﬂow rates through two groups of plates. Subsequently, we show how to compute the heat transfer rate of this mixed-plate PHE. Heat transfer rate (heat duty) for a PHE can be determined by idealizing the two diﬀerent plate groups in a PHE exchanger as two exchangers in parallel coupling (Fig. 9.3). Hence, one needs to determine the individual mass ﬂow rates through these two plate groups ﬁrst. Consider the same core (frictional) pressure drop for each plate group, and neglect manifold and port pressure drops and momentum and elevation change eﬀects; using the core frictional term only [the second term on the right-hand side of Eq. (6.44)], we get fI G2I f G2 ¼ II II De;I De;II

ð9:41Þ

where subscripts I and II denote plate groups I and II and De is the equivalent diameter [see Eq. (8.135) for the deﬁnition]. We consider that the friction factor can be represented as f ¼ a Ren

ð9:42Þ

Combining Eqs. (9.41) and (9.42), and noting that m_ ¼ GAo , the ratio X of the mass ﬂow rates through plate groups I and II can be presented as follows: X¼

m_ I ¼ m_ II

1=ð2nÞ n=ð2nÞ De;I ð1þnÞ=ð2nÞ Ao;I aII II aI I De;II Ao;II

ð9:43Þ

where Ao;I and Ao;II are the total free-ﬂow areas in plate groups I and II, respectively and can readily be calculated with a known number of plates or channels in each plate group. Then, from the mass balance, the total mass ﬂow rate is m_ ¼ m_ I þ m_ II

ð9:44Þ

Therefore, from Eqs. (9.43) and (9.44), m_ I ¼

X m_ 1þX

m_ II ¼ m_ m_ I ¼

m_ 1þX

βlow = 30° βhigh = 60°

Group I

Group II

FIGURE 9.3 Idealized counterﬂow PHE with two plate groups in parallel.

ð9:45Þ

PLATE HEAT EXCHANGERS

637

Once the individual ﬂow rates are determined, the pressure drop for each plate group can be determined from the last three terms on the right-hand side of Eq. (6.44). The manifold and port pressure drops then should be added to get the total pressure drop on each ﬂuid side. If the f –Re correlation is about the same, ideally the total pressure drop on one ﬂuid side of a PHE will be lower for the two diﬀerent plate groups compared to that for only one plate group. This can readily be understood with an electric analogy that an electric circuit having two diﬀerent resistances (such as 4 and 8 ) in parallel will have a lower electric potential than that of an electric circuit having two identical electrical resistances (6 and 6 , a mean value for the two individual resistances of the ﬁrst circuit) in parallel with the same total electric current, despite the fact that the sum of individual resistances is equal. Once the total ﬂow rates in each channel group are determined, the heat transfer analysis is straightforward by considering two exchangers in parallel, as shown in Fig. 9.3, corresponding to two plate groups. The temperature eﬀectiveness of each plate group for a counterﬂow exchanger is given by

P1;I

8 1 exp½NTUI ð1 RI Þ > > > < 1 R exp½NTU ð1 R Þ I I I ¼ > > NTUI > : 1 þ NTUI

for RI 6¼ 1 ð9:46Þ for RI ¼ 1

where NTUI ¼

ðUAÞI ðUAÞI ¼ C1;I ðm_ I cp;I Þ1

RI ¼

C1;I ðm_ I cp;I Þ1 ¼ C2;I ðm_ I cp;I Þ2

ð9:47Þ

and 1 1 1 ^ 1; f þ w þ R ^ 2; f þ 1 ¼ þR kw ðUAÞI AI h1 h2 I

ð9:48Þ

with AI ¼ A1;I ¼ A2;I ¼ Aw;I . Similarly, the temperature eﬀectiveness P1;II of the second plate group can be expressed in terms of NTUII and RII deﬁned in the same manner. The total exchanger heat duty is then given by q ¼ qI þ qII ¼ P1;I C1;I þ P1;II C1;II Th;i Tc;i

9.4.3

ð9:49Þ

Rating a PHE

We now present a rating procedure or determination of heat transfer and pressure drop performance of a PHE that has two plate groups. In group I, all plates have the same chevron angle (such as low ); in group II, two plate geometries (such as having low and high Þ are stacked alternately, thus having a mixed-plate pack (see Fig. 9.3). Since the performance of a given unit is to be determined, the following quantities are speciﬁed: . Exchanger geometry (i.e., plate width and length, channel gap, number of plates, types of plates and how the mixing of plates is achieved in the given exchanger, etc.)

638

HEAT EXCHANGER DESIGN PROCEDURES

. Plate surface pattern with their heat transfer and pressure drop characteristics . Flow arrangement of the two ﬂuids (i.e., the number of passes on each ﬂuid side and overall ﬂuid ﬂow direction) . Mass ﬂow rates, inlet temperatures, ﬂuid physical properties, and fouling resistances for each ﬂuid stream With the foregoing known information, the following is a step-by-step rating procedure. This procedure is outlined for a PHE having two plate groups. If there is only one plate group, use the same procedure with all quantities for the plate group II ignored. 1. Calculate ﬂuid properties ð, , k, and cp ) at the bulk mean temperature for each ﬂuid side. 2. Compute m_ I and m_ II for both ﬂuids from Eq. (9.45). 3. Determine Re for both ﬂuids in each plate group. 4. Calculate hh and hc for both plate groups using the speciﬁed Nu or j vs. Re correlations. 5. Compute (UAÞI using Eq. (9.48). Similarly, compute ðUAÞII . 6. Calculate NTUI and RI using Eq. (9.47). Similarly, calculate NTUII and RII . 7. Determine P1;I using Eq. (9.46). Similarly, determine P1;II . 8. Compute the heat duty q using Eq. (9.49). 9. Calculate f factors from Eq. (9.42). 10. Determine the combined channel pressure drops and other pressure drop components from Eq. (6.44) for both ﬂuid sides. Next we illustrate this procedure with one rating example. Example 9.3 A 1 pass–1 pass counterﬂow water-to-water plate heat exchanger has 47 thermal plates or 48 ﬂuid channels (24 channels for each ﬂuid). On each ﬂuid side, chevron plates of ¼ 308 are used for 8 channels and 308 and 608 mixed chevron plates are used for 16 channels. Assume that eff ¼ 39:88 and the following are empirical correlations for the Nusselt and Reynolds numbers based on De . 0:646 Nu ¼ 0:724 Re0:583 Pr1=3 308 ( f ¼

0:80Re0:25 3:44Re0:25

for ¼ 308 for ¼ 30 and 60 mixed plates

The following process, geometry, and other information are provided. Process Variables Fluid type Mass ﬂow rate (kg/s) Inlet temperature (8C)

Hot Fluid

Cold Fluid

Water 18 40

Water 10 20

639

PLATE HEAT EXCHANGERS

Outlet temperature (8C) Allowable pressure drop (kPa) Plate geometry information: Plate width W (m) Port diameter Dp (m) Equivalent diameter De (m)

30 30 0.5 0.1 7 103

38 20

Plate length (height) L (m) Channel spacing 2a mm Projected area per plate A (m2 )

1.1 3.5 0.55

Fluid properties [use the same constant properties (for simplicity) on both hot- and cold-ﬂuid sides]: Dynamic viscosity ðPa sÞ Thermal conductivity (W/m K)

8:1 104 0.619

Additional information: Total fouling resistance ¼ 4 105 m2 K W

Density (kg/m3 ) Speciﬁc heat (J/kg KÞ

995.4 4177 Pr = 5.47

Plate wall thermal resistance ¼ 3 106 m2 K W

Determine heat transfer and pressure drop performance of this exchanger.

SOLUTION Problem Data and Schematic: The detailed process, geometry, and other data are provided in the problem statement for the PHE. Determine: The heat transfer and pressure drop on each ﬂuid side for this exchanger. Assumptions: The assumptions listed in Section 3.2.1 applicable to a PHE are invoked. Analysis: We follow the steps outlined in Section 9.4.3 for the solution after we compute ﬂow and heat transfer areas as follows: Ao;I ¼ Ao;c;I ¼ Ao;h;I ¼ Wð2aÞNc;I ¼ 0:5 m ð3:5 103 Þ m 8 ¼ 0:014 m2 Ao;II ¼ Ao;c;II ¼ Ao;h;II ¼ Wð2aÞNc;II ¼ 0:5 m ð3:5 103 Þ m 16 ¼ 0:028 m2 Ac;I ¼ Ah;I ¼ 0:55 m2 8 2 ¼ 8:8 m2 Ac;II ¼ Ah;II ¼ 0:55 m2 16 2 ¼ 17:6 m2 Since we have one type of chevron plates ð ¼ 308Þ in group I of the PHE and mixed chevron plates ð ¼ 308 and 608) in group II of the PHE, let us ﬁrst evaluate the ﬂow distribution of each ﬂuid in these sections using Eq. (9.43). For this equation, the ratio of dynamic viscosities and equivalent diameters will be unity from the problem statement. Hence, from Eq. (9.43), we get

1=ð2nÞ n=ð2nÞ De;I ð1þnÞ=ð2nÞ Ao;I II I De;II Ao;II 1=ð20:25Þ 3:44 0:014 m2 ¼ 1 1 ¼ 1:151 0:80 0:028 m2

X¼

aII aI

where the values for aj , j ¼ I or II, and n of Eq. (9.42) are given in the problem formulation. The mass ﬂow rates from Eq. (9.45) are then

640

HEAT EXCHANGER DESIGN PROCEDURES

m_ c;I ¼

X m_ c 1:151 10 kg=s ¼ 1þX 1 þ 1:151

m_ c;II ¼ m_ c m_ c;I ¼ 10 5:351 kg=s ¼ 4:649 kg=s

¼ 5:351 kg=s m_ h;I ¼

X m_ h 1:151 18 kg=s ¼ 1þX 1 þ 1:151

m_ h;II ¼ m_ h m_ h;I ¼ 18 9:632 kg=s ¼ 8:368 kg=s

¼ 9:632 kg=s

The mass velocities are then given by

Gc;I ¼

m_ c;I 5:351 kg=s ¼ Ao;c;I 0:014 m2

¼ 382:21 kg=m2 s Gh;I ¼

m_ h;I 9:632 kg=s ¼ Ao;h;I 0:014 m2

¼ 688:00 kg=m2 s

Gc;II ¼

m_ c;II 4:649 kg=s ¼ Ao;c;II 0:028 m2

¼ 166:04 kg=m2 s Gh;II ¼

m_ h;II 8:368 kg=s ¼ Ao;h;II 0:028 m2

¼ 298:86 kg=m2 s

The Reynolds numbers are determined from the deﬁnition as

Rec;I ¼

Gc;I De 382:21 kg=m2 s 7 103 m ¼ 3303 ¼ 8:1 104 Pa s

Rec;II ¼

Gc;II De 166:04 kg=m2 s 7 103 m ¼ 1435 ¼ 8:1 104 Pa s

Reh;I ¼

Gh;I De 688:00 kg=m2 s 7 103 m ¼ 5946 ¼ 8:1 104 Pa s

Reh;II ¼

Gh;II De 298:86 kg=m2 s 7 103 m ¼ 2583 ¼ 8:1 104 Pa s

Now calculate the heat transfer coeﬃcients on the cold and hot sides for groups I and II using the given correlation for Nu written in terms of heat transfer coeﬃcients. 0:646 k I hc;I ¼ 0:724 Re0:583 Pr1=3 c;I 308 De 0:619 W=m K 308 0:646 ¼ 0:724 ð3303Þ0:583 ð5:47Þ1=3 308 7 103 m ¼ 12,701 W=m2 K

PLATE HEAT EXCHANGERS

641

0:619 W=m K 39:88 0:646 hc;II ¼ 0:724 ð1435Þ0:583 ð5:47Þ1=3 ¼ 9377 W=m2 K 308 7 103 m 0:619 W=m K 308 0:646 hh;I ¼ 0:724 ð5946Þ0:583 ð5:47Þ1=3 ¼ 17,894 W=m2 K 308 7 103 m 0:619 W=m K 39:88 0:646 hh;II ¼ 0:724 ð2583Þ0:583 ð5:47Þ1=3 ¼ 13,210 W=m2 K 308 7 103 m The overall conductance (UA) for each group is computed from Eq. (9.48) as follows: 1 1 1 ^ c; f þ w þ R ^ h; f þ 1 ¼ þR kw ðUAÞI Ac;I hc;I hh;I 1 1 þ 4 105 m2 K=W ¼ 2 8:8 m 12,701 W=m2 K 1 þ 3 106 m2 K=W þ 17,894 W=m2 K ¼ 2:0184 105 K=W or ðUAÞI ¼ 49,544 W=K Similarly, 1 1 1 ¼ þ 4 105 m2 K=W þ 3 106 m2 K=W ðUAÞII 17:6 m2 9377 W=m2 K 1 þ 13,210 W=m2 K ¼ 1:2804 105 K=W or

ðUAÞII ¼ 78,103 W=K

Next, determine NTU and R for groups I and II from their deﬁnitions. NTUI ¼

ðUAÞI 49,544 W=K ¼ ¼ 2:217 ðm_ cp Þc;I 5:351 kg=s 4177 J=kg K

NTUII ¼

ðUAÞII 78,103 W=K ¼ ¼ 4:022 ðm_ cp Þc;II 4:649 kg=s 4177 J=kg K

RI ¼

m_ c;I cp 5:351 kg=s 4177 J=kg K ¼ ¼ 0:556 m_ h;I cp 9:632 kg=s 4177 J=kg K

RII ¼

m_ c;II cp 4:649 kg=s 4177 J=kg K ¼ ¼ 0:556 m_ h;II cp 8:368 kg=s 4177 J=kg K

642

HEAT EXCHANGER DESIGN PROCEDURES

The temperature eﬀectiveness for the two groups of the counterﬂow exchanger, P1;I and P1;II are given by Eq. (I.1.1) of Table 3.6. P1;I ¼

1 exp½NTUI ð1 RI Þ 1 exp½2:217 ð1 0:556Þ ¼ 0:7906 1 RI exp½NTUI ð1 RI Þ 1 0:556 exp½2:217 ð1 0:556Þ

P1;II ¼

1 exp½NTUII ð1 RII Þ 1 exp½4:022 ð1 0:556Þ ¼ 0:9179 1 RII exp½NTUII ð1 RII Þ 1 0:556 exp½4:022 ð1 0:556Þ

Finally, the heat transfer rate from the hot water to cold water in this exchanger is given by q ¼ P1;I m_ c;I cp ðTh;i Tc;i Þ þ P1;II m_ c;II cp ðTh;i Tc;i Þ ¼ ½0:7906 5:351 kg=s 4177 J=kg K ð40 20Þ K þ ½0:9179 4:649 kg=s 4177 J=kg K ð40 20Þ K ¼ 353:4 103 W þ 356:5 103 W ¼ 710 kW

Ans:

To compute the pressure drop, the friction factors for the Reynolds numbers above can be computed from Eq. (9.42) as follows: fc;I ¼ 0:80ð3303Þ0:25 ¼ 0:1055

fc;II ¼ 3:44ð1435Þ0:25 ¼ 0:5589

fh;I ¼ 0:80ð5946Þ0:25 ¼ 0:0911

fh;II ¼ 3:44ð2583Þ0:25 ¼ 0:4825

Now we compute the pressure drop associated within the plate pack on the cold and hot sides using Eq. (6.29) with given m ¼ : pc;I ¼

4fc;I LG2c;I 4 0:1055 1:1 m ð382:21Þ2 ¼ 4866 Pa ¼ 2gc De 2 1 995:4 kg=m3 7 103 m

pc;II ¼

4 0:5589 1:1 m ð166:04Þ2 ¼ 4865 Pa 2 1 995:4 kg=m3 7 103 m

ph;I ¼

4 0:0911 1:1 m ð688:00Þ2 ¼ 13,615 Pa 2 1 995:4 kg=m3 7 103 m

ph;II ¼

4 0:4825 1:1 m ð298:86Þ2 ¼ 13,607 Pa 2 1 995:4 kg=m3 7 103 m

Theoretically, pc;I ¼ pc;II and ph;I ¼ ph;II . As found above, this is true within the round-oﬀ error margins. Thus we consider the following values for the pressure drop associated with this plate pack. pc ¼ 4866 Pa

ph ¼ 13,615 Pa

The other components of the pressure drop are the momentum eﬀect, the elevation change eﬀect, and the inlet and outlet manifolds and ports pressure drops. The ﬁrst two eﬀects are negligible for this case (no change in the density and negligible elevation

PLATE HEAT EXCHANGERS

643

change for 1.1-m-long plates) and are ignored. For the manifold and port pressure drop component, the corresponding mass velocities are Gc;p ¼

10 kg=s m_ c ¼ ¼ 1273 kg=m2 s ð =4ÞD2p ð =4Þð0:1 mÞ2

Gh;p ¼

18 kg=s m_ h ¼ ¼ 2292 kg=m2 s ð =4ÞD2p ð =4Þð0:1 mÞ2

The manifold and port pressure drops for the cold and hot ﬂuid sides are computed from the ﬁrst term on the right-hand side of Eq. (6.44):

pc;p ¼

1:5G2c;p np 1:5 ð1273 kg=m2 sÞ2 1 ¼ ¼ 1221 Pa 2gc i 2 1 995:4 kg=m3

ph;p ¼

1:5 ð2292 kg=m2 sÞ2 1 ¼ 3958 Pa 2 1 995:4 kg=m3

where np ¼ 1 represents the number of passes on the given ﬂuid side. Thus, the total pressure drops on the cold and hot ﬂuid sides are pc ¼ 4866 Pa þ 1221 Pa ¼ 6087 Pa

ph ¼ 13,615 Pa þ 3958 Pa ¼ 17,573 Pa

Ans:

Note that the pressure drops associated with the manifold and port on the cold and hot ﬂuid sides are 20% [¼ ð1221=6087Þ 100 and 22.5% ½¼ ð3958=17,573Þ 100; respectively, of the total pressure drop on individual ﬂuid sides. To compare the eﬀect of mixed-plate performance to that for a single-plate geometry PHE, let us recalculate the performance of a similar PHE of the same number of plates (or channels), but all made from 308 chevron plates. The ﬂow and surface areas on one side of that PHE are Ao;c ¼ Ao;h ¼ Wð2aÞNc ¼ 0:5 m 3:5 103 m 24 ¼ 0:042 m2 Ac ¼ Ah ¼ 0:55 m2 24 2 ¼ 26:4 m2 The mass velocities, Reynolds numbers, and heat transfer coeﬃcients on both ﬂuid sides are as follows: Gc ¼

10 kg=s m_ c ¼ ¼ 238:10 kg=m2 s Ao;c 0:042 m2

Gh ¼

18 kg=s m_ h ¼ ¼ 428:57 kg=m2 s Ao;h 0:042 m2

Rec ¼

Gc De 238:10 kg=m2 s 7 103 m ¼ 2058 ¼ 8:1 104 Pa s

Reh ¼

Gh De 428:57 kg=m2 s 7 103 m ¼ 3704 ¼ 8:1 104 Pa s

644

HEAT EXCHANGER DESIGN PROCEDURES

k 0:619 W=m K ð2058Þ0:583 ð5:47Þ1=3 Re0:583 hc ¼ 0:724 Pr1=3 ¼ 0:724 c De 7 103 m ¼ 9640 W=m2 K k 0:619 W=m K 1=3 hh ¼ 0:724 ð3704Þ0:583 ð5:47Þ1=3 Pr ¼ 0:724 Re0:583 h De 7 103 m ¼ 13,579 W=m2 K The overall thermal conductance UA is given by 1 1 1 ^ c; f þ w þ R ^ h; f þ 1 þR ¼ kw UA Ac hc hh 1 1 ¼ þ 4 105 m2 K=W 26:4 m2 9640 W=m2 K 1 ¼ 8:3476 106 K=W þ 3 106 m2 K=W þ 13,579 W=m2 K or UA ¼ 119,790 W=K Now we compute NTU, R, P, and q for this PHE. NTU ¼

UA 119,790 W=K ¼ ¼ 2:868 ðm_ cp Þc 10 kg=s 4177 J=kg K

R¼

ðm_ cp Þc 10 kg=s 4177 J=kg K ¼ ¼ 0:556 ðm_ cp Þh 18 kg=s 4177 J=kg K

P¼

1 exp½NTUð1 RÞ 1 exp½2:868 ð1 0:556Þ ¼ ¼ 0:8528 1 R exp½NTUð1 RÞ 1 0:556 exp½2:868 ð1 0:556Þ

q ¼ Pðm_ c cp ÞðTh;i Tc;i Þ ¼ 0:8528 10 kg=s 4177 J=kg K ð40 20Þ K ¼ 712 103 W ¼ 712 kW Finally, the friction factors and the pressure drops are 0:25 ¼ 0:1188 fc ¼ a Ren c ¼ 0:80ð2058Þ

fh ¼ 0:80ð3704Þ0:25 ¼ 0:1025

pc;core ¼

4fc LG2c 4 0:1188 1:1 m ð238:10 kg=m2 sÞ2 ¼ 2126 Pa ¼ 2gc De 2 1 995:4 kg=m3 7 103 m

ph;core ¼

4 0:1025 1:1 m ð428:57 kg=m2 sÞ2 ¼ 5944 Pa 2 1 995:4 kg=m3 7 103 m

PLATE HEAT EXCHANGERS

645

Discussion and Comments: This example illustrates how to rate a PHE when it has a plate pack consisting of one type of plates in one group and mixed plates in another group, with the following results:

Group

m_ c (kg/s)

m_ h (kg/s)

q (kW)

pc;core (kPa)

ph;core (kPa)

5.351 4.649

9.632 8.368

353.4 356.5

4.866 4.865

13.615 13.607

I II

We have also compared the performance of this exchanger with the one having all chevron plates of ¼ 308. Following is the comparison of main results: PHE

m_ c (kg/s)

m_ h (kg/s)

q (kW)

pc;core (kPa)

ph;core (kPa)

10 10

18 18

710 712

4.866 2.126

13.615 5.944

Mixed Plates Single Plates

For the present problem, the friction factors for a given Re are over four times larger for the mixed-plate group II. This, in turn, reduces the ﬂow to only 46.5% ½¼ 100 8:368 kg=s=18 kg=s of the total ﬂow in group II despite the ﬂow area being double for group II than for group I. Hence, even though the heat transfer coeﬃcient for the mixed-plate region is larger than that for the single-plate region at a given Re, the reduction in the ﬂow reduces h, and as a result, the overall heat transfer is about the same (710 vs. 712 kW). Hence, it is important to keep in mind that the mixed-plate section should not have excessive friction factors which would otherwise defeat the advantage of having a mixed-plate section.

9.4.4

Sizing a PHE

When sizing a PHE, we have very little choice in the selection of plate dimensions, unlike plate-ﬁn and tube-ﬁn heat exchanger designs, because we cannot arbitrarily select a plate width W or plate length L. Instead, we should select from a relatively small pool of available plate sizes from any manufacturer. As the dies used for forming the plates are extremely expensive, each manufacturer oﬀers only up to about 30 plate sizes. In selecting an appropriate plate size, we may compute the ﬂuid velocity in the port and limit this value to a maximum of 6 m/s (20 ft/sec), as a rough rule of thumb. Further, the manifold and port pressure drops may not be allowed to exceed a certain percentage (typically 10% but up to 30% in rare cases) of the total pressure drop. Most plate sizes are generally available only in two chevron angles. However, by ‘‘mixing’’ plates of diﬀerent chevron angles in various proportions, the designer is able to obtain considerable ﬂexibility in eff for any PHE. Two methods are published in the literature (Shah and Focke, 1988; Shah and Wanniarachchi, 1991) for sizing a PHE. However, we do not describe them here because (1) those procedures are quite involved, (2) engineers in the PHE industry use their own proprietary computer programs with their own data for j and f factors, and (3) it is easy to add or delete some plates if the designed PHE does not perform to the speciﬁcations.

646

HEAT EXCHANGER DESIGN PROCEDURES

Since sophisticated proprietary computer programs are available for rating a PHE which converge quickly, such programs have iterative rating schemes build into them to arrive at a size to meet the speciﬁed heat transfer and/or pressure drop; the size will depend on whether it is a heat transfer–limited design or a pressure drop–limited design, as discussed in Section 9.4.1.

9.5

SHELL-AND-TUBE HEAT EXCHANGERS

Accurate prediction of performance and design characteristics of conventional shell-andtube heat exchangers is more diﬃcult than that for plate-ﬁn and tube-ﬁn exchangers. This is due primarily to the complexity of shell-side ﬂow conditions and the impact of that complexity on heat transfer performance. There are many variables associated with the geometry (i.e., baﬄes, tubes, front- and rear-end heads, etc.) in a shell-and-tube heat exchanger in addition to those for the operating conditions. So complete sizing (with a unique design) of a shell-and-tube exchanger is not possible as for a plate-ﬁn exchanger, described earlier. As a result, the common practice is to presume the complete geometry of the exchanger and perform the rating of the exchanger to determine the tube (shell) length if the heat duty is given, or outlet temperatures if the length is given. In both cases, pressure drops are to be determined. Preliminary sizing (design) of a shell-and-tube heat exchanger is possible based on a number of approximations and the experience of past designs. Once the preliminary design is obtained, the design calculations are essentially a series of iterative rating calculations made on the preliminary design until a satisfactory design is achieved. In this section, we outline the basic steps of (1) a rating procedure with an example, and (2) a preliminary design and subsequent iteration technique for sizing of a shell-and-tube exchanger. It should be added that modern design practices are based almost exclusively on sophisticated commercial or proprietary computer software that takes into account many complex eﬀects on the shell side that are beyond the simpliﬁed methods presented here. In this section, we start with how to compute the shell-side heat transfer and pressure drop by taking into account various ﬂow leakage, bypass, and other eﬀects before providing the rating procedure. These eﬀects are taken into consideration by a widely utilized method in the open literature referred to as the Bell–Delaware method. It was originally reported by Bell (1963) for rating of fully tubed segmentally baﬄed heat exchangers with plain tubes based on the experimental data obtained for an exchanger with geometrical parameters closely controlled. This method has been extended to rate low-ﬁnned-tube E shell, no-tubes-in-window E shell, and F shell heat exchangers (Bell, 1988b; Taborek, 1998). 9.5.1

Heat Transfer and Pressure Drop Calculations

Heat transfer and pressure drop calculations constitute the key part of the rating or design of an exchanger. Tube-side calculations are straightforward. The heat transfer coeﬃcient is computed using available correlations for internal forced convection as presented in Section 7.4, and Eq. (9.20) is used for pressure drop calculations. The shell-side calculations, however, must take into consideration the eﬀect of various leakage streams (A and E streams, Fig. 4.19) and bypass streams (C and F streams, Fig. 4.19) in addition to the main crossﬂow stream B through the tube bundle. Several methods have been in use over the years, but the most accurate method in the open literature is the

SHELL-AND-TUBE HEAT EXCHANGERS

647

Bell–Delaware method. The set of correlations for calculating shell-side heat transfer coeﬃcients and pressure drops discussed next constitutes the core of the Bell–Delaware method (Bell, 1988b). 9.5.1.1 Shell-Side Heat Transfer Coeﬃcient. In the Bell–Delaware method, the shellside heat transfer coeﬃcient hs is determined using Eq. (4.169) by correcting the ideal heat transfer coeﬃcient hid for various leakage and bypass ﬂow streams in a segmentally baﬄed shell-and-tube exchanger. The hid is determined for pure crossﬂow in a rectangular tube bank assuming that the entire shell-side stream ﬂow across the tube bank is at or near the centerline of the shell. It is computed from the Nusselt number correlations of Eq. (7.117) or other appropriate Nu or j vs. Re correlations, modiﬁed for property variation eﬀects as outlined in Section 7.6.1. It is then corrected by ﬁve correction factors as follows: hs ¼ hid Jc J‘ Jb Js Jr

ð9:50Þ

where Jc ¼ correction factor for baﬄe conﬁguration (baﬄe cut and spacing). It takes into account heat transfer in the window and leads to the average for the entire heat exchanger. It is dependent on the fraction of the total number of tubes in crossﬂow between baﬄe tips. Its value is 1.0 for an exchanger with no tubes in the windows and increases to 1.15 for small baﬄe cuts and decreases to 0.65 for large baﬄe cuts. For a typical well-designed heat exchanger, its value is near 1.0. J‘ ¼ correction factor for baﬄe leakage eﬀects, including both tube-to-baﬄe and baﬄeto-shell leakages (A and E streams) with heavy weight given to the latter and credit given to tighter constructions. It is a function of the ratio of the total leakage area per baﬄe to the crossﬂow area between adjacent baﬄes, and also of the ratio of the shell-to-baﬄe leakage area to tube-to-baﬄe leakage area. If the baﬄes are too close, J‘ will be lower, due to higher leakage streams. A typical value of J‘ is in the range 0.7 to 0.8. Jb ¼ correction factor for bundle and pass partition bypass (C and F) streams. It varies from 0.9 for a relatively small clearance between the outermost tubes and the shell for ﬁxed tubesheet construction to 0.7 for large clearances in pull-though ﬂoating head construction. It can be increased from about 0.7 to 0.9 by proper use of the sealing strips in a pull-through bundle. Js ¼ correction factor for larger baﬄe spacing at the inlet and outlet sections compared to the central baﬄe spacing. The nozzle locations result in larger end baﬄe spacing and lower velocities and thus lower heat transfer coeﬃcients. Js usually varies from 0.85 to 1.0. Jr ¼ correction factor for any adverse temperature gradient buildup in laminar ﬂows. This correction applies only for shell-side Reynolds numbers below 100 and fully eﬀective for Res < 20; otherwise, it is equal to 1. The combined correction factor, made up of ﬁve correction factors, in a well-designed shell-and-tube exchanger is about 0.6 (i.e., a reduction of 40% in the ideal heat transfer coeﬃcient). The combined correction factor can be as low as 0.4. Comparison with a large amount of proprietary experimental data indicates that compared to measured values, the shell-side hs predicted from Eq. (9.50) is from 50% too low to 200% too high, with a mean error of 15% low (conservative) at all Reynolds numbers.

648

HEAT EXCHANGER DESIGN PROCEDURES

TABLE 9.2 Correction Factors for the Heat Transfer Coeﬃcient on the Shell Side by the Bell– Delaware Method Correction Factors C’s

Formulas for parameters for Correction Factors

Jc ¼ 0:55 þ 0:72Fc 2:2rlm

J‘ ¼ 0:44ð1 rs Þ þ ½1 0:44ð1 rs Þe

Jb ¼

for Nssþ 1=2

1 Crb ½1ð2Nssþ Þ1=3

for Nssþ 1=2

e

Fc given by Eq. (8.120) Ao;sb Ao;sb þ Ao;tb rs ¼ ; rlm ¼ Ao;sb þ Ao;tb Ao;cr Ao;sb , Ao;tb , and Ao;cr given by Eqs. (8.130), (8.129), and (8.125), respectively Ao;bp N rb ¼ ; Nssþ ¼ ss ; Ao;cr Nr;cc C¼

Js ¼

Nb

ð1nÞ 1 þ ðLþ i Þ Nb 1 þ Lþ i

þ þ

ð1nÞ ðLþ oÞ Lþ o

Lþ i ¼

1:25 for Res > 100

Lb;i Lb;o ; Lþ o ¼ Lbc Lbc

n¼ Jr ¼

1

for Res 100

ð10=Nr;c Þ0:18

for Res 20

1:35 for Res 100

0:6 for turbulent flow 1 3

for laminar flow

Nr;c ¼ Nr;cc þ Nr;cw ; use Eqs. (8.121) and (8.119) for Nr;cc and Nr;cw ; For 20 < Res < 100, linearly interpolate Jr from two formulas

Source: Data from Taborek (1998).

These correction factors were determined from well-controlled experiments, and the results were presented graphically (Bell, 1963). Those correction factors have been curve ﬁtted (Taborek, 1998) and are presented in Table 9.2. The Bell–Delaware method can be used for a low-ﬁnned tube-bundle E shell, notubes-in-window E shell, and F shell exchangers. The modiﬁcations for these extensions are summarized brieﬂy next. . External low-ﬁnned tubes are used when the shell-side heat transfer coeﬃcient is low, such as with viscous liquids. In this case, the ideal heat transfer coeﬃcient for low-ﬁnned tubes is computed from the appropriate correlations, such as Eq. (7.135). Subsequently, the eﬀective shell-side hs is calculated from Eq. (9.50). . The no-tubes-in-window design is used to minimize/eliminate the ﬂow-induced tube vibration problem. In this case, the ﬂow area Ao;w through one window is given by Afr;w of Eq. (8.111) since Afr;t ¼ 0 in Eq. (8.117). The tube count for this exchanger is given by Nt Fc , where Nt is the number of tubes for a fully tubed exchanger. The fraction Fc of the total number of tubes in the crossﬂow section is given by Eq. (8.120). Also, Jc of Eq. (9.50) is unity and Nr;cw ¼ 0 for the calculation of Jr from Table 9.2 for Eq. (9.50). The rest of the procedure remains the same. . For the F shell exchanger, we have two-tube and two-shell passes by the use of a longitudinal baﬄe. If this baﬄe is not welded on both sides to the shell, there will be ﬂuid leakage from the upstream to the downstream pass on the shell side due to

SHELL-AND-TUBE HEAT EXCHANGERS

649

the pressure diﬀerence. Also, there will be heat leakage across the baﬄe by heat conduction from the hotter to colder side of the shell-side pass. These eﬀects may not be negligible in some cases. If we neglect these eﬀects, the Bell–Delaware method remains identical except that all ﬂow and surface areas need to be reduced by half compared to a single shell-side pass. 9.5.1.2 Shell-Side Pressure Drop. Similar to shell-side heat transfer, the shell-side pressure drop is also aﬀected by various leakage and bypass streams in a segmentally baﬄed exchanger. The shell-side pressure drop has three components: (1) pressure drop in the central (crossﬂow) section, pcr ; (2) pressure drop in the window area, pw ; and (3) pressure drop in the shell-side inlet and outlet sections, pi-o (see Section 6.4.2.2). It is assumed that each of the three components is based on the total ﬂow rate, and that it can be calculated correcting the corresponding ideal pressure drops. The ideal pressure drop in the central section, pb;id , assumes pure crossﬂow of the ﬂuid across the ideal tube bundle. This pressure drop should be corrected for (1) the two leakage streams A and E in Fig. 4.19 using the correction factor ‘ , and (2) the bundle and pass partition bypass ﬂow streams C and F in Fig. 4.19 using the correction factor b . The ideal window pressure drop, pw , also has to be corrected for both baﬄe leakage streams. Finally, the ideal inlet and outlet section pressure drops, pio , are based on an ideal crossﬂow pressure drop in the central section. These pressure drops should be corrected for bypass ﬂow (correction factor b ) and for uneven baﬄe spacing in inlet and outlet sections (correction factor s ). Thus, the total shell-side pressure drop, from Eq. (6.43), is given as ps ¼ pcr þ pw þ pio ¼ ½ðNb 1Þpb;id b þ Nb pw;id ‘ Nr;cw þ 2pb;id 1 þ

Nr;cc b s

ð9:51Þ

The formulas for pb;id and pw;id are given by Eqs. (6.37) and (6.39) respectively. The Hagen number for Eq. (6.37) is obtained from Eq. (7.109). Various correction factors of Eq. (9.51) are deﬁned as follows:

‘ ¼ correction factor for tube-to-baﬄe and baﬄe-to-shell leakage (A and E) streams. This factor is related to the same eﬀect as J‘ but is of diﬀerent magnitude. Usually,

‘ 0:4 to 0.5, although lower values are possible with small baﬄe spacing.

b ¼ correction factor for bypass ﬂow (C and F streams). It is diﬀerent in magnitude from Jb and ranges from 0.5 to 0.8, depending on construction type and the number of sealing strips. The lower value will be typical of a pull-through ﬂoating head with one or two sealing strip pairs, and the higher value, if a fully tubed ﬁxed-tubesheet design.

s ¼ correction factor for inlet and outlet sections having diﬀerent baﬄe spacing from that of the central section, in the range 0.5 to 2. These correction factors, originally presented in graphical form (Bell, 1963, 1988b), are given in Table 9.3 in equation form by Taborek (1998). The combined eﬀect of pressure drop corrections reduces the ideal total shell-side pressure drop to 20 to 30% of the pressure drop that would be calculated for ﬂow through the corresponding exchanger without baﬄe leakages and bundle bypass streams (i.e., ps;actual 0:2 to 0.3 ps;id Þ: Comparison with a large number of proprietary experimental data indicate that compared to measured values, the shell-side ps

650

HEAT EXCHANGER DESIGN PROCEDURES

TABLE 9.3 Correction Factors for the Pressure Drop on the Shell Side by the Bell–Delaware Method Correction Factors, ’s

b ¼

expfDrb ½1 ð2Nssþ Þ1=3 g

‘ ¼ exp½1:33ð1 þ rs Þrplm

Lb;c Lb;o

for Nssþ < 12 for Nssþ 12

1

s ¼

Formula for Parameters for Correction Factors

2n 0

þ

Lb;c Lb;i

2n 0

rb and Nssþ defined in Table 9:2 4:5 for Res 100 D¼ 3:7 for Res > 100 rs and rlm deﬁned in Table 9.2 p ¼ ½0:15ð1 þ rs Þ þ 0:8 1:0 for laminar flow n0 ¼ 0:2 for turbulent flow

Source: Data from Taborek (1998).

computed from Eq. (9.51) is from about 5% low (unsafe) at Res > 1000 to 100% high at Res < 10: Despite the facts above, it should be emphasized that the window section contributes high pressure drop [compared to the other components of Eq. (9.51)] with insigniﬁcant contribution to heat transfer. This results in an overall lower heat transfer rate to pressure drop ratio for the segmental baﬄe exchanger than that for grid baﬄe and most newer shell-and-tube heat exchanger designs. 9.5.2

Rating Procedure

The following is a step-by-step rating procedure using the Bell–Delaware method (Bell, 1988b). For the rating problem, the detailed exchanger geometry is speciﬁed and we determine the heat duty, outlet temperatures as well as pressure drops on both ﬂuid sides. We then describe the changes in the solution method if the exchanger length is to be determined. 1. Compute the surface geometrical characteristics on each ﬂuid side. This includes shell-side ﬂow areas in crossﬂow and window zones as well as all leakage ﬂow areas and related information as detailed in Section 8.5. Also compute the tube-side ﬂow area, surface area, ratio of free ﬂow to frontal area, and other pertinent dimensions. 2. Calculate the ﬂuid bulk temperature and ﬂuid thermophysical properties on each ﬂuid side. Since the outlet temperatures are not known for the rating problem, they are guessed initially. Unless it is known from past experience, assume the exchanger effectiveness as 50% for most single and multitube-pass shell-and-tube exchangers, or 60 to 75% for multishell-pass exchangers. For the assumed effectiveness, calculate the ﬂuid outlet temperatures using Eqs. (9.9) and (9.10). Compute ﬂuid mean temperatures on each ﬂuid side, depending on the heat capacity ratio C*, as outlined in Section 9.1. Subsequently, obtain the ﬂuid properties (, cp , k, Pr, and ) from thermophysical property books, handbooks, or Appendix A. 3. Calculate the Reynolds numbers (Re ¼ GDh =) and/or any other pertinent dimensionless groups (from the basic deﬁnitions) needed to determine the nondimensional heat transfer and ﬂow friction characteristics (e.g., j or Nu and f, Eu, or Hg)

SHELL-AND-TUBE HEAT EXCHANGERS

651

of heat transfer surfaces on each ﬂuid side of the exchanger. Subsequently, compute j or Nu and f, Eu or Hg factors. Correct Nu (or j) for variable ﬂuid property effects in the second and subsequent iterations using Eqs. (9.11) and (9.12). 4. From Nu or j, compute the heat transfer coefﬁcients for both ﬂuid streams from the following equations: h ¼ Nu

k Dh

or

h ¼ jGcp Pr2=3

ð9:52Þ

5. Compute various J correction factors for bafﬂe conﬁguration, ﬂow leakage, ﬂow bypass, unequal bafﬂe spacing in the ends, and adverse temperature gradient on the shell side using Table 9.2. Determine the effective or actual shell-side heat transfer coefﬁcient using Eq. (9.50). 6. Also calculate the tube-side heat transfer coefﬁcient, wall thermal resistance, fouling resistances, and the overall heat transfer coefﬁcient. 7. From the known heat capacity rates on each ﬂuid side, compute C* ¼ Cmin =Cmax . From the known UA, determine NTU ¼ UA=Cmin . With the known NTU, C*, and the ﬂow arrangement, determine the exchanger effectiveness " from either closed-form equations (see Table 3.6) or tabular/graphical results. 8. With this ", ﬁnally compute the outlet temperatures from Eqs. (9.9) and (9.10). If these outlet temperatures are signiﬁcantly different from those assumed in step 2, use these outlet temperatures in step 2 and continue iterating steps 2 through 8 until the assumed and computed outlet temperatures converge within the desired degree of accuracy. 9. For the pressure drop calculations, calculate the mean ﬂuid densities on both ﬂuid sides as follows: Use the arithmetic mean value for liquids and harmonic mean value for gases as given by Eq. (9.18). For the shell-side pressure drop, compute various correction factors using the formulas given in Table 9.3, the ideal crossﬂow and window zone pressure drops from Eqs. (6.37) and (6.39), and the shell-side total pressure drop from Eq. (6.43) or (9.51). For the tube-side pressure drop, determine the entrance and exit loss coefﬁcients, Kc and Ke , from Fig. 6.3 for known , Re, and the ﬂow passage entrance geometry. The friction factor on each ﬂuid side is corrected for the variable ﬂuid properties using Eq. (9.11) or (9.12). The core pressure drops on each ﬂuid side are then calculated from Eq. (6.28) or (6.33). If the heat duty and detailed exchanger geometry except for the exchanger (tube) length are given for the TEMA E exchanger, the tube length can be determined as follows by modifying the aforementioned detailed procedure. Follow step 1 except that the surface area is unknown. Since the heat duty is known, outlet temperatures are known, and as a result, the ﬂuid properties mentioned in step 2 can be determined at mean temperatures in the exchanger. Follow steps 3 through 6 to compute the overall shell-side heat transfer coeﬃcient Us . Since all four temperatures are known, calculate the log-mean temperature diﬀerence Tlm using the deﬁnition of Eqs. (3.172) and (3.173). Also compute the temperature eﬀectiveness Ps and heat capacity rate ratio Rs from known four terminal temperatures using Eqs. (3.96) and (3.105) with the subscripts 1 and 2 replaced by s (shell side) and t (tube side). Next, determine the log-mean temperature diﬀerence correction factor F for known Ps , Rs and the exchanger type (ﬂow

652

HEAT EXCHANGER DESIGN PROCEDURES

arrangement). Finally compute the surface area on the shell side from the following equation. As ¼

q Us FTlm

ð9:53Þ

The required eﬀective tube length of the exchanger is then calculated from L ¼ As = do Nt and the number of baﬄes required by using Eq. (8.126). We now illustrate the rating methodology with an example. Example 9.4 Determine heat transfer rate, outlet ﬂuid temperatures, and pressure drops on each ﬂuid side for a TEMA E shell-and-tube heat exchanger with a ﬁxed tubesheet and one shell and two tube passes. The tubes in the bundle are in 458 rotated square arrangement. The ﬂuids are lubricating oil and seawater. Fouling factors for the oil and water sides are 1:76 104 and 8:81 105 m2 K=W, respectively. The geometric dimensions and operating properties are provided as follows. Assume mean ﬂuid temperatures to be 638C and 358C for oil and water, respectively. Shell-side inside diameter Ds ¼ 0:336 m Tube-side outside diameter do ¼ 19:0 mm Tube-side inside diameter di ¼ 16:6 mm Tube pitch pt ¼ 25:0 mm Tube bundle layout ¼ 458 Central baﬄe spacing Lb;c ¼ 0:279 m Inlet baﬄe spacing Lb;i ¼ 0:318 m Outlet baﬄe spacing Lb;o ¼ 0:318 m Baﬄe cut ‘c ¼ 86:7 mm or 25.8% Tube material ¼ admiralty (70% Cu, 30% Ni)

Number of sealing strip pairs Nss ¼ 1 Total number of tubes Nt ¼ 102 Tube length L ¼ 4:3 m Width of bypass lane wp ¼ 19:0 mm Number of tube passes np ¼ 2 Number of pass partitions Np ¼ 2 Diameter of the outer tube limit Dotl ¼ 0:321 m Tube-to-baﬄe hole diametral clearance tb ¼ 0:794 mm Shell-to-baﬄe diametral clearance sb ¼ 2:946 mm Thermal conductivity of tube wall kw ¼ 111 W/m K

Operating conditions: Oil ﬂow rate m_ oil ¼ m_ s ¼ 36:3 kg=s Oil inlet temperature Ts;i ¼ 65:68C Oil side fouling factor ^ o; f ¼ 0:000176 m2 W=K R

Fluid Oil at 638C Seawater at 358C

Water ﬂow rate m_ water ¼ m_ t ¼ 18:1 kg=s Water inlet temperature Tt;i ¼ 32:28C Water side fouling factor ^ i; f ¼ 0:000088 m2 W=K R

Density s (kg/m3 )

Speciﬁc heat cp (J/kg K)

Dynamic Viscosity (Pa s)

849 993

2094 4187

64:6 103 0:723 103

Thermal Prandtl Conductivity Number k (W/m2 KÞ Pr 0.140 0.634

966 4.77

Use the Dittus–Boelter correlation [Eq. (7.80) in Table 7.6] for the tube-side heat transfer coeﬃcient. Use the McAdams correlation [Eq. (7.72) in Table 7.6] for the tubeside friction factor. For the shell-side friction factor and Nusselt numbers, use the follow-

SHELL-AND-TUBE HEAT EXCHANGERS

653

ing correlations: fid ¼ 3:5ð1:33do =pt Þb Re0:476 , where b ¼ 6:59=ð1 þ 0:14 Re0:52 s s Þ; 0:4 0:36 0:25 : Nus ¼ 1:04 Red Prs ðPrs =Prw Þ SOLUTION Problem Data and Schematic: The schematic of the 1–2 TEMA E shell-and-tube heat exchanger is given in Fig. 1.5b with characteristic heat exchanger zones and dimensions shown in Figs. 8.9 and 8.11. All major geometric dimensions, operating conditions, and thermophysical properties of the ﬂuids are given in the problem statement. Determine: This rating problem requires determination of the heat transfer rate, outlet temperatures, and pressure drops for each ﬂuid. Assumptions: The assumptions invoked in Section 3.2.1 applicable to a shell-and-tube exchanger are valid. Analysis: We follow the solution procedure for this rating problem in several steps as outlined preceding this problem. First, all geometric characteristics of the shell side are determined as detailed in Example 8.3. Heat transfer coeﬃcients on both the shell side (as outlined in Section 9.5.1) and the tube side are then calculated. Subsequently, the overall heat transfer coeﬃcient and design parameters for the given operating point are computed. With all these data, the heat transfer rate and outlet temperatures are calculated in a straightforward manner. Finally, determination of pressure drops completes the procedure. Before we outline the details, some required geometrical characteristics are obtained from Example 8.3 as follows: Ao;cr ¼ 0:03275 m2 Ao;bp ¼ 0:00949 m2

Fc ¼ 0:6506 Nr;cc ¼ 9

Ao;sb ¼ 0:001027 m2 Nb ¼ 14

Ao;tb ¼ 0:001995 m2 Ao;w ¼ 0:01308 m2

Shell-Side Heat Transfer Coeﬃcient. We calculate the shell-side velocity, Reynolds number, ideal heat transfer coeﬃcient, and then correct it for various leakage and bypass ﬂow streams. Shell-side mass velocity Gs ¼

36:3 kg=s m_ s ¼ ¼ 1108 kg=m2 s Ao;cr 0:03275 m2

Shell-side Reynolds number Res ¼

Gs do 1108 kg=m2 s 0:0190 m ¼ ¼ 326 s 64:6 103

Now we compute Nus from the given correlation with Red ¼ Res . Note that we have not calculated Tw , so we cannot calculate Prw . So in this iteration, we consider Prs ¼ Prw . 0:36 Nus ¼ 1:04Re0:4 d Prs

hid ¼

Prs Prw

0:25 ¼ 1:04 ð326Þ0:4 ð966Þ0:36 ¼ 125:0 W=m2 K

Nus k w 0:14 125:0 0:140 W=m2 K 0:14 ¼ ¼ 921:0 W=m2 K ð1Þ m do 0:0190 m

654

HEAT EXCHANGER DESIGN PROCEDURES

Baﬄe cut and spacing eﬀect correction factor Jc ¼ 0:55 þ 0:72Fc ¼ 0:55 þ 0:72 0:6506 ¼ 1:018 To calculate the tube-to-baﬄe and baﬄe-to-shell leakage factor J‘ from Table 9.2, we need to calculate rs and rlm as follows:

rs ¼

Ao;sb 0:001027 m2 ¼ ¼ 0:3398 Ao;sb þ Ao;tb 0:001027 m2 þ 0:001995 m2

rlm ¼

Ao;sb þ Ao;tb 0:001027 m2 þ 0:001995 m2 ¼ ¼ 0:0923 Ao;cr 0:03275 m2

J‘ ¼ 0:44ð1 rs Þ þ ½1 0:44ð1 rs Þe2:2rlm ¼ 0:44 ð1 0:3398Þ þ ½1 0:44 ð1 0:3398Þe2:20:0923 ¼ 0:8696 Let us now calculate Jb using the formula from Table 9.2 after we determine C (for Res ¼ 326), rb , and Nssþ as follows: C ¼ 1:25

rb ¼

Ao;bp 0:00949 m2 ¼ ¼ 0:2898 Ao;cr 0:03275 m2

Nssþ ¼

Nss 1 ¼ ¼ 0:1111 Nr;cc 9

Jb ¼ expfCrb ½1 ð2Nssþ Þ1=3 g ¼ expf1:25 0:2898 ½1 ð2 0:1111Þ1=3 g ¼ 0:8669 þ Now we compute Lþ i and Lo for determining unequal baﬄe spacing factor Js from Table 9.2.

Lþ i ¼

Js ¼

Lb;i Lb;o 0:318 m ¼ Lþ ¼ ¼ 1:14 o ¼ Lb;c Lb;c 0:279 m

n ¼ 0:6 for turbulent flow ðRes > 100Þ

ð1nÞ ð1nÞ Nb 1 þ ðLþ þ ðLþ 14 1 þ ð1:14Þ0:4 þ ð1:14Þ0:4 oÞ i Þ ¼ 0:9887 ¼ þ 14 1 þ 1:14 þ 1:14 Nb 1 þ Li þ Lþ o

Finally, the adverse temperature gradient factor Jr ¼ 1 for Res ¼ 326 > 100. Since all correction factors J are determined, the actual shell-side heat transfer coeﬃcient is given by hs ¼ ho ¼ hid Jc J‘ Jb Js Jr ¼ 921:0 W=m2 K 1:018 0:8696 0:8669 0:9887 1 ¼ 698:8 W=m2 K This heat transfer coeﬃcient should be corrected for the ﬂuid property variations as outlined in Section 7.6.1 once the wall temperature is calculated in the next iteration.

SHELL-AND-TUBE HEAT EXCHANGERS

655

Tube-Side Heat Transfer Coeﬃcient Number of tubes per pass Nt; p ¼

Nt 102 ¼ ¼ 51 2 2

Tube-side flow area per pass Ao;t ¼ Tube-side Reynolds number Ret ¼

2 d N ¼ ð0:0166 mÞ2 51 ¼ 0:01104 m2 4 i t; p 4

18:1 kg=s 0:0166 m m_ t di ¼ 37,643 ¼ Ao;t t 0:01104 m2 ð0:723 103 Pa sÞ

Nusselt number Nut ¼ 0:024Re0:8 Pr0:4 ¼ 0:024 ð37,643Þ0:8 ð4:77Þ0:4 ¼ 205:2 Heat transfer coefficient ht ¼ hi ¼

ðNukÞt 205:2 0:634 W=m K ¼ ¼ 7837 W=m2 K 0:0166 m di

Overall Heat Transfer Coeﬃcient. From Eq. (3.31a), 1 1 ^ o; f þ do lnðdo =di Þ þ R ^ i; f do þ 1 do ¼ þR di hi di Uo ho 2kw ¼

1 0:0190 m lnð0:0190 m=0:0166 mÞ þ 0:000176 m2 K=W þ 2 111 W=m K 698:8 W=m2 K 0:0190 m 1 0:0190 m þ þ 0:000088 m2 K=W 0:0166 m 7837 W=m2 K 0:0166 m

¼ ð0:001431 þ 0:000176 þ 0:0000116 þ 0:0001007 þ 0:000146Þ m2 K=W ¼ 0:0018653 m2 K=W or Uo ¼ 536:1 W=m2 K The unit thermal resistance 1=Uo in the calculation above indicates the individual unit thermal resistances as 76.7, 9.5, 0.6, 5.4, and 7.8%. Thus, the largest thermal resistance is on the shell side, and the fouling thermal resistances and wall thermal resistance are of the same order of magnitude as the tube-side thermal resistance. Total tube outside heat transfer area As ¼ At;o ¼ Ldo Nt ¼ 4:3 m 0:0190 m 102 ¼ 26:180 m2 Cmin ¼ Ct ¼ ðm_ cp Þt ¼ 18:1 kg=s 4187 J=kg K ¼ 75,785 W=K Cmax ¼ Cs ¼ ðm_ cp Þs ¼ 36:3 kg=s 2094 J=kg K ¼ 76,012 W=K C* ¼

Cmin 75,785 W=K ¼ ¼ 0:997 1 Cmax 76,012 W=K

656

HEAT EXCHANGER DESIGN PROCEDURES

Number of heat transfer units NTU ¼

Uo At;o Uo At;o 536:1 W=m2 K 26:180 m2 ¼ ¼ ¼ 0:1852 Cmin Ct 75,785 W=K

Heat exchanger effectiveness; using the formula from Table 3:3; is pﬃﬃﬃ pﬃﬃﬃ 2 2 pﬃﬃﬃ ¼ pﬃﬃﬃ pﬃﬃﬃ ¼ 0:1555 " ¼ pﬃﬃﬃ 2 þ cothðNTU= 2Þ 2 þ cothð0:1852= 2Þ

Heat Transfer Rate and Exit Temperatures Heat transfer rate q ¼ "Cmin ðTs;i Tt;i Þ ¼ 0:1555 75,785 W=K ð65:6 32:2Þ8C ¼ 393,600 W ¼ 393:6 kW Oil exit temperature Ts;o ¼ Ts;i "C*ðTs;i Tt;i Þ ¼ 65:68C 0:1555 0:997 ð65:6 32:2Þ8C ¼ 60:48C Water exit temperature Tt;o ¼ Tt;i þ "ðTs;i Tt;i Þ ¼ 32:28C þ 0:1555 ð65:6 32:2Þ8C ¼ 37:48C Mean temperaturesy :

Ts;i þ Ts;o ð65:6 þ 60:4Þ8C ¼ ¼ 63:08C 2 2 Tt;i þ Tt;o ð32:2 þ 37:4Þ8C ¼ ¼ ¼ 34:88C 2 2

Ts;m ¼ Tt;m

Pressure Drop Calculations. To compute the idealized tube bundle pressure drop, we ﬁrst calculate the ideal friction factor using the given formula: d b 1:33 19:0 mm 1:72 fid ¼ 3:5 1:33 o Re0:476 ¼ 3:5 ð326Þ0:476 ¼ 0:2269 s pt 25:0 mm where 6:59 6:59 ¼ ¼ 1:72 0:52 1 þ 0:14Res 1 þ 0:14 ð326Þ0:52 4fid G2s Nr;cc w 0:25 4 0:2269 ð1108 kg=m2 sÞ2 9 0:25 ¼ ¼ ð1Þ ¼ 5906 Pa 2gc s m 2 1 849 kg=m3

b¼ pb;id

To calculate the pressure drop in the crossﬂow section, we ﬁrst compute the correction factors b and ‘ using the expressions from Table 9.3.

b ¼ expfDrb ½1 ð2Nssþ Þ1=3 g ¼ expf3:7 0:2898 ½1 ð2 0:1089Þ1=3 g ¼ 0:6524 y The mean temperatures calculated (638C and 358C for oil and water, respectively) are the same as the values purposely assumed in the problem formulation for thermophysical properties. Consequently, no iterations are required for changes in thermophysical property values for this problem.

SHELL-AND-TUBE HEAT EXCHANGERS

657

using D ¼ 3:7 for Res > 100 and rb ¼ 0:2898 as calculated for Jb earlier.

‘ ¼ exp½1:33ð1 þ rs Þrplm ¼ exp½1:33 ð1 þ 0:3398Þð0:0923Þ0:60 ¼ 0:6527 where p from the formula of Table 9.3 is given by p ¼ ½0:15ð1 þ rs Þ þ 0:8 ¼ ½0:15 ð1 þ 0:3398Þ þ 0:8 ¼ 0:60 Hence, pcr and pw from Eq. (9.51) with pw;id from Eq. (6.39a) are given by pcr ¼ pb;id ðNb 1Þ b ‘ ¼ 5906 Pa ð14 1Þ 0:6524 0:6527 ¼ 32,694 Pa pw ¼ Nb ð2 þ 0:6Nr;cw Þ

G2w ð1754 kg=m2 sÞ2

‘ ¼ 14 ð2 þ 0:6 3Þ 0:6527 2gc s 2 1 849 kg=m3

¼ 62,914 Pa where Gw ¼

m_ s Ao;cr Ao;w

1=2 ¼

36:3 kg=s ð0:03275 m2 0:01308 m2 Þ1=2

¼ 1754 kg=m2 s

Next let us determine pressure drop in inlet–outlet sections using Eq. (9.51) after computing s from Table 9.3.

s ¼

Lb;c Lb;o

2n0 0 Lb;c 2n 0:279 m 20:2 0:279 m 20:2 þ ¼ þ ¼ 1:5803 Lb;i 0:318 m 0:318 m

where n 0 ¼ 0:2

Nr;cw 3 pio ¼ 2pb;id 1 þ

b s ¼ 2 5906 Pa 1 þ 0:6524 1:5803 Nr;cc 9 ¼ 16,237 Pa

Then ps ¼ pcr þ pw þ pio ¼ ð32,694 þ 62,914 þ 16,237Þ Pa ¼ 111,845 Pa ¼ 112 kPa ð29:2%Þ ð56:3%Þ ð14:5%Þ Note that for this problem, the window section pressure drop is more than the crossﬂow section pressure drop, whereas the crossﬂow section provides most of heat transfer. Thus the window section results in excessive pressure drop with insigniﬁcant contribution to heat transfer. Tube-Side Pressure Drop. From Eq. (7.72) in Table 7.6, ¼ 0:046 ð37,643Þ0:2 ¼ 0:005593 f ¼ 0:046Re0:2 t

658

HEAT EXCHANGER DESIGN PROCEDURES

From Fig. 6.3, 2ðpt do Þ 2 ð25:0 19:0Þ mm pﬃﬃﬃ pﬃﬃﬃ ¼ 0:34 ¼ 2pt 2 25:0 mm 4 fL m_ 2t 2 2 pt ¼ þ ð1 þ K Þ ð1 K Þ np c e 2gc t A2o;t di ð18:1 kg=sÞ2 4 0:005593 4:3 m ¼ þ 0:3 þ 0:4 2 0:0166 m 2 1 993 kg=m3 ð0:01104 m2 Þ2 Kc ¼ 0:3

Ke ¼ 0:4

for ¼

¼ 17,582 Pa ¼ 17:58 kPa Here since the inlet and outlet densities for water will not change appreciably and that information is not given, we have considered i ¼ o ¼ m : Discussion and Comments: Despite the seemingly elaborate calculation procedure, this rating problem solution is straightforward. In principle, the calculations must be performed iteratively due to the unknown mean ﬂuid temperatures. In this particular example, the mean temperatures were considered initially to be equal to the values calculated subsequently. To correct for the property variation, the shell-side Nus needs to be calculated again once the thermal resistances are known on both sides to compute Tw and hence Prw . The Tw can be computed using the same procedure as that outlined in Example 9.1. 9.5.3

Approximate Design Method

The objectives of the approximate design (sizing) method for a given service of a shelland-tube heat exchanger are several-fold: (1) quick conﬁguration and size estimation, (2) cost estimation, (3) plant layout, or (4) checking the results of a sophisticated computer program. The basis for this method is Eq. (3.184) rearranged as follows for the shell-side or tube outside surface area: As ¼

q q ¼ Uo Tm Uo F Tlm

ð9:54Þ

Here Uo ¼ Us is the overall heat transfer coeﬃcient based on tube outside or shell-side surface area. By approximately but rapidly estimating q, Uo , F, and Tlm , one can arrive at the approximate surface area requirement and subsequently the size of the exchanger, as discussed next. Since this is a sizing procedure, either the heat duty and inlet temperatures are given or both inlet and outlet temperatures are speciﬁed. They are related by the energy balance of Eq. (3.5) as follows by considering the shell ﬂuid as hot ﬂuid: q ¼ m_ s cp;s ðTs;i Ts;o Þ ¼ m_ cp;t ðTt;o Tt;i Þ

ð9:55Þ

The overall heat transfer coeﬃcient Uo of Eq. (9.54) is calculated from Eq. (3.31a) as 1 1 1 d lnðdo =di Þ d d ¼ þ þ o þ o þ o hi; f di hi di Uo ho ho; f 2kw

ð9:56Þ

Here hi , ho , hi; f ; and ho; f (the subscripts i and o denote tube inside and tube outside or shell side) are selected from Table 9.4. It should be emphasized that the values given in this table are based on the usual velocities or nominally allowable pressure drops; allow-

SHELL-AND-TUBE HEAT EXCHANGERS

659

ance should be made on operating conditions that are quite unusual. Also, care should be exercised as noted in the appropriate footnotes of this table. The log-mean temperature diﬀerence correction factor F should be estimated as follows. The correction factor F ¼ 1 for a counterﬂow exchanger or if one stream changes its temperature only slightly in the exchanger. For a single TEMA E shell with an arbitrary even number of tube-side passes, the correction factor should be F > 0:8 if there is no temperature cross; a rough value would be F ¼ 0:9 unless it can be determined from Fig. 3.13. Consider F ¼ 0:8 when the outlet temperatures of the two streams are equal (thus avoiding the temperature cross). If Ts;o < Tt;o ; there exists a temperature cross (we have assumed here that the shell ﬂuid is hotter than the tube ﬂuid) in the multipass exchanger; and in this case, multiple shells in series should be considered. They can be determined by the procedure outlined in Fig. 3.18. For the known inlet temperatures and given or calculated outlet temperatures, compute the log-mean temperature diﬀerence Tlm from its deﬁnition of Eqs. (3. 172) and (3. 173). Knowing all the parameters on the right-hand side of Eq. (9.54), the tube outside total surface area As (including the ﬁn area, if any) can then be estimated from this equation. 9.5.3.1 Exchanger Dimensions. To relate the As above to the shell inside diameter and the eﬀective tube length, we will use the information shown in Fig. 9.4 (Bell, 1998). It is generated for one of the commonly used fully tubed shell-and-tube heat exchangers that

FIGURE 9.4 Tube outside (shell-side) surface area As as a function of shell inside diameter and eﬀective tube length for a tube bundle having 19.05-mm (34-in.)-outside-diameter plain tubes, 23.8mm (15 16-in.) equilateral triangular tube layout, single tube-side pass, and fully tubed exchanger with ﬁxed tubesheets. (From Bell, 1998.)

660

HEAT EXCHANGER DESIGN PROCEDURES

TABLE 9.4 Typical Film Heat Transfer Coeﬃcients and Fouling Factors for Shell-and-Tube Heat Exchangers Fluid Conditions Sensible heat transfer Waterc Ammonia Light organicsd Medium organicse Heavy organics f

Very heavy organicsg

Gash

Condensing heat transfer Steam, ammonia

Light organicsd

Medium organicse Heavy organics Light multicomponent mixtures, all condensabled Medium multicomponent mixtures, all condensablee Heavy multicomponent mixtures, all condensablef Vaporizing heat transferp;q Waterr

Ammonia Light organicsd

Medium organicse

Heavy organicsf

h (W/m2 KÞa;b

Fouling Resistance (m2 K=WÞa

5,000–7,500 6,000–8,000 1,500–2,000 750–1,500

1–2.5 104 0–1 104 0–2 104 1–4 104

250–750 150–400

2–10 104 2–10 104

100–300 60–150 80–125 250–400 500–800

4–30 103 4–30 103 0–1 104 0–1 104 0–1 104

Pressure 10 kN/m2 abs, no noncondensablesi;j Pressure 10 kN/m2 abs, 1% noncondensablesk Pressure 10 kN/m2 abs, 4% noncondensablesk Pressure 100 kN/m2 abs, no noncondensablesi;j;k;l Pressure 1 MN/m2 abs, no noncondensablesi;j;k;l Pure component, pressure 10 kN/m2 abs, no noncondensablesi Pressure 10 kN/m2 abs, 4% noncondensablesk Pure component, pressure 100 kN/m2 abs, no noncondensables Pure component, pressure 1 MN/m2 abs Pure component or narrow condensing range, pressure 100 kN/m2 absm;n Narrow condensing range, pressure 100 kN/m2 absm;n Medium condensing range, pressure 100 kN/m2 absk;m;o Medium condensing range, pressure 100 kN/m2 absk;m;o Medium condensing range, pressure 100 kN/m2 absk;m;o

8,000–12,000 4,000–6,000 2,000–3,000 10,000–15,000

0–1 104 0–1 104 0–1 104 0–1 104

15,000–25,000

0–1 104

1,500–2,000

0–1 104

750–1,000 2,000–4,000

0–1 104 0–1 104

3,000–4,000 1,500–4,000

0–1 104 1–3 104

600–2,000

2–5 104

1,000–2,500

0–2 104

600–1,500

1–4 104

300–600

2–8 104

Pressure < 0:5 MN/m2 abs, TSH;max ¼ 25 K Pressure < 0:5 MN=m2 abs, pressure < 10 MN/m2 abs, TSH;max ¼ 20 K Pressure < 3MN=m2 abs, TSH;max ¼ 20 K Pure component, pressure < 2 MN=m2 abs, TSH;max ¼ 20 K Narrow boiling range,s pressure < 2 MN/m2 abs, TSH;max ¼ 15 K Pure component, pressure < 2 MN/m2 abs, TSH;max ¼ 20 K Narrow boiling range,s pressure < 2 MN/m2 abs, TSH;max ¼ 15 K Pure component, pressure < 2 MN/m2 abs, TSH;max ¼ 20 K

3,000–10,000 4,000–15,000

1–2 104 1–2 104

3,000–5,000 1,000–4,000

0–2 104 1–2 104

750–3,000

0–2 104

1,000–3,500

1–3 104

600–2,500

1–3 104

750–2,500

2–5 104

Liquid Liquid Liquid Liquid Liquid Heating Cooling Liquid Heating Cooling Pressure 100–200 kN/m2 abs Pressure 1 MN/m2 abs Pressure 10 MN/m2 abs

SHELL-AND-TUBE HEAT EXCHANGERS

661

Heavy organicsg

Narrow boiling range,s pressure < 2 MN/m2 abs, TSH;max ¼ 15 K

400–1,500

28 104

Very heavy organicsh

Narrow boiling range,s pressure < 2 MN/m2 abs, TSH;max ¼ 15 K

300–1,000

2–10 104

Source: Bell, K. J. (1998). a Heat transfer coeﬃcients and fouling resistances are based on area in contact with ﬂuid. Ranges shown are typical, not all encompassing. Temperatures are assumed to be in normal processing range; allowances should be made for very high or low temperatures. b Allowable pressure drops on each side are assumed to be about 50–100 kN/m2 except for (1) low-pressure gas and two-phase ﬂows, where the pressure drop is assumed to be about 5% of the absolute pressure; and (2) very viscous organics, where the allowable pressure drop is assumed to be about 150–250 kN/m2 . c Aqueous solutions give approximately the same coeﬃcients as water. d Light organics include ﬂuids with liquid viscosities less than about 0:5 103 N s=m2 , such as hydrocarbons through C8 , gasoline, light alcohols and ketones, etc. e Medium organics include ﬂuids with liquid viscosities between about 0:5 103 and 2:5 103 N s=m2 , such as kerosene, straw oil, hot gas oil, and light crudes. f Heavy organics include ﬂuids with liquid viscosities greater than 2:5 103 N s=m2 , but not more than 50 103 N s=m2 , such as cold gas oil, lube oils, fuel oils, and heavy and reduced crudes. g Very heavy organics include tars, asphalts, polymer melts, greases, etc., having liquid viscosities greater than about 50 103 N s=m2 . Estimation of coeﬃcients for these materials is very uncertain and depends strongly on the temperature diﬀerence, because natural convection is often a signiﬁcant contribution to heat transfer in heating, whereas congelation on the surface and particularly between ﬁns can occur in cooling. Since many of these materials are thermally unstable, high surface temperatures can lead to extremely severe fouling. h Values given for gases apply to such substances as air, nitrogen, carbon dioxide, light hydrocarbon mixtures (no condensation), etc. Because of the very high thermal conductivities and speciﬁc heats of hydrogen and helium, gas mixtures containing appreciable fractions of these components will generally have substantially higher heat transfer coeﬃcients. i Superheat of a pure vapor is removed at the same coeﬃcient as for condensation of the saturated vapor if the exit coolant temperature is less than the saturation temperature (at the pressure existing in the vapor phase) and if the (constant) saturation temperature is used in calculating the MTD. But see note k for vapor mixtures with or without noncondensable gas. j Steam is not usually condensed on conventional low-ﬁnned tubes; its high surface tension causes bridging and retention of the condensate and a severe reduction of the coeﬃcient below that of the plain tube. k The coeﬃcients cited for condensation in the presence of noncondensable gases or for multicomponent mixtures are only for very rough estimation purposes because of the presence of mass transfer resistances in the vapor (and to some extent, in the liquid) phase. Also, for these cases, the vapor-phase temperature is not constant, and the coeﬃcient given is to be used with the MTD estimated using vapor-phase inlet and exit temperatures, together with the coolant temperatures. l As a rough approximation, the same relative reduction in low-pressure condensing coeﬃcients due to noncondensable gases can also be applied to higher pressures. m Absolute pressure and noncondensables have about the same eﬀect on condensing coeﬃcients for medium and heavy organics as for light organics. For large fractions of noncondensable gas, interpolate between pure component condensation and gas cooling coeﬃcient. n Narrow condensing range implies that the temperature diﬀerence between dew point and bubble point is less than the smallest temperature diﬀerence between vapor and coolant at any place in the condenser. o Medium condensing range implies that the temperature diﬀerence between dew point and bubble point is greater than the smallest temperature diﬀerence between vapor and coolant, but less than the temperature diﬀerence between inlet vapor and outlet coolant. p Boiling and vaporizing heat transfer coeﬃcients depend very strongly on the nature of the surface and the structure of the two-phase ﬂow past the surface in addition to all of the other variables that are signiﬁcant for convective heat transfer in other modes. The ﬂow velocity and structure are very much governed by the geometry of the equipment and its connecting piping. Also, there is a maximum heat ﬂux from the surface that can be achieved with reasonable temperature diﬀerences between surface and saturation temperatures of the boiling ﬂuid; any attempt to exceed this maximum heat ﬂux by increasing the surface temperature leads to partial or total coverage of the surface by a ﬁlm of vapor and a sharp decrease in the heat ﬂux. Therefore, the vaporizing heat transfer coeﬃcients given in this table are only for very rough estimating purposes and assume the sue of plain or low-ﬁnned tubes without special nucleation enhancement. TSH;max is the maximum allowable temperature diﬀerence between surface and saturation temperature of the boiling liquid. No attempt is made in this table to distinguish among the various types of vapor-generation equipment, since the major heat transfer distinction to be made is the propensity of the process stream to foul. Severely fouling streams will usually call for a vertical thermosiphon or a forced-convection (tube-side) reboiler for ease of cleaning. q Subcooling heat load is transferred at the same coeﬃcient as latent heat load in kettle reboilers, using the saturation temperature in the MTD. For horizontal and vertical thermosiphons and forced-circulation reboilers, a separate calculation is required for the sensible heat transfer area, using appropriate sensible heat transfer coeﬃcients and the liquid temperature proﬁle for the MTD. r Aqueous solutions vaporize with nearly the same coeﬃcient as pure water if attention is given to boiling-point elevation, if the solution does not become saturated, and if care is taken to avoid dry wall conditions. s For boiling of mixtures, the saturation temperature (bubble point) of the ﬁnal liquid phase (after the desired vaporization has taken place) is to be used to calculate the MTD. A narrow-boiling-range mixture is deﬁned as one for which the diﬀerence between the bubble point of the incoming liquid and the bubble point of the exit liquid is less than the temperature diﬀerence between the exit hot stream and the bubble point of the exit boiling liquid. Wide-boiling-range mixtures require a case-by-case analysis and cannot be reliably estimated by these simple procedures.

662

HEAT EXCHANGER DESIGN PROCEDURES

has the following geometry: 19.05 mm (34 in.) outside diameter plain tubes, 23.8 mm (15 16 in:) equilateral triangular tube layout, single tube-side pass, and ﬁxed tubesheets. In this ﬁgure, the eﬀective tube length represents the actual tube length between tubesheets for the straight tube exchanger and the length between the tubesheet and the tangent line for the U-tube bundle. The solid black lines indicate the shell inside diameter. From the estimated value of As above, one can calculate a number of combinations of the eﬀective tube length Leff and the shell inside diameter Ds . The desired range of Leff =Ds (shown by dashed lines in Fig. 9.4) is between 3 and 15, with a preferable range between 6 and 10. Leff =Ds 3 results in poor shell-side ﬂow distribution and high p for the inlet and outlet nozzles. Leff =Ds 15 would be diﬃcult to handle mechanically and would require a longer footprint for the tube bundle repair/removal. We now explain how to use Fig. 9.4 for diﬀerent tube diameters and layouts, tube-side multipass construction, and other tube bundle constructions. The eﬀective tube-side surface area for geometry diﬀerent from that for Fig. 9.4 will be designated as As0 . The ordinate of Fig. 9.4 is then renamed As0 : It is related to As calculated from Eq. (9.54) as As0 ¼ As F1 F2 F3

ð9:57Þ

Once we calculate the correction factors F1 , F2 , and F3 as outlined next and As from Eq. (9.54), A0s is computed from Eq. (9.57), and the combination of the eﬀective tube length and the shell inside diameter is then determined from Fig. 9.4 as before. Let us describe how to calculate the correction factors. F1 ¼ correction factor for the tube outside diameter and tube layout. F1 ¼ 1 for 19.05 mm tubes having a 23.8 mm 308 tube layout. For other do and pt , obtain the value from Table 9.5. F2 ¼ correction factor for the number of tube passes. F2 ¼ 1 for a one-tube-pass design. The value of F2 for multiple tube passes (U-tube and ﬂoating head bundles) can be obtained from Table 9.6. F3 ¼ correction factor for various rear-end head designs (shell construction) given in Table 9.7. TABLE 9.5 Values of F1 of Eq. (9.56) for Various Tube Diameters and Layouts Tube Outside Diameter [in. (mm)] 5 8 5 8 3 4 3 4 3 4 3 4

Tube Pitch [in. (mm)]

Layout

F1

(20.6)

!!

0.90

(20.6)

! ^, &

1.04

(23.8)

!!

1.00

(23.8)

! ^, &

1.16

(19.05)

13 16 13 16 15 16 15 16

(19.05)

1 (25.4)

!!

1.14

(19.05)

1 (25.4)

! ^, &

1.31

(31.8)

!!

1.34

(31.8)

! ^, &

1.54

(15.88) (15.88) (19.05)

1 (25.4) 1 (25.4) Source: Data from Bell (1998).

114 114

663

SHELL-AND-TUBE HEAT EXCHANGERS

TABLE 9.6 Values of F2 for Various Numbers of Tube-Passesa F2 for Number of Tube-Side passes Inside Shell Diameter [in. (mm)]

2

4

6

8

Up to 12 (305)

1.20

1.40

1.80

—

1314 to 1714 (337 to 438)

1.06

1.18

1.25

1.50

1914 to 2314 (489 to 591)

1.04

1.14

1.19

1.35

25 to 33 (635 to 838)

1.03

1.12

1.16

1.20

35 to 45 (889 to 1143)

1.02

1.08

1.12

1.16

48 to 60 (1219 to 1524) Above 60 (above 1524)

1.02 1.01

1.05 1.03

1.08 1.04

1.12 1.06

Source: Data from Bell (1998). a Since U-tube bundles must always have at least two passes, use of this table is essential for U-tube bundles estimation. Most ﬂoating-head buindles also require an even number of passes.

TABLE 9.7 Values of F3 for Various Tube Bundle Constructions F3 for Inside Shell Diameter [in. (mm)] Type of Tube bundle Construction

Up to 12 (305)

13–22 (330–559)

23–36 (584–914)

37–48 (940–1219)

Above 48 (above 1219)

Fixed tubesheet (TEMA L, M or N) Split backing ring (TEMA S) Outside packed ﬂoating head (TEMA P) U-tubea (TEMA U) Pull-through ﬂoating head (TEMA T)

1.00

1.00

1.00

1.00

1.00

1.30 1.30

1.15 1.15

1.09 1.09

1.06 1.06

1.04 1.04

1.12 —

1.08 1.40

1.03 1.25

1.01 1.18

1.01 1.15

Source: Data from Bell (1998). a Since U-tube bundles must always have at least two passes, it is also essential to use Table 9.6 for this conﬁguration.

9.5.4

More Rigorous Thermal Design Method

The more rigorous thermal design method includes all elements discussed in preceding sections. In a concise manner, the following is a step-by-step procedure for the design or sizing problem. 1. For given heat transfer duty and ﬂuid streams inlet temperatures, compute the outlet temperatures using overall energy balances and the ﬂuid mass ﬂow rates speciﬁed or selected. If outlet temperatures are given, compute the heat duty requirement. 2. Select a preliminary ﬂow arrangement (i.e., type of shell-and-tube heat exchanger) based on common industry practice (see Section 10.2.1 for selection criteria), mechanical integrity, and maintenance requirements.

664

HEAT EXCHANGER DESIGN PROCEDURES

3. Follow the approximate design method of Section 9.5.3 to arrive at a preliminary size for the exchanger. Select a shell inside diameter, tube diameter, length, pitch and layout, and bafﬂe spacing. Calculate the number of tubes and number of passes. 4. Follow the rating procedure outlined in Section 9.5.2, which employs the Bell– Delaware method (see Section 9.5.1 for heat transfer coefﬁcient calculations), or apply the stream analysis method (Taborek, 1998; see Section 4.4.1.4) or other available rating procedure. 5. Compare the heat transfer and pressure drop performance computed in step 4 with the values speciﬁed. If heat transfer is met and computed pressure drops are within speciﬁcations, the thermal design is ﬁnished. In that case, the mechanical design is pursued in parallel and series to thermal design to ensure structural integrity and compliance with applicable codes and standards. Also, a check for ﬂow-induced vibration (and/or other operating problems) and a cost estimation are performed to ﬁnalize the design. 6. If heat transfer is not met or the computed pressure drop(s) are higher than the speciﬁcations, go to step 3 and select the appropriate shell-and-tube geometry, and iterate through step 5 until thermal, mechanical, and cost estimation criteria are met.

9.6

HEAT EXCHANGER OPTIMIZATION

In the preceding sections, rating problems for extended surface and shell-and-tube heat exchangers are presented, as is a sizing problem for an extended surface exchanger. For the sizing problem, no constraints were imposed on the design except for the pressure drops speciﬁed. The objective of that problem was to optimize the core dimensions to meet the heat transfer required for speciﬁed pressure drops. Heat exchangers are designed for many diﬀerent applications, and hence may involve many diﬀerent optimization criteria. These criteria for heat exchanger design may be minimum initial cost, minimum initial and operating costs, minimum weight or material, minimum volume or heat transfer surface area, minimum frontal area, minimum labor (translated into a minimum number of parts), and so on. When a performance measure has been deﬁned quantitatively and is to be minimized or maximized, it is called an objective function in a design optimization. A particular design may also be subjected to certain requirements, such as required heat transfer, allowable pressure drop, limitations on height, width and/or length of the exchanger, and so on. These requirements are called constraints in a design optimization. A number of diﬀerent surfaces could be incorporated in a speciﬁc design problem, and there are many geometrical variables that could be varied for each surface geometry.y In addition, operating mass ﬂow rates and temperatures could be changed. Thus, a large number of design variables are associated with a heat exchanger design. The question arises as to how one can eﬀectively adjust these design variables within imposed constraints and come up with a design having an optimum objective function. This is what we mean by the optimum component y For a shell-and-tube exchanger, the geometrical variables are those associated with the tube, baﬄes, shell, and front- and rear-end heads. For an extended surface exchanger, the geometrical variable associated with a ﬁn are the ﬁn pitch, ﬁn height, ﬁn thickness, type of ﬁn, and other variables associates with each ﬁn type.

HEAT EXCHANGER OPTIMIZATION

Alternate designs: construction types, flow arrangements, surface selection, etc.

Designe'r constraints and design variables

Problem specifications including customer's constraints and design variables

Total constraints and design variables for the optimization problem

Problem formulation

Heat exchanger design computer programs

Thermophysical properties

Changed geometry and /or operating conditions specified by design variables

Geometrical properties

Fixed operating conditions

Heat transfer and pressure drop evaluation

scaled j and f factors

}

Objective function and constraints evaluation

Optimization package

665

Optimization strategy for redefining the design variables

Optimum solution

FIGURE 9.5 Methodology for heat exchanger optimization. (From Shah et al. 1978.)

design, sometimes also referred to as the most eﬃcient design. If a heat exchanger is part of a system, it could also be optimized based on the system objective function by varying pertinent exchanger design variables as well as system variables in the optimization routine. A complete mathematical component or system-based optimization of heat exchanger design is neither practical nor possible. Many engineering judgments based on experi-

666

HEAT EXCHANGER DESIGN PROCEDURES

ence are involved in various stages of the design. However, once the general conﬁguration and surfaces are selected, an optimized heat exchanger design may be arrived at if the objective function and constraints can be expressed mathematically and if all the variables are changed automatically and systematically on some evaluation criteria basis. A large number of optimization (search) techniques are available in the literature, and quite a lot of commercial optimization software is available. A typical design and optimization procedure for a heat exchanger is summarized here with the ﬂowchart of Fig. 9.5 for completeness. The procedure is referred to as the case study method. In this method, each possible surface geometry and construction type is considered to be an alternative design, as indicated in Fig. 9.5. To make a legitimate comparison of these alternatives, each design must be optimized for the application speciﬁed. Thus there may be several independent optimized solutions satisfying the problem requirements. Engineering judgment, a comparison of objective function values, and other evaluation criteria are then applied to select a ﬁnal optimum solution for implementation. Assume a liquid-to-gas heat exchanger to be required for a speciﬁc application having minimum total cost. From the initial screening of surfaces (see Section 10.3), suppose that two plate-ﬁn constructions (the louver-ﬁn and strip-ﬁn surfaces) and one ﬂat-tube and wavy-ﬁn construction appear to be promising for the gas side. Then, for this problem, there are three alternative designs that need to be optimized. As shown in Fig. 9.5, ﬁrst formulate the total number of constraints for the problem. This includes the customer’s speciﬁed explicit constraints (such as ﬁxed frontal area, the ranges of heat exchanger dimensions) and implicit constraints (such as required minimum heat transfer, allowable maximum pressure drop). Once the basic surface geometry for the design chosen is selected, the designer imposes some additional constraints, such as the minimum and maximum values for the ﬁn height, ﬁn thickness, ﬁn pitch, ﬁn thermal conductivity, ﬂow length, number of ﬁnned passages, gas ﬂow rate, and so on. The designer wants to vary all the design and operating variables within the ranges speciﬁed such that the exchanger will meet the required heat transfer, maximum pressure drop, and other constraints with minimum total cost. To optimize the heat exchanger, the designer starts with one set of heat exchanger surface geometrical dimensions which may not even satisfy all or some of the constraints imposed. Subsequently, the various geometrical properties (such as heat transfer area, free-ﬂow area, hydraulic diameter) and thermal properties are evaluated based on the input operating conditions. The heat transfer rate and pressure drop are then evaluated by the procedure outlined for the rating problem (see, e.g., Sections 9.2.1, 9.4.3, and 9.5.2). Next, the output from heat exchanger calculations is fed to the optimization computer program package, where the constraints and the objective function are evaluated. Subsequently, new values for the design variables are generated and heat exchanger calculations are repeated. The iterations are continued until the objective function is optimized (minimized or maximized as desired) within the accuracy speciﬁed and all constraints are satisﬁed. In some situations, it may not be possible to satisfy all constraints. Engineering judgment is then applied to determine whether or not the optimum design is satisfactory and which constraints to relax. One of the most important but least known inputs for the heat transfer and pressure drop evaluation is the magnitude for scaled-up or scaled-down (modiﬁed from the original) j and f factors. As soon as one of the surface geometrical dimensions is changed (such as the ﬁn pitch, height, or thickness) but others may stay unchanged, the surface is no longer geometrically similar to the original surface for which experimental j and f data are available. In such cases, either theoretical or experimental correlations should be incorporated

REFERENCES

667

in the computer program to arrive at the scaled j and f factors for the new geometry. Some of these correlations are presented in Section 7.5. The designer must use his or her experience and judgment regarding the appropriate correlations to obtain the scaled j and f factors. In addition, care must be exercised to avoid excessive extrapolations. A review of Fig. 9.5 indicates that heat exchanger design (rating) program and optimization software is needed for the optimization. In addition, a system simulation program is added if the heat exchanger optimization has to be done based on the system design approach. Although the foregoing optimization procedure was outlined from a performance and cost point of view, the heat exchanger can also be optimized as a component or as part of a system based on a thermoeconomics point of view. This is discussed further in Section 11.6. SUMMARY The focus of this chapter is to provide step-by-step rating and sizing procedures for major types of heat exchangers. All the information and design theory outlined in previous chapters is applied and extended in formulating the design procedures discussed in this chapter. The detailed thermal and hydraulic design of heat exchangers outlined in this chapter is one of the major objectives of this book. After presenting how to determine the mean temperature on each ﬂuid side in a heat exchanger, we have provided the rating and sizing of extended-surface (plate-ﬁn and tube-ﬁn), plate, and shell-and-tube heat exchangers in depth with an example. Subsequently, we have also provided a general approach to the optimization of a heat exchanger design. REFERENCES Bell, K. J., 1963, Final report of the cooperative research program on shell-and-tube heat exchangers, Univ. Del. Eng. Stn. Bull. No. 5. Bell, K. J., 1988a, Overall design methodology for shell-and-tube exchangers, in Heat Transfer Equipment Design, R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, eds., Hemisphere Publishing, Washington, DC, pp. 131–144. Bell, K. J., 1988b, Delaware method for shell design, in Heat Transfer Equipment Design, R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, eds., Hemisphere Publishing, Washington, DC, pp. 145–166. Bell, K. J., 1998, Approximate sizing of shell-and-tube heat exchangers, in Heat Exchanger Design Handbook, G. F. Hewitt, exec. ed., Begell House, New York, Vol. 3, Sec. 3.1.4. Chiou, J. P., 1980, The advancement of compact heat exchanger theory considering the eﬀects of longitudinal heat conduction and ﬂow nonuniformity, in Compact Heat Exchangers: History, Technological Advancement and Mechanical Design Problems, R. K. Shah, C. F. McDonald, and C. P. Howard, eds., Book G00183, American Society of Mechanical Engineers, New York. Kays, W. M., A. L. London, 1998, Compact Heat Exchangers, reprint 3rd ed., Krieger Publishing, Malabar, FL. Marriot, J., 1977, Performance of an Alfaﬂex plate heat exchanger, Chem. Eng. Prog., Vol. 73, No. 2, pp. 73–78. Raznjevic´, K., 1976, Handbook of Thermodynamic Tables and Charts, McGraw-Hill, New York. Shah, R. K., 1981, Compact heat exchanger design procedures, in Heat Exchangers: ThermalHydraulic Fundamentals and Design, S. Kakac¸, A. E. Bergles and F. Mayinger, eds., Hemisphere Publishing Corp., Washington, DC, pp. 495–536.

668

HEAT EXCHANGER DESIGN PROCEDURES

Shah, R. K., 1988a, Plate-ﬁn and tube-ﬁn heat exchanger design procedures, in Heat Transfer Equipment Design, R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, eds., Hemisphere Publishing, Washington, DC, pp. 255–266. Shah, R. K., 1988b, Counterﬂow rotary regenerator thermal design procedures, in Heat Transfer Equipment Design, R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, eds., Hemisphere Publishing, Washington, DC, pp. 267–296. Shah, R. K., and W. W. Focke, 1988, Plate heat exchangers and their design theory, in Heat Transfer Equipment Design, R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, eds., Hemisphere Publishing, Washington, DC, pp. 227–254. Shah, R. K., and A. D. Giovannelli, 1988, Heat pipe heat exchanger design theory, in Heat Transfer Equipment Design, R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, eds., Hemisphere Publishing, Washington, DC, pp. 609–653. Shah, R. K., and T. Skiepko, 1999, Inﬂuence of leakage distribution on rotary regenerator thermal performance, Appl. Thermal Eng., Vol. 19, pp. 685–705. Shah, R. K., and A. S. Wanniarachchi, 1991, Plate heat exchanger design theory, in Industrial Heat Exchangers, J.-M. Buchlin, ed., Lecture Series 1991-04, von Ka´rma´n Institute for Fluid Dynamics, Belgium. Shah, R. K., K. A. Aﬁmiwala, and R. W. Mayne, 1978, Heat exchanger optimization, Heat Transfer 1978, Proc. 6th Int. Heat Transfer Conf., Vol. 4, pp. 185–191. Taborek, J., 1998, Shell-and-tube heat exchangers: single phase ﬂow, in Handbook of Heat Exchanger Design, G. F. Hewitt, ed., Begell House, New York, pp. 3.3.3-1 to 3.3.11-5.

REVIEW QUESTIONS For each question, circle one or more correct answers. Explain your answers brieﬂy. 9.1

The ﬂuid mean temperature on each ﬂuid side in a gas-to-gas multipass heat exchanger is generally computed as the: (a) arithmetic mean temperature on one ﬂuid side and log-mean average on the other ﬂuid side in each pass (b) arithmetic mean temperature on both ﬂuid sides in each pass (c) can’t tell (d) none of these

9.2

For sizing of a gas-to-gas plate-ﬁn counterﬂow exchanger, we can determine the physical size of the exchanger (with no constraints imposed on the dimensions) such that: (a) The pressure drops on both ﬂuid sides will always be exactly matched. (b) The critical pressure drop on one ﬂuid side can be matched; the pressure drop on the other ﬂuid side will be higher than the speciﬁed value. (c) The critical pressure drop on one ﬂuid side can be matched; the pressure drop on the other ﬂuid side will be lower than the speciﬁed value. (d) can’t tell

9.3

In the following industrial heat exchangers, we can always meet both the heat transfer and pressure drop requirements on at least one ﬂuid side during the design process: (a) shell-and-tube exchanger (c) plate-ﬁn exchanger

(b) gasketed plate exchanger

PROBLEMS

9.4

The following streams do not contribute signiﬁcantly to heat transfer on the shell side of a shell-and-tube heat exchanger: (a) A (d) E

9.5

(b) B (e) F

(c) C

Various leakage and bypass streams on the shell side: (a) (b) (c) (d)

9.6

669

increases heat transfer increases pressure drop decreases heat transfer and pressure drop none of these

A heat exchanger can be optimized for a system using: (a) surfaces selected based on the screening methods (b) performance evaluation criteria (c) commercially available most sophisticated optimization software for heat exchanger optimization (d) all of these (e) none of these

PROBLEMS 9.1

A gas turbine–driven generator is to be installed in a power plant for peaking service. To obtain high system eﬃciency, the hot turbine exhaust gases (4308C and 102.7 kPa pressure) are used to preheat the combustion air, which leaves the compressor at 910 kPa and 1758C. The mass ﬂow rate through the compressor and turbine are 24.3 and 24.7 kg/s, respectively. We have selected an unmixed–unmixed crossﬂow heat exchanger having " ¼ 0:75. This exchanger has 2:0 m 0:9 m frontal area on the air side and 2:0 m 2:0 m frontal area on the gas side. The surface geometries and performance characteristics of the air- and gas-side surfaces are provided below. Louvered Plate-Fin Surface 3/8-6.06 (Fig. 10-38, Kays and London, 1998)

Strip-Fin Plate-Fin Surface 1/2-11.94(D) (Fig. 10-64, Kays and London, 1998)

Fin density ¼ 238:6 m1 Plate spacing ¼ 6:35 mm Louver spacing ¼ 9:525 mm Louver gap ¼ 1:4 mm Fin gap ¼ 2:79 mm Flow passage hydraulic diameter ¼ 4:453 mm Fin metal thickness ¼ 0:152 mm Heat transfer surface area density ¼ 840 m2 =m3 Fin area/total area ¼ 0:640

Fin density ¼ 470 m1 Plate spacing ¼ 6.02 mm Splitter symmetrically located Strip length in ﬂow direction ¼ 12:7mm Flow passage hydraulic diameter ¼ 2:266 mm Fin metal thickness ¼ 0:152 mm Splitter metal thickness ¼ 0:152 mm Heat transfer surface area density ¼ 1521 m2 =m3 Fin area (including splitter)/total area ¼ 0:796

670

HEAT EXCHANGER DESIGN PROCEDURES

Re

j

f

Re

j

f

10,000 8,000 6,000 5,000 4,000 3,000 2,500 2,000 1,500 1,200 1,000 800 600 500

0.00551 0.00593 0.00651 0.00690 0.00738 0.00805 0.00849 0.00900 0.00970 0.0104 0.0112 0.0124 0.0144 0.0160

0.0331 0.0340 0.0354 0.0363 0.0375 0.0394 0.0406 0.0426 0.0461 0.0496 0.0532 0.0587 0.0682 0.0755

8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,500 1,200 1,000 800 600 500 400 300

— 0.00452 0.00471 0.00492 0.00522 0.00575 0.00682 0.00744 0.00830 0.00911 0.01045 0.01255 0.01415 0.0166 0.0205

0.0123 0.0126 0.0131 0.0137 0.0146 0.0162 0.0198 0.0231 0.0265 0.0306 0.0347 0.0429 0.0493 0.0592 0.0758

All materials are stainless steel with k ¼ 20:8 W=m K. The following geometrical properties have been evaluated for the air and gas sides.

Geometrical Properties Minimum free-ﬂow area Ao (m Þ Heat transfer area A (m2 ) L=rh Aw ðm2 Þ 2

Air Side

Gas Side

0.8247 1469 0.455 1781

1.5638 2524 0.398 1614 550.7

(a) At what temperatures will you evaluate ﬂuid properties on each ﬂuid side? Consider the following mean ﬂuid properties: Air side: speciﬁc heat 1:04 kJ=kg K, thermal conductivity 0.0431 W/m K and dynamic viscosity 0:283 104 Pa s; these properties on gas side are: 1.06 kJ=kg K, thermal conductivity 0.0473 W/m K, and dynamic viscosity 0:305 104 Pa s: (b) Evaluate the quality of design by determining: (i) The percentage of the thermal resistance on the air and gas sides. Is this a thermally balanced design? Use f for air and gas sides as 0.674 and 0.773, respectively, and w ¼ 0:1 mm. (ii) The relative pressure drops p=p for each stream. Are the pressure drops balanced? What fraction of the total pressure drop is due to entrance/exit losses? Use the following densities: air-side inlet, outlet, and mean densities: 7.075, 4.962, and 5.833 kg/m3 ; gas-side inlet, outlet, and mean densities: 0.5094, 0.6875, and 0.5852 kg/m3 . 9.2

A waste heat recovery heat exchanger is manufactured in small (0.3 m 0:3 m 0:6 mÞ modules. Each is a single-pass crossﬂow heat exchanger having ﬂuids unmixed on both sides. The 0.6 m dimension corresponds to the

671

PROBLEMS

noﬂow height. Assume that the surface on each ﬂuid side is a plain triangular ﬁn 11.1 of Kays and London (1998). The surface geometry and j and f data are given below. Re 10,000 8,000 6,000 5,000 4,000 3,000 2,500

j 0.00314 0.00333 0.00356 0.00372 0.00390 0.00412 0.00424

f 0.00878 0.00923 0.00971 0.00991 0.0103 0.0112 0.0119

Re

j

f

2,000 1,500 1,200 1,000 800 600 500

0.00436 0.00444 0.00471 0.00515 0.00599 0.00733 0.00840

0.0129 0.0149 0.0169 0.0190 0.0228 0.0294 0.0350

Fin density ¼ 437 m1 , plate spacing b ¼ 6:35 mm, ﬂow passage hydraulic diameter Dh ¼ 3:081 mm, ﬁn metal thickness ¼ 0:15 mm, material ¼ aluminum, heat transfer surface area density ¼ 1204 m2 =m3 , and ﬁn area/total area ¼ 0:756. Assume the plate thickness to be 0:5 mm. Try to accommodate an integer number of ﬁns on each ﬂuid side in an approximately 0:3 m 0:3 m 0:6 m envelope. On one ﬂuid side of the exchanger, process air at 0.40 m3 /s ﬂows with the inlet temperature of 2388C. On the other ﬂuid side, makeup air ﬂows at a 0.26 m3 /s ﬂow rate at 408C inlet temperature. Both ﬂuids are at atmospheric pressure at inlet. Determine the outlet temperatures on each ﬂuid side as well as the pressure drop on each ﬂuid side. Both the ﬁn and plates are made of aluminum having a thermal conductivity of 190.4 W/m K. 9.3

A ﬁnned-tube exchanger is designed to cool water from 528C to 388C with ambient air at 328C to be heated to 438C. The water and air mass ﬂow rates are 2.52 and 13.13 kg/s, respectively. The allowable air-side pressure drop is 149 Pa. Estimate the air-side frontal velocity to satisfy the desired performance. Consider j=f ¼ 0:30, Rair =Rtot ¼ 0:7, and counterﬂow performance. The ﬁnned tube exchanger has Ao =Afr ¼ 0:56 and cp ¼ 1:005 and 4.187 kJ/kg K for air and water, respectively. Use m ¼ 1:136 kg=m3 , Pr ¼ 0:705, and pi ¼ 101:4 kPa for air.

9.4

What will be the exit temperatures of the hot and cold streams and heat duty (heat transfer rate) of a plate heat exchanger if 2.52 kg/s of hot water enters at 1048C and 4.04 kg/s of cold water enters at 168C? These ﬂow rates are expected at some part-load condition, and the plant engineer is interested in determining the temperatures to see how secondary cooling equipment downstream behaves. Use the following data for your solution: the plate exchanger is a 1 pass–1 pass counterﬂow exchanger (see Fig. 1.65a) with 14 ﬂow channels for hot water and 14 ﬂow channels for cold water (total 27 thermal plates). The eﬀective width of plates is 0.457 m and the gasket thickness or gap between plates is 3.0 mm. Note that the hydraulic diameter will be equal to twice the gap dimension. The projected heat transfer area for each plate on one side is 0.28 m2 . Use a fouling factor of 0.0002 m2 K/W for each ﬂuid stream, a plate thickness of 0.9 mm, and wall thermal conductivity of 15.6 W/m K. The j and f factors for the plate surface are given as j ¼ 0:2Re0:25 and f ¼ 0:6Re0:2 . Also, the following properties are provided for water.

672

HEAT EXCHANGER DESIGN PROCEDURES

Property Hot cp (kJ/kg KÞ (Pa sÞ k (W/m KÞ (kg/m3 ) Pr

Hot Water at 888C 4.19 0:320 103 0.68 977 1.97

Cold Water at 248C 4.19 0:922 103 0.61 1001 6.33

9.5

Design a gas-to-gas two-ﬂuid heat exchanger with both ﬂuids assumed to be air. The inlet temperature, pressure, and mass ﬂow rate of one airstream are 487 K, 490 kPa, and 21 kg/s, respectively, and it must leave the heat exchanger at a temperature of 619 K. The mass ﬂow rate of the second airstream is the same as that of the ﬁrst, and its inlet temperature and pressure are 690 K and 103 kPa, respectively. The pressure drops are limited to 4.900 kPa and 2.575 kPa for the cold and hot airstream. Additional data: Flow arrangement is a single-pass cross ﬂow (unmixed–unmixed); heat transfer surface is a plain plate-ﬁn surface with designation 19.86 (Kays and London, 1998). The heat transfer surface material is aluminum with a plate thickness of 1 mm. List all your assumptions and provide proper justiﬁcation for each. If any additional assumption is needed, provide explicit reasons for it.

9.6

Analyze a three-pass two-ﬂuid overall cross-counterﬂow heat exchanger of a gasturbine plant with all three passes as equal unmixed–unmixed crossﬂow units (see the right-hand side of Fig. 1.58a as a schematic). The desired overall heat exchanger eﬀectiveness must be 0.766 in order to reach the required plant eﬀectiveness. The air mass ﬂow rate is 21 kg/s and the inlet temperature 487 K. The mass ﬂow rate of the gas is the same, but the inlet temperature is 690 K. Determine the outlet temperatures of both air and gas. Perform calculations involving determination of thermophysical properties of ﬂuids at the true mean (integral) temperatures.

10

Selection of Heat Exchangers and Their Components

As described in Chapter 1, a variety of heat exchangers are available, and the question becomes which one to choose for a given application. In addition, for each type (core construction), either a large number of geometrical variables (such as those associated with each component of the shell-and-tube exchanger) or a large number of surface geometries (such as those for plate, extended surface, or regenerative exchangers) are available for selection. Again the question involves which set of geometries/surfaces will be most appropriate for a given application. There is no such thing as a particular heat exchanger or the selection of a particular heat transfer surface that is best (i.e., optimum) for a given application. Near-optimum heat exchanger designs involve many trade-oﬀs, since many geometrical and operating variables are associated with heat exchangers during the selection process. For example, a cheaper exchanger may be obtained if one wants to give up some performance or durability. One can get higher performance if the exchanger is heavier or costs a little more. A heat exchanger can be made smaller if we accept a little lower performance or provide more pumping power for higher ﬂuid ﬂow rates. The heat exchanger design team must consider the trade-oﬀs and arrive at the optimum exchanger for a given application to meet the design requirements and constraints. In this chapter we discuss qualitative and quantitative criteria/methods used to select exchanger type and surface geometry for a given application for engineers not having prior design/operational experience. If one or more exchangers are already in service for similar applications, this prior experience is the best guide for the selection and design of a heat exchanger for a given application. We ﬁrst describe important qualitative selection criteria for heat exchangers in two categories: (1) criteria based on important operating variables of exchangers in Section 10.1, and (2) general guidelines on major heat exchanger types in Section 10.2. Next, we describe some quantitative criteria for selection of extended heat exchanger surfaces (screening methods) in Section 10.3.1 and for selection of tubular exchangers (performance evaluation criteria) in Section 10.3.2. These quantitative criteria are energy-based (the ﬁrst law of thermodynamics) for a heat exchanger as a component. Criteria based on the second law of thermodynamics can also be devised as indicated in Section 10.3.3, with the details presented in Section 11.7. Finally, selection based on cost criteria is presented brieﬂy in Section 10.3.4 and discussed further in Section 11.6.6. Except for some qualitative discussion, it is not easy to present a method for system-based selection and optimization of a heat exchanger in a textbook since there are many systems in which heat exchangers are used, and each is diﬀerent, Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

673

674

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

depending on the process. Thus, the overall objective of this chapter is to provide a good understanding of heat exchanger selection in general as a component based on qualitative and quantitative criteria. However, system-based heat exchanger optimization is the current industrial practice.

10.1 SELECTION CRITERIA BASED ON OPERATING PARAMETERS A large number of heat exchangers are described in Chapter 1, providing understanding of their functions, the range of operating parameters, reasons why they are used in certain applications, and so on. With such thorough understanding, one would have a good idea of which types of exchangers to use in given applications. Refer to Table 10.1 for operating conditions and principal features for many heat exchanger types. Here we highlight heat exchanger selection criteria based on major operating parameters.

10.1.1

Operating Pressures and Temperatures

The exchanger in operation must withstand the stresses produced by the operating pressure and the temperature diﬀerences between two ﬂuids. These stresses depend on the inlet pressures and temperatures of the two ﬂuids. The most versatile exchangers for a broad range of operating pressures and temperatures are shell-and-tube exchangers for medium- to high-heat duties and double-pipe exchangers for lowerheat duties. They can handle from high vacuum to ultrahigh ﬂuid pressures [generally limited to 30 MPa (4350 psi) on the shell or annulus side and 140 MPa (20,000 psi) on the tube side]. Coupled with high pressures, shell-and-tube exchangers can withstand high temperatures, limited only by the materials used; however, the inlet temperature diﬀerence is limited to 508C (1208F) from the thermal expansion point of view when the exchanger design allows only limited thermal expansion, such as in the E-shell design. These exchangers are used for gas, liquid, and phase-change applications. For liquid–liquid or liquid–phase change applications, if the operating pressures and temperatures are modest (less than about 2.5 MPa and 2008C), gasketed or semiwelded plate exchangers should be considered. For somewhat higher pressures and temperatures, fully welded or brazed plate exchangers may be the choice, depending on other design criteria. The plate-ﬁn extended surface exchanger is designed for low-pressure applications, with the operating pressures on either side limited to about 1000 kPa (150 psig), except for cryogenics applications, where the operating pressure is about 9000 kPa gauge (1300 psig). The maximum operating temperature for plate-ﬁn exchangers is below 6508C (12008F) and usually below 1508C (3008F), to avoid the use of expensive materials. There is no limit on the minimum operating temperature; plate-ﬁn exchangers are commonly used in cryogenic applications. Fins in a plate-ﬁn exchanger act as a ﬂow-mixing device for highly viscous liquids, and if properly designed, add surface area for heat transfer with a reasonably high ﬁn eﬃciency. In a plate-ﬁn exchanger, ﬁns on the liquid side are used primarily for pressure containment and rigidity. Fins on the gas side are used for added surface area for heat transfer, with ﬁn eﬃciencies usually greater than 80%.

SELECTION CRITERIA BASED ON OPERATING PARAMETERS

675

The tube-ﬁn exchanger is used to contain the high-pressure ﬂuid on the tube side if only one ﬂuid is at a high pressure. Fins on the liquid or phase-change side generally have ‘‘low’’ heights, to provide reasonably high ﬁn eﬃciencies. Turbulators may be used within tubes for ﬂow mixing. Tube-ﬁn exchangers with or without shells are designed to cover the operating temperature range from low cryogenic temperatures to about 8708C (16008F). For ultrahigh temperature [870 to 20008C (1600 to 36008F)] and near-atmospheric pressures, as in high-temperature waste heat recovery, either rotary regenerators (870 to 11008C) or ﬁxed-matrix regenerators (up to 20008C) are used. 10.1.2

Cost

Cost is a very important factor in the selection of the heat exchanger construction type. The cost per unit of heat transfer surface area is higher for a gasketed plate exchanger than for a shell-and-tube exchanger. However, from the total cost (capital, installation, operation, maintenance, etc.) point of view, PHEs are less expensive than shell-and-tube exchangers when stainless steel, titanium, and other higher quality alloys are used. Since tubes are more expensive than extended surfaces or a regenerator matrix, shell-and-tube (or broadly, tubular) exchangers are in general more expensive per unit of heat transfer surface area. In addition, the heat transfer surface area density of a tubular core is generally much lower than that of an extended surface or regenerative exchanger. Rotary regenerators made of paper or plastic are in general the least expensive per unit of heat transfer surface area. 10.1.3

Fouling and Cleanability

Fouling and cleanability are among the most important design considerations for liquid-to-liquid or phase-change exchangers and for some gas-to-ﬂuid exchangers. Fouling should be evaluated for both design and oﬀ-design points. Periodic cleaning and/or replacement of some exchanger components depend on the fouling propensity of the ﬂuids employed. In applications involving moderate to severe fouling, either a shell-and-tube or a gasketed plate heat exchanger is used, depending on the other operating parameters. In a shell-and-tube exchanger, the tube ﬂuid is generally selected as the heavily fouling ﬂuid since the tube side may be cleaned more easily. A plate heat exchanger is highly desirable in those relatively low temperature applications [ 400

70

6

> 400

200 to þ900 up to 150s

Chemical

> 200

up to 500

Excellent

Good

Mechanicald;o , Chemicalu Chemical

Excellent

Excellent

Excellent

Good

Water wash

Chemical

Chemical

90

Cryogenic to þ650

Source: Data from Lancaster (1998). a Two-phase includes boiling and condensing duties. b s/s, stainless steel; Ti, titanium; Ni, nickel; Cu, copper. Alloys of these materials and other special alloys are frequently available. c The maximum pressure capability is unlikely to occur at the higher operating temperatures, and assumes no pressure/stress-related corrosion. d Can be dismantled. e Function of gasket as well as plate material. f Not common. g On gasket side. h On welded side. i Ensure compatibility with copper braze. j Function of braze as well as plate material. k Not in a single unit. l On tube side. m Only when ﬂanged access provided; otherwise, chemical cleaning. n Five ﬂuids maximum. o On shell side. p Condensing on gas side. q Polyvinylidene diﬂuoride. r Polypropylene. s PEEK (polyetheretherketone) can go to 2508C t Shell may be composed of polymeric material. u On plate side.

up to 10; 000

700–800

Diﬀusionbonded plate-ﬁn

Marbond

800–1500

Brazed plate-ﬁn

Yes

No

No

Yes

Yes

Yes

Yes

Yes

Not usually

Yes

Yes

Yes

678

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

regenerator, so that the impact of fouling is reduced, or cleaning by one of the methods described in Section 13.4 may be employed. 10.1.4

Fluid Leakage and Contamination

Whereas in some applications, ﬂuid leakage from one ﬂuid side to the other ﬂuid side is permissible within limits, in other applications ﬂuid leakage is absolutely not allowed. Even in a good leak tight design, carryover and bypass leakages from the hot ﬂuid to the cold ﬂuid (or vice versa) occur in regenerators. Where these leakages and subsequent ﬂuid contamination is not permissible, regenerators are not used. The choices left are either a tubular, extended surface, or some plate type heat exchangers. Gasketed plate exchangers have more probability of ﬂow leakage than do shell-and-tube exchangers. Plate-ﬁn and tube-ﬁn exchangers have potential leakage problems at the joint between the corrugated ﬁn passage and the header or at the tube-to-header joint. Where absolutely no ﬂuid contamination is allowed (as in the processing of potable water), a double-wall tubular or shell-and-tube exchanger or a double-plate PHE is used.

10.1.5

Fluids and Material Compatibility

Materials selection and compatibility between construction materials and working ﬂuids are important issues, in particular with regard to corrosion (see Section 13.5) and/or operation at elevated temperatures. While a shell-and-tube heat exchanger may be designed using a variety of materials, compact heat exchangers often require preferred metals or ceramics. For example, a requirement for low cost, light weight, high conductivity, and good joining characteristics for compact heat exchangers often leads to the selection of aluminum for the heat transfer surface. On the other side, plate exchangers require materials that are either used for food ﬂuids or require corrosion resistance (e.g., stainless steel). In general, one of the selection criteria for exchanger material depends on the corrosiveness of the working ﬂuid. In Table 10.2, a summary of some materials used for noncorrosive and/or corrosive services is presented. More details about the selection of materials are provided in specialized literature of TEMA (1999) and the ASME (1998) codes.

10.1.6

Fluid Type

A gas-to-gas heat exchanger requires a signiﬁcantly greater amount of surface area than that for a liquid-to-liquid heat exchanger for a given heat transfer rate. This is 1 1 because the heat transfer coeﬃcient for the gas is 10 to 100 that of a liquid. The increase in surface area is achieved by employing surfaces that have a high heat transfer surface area density . For example, ﬁns are employed in an extended surface heat exchanger, or a small hydraulic diameter surface is employed in a regenerator, or small-diameter tubes are used in a tubular heat exchanger. Plate heat exchangers (of the type described in Section 1.5.2) are generally not used in a gas-to-gas exchanger application because they produce excessively high pressure drops. All prime surface heat exchangers with plain (uncorrugated) plates are used in some waste heat recovery applications. The ﬂuid pumping power is generally signiﬁcant and a controlling factor in designing gas-to-gas exchangers.

SELECTION CRITERIA BASED ON OPERATING PARAMETERS

679

TABLE 10.2 Materials for Noncorrosive and Corrosive Service Material

Heat Exchanger Type or Typical Service Noncorrosive Service

Aluminum and austenitic chromium–nickel steel 3 12 Ni steel Carbon steel (impact tested) Carbon steel Refractory-lined steel

Any heat exchanger type, T < 1008C Any heat exchanger type, 100 < T < 458C Any heat exchanger type, 45 < T < 08C Any type of heat exchanger, 0 < T < 5008C Shell-and-tube, T > 5008C

Corrosive Service Carbon steel Ferritic carbon–molybdenum and chromium–molybdenum alloys Ferritic chromium steel

Austenitic chromium–nickel steel Aluminum Copper alloys: admiralty, aluminum brass, cupronickel High nickel–chromium–molybdenum alloys Titanium Glass Carbon Coatings: aluminum, epoxy resin Linings: lead and rubber Linings: austenitic chromium–nickel steel

Mildly corrosive ﬂuids; tempered cooling water Sulfur-bearing oils at elevated temperatures (above 3008C); hydrogen at elevated temperatures Tubes for moderately corrosive service; cladding for shells or channels in contact with corrosive sulfur bearing oil Corrosion-resistant duties Mildly corrosive ﬂuids Freshwater cooling in surface condensers; brackish and seawater cooling Resistance to mineral acids and Cl-containing acids Seawater coolers and condensers, including PHEs Air preheaters for large furnaces Severely corrosive service Exposure to sea and brackish water Channels for seawater coolers General corrosion resistance

Source: Data from Lancaster (1998).

In liquid-to-liquid exchanger applications, regenerators are ruled out because of the associated ﬂuid leakage and carryover (contamination). Fluid pumping power is, however, not as critical for a liquid-to-liquid heat exchanger as it is for a gas-to-gas heat exchanger. 1 1 In a liquid-to-gas heat exchanger, the heat transfer coeﬃcient on the gas side is 10 to 100 y of that on the liquid side. Therefore, for a ‘‘thermally balanced’’ design (i.e., having hA of the same order of magnitude on each ﬂuid side of the exchanger), ﬁns are employed to increase the gas-side surface area. Thus, the common heat exchanger constructions used for a liquid-to-gas heat exchanger are the extended surface and tubular; plate-type and regenerative constructions are not used. For phase-change exchangers, the condensing or evaporating ﬂuid has a range of heat transfer coeﬃcients that vary from low values approximating those for gas ﬂows to high values approximating those for high liquid ﬂows and higher. Therefore, the selection of

{ A thermally balanced design usually results in an optimum design from the cost viewpoint since the cost of the extended surface per unit surface area is less than that of the prime surface, either tubes or plates.

680

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

exchanger type for phase-change exchangers parallels the guidelines provided for the gas or liquid side of the exchanger.

10.2 GENERAL SELECTION GUIDELINES FOR MAJOR EXCHANGER TYPES A large number of heat exchangers are described in Chapter 1. That information, complemented by the material presented in this section, will provide a good understanding of the selection of heat exchanger types. 10.2.1

Shell-and-Tube Exchangers

More than 65% of the market share (in the late 1990s) in process and petrochemical industry heat exchangers is held by the shell-and-tube heat exchanger, for the following reasons: its versatility for handling a wide range of operating conditions with a variety of materials, design experience of about 100 years, proven design methods, and design practice with codes and standards. The selection of an appropriate shelland-tube heat exchanger is achieved by a judicious choice of exchanger conﬁguration, geometrical parameters, materials, and the ‘‘right’’ design. Next we summarize some guidelines on all these considerations qualitatively to provide the feel for the right design for a given application. The major components of a shell-and-tube exchanger are tubes, baﬄes, shell, front-end head, read-end head, and tubesheets. Depending on the applications, a speciﬁc combination of geometrical variables or types associated with each component is selected. Some guidelines are provided below. For further details on geometrical dimensions and additional guidelines, refer to TEMA (1999). 10.2.1.1 Tubes. Since the desired heat transfer in the exchanger takes place across the tube surface, the selection of tube geometrical variables is important from a performance point of view. In most applications, plain tubes are used. However, when additional surface area is required to compensate for low heat transfer coeﬃcients on the shell side, low ﬁnned tubing with 250 to 1200 ﬁns/m (6 to 30 ﬁns/in.) and a ﬁn height of up to 6.35 mm (14 in.) is used. While maintaining reasonably high ﬁn eﬃciency, low-height ﬁns increase surface area by two to three times over plain tubes and decrease fouling on the ﬁn side based on the data reported. The most common plain tube sizes have 15.88, 19.05, and 25.40 mm (58, 34, and 1 in.) tube outside diameters. From the heat transfer viewpoint, smaller-diameter tubes yield higher heat transfer coeﬃcients and result in a more compact exchanger. However, larger-diameter tubes are easier to clean and more rugged. The foregoing common sizes represent a compromise. For mechanical cleaning, the smallest practical size is 19.05 mm (34 in.). For chemical cleaning, smaller sizes can be used provided that the tubes never plug completely. The number of tubes in an exchanger depends on the ﬂuid ﬂow rates and available pressure drop. The number of tubes is selected such that the tube-side velocity for water and similar liquids ranges from 0.9 to 2.4 m/s (3 to 8 ft/sec) and the shell-side velocity from 0.6 to 1.5 m/s (2 to 5 ft/sec). The lower velocity limit corresponds to limiting the fouling, and the upper velocity limit corresponds to limiting the rate of erosion. When sand and silt are present, the velocity is kept high enough to prevent settling.

GENERAL SELECTION GUIDELINES FOR MAJOR EXCHANGER TYPES

681

The number of tube passes depends on the available pressure drop. Higher velocities in the tube result in higher heat transfer coeﬃcients, at the expense of increased pressure drop. Therefore, if a higher pressure drop is acceptable, it is desirable to have fewer but longer tubes (reduced ﬂow area and increased ﬂow length). Long tubes are accommodated in a short shell exchanger by multiple tube passes. The number of tube passes in a shell generally range from 1 to 10 (see Fig. 1.61). The standard design has one, two, or four tube passes. An odd number of passes is uncommon and may result in mechanical and thermal problems in fabrication and operation. 10.2.1.2 Tube Pitch and Layout. The selection of tube pitch is a compromise between a close pitch (small values of pt =do ) for increased shell-side heat transfer and surface compactness, and an open pitch (large values of pt =do ) for decreased shell-side plugging and ease in shell-side cleaning. In most shell-and-tube exchangers, the ratio of the tube pitch to tube outside diameter varies from 1.25 to 2.00. The minimum value is restricted to 1.25 because the tubesheet ligamenty may become too weak for proper rolling of the tubes and cause leaky joints. The recommended ligament width depends on the tube diameter and pitch; the values are provided by TEMA (1999). Two standard types of tube layouts are the square and the equilateral triangle, shown in Fig. 10.1. The equilateral pitch can be oriented at 308 or 608 angle to the ﬂow direction, and the square pitch at 458 and 908.z Note that the 308, 458 and 608 arrangements are staggered, and 908 is inline. For the identical tube pitch and ﬂow rates, the tube layouts in decreasing order of shell-side heat transfer coeﬃcient and pressure drop are: 308, 458, 608, and 908. Thus the 908 layout will have the lowest heat transfer coeﬃcient and the lowest pressure drop. The square pitch (908 or 458) is used when jet or mechanical cleaning is necessary on the shell side. In that case, a minimum cleaning lane of 14 in. (6.35 mm) is provided. The square pitch is generally not used in the ﬁxed tubesheet design because cleaning is not feasible. The triangular pitch provides a more compact arrangement, usually resulting in a smaller shell, and the strongest header sheet for a speciﬁed shell-side ﬂow area. Hence, it is preferred when the operating pressure diﬀerence between the two ﬂuids is large. If

FIGURE 10.1 Tube layout arrangements.

{ The ligament is a portion of material between two neighboring tube holes. The ligament width is deﬁned as the tube pitch minus the tube hole diameter, such as the distance a shown in Fig. 10.1. { Note that the tube layout angle is deﬁned in relation to the ﬂow direction and is not related to the horizontal or vertical reference line. Refer to Table 8.1 for the deﬁnitions of tube layouts and associated geometrical variables.

682

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

designed properly, it can be cleaned from six directions instead of four as in the square pitch arrangement. When mechanical cleaning is required, the 458 layout is preferred for laminar or turbulent ﬂow of a single-phase ﬂuid and for condensing ﬂuid on the shell side. If the pressure drop is limited on the shell side, the 908 layout is used for turbulent ﬂow. For boiling applications, the 908 layout is preferred, because it provides vapor escape lanes. However, if mechanical cleaning is not required, the 308 layout is preferred for single-phase laminar or turbulent ﬂow and condensing applications involving a high T rangey (a mixture of condensables). The 608 layout is preferred for condensing applications involving a low T range (generally, pure vapor condensation) and for boiling applications. Horizontal tube bundles are used for shell-side condensation or vaporization. 10.2.1.3 Baﬄes. As presented in Section 1.5.1.1, baﬄes may be classiﬁed as either longitudinal or transverse type. Longitudinal baﬄes are used to control the overall ﬂow direction of the shell ﬂuid. Transverse baﬄes may be classiﬁed as plate baﬄes or grid baﬄes. Plate baﬄes are used to support the tubes, to direct the ﬂuid in the tube bundle at approximately right angles to the tubes, and to increase the turbulence and hence the heat transfer coeﬃcient of the shell ﬂuid. However, the window section created by the plate baﬄes results in excessive pressure drop with insigniﬁcant contribution to heat transfer; ﬂow normal to the tubes in crossﬂow section may create ﬂowinduced vibration problems. The rod baﬄes, a most common type of grid baﬄes, shown in Fig. 1.11, are used to support the tubes and to increase the turbulence. Flow in a rod baﬄe heat exchanger is parallel to the tubes, and hence ﬂow-induced vibration is virtually eliminated by the baﬄe support of the tubes. The choice of baﬄe type, spacing, and cut are determined largely by the ﬂow rate, required heat transfer, allowable pressure drop, tube support, and ﬂow-induced vibration. The speciﬁc arrangements of baﬄes in various TEMA shells are shown in Fig. 10.3. Plate Baﬄes. Two types of plate baﬄes, shown in Fig. 1.10 are segmental, and disk and doughnut. Single and double segmental baﬄes are used most frequently. The single segmental baﬄe is generally referred to simply as a segmental baﬄe. The practical range of single segmental baﬄe spacing is 15 to 1 shell diameter, although optimum could be 25 to 12. The minimum baﬄe spacing for cleaning the bundle is 50.8 mm (2 in.) or 15 shell diameter, whichever is larger. Spacings closer than 15 shell diameter provide added leakagez that nulliﬁes the heat transfer advantage of closer spacings. If the foregoing limits on the baﬄe spacing do not satisfy other design constraints, such as pmax or tube vibration, no-tubes-in-window or pure crossﬂow design should be tried. The segmental baﬄe is a circular disk (with baﬄe holes) with one disk segment removed. The baﬄe cut varies from 20 to 49% (the height ‘c in Fig. 8.9 given as a percentage of the shell inside diameter), with the most common being 20 to 25%. At larger spacings, it is 45 to 50%, to avoid excessive pressure drop across the windows as compared to the bundle. Large or small spacings coupled with large baﬄe cuts are undesirable because of the increased potential of fouling associated with stagnant ﬂow

{ Here the T range represents the diﬀerence in condensing temperature at the inlet minus condensing temperature at the outlet of an exchanger. { These are tube to baﬄe hole, baﬄe to shell, bundle to shell, and the tube pass partition leakages or bypasses described in Section 4.4.1.1.

GENERAL SELECTION GUIDELINES FOR MAJOR EXCHANGER TYPES

683

areas. If fouling is a primary concern, the baﬄe cut should be kept below 25%. The baﬄe cut and spacing should be designed such that the ﬂow velocity has approximately the same magnitude for the cross ﬂow and window ﬂow sections. Alternate segmental baﬄes are arranged 1808 to each other, which cause shell-side ﬂow to approach crossﬂow in the central bundle regiony and axial ﬂow in the window zone. All segmental baﬄes shown in Fig. 1.10 have horizontal baﬄe cuts. The direction of the baﬄe cut is selected as follows for shell-side ﬂuids: Either horizontal or vertical for a single-phase ﬂuid (liquid or gas), horizontal for better mixing for very viscous liquids, and vertical for the following shellside applications: condensation (for better drainage), evaporation/boiling (for no stratiﬁcation and for providing disengagement room), entrained particulates in liquid (to provide least interference for solids to fall out), and multishell pass exchanger, such as those in Fig. 1.62 and the F shell. Since one of the principal functions of the plate baﬄe is to support the tubes, the terms baﬄe and support plate are sometimes used interchangeably. However, a support plate does not direct the ﬂuid normal to the tube bank, it may be thicker than a baﬄe, it has less tube-to-baﬄe hole clearance, and it provides greater stiﬀness to the bundle. Support plates with single-segmental baﬄes are cut approximately at the centerline and spaced 0.76 m (30 in.) apart. This results in an unsupported tube span of 1.52 m (60 in.) because each plate supports half the number of tubes. The double-segmental baﬄe (Fig. 1.10), also referred to as a strip baﬄe, provides lower shell-side pressure drop (and allows larger ﬂuid ﬂows) than that for the single segmental baﬄe for the same unsupported tube span. The baﬄe spacing for this case should not be too small; otherwise, it results in a more parallel (longitudinal) ﬂow (resulting in a lower heat transfer coeﬃcient) with signiﬁcant zones of ﬂow stagnation. Triple-segmental baﬄes have ﬂows with a strong parallel ﬂow component, provide lower pressure drop, and permit closer tube support to prevent tube vibrations. The lower allowable pressure drop results in a large baﬄe spacing. Since the tubes in the window zone are supported at a distance of two or more times the baﬄe spacing, they are most susceptible to vibration. To eliminate the possibility of tube vibrations and to reduce the shell-side pressure drop, the tubes in the window zone are removed and support plates are used to reduce the unsupported span of the remaining tubes. The resulting design is referred to as the segmental baﬄe with no-tubes-in-window, shown in Fig. 1.10. The support plates in this case are circular and support all the tubes. The baﬄe cut and number of tubes removed varies from 15 to 25%. Notice that low-velocity regions in the baﬄe corners do not exist, resulting in good ﬂow characteristics and less fouling. Thus the loss of heat transfer surface in the window section is partially compensated for. However, the shell size must be increased to compensate for the loss in the surface area in the window zone, which in turn may increase the cost of the exchanger. If the shell-side operating pressure is high, this no-tubes-in-window design is very expensive compared to a similar exchanger having tubes in the window zone. The disk-and-doughnut baﬄe is made up of alternate disks and doughnut-shaped baﬄes, as shown in Fig. 1.10. Generally, the disk diameter is somewhat greater than the half-shell diameter, and the diameter of the hole of the doughnut is somewhat smaller than the half-shell diameter. This baﬄe design provides a lower pressure drop compared to that in a single-segmental baﬄe for the same unsupported tube span and eliminates the tube bundle-to-shell bypass stream C. The disadvantages of this design are that (1) all the {

Various allowable clearances required for construction of a tube bundle with plate baﬄes are provided by TEMA (1999).

684

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

tie rods to hold baﬄes are within the tube bundle, and (2) the central tubes are supported by the disk baﬄes, which in turn are supported only by tubes in the overlap of the largerdiameter disk over the doughnut hole. Rod Baﬄes. Rod baﬄes are used to eliminate ﬂow-induced vibration problems. For certain shell-and-tube exchanger applications, it is desirable to eliminate the cross ﬂow and have pure axial (longitudinal) ﬂow on the shell side. For the case of high shell-side ﬂow rates and low-viscosity ﬂuids, the rod baﬄe exchanger has several advantages over the segmental baﬄe exchanger: (1) It eliminates ﬂow-induced tube vibrations since the tubes are rigidly supported at four points successively; (2) the pressure drop on the shell side is about one-half that with a double segmental baﬄe at the same ﬂow rate and heat transfer rate. The shell-side heat transfer coeﬃcient is also considerably lower than that for the segmental baﬄe exchanger. In general, the rod baﬄe exchanger will result in a smaller-shell-diameter longer-tube unit having more surface area for the same heat transfer and shell-side pressure drop; (3) there are no stagnant ﬂow areas with the rod baﬄes, resulting in reduced fouling and corrosion and improved heat transfer over that for a plate baﬄe exchanger; (4) since the exchanger with a rod (grid) baﬄe design has a counterﬂow arrangement of the two ﬂuids, it can be designed for higher exchanger eﬀectiveness and lower mean (or inlet) temperature diﬀerences than those of an exchanger with a segmental baﬄe design; and (5) a rod baﬄe exchanger will generally be a lower-cost unit and has a higher exchanger heat transfer rate to pressure drop ratio overall than that of a segmental baﬄe exchanger. If the tube-side ﬂuid is controlling and has a pressure drop limitation, a rod baﬄe exchanger may not be applicable. Refer to Gentry (1990) for further details on this exchanger. Impingement Baﬄes. Impingement baﬄes or plates are generally used in the shell side just below the inlet nozzle. Their purpose is to protect the tubes in the top row near the inlet nozzle from erosion, cavitation, and/or vibration due to the impact of the highvelocity ﬂuid jet from the nozzle to the tubes. One of the most common forms of this baﬄe is a solid square plate located under the inlet nozzle just in front of the ﬁrst tube row, as shown in Fig. 10.2. The location of this baﬄe is critical within the shell to minimize the associated pressure drop and high escape velocity of the shell ﬂuid after the baﬄe. For this purpose, adequate areas should be provided both between the nozzle and plate and between the plate and tube bundle. This can be achieved either by omitting some tubes from the circular bundle as shown in Fig. 10.2 or by modifying the nozzle so that it has an expanded section (not shown in Fig. 10.2). Also, proper positioning of this plate in the ﬁrst baﬄe space is important for eﬃcient heat transfer.

FIGURE 10.2 Impingement baﬄes at the shell-side inlet nozzle. (From Bell, 1998.)

GENERAL SELECTION GUIDELINES FOR MAJOR EXCHANGER TYPES

Shell Type

Fixed Tubesheet and Floating Head Bundles

685

U-Tube Bundles

TEMA E

TEMA F

TEMA G

TEMA H

TEMA J single nozzle entry

TEMA J double nozzle entry

L longitudinal flow

TEMA X cross flow

FIGURE 10.3 Shell-side ﬂow arrangement for various shell types (Courtesy of Heat Transfer Research, Inc., College Station, Texas).

686

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

Enough space should be provided between the tip of the plate and the tubesheet and between the tip of the plate and the ﬁrst segmental baﬄe. The most common cause of tube failure is improper location and size of the impingement plate. 10.2.1.4 Shells. Seven types of shells, as classiﬁed by TEMA (1999), are shown in Fig. 1.6; they are also shown in Fig. 10.3 with baﬄes. The E shell, the most common due to its low cost and relative simplicity, is used for single-phase shell ﬂuid applications and for small condensers with low vapor volumes. Multiple passes on the tube side increase the heat transfer coeﬃcient h (if corresponding more increased p is within allowed limits). However, a multipass tube arrangement can reduce the exchanger eﬀectiveness or F factor compared to that for a single-pass arrangement (due to some tube passes being in parallelﬂow) if the increased h and NTU do not compensate for the parallelﬂow eﬀect. Two E shells in series (in overall counterﬂow conﬁguration) may be used to increase the exchanger eﬀectiveness ". As an alternative, a counterﬂow arrangement is desirable (i.e., high ") for a two-tubepass exchanger. This is achieved by the use of an F shell having a longitudinal baﬄe, resulting in two shell passes. However, a TEMA F shell is rarely used in practice because of heat leakage across the longitudinal baﬄe and potential ﬂow leakage that can occur if the area between the longitudinal baﬄe and the shell is not sealed properly. Also, the F shell presents additional problems of fabrication and maintenance, and it is diﬃcult to remove or replace the tube bundle. If one needs to increase the exchanger eﬀectiveness, multiple shells in series are preferred over an F shell. The TEMA G and H shells are related to the F shell but have diﬀerent longitudinal baﬄes. Hence, when the shell-side p is a limiting factor, a G or H shell can be used; however, " or F will be lower than that of a counterﬂow exchanger. The split-ﬂow G shell has horizontal baﬄes with the ends removed; the shell nozzles are 1808 apart at the midpoint of the tubes. The double-split-ﬂow H shell is similar to the G shell, but with two inlet and two outlet nozzles and two longitudinal baﬄes. The G and H shells are seldom used for shell-side single-phase applications, since there is no advantage over E or X shells. They are used as horizontal thermosiphon reboilers, condensers, and other phase-change applications. The longitudinal baﬄe serves to prevent ﬂashing of the lighter components of the shell ﬂuid, helps ﬂush out noncondensables, provides increased mixing, and helps distribute the ﬂow. Generally, T and p across longitudinal baﬄes are small in these applications, and heat transfer across the baﬄe and ﬂow leakages at the sides have insigniﬁcant inﬂuence on the performance. The H shell approaches the crossﬂow arrangement of the X shell, and it usually has low shell-side p compared to the E, F, and G shells. For high-inlet-velocity applications, two nozzles are required at the inlet, hence the H or J shell is used. The divided-ﬂow TEMA J shell has two inlets and one outlet or one inlet and two outlet nozzles (a single nozzle at the midpoint of the tubes and two nozzles near the tube ends). The J shell has approximately one-eighth the pressure drop of a comparable E shell and is therefore used for low-pressure-drop applications such as in a condenser in vacuum. For a condensing shell ﬂuid, the J shell is used, with two inlets for the gas phase and one central outlet for the condensate and residue gases. The TEMA K shell is used for partially vaporizing the shell ﬂuid. It is used as a kettle reboiler in the process industry and as a ﬂooded chiller (hot liquid in tubes) in the refrigeration industry. Usually, it consists of an overall circular-cross-section horizontal bundle of U tubes placed in an oversized shell with one or more vapor nozzles on the top side of the shell (see one vapor nozzle in Fig. 1.6) to reduce liquid entrainment. The tube

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687

FIGURE 10.4 An X shell exchanger with omission of top tube rows for better ﬂow distribution for a condensing application. (From Bell, 1998.)

bundle diameter ranges 50 to 70% of the shell diameter. The liquid (to be vaporized) enters from below near the tubesheet through the left-hand nozzle and covers the tube bundle. Pool and some convective boiling takes place on the shell side without forced ﬂow of the vaporizing ﬂuid outside the tubes on the shell side. The vapor occupies the upper space in the shell without the tubes. The large empty space in the shell acts as a vapor disengaging space; and if properly sized, almost dry vapor exits from the top nozzle, thus eliminating the need for an external vapor–liquid separator. Hence, it is commonly used, although it is more expensive to fabricate, particularly, for highpressure applications. Generally, the kettle reboiler is considered as a pool boiling device; however, convective (ﬂow) boiling prevails in the tube bundle. For a given ﬂow rate and surface area, the crossﬂow TEMA X shell has the lowest shell-side pressure drop of all (except for K) shell conﬁgurations. Hence, it is used for gas heating and cooling applications with or without ﬁnned tubes and for vacuum condensing applications. It is also used for applications having large shell ﬂows. No transverse baﬄes are used in the X shell; however, support plates are used to suppress the ﬂowinduced vibrations. Flow distributions on the shell side could be a serious problem unless proper provision has been made to feed the ﬂuid uniformly at the inlet. This could be achieved by a bathtub nozzle, multiple nozzles, or by providing a clear lane along the length of shell near the nozzle inlet as shown in Fig. 10.4. The type of shell described in Fig. 1.6 has either one or two shell passes in one shell. The cost of the shell is much more than the cost of tubes; hence, a designer tries to accommodate the required heat transfer surface in one shell. Three or four shell passes in a shell could be made by the use of longitudinal baﬄes.y Multipassing on the shell side with longitudinal baﬄes will reduce the ﬂow area per pass compared to a single pass on the shell side in a single shell, resulting in a possibly higher shell-side pressure drop. Multiple shells in series are also used for a given application for the following reasons: . They increase the exchanger eﬀectiveness ", or reduce the surface area for the same ". For the latter case, a subsequent reduction in tubing cost may oﬀset the cost of an additional shell and other components. {

Positive or tight sealing between the longitudinal baﬄes and the shell is essential to maintain the high exchanger eﬀectivenesses predicted.

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SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

. For an exchanger requiring high eﬀectiveness, multipassing is the only alternative. . For part-load operation and where a spare bundle is essential, multiple shells (may be smaller in size) will result in an economical operation. . Shipping and handling may dictate restrictions on the overall size or weight of the unit, resulting in multiple shells for an application. In heat recovery trains and some other applications, up to six shells in series are commonly used. The limitation on the number of shells in such applications is the pressure drop limit on one of the ﬂuid streams. 10.2.1.5 Front-End Heads. The front- and rear-end head types, as classiﬁed by TEMA (1999), are shown in Fig. 1.6. The front-end head is stationary, while the rear-end head can be either stationary or ﬂoating, depending on the allowed thermal stresses between the tubes and the shell. The major criteria for the selection of front- and rear-end heads are the thermal stresses, operating pressures, cleanability, hazards, and cost. The front-end heads are primarily of two types, the channels and the bonnet. The bonnet head B is cast in one piece and has either a side- or an end-entering nozzle.y Although the bonnet head is less expensive, inspection and maintenance requires breaking the pipe joints and removing the bonnet. Hence, the bonnet head is generally used for clean tube-side ﬂuids. The channel head can be removable, as in the TEMA A head, or can be integral with the tubesheet, as in TEMA C and N heads. There is a removable channel cover in these front-end heads for inspection and maintenance without disturbing the piping. The nozzles are side entering in these types. Notice that while the shell is welded onto the TEMA N head, it is ﬂanged to the TEMA C head. In the TEMA N head, no mechanical joint exists (all welded joints) between the channel and tubesheet and between the tubesheet and the shell, thus eliminating leakage between the shell and the tubes. The TEMA D head has a special high-pressure closure and is used for applications involving 2100 kPa gauge (300 psig) for ammonia service and higher pressures for other applications. 10.2.1.6 Rear-End Heads. In a shell-and-tube exchanger, the shell is at a temperature diﬀerent from that of the tubes because of heat transfer between the shell and tube ﬂuids. This results in a diﬀerential thermal expansion and stresses among the shell, tubes, and the tubesheet. If proper provisions are not made, the shell or tubes can buckle, or tubes can be pulled apart or out of the tubesheet. Provision is made for diﬀerential thermal expansion in the rear-end heads. They may be categorized as ﬁxed or ﬂoating rear-end heads, depending on whether there are no provisions or some provisions for diﬀerential thermal expansion. A more commonly used third design that allows tube expansion freely is the exchanger with U tubes having the frontand rear-end heads ﬁxed; it is included in the ﬂoating rear-end-head category in the discussion to follow. The design features of shell-and-tube exchangers with various rear-end heads are summarized in Table 10.3. A heat exchanger with a ﬁxed rear-end head L, M, or N has a ﬁxed tubesheet on that side. Hence, the overall design is rigid. The tube bundle-to-shell clearance is least among the designs, thus minimizing the bundle-to-shell bypass stream C. Any number of tube { Notice that the nozzles on the front- and rear-end heads are for the tube ﬂuid. The nozzles for the shell ﬂuid are located on the shell itself.

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TABLE 10.3 Design Features of Shell-and-Tube Heat Exchangers

Design Feature

Fixed Tubesheet

Outsidepacked Stuﬃng box

Return Bend (U-Tube)

OutsidePacked Latern Ring

Inside PullSplit Through Backing Bundle Ring

TEMA Rear-Head Type: L, M, N

U

P

W

T

S

Tube bundle removable Spare bundles used Provides for diﬀerential movement between shell and tubes Individual tubes can be replaced Tubes can be chemically cleaned, both inside and outside Tubes can be mechanicall cleaned on inside Tubes can be mechanically cleaned on outside Internal gaskets and bolting are required Double tubesheets are practical Number of tubesheet passes available Approximate diametral clearance (mm) (Shell ID, Dotl Þ Relative costs in ascending order, (least expensive ¼ 1Þ

No No Yes, with bellows in shell Yes

Yes Yes Tes

Yes Yes Yes

Yes Yes Yes

Yes Yes Yes

Yes Yes Yes

Yesa

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

With special tools Yesb

Yesb

Yesb

Yesb

Yesb

No

No

No

No

Yes

Yes

Yes

Yes

Yes

No

No

No

Any

Anyc

Anye

25–50

One or twod 15–35

Anye

11–18

Any even number 11–18

95–160

35–50

2

1

4

3

5

6

Source: Data from Shah (1995). a Only those in outside rows can be replaced without special designs. b Outside mechanical cleaning possible with square or rotated square pitch, or wide triangular pitch. c Axial nozzle required at rear end for odd number of passes. d Tube-side nozzles must be at stationary end for two passes. e Odd number of passes requires packed or bellows at ﬂoating head.

passes can be employed. The TEMA L, M, and N rear-end heads are the counterparts of TEMA A, B, and N front-end heads. The major disadvantages of the ﬁxed tubesheet exchanger are (1) no relief for thermal stresses between the tubes and the shell, (2) the impossibility of cleaning the shell side mechanically (only chemical cleaning is possible), and (3) the impracticality of replacing the tube bundle. Fixed tubesheet exchangers are thus used for applications involving relatively low temperatures [3158C (6008F) and lower] coupled with low pressures [2100 kPa gauge (300 psig) and lower]. As a rule

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SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

of thumb, the ﬁxed tubesheet design is used for an inlet temperature diﬀerence between the two ﬂuids that is less than about 50 to 608C (1008F). If an expansion bellows is used, this temperature diﬀerence can be increased to about 80 to 908C (1508F). Expansion bellows are generally uneconomical for high pressures [>4150 kPa gauge (600 psig)]. The ﬁxed tubesheet exchanger is a low-cost unit ranked after the U-tube exchanger. The diﬀerential thermal expansion can be accommodated by a ﬂoating rear-end head in which tubes expand freely within the shell, thus eliminating thermal stresses. Also, the tube bundle is removable for mechanical cleaning of the shell side. Basically, there are three types of ﬂoating rear-end heads: U-tube heads, internal ﬂoating heads (pullthrough/split-ring heads), and outside packed ﬂoating heads. In the U-tube bundle, the thermal stresses are signiﬁcantly reduced, due to free expansion of the U-tubes, and the rear-end head has an integral cover which is the least expensive among rear-end heads. The exchanger construction is simple, having only one tubesheet and no expansion joints, and hence it is the lowest-cost design, particularly at high pressures. The tube bundle can be removed for shell-side cleaning; however, it is diﬃcult to remove a U tube from the bundle except in the outer row, and it is also diﬃcult to clean the tube-side bends mechanically. So a U-tube exchanger is used with clean ﬂuids inside the tubes unless the tube side can be cleaned chemically. Flowinduced vibration can also be a problem for the tubes in the outermost row because of a long unsupported span, particularly in large-diameter bundles. The next-simplest ﬂoating head is the pull-through head T shown in Fig. 10.5. On the ﬂoating-head side, the tubesheet is small, acts as a ﬂange, and ﬁts in the shell with its own bonnet head. The tube bundle can easily be removed from the shell by ﬁrst removing the

FIGURE 10.5 Two-pass exchanger (BET) with a pull-through (T) rear-end head. (Courtesy of Patternson-Kelley Co., Division of HARSCO Corporation, East Stroudsburg, Pennsylvania.)

FIGURE 10.6 (a) Sealing strips; (b) dummy tubes or ties rods; (c) sealing strips, dummy tubes, or tie rods covering the entire length of the tube bundle.

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691

front-end head. Individual tubes or the tube bundle can also be replaced if required. Due to the ﬂoating-head bonnet ﬂange and bolt circle, many tubes are omitted from the tube bundle near the shell. This results in the largest bundle-to-shell circumferential clearance or a signiﬁcant bundle-to-shell bypass stream C. So as not to reduce exchanger performance, sealing strips (or dummy tubes or tie rods with spacers) in the bypass area are essential, as shown in Fig. 10.6. They are placed in pairs every ﬁve to seven tube pitches between the baﬄe cuts. They force the ﬂuid from the bypass stream back into the bundle. However, localized high velocities near the sealing strips could cause ﬂowinduced tube vibration; hence, proper care must be exercised for the design. Since this design has the least number of tubes in a bundle for a given shell diameter compared to other ﬂoating-head designs, the shell diameter is somewhat larger, to accommodate a required amount of surface area. One of the ideal applications of the TEMA T head design is in the kettle reboiler, for which there is ample space on the shell side and the ﬂow bypass stream C is of no concern. The large bundle-to-shell clearance can be minimized by bolting the ﬂoating-head bonnet to a split backing ring (ﬂange) as shown in Fig. 10.7. It is referred to as the TEMA S rear-end head. The shell cover over the tube ﬂoating head has a diameter larger than that of the shell. As a result, the bundle-to-shell clearances are reasonable and sealing strips are generally not required. However, both ends of the exchanger must be disassembled for cleaning and maintenance. In both TEMA S and T heads, the shell ﬂuid is held tightly to prevent leakage to the outside. However, internal leakage is possible due to the failure of an internal hidden gasket and is not easily detectable. The TEMA T head has more positive gasketing between the two streams than does the S head. Both TEMA S and T head conﬁgurations are used for the tube-side multipass exchangers; the singlepass construction is not feasible if the advantages of the positive sealing of TEMA S and T heads are to be retained. The cost of TEMA S and T head designs is relatively high

FIGURE 10.7 Two-pass exchanger (AES) with a split-ring (S) ﬂoating head. (Courtesy of Patternson-Kelley CO., Division of HARSCO Corporation, East Stroudsburg, Pennsylvania.)

FIGURE 10.8 Two-pass exchanger (AEP) with an outside packed (P) ﬂoating head. (Courtesy of Patternson-Kelley Co., Division of HARSCO Corporation, East Stroudsburg, Pennsylvania.)

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SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

FIGURE 10.9 Two-pass exchanger (BEW) having a packed ﬂoating head (W) with lantern rings. (Courtesy of Patternson-Kelley Co., Division of HARSCO Corporation, East Stroudsburg, Pennsylvania.)

compared to U-tube or ﬁxed-tubesheet units. The cost for the TEMA S head is higher than for the TEMA T head. The split backing ring ﬂoating head is used extensively in the petroleum industry for moderate operating pressures and temperatures. For very high operating pressures and temperatures, the TEMA S head design has a special test ring (TEMA, 1999). In the outside-packed ﬂoating-head TEMA P design of Fig. 10.8, the stuﬃng box provides a seal against the skirt of the ﬂoating head and prevents shell-side ﬂuid leakage to the outside. This skirt (and the tube bundle) is free to move axially against the seal to take thermal expansion into account. A split-ring ﬂange near the end of the skirt seals the back end of the chamber. Because of the speciﬁc design of this ﬂoating head, any leak (from either the shell side or the tube side) at the gaskets is to the outside. Hence, the TEMA P head is generally not used with very toxic ﬂuids. Also, the inlet and outlet nozzles must be located at the stationary end; hence, this design could have only an even number of tube passes. In this design, the bundle-to-shell clearance is large [about 38 mm (1.5 in.)]; as a result, sealing strips are required. The TEMA P head exchanger is more expensive than the TEMA W head exchanger. The packed ﬂoating head with lantern ring or TEMA W head is shown in Fig. 10.9. Here a lantern ring rests on the machined surface of the tubesheet and provides an eﬀective seal between the shell- and tube-side ﬂanges. Vents are usually provided in the lantern ring to help locate any leaks in the seals before the shell-side and tube-side ﬂuids mix. Although a single-pass design is possible on the tube side, generally an even number of tube passes is used. The TEMA W head exchanger is the lowest-cost design of all ﬂoating heads. Although its cost is higher than that of the U-tube bundle, this higher cost is oﬀset by the accessibility to the tube ends (by opening both rear- and front-end heads) for cleaning and repair; consequently, this design is sometimes used in the petrochemical and process industries. A large number of combinations of front- and rear-end heads with diﬀerent shell types of Fig. 1.6 are possible, depending on the application and the manufacturer. Some common types of combinations result in the following shell-and-tube heat exchangers: AEL, AES, AEW, BEM, AEP, CFU, AKT, and AJW. In light of the availability of diﬀerent types of front- and rear-end heads, the tube bundle of a shell-and-tube exchanger may simply be classiﬁed as a straight-tube or Utube bundle. Both have a ﬁxed tubesheet at the front end. The U-tube bundle has a shell with a welded shell cover on the U-bend end. The straight tube bundle has either a ﬁxed

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693

FIGURE 10.10 Inﬂuence of various geometrical parameters of a shell-and-tube exchanger on heat transfer and pressure drop.

tubesheet or a ﬂoating head at the rear end. The former is referred to as a ﬁxed-tubesheet bundle; the latter is referred to as a ﬂoating-head bundle. With this background, two ﬂowcharts are presented in Fig. 10.10 to increase the overall heat transfer rate or decrease the pressure drop on either the tube or shell side during various stages of designing a shell-and-tube exchanger. 10.2.2

Plate Heat Exchangers

The chevron plate is the most common in PHEs. Hence, we will not discuss the reasoning behind why various other plate geometries have been used in PHEs. As described in Section 1.5.2.1, PHEs have a number of advantages over shell-and-tube heat exchangers, such as compactness, low total cost, less fouling, accessibility, ﬂexibility in changing the number of plates in an exchanger, high q and ", and low ﬂuid residence time. Because of these advantages, they are in second place to shell-and-tube heat exchangers for market share in liquid-to-liquid and phase-change applications. The main reason for their limited versatility involves the pressure and temperature restrictions imposed by the gaskets. Replacing gaskets on one or both sides by laser welding of the plates (as in

694

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

a welded PHE) increases both the operating pressure and temperature limits of the gasketed PHEs, and it allows the PHE to handle corrosive ﬂuids compatible with the plate material. For low-heat duties (that translate into a total surface area up to 10 m2 ), a more compact brazed PHE can replace the welded PHE, thus eliminating the frame, guide bars, bolts, and so on, of welded or gasketed PHEs. A variety of other PHEs have been developed for niche applications to cover speciﬁc operating conditions that cannot be handled by the above-described PHEs. Some of these PHEs are described brieﬂy at the end of Section 1.5.2.2. Although the shell-and-tube heat exchanger is versatile and can handle all kinds of operating conditions, it is not compact, not ﬂexible in design, requires a large footprint, and is costly (total cost) compared to PHEs and other compact heat exchangers. Hence, PHEs and many other heat exchanger designs have been invented to replace shell-andtube heat exchangers in individual narrow operating ranges. Refer to Sections 1.5.2 and 1.5.3 for details of these exchangers. Also, an excellent source of information on compact heat exchangers for liquid-to-liquid and phase-change applications is a recent monograph by Reay (1999).

10.2.3

Extended Surface Exchangers

10.2.3.1 Plate-Fin Exchanger Surfaces. The plate-ﬁn construction is commonly used in gas–to–gas or gas–to–phase change exchanger applications where either the heat transfer coeﬃcients are low or an ultrahigh exchanger eﬀectiveness is desired. It oﬀers high surface area densities [up to about 6000 m2 /m3 (1800 ft2 /ft3 )] and a considerable amount of ﬂexibility. The passage height on each ﬂuid side could easily be varied and diﬀerent ﬁns can be used between plates for diﬀerent applications. On each ﬂuid side, the ﬁn thickness and number of ﬁns can be varied independently. If a corrugated ﬁn (such as the plain triangular, louver, perforated, or wavy ﬁn) is used, the ﬁn can be squeezed or stretched to vary the ﬁn pitch, thus providing added ﬂexibility. The ﬁns on each ﬂuid side could easily be arranged such that the overall ﬂow arrangement of the two ﬂuids can result in crossﬂow, counterﬂow, or parallelﬂow. Even the construction of a multistream plate-ﬁn exchanger is relatively straightforward with the proper design of inlet and outlet headers for each ﬂuid (ALPEMA, 2000). Plate-ﬁn exchangers are generally designed for low-pressure applications, with operating pressures limited to about 1000 kPa gauge (150 psig). However, cryogenic plate-ﬁn exchangers are designed for operating pressures of 8300 kPa (1200 psig). With modern manufacturing technology, they can be designed for very high pressures; for example, the gas cooler for an automotive air-conditioning system with CO2 as the refrigerant has an operating pressure of 12.5 to 15.0 MPa (1800 to 2200 psig). The maximum operating temperatures are limited by the type of ﬁn-to-plate bonding and the materials employed. Plate-ﬁn exchangers have been designed from low cryogenic operating temperatures ½2008C ð4008FÞ to about 8008C (15008F). Fouling is generally not as severe a problem with gases as it is with liquids. A plate-ﬁn exchanger is generally not designed for applications involving heavy fouling since there is no easy method of cleaning the exchanger unless chemical cleaning can be used. If an exchanger is made of small modules (stacked in the height, width, and length directions), and if it can be cleaned with a detergent, a high-pressure air jet, or by baking it in an oven (as in a paper industry exchanger), it could be designed for applications having considerable fouling. Fluid

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695

contamination (mixing) is generally not a problem in plate-ﬁn exchangers since there is practically zero ﬂuid leakage from one ﬂuid side of the exchanger to the other. Selection of a ﬁn surface depends on the operating temperature, with references to bonding of the ﬁns to plates or tubes and a choice of material. For low-temperature applications, a mechanical joint, or soldering or brazing may be adequate. Fins can be made from copper, brass, or aluminum, and thus maintain high ﬁn eﬃciency. For hightemperature applications, only special brazing techniques and welding may be used; stainless steel and other expensive alloys may be used to make ﬁns but with a possibly lower ﬁn eﬃciency, due to their relatively lower thermal conductivities unless proper lower ﬁn height is selected. Consequently, suitable high-performance surfaces may be selected to oﬀset the potential reduction in ﬁn eﬃciency unless the proper ﬁn height is selected. Brazing would require added capital and maintenance cost of a brazing furnace, cost of brazing, and process expertise (Sekulic´ et al., 2003). Cost is a very important factor in the selection of exchanger construction type and surface. The plate-ﬁn surface in general is less expensive than a tube-ﬁn surface per unit of heat transfer surface area. In most applications, one does not choose a high-performing surface, but rather, the least expensive surface, if it can meet the performance criteria within speciﬁed constraints. For example, if a plain ﬁn surface can do the job for a speciﬁed application, the higher-performance louver or oﬀset strip ﬁn surface is not used because it is more expensive to manufacture. We now discuss qualitatively the construction and performance behavior of plain, wavy, oﬀset strip, louver, perforated, and pin ﬁn plate-ﬁn surfaces. Plain Fin Surfaces. These surfaces are straight ﬁns that are uninterrupted (uncut) in the ﬂuid ﬂow direction. Although triangular and rectangular passages are more common, any complex shape desired can be formed, depending on how the ﬁn material is folded. Although the triangular (corrugated) ﬁn (e.g., Fig. 1.29a, e and f ) is less expensive, can be manufactured at a faster rate, and has the added ﬂexibility of having an adjustable ﬁn pitch, it is generally not structurally as strong as the rectangular ﬁn (e.g., Fig. 1.29b and d) for the same passage size and ﬁn thickness. The triangular ﬁns can be made in very low to ultrahigh ﬁn densities [40 to 2400 ﬁns/m (1 to 60 ﬁn/in.)]. Plain ﬁns are used in applications where the allowed pressure drop is low and the augmented interrupted surfaces cannot meet the design requirement of allowed p for a desired ﬁxed frontal area. Also, plain ﬁns are preferred for very low Reynolds numbers applications. This is because with interrupted ﬁns, when the ﬂow approaches the fully developed state at such low Re, the advantage of the high h value of the interrupted ﬁns is diminished while cost remains high, due to making interruptions. Plain ﬁns are also preferred for high-Reynolds-number applications where the p for interrupted ﬁns become excessively high. Wavy Fin Surfaces. These surfaces also have uncut surfaces in the ﬂow direction, and have cross-sectional shapes similar to those of plain surfaces (see Fig. 1.29c). However, they are wavy in the ﬂow direction, whereas the plain ﬁns are straight in the ﬂow direction. The waveform in the ﬂow direction provides eﬀective interruptions to the ﬂow and induces very complex ﬂows. The augmentation is due to Go¨rtler vortices, which form as the ﬂuid passes over the concave wave surfaces. These are counterrotating vortices, which produce a corkscrewlike pattern. The heat transfer coeﬃcient for a wavy ﬁn is higher than that for an equivalent plain ﬁn. However, the heat transfer coeﬃcient for wavy ﬁns is lower than that for interrupted ﬁns such as oﬀset or louver

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SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

ﬁns. Since there are no cuts in the surface, wavy ﬁns are used in those applications where an interrupted ﬁn might be subject to a potential fouling or clogging problem due to particulates, freezing moisture, bridging louvers due to condensate, and so on. Oﬀset Strip Fins. This is one of the most widely used enhanced ﬁn geometries in plateﬁn heat exchangers (see Fig. 1.29d). The ﬁn has a rectangular cross section, and is cut into small strips of length ‘s . Every alternate strip is displaced (oﬀset) by about 50% of the ﬁn pitch in the transverse direction. In addition to the ﬁn spacing and ﬁn height, the major variables are the ﬁn thickness and strip length in the ﬂow direction. The heat transfer coeﬃcients for the oﬀset strip ﬁns are 1.5 to 4 times higher than those of plain ﬁn geometries. The corresponding friction factors are also high. The ratio of j=f for an oﬀset strip ﬁn to j=f for a plain ﬁn is about 80%. Designed properly, the oﬀset strip ﬁn exchanger would require a substantially lower heat transfer surface area than that of plain ﬁns at the same p. The heat transfer enhancement for an oﬀset strip ﬁn is caused mainly by the redeveloping laminar boundary layers for Re 10,000. However, at higher Re, it acts as a rough surface (a constant value of f with decreasing j for increasing Re). Oﬀset strip ﬁns are used in the approximate Re range 500 to 10,000, where enhancement over the plain ﬁns is substantially higher. For speciﬁed heat transfer and pressure drop requirements, the oﬀset strip ﬁn requires a somewhat higher frontal area than a plain ﬁn, but a shorter ﬂow length and overall lower volume. Oﬀset strip ﬁns are used extensively by aerospace, cryogenic, and many other industries where higher heat transfer performance is required. Louver Fins. Louvers are formed by cutting the metal and either turning, bending, or pushing out the cut elements from the plane of the base metal (see Fig. 1.29e). Louvers can be made in many diﬀerent forms and shapes. The louver ﬁn gauge is generally thinner than that of an oﬀset strip ﬁn. Louver pitch (also referred to as louver width) and louver angle (in addition, the ﬁn spacing and ﬁn height) are the most important geometrical parameters for the surface heat transfer and ﬂow friction characteristics. On an absolute level, j factors are higher for louver ﬁns than for the oﬀset strip ﬁn at the same Reynolds number, but the f factors are even higher than those for the oﬀset strip ﬁn geometry. Since the louver ﬁn is triangular (or corrugated), it is generally not as strong as an oﬀset strip ﬁn; the latter ﬁn has a relatively large ﬂat area for brazing, thus providing strength. The louver ﬁns may have a slightly higher potential for fouling than oﬀset strip ﬁns. Louver ﬁns are amenable to high-speed mass production manufacturing technology, and as a result, are less expensive than oﬀset strip ﬁns and other interrupted ﬁns when produced in very large quantities. The ﬁn spacing desired can be achieved by squeezing or stretching the ﬁn; hence it allows some ﬂexibility in ﬁn spacing without changes in tools and dies. This ﬂexibility is not possible with the oﬀset strip ﬁn. A wide range in performance can be achieved by varying the louver angle, width, and form. The operating Reynolds number range is 100 to 5000, depending on the type of louver geometry employed. Modern multilouver ﬁns have higher heat transfer coeﬃcients that those for oﬀset strip ﬁns, but with somewhat lower j=f ratios. However, the performance of a well-designed multilouver ﬁn exchanger can approach that of an oﬀset strip exchanger, possibly with increased surface compactness and reduced manufacturing cost. Multilouver ﬁns (see Figs. 1.27, 1.28, and 1.29e) are used extensively by the automotive industry.

GENERAL SELECTION GUIDELINES FOR MAJOR EXCHANGER TYPES

697

Perforated Fins. A perforated ﬁn has either round or rectangular perforations with the size, shape, and longitudinal and transverse spacings as major perforation variables (see Fig. 1.29f ). The perforated ﬁn has either triangular or rectangular ﬂow passages. When used as a plate-ﬁn surface, it is generally brazed. The holes interrupt the ﬂow and may increase h somewhat, but considerable surface area may also be lost, thus nullifying the advantage. Perforated ﬁns are now used only in a limited number of applications. They are used as turbulators in oil coolers for mixing viscous oils, or as a high-p ﬁn to improve ﬂow distribution. Perforated ﬁns were once used in vaporizing cryogenic ﬂuids in air separation exchangers, but oﬀset strip ﬁns have now replaced them. Pin Fins. These can be manufactured at very high speed continuously from a wire of proper diameter. After the wire is formed into rectangular passages (e.g., rectangular plain ﬁns), the top and bottom horizontal wire portions are ﬂattened for brazing or soldering with the plates. Pins can be circular or elliptical in shape. Pin ﬁn exchanger performance is considerably lower, due to the parasitic losses associated with round pins in particular and to the inline arrangement of the pins (which results from the highspeed manufacturing techniques). The surface compactness achieved by pin ﬁn geometry is much lower than that of oﬀset strip or louver ﬁn surfaces. Due to vortex shedding behind the round pins, noise- and ﬂow-induced vibration may be a problem. Finally, the cost of a round wire is generally more than the cost of a ﬂat sheet, so there may not be a material cost advantage. The potential application for pin ﬁns is at very low Reynolds number (Re < 500Þ, for which the pressure drop is of no major concern. Pin ﬁns are used in electronic cooling devices with generally free convective ﬂows over the pin ﬁns. 10.2.3.2 Tube-Fin Surfaces. When an extended surface is needed on only one ﬂuid side (such as in a gas-to-liquid exchanger) or when the operating pressure needs to be contained on one ﬂuid side, a tube-ﬁn exchanger (see Section 8.2) may be selected, with the tubes being round, ﬂat, or elliptical in shape. Also, when minimum cost is essential, a tube-ﬁn exchanger is selected over a plate-ﬁn exchanger since the ﬁns are not brazed but are joined mechanically to the tubes by mechanical expansion. Flat or elliptical tubes, instead of round tubes, are used for increased heat transfer in the tube and reduced pressure drop outside the tubes; however, the operating pressure is limited compared to that for round tubes. Tube-ﬁn exchangers usually have lower heat transfer surface compactness than a plate-ﬁn unit, with a maximum heat transfer surface area density of about 3300 m2 /m3 (1000 ft2 /ft3 ). A tube-ﬁn exchanger may be designed for a wide range of tube ﬂuid operating pressures [up to about 3000 kPa gauge (450 psig) or higher] with the other ﬂuid being at low pressure [up to about 100 kPa (15 psig)]. The highest operating temperature is again limited by the type of bonding and the materials employed. Tube-ﬁn exchangers are designed to cover the operating temperature range from low cryogenic temperatures to about 8708C (16008F). Reasonable fouling can be tolerated on the tube side if the tubes can be cleaned. Fouling is generally not a problem on the gas side (ﬁn side) in many applications; plain uninterrupted ﬁns are used when ‘‘moderate’’ fouling is expected. Fluid contamination (mixing) of the two ﬂuids is generally not a problem since there is essentially no ﬂuid leakage between them. Since tubes are generally more expensive than extended surfaces, the tube-ﬁn exchanger is in general more expensive. In addition, the heat transfer surface area density of a tube-ﬁn core is generally lower than that of a plate-ﬁn exchanger, as mentioned earlier.

698

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

The tube-ﬁn construction is generally used in liquid-to-gas or phase-change ﬂuid-togas heat exchanger applications with liquid, condensing ﬂuid, or evaporating ﬂuid on the tube side. Fins are generally used on the outside of tubes (on the gas side), although depending on the application, ﬁns or turbulators may also be used inside the tubes. Round and ﬂat tubes (rectangular tubes with rounded or sharp corners) are most common; however, elliptical tubes are also used. Round tubes are used for higher-pressure applications and also when considerable fouling is anticipated. Parasitic form drag is associated with ﬂow normal to round tubes. In contrast, the ﬂat tubes yield a lower pressure drop for ﬂow normal to the tubes, due to lower form drag, and thus avoid the low-performance wake region behind the tubes. Also, the heat transfer coeﬃcient is higher for ﬂow inside ﬂat tubes than for circular tubes, particularly at low Re. The use of ﬂat tubes is limited to low-pressure applications, such as automotive radiators, unless the tubes are extruded with ribs inside (see the multiport tube in Fig. 1.27, also referred to as microchannels) or with integral ﬁns outside. Flat Fins on a Tube Array. This type of tube-ﬁn geometry (shown in Fig. 1.31b) is most commonly used in air-conditioning and refrigeration exchangers in which high pressure needs to be contained on the refrigerant side. As mentioned earlier, this type of tube-ﬁn geometry is not as compact (in terms of surface area density) as the plate-ﬁn geometries, but its use is becoming widespread due to its lower cost. This is because the bond between the ﬁn and tube is made by mechanically or hydraulically expanding the tube against the ﬁn instead of soldering, brazing, or welding the ﬁn to the tube. Because of the mechanical bond, the applications are restricted to those cases in which the diﬀerential thermal expansion between the tube and ﬁn material is small, and preferably, the tube expansion is greater than the ﬁn expansion. Otherwise, the loosened bond may have a signiﬁcant thermal resistance. Many diﬀerent types of ﬂat ﬁns are available (see some examples in Fig. 1.33). The most common are the plain, wavy, and interrupted. The plain ﬂat ﬁns are used in those applications in which the pressure drop is critical (quite low), although a larger amount of surface area is required on the tube outside for the heat transfer speciﬁed than with wavy or interrupted ﬁns. Plain ﬂat ﬁns have the lowest pressure drop than that of any other tube-ﬁn surfaces at the same ﬁn density. Wavy ﬁns are superior in performance to plain ﬁns and are more rugged. Wavy ﬁns are used most commonly for air-conditioning condensers and other commercial heat exchangers. A variety of louver geometries are possible on interrupted ﬂat ﬁns. A well-designed interrupted ﬁn would have even better performance than a wavy ﬁn; however, it may be less rugged, more expensive to manufacture, and may have a propensity to clog. Individually Finned Tubes. This tube-ﬁn geometry (shown in Fig. 1.31a) is generally much more rugged than continuous ﬁn geometry but has lower compactness (surface area density). Plain circular ﬁns are the simplest and most common. They are manufactured by tension wrapping the ﬁn material around a tube, forming a continuous helical ﬁn or by mounting circular disks on the tube. To enhance the heat transfer coeﬃcient on the ﬁns, a variety of enhancement techniques have been used (see Fig. 1.32). Segmented or spine ﬁns are the counterpart of the strip ﬁns used in plate-ﬁn exchangers. A segmented ﬁn is generally rugged, has heavy-gauge metal, and is usually less compact than a spine ﬁn. A studded ﬁn is similar to a segmented ﬁn, but individual studs are welded to the tubes. A slotted ﬁn has slots in the radial direction; when radially slitted material is wound on a tube, the slits open, forming slots whose width

SOME QUANTITATIVE CONSIDERATIONS

699

increases in the radial direction. This ﬁn geometry oﬀers an enhancement over tensionwound plain ﬁns; however, segmented or spine ﬁns would yield a better performance. The wire loop ﬁn is formed by spirally wrapping a ﬂattened helix of wire around the tube. The wire loops are held to the tube by a tensioned wire within the helix or by soldering. The enhancement characteristic of small-diameter wires is important at low ﬂows, where the enhancement of other interrupted ﬁns diminishes. 10.2.4

Regenerator Surfaces

Regenerators, used exclusively in gas-to-gas heat exchanger applications, can have a higher surface area density (a more compact surface) than that of plate-ﬁn or tubeﬁn surfaces. While rotary regenerators have been designed for a surface area density of up to about 8800 m2 /m3 (2700 ft2 /ft3 ), the ﬁxed-matrix regenerators have been designed for of up to about 16,000 m2 /m3 (5000 ft2 /ft3 ). Regenerators are usually designed for low-pressure applications, with operating pressures limited to nearatmospheric pressures for rotary and ﬁxed-matrix regenerators; an exception is the gas turbine rotary regenerator, having an inlet pressure of 615 kPa gauge or 90 psig on the air side. The regenerators are designed to cover an operating temperature range from low cryogenic to very high temperatures. Metal regenerators are used for operating temperatures up to about 8708C (16008F); ceramic regenerators are used for higher temperatures, up to 20008C (36008F). The maximum inlet temperature for paper and plastic regenerators is 508C (1208F). Regenerators have self-cleaning characteristics because hot and cold gases ﬂow in opposite directions periodically through the same passage. As a result, compact regenerators have minimal fouling problems and usually have very small hydraulic diameter passages. If severe fouling is anticipated, rotary regenerators are not used; ﬁxed-matrix regenerators with large hydraulic diameter ﬂow passages [50 mm (2 in.)] could be used for very corrosive/fouled gases at ultrahigh temperatures [925 to 16008C (1900 to 29008F)]. Carryover and bypass leakages from the hot ﬂuid to the cold ﬂuid (or vice versa) occur in the regenerator. Where this leakage and subsequent ﬂuid contamination is not permissible, regenerators are not used. Hence, they are not used with liquids. The cost of the rotary regenerator surface per unit of heat transfer surface area is generally substantially lower than that of a plate-ﬁn or tube-ﬁn surface.

10.3 SOME QUANTITATIVE CONSIDERATIONS As presented in Fig. 1.1, heat exchangers can be broadly classiﬁed according to construction as tubular, plate type, extended surface, and regenerative. A large variety of high-performance surfaces are used on the gas side of extended surface and regenerative exchangers. A large number of enhanced tube geometries are available for selection in tubular exchangers. For the general category of enhanced tubes, internally ﬁnned tubes, and surface roughness, Webb and Bergles (1983) have proposed a number of performance evaluation criteria (PEC) to assess the performance of enhanced surfaces compared to similar plain (smooth) surfaces. In plate-type exchangers (used primarily with liquids), although many diﬀerent types of construction are available, the number of surface geometries used in modern exchangers is limited to high-performance chevron plate geometry, which is most commonly used in PHEs. As a result, in this chapter we focus on quantitative screening methods for

700

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

FIGURE 10.11 Comparison of surface basic characteristics of two heat exchanger surfaces. (From Shah, 1983.)

gas ﬂows in compact heat exchangers and performance evaluation criteria (PECs) for tubular surfaces. For extended surfaces, particularly the plate-ﬁn type, selection of the surfaces on both ﬂuid sides are independent of each other, and generally, one ﬂuid side is critical from the pressure drop requirement. Hence, we consider only one ﬂuid side for the surface selection for plate-ﬁn surfaces. 10.3.1

Screening Methods

Surface selection is made by comparing the performance of various heat exchanger surfaces and choosing the best under some speciﬁed criteria (objective function and constraints) for a given heat exchanger application. Consider the j and f characteristics of surfaces A and B in Fig. 10.11. Surface A has both j and f higher than those for surface B. Which is a better surface? This question is meaningless unless one speciﬁes the criteria for surface comparison. If the pressure drop is of less concern, surface A will transfer more heat than surface B for the same heat transfer surface area for a given application. If the pressure drop is critical, one cannot in general say that surface A is better than surface B. One may need to determine, for example, the volume goodness factors for comparison (see Section 10.3.1.2); or one may even need to carry out a complete exchanger optimization after selecting the surface on the other ﬂuid side of a two-ﬂuid exchanger. A variety of methods have been proposed in the literature for surface performance comparisons. These methods could be categorized as follows: (1) direct comparisons of j and f, (2) a comparison of heat transfer as a function of ﬂuid pumping power, (3) miscellaneous direct comparison methods, and (4) performance comparisons with a reference surface. Over 30 such dimensional or nondimensional comparison methods have been reviewed critically by Shah (1978), and many more methods have been published since then. It should be emphasized that most of these comparisons are for the surfaces only on one ﬂuid side of a heat exchanger. When a complete exchanger design is considered that does not lend itself to having one ﬂuid side as a strong side (i.e., having high o hA), the best surface selected by the foregoing methods may not be an optimum surface for a given application. This is because the selection of the surface for the other ﬂuid side and its thermal resistance, ﬂow arrangement, overall exchanger envelope, and other criteria

SOME QUANTITATIVE CONSIDERATIONS

701

(not necessarily related to the surface j and f vs. Re characteristics) inﬂuence the overall performance of a heat exchanger. In addition, if the exchanger is considered as part of an (open or closed system), the exchanger surface (and/or other variables) may be selected based on the system as a whole rather than based on the optimum exchanger as a component. Current methods of surface selection for an optimum heat exchanger for a system include the use of sophisticated computer programs that take into account many possible eﬀects. Such selection is not possible in simpliﬁed approaches presented in the open literature. We focus on considering simple but important quantitative screening methods for surface selection on the gas side of compact heat exchangers since these exchangers employ a large variety of high-performance surfaces. The selection of a surface for a given application depends on exchanger design criteria. For a speciﬁed heat transfer rate and pressure drop on one ﬂuid side, two important design criteria for compact exchangers (which may also be applicable to other exchangers) are the minimum heat transfer surface area requirement and the minimum frontal area requirement. Let us ﬁrst discuss the signiﬁcance of these criteria. To understand the minimum frontal area requirement, let us ﬁrst review how the ﬂuid pressure drop and heat transfer are related to the ﬂow area requirement, the exchanger ﬂow length, and the ﬂuid velocity. The ﬂuid pressure drop on one ﬂuid side of an exchanger, neglecting the entrance/exit and ﬂow acceleration/deceleration losses, is given from Eq. (6.29) as p ¼

4fLG2 2gc Dh

ð10:1Þ

Since predominantly developed and/or developing laminar ﬂows prevail in compact heat exchangers, the friction factor is related to the Reynolds number as follows (see Sections 7.4.1.1 and 7.4.2.1): ( f ¼

C1 Re1 C2 Re0:5

for fully developed laminar flow for developing laminar flow

ð10:2Þ

where C1 and C2 are constants. Substituting Eq. (10.2) into Eq. (10.1) and noting that Re ¼ GDh =, we get p /

LG LG1:5

for fully developed laminar flow for developing laminar flow

ð10:3Þ

Here G ¼ m_ =Ao : Therefore, for a speciﬁed constant ﬂow rate m_ , the pressure drop is proportional to the ﬂow length L and inversely proportional to the ﬂow area Ao or A1:5 o . The Nusselt number for laminar developed and developing temperature proﬁles and developed velocity proﬁles is given by (see Sections 7.4.1.1 and 7.4.3.1) Nu ¼

C3 C4 ðDh Pr Re=LÞ1=3

for thermally developed laminar flow for thermally developing laminar flow

ð10:4Þ

702

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

where C3 and C4 are constants. Thus, the heat transfer coeﬃcient is independent of the mass ﬂow rate m_ or mass velocity G for the thermally developed laminar ﬂow and is proportional to G1=3 for the thermally developing laminar ﬂow. We have not considered simultaneously developing laminar ﬂow (in which Nu / G1=2 ) since thermally developing ﬂow provides a conservative estimate of Nu. Considering p and h simultaneously, a decrease in G will reduce p linearly without reducing h for fully developed laminar ﬂow; for developing laminar ﬂows, a reduction in G will reduce p as in Eq. (10.3), with a slight decrease in h as given by Eq. (10.4). Now as discussed earlier, there are a variety of enhanced surfaces available for selection. An undesirable consequence of the heat transfer enhancement is an increase in the friction factor, which results in a higher pressure drop for an exchanger having a ﬁxed frontal area and a constant ﬂow rate. As noted in the preceding paragraph, reducing G can reduce the pressure drop in compact exchangers without signiﬁcantly reducing the heat transfer coeﬃcient h. For a speciﬁed constant ﬂow rate, a reduction in G means an increase in the ﬂow area Ao for constant m_ and approximately constant (the ratio of free-ﬂow to frontal area). So as one employs the more enhanced surface, the required ﬂow area (and hence frontal area) increases accordingly to meet the heat transfer and pressure drop requirements speciﬁed. Thus, one of the characteristics of highly compact surfaces is that the resulting shape of the exchanger becomes more like a pancake, having a large frontal area and a short ﬂow length (e.g., think of the shape of an automotive radiator in contrast to a shell-and-tube exchanger; see also Example 10.3 to show the increase in free-ﬂow area when employing a higher-performance surface). Hence, it is important to determine which of the compact surfaces will meet a minimum frontal area requirement. The surface having the highest heat transfer coeﬃcient at a speciﬁed ﬂow rate will require the minimum heat transfer surface area. However, the allowed pressure drop is not unlimited. Therefore, one chooses the surface having the highest heat transfer coeﬃcient for a speciﬁed ﬂuid pumping power. The exchanger with the minimum surface area will have the minimum overall volume requirement. From the foregoing discussion, two major selection criteria for compact surfaces with gas ﬂows are (1) a minimum frontal area requirement and (2) a minimum volume requirement. For this purpose, the surfaces are evaluated based on the surface ﬂow area and volume goodness factors. We discuss these comparison methods after the following example. Example 10.1 Consider a gas turbine rotary regenerator (Fig. E10.1) having compact triangular ﬂow passages operating at Re ¼ 1000 and the pressure drop on the highpressure air side as 10 kPa. Determine the change in heat transfer and pressure drop if this regenerator is operated at Re ¼ 500. The following data are provided for the analysis: j¼

3:0 Re

f ¼

14:0 Re

Pr ¼ 0:7

ðhAÞh ¼ ðhAÞc

flow split ¼ 50 : 50

Ignore the eﬀect of the ﬂow area blockage by the hub and the radial seals to determine the required changes. Assume that the mass ﬂow rate of air (and hence gas) does not change when Re is reduced. How would you achieve the reduction in Re when the L and Dh values of the regenerator surface are being kept constant?

SOME QUANTITATIVE CONSIDERATIONS

703

Hot gas

Cold air FIGURE E10.1

SOLUTION Problem Data and Schematic: The data are given in the problem statement for j, f , Pr, and thermal conductances and ﬂow split. Also, L and Dh are to be kept constant. Determine: Determine the change in heat transfer and pressure drop for this regenerator when the air-side (and hence gas-side) Re is reduced from 1000 to 500. How is the reduction in Re achieved when the mass ﬂow rate of the air is kept constant? Assumptions: The ﬂow is fully developed (thermally and hydrodynamically) laminar. The wall thermal resistance and fouling resistance are negligible. The ﬂuid properties do not change with the change in Re; and with a ﬂow split of 50 : 50, we expect the same eﬀects on the air and gas sides with a change in Re. Analysis: Since the ﬂow is assumed to be fully developed laminar, Nu will be constant (see Section 7.4.1.1) and is given by its relationship to the j factor by Eq. (7.33) and input data j Re ¼ 3:0 as Nu ¼ j Re Pr1=3 ¼ 3:0ð0:7Þ1=3 ¼ 2:66 Using the deﬁnition of Nu ¼ hDh =k and constant Dh for the change of Re from 1000 to 500, we get Nu2 h1 ¼ ¼1 Nu1 h2 Here subscripts 1 and 2 denote the cases for Re ¼ 1000 and 500, respectively. Thus, the heat transfer coeﬃcient does not change with Re. This will also be the case for h on the gas side. Thus, UA and hence the heat transfer rate q will not change.

704

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

When Re ð¼ GDh =Þ is changed from 1000 to 500 without changing the mass ﬂow rate and the regenerator geometry (L and Dh ), we get Re2 G2 500 1 ¼ ¼ ¼ Re1 G1 1000 2 Now let us evaluate the pressure drop. From Eq. (6.29), p /

L Gð f ReÞ D2h

Since L, Dh , and f Re are constant for this regenerator, we can evaluate p2 with Re ¼ 500 from p2 G2 1 ¼ ¼ p1 G1 2

or

p2 ¼ 0:5 10 kPa ¼ 5kPa

Ans:

Hence, the pressure drop will be reduced to 50% by decreasing Re by 50%. As we ﬁnd above, the change in Re is accomplished by the corresponding change in G for constant m_ ¼ GAo : This means the ﬂow area Ao has to be doubled to reduce G by 50%, and hence the frontal area of the disk has to be doubled. By neglecting the eﬀect of the area blockage by hub and radial seals, we obtain r22 Af ;2 Ao;2 = ¼2 ¼ ¼ r21 Af ;1 Ao;1 = Therefore, r2 pﬃﬃﬃ ¼ 2 ¼ 1:41 r1 where is the ratio of free-ﬂow area to frontal area, and r2 and r1 are disk radii for Re ¼ 1000 and 500, respectively. Hence, the disk radius or diameter of this regenerator will need to be increased by 41%. Ans. Discussion and Comments: As discussed in Section 7.4.1.1, this problem clearly indicates that in fully developed laminar ﬂow, the heat transfer coeﬃcient is not reduced by reducing the ﬂuid velocity ðum ¼ G=Þ, whereas the pressure drop is linearly reduced with the reduction in the ﬂow velocity. Hence, a reduction in ﬂow velocity without a reduction in the mass ﬂow rate can be achieved by increasing the ﬂow area and hence the frontal area of the exchanger. 10.3.1.1 Surface Flow Area Goodness Factor Comparison. London (1964) deﬁned the ratio j=f as the surface ﬂow area goodness factor. Using the deﬁnitions of j, Nu, f , and Re, we get the ratio j=f as j Nu Pr1=3 1 ¼ 2 ¼ f Re f Ao o

Pr2=3 ntu m_ 2 2gc p

! ð10:5Þ

SOME QUANTITATIVE CONSIDERATIONS

705

The term in parentheses on right-hand side of Eq. (10.5) is dependent only on the operating parameters and is independent of the geometry and heat transfer surface involved. Equation (10.5) can be rearranged as A*o ¼ Afr* ¼

Ao ½ðPr

2=3

=2gc Þðntu

m_ 2 =pÞ1=2

Afr ½ðPr2=3 =2gc Þðntu m_ 2 =pÞ1=2

¼ ¼

1 ½o ð j=f Þ1=2 1 ½o ð j=f Þ1=2

ð10:6aÞ ð10:6bÞ

The left-hand sides of Eq. (10.6a) and (10.6b) are the dimensionless free-ﬂow area A*o and frontal area Afr*, respectively. Equations (10.5) and (10.6) show the signiﬁcance of j=f as being inversely proportional to A2o ðAo is the surface minimum free-ﬂow area) for speciﬁed operating conditions and o as constant. A surface having a higher j=f factor is good because it will require a lower free-ﬂow area and hence a lower frontal area for the exchanger. The dimensionless j and f factors are independent of the length scale of the geometry (i.e., the hydraulic diameter).y Thus, the ﬂow area Ao is independent of the hydraulic diameter, but dependent on the operating conditions (m_ and pÞ, design condition (ntu), and ﬂuid type (Pr). Note that for many compact surfaces, no signiﬁcant variation is found in the j=f ratio over the reported test Reynolds number range, and hence Ao and Afr are not a strong function of the surface type. For fully developed laminar ﬂow through simple geometries of Table 7.3, we ﬁnd jH1 =f ranging from 0.263 for the equilateral triangular duct to 0.386 for the parallelplate duct. Thus, the parallel-plate duct relative to the triangular duct has 47% ð0:386=0:263 1Þ higher j=f . Then from Eq. (10.6), we get Ao;1 =Ao;2 ¼ ð0:263=0:386Þ1=2 ¼ 0:825 (since o;1 ¼ o;2 ¼ 1Þ; where the subscripts 1 and 2 are for the parallel-plate and triangular passages, respectively. Thus a parallel-plate exchanger would have a 17.5% smaller free-ﬂow area requirement. In the free-ﬂow area goodness factor comparison, no estimate of total heat transfer area or volume can be inferred. Such estimates may be derived from the core volume goodness factors described next. 10.3.1.2 Core Volume Goodness Factor Comparisons. Two types of core volume goodness factor comparisons are suggested: hstd vs. Estd and o hstd vs. Estd . In the ﬁrst method, a comparison is made for surfaces having the same hydraulic diameter. In the second method, a comparison is made of the actual performance of surfaces having equal or diﬀerent hydraulic diameters. The heat transfer rate per unit temperature diﬀerence and per unit surface area ½q=AðTw Tm Þz , and the ﬂuid pumping power due to friction per unit of surface area are expressed as h¼

{

cp

1 j Re Pr2=3 Dh

ð10:7Þ

As long as the surface geometrical similarity is maintained (i.e., the surface geometry is enlarged or reduced in size by changing the hydraulic diameter), and j and f vs. Re characteristics remain the same. Hence, the j=f ratio is independent of Dh at a given design Re. { The term heat transfer power is sometimes used to denote heat transfer rate q, watts.

706

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

E¼

P 1 3 1 f Re3 ¼ A 2gc 2 D3h

ð10:8Þ

In either of the volume goodness factor comparisons, it is assumed that the surfaces under comparison will provide the same performance. It means that the following quantities are kept constant: (1) the same heat transfer rate, (2) the same pressure drop, (3) the same temperature diﬀerence between the wall and the ﬂuid, and (4) the same ﬂuid ﬂow rate. Remember that we want to arrive at a ‘‘best’’ surface on one ﬂuid side of the exchanger from the minimum-volume-requirement point of view. The heat transfer rate q and ﬂuid pumping power P due to friction on one ﬂuid side are q ¼ o hAðTw Tm Þ ¼ o hVðTw Tm Þ P ¼ EA ¼ EV ¼

m_ p

ð10:9Þ ð10:10Þ

where E is the ﬂuid pumping power per unit surface area. hstd vs. Estd Comparisons. For a rotary regenerator application in which all the surface is prime surface ðo ¼ 1Þ, the comparison of performance is made at the same ﬂuid pumping power due to friction per unit surface area (E) and the same compactness or hydraulic diameter Dh ð¼ 4=Þ assuming that remains constant. The same Dh eliminates the scale or size variable of the passages. Then for the same q, Tw Tm , and , from Eq. (10.9), h/

1 V

ð10:11Þ

Thus the higher hy means a lower overall core volume requirement. Then the excellence of a particular surface geometry in terms of the core volume is characterized by a high position on a dimensional plot of h vs. E, as suggested by London and Ferguson (1949). Considering the gas turbine application, they evaluated all ﬂuid properties for dry air at standard conditions (‘‘std’’) of 2608C (5008F) and 1 atm pressure. However, for other applications, these standard conditions for ﬂuid property evaluation could be changed to any conditions for the desired ﬂuid for the hstd vs. Estd plot. Since the regenerator is used for a gas-to-gas application, it is generally a thermally balanced heat exchanger. In this case, the thermal resistances on both sides are of the same order of magnitude and hence UA hA=2. Thus the comparison of hstd with Estd is a realistic comparison for regenerators. The hstd vs. Estd plot for fully developed ﬂow with constant ﬂuid properties through some constant cross-sectional ducts is presented in Fig. 10.12 for Dh ¼ 0:5 mm (0.0016 ft). From this ﬁgure, it is found that hstd varies from 256.4 to 700.6 W/m2 K, a factor of 2.7, with hstd ¼ 264:7 W=m2 K for the equilateral triangular duct. When made for a ﬁxed Dh , a plot of Fig. 10.12, clearly shows the inﬂuence of the passage shape. The parallel plate heat exchanger may prove impractical, but it is clear that there are several other conﬁgurations that possess signiﬁcant advantages over the triangular and sine duct geometries. Based on this plot, the development and use of { Note that the heat transfer coeﬃcient h is the same as the heat transfer power per unit temperature diﬀerence and per unit surface area.

SOME QUANTITATIVE CONSIDERATIONS

707

FIGURE 10.12 Theoretical laminar volume goodness factors for some simple duct geometries (From Shah, 1983.)

rectangular passage geometry is being continued for applications that involve fully developed laminar ﬂows. From Eqs. (10.7) and (10.8), it is evident that the dimensional hstd vs. Estd performance is strongly dependent on the length scale of the surface geometry (i.e., Dh ). Thus, this comparison method reveals the beneﬁt of increased performance (reduced surface area requirement) by going to a smaller Dh surface. This will also result in a much more compact surface. For the foregoing reasons, the plot of hstd vs. Estd is recommended for selection of a heat exchanger surface for a new application for which there are no signiﬁcant system or manufacturing constraints. o hstd vs. Estd Comparisons. The preceding method of comparison was for surfaces having the same Dh , , and o value (if there are any ﬁns). When one wants to compare the performance of extended surfaces, for which j and f data are available, one may be interested in comparing the surfaces as they are. This is because we may not be able to manufacture a surface whose geometry is scaled up or scaled down. Such a comparison of actual surfaces could be made by a plot of o hstd vs. Estd . Here is the surface area density or compactness, o hstd represents the heat transfer power per unit temperature diﬀerence and unit core volume, and Estd represents the friction power expenditure per unit core volume. Note that this plot is modiﬁed from the hstd vs. Estd recommended by Kays and London (1998) by including the eﬀect of overall ﬁn eﬃciency o of the secondary surface. This eﬀect is important for extended surface heat exchanger applications. The foregoing variables, for a given set of surfaces, are evaluated from the following equations with ﬂuid properties determined at some standard conditions: o hstd ¼ Estd ¼

cp Pr2=3

o

4 j Re D2h

3 4 f Re3 2gc 2 D4h

ð10:12Þ ð10:13Þ

708

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

where ¼ 4=Dh and o ¼ 1 ðAf =AÞð1 f Þ: These equations have been derived from the deﬁnitions of j, f , and P. From Eq. (10.9), o hstd / 1=V for a given q and Tw Tm . Hence for constant Estd , a surface having a high plot of o hstd vs. Estd is characterized as the best from the viewpoint of heat exchanger volume. Example 10.2 Consider the rotary regenerator of Example 10.1 operating at Re ¼ 1000 ðp ¼ 10 kPa, q ¼ some given value). As noted in Table 7.3, the rectangular passage of aspect ratio 18 has higher Nu and f Re than those for the equilateral triangular ﬂow passages. What would be the change in disk diameter, ﬂow length, and volume of the regenerator if the ﬂow passages are changed from triangular to rectangular with the same hydraulic diameter Dh , porosity , p, q, and air and gas mass ﬂow rates. The following data are provided. Equilateral triangular: j Re ¼ 3:0 and f Re ¼ 14:0; rectangular: j Re ¼ 5:2 and f Re ¼ 22:0. SOLUTION Problem Data and Schematic: In addition to the data provided in Example 10.1, the following data are provided for the rectangular ﬂow passages: j Re ¼ 5:2 and f Re ¼ 22:0. The hydraulic diameters of triangular and rectangular passages are identical. The regenerator sketch is shown in Fig. E10.1. Determine: The change in the disk diameter, disk ﬂow depth, regenerator core volume, and operating Reynolds number when changing the ﬂow passages from triangular to rectangular. Assumptions: The assumptions are the same as those in Example 10.1. Analysis: Let us ﬁrst evaluate the change in ﬂow area due to the change in passage geometry. From Eq. (10.5), Ao;2 ð j=f Þ1=2 ð3:0=14:0Þ1=2 1 ¼ ¼ ¼ 0:952 1=2 Ao;1 ð j=f Þ ð5:2=22:0Þ1=2 2 where the subscripts 1 and 2 are for triangular and rectangular ﬂow passages. Thus, a rectangular passage exchanger will require 4.8% (0.048) smaller ﬂow area and hencepfrontal ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ area for the same porosity. This translates approximately into 2.4% ð1 0:952Þ reduction in the disk diameter. From Eqs. (10.11) and (10.12), the core volume ratio is V2 h1 Nu1 ð j Re Pr1=3 Þ1 3:0 ¼ ¼ ¼ ¼ ¼ 0:577 V1 h2 Nu2 ð j Re Pr1=3 Þ2 5:2 Hence the reduction in the core volume is 42.3% ð1 0:577Þ:

Ans.

y

Since V ¼ LAfr ¼ LAo =, we get L2 V2 Ao;1 V2 =V1 0:577 ¼ ¼ ¼ ¼ 0:606 L1 Ao;2 V1 Ao;2 =Ao;1 0:952 {

As noted just before, Eq. (10.11) is valid for 1 ¼ 2 ¼ . Hence, we have considered here as constant. This means that the wall thickness for rectangular and triangular passages will be diﬀerent. They can be calculated using the formulas of Table 8.2 by equating hydraulic diameters, and for the same computing b for rectangular passages for the known b for equilateral triangular passages.

SOME QUANTITATIVE CONSIDERATIONS

709

Thus, the regenerator disk thickness or ﬂow length will be reduced by 39.4% ð1 0:606Þ. Ans. Finally, the operating Reynolds number will be changed as follows: Re2 ðm_ Dh =Ao Þ2 Ao;1 1 ¼ ¼ ¼ ¼ 1:05 Re1 ðm_ Dh =Ao Þ1 Ao;2 0:952 Thus, the operating Reynolds number will increase by 5% or will be 1050 ð1000 1:05Þ: Ans.

Discussion and Comments: As shown in this example, by going from triangular ﬂow passages to rectangular ﬂow passages of low aspect ratio, substantial savings in the regenerator volume and hence mass can be achieved along with lower packaging. However, it is a challenge to manufacture more diﬃcult rectangular ﬂow passages than triangular ﬂow passages. Example 10.3 Select an oﬀset strip ﬁn versus a plain ﬁn on the air side of an exchanger for which the heat transfer rate and inlet temperatures are speciﬁed. The frontal area and surface area comparisons are to be done at the same airﬂow rates, hA, and ﬂuid pumping powers. The j and f data for these ﬁns at the same hydraulic diameter are provided in Fig. E10.3. The design Reynolds number for the plain ﬁn is 3000. Determine the frontal area, surface area, ﬂow length, and volume requirements for the oﬀset strip ﬁn compared to those for the plain ﬁn.

FIGURE E10.3 The j and f data comparison for an oﬀset strip ﬁn and plain ﬁn at Dh ¼ 3:51 mm. (From Webb, 1994.)

710

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

SOLUTION Problem Data and Schematic: The j and f data are given in Fig. E10.3 for the oﬀset strip ﬁn and plain ﬁn of Fig. 1.29d and b respectively. For the plain ﬁn, Re ¼ 3000. Determine: For identical m_ , hA, and P, determine As =Ap and Ao;s =Ao; p where the subscripts s and p denote the values for strip ﬁn and plain ﬁn surfaces, respectively. Assumptions: Assume constant ﬂuid properties and only one ﬂuid side of the exchanger. Analysis: Since the j and f factors are higher for the oﬀset strip ﬁn, the operating Reynolds number for the oﬀset strip ﬁn will be lower than that for the plain ﬁn for the same m_ , hA, P, q, Th;i , and Tc;i . This Reynolds number needs to be evaluated iteratively from the following relationship, which is derived from Eq. (10.5) with constant ntu, m_ and p, and, the deﬁnition of Re½¼ ðm_ =Ao ÞDh = with m_ and Dh constant]: Res Ao; p ð j=f Þ1=2 s ¼ ¼ Rep Ao;s ð j=f Þ1=2 p

ð1Þ

From Fig. E10.3, the j and f factors at Re ¼ 3000 for the plain ﬁn are jp ¼ 0:0038

fp ¼ 0:011

We need to assume a value of Res such that the corresponding js and fs satisfy Eq. (1) for the values of Rep , jp , and fp above. Let us assume that Rep =Res ¼ 1:20. Hence, Res ¼

3000 ¼ 2500 1:20

From Fig. E10.3, the j and f factors at Re ¼ 2500 for the strip ﬁn are js ¼ 0:010

fs ¼ 0:042

Substituting values of j’s and f ’s in Eq. (1) yields Res ð0:010=0:042Þ1=2 ¼ ¼ 0:83 Rep ð0:0038=0:011Þ1=2 or Rep =Res ¼ 1:20. Therefore, our guess of Rep =Res is correct (which was obtained based on iterations) and we don’t need to iterate further. Otherwise, continue guessing the values of Rep =Res until Eq. (1) is satisﬁed. From Eq. (10.8), the ﬂuid pumping power P is given by P¼

1 3 1 f Re3 A 2gc 2 D3h

Applying this equation, for equal pumping power (P p ¼ P s Þ and Dh;p ¼ Dh;s , we get 3 As ð f Re Þp 0:011 ¼ ¼ ð1:20Þ3 ¼ 0:453 Ap ð f Re3 Þs 0:042

Ans:

SOME QUANTITATIVE CONSIDERATIONS

711

where the numerical values computed earlier are substituted. The minimum free-ﬂow area ratio, from Eq. (10.5), is given by Ao;s ð j=f Þ1=2 p ¼ ¼ 1:20 Ao; p ð j=f Þ1=2 s

Ans:

The oﬀset strip ﬁn will require 20% higher frontal area (for the same ) and 54.7% ð1 0:453Þ less surface area than will the plain ﬁn for this problem. Using the deﬁnition of the hydraulic diameter Dh ¼ 4Ao L=A, we get Ls As Ao; p Dh;s 1 ¼ ¼ 0:453 1 ¼ 0:378 Lp Ap Ao;s Dh; p 1:20 since Dh;s ¼ Dh; p is given. Now if we assume the porosity is the same for the two surfaces, the volume ratio is given by Ao;s Ls Vs ¼ ¼ 1:20 0:378 ¼ 0:454 Vp Ao; p Lp Discussion and Comments: This example demonstrates that by employing higherperforming surface geometry, in general, one ends up with somewhat larger freeﬂow and frontal areas, but overall signiﬁcantly lower surface area, smaller ﬂow length, lower volume, and lower mass of the exchanger. Thus if packaging allows somewhat higher frontal area, the material cost of the exchanger surface will be lower with a higher-performing surface. 10.3.1.3 General Relationships for Compact Heat Exchanger Surfaces. There is a continued technology drive toward higher compactness as practiced in the development of automotive, aerospace, and other compact heat exchangers. One of the common ways to describe the surface compactness is to characterize the surface with its hydraulic diameter as outlined in Section 1.4. In plate-ﬁn exchangers, the heat transfer surface on each ﬂuid side can often be selected independently of the shape and size of the resultant exchanger. There are other reasons (e.g., one ﬂuid side is critical from a pressure drop requirement, a retroﬁt application, improved manufacturing process) why replacement of one heat transfer surface with the other may be an option. Hence, we provide general relationships that include geometry and surface performance characteristics ( j, f, and related factors) when we change the hydraulic diameter of a compact surface. These relations will include the free-ﬂow area, frontal area, volume, ﬂow length surface area, and Reynolds number as functions of Dh and j, f , Re, o , , and/or . Once the appropriate surface is selected, eventually we need to consider both ﬂuid sides for the exchanger heat transfer performance and pressure drop calculations. We now provide these relations based on the ﬂow area and volume goodness factor comparisons provided in Sections 10.3.1.1 and 10.3.1.2, with all ﬂuid properties as constant. These relationships will be useful for comparing two heat transfer surfaces on one ﬂuid side of the exchanger (such as in plate-ﬁn exchangers or rotary regenerators), if the surface on that ﬂuid side can be selected or changed independently. They are not useful for the PHEs since the surfaces on both ﬂuid sides cannot be selected independently (refer to Section 9.4.1). The following variables are kept constant: ﬂuid ﬂow rate, heat transfer

712

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

rate (heat duty), and pressure drop (or ﬂuid pumping power) for the derivation of these relationships. The ratios of free-ﬂow area and frontal areas on one ﬂuid side for surface 1 to surface 2 are as follows using Eq. (10.6): Ao;2 o;1 ð j=f Þ1 1=2 ¼ Ao;1 o;2 ð j=f Þ2

Afr;2 1 o;1 ð j=f Þ1 1=2 ¼ Afr;1 2 o;2 ð j=f Þ2

ð10:14Þ

Note that this equation is valid for any (the same or diﬀerent) hydraulic diameter of the two surfaces and satisﬁes the heat transfer and pressure drop requirements speciﬁed for a given ﬂuid ﬂow rate (i.e., q, p, and m_ are constant for these relationships). For an equal ﬂuid pumping power requirement [i.e., P ¼ constant for Eq. (10.10)] and equal heat duty q, from Eq. (10.9) we get the following equation for volume between plates on one ﬂuid side with use of the deﬁnitions Nu ¼ hDh =k; Dh ¼ 4=, and Nu ¼ j Re Pr1=3 : V2 ðo hÞ1 o;1 1 Dh;2 Nu1 o;1 1 ð j ReÞ1 Dh;2 2 ¼ ¼ ¼ V1 ðo hÞ2 o;2 2 Dh;1 Nu2 o;2 2 ð j ReÞ2 Dh;1

ð10:15Þ

Using Dh from Re ¼ ðm_ =Ao ÞDh =, substituting it in Eq. (10.15), and using Eq. (10.14), we get an alternative form of Eq. (10.15) as follows: V2 ð f ReÞ2 o;1 1 ¼ V1 ð f ReÞ1 o;2 2

2 j1 j2

ð10:16Þ

The ﬂow length ratio on one ﬂuid side can be computed from L ¼ V=Afr as follows using Eq. (10.15) and the deﬁnition of Re ¼ ðm_ =Afr ÞDh =: L2 o;1 j1 Dh;2 ¼ L1 o;2 j2 Dh;1

ð10:17Þ

The heat transfer surface area ratio A2 =A1 on one ﬂuid side is then calculated from the deﬁnition A ¼ 4Ao L=Dh and Re / Dh =Ao , Eqs. (10.14) and (10.17), as follows: A2 Ao;2 L2 Dh;1 L2 Re1 ð j=f Þ1 1=2 o;1 3=2 j1 ¼ ¼ ¼ A1 Ao;1 L1 Dh;2 L1 Re2 ð j=f Þ2 o;2 j2

ð10:18Þ

Finally, the ratio of operating Reynolds numbers for the two surfaces can be determined from the deﬁnition Re ¼ ðm_ =Ao ÞDh = and the use of Eq. (10.14) as follows: Re2 ð j=f Þ2 1=2 Dh;2 ¼ Re1 ð j=f Þ1 Dh;1

ð10:19Þ

Since the relationships above are derived for constant q, p, m_ , and P, they are not included in the list of operational parameters. Based on Eqs. (10.14)–(10.18), we ﬁnd that the ratios for Ao , Afr , and A are independent of Dh , and the ratios for V and L are proportional to Dh for fully developed laminar ﬂow. However, for other ﬂows

SOME QUANTITATIVE CONSIDERATIONS

713

(such as turbulent, transition, and developing laminar ﬂows), one more constraint (in addition to constant q, p, m_ ; P) needs to be speciﬁed by keeping constant the lefthand side of one of the equations (10.14)–(10.19). While the foregoing relationships are obtained for the case of keeping q, p and m_ constant, similar relationships can be obtained by keeping some variables constant and others varying from the following set: Afr , V, L, A, Dh , Re, q, p, m_ ; P, and so on. Cowell (1990) presents a number of such relationships. Several important observations can be made from the foregoing relationships keeping q, p and m_ constant. These are useful when we have diﬀerent surfaces for selection and we need to decide which to choose for a compact surface having fully developed laminar ﬂow. . Since there is no hydraulic diameter involved in the ﬂow area ratio relationship of Eq. (10.14), the ﬂow area on one ﬂuid side is independent of Dh for fully developed laminar ﬂow. Changing Dh without changing the wall thickness would result in a slight change in the porosity and hence in Afr . . From Eq. (10.15), V / Dh =j. Hence, the heat exchanger volume on one ﬂuid side decreases with increasing porosity , increasing Colburn factor j and reducing hydraulic diameter Dh . . Based on Eq. (10.17), the ﬂow length on one ﬂuid side decreases with increasing Colburn factor j and decreasing hydraulic diameter Dh . . Based on Eq. (10.18), the heat transfer surface area on one ﬂuid side decreases with increasing j, Re, or Nu (¼ j Re Pr1=3 ), and decreasing Dh . Note that j and f factors for any surface are independent of the hydraulic diameter as long as the surface is geometrically scaled up or down. To emphasize the foregoing points, let us consider a case of a rotary regenerator having a 50 : 50% split for air and gas ﬂows and having fully developed laminar ﬂows, or one ﬂuid side of a plate-ﬁn exchanger with fully developed laminar ﬂow. If the hydraulic diameter is reduced to one-half of the original value on one ﬂuid side, one can show from Eqs. (10.14)–(10.19) that Ao and A remain constant, V reduces to 12, L reduces to 12, and Re reduces to 12, all changes for constant q, p, and m_ on one ﬂuid side.

10.3.2

Performance Evaluation Criteria

Webb (1994) has presented a number of performance evaluation criteria (PECs), shown in Table 10.4, to assess performance merits of enhanced heat transfer surfaces relative to plain surfaces in single-phase ﬂow. When two ﬂuid sides are not independent of each other, these PECs are applied in general; otherwise, use the methods of Section 10.3.1. These PECs have been developed for the following enhancement types: surface roughness, internally ﬁnned tubes, and enhanced tubes; and they can also be applied to plate-ﬁn surfaces. These PECs are actually screening methods since they consider only the thermal and hydraulic performance and do not consider nonthermal performance considerations in the design and optimization of a heat exchanger. A PEC is established by selecting one of the operating variables for the performance objective subject to design constraints on the remaining variables. Operating variables considered are geometry (number of tubes nt in a pass and tube

714

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

TABLE 10.4 Performance Evaluation Criteria for Enhanced Surfaces (with Constant di ) for SinglePhase Heat Exchangers Fixed PEC Case FG-1a FG-1b FG-2a FG-2b FG-3 FN-1 FN-2 FN-3 VG-1 VG-2a VG-2b VG-3

Geometry

m_

nt ; L nt ; L nt ; L nt ; L nt ; L nt nt nt

nt L nt L nt L

P

q

Tmax

Objective

Increase q Decrease Tmax Increase q Decrease Tmax Decrease P Decrease L Decrease L Decrease P Decrease nt L Increase q Decrease Tmax Decrease P

Source: Data from Webb (1994.)

length L), ﬂow rate m_ , ﬂuid pumping power P, heat transfer rate q, and ﬂuid inlet temperature diﬀerence Tmax . For a given PEC, the ratio of the design objective for a surface of interest to that for a reference surface is then calculated as a function of a similar ratio of a design variable. The heat transfer rate in an exchanger is given by q ¼ UA Tm ¼ PRðTh;i Tc;i Þ ¼ PR Tmax

ð10:20Þ

Here P is the exchanger temperature eﬀectiveness and R is the heat capacity rate ratio. Reviewing this equation, the design objectives for the use of enhanced surface may be as follows: 1. Increased UA for equal pumping power P and ﬁxed geometry [frontal area (represented by nt ) and length L]. A higher UA means (a) higher q for a given Tmax or Tm , and (b) lower Tmax or Tm for a given q. 2. For a ﬁxed ﬂow area (i.e., ﬁxed nt in Table 10.4), (a) reduce the tube length L (and hence A) for equal q and m_ or P, and (b) reduce P for equal q and m_ : In all cases, Tmax or Tm is ﬁxed. 3. Reduce the surface area A and hence the volume and mass of the exchanger for ﬁxed m_ and speciﬁed q (or Tmax Þ and P. Based on the foregoing objectives, three major categories of performance evaluation criteria are developed: Fixed geometry (FG), ﬁxed ﬂow area (FN), and variable geometry (VG) criteria. For ﬁxed-geometry criteria, a plain surface is replaced by an enhanced surface of equal length, a retroﬁt application, resulting in higher q (or reduced Tmax ) and higher P. For ﬁxed-ﬂow-area criteria, either P (and L) is reduced at constant q and m_ or m_ (and L) is reduced at constant P and q, employing an enhanced surface. For

SOME QUANTITATIVE CONSIDERATIONS

715

variable-geometry criteria, the surface area A (/ nt L) is reduced and frontal area Afr ð/ nt Þ is increased for an enhanced surface for ﬁxed q, m_ , and P. Several PECs are formulated based on these major criteria, as noted in various FG, FN, and VG cases in Table 10.4. The advantages of PEC comparison methods are that (1) the designer can select his or her own criteria for comparison; (2) he or she can then compare the performance of a surface to that of a reference surface directly, and (3) he or she does not need to evaluate the ﬂuid properties since they drop out in computing the performance ratios. The performance comparisons can include the eﬀect of the thermal resistances of the wall and of fouling and convection on the other ﬂuid side. The optimum surface selected by this method may not be optimum in a two-ﬂuid heat exchanger when non-performancerelated overall heat exchanger constraints are imposed. These aspects are considered in heat exchanger optimization, discussed in Section 9.6. The algebraic relations for the PECs will now be summarized for comparing an enhanced tubular surface to the corresponding plain tube for a shell-and-tube heat exchanger having the tube length L per pass, tube diameters di and do , number of tubes nt in each pass, and number of passes np . For this exchanger, the ﬂow area and surface area on the tube side are given by Ao ¼

2 d n 4 i t

A ¼ di Lnt np

ð10:21Þ

The frontal area Afr on the tube side is then related to Ao for speciﬁed tube layout and pitches. Heat transfer and ﬂow friction characteristics are needed for comparing the performance of an enhanced surface to the corresponding plain surface. We will use a subscript p to designate the quantities for the plain surface (except for cp ), and no subscript for the enhanced surface. From the deﬁnition of the j factor, h ¼ jGcp Pr2=3

ð10:22Þ

To evaluate the heat duty q of the enhanced surface, we need to compare hA of the enhanced surface to ðhAÞp for the plain surface. From Eq. (10.22), hA j G A ¼ ðhAÞp jp Gp Ap

ð10:23Þ

Note that here and in this section Ap represents the surface area of the plain surface. Using Eq. (6.30), the ﬂuid pumping power ratio is given by P f A G 3 ¼ P p fp Ap Gp

ð10:24Þ

A relationship between hA and P for enhanced and plain surfaces is obtained by eliminating the G=Gp term from Eqs. (10.23) and (10.24): hA=ðhAÞp ðP=P p Þ1=3 ðA=Ap Þ2=3

¼

j=jp ð f =fp Þ1=3

ð10:25Þ

716

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

This equation is used for a comparison of two heat transfer surfaces under various criteria of Table 10.4 when two of the three ratios on the left-hand side of Eq. (10.25) are given and the third is to be determined. However, when comparing an enhanced surface with a plain surface in a heat exchanger, the ratio needed for heat transfer performance is UA=ðUAÞp , which takes into account wall thermal resistance, fouling resistance, and convection resistance on the second ﬂuid side. Using Eq. (3.20) or (3.24) for UA and ðUAÞp , one can arrive at the following expression modiﬁed from Webb (1981): 1 þ R*p 1 þ R*p 1 þ R*p UA ¼ ¼ ¼ " #1=3 jp Gp Ap ðUAÞp ðhAÞi; p jp f P p A p 2 þ R* þ R* þ R* j G A hA j fp P A

ð10:26Þ

where R* and R*p are the total thermal resistances (excluding tube inside thermal resistance) for enhanced and plain surfaces, respectively, normalized with respect to the plain tube inside thermal resistance ½1=ðhAÞi; p . They are given explicitly in Table 10.5. Note that Ai; p is simply designated as Ap (plain tube inside surface area) in the last two equalities of Eq. (10.26) and throughout in Table 10.6. In Eq. (10.26), R* includes Ap =A. Hence, R* can be presented as a function of Ap =A as follows (a formula alternative to that in Table 10.5) when Ap =A is not unity, but is a variable: ^ *o þ R ^ o;*f Ap ¼ hi; p þ hi; p w þ hi; p Ap ^ * Ao þ R ^ * Ap ¼ R ð10:27Þ R* ¼ R A Aw A ho kw ho; f A Using Eq. (3.94), the ratio of the heat transfer rate in an enhanced to the plain surface is given by q m_ P Tmax ¼ qp m_ p Pp Tmax; p

ð10:28Þ

Here we have assumed the ﬂuid cp to be the same for enhanced and plain surfaces (and hence C=Cp ¼ m_ =m_ p Þ, P is the temperature eﬀectiveness, and Tmax ð¼ Th;i Tc;i Þ is the inlet temperature diﬀerence. Since the temperature eﬀectiveness P is dependent on NTU, the NTU of the enhanced surface is related to that for the plain surface by NTU ¼ NTUp

UA m_ p ðUAÞp m_

ð10:29Þ

TABLE 10.5 Dimensionless Thermal Resistances for Eq. (10.26) Reference (Plain) Heat Exchanger

Enhanced Heat Exchanger

Outer surface thermal resistance

Ro;*p ¼ ðhAÞi; p =ðhAÞo; p

Ro* ¼ ðhAÞi; p =ðhAÞo

Wall thermal resistance

Rw;* p ¼ ðhAÞi; p w =kw Aw

R*w ¼ ðhAÞi; p w =kw Aw

Fouling resistance

Rf*; p ¼ ðhAÞi; p ðRi; f ; p þ Ro; f ; p Þ

R*f ¼ ðhAÞi; p ðRi; f þ Ro; f Þ

Deﬁnition

Combined resistance

* p þ Rf*; p R*p ¼ Ro;*p þ Rw;

R* ¼ R*o þ Rw* þ R*f

717

Tmax Tmax; p

q qp

FG-1b

FG-2a

Tmax ¼1 Tmax; p

q ¼1 qp

m_ G ¼ ¼1 m_ p Gp

P ¼ 1; Pp

L A ¼ ¼1 Lp Ap

Tmax ¼1 Tmax; p

nt A ¼ fr ¼ 1 nt; p Afr; p

In this case, Tmax will go down and P will go up for the enhanced surface

L A ¼ ¼1 Lp Ap

nt A ¼ fr ¼ 1 nt; p Afr; p

In this case, both q and P will go up for the enhanced surface

m_ G ¼ ¼1 m_ p Gp

nt A ¼ fr ¼ 1 nt; p Afr; p

q qp

FG-1a

L A ¼ ¼1 Lp Ap

Information or Constraints Given and Comments

Determine:

Case

TABLE 10.6 Algebraic Formulas to Evaluate PEC of Table 10.4a

1 þ R*p UA ¼ ðUAÞp ð jp =jÞ þ R* since

P p fp ¼ P f

1 þ R*p UA ¼ ðUAÞp ð jp =jÞ þ R*

since

P p fp ¼ P f

Solve G=Gp iteratively from the ﬁrst equation knowing fp vs. Rep and f vs. Re characteristics. Compute jp and j from the known jp vs. Rep and j vs. Re characteristics. Calculate UA=ðUAÞp from the foregoing equation. Then follow the procedure of FG-1a to get q=qp from NTU. continued over

From Eq. (10.24) and (10.26), for this case 1=3 fp 1 þ R*p G UA ¼ and ¼ f Gp ðUAÞp ð jp =jÞð f =fp Þ1=3 þ R*

Pp m_ p Tmax ¼ Tmax; p P m_

Compute NTU from Eq. (10.29) and subsequently, P, for the given ﬂow arrangement. Then, from Eq. (10.28),

From Eq. (10.26),

q P ¼ qp Pp

Compute NTU from Eq. (10.29) and subsequently P for the given ﬂow arrangement. Then, from Eq. (10.28),

From Eq. (10.26),

Resultant Formulas to Compute the Objective of the PEC

718

FN-1

L Lp

m_ G ¼ m_ p Gp P q Tmax ¼ ¼ ¼1 P p qp Tmax; p

A L ¼ Ap Lp

m_ G ¼ m_ p Gp

q Tmax ¼ ¼1 qp Tmax; p

nt A ¼ fr ¼ 1 nt; p Afr; p

L A ¼ ¼1 Lp Ap

nt A ¼ fr ¼ 1 nt; p Afr; p

q ¼1 qp

P ¼1 Pp

P Pp

L A ¼ ¼1 Lp Ap

nt A ¼ fr ¼ 1 nt; p Afr; p

Tmax Tmax; p

FG-2b

FG-3

Information or Constraints Given and Comments

Determine:

Case

TABLE 10.6 Continued

ð3Þ

ð1Þ

P f ¼ P p fp

G Gp

3 ð2Þ

P f L ¼1¼ Pp fp Lp

G Gp

3 ;

q G P ¼1¼ qp Gp Pp

1 þ R*p UA ¼ ðUAÞp ð jp =jÞð f= fp Þ1=3 ðLp =LÞ2=3 þ R*ðLp =LÞ

From Eqs. (10.26), (10.24), and (10.28)

continued over

From these equations, compute G=Gp as follows: (1) Assume G=Gp . (2) Solve for P=P p from Eq. (2). (3) Compute UA=ðUAÞp from Eq. (1). (4) Calculate NTU from Eq. (10.28). (5) Determine P and Pp for the exchanger ﬂow arrangement given. (6) compute q=qp from Eq. (3) which would be diﬀerent from unity until convergence. Iterate on G=Gp to obtain q=qp ¼ 1: Using converged value of G=Gp ; calculate P=Pp from Eq. (2).

q G P ¼1¼ qp Gp Pp

1 þ R*p UA ¼ ðUAÞp ð jp =jÞ½ð f = fp ÞðP p =PÞ1=3 þ R*

From Eqs. (10.26), (10.24), and (10.28)

Pp m_ p Tmax ¼ Tmax; p P m_

Compute UA=ðUAÞp as outlined for the FG-2a case. Compute NTU from Eq. (10.29) and subsequently, P, for the given ﬂow arrangement. Then from Eq. (10.28),

Resultant Formulas to Compute the Objective of the PEC

719

L Lp

P Pp

FN-2

FN-3

q Tmax ¼ ¼1 qp Tmax; p

m_ G ¼ ¼1 m_ p Gp

Information here identical to that of the FN-2 case

q P NTU ¼ ¼1) ¼1 Since qp Pp NTUp

A L ¼ Ap Lp

nt A ¼ fr ¼ 1 nt; p Afr; p

ð4Þ

Then

Up L ¼ U Lp

P f L ¼ P p fp L p

continued over

All formulas and the procedure for the FN-2 case apply here. After computing the value of L=Lp , calculate P=P p from the equation

1 þ R*p U ¼ Up ð jp =jÞ þ R*

Since G=Gp ¼ 1 the j and f factors are directly calculable. From Eq. (10.24), P=P p ¼ ð f =fp ÞðL=Lp Þ. From Eq. (10.28), UA=ðUAÞp ¼ 1 or L=Lp ¼ Up =U. Substitution of these equations and given information into Eq. (10.26) yields

From these equations, compute G=Gp as follows: (1) Assume G=Gp . (2) Solve for UA=ðUAÞp from Eq. (4). (3) Calculate NTU from Eq (10.29). (4) Determine P and Pp for the exchanger ﬂow arrangement given. (5) Compute q=qp from Eq. (3), which would be diﬀerent from unity until convergence. Iterate on G=Gp to obtain q=qp ¼ 1: Using converged value of G=Gp , calculate L=Lp from the second P=P p equation above.

1 þ R*p UA ¼ ðUAÞp ð jp =jÞð f=fp ÞðG=Gp Þ2 þ R*ð f=fp ÞðG=Gp Þ3

Eliminating Lp =L from the foregoing ﬁrst and second equations, we get

720

Tmax Tmax; p

P Pp

VG-2b

VG-3

L A ¼ ¼1 Lp Ap

q ¼1 qp

m_ ¼1 m_ p

P ¼1 Pp

nt A ¼ fr nt; p Afr; p

G Gp

3

All formulas and the procedure of the FG-2b case apply here. The comment of case VG-2a is also valid here.

The procedure outlined for the FG-2a case applies here. For this case, the mass ﬂow rates are equal. This results in diﬀerent nt or frontal area because G’s are diﬀerent. In case FG-2a, nt or frontal areas are the same and result in diﬀerent mass ﬂow rates. Otherwise, all formulas and the procedure of the FG-2a case are identical for this case.

q Tmax ¼ ¼ 1 U=A=ðUAÞp ¼ 1 since q=qp ¼ P=Pp ¼ 1: Then, we ﬁnd that qp Tmax; p G=Gp ¼ ð j=jp Þð1 þ R*p R*Þ from Eq. (10.26). Once G=Gp is calculated from the foregoing equation from the known j vs. Re correlations, f =fp can be computed for G ¼ 6¼ 1 known G=Gp . Subsequently, compute P=P p from Eq. (10.24) for A=Ap ¼ 1: Gp

m_ ¼1 m_ p

Tmax ¼1 Tmax; p

P ¼1 Pp

A ¼1 Ap

L A ¼ ¼1 Lp Ap

m_ ¼1 m_ p

Substitute Ap =A from Eq. (6) into Eq. (5). The resultant equation has only one unknown G=Gp since j=jp and f=fp are dependent on the value of G=Gp . Solve for G=Gp from this implicit equation. Then A=Ap is known from Eq. (6). Subsequently, compute nt =nt; p (or Afr =Afr; p Þ and L=Lp from Eq. (10.30).

P f A ¼1¼ Pp fp Ap

1 þ R*p UA ¼1¼ ðUAÞp ð jp =jÞ½ð f =fp ÞðAp =AÞ2 1=3 þ R*ðAp =AÞ

Since, P=Pp ¼ 1; we get NTU/NTUp ¼ 1 and hence UA=ðUAÞp ¼ 1. Thus, Eqs. (10.26) and (10.24) reduce to

Resultant Formulas to Compute the Objective of the PEC

ð6Þ

ð5Þ

a

When appropriate, it is assumed that the correlation for j vs. Re (and hence G) for the enhanced and plain surfaces have the same form [i.e., j=jp ¼ ðG=Gp Þn where n is an exponent]. Similarly, it is assumed that f =fp ¼ ðG=Gp Þm , where m is some other exponent, and that R* and R*p are known.

q qp

L Lp

Hence, P=Pp ¼ 1

Afr Afr; p

P q Tmax ¼ ¼ ¼1 P p qp Tmax; p

m_ ¼1 m_ p

nt nt; p

Information or Constraints Given Determine and Comments

VG-2a

VG-1

Case

TABLE 10.6 (Continued)

721

SOME QUANTITATIVE CONSIDERATIONS

Thus, knowing the ratio UA=ðUAÞp from Eq. (10.26), the NTU of the enhanced surface can be calculated using Eq. (10.29) for a given ﬂow rate, and subsequently, the temperature eﬀectiveness for the given exchanger ﬂow arrangement can be found from the formulas of Table 3.6. The heat transfer rate is then computed from Eq. (10.28). Using Eq. (10.21) and m_ ¼ Ao G, the ratios of surface areas and ﬂow rates for enhanced and plain tubular surface for the same di are given by A n L ¼ t Ap nt; p Lp

n G m_ ¼ t m_ p nt; p Gp

ð10:30Þ

Using the foregoing equations, speciﬁc algebraic formulas for the PECs of Table 10.4 are summarized in Table 10.6. It should be emphasized that the formulas of Table 10.6 can also be used to compare two diﬀerent surfaces 1 and 2 for the speciﬁc PEC by adding the subscript 1 to the enhanced surface values and replacing the subscript p by 2 as a base surface. Also, the values R* and R*p in Eq. (10.26) are normalized with respect to the plain tube inside thermal resistance. They can be normalized consistently with respect to any ﬂuid-side thermal resistance. Example 10.4 Would a selection of the following heat transfer surface provide better performance than the existing design in which a given plain plate-ﬁn surface (i.e., surface 11.1, Table 10-3, Fig. 10-26, Kays and London, 1998) is used? The comparison is based on the requirements of ﬁxed (1) ﬂow rate, (2) ﬂuid pumping power, (3) heat transfer rate, and (4) inlet temperature diﬀerence between the hot- and cold-ﬂuid streams. The argument is supposed to be valid for any Reynolds number. The new heat transfer surface has the following Colburn and Fanning friction factors: j ¼ expða0 þ a1 r þ a2 r2 þ a3 r3 þ a4 r4 þ a5 r5 þ a6 r6 Þ and f ¼ expðb0 þ b1 r þ b2 r2 þ b3 r3 þ b4 r4 þ b5 r5 þ b6 r6 Þ where r ¼ ln Re and the numerical values of the coeﬃcients are as follows: Coeﬃcient

Numerical Value

Coeﬃcient

Numerical Value

a0 a1 a2 a3 a4 a5 a6

0.1624564980E þ 04 0.1404062382E þ 04 0.4999486289E þ 03 0.9400171748E þ 02 0.9835078386E þ 01 0.5428407378E þ 00 0.1235104592E 01

b0 b1 b2 b3 b4 b5 b6

0.1242054696E þ 04 0.1077301312E þ 04 0.3855707180E þ 03 0.7291091700E þ 02 0.7674456065E þ 01 0.4263025064E þ 00 0.9766547339E 02

SOLUTION Problem Data and Schematic: Two surfaces should be compared under the ﬁxed ﬂow rate, pumping power, heat transfer rate, and inlet temperature diﬀerence between the hot- and cold-ﬂuid streams. The entire range of applicable Re numbers should be considered. The correlations for Colburn and Fanning friction factors for a new heat

722

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

transfer surface are provided in the problem formulation. The Colburn and Fanning factors for the plain plate-ﬁn geometry 11.1 are given in Kays and London (1998) in tabular form. A curve ﬁtting of these data (in the same form as the ones given in the problem formulation) leads to the following set of coeﬃcients: Coeﬃcient

Numerical Value

Coeﬃcient

Numerical Value

a0; p a1; p a2; p a3; p a4; p a5; p a6; p

0:1305722226E þ 05 0:1014346095E þ 05 0:3268456896E þ 04 0:5590860182E þ 03 0:5355816661E þ 02 0:2724931767E þ 01 0:5753732235E 01

b0; p b1; p b2; p b3; p b4; p b5; p b6; p

0:6188627536E þ 04 0:4958849985E þ 04 0:1647882636E þ 04 0:2906965749E þ 03 0:2872245480E þ 02 0:1507451536E þ 01 0:3283617030E 01

Determine: Determine which of the two heat transfer surfaces has better heat transfer/pressure drop performance. Assumptions: The assumptions invoked in Section 3.2.1 are valid. Analysis: We are given the information on only one ﬂuid side of the exchanger in this problem. The problem formulation requires that the following relations be satisﬁed. P hA Tmax m_ ¼ ¼ ¼ ¼1 m_ p P p ðhAÞp Tmax; p where the symbols without a subscript denote the variables for the new surface and the symbols with the subscript p denote the plain plate-ﬁn surface (surface 11.1 from Kays and London, 1998). This is the VG-1 PEC of Table 10.4 on one ﬂuid side of the exchanger. According to Eq. (10.24) or Eq. (6) of Table 10.6, the pumping power ratio (which is equal to unity, as indicated above) must be equal to P f A G 3 ¼1¼ Pp fp Ap Gp

ð1Þ

Similarly, from Eq. (10.25), we get hA=ðhAÞp ðP=P p Þ1=3 ðA=Ap Þ2=3

¼

2=3 Ap j=jp ¼ 1=3 A f =fp

ð2Þ

From Eqs. (1) and (2), we can eliminate the surface area ratio, A=Ap . Hence, the ratio of mass velocities for the two surfaces must satisfy the relation G ¼ Gp

fp j f jp

1=2

SOME QUANTITATIVE CONSIDERATIONS

723

Note that G=Gp ¼ Re=Rep . Therefore, Re ¼ Rep

fp Rep j Re f Re jp Rep

1=2 ð3Þ

This relation indicates that we can determine Re for which the new surface would satisfy the aforementioned constraints for any given Rep for ﬂow through the plain plate-ﬁn surface.y The pair of Re numbers would then allow calculation of the corresponding Colburn and Fanning factors, and subsequently, the evaluation of heat transfer surface area reduction (or increase, if the new surface is not better than the old one) that would be accomplished with the proposed change of geometry. Therefore, we should determine these pairs of Re numbers, say for a low Re number range (say, around 500) and for a large Re number range (say, around 5000), and compare the heat transfer area changes. It should be noted that the determination of these Re numbers may be a tedious iterative job, in particular if one uses the graphical presentations of j and f factors as those given in Kays and London (1998). In our case, we perform the calculations numerically. This calculation leads to the following results. For Rep ¼ 500, a very little change in Re would occur (i.e., Re ¼ 505), but A=Ap ¼ 0:93. That means that the new surface would require only 7% less heat transfer area for the same performance as the old one. For Rep ¼ 5000, however, Re becomes 4870 and A=Ap ¼ 1:13! The proposed new surface would require not less but 13% more heat transfer surface than the old one. So it can be concluded that the proposed solution does not bring a signiﬁcant beneﬁt for the low-Re range and may actually be worse for the high-Re range. Discussion and Comments: The performance evaluation criteria may provide a useful tool for assessing the performance of a selected heat transfer surface as well as for the comparison of various solutions. In this case, the claim that a new design is better is not substantiated. Our conclusion is based on the use of a variable-geometry criterion (VG-1; see Table 10.6). It should be pointed out, though, that highly augmented heat transfer surfaces may reduce the required heat transfer area signiﬁcantly (50% and more) compared with plain plate-ﬁn heat transfer surfaces for the VG-1 criterion. 10.3.3

Evaluation Criteria Based on the Second Law of Thermodynamics

All performance evaluation criteria discussed so far are based exclusively on the ﬁrst law of thermodynamics. These criteria were devised utilizing mass and energy balances without involving the thermodynamic quality of the energy ﬂows. However, heat transfer and friction characteristics of heat transfer surfaces may easily be related to the quality level of energy ﬂows deﬁned by the second law of thermodynamics.z That becomes very important in any system analysis, and feedback from {

For a given Rep , jp , and fp are determined from the given surface data. Now one assumes a value of Re, determines corresponding j and f factors and computes Re from Eq. (3). If this Re does not match with the assumed Re, iterations are carried out with new values of Re, until Eq. (3) is satisﬁed, and corresponding j and f factors are computed to be used subsequently in Eq. (2). { The body of knowledge usually called the second law of thermodynamics analysis always involves both the ﬁrst and second laws of thermodynamics. However, it is customary to name the product of such an analysis by indicating the second law of thermodynamics only (Bejan, 1988).

724

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

a system engineer may indicate a need for a change of design based on the second law of thermodynamics. This approach is considered in Section 11.7, with an additional performance evaluation criterion. 10.3.4

Selection Criterion Based on Cost Evaluation

The cost of an exchanger is usually an important selection criterion for a user. Let us assume that all other pertinent evaluation criteria, if not already incorporated in the cost evaluation/optimization routine, are satisﬁed. In that case, existing design options will be selected based on the most cost-eﬀective design. So a methodology for cost estimation must be developed. Most heat exchanger manufacturers have their own proprietary methods for cost estimation. Some approaches to this problem have been reported in the literature in the past. We present here a simple procedure of the ESDU (1994) used in some industries. Heat exchanger cost may be related either to the heat transfer surface area of an exchanger or to the heat exchanger duty required. In Fig. 10.13, this dilemma is presented schematically as two options that may be perceived from the perspective of a heat exchanger designer on one side and of a process engineer on the other. The simple logic implied by Fig. 10.13 has been used to deﬁne cost estimation evaluation. The proposed methodology is based on empirical cost data compiled to evaluate all the various feasible heat exchanger types. The decision variable is the cost of a heat exchanger per unit of its thermal size, that is, per unit of the product UA ð¼ q=Tm Þ. In Fig. 10.13, this quantity is denoted as CUA . An alternative solution would be the cost of an exchanger per unit heat transfer surface area, CA of Fig. 10.13. The latter is less attractive because it does not explicitly take into account the heat transfer duty, or what is equivalent, the relevant thermal size of the exchanger, UA. It must be clear that the overall heat transfer coeﬃcient, determined for a particular design, must inﬂuence the cost. In Table 10.7, a selection of the cost data, represented by the values of CUA , is compiled. This table is prepared for a particular exchanger purpose, namely, for an application in which the heat exchange is accomplished between gas as a hot ﬂuid at

FIGURE 10.13 Cost of a heat exchanger vs. heat transfer area and/or heat duty.

SOME QUANTITATIVE CONSIDERATIONS

725

TABLE 10.7 Cost Data CUA vs UA for Various Heat Exchanger Typesa CUA [$/(W/K)] q=Tm or UA (W/K)

Shell- andTube, U ¼ 484 (W/m2 KÞ

Double Tube, U ¼ 484 (W/m2 KÞ

Printed Circuit U ¼ 1621 (W/m2 KÞ

Plate-Fin, U ¼ 491 (W/m2 KÞ

Welded Plate U (W/m2 KÞ

CUA [$/(W K)]

103 5 103 3 104 105 106

3.98 1.00 0.29 0.17 0.106

2.5 0.75 0.31 0.31 0.31

12 2.4 0.6 0.42 0.28

— 3.1 0.513 0.210 0.115

349 1187 1068 1112 1173

4.9 1.22 0.42 0.28 0.22

Source: Data from ESDU (1994). a The hot ﬂuid is medium-pressure gas and the cold ﬂuid-treated water. The original ESDU cost data in the British pound are approximately in the US dollar value in 2000.

medium pressure (say, 20 bar) and cold ﬂuid as treated water. Five diﬀerent heat exchanger types may be used for this particular combination of working ﬂuids (see Chapters 1 and 2 for a description of each type and the assessment of the feasibility of the possible selections): (1) shell-and-tube heat exchanger, (2) double-pipe heat exchanger, (3) printed-circuit heat exchanger, (4) plate-ﬁn exchanger, and (5) welded plate exchanger. Depending on the magnitude of q=Tm , diﬀerent cost CUA values can be determined for each of the heat exchanger types. An extensive set of CUA data for various heat exchangers is compiled by ESDU (1994), and partially summarized in Appendix D; they can be used for this purpose. From Table 10.7, it is clear that the cost of a heat exchanger per unit of its thermal size (i.e., per unit of UA), CUA , decreases with an increase in the heat load ðq=Tm Þ or heat exchanger size (UA). The procedure for evaluation of a heat exchanger type based on the given cost criterion is as follows: 1. Estimate the heat duty q from a heat balance using Eq. (2.1). 2. Determine q=Tm for the heat exchanger under consideration (a) by computing Tm ¼ F Tlm with Tlm from Eq. (3.172) and the best estimate of F for (ESDU, 1994), or (b) from a known NTU and Cmin using q=Tm ¼ Cmin NTU [see Eqs. (3.12) and (3.59)]. 3. Repeat step 2 for each heat exchanger type. 4. From empirical data ðCUA vs. q=Tm , see Appendix D), estimate the CUA factor. 5. Calculate the cost of a particular heat exchanger type by multiplying CUA and q=Tm : 6. Compare the costs for various heat exchanger types. If one of the types is much less expensive than the other (by a factor of 1.5 to 2.0 or more), that design should be selected. If the costs for all solutions are close to each other, a more detailed analysis of each individual cost must be performed. The procedure outlined above is utilized in a simpliﬁed way in Example 2.4. A more elaborate analysis is the subject of Problem 10.8.

726

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

SUMMARY Heat exchangers are designed for a variety of applications under varied operating conditions. As a result, an optimum heat exchanger will be diﬀerent depending on the application. The most important selection criteria for a heat exchanger are summarized next. . The heat exchanger must function as designed for performance, durability, and other criteria during its design life. As a result, the operating environment (i.e., pressure, temperature, fouling potential, ﬂuid leakage and contamination, material compatibility, etc.), cost packaging, maintenance, and so on, are very important variables. Based on these operating and design conditions, an engineer can select an appropriate exchanger from Table 10.1 with additional considerations of the cost, manufacturability, and other requirements. . A large number of geometric variables are associated with shell-and-tube exchangers. Considerable discussion is provided in Section 10.2.1 for the choice of speciﬁc geometrical variables. Similarly, for extended surface exchangers, a variety of ﬁn geometries is available, and a qualitative discussion on the selection of particular geometries is presented in Section 10.2.3. Surface selection for plate heat exchangers and regenerators is discussed brieﬂy in Sections 10.2.2 and 10.2.4. In many applications, the heat exchanger operates in a system or a thermodynamic cycle. Therefore, quantitative criteria for component design and optimization have less meaning since the heat exchanger should be designed for optimum system performance. Hence, quantitative methods are presented in the text for screening various surfaces to select the most appropriate ones as components. In this regard, two categories of quantitative methods are summarized: . The surface ﬂow area and core volume goodness factor comparisons are presented to screen and arrive at higher-performing extended surfaces. Geometrical scaling laws are then summarized for a compact heat exchanger surface on one ﬂuid side for changes in ﬂow area, volume, surface area, and length of that surface for the case of constant q, p, and m_ . . Performance evaluation criteria are employed to compare the performance of an enhanced tubular to a plain tubular exchanger surface. REFERENCES ALPEMA, 2000, The Standards of the Brazed Aluminum Plate-Fin heat Exchanger Manufacturer’ Association (ALPEMA), 2nd Edition, AEA Technology plc, Didcot, Oxon, UK. ASME, 1998, ASME Boiler and Pressure Vessel Code, 1998: Rules for Construction of Pressure Vessels, Sec. VIII, Div. 1, American Society of Mechanical Engineers, New York. Bejan, A., 1988, Thermodynamic Design in Advanced Engineering Thermodynamics, Wiley, New York, pp. 594–669. Bell, K. J., 1981, Construction features of shell-and-tube heat exchangers, in Heat Exchangers: Thermal-Hydraulic Fundamentals and Design, edited by S. Kakac¸, A. E. Bergles and F. Mayinger, Hemisphere Publishing Corp., Washington, DC, pp. 721–763. Bell, K. J., 1998, Approximate sizing of shell-and-tube heat exchangers, in Heat Exchanger Design Handbook, G. F. Hewitt, exec. ed., Begell House, New York, Vol. 3, Sec. 3.1.4. Cowell, T. A., 1990, A general method for the comparison of compact heat transfer surfaces, ASME J. Heat Transfer, Vol. 112, pp. 288–294.

REVIEW QUESTIONS

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ESDU, 1994, Selection and costing of heat exchangers, Engineering Science Data, Item 92013, ESDU, Int., London, UK. Gentry, C. G., 1990, RODbaﬄe heat exchanger technology, Chem. Eng. Prog. July, pp. 48–57. Kays, W. M., and A. L. London, 1998, Compact Heat Exchangers, reprint 3rd ed., Krieger Publishing, Malabar, FL. Lancaster, J. F., 1998, Materials of construction, in Handbook of Heat Exchanger Design, G. F. Hewitt, ed., Begell House, New York, Sec. 4.5. London, A. L., 1964, Compact heat exchangers, Part 2, Surface geometry, Mech. Eng., Vol. 86, June, pp. 31–34. London, A. L., and C. K. Ferguson, 1949, Test results of high-performance heat exchanger surfaces used in aircraft intercoolers and their signiﬁcance for gas-turbine regenerator design, Trans. ASME, Vol. 71, pp. 17–26. Reay, D. A., 1999, Learning from Experiences with Compact Heat Exchangers, CADDET Analyses Series 25, Centre for the Analysis and Dissemination of Demonstrated Energy Technologies, Sittard, The Netherlands. Sekulic´, D. P., A. J. Salazar, F. Gao, J. S. Rosen, and H. S. Hutchins, 2003. Local transient behavior of compact heat exchanger core during brazing, Int. J. Heat Exchangers, Vol. 4, No. 1. Shah, R. K., 1978, Compact heat exchanger surface selection methods, Heat Transfer 1978, Proc. 6th Int. Heat Transfer Conf., Vol. 4, pp. 193–199. Shah, R. K., 1983, Compact heat exchanger surface selection optimization and computer-aided thermal design, in Low Reynolds Number Flow Heat Exchangers, edited by S. Kakac¸, R. K. Shah and A. E. Bergles, pp. 845–874, Hemisphere Publishing Corp., Washington, DC. Shah, R. K., 1995, Heat exchangers, in Encyclopedia of Energy Technology and the Environment, A. Bisio and S. G. Boots, eds. Wiley, New York, Vol. 3, pp. 1651–1670. TEMA, 1999, Standard of the Tubular Exchanger Manufacturers Association, 8th ed. Tubular Exchange Manufacturers Association, New York. Webb, R. L., 1981, Performance evaluation criteria for use of enhanced heat transfer surfaces in heat exchanger design, Int. J. Heat Mass Transfer, Vol. 24, pp. 715–726. Webb, R. L., 1994, Principles of Enhanced Heat Transfer, Wiley, New York. Webb, R. L., and A. E. Bergles, 1983, Performance evaluation criteria for selection of heat transfer surface geometries used in low Reynolds number heat exchangers, in Low Reynolds Number Flow Heat Exchangers, S. Kakac¸, R. K. Shah, and A. E. Bergles, eds., Hemisphere Publishing, Washington, DC, pp. 735–752.

REVIEW QUESTIONS Where multiple choices are given, circle one or more correct answers. Explain your answers brieﬂy. 10.1

For liquid-to-gas exchangers, the commonly used exchanger constructions are: (a) shell-and-tube (b) plate-type (c) extended surface (d) regenerators

10.2

For fouling ﬂuids, the commonly used exchanger constructions are: (a) shell-and-tube (b) plate-type (c) extended surface (d) regenerators

10.3

Regenerators are exclusively used as: (a) gas-to-liquid exchangers (b) gas-to-gas exchangers (c) condensing ﬂuid-to-gas exchangers (d) gas-to-evaporating ﬂuid exchangers

10.4

Plate-ﬁn exchangers are commonly used for the application having:

728

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

(a) 0 to 20 MPa operating pressures (b) 200 to 5408C operating temperatures (c) heavy fouling ﬂuids (d) highly corrosive ﬂuids

(e) none of these

10.5

To cool concentrated hydrochloric acid with a heavy oil (low h), the following exchanger(s) would be feasible: (a) plate exchanger with titanium plates (b) plate-ﬁn exchanger with ﬁns on oil side (c) TEMA AEW with oil in the shell (d) TEMA CEN with oil in the shell.

10.6

In a shell-and-tube exchanger, low-ﬁnned tubes result in the following on the shell side: (a) increase in the heat transfer coeﬃcient (b) increase in surface area (c) increase in pressure containment and rigidity (d) better ﬂow mixing (e) reduction in fouling

10.7

The tube-side turbulent ﬂow heat transfer coeﬃcient for a given ﬂow rate is increased by: (a) increasing the number of tubes with decreased tube length (b) decreasing the number of tubes with increased tube length (c) increasing the number of tube passes (d) increasing the number of shell passes (e) decreasing the tube gauge

10.8

The shell-side heat transfer coeﬃcient is increased by: (a) increasing the number of baﬄes (b) decreasing the baﬄe cut (c) increasing the tube pitch (d) increasing number of tube passes (e) increasing tube ﬂuid velocity

10.9

In general, all shell-and-tube exchangers have transverse plate baﬄes except for the following TEMA shell: (a) G (b) J (c) K (d) X

10.10

The horizontal baﬄe cut is used for the following shell-side ﬂuids: (a) single-phase ﬂuids (b) condensing ﬂuids (c) evaporating ﬂuids (d) very viscous liquids (e) slurries

10.11

Some consequences of using the transverse plate baﬄes are: (a) reduce fouling on the shell side (b) reduce pressure drop on the shell side (c) minimize tube-to-tube temperature diﬀerentials (d) eliminate ﬂow-induced tube vibrations (e) none of these

10.12

The use of rod baﬄes in shell-and-tube exchangers results in: (a) increased shell-side heat transfer coeﬃcient compared to a plate baﬄe exchanger at the same mean velocity

REVIEW QUESTIONS

729

(b) reduced number of baﬄes in the exchanger compared to an equivalent segmental baﬄe exchanger (c) increased rigidity of the tube bundle (d) reduced shell-side fouling (e) reduced shell-side pressure drop 10.13

Impingement baﬄes are used to: (a) increase shell-side heat transfer coeﬃcient (b) support the tubes (c) protect the tube damage in the inlet region (d) provide nonuniform ﬂow distribution at inlet on the shell side

10.14

The following shell types are commonly used for single-phase ﬂuids on the shell side: (a) E (b) F (c) G (d) H (e) J (f ) K (g) X

10.15

The following shell types are commonly used for two-phase or multiphase ﬂuids on the shell side: (a) E (b) F (c) G (d) H (e) J (f ) K (g) X

10.16

In a single tubesheet design, the least likelihood of leakage between the shell and tube ﬂuids is with the following front-end heads: (a) A (b) B (c) C (d) N

10.17

The most important criteria for the selection of rear-end heads are: (a) to control the ﬂuid velocities (b) operating pressures (c) thermal stresses between tubes and shell (d) shell- or tube-side cleaning requirement (e) high shell-side heat transfer coeﬃcient (f) can’t tell

10.18

The bundle-to-shell bypass stream C could be signiﬁcant in the following rearend head constructions, and as a result, sealing strips are usually required: (a) L (b) M (c) N (d) P (e) S (f ) T (g) U (h) W

10.19

The ﬂow-induced tube vibration could often be reduced in single-segmental exchangers by: (a) adding sealing strips (b) decreasing the number of tubes in the window area (c) increasing the shell (and tube bundle) diameter (d) decreasing the baﬄe cut (e) none of these

10.20

Arrange the following shell types from high to low p on the shell side for a liquid at a speciﬁed ﬂow rate (assume turbulent ﬂow) and inlet temperature, for the same total surface area, and for single-segmental baﬄes/support plates at the same spacing: (a) E (b) F (c) G (d) H (e) J (f ) X

730

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

10.21

Arrange the following rear-end heads in order of the most easy to the most diﬃcult for cleaning and inspection of the shell side: (a) M (b) P (c) S (d) T (e) U (f ) W

10.22

Arrange the following rear-end heads from the least-cost to highest-cost designs: (a) P (b) S (c) T (d) U (e) W

10.23

Circle the following statements as true or false. (a) T F Diﬀerent constructions of shell types, front-end heads, and rearend heads are available that cannot be identiﬁed clearly by the TEMA designation scheme. (b) T F From the high- heat-transfer point of view, tubes with small diameters are preferred, but from the cleaning-requirement point of view, tubes of large diameters are preferred. (c) T F Shell diameters of 4 m (160 in.) or tube lengths of 18.3 m (60 ft) in shell-and-tube exchangers are not possible because they do not represent values mentioned in TEMA standards. (d) T F The commonly used ratio of the tube pitch to tube diameter in shell-and-tube exchangers is below 1.2. (e) T F A square tube layout is generally used in a ﬁxed tubesheet exchanger. (f) T F A 60% baﬄe cut is used for single-segmental baﬄes in some applications. (g) T F A bonnet head is generally used for ease in inspection and cleaning of the tubes. (h) T F Improper location and size of the impingement baﬄe is a common cause of tube failure.

10.24

Circle the following statements as true or false for preferred tube layout when shell-side mechanical cleaning is required. (a) T F A 608 tube layout is used for shell-side laminar ﬂow. (b) T F A 458 tube layout is used for shell-side turbulent ﬂow. (c) T F A 458 tube layout is used for condensing ﬂuid on the shell side. (d) T F A 608 tube layout is used for boiling ﬂuid on the shell side.

10.25

Fill in the blanks: (a) Flow-induced tube vibrations can be particularly serious in U-tube bundles due to ________________ . (b) The ________________ design is used when thermal stresses in the tubes must be kept to a minimum. (c) As an alternative to the ﬂoating heads or U-tubes, _______________ can be used with E shell to allow increased inlet temperature diﬀerence between two ﬂuids. (d) Other than heat transfer, two of the main reasons for selecting tube orientation and pitch are ________________ and ________________.

10.26

Double tubesheets are required in shell-and-tube exchangers to: (a) relieve thermal stresses between the tubes and shell (b) make overall design rigid (c) prevent leakage from one ﬂuid to the other ﬂuid

REVIEW QUESTIONS

731

10.27

For a speciﬁed heat transfer and pressure drop requirement, strip ﬁns require the following compared to plain ﬁns: (a) larger frontal area (b) larger frontal area and core volume (c) shorter ﬂow length but larger core volume

10.28

In a plain ﬁn compact heat exchanger, the pressure drop on the side of concern is 500 Pa. If the ﬂow is developing laminar, what is the approximate p if the ﬂow length is doubled? (a) 1000 Pa (b) 2000 Pa (c) 700 Pa (d) can’t tell

10.29

In fully developed laminar ﬂow, increasing G will: (a) increase p and h (b) decrease p and h (c) increase p but h remains constant

10.30

Rotary regenerators: (a) employ interrupted surfaces (b) have surface area densities greater than 400 m2 /m3 (c) are used for moderately fouling gases (d) are more expensive per unit surface area compared to plate-ﬁn and tube-ﬁn surfaces

10.31

What type of ﬁn is preferred in a plate-ﬁn exchanger at high Re for low pressure drops? (a) oﬀset strip (b) louver (c) plain (d) perforated

10.32

If the j and f characteristics for surface P are both 20% higher than those of surface Q, which surface would you select for your heat exchanger? Why? (a) surface P (b) surface Q (c) can’t tell

10.33

Usually, the maximum operating conditions for design of a metal plate-ﬁn unit is 1000 kPa or 8008C due to: (a) joining techniques between the ﬁns and plates (b) manufacturing technology limitations (c) cost factors

10.34

Which compact heat exchanger has the highest heat transfer surface area density and the lowest cost per unit surface area? (a) shell-and-tube (b) plate (c) plate-ﬁn (d) rotary regenerator

10.35

A long slender compact exchanger with triangular passages would have: (a) a higher (b) a lower (c) the same j=f factor than that of a similar unit with rectangular passages? Consider air ﬂowing in both cases.

10.36

In a compact heat exchanger having fully developed laminar ﬂow, the following relationships exist when we compare two surfaces. Consider q, m_ , and p as given and constant for these comparisons in parts (a), (c), (d), and (f). Circle the following statements as true or false. (a) T F The ﬂow length is inversely proportional to the hydraulic diameter and directly proportional to the j factor. (b) T F The ﬂow areas for two surfaces are the same if the L=Dh ratio and f factors are the same for the given ﬂuid, ﬂow rate, and pressure drop.

732

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

(c) T F (d) T F

(e) T F (f ) T F

The ﬂow area is practically independent of the surface hydraulic diameter. A compact heat exchanger having a 1-mm hydraulic diameter on one ﬂuid side will have the identical performance ðq; p, and m_ ) of a noncompact exchanger having a 20-mm hydraulic diameter on the same ﬂuid side at the same Reynolds number. For a given compact surface, as one reduces the hydraulic diameter (truly scaled down geometry), the j factor also reduces in the same proportion at a speciﬁed Reynolds number. Reducing the hydraulic diameter on one ﬂuid side, its volume and surface area reduce in the same proportion.

PROBLEMS 10.1

To select an appropriate shell type for a given application (having single-phase ﬂuids on both sides), we want to evaluate all major shell types: E, F, G, H, J, and X. Because of the thermal stress considerations, we will use U tubes (two tube passes) in a single-shell exchanger with the design total NTUt ¼ 1:4 and Rt ¼ 0:8: Consider the shell ﬂuid mixed for all shell types except for the X shell, in which it is unmixed. For simple analysis purposes, consider the tube ﬂuid as unmixed in each pass and mixed between passes for the X shell with an overall counterﬂow arrangement. Assume an overall counterﬂow arrangement for 1–2 TEMA G and H exchangers. (a) Tabulate the exchanger eﬀectiveness for the above shell types using the appropriate ﬁgures and equations from Chapter 3. (b) For the identical tube ﬂuid ﬂow rate, will the pressure drop on the tube side be the same or diﬀerent in the above types of shells? Why? Assume all ﬂuid properties to be constant. (c) Estimate the shell-side pressure drop for each shell type as a function of um and L, where um is the mean shell-side velocity in the E shell, and L is the shell length. Note that the shell-side ﬂow rate is the same for all shell types and each has single-segmental baﬄes/support plates at the same spacing. The shell and tube diameters and number of tubes are the same for all shell types. The shell length is much greater than the shell diameter. Don’t forget to add p’s due to the ﬂuid turning 1808 on the shell side in some shell types. (d) Select a shell type for (i) the highest heat transfer performance, and (ii) the lowest shell-side pressure drop.

10.2

You have designed two heat exchangers to do the same thermal ‘‘job’’ [i.e., they have the same q and total tube-side mass ﬂow rate m_ t (kg/s)]. A comparison of these heat exchangers shows the following:

Property Heat transfer area A ðm Þ Number of tubes Nt 2

Smooth Tube

Rough Tube

13.62 100

9.80 120

PROBLEMS

Length of tubes L (m) Tube inside diameter di (mm) Tube-side friction factor f

2.5 18 0.008

733

1.25 18 0.016

Assume constant and identical ﬂuid properties for both heat exchangers. (a) Calculate P=P s , where the subscript s is for the smooth tube. (b) Calculate p=ps . (c) Suppose that P=P s is 0.8; how would you alter the rough tube exchanger geometry to make P=P s ¼ 1, keeping the same rough tube length? Give quantitative estimates considering no change in the friction factor. (d) How would you expect UA to change for part (c)? Give explicit reasons whether it will increase, decrease, or will have no change. 10.3

An aircraft oil cooler has oﬀset strip ﬁns on the air side with 790 ﬁns/m, a 0.15 mm ﬁn thickness, a 0.25 mm plate thickness, and a 9.5 mm plate spacing. Air enters the heat exchanger with 5 m/s frontal velocity ð ¼ 1:579 105 m2 =s, ¼ 1:20 kg=m3 Þ. Assume that you have an option of using 3- and 12-mm strip lengths for the ﬁn. Compare the performance of these two ﬁn geometries. (a) Calculate Dh , *, and . Hint: Draw a unit cell and use associated simple geometrical relationship. (b) Compute j and f for each surface. Hint: Use Manglik and Bergles correlations, Eqs. (7.124) and (7.125). (c) Determine h2 =h1 , E2 =E1 , and ð j=f Þ2 =ð j=f Þ1 . Here the subscript 1 refers to the 12 mm strip length. (d) For both strip ﬁns, compare the values of E and P for the same Afr and the same hA. What would be the ratio of surface areas, A2 =A1 ? (e) What would be the approximate value of u1 for the 3 mm strip ﬁn exchanger for the same hA and same P? Assume that fold ¼ fnew for the 3 mm strip ﬁn. Hint: Use the functional relationship of P from Eq. (10.8). Knowing the new value of u1 , compare it with the old value of u1 and estimate the new value of h and subsequently the new value of A for the same hA. (f) Discuss the results of parts (d) and (e).

10.4

As a heat exchanger designer, you have designed a plain plate-ﬁn heat exchanger to transfer the heat speciﬁed within the pressure drop allowed. The design point for the plain plate-ﬁn corresponds to Re ¼ 1000. You have an option to employ an oﬀset strip ﬁn of the same hydraulic diameter as an alternative. These surfaces are shown in Fig. 10.3. (a) Determine the design Reynolds number for the oﬀset strip ﬁn for equal ﬂuid pumping power per unit surface area (equal E). (b) Determine h2 =h1 for equal E, where subscripts 1 and 2 designate plain and oﬀset strip ﬁns, respectively. Hint: Use the deﬁnition of j and appropriate data. (c) Based on the answer for part (b), how much reduction is achieved in the core volume on the one ﬂuid side under consideration using the oﬀset strip ﬁn? (d) Determine the area goodness factor ratio for these surfaces. Can you tell which surface will require a higher frontal area? Why?

734

SELECTION OF HEAT EXCHANGERS AND THEIR COMPONENTS

10.5

Consider the triangular ﬂow passage rotary regenerator of Example 10.1. Determine the change in the disk diameter, disk depth, and volume of the regenerator if the triangular ﬂow passage hydraulic diameter is reduced by 50%.

10.6

A compact air-to-water heat exchanger is to be designed with an air-side mass ﬂow rate of 0.83 kg/s. The required NTU for the exchanger is 1. We would like to design for an air-side Reynolds number ðGDh =Þ of 3000 for which j ¼ 0:018. The following additional data are available: Airside: Dh ¼ 3:47 mm, ¼ 0:48, ¼ 558 m2 =m3 , cp ¼ 1004:9 J=kg K, ¼ 2:07 105 Pa s, Pr ¼ 0:7, o ¼ 0:80. Waterside: h ¼ 1:7 kW=m2 K, ¼ 32:8 m2 =m3 , o ¼ 1. Determine (a) the air-side frontal area, (b) the air-side heat transfer surface area, and (c) the air-side ﬂow length. Neglect wall resistance and fouling on both sides. The air side is the minimum heat capacity rate side.

10.7

In automotive radiators, a corrugated multilouver ﬁn (see Fig. 7.29) is used on the air side. Under consideration is replacing it with an oﬀset strip ﬁn geometry of Fig. 8.7. The following geometrical information is provided for the air-side ﬁn geometry. Fin Geometry Fin density (ﬁns/m) Louver pitch or oﬀset strip length (mm) Vertical ﬁn height , (mm) Louver cut length along the vertical height (mm) Fin thickness (mm) Louver angle (deg) Tube width (mm) Tube pitch (mm) Tube height (mm) Radiator core depth (mm)

Multilouver Fin 800 1.25 6 4.8 0.075 30 24 9.5 3.5 24

Oﬀset Strip Fin 800 1.25 6 6 0.075 Not applicable 24 9.5 3.5 24

Compute the change in the heat transfer surface and the air pumping power requirements by replacing the louver ﬁn with the oﬀset strip ﬁn under the ﬁxed ﬂow area (FN-2) performance evaluation criterion. Assume the mean air velocity through both ﬁn geometries to be 10 m/s. Use the appropriate correlations for the j and f factors from Chapter 7. Assume that the thermal resistances of the wall and the coolant are zero. Ignore fouling on both ﬂuid sides. Use the following properties of air: ¼ 1:058 kg=m3 ; cp ¼ 1:008 kJ=kg K, k ¼ 0:0288 W=m K, and ¼ 20:4 106 Pa s. 10.8

A high viscosity liquid with cp ¼ 1:9 kJ=kg K and mass ﬂow rate 0.6 kg/s enters a heat exchanger at 758C having an inlet pressure of 3.1 MPa. This liquid is cooled to 358C by water having an inlet temperature of 188C and a mass ﬂow rate of 2.1 kg/s. No signiﬁcant fouling should be expected. Select the most feasible heat exchanger type using a cost estimate.

11

Thermodynamic Modeling and Analysis

The main objectives of this chapter are twofold: (1) to present and discuss important factors that aﬀect heat exchanger performance, and (2) to introduce a basic analysis for the thermodynamic design and optimization of heat exchangers. A quest for answers regarding the ﬁrst objective will help us to identify the important factors that aﬀect heat exchanger eﬀectiveness, to quantify the eﬀects of these factors, and to provide guidelines for a qualitative assessment of the eﬀectivenesses of the exchangers with diﬀerent ﬂow arrangements but with a given, identical design task. The second objective is to deﬁne a ﬁgure of merit for assessing the thermodynamic eﬃciency of a heat exchanger and to present an approach to thermoeconomic considerations. In Section 11.1, the diﬀerences between a heat exchanger as a component and as part of a system are identiﬁed. In Section 11.2, a detailed modeling of a heat exchanger using energy balances only (i.e., the ﬁrst law of thermodynamics) is provided for the determination of heat exchanger eﬀectiveness and temperature distributions. In Section 11.3, a combined approach based on both the ﬁrst and second laws of thermodynamics is introduced to quantify inherent irreversibilities in a heat exchanger. The most important source of irreversibility is heat transfer across the ﬁnite temperature diﬀerences, which is discussed ﬁrst. Fluid mixing and ﬂuid friction, as additional sources of irreversibility, are studied next. A temperature cross phenomenon is then discussed in detail in Section 11.4 by evaluating entropy generation in a 1–2 TEMA J shell-and-tube heat exchanger. Using all the analysis tools presented in the ﬁrst four sections, a heuristic approach to an assessment of heat exchanger eﬀectiveness is developed in Section 11.5. In Section 11.6, energy, exergy, and cost balances important for analysis and optimization of heat exchangers are presented. Finally, a thermodynamic criterion for evaluation/selection of heat transfer surfaces is summarized in Section 11.7.

11.1 INTRODUCTION Traditionally, modeling of a heat exchanger is based on energy balances (i.e., on the consequences of both the ﬁrst law of thermodynamics and the mass conservation principle), so only the concepts of heat transfer rate and enthalpy rate change would Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

735

736

THERMODYNAMIC MODELING AND ANALYSIS

suﬃce for such an analysis.{ For an adiabatic heat exchanger (see the assumptions in Section 3.2.1), the enthalpy rate change of one ﬂuid stream must be equal to the enthalpy rate change of the other, being at the same time equal to the exchanger heat transfer rate. This simple energy balance statement will be used in subsequent sections in the diﬀerential form to model spatial distributions of temperatures of both ﬂuid streams. In Chapter 3, we also used the energy balances (both diﬀerential and overall), but only to determine the heat exchanger eﬀectiveness without determining the temperature distributions. Let us start with a review of the analysis of heat exchanger design methods discussed in Chapter 3 that relies on a relationship that can be presented in generalized form as follows:

heat transfer rate q

0

1 0 1 effectiveness= heat capacity temperature @ A @ A ¼ correction rate or thermal difference factor conductance 8 "Cmin Tmax in "-NTU method > > > > > < P1 C1 Tmax in P-NTU method ¼ ð11:1Þ > FUA Tlm in LMTD method > > > > : UA T max in -NTU method

Equation (11.1) is based on energy balances formulated as consequences of the ﬁrst law of thermodynamics. It is important to note that each of the methods implied by Eq. (11.1) assumes the determination of either an eﬀectiveness factor (heat exchanger eﬀectiveness " or temperature eﬀectiveness P1 ) or a correction factor (F or ) as a function of design parameters, e.g., NTU and C*. We use the expression factor as a generic term to indicate a common ﬁrst law of thermodynamics origin for both the eﬀectiveness and correction factors. Of course, the correction factors do not have the same physical meaning as the eﬀectiveness factors. Each relationship in Eq. (11.1) involves a temperature diﬀerence, either the maximum imposed temperature diﬀerence Tmax or the logarithmic-mean temperature diﬀerence Tlm . It has been demonstrated in Sections 3.3 and 3.5 that relationships between eﬀectiveness factors and the pertinent design parameters can be devised from the heat transfer model of a heat exchanger. In some cases, these relationships can even be obtained without a detailed study of internal heat transfer interactions. Moreover, a designer who already has these relationships would be able to calculate the eﬀectiveness for the given set of parameters and execute a design method procedure without a need to study temperature distributions of a selected ﬂow arrangement as outlined in Sections 3.9 and 9.2 through 9.5. So one would treat the heat exchanger as a black box for determination of the overall heat transfer surface area or heat transfer performance of an exchanger as a component. In such a case, there is no need to know temperature distributions. For example, for any exchanger (for which the {

In our analysis, the enthalpy rate change is the rate change of enthalpy of a ﬂuid stream caused by heat transfer interaction between the two ﬂuid streams. From the thermodynamic point of view, note that the heat transfer rate is an energy interaction (not a change in the ﬂuid property), while the enthalpy rate change represents a change of ﬂuid property caused by existing heat interaction. This distinction is important for understanding of the interpretation of heat exchanger performance and will be emphasized in more detail later through the introduction of several advanced concepts of thermodynamics.

INTRODUCTION

737

eﬀectiveness relationships are already known or can be determined using the matrix formalism mentioned in Section 3.11.4), an engineer needs only a relationship between the eﬀectiveness/correction factor and design parameters, without detailed insight into local temperature distributions. Thus, a misleading conclusion may be reached: that the only information a designer should possess concerning a ﬂow arrangement is the relationship between an eﬀectiveness factor and design parameters (e.g., P1 -NTU1, "-NTU, or F-P relationship). Analysis presented so far does suﬃce for a design procedure for a heat exchanger with an already deﬁned eﬀectiveness relationship. However, a very important and still unanswered question should be addressed as well. Why does an eﬀectiveness factor (say, heat exchanger eﬀectiveness) have a high (or low) value for a given ﬂow arrangement (especially for a complex one) compared to the corresponding value for another ﬂow arrangement (for the same set of design parameters)? For example, we do know that a crossﬂow heat exchanger has less exchanger eﬀectiveness than for a counterﬂow exchanger (for the same set of design parameters NTU and C*). The only rational explanation that we can oﬀer at this point (in addition to the intuitive ones) is that the eﬀectiveness relationship for a crossﬂow exchanger simply provides a smaller numerical value for " or P for the given heat capacity rate ratio and NTU than does " or P for a counterﬂow exchanger. In addition, "/NTU for a ﬁxed heat capacity rate ratio C* is diﬀerent for counterﬂow than for crossﬂow, for NTUmin < NTU < 1. For NTU 4, this gradient is almost identical. For NTUmin 0:4, all ﬂow arrangements provide almost identical eﬀectiveness values for a given set of design parameters [see Eq. (3.89)]. Why is that so? The reasoning will become clear when we present an astonishingly simple heuristic approach based on the second law analysis of exchanger ﬂow arrangements. Also, we present a thermodynamic performance ﬁgure of merit, the eﬃciency of a heat exchanger from a system viewpoint. Consequently, these analysis tools will help us in assessing relative magnitudes of exchanger eﬀectiveness for complex ﬂow arrangements for the selection of an appropriate ﬂow arrangement for a speciﬁed task. This understanding will also become valuable in ﬁnding an optimum heat exchanger design from a system viewpoint.

11.1.1

Heat Exchanger as Part of a System

Heat exchangers in numerous engineering applications are only one of many components of a system. Thus, the design of a heat exchanger is inevitably inﬂuenced by system requirements and should be based on system optimization rather than component optimization. An objective function for such system-based optimization is inﬂuenced by the main features of heat exchanger operation. For a given set of input data (e.g., ﬂow rates and inlet temperatures), exchanger geometry, and other pertinent information, the output data (e.g., the outlet temperatures) will depend on heat transfer and ﬂuid ﬂow phenomena that take place within the boundaries of the heat exchanger. So even though one seeks a system optimum, in the process of determining that optimum, one must fully understand the features of the exchanger as a component. Since heat exchangers are used in many systems, we do not attempt any speciﬁc system analysis or process integration. We discuss only the basic thermodynamic aspects. Despite exact mathematical/numerical results obtained through system-based optimization, the designer should know that the heat exchanger design (sizing) problem studied is a complex problem that has no single exact solution at all. In all but trivial cases, a

738

THERMODYNAMIC MODELING AND ANALYSIS

designer must deal with uncertainty margins of the input data, in addition to numerous assumptions. Usually, a range of data (say, for a cost analysis in an optimization routine), and not a single set of parameters, must be considered. As shown in Fig. 2.1, for every case considered in the top left box of the problem speciﬁcation, one arrives at an optimum solution at the end of the process of Fig. 2.1. Hence, there can be many (and not only one) optimum solution for a given exchanger sizing problem as provided by diﬀerent heat exchanger manufacturers. With that in mind, the reader should understand the limitations of results obtained by modern computer software for design, optimization, and system integration. 11.1.2

Heat Exchanger as a Component

Before a system-based optimization can be carried out, a good understanding of the exchanger as a component must be gained. In addition to rating or sizing, this may include information about temperature distributions, local temperature diﬀerences, hot and cold spots, pressure drops, and sources of local irreversibilities—all as functions of possible changes of design and/or process variables and/or parameters. The outlet state variables of the ﬂuids depend on the eﬃciency of the heat transfer process inﬂuenced by ﬂuid ﬂow and heat transfer phenomena within the exchanger. A measure of this eﬃciency is not deﬁned exclusively by heat exchanger or temperature eﬀectivenesses because it gives relevant but limited information about heat exchanger performance since the inﬂuence of irreversibility, as discussed in Section 11.3 is not included. Thus, the key questions to be answered involve how to deﬁne exchanger eﬃciency, and how the heat transfer and ﬂuid ﬂow processes (manifested within the heat exchanger boundaries) aﬀect the exchanger eﬀectiveness and thermodynamic eﬃciency (see Section 11.6.5). To answer these questions, we ﬁrst identify the important heat transfer/ﬂuid ﬂow phenomena in the operation of a heat exchanger having an arbitrary ﬂow arrangement in Section 11.3. Design of a heat exchanger as a component is to a large extent an engineering art. So, despite high sophistication in heat exchanger thermal modeling, some of the ﬁnal decisions (in particular those related to optimization) are based on qualitative judgments due to nonquantiﬁable variables associated with exchanger manufacturing and other evaluation criteria. Still, analytical modeling—a very valuable tool—is crucial to understanding the relevant thermal–hydraulic phenomena and design options and various venues for design improvements. In structuring this chapter, special attention is devoted to a balanced use of both rigorous mathematical modeling (Sections 11.2 through 11.4) and qualitative analysis and heuristic judgments (Section 11.5). The results based on mathematical modeling, although elegant and transparent concerning the inﬂuences involved, always carry within them all consequences of numerous assumptions and often simpliﬁcations. So the primary purpose of our study in this chapter is to gain a good understanding of the factors that aﬀect exchanger performance, not necessarily to provide new tools for design and system-based optimization of a heat exchanger.

11.2 MODELING A HEAT EXCHANGER BASED ON THE FIRST LAW OF THERMODYNAMICS An important objective of the material presented in this section is to learn how to model a heat exchanger to determine temperature distributions. In Chapter 3, we focused on the

MODELING A HEAT EXCHANGER ON THE FIRST LAW OF THERMODYNAMICS

739

determination of the exchanger eﬃciency factor ", P, and F [see Eq. (11.1)], through various heat exchanger basic design methods; we did not pay any particular attention to ﬂuid temperature distributions and their relationship to exchanger performance. Let us now consider the distribution of local temperatures and temperature diﬀerences in a heat exchanger having simple counterﬂow and parallelﬂow arrangements. Both these arrangements correspond to two limiting cases of the same geometrical situation: The two ﬂuid streams are ﬂowing in geometrically parallel orientation but in opposite or same directions to each other, thus providing the largest and smallest heat exchanger eﬀectiveness values (see Figs. 3.7 and 3.8 and Table 3.3). It is assumed, by deﬁnition, that both ﬂuids change their respective temperatures only in ﬂow directions (i.e., the local temperature distribution is uniform for a ﬂuid across a ﬂow cross section). Subsequently, we consider a more complex situation with a cross ﬂow of working ﬂuids and the possibility of local mixing along the ﬂow direction. 11.2.1

Temperature Distributions in Counterﬂow and Parallelﬂow Exchangers

In Fig. 11.1 a schematic of a counterﬂow heat exchanger is presented. The assumptions formulated in Section 3.2.1 are invoked here. In general, ﬂow directions may be either in the positive (in Fig. 11.1, from left to right) or negative direction in relation to the axial coordinate (i.e., counterﬂow/parallelﬂow). The energy balances for the respective control volumes can be written as follows using the ﬁrst law of thermodynamics and following rigorously the standard sign convention for heat/enthalpy rate ﬂows across the control volume boundary (positive if entering into the system and negative if leaving the system). For ﬂuid 1 only (the elementary control volume in Fig. 11.1a): dT i1 ðm_ cp Þ1 T1 i1 ðm_ cp Þ1 T1 þ 1 dx UðT1 T2 Þ dA ¼ 0 dx

ð11:2Þ

Adiabatic wall

. mc

Fluid 1

dq

1 p,1

x

x

x + dx

.

m 2 c p, 2

Fluid 2

Adiabatic wall

.

.

.

dq

.

.

.

H1 + dH1

H1

H1 + dH1

H1

.

.

.

H 2 + dH 2

H2

dq (a)

.

.

.

H2

H 2 + dH2 (b)

(c)

FIGURE 11.1 Energy balance control volumes for a counterﬂow arrangement. The control volumes are represented by rectangular areas. (a) Control volume for ﬂuid 1 diﬀerential energy balance; (b) control volume for ﬂuid 2 diﬀerential energy balance; (c) control volume for both ﬂuids 1 and 2 diﬀerential energy balance. Enthalpy rates H_ j ¼ ðm_ cp Þj Tj , j ¼ 1 or 2.

740

THERMODYNAMIC MODELING AND ANALYSIS

where i1 ¼ þ1 or 1, for the same or opposite (positive or negative) direction of ﬂuid 1 with respect to the positive direction of the x axis, respectively. For ﬂuid 2 only (the elementary control volume in Fig. 11.1b): dT i2 ðm_ cp Þ2 T2 i2 ðm_ cp Þ2 T2 þ 2 dx þ UðT1 T2 Þ dA ¼ 0 dx

ð11:3Þ

where i2 ¼ þ1 or 1, for the same or opposite (positive or negative) direction of ﬂuid 2 with respect to the positive direction of the x axis, respectively. For both ﬂuids 1 and 2 (the elementary control volume in Fig. 11.1c): dT dT þ i2 ðm_ cp ÞT ðm_ cp Þ T þ ¼0 i1 ðm_ cp ÞT ðm_ cp Þ T þ dx dx dx dx 1 2

ð11:4Þ

Note that an assumption of uniform distribution of the total heat transfer surface area A along the ﬂow length L means that dA=dx ¼ A=L; that is, U dA ¼ ðUA=LÞ dx. Let us, without restricting the model generality, ﬁx the direction of ﬂuid 1 to be in a positive axial direction (i1 ¼ þ1) while i2 ¼ 1 [i.e., i2 ¼ 1 for counterﬂow (see Fig. 11.1), or i2 ¼ þ1 for parallelﬂow]). Rearranging Eqs. (11.2)–(11.4), we obtain dT1 UA ¼ ðT2 T1 Þ dx L dT2 UA i2 ðm_ cp Þ2 ¼ ðT1 T2 Þ dx L dT1 dT ðm_ cp Þ1 þ i2 ðm_ cp Þ2 2 ¼ 0 dx dx ðm_ cp Þ1

ð11:5Þ ð11:6Þ ð11:7Þ

Note that only two of the three balance equations are suﬃcient to deﬁne the two temperature distributions. For example, either Eqs. (11.5) and (11.6) or Eqs. (11.5) and (11.7) can be utilized. Subsequently, distribution of the temperature diﬀerence along the heat exchanger can be determined. To close the problem formulation, a set of boundary conditions is required at the heat exchanger terminal points. For a parallelﬂow exchanger, the inlet temperatures for both ﬂuids are known at x ¼ 0. For the counterﬂow exchanger, the known inlet temperatures are on the opposite sides of the exchanger, at x ¼ 0 and x ¼ L, respectively. Explicitly, these conditions are T1 ¼ T1;i

at x ¼ 0

T2 ¼ T2;i at

x ¼ 0 for parallelflow x ¼ L for counterflow

ð11:8Þ

Equations (11.5)–(11.8) are made dimensionless with the following variables: ¼

T T1;i T2;i T1;i

¼

x L

ð11:9Þ

MODELING A HEAT EXCHANGER ON THE FIRST LAW OF THERMODYNAMICS

741

and design parameters NTU1 and R1 , as deﬁned by Eqs. (3.101) and (3.105). Hence, d1 þ NTU1 ð1 2 Þ ¼ 0 d

ð11:10Þ

d2 i2 NTU1 R1 ð1 2 Þ ¼ 0 d

ð11:11Þ

d1 d2 þ i2 ¼0 d d

ð11:12Þ

R1 The boundary conditions are as follows: 1 ¼ 0 2 ¼ 1

at ¼ 0 ( ¼ 0 for parallelflow at ¼ 1 for counterflow

ð11:13Þ ð11:14Þ

The set of relationships given by Eqs. (11.5)–(11.8) or (11.10)–(11.14) deﬁne a mathematical model of the heat transfer process under consideration in terms of temperature distributions for both ﬂuids. For example, one can solve Eqs. (11.5), (11.7), and (11.8) to obtain temperature distributions (as presented in Figs. 1.50 and 1.52). This can be done for virtually any combination of design parameters (NTU1 and R1 ) for both parallelﬂow and counterﬂow arrangements without a need for separate mathematical models. Some mathematical aspects of the solution procedure and thermodynamic interpretation of the results will be addressed in the examples that follow. A rigorous and uniﬁed solution of the parallelﬂow heat exchanger problem deﬁned above is provided in Example 11.1 (Sekulic´, 2000). The relation between the heat exchanger and/or temperature eﬀectiveness as a dimensionless outlet temperature of one of the ﬂuids and a thermodynamic interpretation of these ﬁgures of merit is emphasized in Example 11.2. An approach to modeling more complex situations, such as a 1–2 TEMA J shell-and-tube heat exchanger or various crossﬂow arrangements, is left for an individual exercise (see Problems 11.1 through 11.10 at the end of the chapter). Example 11.1 Determine temperature distributions of two parallel ﬂuid streams in thermal contact. The ﬂuid streams have constant mass ﬂow rates and constant but diﬀerent inlet temperatures. Show that a uniﬁed solution procedure can be formulated for both parallelﬂow and counterﬂow arrangements. SOLUTION Problem Data and Schematic: The two ﬂuid streams ﬂow in a parallel geometric orientation as presented in Fig. E11.1A. Both parallelﬂow and counterﬂow arrangements are considered (i.e., ﬂuid 2 can be in either one of two opposite directions). Inlet temperatures, mass ﬂow rates, and ﬂuid properties are known. All the geometric characteristics of the ﬂow passages are deﬁned as well. Determine: The local temperatures of both ﬂuids as functions of the axial distance along the ﬂuid ﬂow direction.

742

THERMODYNAMIC MODELING AND ANALYSIS

Fluid 1

Fluid 2 (parallelflow) i 2 = +1

x ξ = x/L

Fluid 2 x = L (counterflow) i2 = – 1 ξ=1

FIGURE E11.1A Fluid ﬂow orientations in counterﬂow and parallelﬂow heat exchangers.

Assumptions: It is assumed that thermal interaction between the ﬂuids takes place under the assumptions described in Section 3.2.1 Analysis: Any two of three diﬀerential balances given by Eqs. (11.10)–(11.12) together with the boundary conditions given by Eqs. (11.13) and (11.14) describe the theoretical model for analysis. Let us deﬁne the model of this heat transfer process using Eqs. (11.10) and (11.12)–(11.14). A general solution will be obtained utilizing the Laplace transforms method (Sekulic´, 2000), although several other methods can be used as well (see Section 3.11). The rationale for using this particular method is that it can be applied eﬃciently to a number of more complex situations, such as for a crossﬂow arrangement with both ﬂuids unmixed (see Problem 11.2). Applying Laplace transforms to Eq. (11.10) yields d1 l ¼0 þ NTU1 ð1 2 Þ d !s

ð1Þ

Using Laplace transforms rules, Eq. (1) is reduced to 2 ðsÞ ¼ 0 1 ðsÞ 1 ð0Þ þ NTU1 1 ðsÞ s

ð2Þ

j ðsÞ, j ¼ 1, 2, in Eq. (2) represent the Laplace transforms of the yet Variables unknown temperature distributions j ðÞ, j ¼ 1, 2. A complex independent variable denoted as s replaces the original independent variable . Knowing the inlet boundary condition at ¼ 0 [i.e., 1 ð0Þ ¼ 0], Eq. (2) is solved with respect to the Laplace transform 1 ðsÞ as follows: 1 ðsÞ ¼

NTU1 ðsÞ s þ NTU1 2

ð3Þ

2 ðsÞ still has to be determined. This can be done In Eq. (3), an explicit form of involving the other diﬀerential equation of the mathematical model [i.e., Eq. (11.12)]. The same procedure as the one just outlined is applied. Hence, d1 d2 l R1 þ i2 ¼0 d d

ð4Þ

MODELING A HEAT EXCHANGER ON THE FIRST LAW OF THERMODYNAMICS

743

and 1 ðsÞ 1 ð0Þ þ i2 ½s 2 ðsÞ 2 ð0Þ ¼ 0 R1 ½s

ð5Þ

2 ðsÞ and utilizing Eq. (3) for 1 ðsÞ, we get Solving Eq. (5) for 2 ðsÞ ¼

s þ NTU1 2 ð0Þ s2 þ sNTU1 ð1 þ i2 R1 Þ

ð6Þ

1 ðsÞ explicitly in terms of s. Substitute Eq. (6) into Eq. (3) to get Now, applying inverse Laplace transforms on Eqs. (3) and (6), we get (for R1 6¼ 1 if i2 ¼ 1) NTU1 ð1þi2 R1 Þ 1 ðsÞg ¼ 2 ð0Þ 1 e 1 ðÞ ¼ l1 f 1 þ i2 R 1

ð7Þ

and 2 ðÞ ¼ l1 f2 ðsÞg ¼ 2 ð0Þ

1 þ i2 R1 eNTU1 ð1þi2 R1 Þ 1 þ i2 R 1

ð8Þ

Parameter 2 ð0Þ in Eqs. (7) and (8) depends on both design parameters (NTU1, R1 ) and the value of i2 . The value of 2 ð0Þ can be determined for the parallelﬂow arrangement (i2 ¼ þ1) directly from the boundary condition at the ﬂuid 2 inlet [i.e., 2 ð0Þ ¼ 2;i ¼ 1]. For the counterﬂow arrangement (for i2 ¼ 1), the value of 2 ð0Þ can be obtained by collocating Eq. (8) at the ﬂuid 2 inlet (i.e., at ¼ 1), and solving for 2 ð0Þ. Consequently, 2 ð0Þ ¼

1 R1 1 R1 eNTU1 ð1R1 Þ

for i2 ¼ 1

ð9Þ

By inspection of Eq. (9), a generalized algebraic expression for the parameter 2 ð0Þ can be formulated to extend the validity of that equation to include parallelﬂow as follows: 1 R1 1 R1 eð1=2Þð1i2 ÞNTU1 ð1R1 Þ 8 1 for i2 ¼ þ1 ðparallelflowÞ > < ¼ 1 R1 > for i2 ¼ 1 ðcounterflowÞ : 1 R1 eð1R1 ÞNTU1

2 ð0Þ ¼

ð10Þ

In Eq. (10), one should ﬁrst deﬁne the ﬂuid stream direction parameter (i.e., i2 ¼ 1), and then select the numerical values for design parameters. Finally, combining Eqs. (7), (8), and (10), the general solution for temperature distributions for both parallelﬂow and counterﬂow exchangers can be written as follows:

744

THERMODYNAMIC MODELING AND ANALYSIS

1 ðÞ ¼

1 R1 1 eNTU1 ð1þi2 R1 Þ 1 þ i2 R1 1 R1 eð1=2Þð1i2 ÞNTU1 ð1R1 Þ

ð11Þ

1 R1 1 þ i2 R1 eNTU1 ð1þi2 R1 Þ 2 ðÞ ¼ 1 þ i2 R1 1 R1 eð1=2Þð1i2 ÞNTU1 ð1R1 Þ

Inserting i2 ¼ þ1 for parallelﬂow and i2 ¼ 1 for counterﬂow into Eq. (11), we get the following temperature distributions: Flow Arrangement

Flow Indicator i2

1 ðÞ

2 ðÞ

Parallelﬂow

þ1

1 eNTU1 ð1þR1 Þ 1 þ R1

1 þ R1 eNTU1 ð1þR1 Þ 1 þ R1

Counterﬂow

1

1 eNTU1 ð1R1 Þ 1 R1 eNTU1 ð1R1 Þ

1 R1 eNTU1 ð1R1 Þ 1 R1 eNTU1 ð1R1 Þ

The temperature distributions for both parallelﬂow and counterﬂow arrangements and for several sets of parameters are presented in Fig. E11.1B. All the results presented so far for i2 ¼ 1 require that 0 R1 < 1 (i.e., R1 6¼ 1). In the case of a balanced counterﬂow heat exchanger, R1 ¼ 1 and i2 ¼ 1, the original mathematical model given by the set of equations (11.10) and (11.12) transforms into d1 d2 ¼ ¼ NTU1 ð2 1 Þ d d

1.00

R1= 0

1.00

Θ2

0.60 Θ1 Θ2

R1 = 0 Θ2

R1 0

0.80

0.80

0.6

Θ1

ð12Þ

R1 = 0 0.6

0.6

0.6

1.0

1

0.60

Θ1

Θ1

1

Θ2 0.40

0.40

0.20

1.0

0.20

NTU1 = 2.0 Counterflow 0.00 0.00

0.20

0.40

ξ (a)

0.60

0.80

1.00

NTU1 = 2.0 Parallelflow 0.00 0. 00

0.20

0.40

0.60

0.80

1.00

ξ (b)

FIGURE E11.1B Temperature distributions in (a) counterﬂow and (b) parallelﬂow heat exchangers.

MODELING A HEAT EXCHANGER ON THE FIRST LAW OF THERMODYNAMICS

745

with the boundary conditions as given by Eqs. (11.13) and (11.14) for counterﬂow. The solution of this problem leads to linear dimensionless temperature distributions. The temperature distributions for ﬂuids 1 and 2 are 1 ðÞ ¼

NTU1 1 þ NTU1

2 ðÞ ¼

NTU1 þ 1 1 þ NTU1

ð13Þ

Discussion and Comments: In this example, it has been shown how to ﬁnd temperature distributions in a heat exchanger with parallel streams (in geometrical sense). This example demonstrates that both parallelﬂow and counterﬂow arrangements represent the two subproblems of a single heat transfer problem that diﬀer only in the stream direction. Still, the character of temperature diﬀerence distributions in these two situations is radically diﬀerent, as shown in Fig. 11.2. Note that this modeling is based on energy balances performed on two control volumes selected arbitrarily from a total of three balances (either for one or the other ﬂuid, or for both ﬂuids). All results of this modeling are the consequence of conservation principles. 11.2.2

True Meaning of the Heat Exchanger Eﬀectiveness

In Section 3.3.1, heat exchanger eﬀectiveness and the maximum possible heat transfer rate qmax are introduced by deﬁnition [see Eqs. (3.37) and (3.42)]. This has provided the basis for the formulation of heat exchanger eﬀectiveness in terms of terminal temperatures of the ﬂuids and their heat capacity rates as in Eq. (3.44). However, that approach requires a priori deﬁnition of a hypothetical inﬁnite surface area of the heat exchanger. On the other hand, by knowing the temperature distributions of a given heat exchanger, we can devise the concept of heat exchanger eﬀectiveness without invoking the concept of a hypothetical counterﬂow heat exchanger of inﬁnite surface. We can show that the deﬁnition of heat exchanger eﬀectiveness is obtained using the ﬁrst law of thermodynamics only (Sekulic´, 2000), without invoking explicitly the second law of thermodynamics. The true meaning of heat exchanger eﬀectiveness as a dimensionless outlet temperature of the ﬂuid stream having the smaller heat capacity rate is a direct consequence of this interpretation. Moreover, the maximum possible heat transfer rate

TABLE 11.1 Interpretations of the Meaning of Heat Exchanger Eﬀectiveness Ch ðTh;i Th;o Þ q ¼ Traditional " ¼ Based on a comparison of Deﬁned utilizing the ﬁrst qmax Cmin ðTh;i Tc;i Þ meaning the actual heat transfer law of thermodynamics rate exchanged in the explicitly, and the Cc ðTc;o Tc;i Þ ¼ heat exchanger to that second law of Cmin ðTh;i Tc;i Þ exchanged in an ideal, thermodynamics hypothetical heat implicitly exchanger having UA ! 1 True meaning

"¼

T1;o T1;i T2;i T1;i

C1 ¼ Cmin

Dimensionless outlet temperature of a ﬂuid with smaller heat capacity rate, C1 < C2

The ﬁrst law of thermodynamics is used explicitly

746

THERMODYNAMIC MODELING AND ANALYSIS

exchanged in a heat exchanger can subsequently be derived, not postulated. The two interpretations are summarized in Table 11.1. In Example 11.2, these important thermodynamic consequences of the analysis of temperature distributions in a heat exchanger will be illustrated using both parallelﬂow and counterﬂow heat exchangers as an example. It should be reiterated that the interpretation given is universally valid, regardless of the complexity of the ﬂow arrangement involved. Example 11.2 Show that heat exchanger eﬀectiveness and/or temperature eﬀectiveness represent the nondimensional outlet temperature of one of the two ﬂuid streams of a heat exchanger. Use the parallelﬂow and counterﬂow arrangements as examples. Demonstrate that heat exchanger eﬀectiveness can be interpreted as a ratio of actual heat transfer rate to the heat transfer rate of a hypothetical exchanger with an inﬁnitely large thermal size (NTU ! 1), as emphasized in Chapter 3, but without invoking explicitly the second law of thermodynamics. SOLUTION Problem Data and Schematic: As presented in Example 11.1. Determine: Demonstrate the equivalence between the heat exchanger eﬀectiveness deﬁnition and the dimensionless outlet temperature of one of the ﬂuids in a heat exchanger. Assumptions: As invoked in Example 11.1. Analysis: The outlet temperature of ﬂuid 1 for parallelﬂow and counterﬂow can be obtained from Eq. (11) of Example 11.1 at ¼ 1 (the outlet of ﬂuid 1). 1 ð1Þ ¼

1 R1 1 eNTU1 ð1þi2 R1 Þ 1 þ i2 R1 1 R1 eð1=2Þð1i2 ÞNTU1 ð1R1 Þ

ð1Þ

Substituting i2 ¼ þ1 for parallelﬂow, we obtain 1 ð1Þ ¼ P1 ¼

1 eNTU1 ð1þR1 Þ 1 þ R1

ð2Þ

Substituting i2 ¼ 1 for counterﬂow, we obtain 1 ð1Þ ¼ P1 ¼

1 eNTU1 ð1R1 Þ 1 R1 eNTU1 ð1R1 Þ

ð3Þ

Equations (2) and (3) are identical to Eqs. (I.2.1) and (I.1.1) of Table 3.6. So the dimensionless outlet temperatures are equal to parallelﬂow ("pf ) and counterﬂow ("cf ) heat exchanger eﬀectivenesses, respectively. We have obtained expressions for heat exchanger eﬀectiveness without invoking the concept of an ‘‘ideal’’ heat exchanger. Hence, the true meaning of the eﬀectiveness is simply the dimensionless outlet temperature of the ﬂuid with the smaller heat capacity rate (note that R1 ¼ C* in this case). This conclusion is general and valid for any ﬂow arrangement.

MODELING A HEAT EXCHANGER ON THE FIRST LAW OF THERMODYNAMICS

747

To devise a traditional deﬁnition of heat exchanger eﬀectiveness, let us ﬁrst determine the outlet temperature of a counterﬂow heat exchanger in the limit of an inﬁnitely large heat exchanger with thermal size NTU1 ! 1:

lim

NTU1 !1

1 ð1Þ ¼

lim

NTU1 !1

1 eNTU1 ð1R1 Þ 10 ¼ ¼1 NTU ð1R Þ 1 1 10 1 R1 e

ð4Þ

Invoking the deﬁnition of the dimensionless temperature from Eq. (11.9), the following result can be obtained from Eq. (4): lim ðT1;o Þ ¼ T2;i

NTU1 !1

ð5Þ

As indicated above, R1 ¼ C * 1, C1 C2 by deﬁnition (i.e., ﬂuid 1 has the smaller heat capacity rate, C1 ¼ Cmin ). In that case, from Eq. (3.103), NTU1 ¼ NTU. Now the heat transfer rate in a countercurrent heat exchanger of NTU1 ! 1 is given as follows using Eq. (5): lim q ¼

NTU!1

lim ½ðm_ cp Þ1 ðT1;o T1;i Þ ¼ ðm_ cp Þ1 ðT2;i T1;i Þ ¼ qmax

NTU!1

ð6Þ

The actual heat transfer rate in a two-ﬂuid single-phase heat exchanger of any ﬂow arrangement is as follows: q ¼ ðm_ cp Þ1 T1;o T1;i

ð7Þ

Finally, dividing the right-hand side of Eq. (7) with qmax from Eq. (6), and comparing the result with the deﬁnition of the outlet temperature of ﬂuid 1, one can obtain T1;o T1;i q ¼"¼ ¼ 1 ð1Þ ¼ P1 T2;i T1;i qmax

ð8Þ

This constitutes the proof required. Note that for a counterﬂow arrangement with R1 ¼ C * ¼ 1, the heat exchanger eﬀectiveness becomes " ¼ 1 ð1Þ ¼ NTU=ð1 þ NTUÞ; see Example 11.1 for the corresponding temperature distributions. Discussion and Comments: The heat exchanger/temperature eﬀectiveness has its true meaning as a dimensionless outlet temperature of the ﬂuid with the smaller heat capacity rate (heat exchanger eﬀectiveness, ") or the dimensionless outlet temperature of a given ﬂuid (temperature eﬀectiveness, say P1 for ﬂuid 1). The numerical value of the heat exchanger eﬀectiveness is between 0 and 1 and indicates how close the outlet temperature of one ﬂuid can approach the inlet temperature of the other ﬂuid. The traditional meaning of the heat exchanger eﬀectiveness [although a thermodynamic interpretation based on Eq. (8) is perfectly valid and insightful] involves the concept of a hypothetical ‘‘inﬁnitely large counterﬂow heat exchanger.’’ Refer to Sekulic´ (2000) for a detailed discussion concerning an analysis of this approach; discussion of a concept of the thermodynamic eﬃciency for a heat exchanger is given in Section 11.6.5.

748

THERMODYNAMIC MODELING AND ANALYSIS

11.2.3 Temperature Diﬀerence Distributions for Parallelﬂow and Counterﬂow Exchangers Let us now address the magnitude of the local temperature diﬀerence between ﬂuids 1 and 2 (T ¼ jT1 T2 j) in either a parallelﬂow or counterﬂow exchanger.{ We need this information to better understand the inﬂuence of temperature distributions on ", P, or F. The distribution of local temperature diﬀerences for parallelﬂow and counterﬂow heat exchangers can be determined in a general form by utilizing the corresponding temperature distributions: for example, Eq. (11) of Example 11.1. The temperature diﬀerence distribution is as follows (see Problem 11.10 for details): ðÞ ¼ j1 ðÞ 2 ðÞj ¼

ð1 R1 Þ exp½NTU1 ð1 þ i2 R1 Þ 1 R1 exp½ð1=2ÞNTU1 ð1 R1 Þð1 i2 Þ

ð11:15Þ

Note that Eq. (11.15) is valid for both parallelﬂow (i2 ¼ þ1) and counterﬂow (i2 ¼ 1) arrangements for 0 R1 < 1. In Fig. 11.2, a graphical representation of Eq. (11.15) is given for both counterﬂow and parallelﬂow arrangements and for several

1.0

Parallelflow

NTU1 0.4

Counterflow

0.8

NTU1 0.4

0.8 1.2

0.6

0.8

∆Θ

1.2 0.4

0. 2

5.0

5.0 0.0 0.0

0.2

0.6

0.4

0.8

1.0

ξ FIGURE 11.2 Temperature diﬀerence distributions for a counterﬂow exchanger and a parallelﬂow exchanger with R1 ¼ 0:6. {

The terms parallelﬂow and counterﬂow are used throughout the book following the practice by Kays and London (1998), which has been extensively used throughout the world for almost last ﬁve decades. As demonstrated in this section using a straightforward ﬁrst law analysis, a more favorable terminology from a semantic point of view would be unidirectional and bidirectional ﬂows (i.e., in both arrangements, ﬂuid streams are ﬂowing parallel in a geometric sense but oriented in the same or opposite directions).

MODELING A HEAT EXCHANGER ON THE FIRST LAW OF THERMODYNAMICS

749

values of NTU1 and R1 ¼ 0:6. Refer to Sekulic´ (2000) for more detailed data. A study of these temperature diﬀerence distributions reveals the following conclusions. . As we have learned in Chapter 3, counterﬂow is the best and parallelﬂow the worst ﬂow arrangement from the eﬀectiveness viewpoint for given NTU and C* (or NTU1 and R1 ). For the same heat capacity rate ratio (say, R1 ¼ 0:6), and in particular at large NTU1 values, the magnitude of temperature diﬀerence change along the ﬂow direction () is substantially larger for parallelﬂow than it is for counterﬂow (compare corresponding curves in Fig. 11.2). The two temperature distributions are identical for R1 ¼ 0 (not shown in Fig. 11.2) and diﬀer more at large heat capacity rate ratios. . For a given NTU for counterﬂow, features a more pronounced variation along as R1 (or C*) decreases from 1 to 0. This means that there will be larger local temperature diﬀerences with decreasing R1 (or C*) if all other parameters remain the same. If the temperature diﬀerence distributions are close to each other or identical, they will have similar or identical eﬀectiveness. . If the temperature diﬀerence distributions diﬀer substantially, the heat exchanger eﬀectiveness will diﬀer considerably as well. In the limiting case of R1 ¼ 1, the temperature diﬀerence distribution for counterﬂow arrangement is uniform throughout the exchanger, ¼ 2 1 ¼ 1=ð1 þ NTU1 Þ, as derived in Example 11.1. For the same heat capacity rate ratio in the parallelﬂow arrangement, the change in local temperature diﬀerence is the largest possible. Comparing the corresponding heat exchanger eﬀectiveness (see Figs. 3.7 and 3.8) for these conditions, one can easily conclude that the temperature eﬀectiveness of these two ﬂow arrangements diﬀers the most. The distribution of local temperature diﬀerences has a profound inﬂuence on exchanger eﬀectiveness (either " or P1 ). A ﬁnite temperature diﬀerence between the two ﬂuid streams is the driving potential for heat transfer, but large temperature diﬀerences lower the exchanger or temperature eﬀectiveness, and ultimately may contribute to a lower system eﬃciency, as we demonstrate in subsequent sections.

11.2.4

Temperature Distributions in Crossﬂow Exchangers

A model of a crossﬂow arrangement with/without mixing provides a good example of how a simple geometric conﬁguration of two ﬂuid streams in thermal contact may lead to two-dimensional temperature ﬁelds with a very complex relationship between the design parameters. In addition, a study of temperature distributions within a crossﬂow heat exchanger illustrates how mixing may inﬂuence an outcome of the heat transfer process. This insight is very important for an assessment of the inﬂuence of mixing on a reduction in exchanger thermodynamic performance, as demonstrated in Section 11.5. In Fig. 11.3, a schematic of the main features of the geometry and control volumes in a crossﬂow heat exchanger is presented. Fluids 1 and 2 ﬂow perpendicular to each other over a heat transfer surface that separates them. This ﬂow arrangement is discussed in Section 1.6.1.3 (see Fig. 1.53). All the assumptions of Section 3.2.1 are invoked here. Due to heat transfer across the heat transfer surface, both ﬂuids will change their temperatures in either one or both ﬂow directions, depending on the presence or absence of mixing, respectively.

750

THERMODYNAMIC MODELING AND ANALYSIS

Four distinct situations are possible with respect to mixing of the ﬂuids, as emphasized in Section 1.6.1.3. These are also shown graphically in sketches for Eqs. (II.1)–(II.4) of Table 3.6. In Table 11.2, various temperature distributions are shown to be either oneor two-dimensional ½j ¼ f ð or Þ or j ¼ f ð; Þ, j ¼ 1, 2, depending on ﬂuid mixing. Our goal now is to show how one can formulate the models and subsequently solve them to ﬁnd these temperature ﬁelds and/or corresponding outlet temperatures. As a byproduct of this analysis, one can easily devise temperature and/or heat exchanger eﬀectiveness. In Tables 3.3 and 3.6, the formulas for the heat exchanger eﬀectiveness are listed. Here we discuss the analytical models for determining both temperature ﬁelds and eﬀectiveness. This will provide an insight into the inﬂuence of ﬂuid mixing on heat exchanger eﬀectiveness, discussed in Section 11.3. Referring to Fig. 11.3, one can write energy balances for control volumes as follows: d m_ 1 cp;1 T1 |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} fluid enthalpy rate into the control volume

@T1 _ dx d m1 cp;1 T1 þ dq ¼0 |{z} @x |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} heat transfer rate

ð11:16Þ

from fluid to wall

fluid enthalpy rate out of the control volume

and d m_ 2 cp;2 T2 |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} fluid enthalpy rate into the control volume

þ

@T d m_ 2 cp;2 T2 þ 2 dy ¼ 0 @y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} heat transfer rate in dq |{z}

from wall to fluid

ð11:17Þ

fluid enthalpy rate out of the control volume

Note that d m_ j cp; j ¼ dCj , j ¼ 1, 2, in Eqs. (11.16) and (11.17) implies that constant thermophysical properties assumption is invoked. It is assumed that no mixing takes place on either side of the heat transfer surface. Consequently, both ﬂuids will have twodimensional temperature ﬁelds. In Eqs. (11.16) and (11.17), dq represents heat transfer by convection from the hot ﬂuid to the wall; and in the steady-state formulation, that heat will be transferred by conduction through the wall and by convection to the cold ﬂuid.

L1

L1

.

Control volumes dx

L2

y+dy y

dy

Fluid 1

. dm2cp,2[T2 + (∂T2 / ∂y)dx] dm1cp,1T1 .

.

dm2cp,2T2 x x+dx

y h1

Heat transfer surface, Tw

h2 x

L2

dm1cp,1[T1 + (∂T1/ ∂x)dx]

Fluid 2

FIGURE 11.3 Energy balance control volumes for crossﬂow arrangements.

751

1 ¼ 1 ðÞ 2 ¼ 2 ð; Þ d1 1 þ 1 ¼ d C* NTU @2 þ 2 ¼ 1 @ 0 NTU 0 C* NTU 1 ð0Þ ¼ 1 2 ð; 0Þ ¼ 0

1 ¼ 1 ð; Þ 2 ¼ 2 ð; Þ

@1 þ 1 ¼ 2 @ @2 þ 2 ¼ 1 @

0 NTU 0 C* NTU

1 ð0; Þ ¼ 1 2 ð; 0Þ ¼ 0

Temperature ﬁeld

Independent variables

Boundary conditions

0

ð C NTU 2 d

1 ð0; Þ ¼ 1 2 ð0Þ ¼ 0

0 NTU 0 C* NTU

@1 þ 1 ¼ 2 @ ð NTU d2 1 1 d þ 2 ¼ d NTU 0

1 ¼ 1 ð; Þ 2 ¼ 2 ðÞ

Cmin Fluid Unmixed

Model

1 ð0Þ ¼ 1 2 ð0Þ ¼ 0

0 NTU 0 C * NTU

ð C NTU d1 1 þ 1 ¼ 2 d d C* NTU 0 ð NTU d2 1 1 d þ 2 ¼ d NTU 0

1 ¼ 1 ðÞ 2 ¼ 2 ðÞ

Mixed–Mixed

a

The dimensionless variables are deﬁned as follows: j ¼ ðTj T2;i Þ=ðT1;i T2;i Þ, ¼ ðx=L1 ÞNTU, and ¼ ðy=L2 ÞC * NTU. Horizontal and/or vertical lines in schematic ﬁgures symbolize ﬂow of the respective ﬂuid through the ﬂow areas characterized with one-dimensionality (along the ﬂow direction). The absence of lines in the direction of ﬂow symbolizes ﬂuid mixing in the direction transverse to the ﬂow direction. Fluid 1 is assumed to have a smaller heat capacity rate Cmin .

Schematic

Diﬀerential equations

Cmax Fluid Unmixed

Unmixed–Unmixed

Information

TABLE 11.2 Models of Crossﬂow Arrangementsa

752

THERMODYNAMIC MODELING AND ANALYSIS

Using the rate equations for convection and conduction, dq from the hot ﬂuid to the cold ﬂuid can be expressed as follows: Tw;1 Tw;2 dq ¼ o;1 h1 ðT1 Tw;1 Þ dx dy ¼ kw dx dy ¼ o;2 h2 ðTw;2 T2 Þ dx dy w |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} convection from hot fluid to wall convection from wall to cod fluid conduction within the wall

ð11:18Þ Equation (11.18) indicates that there is neither energy generation nor axial conduction through the separating wall, as idealized in Section 3.2.1. The products hj o; j Tj , j ¼ 1, 2 of Eq. (11.18) represent the heat transfer rates exchanged per unit heat transfer area between ﬂuids 1 or 2 and the wall separating the ﬂuids. In the case where the thermal resistance of the separating wall is neglected, only convective terms exist in Eq. (11.18), with Tw;1 ¼ Tw;2 ¼ Tw (i.e., the heat transfer surface has a uniform wall temperature orthogonal to the ﬂow directions). Note that the balances presented by Eqs. (11.16)– (11.18) do not involve the overall heat transfer coeﬃcient. They are formed by applying the thermodynamic convention for each of the thermal energy ﬂow rates (positive if entering the system, otherwise negative). Note also that the assumptions of uniform distribution of the heat transfer area and uniform wall thermal resistance are invoked. Equation (11.18) is rewritten as follows: T1 Tw;1 ¼

dq o;1 h1 dx dy

Tw;1 Tw;2 ¼

dq ðkw =w Þ dx dy

Tw;2 T2 ¼

dq o;2 h2 dx dy ð11:19Þ

Adding up the temperature diﬀerences from three equations of Eq. (11.19), deﬁning dA ¼ dx dy, and using Eq. (3.18) for deﬁnition of the overall heat transfer coeﬃcient U, but neglecting the fouling thermal resistances [since we did not include them in the formulation of Eq. (11.18), which we could have included readily if desired], we get dq ¼ U dAðT1 T2 Þ

ð11:20Þ

Substituting Eq. (11.20) into Eqs. (11.16) and (11.17) and simplifying, we can get the following partial diﬀerential equations: @1 þ 1 ¼ 2 @

@2 þ 2 ¼ 1 @

ð11:21Þ

where j ¼ ðTj T2;i Þ=ðT1;i T2;i Þ, with j ¼ 1, 2, ¼ ðx=L1 ÞNTU, and ¼ ðy=L2 ÞC* NTU. The number of heat transfer units NTU is based on the heat transfer surface between the ﬂuids deﬁned (for the sake of clarity) as the product of L1 and L2 (see Fig. 11.3). Note that UA is distributed uniformly throughout the exchanger due to assumptions of uniformity of heat transfer surface and thermal resistances. The same holds for the heat capacity rates. Hence, NTU is based on any (hot or cold ﬂuid side) heat transfer surface on which U is deﬁned. The deﬁnition of dimensionless temperatures j is complementary to the deﬁnition of the dimensionless temperature of Eq. (11.9). This ﬂexibility in deﬁning dimensionless temperature allows an analyst to deﬁne the heat exchanger eﬀectiveness as either a dimensionless outlet temperature of the ﬂuid

MODELING A HEAT EXCHANGER ON THE FIRST LAW OF THERMODYNAMICS

753

with smaller heat capacity rate or as its complementary value. Note also that both dimensionless temperatures j are assumed to be locally dependent on both independent coordinates and : 1 ¼ 1 ð; Þ

2 ¼ 2 ð; Þ

ð11:22Þ

where the ranges of independent variables take the values 0 NTU

0 C* NTU

ð11:23Þ

Two boundary conditions (for uniform inlet temperatures) accompany the set of Eqs. (11.21): 1 ð0; Þ ¼ 1

2 ð; 0Þ ¼ 0

ð11:24Þ

The set of equations Eqs. (11.21) and (11.24) represents the mathematical model of a crossﬂow heat exchanger. Four particular cases of crossﬂow (see Section 1.6.1.3, and Tables 3.6 and 11.2) diﬀer from each other with respect to the presence or absence of ﬂuid mixing on each ﬂuid side within the heat exchanger core (see Problems 11.5 and 11.6). In Table 11.2, a summary containing all four models, deduced from Eqs. (11.21)–(11.24), is presented. Each of these models can be solved and closed-form analytical solutions can be obtained using various solution methods (see Section 3.11). The solution of the general unmixed–unmixed case is asked in Problem 11.2. A particular case of an unmixed–mixed crossﬂow arrangement is considered in detail in the following example. The mixed–mixed case is considered in Problem 11.7. Example 11.3 Determine temperature diﬀerence ﬁelds in a heat exchanger with a mixed–unmixed crossﬂow arrangement. Assume that the ﬂuid with the smaller heat capacity rate is mixed. SOLUTION Problem Data and Schematic: The ﬂow arrangement under consideration corresponds to the model and schematic given in the third column of Table 11.2 (Cmax ﬂuid unmixed). Determine: Temperature diﬀerence as a function of both axial and transverse coordinates ( and , as deﬁned in Table 11.2). Assumptions: The assumptions are as presented in Section 3.2.1. Analysis: We ﬁrst determine the temperature ﬁelds for both ﬂuids followed by a temperature diﬀerence distribution relationship within the heat exchanger core. The analytical model consists of two equations, one partial and one ordinary diﬀerential equation, as presented in the third column of Table 11.2 (the details of the model development are the subject of Problem 11.6). Uniform temperatures are considered at inlets as corresponding boundary conditions. The solution to this analytical model will provide the desired temperature ﬁelds. Let us ﬁrst solve the partial diﬀerential equation for ﬂuid 2. Subsequently, using the temperature ﬁeld solution for ﬂuid 2 and replacing its explicit form in the ordinary diﬀerential equation for ﬂuid 1, we will ﬁnd the temperature

754

THERMODYNAMIC MODELING AND ANALYSIS

distribution for ﬂuid 1. Finally, the diﬀerence between the two ﬂuid temperatures will provide the solution of the problem. The solution of the partial diﬀerential equation for ﬂuid 2 from Table 11.2 can be obtained as follows using the Laplace transforms technique: l

@2 þ 2 ¼ lf1 g!s @ !s

s2 ð; sÞ þ 2 ð; 0Þ þ 2 ð; sÞ ¼

1 ðÞ s

ð1Þ ð2Þ

where s represents a complex variable that replaces . Rearranging Eq. (2) with 2 ð; 0Þ ¼ 0 from Table 2, we get ðÞ 2 ð; sÞ ¼ 1 sðs þ 1Þ

ð3Þ

An inverse Laplace transform of Eq. (3) provides the temperature ﬁeld for ﬂuid 2: l1 2 ð; sÞgs! ¼ 2 ð; Þ ¼ 1 ðÞð1 e Þ

ð4Þ

Note that the explicit form of 1 ðÞ still has to be determined. The ordinary diﬀerential equation for ﬂuid 1 of Table 11.2 can now be written as follows: d1 ðÞ 1 þ 1 ðÞ ¼ d C * NTU

ð C NTU 0

1 ðÞð1 e Þ d

ð5Þ

After determining the integral on the right-hand side of Eq. (5) and rearranging, the diﬀerential equation for the ﬂuid 1 temperature distribution becomes d1 ðÞ þ k1 ðÞ ¼ 0 d

ð6Þ

where k ¼ ½1 expðC * NTUÞ=ðC * NTUÞ. The boundary condition for Eq. (6) is 1 ð0Þ ¼ 1

ð7Þ

The solution of a simple problem deﬁned by Eqs. (6) and (7) is 1 ðÞ ¼ ek

ð8Þ

So the temperature ﬁeld of ﬂuid 2 can be obtained by introducing the temperature distribution of ﬂuid 1 given by Eq. (8) into Eq. (4): 2 ð; Þ ¼ ð1 e Þek

ð9Þ

Finally, the relationship for the temperature diﬀerence distribution can easily be determined from Eqs. (8) and (9) as ð; Þ ¼ 1 ðÞ 2 ð; Þ ¼ eðkþÞ

ð10Þ

IRREVERSIBILITIES IN HEAT EXCHANGERS

755

Discussion and Comments: As expected, the temperature distribution for unmixed ﬂuid 2 is two-dimensional and that for mixed ﬂuid 1 is one-dimensional, both dependent on k, which in turn depends on NTU and C*. Knowing the temperature distribution of the ﬂuid with the smaller heat capacity rate (ﬂuid 1 in this case), one can easily determine heat exchanger eﬀectiveness (solve Problem 11.3 for understanding the details). Similarly, the analysis of a crossﬂow arrangement for the mixed ﬂuid having a larger heat capacity rate can be performed, and heat exchanger eﬀectiveness can subsequently be determined (see Problem 11.4). Even more complex situations with nonuniform inlet temperatures are the subject of Problems 11.8 and 11.9.

11.3 IRREVERSIBILITIES IN HEAT EXCHANGERS Important phenomena that shape the heat transfer and ﬂow characteristics within a heat exchanger are (1) heat transfer at ﬁnite temperature diﬀerences, (2) mixing and/or splitting of the ﬂuid streams, and (3) ﬂuid ﬂow friction phenomena; additional phenomena when present are phase change, ﬂow throttling, and so on. The ﬁrst two phenomena inﬂuence temperature distributions, and the third the ﬂow friction characteristics on each ﬂuid side of a heat exchanger. Thermodynamics teaches us (Bejan, 1988) that these processes are accompanied by entropy generation, an indicator of undesirable thermodynamic irreversibilities that diminish the thermal performance. So thermodynamic irreversibility is an inevitable by-product of these processes and a principal cause of exchanger/system performance deterioration. Some of the irreversibilities associated with heat transfer and ﬂuid ﬂow are (Gregorig, 1965; Sontag and Van Wylen, 1982): . Heat transfer across a ﬁnite temperature diﬀerence (including both heat transfer between the ﬂuids and heat transfer across the heat exchanger boundary, i.e., heat leak and/or gain to/from surroundings) . Mixing of ‘‘dissimilar’’ ﬂuids (dissimilar with respect to p, T, and/or composition) . Fluid friction and ﬂow impact . Phase change where initial conditions are not in equilibrium . Flow throttling In this section we focus on identiﬁcation and quantitative evaluation of the three dominant irreversibilities in a heat exchanger: (1) irreversibility caused by a ﬁnite temperature diﬀerence, (2) irreversibility caused by ﬂuid mixing, and (3) irreversibility caused by ﬂuid friction. We evaluate these irreversibilities in terms of entropy generation. This analysis will assist us in assessing the quality of heat transfer and associated phenomena in heat exchangers that cannot be evaluated and explained by the analysis presented in Sections 3.3 through 3.8. This analysis requires simultaneous use of both the ﬁrst and second laws of thermodynamics and introduction of the concept of exergy (see Section 11.6.4). Let us ﬁrst review very brieﬂy the concepts of irreversibility, entropy, entropy generation, and exergy before we start the foundation of thermodynamic analysis of heat exchangers. For further details on these concepts and related thermodynamics background, refer to Bejan (1988) and Moran and Shapiro (1995). Thermodynamic irreversibility (simply referred to as irreversibility) is a term used to describe a natural tendency of any real system not to be able to revisit the same sequence

756

THERMODYNAMIC MODELING AND ANALYSIS

of states during a reverse change of state from the ﬁnal to the initial state without additional energy interaction(s). An additional energy interaction is due to an absence of reversibility in a real world when thermal phenomena are involved. In practical terms, this means that the presence of irreversibilities is accompanied by thermodynamic losses, ultimately leading to poorer thermal performance than predicted by an idealized reversible process. In a narrower sense, the same term is used to describe the losses in energy terms I_irr caused by the presence of irreversibilities. The value of irreversibility cannot be negative and it is not a system property as discussed next. Irreversibility can be expressed in energy terms as a product of entropy generation S_irr and a temperature weighting factor To (i.e., I_irr ¼ To S_ irr ). It can be shown that in many engineering applications the weighting factor can be interpreted as the temperature of the surroundings, which is identiﬁed as a thermodynamic reference state for measuring the thermal energy potential of the system at hand. It should be noted that entropy generation is not a property of a system, while the entropy S is. Entropy is deﬁned as a system property by a statement Ð that its change in an ideal, reversible process must be equal to the transfer of an entity dq=T that accompanies any heat transfer dq across the system boundary where the local temperature is T. Hence, this abstract system property indicates that heat transfer must be accompanied by an entropy change. As a consequence, a reversible adiabatic process can be identiﬁed by zero entropy change. If a process is not reversible (as with any heat transfer across a ﬁnite temperature diﬀerence), the situation is radically diﬀerent. Entropy change S is either equal (reversible process) or larger Ð (irreversible process) than the entropy transfer ( dq=T, nonproperty){ that accompanies heat transfer dq, the diﬀerence being attributed to entropy generation S_ irr [see Eq. (11.36)]. The amount of entropy generation is the quantitative measure of the quality level of energy transfer. Entropy generation of zero corresponds to the highest quality of energy transfer and/or energy conversion (a reversible process), and entropy generation greater than zero represents poorer quality. All real processes are characterized by entropy generation greater than zero. The concept of exergy or available energy e is introduced to describe the maximum available energy that can be obtained from a system in a given state. Each ﬂuid stream that enters or leaves a heat exchanger carries exergy rate. Due to irreversible processes in a system (e.g., a heat exchanger), the available energy of a ﬂuid decreases and the diﬀerence between the input and output exergy rates is equal to the lost exergy (lost available energy), which in turn is identical to the irreversibility in energy terms (a Guy– Stodola theorem, i.e., exergy destroyed ¼ lost available energy ¼ temperature weighting factor entropy generation; that is, e_ ¼ W_ lost ¼ To S_ irr Þ: 11.3.1

Entropy Generation Caused by Finite Temperature Diﬀerences

Temperature and temperature diﬀerence distributions within the heat exchanger inﬂuence thermodynamic irreversibility. Thermodynamics teaches us that a measure of the eﬃciency of any thermal process can be assessed by gaining an insight into the irreversibility level incurred in an associated heat transfer process (Bosˇ njakovic´, 1965). This irreversibility can be identiﬁed by determining the corresponding entropy generation. So there is a deeper rationale for turning our attention toward temperature diﬀerences within a heat exchanger to determine heat transfer performance behavior. The driving potential for heat transfer in a heat exchanger is the ﬁnite local temperature diﬀerence {

Note that the entropy is a system property and entropy transfer or entropy generation is not a system property.

IRREVERSIBILITIES IN HEAT EXCHANGERS

757

between the ﬂuids exchanging heat, and we should expect that it greatly inﬂuences the exchanger eﬀectiveness as well. Consequently, it is plausible to conclude that these temperature diﬀerences are related to both the exchanger eﬀectiveness and thermodynamic eﬃciency of an exchanger (one such thermodynamic ﬁgure of merit is deﬁned in Section 11.6.5). How these temperature diﬀerences inﬂuence the irreversibility level is explained next in terms of entropy generation. The thermodynamic irreversibility manifested within a heat exchanger as an adiabatic open system can be identiﬁed in terms of entropy generation by total entropy change (entropy measure of irreversibility, i.e., entropy generation rate S_ irr ) of both ﬂuid streams: S_irr ¼ S_ ¼ m_ 1 s1 þ m_ 2 s2

ð11:25Þ

Now we will evaluate S_ irr only due to ﬁnite temperature diﬀerences, considering the ﬂuids as pure simple single-phase compressible substances. Since ds ¼ dh=T (where h is the speciﬁc enthalpy), the entropy rate change for ﬂuid 1 in a heat exchanger operating under steady-state conditions for an ideal gas or an incompressible liquid is given by m_ 1 s1 ¼

ðo i

ðo m_ cp dT T1;o m_ dh ¼ ¼ ðm_ cp Þ1 ln T1;i T 1 T i 1

ð11:26Þ

Similarly, the entropy rate change for ﬂuid 2 in the exchanger will be m_ 2 s2 ¼ ðm_ cp Þ2 ln

T2;o T2;i

ð11:27Þ

Note that we do not need to distinguish at this point whether the ﬂuid is hot or cold. Moreover, this distinction is not necessarily relevant for calculating the entropy measure of irreversibility. What matters, though, is that the two ﬂuids have diﬀerent temperatures. Hence, the concepts of hot and/or cold will not necessarily be used here. Consequently, Eq. (11.25) can be rewritten as follows: S_irr ¼

2 X j¼1

m_ j sj ¼ m_ 1 cp;1 ln

T1;o T2;o þ m_ 2 cp;2 ln T1;i T2;i

ð11:28Þ

The two terms in Eq. (11.28) have the opposite signs since the ﬂuids have diﬀerent temperatures (T1;i 6¼ T2;i , i.e., either T1;o T1;i and T2;o T2;i or T1;o T1;i and T2;o T2;i ). Two important thermodynamic points have to be reiterated here. First, the fact that the two ﬂuids have diﬀerent temperatures is of far more importance than that one may conveniently be described as hot and the other as cold. This is because the hot/cold dichotomy is introduced by convention. In a heat exchanger, as will be demonstrated later (see Section 11.4.3), the same ﬂuid may change the role of a hot/cold ﬂuid side over some ﬂow length! So, in this chapter, we will, as a rule, refer to a ﬂuid as ﬂuid 1 or ﬂuid 2 whenever a general case has to be considered in which any of the two ﬂuids may either be hot or cold. If a ﬂuid is identiﬁed as having higher/lower temperature (such as in a particular given example), we denote as the hot ﬂuid that has a temperature at its inlet port higher than the temperature of other ﬂuid at its inlet port. Second, a more advanced thermodynamic analysis advocated in this chapter involves the concept of entropy; hence

758

THERMODYNAMIC MODELING AND ANALYSIS

it involves not only temperature diﬀerences but also temperature ratios and the products of absolute temperature and entropy diﬀerences [see, e.g., Eq. (11.28) or (11.53)]. As a consequence, proper care must be taken regarding the use of absolute temperatures (K or 8R) for all temperatures associated with entropy and also exergy later. To emphasize this fact, we use in this chapter, as a rule, temperatures on the absolute Kelvin (or Rankine) scale and not on the commonly used degree Celsius (or Fahrenheit) scale. The heat transfer rate between the two ﬂuid streams in thermal contact under adiabatic conditions is equal to the respective enthalpy rate changes (see Chapters 2 and 3): q ¼ m_ 1 h1 ¼ m_ 2 h2

ð11:29Þ

For better clarity, we consider ﬂuid 1 as the hot ﬂuid and ﬂuid 2 as the cold ﬂuid. Hence, the enthalpy rate changes for the hot and cold ﬂuids are hh ¼ cp;h ðTh;i Th;o Þ, hh ¼ cp;h ðTc;o Tc;i Þ for Eq. (11.29). Changing the subscripts 1 and 2 of Eqs. (11.28) and (11.29) to h and c, combining them, and rearranging, we get, Th;lm Tc;lm S_ irr 1 1 ¼ þ ¼ q Th;lm Tc;lm Th;lm Tc;lm

ð11:30Þ

where Th;lm ¼

Th;i Th;o lnðTh;i =Th;o Þ

Tc;lm ¼

Tc;o Tc;i lnðTc;o =Tc;i Þ

ð11:31Þ

Here Th;lm represents the log-mean temperature of the hot ﬂuid as deﬁned using inlet and outlet temperatures Th;i and Th;o . The Tc;lm is deﬁned similarly. In contrast, the arithmetic mean temperatures of the hot and cold ﬂuids are Th;m ¼ ðTh;i þ Th;o Þ=2 and Tc;m ¼ ðTc;i þ Tc;o Þ=2, and the log-mean temperature diﬀerence between hot and cold ﬂuids in a heat exchanger is given by Eq. (3.172). The entropy generation is related to the diﬀerence in the ﬂuid temperatures. Equation (11.30) is written for the exchanger as a whole. On the local level, entropy generation is related to local temperature diﬀerences [such as Eq. (11.15)]. Hence, the diﬀerence between mean temperatures of two ﬂuids [the numerator of Eq. (11.30)] directly inﬂuences the entropy measure of the irreversibility manifested within the heat exchanger. As a consequence, the heat exchanger irreversibility for a given heat transfer rate can be reduced by reducing temperature diﬀerences between the ﬂuids, which in turn will increase the exchanger eﬀectiveness ".{ A heat exchanger characterized by smaller temperature diﬀerences between the ﬂuids generates a smaller irreversibility in a given system compared to a heat exchanger (for the same heat transfer rate) that has larger temperature diﬀerences between the ﬂuids. Since the entropy measure of irreversibility is related directly to thermodynamic system eﬃciency (see Section 11.6.5), this statement leads to an anticipated conclusion about the possible detrimental inﬂuence of this source of irreversibility on the overall system eﬃciency. Thermodynamic irreversibility represented by entropy generation as in Eq. (11.28) can be formulated in terms of heat exchanger thermal design parameters. Using the { Note that the log-mean temperature diﬀerence Tlm of Eq. (3.172) is proportional to (Th;lm Tc;lm ). Hence, a smaller value of Tlm means a larger value of " for an exchanger.

IRREVERSIBILITIES IN HEAT EXCHANGERS

759

deﬁnitions of heat exchanger eﬀectiveness and heat capacity rate ratio Eqs. (3.44) and (3.56)] and considering C1 ¼ Cmin , one can show that T1;o ¼ 1 þ "ð#1 1Þ ¼ 1 P1 ð#1 1Þ T1;i

T2;o ¼ 1 þ C*"ð# 1Þ ¼ 1 þ R1 P1 ð# 1Þ T2;i ð11:32Þ

where # ¼ T1;i =T2;i represents the inlet temperature ratio. Substituting these expressions in Eq. (11.28) results in S_irr ¼ S* ¼ C* ln½1 þ "ð#1 1Þ þ ln½1 þ C *"ð# 1Þ Cmax S_ irr ¼ S* ¼ R1 ln½1 þ P1 ð#1 1Þ þ ln½1 þ R1 P1 ð# 1Þ C2

ð11:33aÞ ð11:33bÞ

where S_ irr 6¼ 0 for # 6¼ 1 and S_ irr ¼ 0 for # ¼ 1. Note that normalizing S_ irr by Cmax or C2 , as indicated in the leftmost term of Eq. (11.33), is a matter of arbitrary choice. The entropy generation is equal to zero for the inlet temperature ratio equal to 1. Entropy generation [Eq. (11.33)] for diﬀerent ﬂow arrangements is diﬀerent for the same inlet temperature ratio, heat capacity rate ratio C * or R1 , and NTU (Sekulic´, 1990b). This is because of diﬀerent heat exchanger eﬀectivenesses, " ¼ " (NTU, C*), for diﬀerent ﬂow arrangements [and ﬁxed values of NTU and C* (or NTU1 and R1 )]. It should be emphasized that the control volume for the entropy generation of Eq. (11.33) [see Eq. (11.25)] is drawn outside the exchanger core or matrix through inlet/outlet ports. Hence, the S* expression of Eq. (11.33) is valid for an exchanger with any ﬂow arrangement by employing its appropriate "-NTU or P-NTU formula. Some additional features of Eq. (11.33) are discussed in Section 11.4.1. 11.3.2

Entropy Generation Associated with Fluid Mixing

Fluid mixing in a heat exchanger causes thermodynamic irreversibility and generates entropy, leading to a reduction in the thermodynamic eﬃciency of the heat transfer process, thus reducing the heat exchanger eﬀectiveness. In general, the mixing of ﬂuids that are dissimilar with respect to their composition and/or state variables is an irreversible process. These dissimilarities may be mechanical (pressure gradients), thermal (temperature gradients), and/or chemical (chemical potentials). Entropy generation associated with mixing depends on the degree of dissimilarity between mixing ﬂuids. The irreversibility associated with a mixing process is due to (1) a process of intermingling molecules of diﬀerent substances, (2) energy interchange between the same or diﬀerent substances or within the mixing substances, (3) heat transfer between surroundings and mixing substances, and (4) viscous dissipation eﬀects. For the analysis of many heat exchangers, thermal dissimilarities (due to temperature gradients on each ﬂuid side in the transverse direction) of two or more ﬂuid streams during mixing are of primary interest. For example, in a crossﬂow heat exchanger, heat transfer between two ﬂuids causes the presence of local temperature nonuniformities in any given ﬂow cross section. However, a ﬂuid ﬂowing through a nonpartitioned ﬂow passage (i.e., a mixed ﬂuid stream side) is characterized by an important feature. The unrestrained mixing attenuates temperature nonuniformities at a given cross section of

760

THERMODYNAMIC MODELING AND ANALYSIS

.

A virtual stream

.

mo = mi

.

mi =

n

.

Σj=1 m j i ,

(a) Fluid 1: Uniform or nonuniform temperature inlet, Manifold partitioned

Stream 1

Fluid 2 flow passages (a partitioned manifold), fluid unmixed

Uniform temperature at outlet. Manifold is not partitioned

Stream n (b)

(c)

Fluid 1 flow passage (not partitioned manifold), fluid mixed

FIGURE 11.4 Flow passages with ﬂuid mixing: (a) passage or duct with ﬂuid mixing; (b) outlet manifold; (c) two adjacent ﬂow passages with/without ﬂuid mixing.

the mixed ﬂuid and the available thermal potential is destroyed. This is certainly an irreversible phenomenon that leads to a corresponding entropy increase. A practical consequence of this thermodynamically detrimental process is the equalization of ﬂuid local temperatures across the ﬂow passage cross section, ultimately leading to a reduced heating/cooling manifested at the ﬂuid exits. As a consequence, the respective temperature eﬀectiveness (or the heat exchanger eﬀectiveness) is also reduced. Let us consider a ﬂuid stream being mixed while ﬂowing right to left through a duct, as shown in Fig. 11.4a. The objective of the analysis is to determine entropy generation associated with ﬂuid mixing using a very simpliﬁed approach. This situation is often present in heat exchangers. For example, in the outlet header, mixing is accomplished between diﬀerent streams of the same ﬂuid ideally with no heat transfer with the surroundings as shown in Fig. 11.4b. In the heat exchanger core region, a mixed ﬂuid exchanges heat with the other ﬂuid, either mixed or unmixed. A signiﬁcant simultaneous heat transfer and mixing (see Fig. 11.4c for the unmixed–mixed case) prevent us from determining the sole contribution of mixing to the total irreversibility for a control volume under consideration. That is the reason why we formulate a more general case but consider a mixing-only case as follows. A mixed ﬂuid (ﬂuid 1 in Fig. 11.4a) ﬂows through the passage while simultaneously being mixed in a direction transverse to the overall ﬂow direction. For the sake of clarity (but with an inevitable loss of rigor), let us assume that resultant heat exchange between the ﬂuid and the environment can be modeled as heat transfer between each of n virtual streams of the mixed ﬂuid at the inlet that merge into one mixed stream at the outlet. Mass rate, energy/enthalpy rate, and entropy rate balances (continuity, energy, and entropy equations) for the control volume of the bulk ﬂow of ﬂuid 1 are as follows:

IRREVERSIBILITIES IN HEAT EXCHANGERS

m_ o

Continuity equation:

n X

m_ j;i þ

j¼1

Energy equation:

m_ o ho

n X

m_ j;i hj;i

j¼1

Entropy equation:

ðm_ sÞo

n X

n X

qj þ

j¼1

ðm_ sÞj;i

j¼1

n X q j¼1

T

þ

j

761

dmcv ¼0 d

ð11:34Þ

dEcv ¼0 d

ð11:35Þ

dScv ¼ S_ irr > 0 d

ð11:36Þ

where qj , j ¼ 1, n, represent the equivalent heat transfer rates between virtual streams (having average individual bulk temperatures Tj along the respective ﬂow paths) and the surroundings (the other ﬂuid side). Note that S_ irr > 0 in Eq. (11.36) is a consequence of the second law of thermodynamics for a real system. For a steady-state ﬂow, Eqs. (11.34)–(11.36) reduce to{ m_ o ¼

n X

m_ j;i

ð11:37Þ

j¼1

m_ o ho ¼

n X

m_ j;i hj;i þ

j¼1

m_ o so ¼

n X

n X

ð11:38Þ

qj

j¼1

ðm_ sÞj;i þ

j¼1

n X q j¼1

T

þ S_ irr

ð11:39Þ

j

Taking into account the change of local temperatures along the ﬂow paths would, indeed, require writing the balances in a diﬀerential form and integrating them along the ﬂow path. The form of these relationships would depend on actual heat transfer conditions. For isolating the mixing eﬀect only, we consider an adiabatic mixing case. The simplest physical situation of practical interest corresponds to the conditions encountered in the headers/manifolds or in the parts of a heat exchanger where the heat transfer q can be neglected (as is truly for example, in the exit zone of a TEMA J heat exchanger; Fig. 11.6). In such an adiabatic situation, Eqs. (11.38) and (11.39) can be simpliﬁed as follows: m_ o ho ¼

n X

m_ j;i hj;i

ð11:40Þ

j¼1

S_irr ¼ m_ o so

n X

ðm_ sÞj;i ¼

j¼1

n X

ðm_ sÞj

ð11:41Þ

j¼1

For a simple compressible substance, using the expression of Eq. (11.26) for m_ j sj for each stream, S_ irr of Eq. (11.41) reduces to S_ irr ¼

n X j¼1

{

qj is not shown in Fig. 11.4.

ðm_ cp Þj ln

To Tj;i

ð11:42Þ

762

THERMODYNAMIC MODELING AND ANALYSIS

Hence, S_ irr ¼ 0 for uniform inlet temperature ðTj;i ¼ Ti ¼ To , j ¼ 1, n) and S_irr 6¼ 0 when thermal dissimilarity is present (i.e., nonuniform temperature Tj;i 6¼ To ). For example, mixing the two streams of ﬂuid 1 at the exit of a 1–2 TEMA J shell-and-tube heat 0 00 exchanger (see Fig. 11.6 where To ¼ T1;o , and Tj;i are denoted as T1;o and T1;o ) is a source of irreversibility due to the fact that these outlet temperatures of the shell ﬂuid (ﬂuid 1) leaving the two zones A and B are not the same. From Eq. (11.42), it is obvious that entropy generation caused by mixing would never be equal to zero if thermal dissimilarity is present while mixing the streams, even for a single ﬂuid. The mixing process actually eliminates the presence of local temperature diﬀerences; hence it is an inherently irreversible process. This statement holds true regardless of the presence or absence of heat transfer to the environment (or other ﬂuid) during mixing. Consequently, it is expected that a heat exchanger featuring mixing of either ﬂuid within the heat exchanger core or within the header zones should have less eﬀectiveness compared to an exchanger with the same design parameters but without mixing. 11.3.3

Entropy Generation Caused by Fluid Friction

The importance of ﬂuid pressure drop and ﬂuid pumping power P in the heat exchanger is discussed in Chapter 6. One of the important components of the ﬂuid pressure drop in the heat exchanger is the ﬂuid friction associated with ﬂow over the heat transfer surface. We derive the irreversibility associated with this ﬂuid friction in this section. Since the control volume is drawn at the inlet and outlet pipes/tanks, this analysis does take into account the contribution of both skin friction and form drag that is important in many exchangers. To identify the irreversibility caused only by ﬂuid friction, let us assume a ﬂuid ﬂowing through a ﬂow passage of an arbitrary cross section. The ﬂow is caused solely by the pressure diﬀerence between the two points along the ﬂuid path. The entropy generated with such a ﬂow is equal to the entropy change between the two points along the ﬂow path, say between inlet and outlet. If the enthalpy change contribution to entropy change can be neglected (steady and adiabatic ﬂow), the entropy change is as follows using the T ds relationship: dh ¼ 0 ¼ T ds þ v dp (where s and v are speciﬁc enthalpy and speciﬁc volume, respectively): ðo i

d S_ ¼

ðo i

m_ ds ¼

ðo i

m_

v dp T

ð11:43Þ

For an ideal gas ﬂow, Eq. (11.43) reduces to ðo dp p p p ¼ m_ R~ ln 1 þ ¼ m_ R~ ln o ¼ m_ R~ ln 1 S_irr ¼ S_ ¼ m_ R~ pi pi po i p ð11:44Þ where pressure drop p ¼ pi po 0: For an incompressible ﬂuid (liquid) ﬂow, under nonadiabatic conditions entropy generation caused by ﬂuid friction is as follows as discussed by Roetzel in London and Shah (1983).

S_ irr ¼

m_

ðo v dp i

Tlm

¼

p lnðTo =Ti Þ m_ To Ti

ð11:45Þ

THERMODYNAMIC IRREVERSIBILITY AND TEMPERATURE CROSS PHENOMENA

763

In both Eqs. (11.44) and (11.45), we have S_ irr ¼ 0 for p ¼ 0 and S_ irr 6¼ 0 for p 6¼ 0. Hence, the entropy generation caused by ﬂuid friction is never equal to zero for p > 0, as in a heat exchanger. In a heat exchanger with two ﬂuids, the irreversibility contribution of each of the two ﬂuids has to be included [i.e., two terms of the form of Eq. (11.44) or (11.45) have to be calculated].

11.4 THERMODYNAMIC IRREVERSIBILITY AND TEMPERATURE CROSS PHENOMENA In Section 11.3.1 we have demonstrated that heat transfer, accomplished at ﬁnite temperature diﬀerences in a heat exchanger, must be accompanied by entropy generation. This entropy generation is a function of heat exchanger design parameters [see Eq. (11.33) and Problem 11.11]. Let us now explore this relationship to relate thermodynamic performance of a heat exchanger to its heat transfer and design parameters.

11.4.1

Maximum Entropy Generation

Let us rewrite Eq. (11.33) in a symbolic form as a function of relevant design parameters as follows: S_ irr ¼ S * ¼ f ðC*; "; #Þ ¼ f ðC *; NTU; #; flow arrangementÞ Cmax S_irr ¼ S * ¼ f ðR1 ; P1 ; #Þ ¼ f ðR1 ; NTU1 ; #; flow arrangementÞ C2

ð11:46aÞ ð11:46bÞ

The second equality in Eq. (11.46) is written by taking into account Eq. (3.50). So S* is a function of the heat capacity rate ratio, NTU, inlet temperature ratio, and ﬂow arrangement. In Fig. 11.5, this relationship is presented for counterﬂow and parallelﬂow arrangements for ﬂuids having equal heat capacity rates (C* ¼ 1), and an inlet temperature ratio equal to 0.5 (i.e., # ¼ 0:5). It can be shown that corresponding curves for numerous other ﬂow arrangements will be located between the two limiting cases presented in Fig. 11.5 (Sekulic´, 1990b). It should be emphasized that these curves (except for parallelﬂow) have at least one distinct maximum, as explained next. For an exchanger at small NTU values, when NTU ! 0, the magnitude of entropy generation will tend to zero (i.e., S* ! 0Þ. This is certainly an expected result because at NTU ¼ 0 there is no heat transfer since UA ¼ 0 despite the temperature potential for it (represented by the given inlet temperature diﬀerence). On the other side, if NTU ! 1, the temperature diﬀerences along the heat exchanger tend to their minimum possible values (e.g., T ¼ 0 for C* ¼ 1 for a counterﬂow exchanger). Consequently, S* decreases to a limiting asymptotic value (equal to zero for C* ¼ 1 for a counterﬂow exchanger). Hence, a curve having minimum values at both ends will have at least one maximum value in between 0 < NTU < 1. This analysis thus provides the following conclusions for many ﬂow arrangements having only one maximum for S* [including counterﬂow but excluding parallelﬂow (Sekulic´, 1990a); see Section 11.4.3 for an exception; refer to Shah and Skiepko (2002) for other exceptions]:

764

THERMODYNAMIC MODELING AND ANALYSIS

Parallelflow arrangement

0.12

0.10

0.08

S* 0.06

Counterflow arrangement

0.04

C*= 1.0 Inlet temperature ratio = 0.5

0.02

0.00 0

1

2

3

4

5

6

7

8

9

10

NTU FIGURE 11.5 Entropy generation in parallelﬂow and counterﬂow exchangers with C* ¼ 1.

8 >0 > > > > >

¼0 > > > > : 0 at NTU ¼ NTU* lim S* ¼ S*min;1 0

ð11:47Þ

NTU!1

at NTU* < NTU < 1

The most interesting feature of the heat transfer irreversibility behavior implied by Eq. (11.47) is an existence of at least one maximum of S* for a ﬁnite-size heat exchanger at NTU* (or NTU*1 ). Let us determine the value of NTU* and the corresponding eﬀectiveness at that operating point (the same can be done for the number of transfer units deﬁned as NTU1 and the corresponding temperature eﬀectiveness P1 ). The entropy generation maximum is characterized by{ @S* ¼ 0 at @NTU

NTU

Smax

¼ NTU*

ð11:48Þ

Performing the calculation as indicated in Eq. (11.48) on Eq. (11.33), one can show (see Problem 11.13) that the maximum of entropy generation corresponds to{

{

It can be shown that @ 2 S *=@NTU2 < 0 at NTU ¼ NTU* (Sekulic´, 1990a). Explicit expressions for NTU in terms of " and C* are available only for a limited number of ﬂow arrangements * ) in terms of C* can be as outlined in Table 3.4. For these arrangements, explicit formulas for NTU* (NTU at Smax * from Eq. (11.49). For the related formulas for countercurrent and crossﬂow obtained by substituting " at Smax exchangers, see Sekulic´ (1990b). {

THERMODYNAMIC IRREVERSIBILITY AND TEMPERATURE CROSS PHENOMENA

"

Smax

¼

1 1 þ C*

or

P1

Smax

¼

1 1 þ R1

765

ð11:49Þ

Note that Eq. (11.49) is identical to the corresponding relationships in Eq. (3.114). Thus, the number of heat transfer units at the maximum irreversibility in a heat exchanger is exactly the same as the limiting value of NTU (¼ NTU*) at the onset of an external temperature cross. At that operating point, the outlet temperatures of both ﬂuids are equal. Hence, beyond this NTU > NTU*, there will be a temperature cross, and the hotﬂuid outlet temperature will become lower than the cold-ﬂuid outlet temperature. As deﬁned in Sections 3.2.3 and 3.6.1.2, the temperature cross derives its name from ﬁctitious or actual crossing of the temperature distributions of the hot and cold ﬂuids in an exchanger. If there is no actual crossing of hot- and cold-ﬂuid temperature distributions within an exchanger and Tc;o > Th;o , we refer to it as an external temperature cross as found in the temperature distributions of a counterﬂow exchanger in Example 3.2 or the 1–2 TEMA E shell-and-tube heat exchanger of Fig. 3.17a for the high-NTU case (the solid lines). We will call it an internal temperature cross if there is an actual crossing of hot- and cold-ﬂuid temperature distributions within an exchanger. There are two possibilities for the internal temperature cross: (1) Tc;o > Th;o (as in the 1–2 TEMA E exchanger of Fig. 3.17b at high NTU, or (2) Tc;o;local > Th;o;local , where the subscript ‘‘local’’ means one of the multiple outlets on one or both ﬂuid sides of an exchanger (see the temperature distributions in Fig. 11.7 for the 1–2 TEMA J shell-and-tube heat exchanger of Fig. 11.6). Note that there can be an external temperature cross without an internal temperature cross (such as in a counterﬂow exchanger, as shown in the temperature distribution of Example 3.2). Let us discuss the implications of the external and internal temperature crosses further in the following two subsections.

11.4.2

External Temperature Cross and Fluid Mixing Analogy

Results obtained in Section 11.4.1 [i.e., an interpretation of the physical meaning of Eq. (11.49), and the relation between the equalization of outlet temperatures and maximum entropy generation] lead to yet another interesting analogy. This analogy can be explained by studying Eq. (11.49) in a dimensional form. First, let us reconﬁrm (see also Section 3.6.1.2) the statement of equality of the outlet temperatures at the operating point indicated by Eq. (11.49). By invoking deﬁnitions of exchanger eﬀectiveness [Eq. (3.44)] and heat capacity rate ratio [Eq. (3.56)], Eq. (11.49) can be rewritten as follows: T1;o T1;i 1 ¼ T2;i T1;i 1 þ ðT2;i T2;o Þ=ðT1;o T1;i Þ

ð11:50aÞ

Simplifying this equation leads to T1;o ¼ T2;o ¼ To

ð11:50bÞ

So Eq. (11.50) conﬁrms that Eq. (11.49) corresponds to an equality of outlet temperatures of the two ﬂuids of a heat exchanger. Now, using the conclusion reached by Eq. (11.50), let us rewrite Eq. (11.49) this time keeping the heat capacity

766

THERMODYNAMIC MODELING AND ANALYSIS

rate ratio in its explicit form: To T1;i 1 ¼ T2;i T1;i 1 þ C1 =C2

ð11:51aÞ

ðC1 þ C2 ÞTo ¼ C1 T1;i þ C2 T2;i

ð11:51bÞ

Therefore,

Equation (11.51) clearly indicates an identical result as before, but this time it can be interpreted as having been obtained for a quite diﬀerent physical situation of adiabatic mixing of the two ﬂuid streams (with heat capacity rates C1 and C2 and inlet temperatures T1;i and T2;i ). Such imaginary mixing will lead to the outlet temperature To for the mixture of the two given streams (C1 þ C2 ). From thermodynamics, we know that adiabatic mixing leads to total destruction of the available thermal energy potential (implied by the diﬀerence of the temperatures of the ﬂuids) that exists at the onset of mixing. Thus, this process must be characterized by maximum entropy generation. In conclusion, there is an analogy between a heat exchange process at the operating point that corresponds to the condition of an equalization of outlet temperatures and the adiabatic mixing process of the same two ﬂuids. This analogy demonstrates that entropy generation in a heat exchanger at that operating point must be the maximum possible. This constitutes an additional physical explanation of the thermodynamic signiﬁcance of an external temperature-cross operating point in a heat exchanger. A clear thermodynamic meaning of the result given by Eq. (11.49) can easily be conﬁrmed by comparing Eqs. (11.28) and (11.42), that is, comparing the entropy generations obtained for two completely diﬀerent processes: (1) a heat transfer process in a heat exchanger characterized with equal outlet temperatures of the ﬂuids involved [i.e., T1;o ¼ T2;o ¼ To ; see Eq. (11.28)], and (2) a mixing process of the two ﬂuids [Tj ¼ Tj;i ; j ¼ 1, 2; see Eq. (11.42)]. The entropy generation rates for two physically quite diﬀerent processes are found to be identical. Analysis provided so far shows clearly how the facts related to the detrimental inﬂuence of ﬂuid mixing on heat exchanger performance ﬁt nicely into the consistent thermodynamic picture. Hence, the results for heat exchanger eﬀectiveness presented in Chapter 3 (Tables 3.3 and 3.6) have deeper physical explanations. Let us now show how this kind of thermodynamic analysis can be used to understand the behavior of a relatively complex heat exchanger ﬂow arrangement. Moreover, we demonstrate why such analysis has importance for practical design considerations.

11.4.3

Thermodynamic Analysis for 1–2 TEMA J Shell-and-Tube Heat Exchanger

It has been emphasized (see Section 3.6.1.2) that contrary to a general design requirement to transfer heat only from one ﬂuid to the other, and not vice versa, in some heat exchangers, reverse heat transfer takes place. For example, consider the 1–2 TEMA J shell-and-tube heat exchanger of Fig. 11.6. The existence of a temperature cross leads to a situation in which an addition of more surface area in the second tube pass does not contribute a signiﬁcant increase in heat transfer because of reverse heat transfer taking place in the second pass. Note that we reached this conclusion in Section 3.6.1.2 based on assumed (i.e., at that point not analytically derived) temperature distributions. In this

THERMODYNAMIC IRREVERSIBILITY AND TEMPERATURE CROSS PHENOMENA

767

T1, o = (T′1,o + T′′1,o )/2

Mixing exit zone

T′1, o

T2, i

T′′1,o

Tube segment b

Tube segment d

x

x = - L/2 T2, o

x=0 Tube segment a

Tube segment c

Shell zone A

Shell zone B T1, i L

FIGURE 11.6 Schematic of a 1–2 TEMA J shell-and-tube heat exchanger.

chapter (see Section 11.4.2), we provided a thermodynamic interpretation of this undesirable phenomenon through temperature distributions. We elaborated in detail how one can determine temperature distributions and subsequently evaluate the inﬂuence of local temperature diﬀerences and a mixing process on heat exchanger performance. Let us now show how the internal temperature cross can cause peculiar behavior in the P1 -NTU1 or "-NTU results. For a given heat capacity rate ratio, with increasing NTU1, the temperature eﬀectiveness P1 reaches a maximum beyond which an increase in NTU1 causes P1 to decrease rather than increase monotonically as may be expected.{ This behavior is illustrated in Fig. 3.16. Figures 11.7 and 11.8 summarize the results of a thermodynamic analysis of this exchanger. This analysis includes both the ﬁrst and second laws of thermodynamics. The approach is straightforward. It starts with a determination of temperature and heat transfer rate distributions obtained through an application of the ﬁrst law of thermodynamics (Figs. 11.7 and 11.8a; model formulations in Problem 11.1). Subsequently, entropy generation is determined utilizing both the ﬁrst and second laws of thermodynamics [see Section 11.3.1 and Eq. (11.33)]. Figure 11.7 shows three temperature distributions for NTU1 ¼ 0:87, 1.83, and 5.0 for R1 ¼ 2. Figure 11.8 provides data regarding the corresponding distribution of dimensionless heat transfer rates [the total, and those determined at diﬀerent tube sections (a, b, c, and d) and shell zones (A and B) of the Fig. 11.6 exchanger] and entropy generations. To demonstrate this analysis, let us consider a numerical example from the results of Kmecko (1998). Example 11.4 For a 1–2 TEMA J shell-and-tube heat exchanger, establish the number of temperature crosses and explain the meaning of the existence of a maximum eﬀective{

There are also diﬀerent and unexpected P1 vs. NTU1 curve behaviors for n-pass n-pass (n > 2) plate heat exchangers and complex ﬂow arrangements (Shah and Skiepko, 2002).

768

THERMODYNAMIC MODELING AND ANALYSIS

1

1

ΘIV2 ,d 0.8

0.8

Θ2I V2 ,d

Θ2,b II

2

0.6

0.6

Θ

Θ 0.4

ITC

Θ 2 ,o 0.2

III

Θ22 ,c

⊗

0.4

d

0.2

ITC

Θ1 1, A

0 -0.2

0

ξ

⊗

ETC

Θθ11, B -0.4

Θ 2 ,b

Θ2 2 , a

0. 2

Θ2 ,a

Θ 2, c

Θ1,A

Θ11 ,B

0

0.4

-0.4

-0.2

0

ξ

0.2

0.4

(b)

(a) 1 0.8

θd Θ 2 ,d2

0.6

Θ 0.4

Θ1,B

⊗ ETC

Θ22II,b

0.2

Θ 2, c

θ 22È,a 2 ,a Θ1,AΘ I

ITC

0 -0.4

-0.2

0

ξ

0. 2

0.4

(c)

FIGURE 11.7 Temperature distributions in a 1–2 TEMA J shell-and-tube heat exchanger; heat capacity rate ratio R1 ¼ 2: (a) NTU1 ¼ 0:87; (b) NTU1 ¼ 1:83; (c) NTU1 ¼ 5:00. Note that the scale at the abscissa is extended slightly beyond 0:5 (physical terminal ends) to show clearly the ends of the curves separate from the ordinates. (From Kmecko, 1998.)

ness at ﬁnite NTUs. The range of operating points under consideration is deﬁned with R1 ¼ 2 and a range of NTU values between NTU1 ¼ 0:87 and 5.0. The inlet temperature ratio is equal to 2. SOLUTION Problem Data and Schematic: The schematic of a 1–2 TEMA J shell-and-tube heat exchanger is given in Fig. 11.6. The heat capacity rate ratio R1 ¼ 2. The range of NTU values is 0.87 to 5.0. The inlet temperature ratio # ¼ 2:0. Determine: The number of temperature crosses for this exchanger and explain the existence of maximum eﬀectiveness at ﬁnite NTUs.

THERMODYNAMIC IRREVERSIBILITY AND TEMPERATURE CROSS PHENOMENA

769

1

q* d

q*

0.4

R1 = 2 0.2 q*

q*

b

0

R1 = 1

* S */S max

q *B q*A

0.99

q*a q*c

-0.2 0

2

4

6

8 10 NTU1 (a)

12

14

0.98

0

2

4

6 NTU1 (b)

8

10

FIGURE 11.8 (a) Heat transfer rate (R1 ¼ 2), and (b) dimensionless entropy generation rate (R1 ¼ 1 and 2, # ¼ 2) in a 1–2 TEMA J shell-and-tube heat exchanger. Note that the NTU1 axis is extended slightly beyond 0 to show clearly the ends of the curves separate from the y axis. (From Kmecko, 1998.)

Assumptions: The assumptions are those listed in Section 3.2.1. Analysis: The solution of the mathematical model of this heat exchanger (for the model formulation, study Problem 11.1) provides temperature distributions for both shell and tube ﬂuids. The solution method may be the Laplace transforms technique, as discussed earlier in this chapter, or any other pertinent method as mentioned in Section 3.11. These distributions are given in graphical form in Fig. 11.7 for a ﬁxed heat capacity rate ratio (R1 ¼ 2) and three values of NTU1. The NTU1 values correspond to the operating points for the following three speciﬁc cases: (1) the equalization of exit temperatures at small NTU (the ﬁrst external temperature cross at NTU1 ¼ 0:87 at the exchanger outlets of two ﬂuids (T1;o ¼ T2;o ), (2) the maximum temperature eﬀectiveness (P1;max at NTU1 ¼ 1:83), and (3) the equalization of exit temperatures at a large NTU value (the second external temperature cross at NTU1 ¼ 5:0 at the exchanger outlet, T1;o ¼ T2;o ). In all three cases, we have internal temperature crosses. The temperature distribution in Fig. 11.7a corresponds to an operating point NTU1 ¼ 0:87 and R1 ¼ 2:0. The ﬂuid 2 (tube ﬂuid) dimensionless temperature decreases along both tube passes (tube segments d–b–a–c in Fig. 11.6 and 2;d ! 2;b ! 2;a ! 2;c in Fig. 11.7a) reaching the ﬂuid exit with the dimensionless temperature 2;o (its corresponding dimensional value is T2;o ). The outlet temperature of ﬂuid 2 at that operating point is exactly equal to the mixed mean temperature of ﬂuid 1 obtained 0 00 after mixing both exit streams of ﬂuid 1 [i.e., T1;o ¼ ðT1;o þ T1;o Þ=2, or in dimensionless form 1;o ¼ ð1;A;o þ 1;B;o Þ=2]. To emphasize, 2;o ¼ 1;o or T2;o ¼ T1;o for NTU1 ¼ 0:87 and R1 ¼ 2 for Fig. 11.7a. Note that ﬂuid 1 (shell ﬂuid) dimensionless temperature changes (increases) along both shell zones (A and B) following the distributions presented in Fig. 11.7a (1;A and 1;B ). Therefore, this operating point corresponds to an equalization of exit temperatures, that is, an onset of an external temperature cross (ETC). In addition, an internal (or actual) temperature cross (ITC) is present in zone B at

770

THERMODYNAMIC MODELING AND ANALYSIS

the ITC point 1;B ¼ 2;c . Note that the temperature eﬀectiveness of ﬂuid 1 is P1 ¼ 1;o , and P2 can then be calculated from P2 ¼ P1 R1 . If NTU1 is increased from 0.87 to 1.83, a new operating point will be reached. Under that condition, temperature distributions will become as given in Fig. 11.7b. This ﬁgure reveals a peculiar increase of the ﬂuid 2 dimensionless temperature (2;c ) in the second part of the tube segment c: instead of decreasing, the original hot tube ﬂuid 2 dimensionless (or dimensional) temperature increases. This is due to reverse heat transfer taking place to the left of the ITC point in Fig. 11.7b. As expected, the resulting exit temperature of ﬂuid 1 is higher than that for the operating point presented in Fig. 11.7a. This means that the temperature eﬀectiveness of ﬂuid 1 has increased from P1;a ð¼ 1;o;a Þ to P1;b ð¼ 1;o;b Þ (the subscripts a and b denote the cases from Fig. 11.7a and b, with NTU ¼ 0:87 and 1.83, respectively). If we increase the number of transfer units even further, say from 1.83 to 5.0 (see Fig. 11.7c), the increase in the ﬂuid 2 dimensionless temperature 2;c in the tube segment c becomes very pronounced (see, e.g., 2;c at ¼ 0:5 in Fig. 11.7c). The shell ﬂuid temperature still continues to increase in both shell zones, but much more so in zone B, 1B in Fig. 11.7c. However, 1;o ½¼ ð1;A;o þ 1;B;o Þ=2 of Fig. 11.7c is less than 1;o of Fig. 11.7b. As a result, the temperature eﬀectiveness of ﬂuid 1 will become smaller than the corresponding value of Fig. 11.7b. This means that with an increase of NTU1 starting from 1.83, the temperature eﬀectiveness decreases from its maximum value at NTU1 ¼ 1:83 and R1 ¼ 2:0, as shown in Fig. 11.8a, in the q* ð¼ P1 Þ vs. NTU1 curve. To understand this peculiar behavior, let us now consider the distribution of heat transfer rates within the heat exchanger in various shell zones and tube segments, as presented in Fig. 11.8a. It is interesting to notice from Fig. 11.8a that with an increase in NTU1, an initial increase in total dimensionless heat transfer rate, q* ¼ q=½ðm_ cp Þ1 ðT2;i T1;i Þ ¼ P1 , is followed by a maximum at NTU1 ¼ 1:83 and decrease at higher NTU1. Adding more heat transfer area beyond that for the maximum eﬀectiveness decreases the heat transfer rate exchanged between the two ﬂuids. How could that be possible? The components of this heat transfer rate are presented in the same diagram (Fig. 11.8a) for both the shell ﬂuid (zones A and B) and the tube ﬂuid (tube segments a to d) having appropriate subscripts with q*. From this ﬁgure, it is clear that only the tube segment d (the inlet segment of the ﬁrst tube pass) contributes to the heat exchanger performance by its positive dimensionless heat transfer rate slope in the direction of increasing NTU1. The heat transfer contribution in tube segments a and b (and correspondingly, in the shell zone A) is diminishing rapidly, while the heat transfer in tube segment c becomes reversed and is increasing in the negative direction. Consequently, a large portion of the heat exchanger actually fails to contribute to the original design goal of increasing q with P1 at large NTU1 values, beyond NTU1 ¼ 1:83 for this case! Discussion and Comments: The behavior of the analyzed heat exchanger can be interpreted in terms of entropy generation. For this exchanger, S_ irr can be calculated using Eq. (11.33) for # ¼ T1;i =T2;i ¼ 2:0 and the P-NTU formula of Eq. (III.11) of Table 3.6 for given NTU1 and R1 ¼ 1 and 2 as shown in Fig. 11.8b. For this exchanger, the entropy generation features two maximums and one local minimum. The two equalizations of outlet temperatures (ETCs, one at smaller and the other at larger NTU1) correspond to two maximums in entropy generation for the case of R1 ¼ 1 and 2. In this situation, the ﬁrst is at NTU1 ¼ 0:87 and the second at NTU1 ¼ 5:0, as shown in Fig. 11.8b. At the operating point characterized by the maximum temperature eﬀectiveness, the heat

A HEURISTIC APPROACH ASSESSMENT OF HEAT EXCHANGER EFFECTIVENESS

771

exchanger operates at a local minimum entropy generation. Note that the existence of the second maximum at large NTU1 is not of great practical importance per se (a 1–2 TEMA J shell-and-tube heat exchanger is never designed as a single-pass unit at such large NTU1). Rather, it indicates that the entropy generation between NTU1 ¼ 0:87 and 5.0 may both decrease and increase. In other words, it makes no thermodynamic sense (or practical sense either) to design a 1–2 TEMA J heat exchanger for a large NTU (in the analyzed example, a large NTU means that NTU > 1:83). The foregoing discussion demonstrates why study of a heat exchanger design should be accompanied by a study of internal heat transfer intricacies and a thermodynamic analysis. A very practical conclusion has been reached. The tube segments a and b are contributing virtually nothing to the heat exchange process at high NTUs, and the tube segment c has reverse heat transfer. This conclusion would never have been reached if study of this design were conducted looking at the heat exchanger as a black box. Of course, shell-and-tube heat exchangers are hardly ever designed for NTU > 1:5 with a single shell pass. We ﬁnd the following interesting observations from a detailed review of some "-NTU curves of exchangers presented in Chapter 3 and many "-NTU curves for other ﬂow arrangements from Shah and Pignotti (1989). . When there is an external temperature cross only (as in a counterﬂow exchanger) or only an external temperature cross can be found by modifying the original ﬂow arrangement (such as modifying the ﬂow arrangement of Fig. 3.17b to 3.17a), the exchanger eﬀectiveness will continue to increase monotonically with increasing NTU. . When an internal temperature cross cannot be eliminated [due to the exchanger geometry (such as the 1–2 TEMA J exchanger of Fig. 11.6) or even if by modifying the ﬂow arrangement], the exchanger eﬀectiveness will decrease with increasing NTU beyond the point of S*min at NTU > 0. For such exchangers, the S_ irr versus NTU curve has at least two maximums, and the minimum value of S* occurs between the two maximums (see, e.g., Fig. 11.8b). The following exchangers have an internal temperature cross and the exchanger eﬀectiveness decreases with increasing NTU beyond the point of S*min at NTU > 0 based on the extensive P-NTU results of Shah and Pignotti (1989): a crossﬂow exchanger with both ﬂuids mixed, 2–2, 2–3, and 2–4 overall parallelﬂow PHEs, and the following TEMA shell-and-tube heat exchangers: 1–3 E (two passes in parallelﬂow) , 1–4 E, 1–2 G (overall parallelﬂow), 1–2 H (overall parallelﬂow), 1–2 J, and 1–4 J.

11.5 HEURISTIC APPROACH TO AN ASSESSMENT OF HEAT EXCHANGER EFFECTIVENESS As demonstrated in previous sections, thermodynamics provides an insight into the relationship between the irreversibility level (entropy generation as an indicator of heat transfer quality performance) and the heat transfer and ﬂuid ﬂow features of a ﬂow arrangement. For a detailed quantitative analysis, ﬂuid temperature distributions within a heat exchanger need to be determined. However, one can use the conclusions reached through a qualitative study of temperature diﬀerence distributions, ﬂuid mixing,

772

THERMODYNAMIC MODELING AND ANALYSIS

and ﬂow friction phenomena to assess the heat exchanger performance, even without a detailed quantitative analysis. We use this argument to show how a simple heuristic approach can be used to assess heat exchanger performance (Sekulic´, 2003). Our goal is to compare the pairs of ﬂow arrangements having the same #, NTU, and R. The objective is to predict through a heuristic analysis which of the two arrangements in a pair has better eﬀectiveness without computing the eﬀectiveness. Let us compare two single-pass crossﬂow arrangements: (1) the arrangement with ﬂuid 1 unmixed and connected in identical order between the rows A and B and ﬂuid 2 split into two equal individually mixed streams as presented in Fig. 11.9a, and (2) the conﬁguration with an inverted order coupling of ﬂuid 1 shown in Fig. 11.9b. Thus the only diﬀerence between the two arrangements is the coupling of ﬂuid 1 streams between the two rows of ﬂuid 2. Let us also assume that the P1 -NTU1 relationships for the two arrangements are not known but that the two exchangers have identical NTU1 and R1 . The question is as follows: Which of the two arrangements has better performance? To attempt to answer the question, let us recall that the presence of larger temperature diﬀerences and/or ﬂuid mixing in an exchanger inevitably increases thermodynamic irreversibility (i.e., entropy generation), and correspondingly decreases the exchanger eﬀectiveness, if all other conditions remain the same. As far as mixing in individual rows is concerned, the two arrangements are identical. However, the local temperature diﬀerences in the two ﬂow arrangements are diﬀerent due to overall ﬂow conﬁgurations. Consequently, the two exchangers must have diﬀerent eﬀectivenesses. Hence, let us compare qualitatively the magnitude of the local temperature diﬀerences in these two arrangements by concentrating on the two lateral streams S1 and S2 of ﬂuid 1 in Fig. 11.9a and b. In the case of an identical order coupling (Fig. 11.9a), the stream S1 leaves the ﬁrst row A of the ﬂuid 2 from a zone where ﬂuid 2 enters that row, and subsequently enters the second row B of the same ﬂuid again in the zone where ﬂuid 2 enters that row (that is why we call this coupling identical). Stream S2 leaves the ﬁrst row A of ﬂuid 2 with a temperature that has been changed less when compared to the change experienced by stream S1 (due to the fact that ﬂuid 2 has already experienced some heat transfer through row A before it meets stream S2 ). Moreover, stream S2 enters the second row B of ﬂuid 2 in the zone where the exit of ﬂuid 2 is located in that row. The heat transfer will be accomplished at established temperature diﬀerences, deﬁned by the given operating and design conditions. In the case of the inverted order, though (Fig. 11.9b), stream S1 from the exit of the ﬁrst row A of ﬂuid 2, from a zone where ﬂuid 2 enters the row, is led into the

S2

S2

Ao Bi

S2

Ao Bi

S1

Fluid 1 B

A S1

Fluid 2 (a)

S2

Fluid 1 S1

A

B Fluid 2

S2

(b)

FIGURE 11.9 Single-pass crossﬂow: ﬂuid 2 split into two equal streams individually mixed and ﬂowing through rows A and B: (a) ﬂuid 1 unmixed and connected in identical order; (b) ﬂuid 1 unmixed and connected in inverted order. The outlet and inlet zones of the two ﬂuid 2 stream rows are denoted by Ao and Bi .

A HEURISTIC APPROACH ASSESSMENT OF HEAT EXCHANGER EFFECTIVENESS

773

second row B of ﬂuid 2 to the zone from which ﬂuid 2 leaves that row. Similarly, stream S2 is led from the ﬁrst row A of ﬂuid 2, from a zone where ﬂuid 2 leaves the row, to the second row B of ﬂuid 2, into the zone where ﬂuid 2 enters the row. So stream S1 , the stream that has experienced a larger temperature change in the ﬁrst row A of ﬂuid 2 than stream S2 , will meet ﬂuid 2 in two diﬀerent zones of the second row B of ﬂuid 2 for the two analyzed arrangements. As a consequence, in the case of the identical order coupling, the heat transfer between ﬂuids 1 and 2 will be accomplished in the second row B of ﬂuid 2 at smaller temperature diﬀerences at the entrance than that in the case of the inverted order. Based on this observation of the smaller temperature diﬀerence at the inlet of the second tube row or a pass, we could infer that it will have lower temperature diﬀerence irreversibility, which means higher exchanger eﬀectiveness. This conclusion has been reached without any knowledge about the corresponding eﬀectiveness relationships; it is based on the qualitative analysis of the heat transfer process involved. Calculating the eﬀectivenesses for the two arrangements at the same inlet operating conditions, using appropriate P-NTU relationships from Shah and Pignotti (1989), can easily provide a simple proof. For example, if NTU1 ¼ 3 and R1 ¼ 0:8, the temperature eﬀectiveness of the identical order and inverted order arrangements are 0.7152 and 0.6668, respectively. The aforementioned conclusion has been veriﬁed for pairs of heat exchangers that diﬀer only with respect to a particular source of irreversibility (a ﬂuid coupling between the passes, mixing, ﬁnite temperature diﬀerence magnitude at a location where they may be at maximum). A systematic analysis of the applicability of this heuristic approach can be conducted for other heat exchangers. For crossﬂow arrangements, it was shown that even in the cases for which the closed-form expressions of the eﬀectiveness relationship does not exist, the simple heuristic approach usually works. In Fig. 11.10, two two-pass arrangement pairs are compared.{ For both the two-pass cross-parallelﬂow exchangers and the two-pass cross-counterﬂow arrangements, the identical order coupled passes (Fig. 11.10a and c) have larger eﬀectiveness values compared to the corresponding inverted-order coupled passes (Fig. 11.10b and d). Finally, let us consider an exchanger with the most complex two-pass cross-counterﬂow arrangement, with both ﬂuids unmixed and connected either in identical (Fig. 11.11a) or inverted order (Fig. 11.11b). For these ﬂow arrangements, there are no closed-form analytical solutions for eﬀectiveness–NTU relationships. The solutions for NTU1 ¼ 3 and R1 ¼ 0:8 for the two cases are (1) both ﬂuids coupled in identical order (Fig. 11.11a), P1 ¼ 0:757, and (2) both ﬂuids coupled in inverted order (Fig. 11.11b), P1 ¼ 0:736. Consequently, the identical order arrangement has obviously higher eﬀectiveness. We can reach the same conclusion without utilizing the complex semianalytical relationship if we simply use the heuristic approach described above. This is presented in Example 11.5. Example 11.5 Provide a heuristic argument that a two-pass cross-counterﬂow arrangement with both ﬂuids unmixed and connected in identical order has a larger ﬂuid 1 { Note that the identical and inverted orders between rows/passes appear to be deﬁned diﬀerently sometimes (see, e.g., Fig. 11.9b vs. Fig. 11.10b, both having inverted order). However, they are deﬁned consistently as identical order if both ﬂuid streams enter the second row/pass in the same order as they enter the ﬁrst row/pass; if they enter the second row/pass in a diﬀerent order (one of the streams enters from the other end of the row/pass), we refer to it as an inverted order. Review Sections 1.6.1.3 and 1.6.2.1 for further clariﬁcation.

774

THERMODYNAMIC MODELING AND ANALYSIS

Fluid 1

Fluid 1

Fluid 2

Fluid 2 (a)

(b)

Fluid 1

Fluid 1

Fluid 2

Fluid 2 (c)

(d)

FIGURE 11.10 Two-pass cross-parallelﬂow and cross-counterﬂow arrangements: ﬂuid 2 mixed: (a) two-pass cross-parallelﬂow arrangement with ﬂuid 1 unmixed and coupled in identical order; (b) two-pass cross-parallelﬂow arrangement with ﬂuid 1 unmixed and coupled in inverted order; (c) two-pass cross-counterﬂow exchanger with ﬂuid 1 unmixed and coupled in identical order; (d) two-pass cross-counterﬂow with ﬂuid 1 unmixed and coupled in inverted order.

temperature eﬀectiveness than that of a two-pass cross-counterﬂow arrangement with both ﬂuids unmixed and connected in inverted order.

SOLUTION Problem Data and Schematic: Schematics of the two ﬂow arrangements are given in Fig. 11.11. Determine: Which of the two ﬂow arrangements has the higher temperature eﬀectiveness for the same operating point. Assumptions: The assumptions are as discussed in Section 3.2.1.

S2

A

S2

A

B

B S1

S1 Fluid 1

Fluid 1 Fluid 2 (a)

Fluid 2 (b)

FIGURE 11.11 Two-pass cross-counterﬂow exchangers with both ﬂuids unmixed and connected in (a) identical order and (b) inverted order.

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

775

Analysis: Assume that ﬂuid 1 is the hot ﬂuid. In that case, the ﬂuid 1 stream S1 leaves pass A with its temperature higher than the temperature of stream S2 . In the case of an identical order coupling (Fig. 11.11a), ﬂuid 1 stream S1 (the warmer of the two) is channeled to meet ﬂuid 2 at the exit of pass B, at the point where the corresponding ﬂuid 2 stream has already been heated by ﬂowing through that pass. The ﬂuid 1 stream S2 (the colder of the two), however, is guided to meet ﬂuid 2 at its inlet into pass B, where it has the lowest temperature (ﬂuid 2 is assumed to be the cold ﬂuid). So these temperature diﬀerences would be smaller for the identical order coupling (Fig. 11.11a) than for the case of an inverted order coupling (Fig. 11.11b). This is because in the inverted order coupling of Fig. 11.11b, streams S1 and S2 of ﬂuid 1 are channeled to meet the corresponding streams of ﬂuid 2 in pass B at locations characterized with opposite inlet/outlet sections when compared to pass A. The same reasoning can be conducted following the two bordering streams of ﬂuid 2. So we should expect the identical order coupling to be more favorable than the inverted, the result already indicated above. Discussion and Comments: A note regarding the presence of mixing is warranted at this point. It seems to be more obvious to assess two ﬂow arrangements with respect to ﬂuid mixing than with respect to local temperature diﬀerences. A good example is the sequence of decreasing eﬀectiveness values for, say, single-pass crossﬂow arrangements given in Table 11.2, for the given NTU and R. For example, for the set of values NTU1 ¼ 3 and R1 ¼ 0:8, the temperature eﬀectiveness P1 takes the values 0.7355, 0.6791, 0.6655, and 0.6254, for unmixed–unmixed, unmixed (ﬂuid 2)–mixed (ﬂuid 1), unmixed (ﬂuid 1)–mixed (ﬂuid 2), and mixed–mixed ﬂow arrangements, respectively. The increasing inﬂuence of mixing is obvious. Note that for unmixed–mixed and mixed– unmixed arrangements, ﬂuid 1/ﬂuid 2 is either mixed or unmixed, respectively, and because R1 ¼ 0:8 < 1, it is clear that mixing of the ﬂuid with the larger heat capacity rate contributes more to performance deterioration, as expected. It should be mentioned that the heuristic approach might become more diﬃcult to conduct in complex multipass heat exchangers. However, if performed correctly, it should provide at least guidance regarding the actual performance.

11.6 ENERGY, EXERGY, AND COST BALANCES IN THE ANALYSIS AND OPTIMIZATION OF HEAT EXCHANGERS As emphasized in the introduction to this chapter, a heat exchanger is always part of a system. From the system point of view, heat exchanger design must be based on design speciﬁcations that are in full accord with an optimization objective deﬁned for the system as a whole. The optimization objective may be formulated using energy rate and cost balances. When combined with the thermodynamic irreversibility analysis, the approach is called thermoeconomics (Bejan et al., 1995). Thus, a heat exchanger designer must be aware of (but not limited to) energy, cost, and exergy balances of the system. Therefore, this type of analysis requires not only energy balances based on the ﬁrst law of thermodynamics, but also insights based on the second law of thermodynamics, as well as economic considerations. In this section, we address brieﬂy a link between the thermal size of a heat exchanger (and accordingly, temperature distributions, including other related features of the heat exchanger performance) as well as energy, exergy, and cost balances. A more detailed analysis (and one that is less tightly related to the design theory

776

THERMODYNAMIC MODELING AND ANALYSIS

of a heat exchanger) can be found in a number of books devoted to thermodynamic design and optimization of thermal systems, but such a discussion is beyond the scope of this book. Those interested in the subject should consult the literature provided by Linnhoﬀ et al. (1982) and Bejan et al. (1995). Heat exchanger optimization summarized in Section 9.6 was focused on optimization of a heat exchanger as a component. In this section, we discuss heat exchanger optimization when it is part of a system or when the system-imposed constraints during operation dictate the thermal design of a heat exchanger. However, it should be noted that we will not study the comprehensive system optimization, only some aspects of the analysis for system optimization with focus on the heat exchanger in the system. The system speciﬁed is ﬁrst optimized based on a variety of criteria, including energy, entropy, economy criteria, and/or packaging. As a result, the performance and packaging requirements for individual heat exchangers in the system are deduced. Such optimization methods work and are usually straightforward when heat exchangers are part of a relatively simple system such as gas turbine or steam power plant, vapor compression air-conditioning, and so on, where we deal with a single or just a few working ﬂuids to accomplish the system objectives. However, in the process industry, we deal with many process ﬂuid streams to heat, cool, condense, vaporize, distill, concentrate, and so on, as mentioned in the beginning of Section 1.1. When a number of heat exchangers in a network are used to heat, cool, or change of phase of the process streams with available utilities, the analysis is often conducted based on the pinch analysis or pinch technology to ensure that all exchangers in a system meet the requirements of the process streams based on performance targets (Linnhoﬀ et al., 1982). Practical thermodynamic performance targets must be deﬁned before the actual design is carried out. These performance targets are either for energy performance of the system or for a realistic number of units to be used in a system. The actual heat exchanger performance and packaging requirements are deduced as a result of process integration. Subsequently, the methodology of Section 9.6 can be utilized for heat exchanger optimization as a component, which will also result in optimum individual heat exchangers in an optimum system. With the currently available very sophisticated commercial software for optimization, the methodologies of Section 9.6 and this section can be combined and the heat exchanger optimum dimensions and/or operating conditions can be obtained directly for an optimum system being characterized by the least cost, least energy consumption, or other set of criteria for optimization. Now we concentrate on the analysis and optimization of a process industry application. First, we explain how the thermal size of a heat exchanger may become an object of optimization study based only on a ﬁrst law of thermodynamics inquiry. In approaching that problem, we utilize energy rate balance considerations only. Subsequently, we extend our inquiry to include the second law of thermodynamics (exergy rate balance) and economic considerations (cost rate balance). 11.6.1

Temperature–Enthalpy Rate Change Diagram

Let us assume that the particular design requirements have been formulated for a process industry application. Since we are dealing here with a speciﬁc application, we identify the ﬂuids as hot and cold instead of using more general terminology. The temperature of a hot-ﬂuid stream in a process has to be reduced from Th;i to Th;o while simultaneously a cold-ﬂuid stream temperature in the same process has to be increased from Tc;i to Tc;o , as

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

777

shown in Fig. 11.12.{ We specify Th;o < Tc;i as shown in Fig. 11.12. Note that it is possible that in some applications Tc;o may be desired to be even greater than Th;i . For the sake of simplicity, both ﬂuids are considered to be simple substances and have heat capacity rates Ch and Cc . In addition, cold and hot thermal sources (usually called utilities—the heat sources or sinks, or yet other process streams) are available if the required enthalpy changes cannot be accomplished utilizing the cold stream alone. Note that the general validity of this analysis is not restricted by adopting an assumption that the ﬂuids are simple compressible substances within the given temperature ranges and away from the onset of phase-change phenomena. Utilizing Eq. (2.1), and writing the outlet temperatures explicitly for the two streams leaving the process, one obtains Tj;o ¼ Tj;i

q H_ ¼ Tj;i Cj Cj

ð11:52Þ

where j ¼ h or c. The enthalpy rate change, H_ , equals the heat transfer rate q in the exchanger. Relationships deﬁned by Eq. (11.52) can be interpreted as line segments in a (T; H_ ) graph, each with a slope equal to the reciprocal value of the corresponding heat capacity rate. These graphs are presented in Fig. 11.12 (for a selection of given inlet and outlet temperatures and heat capacity rates) for the three coupled exchangers shown underneath. They are cold utility, main, and hot utility exchangers.{ Note that equalization of heat capacity rates would lead to parallel temperature–enthalpy rate change lines. In Fig. 11.12, the temperature–enthalpy rate change lines are not parallel, therefore, the corresponding heat capacity rates are not equal. The greater the imbalance between the heat capacity rates, the more pronounced is the nonuniform temperature diﬀerence (Th;hex Tc;hex ) distribution along the enthalpy rate change axis. The situation presented in Fig. 11.12 can be interpreted as a general case situated between the two limiting designs—to accomplish the design goal (to reduce the hot-ﬂuid enthalpy and to increase the cold-ﬂuid enthalpy)—a designer may distribute the total heat load between (1) the main heat exchanger, (2) the cold utility heat exchanger, and (3) the hot utility heat exchanger, shown in Fig. 11.12. This may be done in such a way as to exploit the heat recovery in the overlapping region of the temperature–enthalpy rate change diagram (note: Th;o < Tc;i and Tc;o < Th;i ). An important question for the exchanger designer is the following: What would be the optimum thermal size, UA, of the main heat exchanger? In a limiting case, the size of the main heat exchanger may be very large, say close to an inﬁnitely large thermal size (i.e., NTU ! 1). The outlet temperature of the hot ﬂuid from that exchanger (Th;o;hex ) will become very close to the inlet temperature of the cold stream (Tc;i ). That is, the minimum temperature diﬀerence between the ﬂuids Tmin ¼ Th;o;hex Tc;i;hex ¼ Th;o;hex Tc;i (the so-called ‘‘pinch’’) becomes very close to zero. Note that an increase in size of the main heat exchanger in the temperature– enthalpy diagram (Fig. 11.12) may easily be presented by shifting horizontally the {

It should be emphasized that the required heating and cooling of cold- and hot-ﬂuid streams of Fig. 11.12 may need multiple (and not one) heat exchangers. In Fig. 11.12, the main exchanger has the hot process stream on one ﬂuid side and the cold process stream on the other ﬂuid side, so that thermal energy is recovered from the hot process stream, heating the cold process stream (as desired) without any additional expense of utility streams. The cold utility exchanger cools the hot stream coming out of the main exchanger and the hot utility exchanger heats the cold stream coming out of the main exchanger. The utility streams are water, steam, or air in general. {

778

THERMODYNAMIC MODELING AND ANALYSIS

Hot utility C * or R1

ε or P1

⊕

Th,i

NTU or NTU1 Tc,o

Hot fluid T

Th,o,hex

Tc,o,hex

∆T min Cold fluid

Tc,i = Tc,i,hex Th,o Cold utility qm,hex

qcu,ex

qhu,ex

.

H Th,i

Cold utility exchanger

Main exchanger

System control volume Tc, o

Th,o

Tc,i Component control volumes

Hot utility exchanger

FIGURE 11.12 Heat exchanger temperature–enthalpy rate diﬀerence diagram. The energy balance control volumes (CVs) denote system component boundaries. The subscript ‘‘hex’’ denotes the main heat exchanger under consideration.

temperature curves toward each other until Tmin ¼ 0 (note that qm;hex on the abscissa becomes the largest possible). This hypothetical design would reduce the requirement for utilities to a minimum (qhu;ex and qcu;ex become the smallest possible). Note that this idealized design cannot eliminate the need for either a hot or cold utility (the hot and cold ﬂuid lines will not necessarily overlap entirely) for the given Cc and Ch , but it would reduce both utilities to the minimum. Still, this design is not a realistic one (i.e., it would not be a workable design) because it requires an inﬁnitely large main heat exchanger. Another limiting case would be to invest virtually nothing into the design of the main heat exchanger and to accomplish the required task by (1) cooling the hot ﬂuid exclusively with a cold utility, and (2) heating the cold ﬂuid with a hot utility. The temperature–enthalpy diagram in that case would feature the absence of the overlapping of the TH_ lines, such as by shifting the TH_ line for the cold ﬂuid to the right until Tc;i is vertically in the same location as Th;i . This solution would require a maximum possible energy cost (no heat recovery at all) but at the same time would reduce the investment in the main heat exchanger to zero (qm;hex ¼ 0). For this example, it is assumed that hot and cold utilities would be available without an additional need for capital investment. This design is clearly workable (see Example 11.6 on the next page).

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

779

It is obvious that an optimum design must be somewhere between these two limiting cases. The objective functions may be, say, the total cost (involving the energy cost and number and size of the units required), the physical size of the exchanger, the exergy losses (see the following sections), and the like. Our intention in this section, however, is not to study thermoeconomic optimization and/or to extend the analysis considering a whole system to perform an integration of a heat exchanger network using either pinch analysis or exergy (or entropy generation) analysis. Our goal is less ambitious—to formulate the design optimization objective problem related to a single exchanger or a couple of them, and to show how an optimization problem arises. In a more general case, most notably for the case of a heat exchanger network, the temperature–enthalpy rate change diagram becomes a valuable tool in constructing composite curves. Namely, instead of presenting temperature–enthalpy rate change lines for only two streams of a single heat exchanger, one composite TH_ curve for all hot ﬂuid streams and another such composite curve for all cold ﬂuid streams within the corresponding temperature ranges can be depicted as well. Each such line has a slope related to the sum of all heat capacity rates of the involved streams within the given temperature range. This analysis, though, is beyond the scope of our interest. More details about the procedure of constructing the composite curves, very useful for the pinch technology method, can be found in the literature devoted to process synthesis and integration (e.g., Linnhoﬀ et al., 1982; Gunderson and Naess, 1988; Sama, 1995a). 11.6.2

Analysis Based on an Energy Rate Balance

As emphasized in Chapter 9, an optimum design, or in terms of a practically achievable design solution, a nearly optimum design, is an engineering goal imposed by real-world requirements and constraints. For example, a heat exchanger positioned at a certain location in a large chemical engineering plant may fully satisfy the system purpose. Diﬀerent integration and/or heat exchanger redesign may lead to a system that would be even more eﬃcient or cost eﬀective. However, the design requirements are, as a rule, conﬂicting propositions. For example, a smaller temperature diﬀerence in a heat exchanger leads to a larger system thermal eﬃciency but requires larger heat exchanger size and often more pumping power, which may, in turn, cause a signiﬁcant system objective deterioration. An optimization of a heat exchanger as a component integrated into a system may involve not only the traditional tools of thermal design (based almost exclusively on the ﬁrst law of thermodynamics) but also an integrated approach known as thermoeconomics. This approach includes simultaneous analysis based on (1) energy rate balances, (2) exergy rate balances and/or entropy generation calculations, and (3) cost balances. To illustrate a problem that involves determination of the physical size of an exchanger for which only energy rate balances are suﬃcient, we present Example 11.6. This example is based on a study of an optimum area allocation in a system of coupled heat exchangers as reported by Alt-Ali and Wilde (1980). This example is similar to the problem of Fig. 11.12, but without the main heat exchanger. Example 11.6 A ﬂuid stream with heat capacity rate of 52,740 W/K needs to be heated from 388C (311 K) to 2608C (533 K). Two hot-ﬂuid streams are available for accomplishing this task. These hot-ﬂuid streams have the same heat capacity rates as that of the cold-ﬂuid stream. The inlet temperatures of two hot ﬂuids are 1498C (422 K) and 3168C

780

THERMODYNAMIC MODELING AND ANALYSIS

(589 K). The task can be accomplished either by using only one two-ﬂuid heat exchanger (and utilizing only the stream with the higher temperature) or using two two-ﬂuid heat exchangers connected in a two-stage array with the cold ﬂuid ﬂowing through both exchangers. The available two hot streams (the utilities) should be used in the two exchangers. Determine the optimal distribution of heat transfer surface areas of the two exchangers to get the minimum total heat transfer area suﬃcient to heat the cold stream to the desired temperature. Compare the two heat exchanger solution with the design having one heat exchanger. Assume constant and uniform overall heat transfer coeﬃcients in both units to be equal to 454 W=m2 K. SOLUTION Problem Data and Schematic: Two heat exchangers arranged in a two-stage array with only one cold stream ﬂowing through both units is shown in Fig. E11.6. For a minimum surface area requirement, both exchangers are considered to be counterﬂow. The input data are given in Fig. E11.6.

Tc,A,o = Tc,B,i Tc,B,o = 533 K Tc,A,i = 311 K

B

A Hot fluid, stage A

Th,A,o

Hot fluid, stage B

Th,A,i = 422 K Th,B,o

Th,B,i = 589 K

Cold fluid with Cc,A = Cc,B = Cc = 52,740 W/K Ch,A = Ch,B = Cc UA = UB = 454 W/m2. K FIGURE E11.6 Two-stage heat exchanger system with operating conditions.

Determine: The heat transfer surface area of each of the two exchangers for a minimum total heat transfer surface area. Compare that solution to a single-exchanger solution. Assumptions: All appropriate assumptions for modeling each of the exchangers of Section 3.2.1 are invoked. Additionally, it is assumed that both units have equal overall heat transfer coeﬃcients. Analysis: We need to formulate a functional relationship between total surface area Atot and cold-ﬂuid interstage temperature T ¼ Tc;A;o ¼ Tc;B;i (shown in Fig. E11.6) for a minimum total surface area: Find minfAtot ¼ f ðTÞg

ð1Þ

where Atot ¼ AA þ AB

and

T ¼ Tc;A;o ¼ Tc;B;i

ð2Þ

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

781

Therefore, the problem is characterized with the following variables: (1) the interstage temperature T of the cold ﬂuid is a decision variable, (2) the outlet temperatures (Th;A;o and Th;B;o ) of the two hot ﬂuids are dependent variables, and (3) the independent variables (i.e., parameters of optimization) are: the inlet and outlet temperatures (Tc;A;i and Tc;B;o ) of the cold ﬂuid, the inlet temperatures (Th;A;i and Th;B;i ) of the hot ﬂuids, the heat capacity rates of both ﬂuids, and the overall heat transfer coeﬃcients of both units. To deﬁne the objective function, Atot ¼ f ðTÞ, ﬁrst let us compute the surface areas AA and AB . To accomplish that task we have to utilize the ﬁrst law of thermodynamics by formulating energy rate balances (i.e., equating the heat transfer rates and ﬂuid enthalpy rates imposed by the ﬁrst law of thermodynamics for each ﬂuid side): ðUAÞj Tlm; j ¼ Cc; j ðT Tc; j;i Þ

ð3Þ

where j ¼ A or B. Hence, the individual surface areas AA and AB from Eq. (3) are AA ¼

Cc;A ðT Tc;A;i Þ UA Tlm;A

AB ¼

Cc;B ðTc;B;o TÞ UB Tlm;B

ð4Þ

As emphasized above, Eq. (3) is based on energy and mass conservation principles. Hence, all the subsequent results are the consequences of the ﬁrst law of thermodynamics (energy equation) and the continuity equation only. The total area is deﬁned as follows: Atot ¼ AA þ AB ¼ where Tlm;A ¼

Cc;A ðT Tc;A;i Þ Cc;B ðTc;B;o TÞ þ UA Tlm;A UB Tlm;B

ðTh;A;i TÞ ðTh;A;o Tc;A;i Þ ln½ðTh;A;i TÞ=ðTh;A;o Tc;A;i Þ

Tlm;B ¼

ð5Þ

ðTh;B;i Tc;B;o Þ ðTh;B;o TÞ ln½ðTh;B;i Tc;B;o Þ=ðTh;B;o TÞ ð6Þ

In Eqs. (5) and (6), the outlet temperatures Th;A;o and Th;B;o are dependent on the interstage temperature T and can be found again from energy conservation principles. This time, however, we should formulate enthalpy rate balances for ﬂuids in both units; that is, for units A and B, Cc;A ðT Tc;A;i Þ ¼ Ch;A ðTh;A;i Th;A;o Þ

Cc;B ðTc;B;o TÞ ¼ Ch;B ðTh;B;i Th;B;o Þ ð7Þ

Therefore, Th;A;o ¼ Th;A;i

Cc;A ðT Tc;A;i Þ Ch;A

Th;B;o ¼ Th;B;i

Cc;B ðT TÞ Ch;B c;B;o

ð8Þ

Substitution of Th;A;o and Th;B;o of Eq. (8) in Eqs. (5) and (6) yields an objective function Atot ¼ f ðTÞ, where T is a decision variable. Formally, to solve the problem, we should ﬁnd an extremum of this function with respect to the decision variable (i.e., @Atot =@T ¼ 0). Performing this partial diﬀerentiation, we get (Alt-Ali and Wilde, 1980) Cc;B @Atot Cc;A A 1 ¼ ¼0 @T UA ðA A ÞðA RA A Þ UB B RB B

ð9Þ

782

THERMODYNAMIC MODELING AND ANALYSIS

where A ¼ T Tc;A;i

B ¼ Tc;B;o T

A ¼ Th;A;i Tc;A;i

B ¼ Th;B;i T

RA ¼

Cc;A Ch;A

RB ¼

Cc;B Ch;B

ð10Þ

Temperature diﬀerences denoted as ’s represent the temperature diﬀerences between terminal temperatures of the cold ﬂuid of concern for each of the two exchangers, while ’s represent inlet temperature diﬀerences for the ﬂuids in both heat exchangers, respectively. So the ﬁnal results will depend not only on the individual temperature changes accomplished within the two exchangers, but also on the magnitudes of the inlet temperature diﬀerences. Rearranging Eq. (9) leads to Th;A;i Topt Th;A;o Tc;A;i Cc;A =UA A A A RA A ¼ ¼ Cc;B =UB A B RB B Th;A;i Tc;A;i Th;B;o Topt

ð11Þ

where Topt represents the value of the interstage temperature of the cold ﬂuid for @Atot =@T ¼ 0 (i.e., the optimum interstage temperature). Finally, the optimum interstage temperature of the cold ﬂuid Topt can be obtained from Eq. (11) after introducing the deﬁnitions of , , and R’s as follows:

Topt

Cc;A =UA ¼ Th;A;i ðTh;A;i Tc;A;i ÞðTh;B;i Tc;B;o Þ Cc;B =UB

1=2 ð12Þ

For the given problem, inserting the numerical values of the variables in Eq. (12), we get "

Topt

ð52,740 W=KÞ=ð454 W=m2 KÞ ¼ 422 K ð422 311Þ K ð589 533Þ K ð52,740 W=KÞ=ð454 W=m2 KÞ ¼ 344 K

#1=2

ð13Þ

Now, replacing T ¼ Topt ¼ 344 K into Eq. (4) and introducing other known variables, the heat transfer surface areas become AA ¼ 48:1 m2 and AB ¼ 395:5 m2 . Total heat transfer area is Atot ¼ AA þ AB ¼ 444:6 m2 . In a trivial case, if only one heat exchanger is used (with the hot ﬂuid having an inlet temperature of 589 K), the heat transfer surface area can easily be calculated from Eq. (3) for j ¼ 1. In such a case, this area would be 464.5 m2, which is 4.5% more than Atot ¼ 444:6 m2 for the optimum area for two exchangers. Discussion and Comments: This example demonstrates how a proper distribution of heat transfer surface area can lead to an optimal design. This conclusion has been reached utilizing only energy and mass ﬂow rate balances. The fact that a single heat exchanger would have a larger heat transfer surface area (compared to a two-heat-exchanger solution) does not mean that the solution with two exchangers would be the desired design solution. Cost (in addition to other considerations) may be a key decision factor for the best achievable design. Note that temperature diﬀerences between the ﬂuids at the terminal points of both heat exchangers have an important role in the determination of the optimum solution. However, the temperature diﬀerences deﬁned only on the basis of

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

783

total area minimization (i.e., utilizing only the ﬁrst law of thermodynamics and mass balances) do not necessarily minimize the entropy generation in the assembly or lead to the most economic solution. For this to be accomplished, a combined thermodynamic and economic analysis has to be performed as discussed in the following section. 11.6.3

Analysis Based on Energy/Enthalpy and Cost Rate Balancing

Let us now illustrate how cost rate balances may inﬂuence the design solution. Of course, the simultaneous use of energy balances is mandatory as well. This time, the objective function will be the total annual cost as a function of the cold-ﬂuid exit temperature, keeping the heat duty ﬁxed in a heat exchanger. Obviously, variation of the exit temperature will change exchanger " or P1 as well as cause a change in the coolant mass ﬂow rate for ﬁxed q. Thus, C* or R1 will change. This will yield a diﬀerent NTU and hence A. Consequently, a search for the most favorable design based on an economic criterion would require resizing the exchanger. Both economics and thermodynamics will inﬂuence the solution. It is important to notice that the corresponding thermodynamic part of the analysis would imply only the ﬁrst law of thermodynamics. How that has to be accomplished can be illustrated best by using an example. The example represents a slightly modiﬁed problem presented by Peters and Timmerhaus (1980). Example 11.7 Design a condenser in a distillation unit to operate at optimum total annual cost. The particular unit under consideration uses cooling water to condense vapor. It operates at a minimum total annual cost if water leaves the condenser at 528C (325 K), and at the given inlet ﬂuid conditions and with installed optimal heat transfer area. The outlet temperature of the cooling water, however, may increase to 578C (330 K) due to changing environmental considerations with a corresponding decrease in the cooling-water mass ﬂow rate and appropriate resizing of the exchanger. Assuming that all inlet variables (except for the mass ﬂow rate of water) must stay unchanged, determine how large the change of total annual cost would be in excess of the established optimum value for the given design, if the mentioned changes take place. The additional information is as follows. The heat exchanger condenses m_ ¼ 2000 kg/h of vapor, which has an enthalpy of phase change of 4 105 J/kg. Condensation occurs at 778C (350 K). The inlet temperature of the cooling water is 178C (290 K), and the speciﬁc heat of water at constant pressure is 4:2 103 J=kg K. The overall heat transfer coeﬃcient is 280 W/m2 K. The distillation unit must operate for o ¼ 6500 hr/y. The unit cost of cooling water is 2 105 c/kg, where c is the monetary unit. The unit cost for the heat exchanger per installed unit of heat transfer area is 300 c/m2. The annual cost of heat exchanger operation is 20% of the cost of installed heat exchanger area.

SOLUTION Problem Data and Schematic: In the condenser, the cooling water increases its temperature along the ﬂow path, and the condensing stream has a constant temperature, as shown in Fig. E11.7A. All pertinent data are provided in this ﬁgure. Determine: The increase in total cost in excess of the optimum value when the exit temperature of the cooling water increases by 5 K.

784

THERMODYNAMIC MODELING AND ANALYSIS

Th = T h,i = T h,o = 350 K Tc,o = 325 K original Tc,o = 330 K changed

Condensing vapor Cooling water Tc,i = 290 K

.

cp,w = 4200 J/kg .K, hᐉg = 4 × 10 5 J/kg, U = 280 W/m2 . K m h = 2000 kg/h τo = 6500 h/yr, Cw = 2 ×

10–5

C /kg, CA = 300 C

/m2,

cA = 0.2/yr

FIGURE E11.7A Schematic of temperature distributions in a condenser.

Assumptions: It is assumed that Eq. (3.12) is valid. The mean temperature diﬀerence in that equation is assumed to be equal to the log-mean temperature diﬀerence determined using terminal temperature diﬀerences. Assume the total annual cost of heat exchanger operation to be Ctot ¼ Cw þ CA , where Cw is the cost of the total amount of cooling water, and CA is the ﬁxed cost of the heat exchanger installed. Analysis: Let us start with evaluating the heat transfer rate q: q ¼ m_ h h‘g ¼ ð2000 kg=hÞð4 105 J=kgÞ ¼ 8 108 J=h The total annual cost of the condenser is equal to Ctot ¼ Cw þ CA

ð1Þ

Therefore, q q þ c A CA cp ðTc;o Tc;i Þ U Tm " # 8 108 J=h c=kgÞ ð4:2 103 J=kg KÞðTc;o 290 KÞ

Ctot ¼ o Cw m_ 2 þ cA CA A ¼ o Cw ¼ ð6500 h=yrÞð2 105

" # 8 1 2 ð8 10 J=hÞð1 h=3600 sÞ þ 0:2 300 c=m yr ð280 W=m2 KÞ Tm ¼

24,762 c 0:4762 105 c þ Tm ðTc;o 290 KÞ yr yr

ð2Þ

where Tm ¼ Tlm ¼

ðTh Tc;i Þ ðTh Tc;o Þ Tc;o Tc;i T 290 K ¼ c;o ¼ Th Tc;i Th Tc;i ð350 290Þ K ln ln ln Th Tc;o Th Tc;o 350 K Tc;o

ð3Þ

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

785

The relationship of Eq. (2) is presented in Fig. E11.7B. The minimum value of total cost, Ctot; min ¼ 1898 c=yr, is at Tc;o ¼ 325 K, as suggested in the problem formulation. This can be conﬁrmed easily by ﬁnding the ﬁrst derivative of the objective function, Eq. (2), with respect to the coolant exit temperature, Tc;o (i.e., @Ctot =@Tc;o ¼ 0). This minimum point (see Fig. E11.7B), corresponds to the designed heat exchanger heat transfer area of 20 m2 [calculated from the rate equation, Eq. (2.2)]. If the outlet temperature of the coolant increases to 330 K, the total cost will increase to Ctot ¼ 1927 c=yr, a 1.5% higher value than the value at the optimal point. To keep the duty unchanged, the heat transfer area changes to approximately 22 m2, and the coolant mass ﬂow rate will change from 1.512 kg/s to 1.323 kg/s. Discussion and Comments: In this example, the objective function is the total annual cost of heat exchanger operation. The optimum (minimum) total cost corresponds to the outlet coolant temperature equal to 325 K. Any change in this temperature incurred through a change of heat exchanger size (area) causes some increase in the total cost. At the same time, both the coolant mass ﬂow rate and the exchanger heat transfer area must change to keep the heat load ﬁxed. In this analysis, we utilized only the ﬁrst law of thermodynamics to formulate energy balances (used implicitly for setting up the cost rate balance). This example illustrates how an optimum design may lead to an economic penalty if a heat exchanger operates at oﬀ-design. In this case, only the exchanger surface area and the cooling water utility are considered for cost. The optimization problem required a minimum annual variable cost as a function of the exit temperature. Equations (2.1) and (2.2) deﬁned the heat exchanger model. The temperature at optimum (i.e., at the minimum cost) is conﬁrmed to be 325 K from Fig. E11.7B. The oﬀ-design condition requires a heat exchanger size change, as a result of a change in the exit temperature (for other variables constant except for the mass ﬂow rate). Note that this economic optimum was not related in any way to a thermodynamic irreversibility minimization.

1 960 1 950

Ctot,modif.

C tot

1 940 1 930 1 920

Ctot,min

1 910 1 900 1 890 320

322

324

326

328

330

332

Tc,o (K) FIGURE E11.7B Total annual variable operating cost of a heat exchanger vs. coolant exit temperature.

786

THERMODYNAMIC MODELING AND ANALYSIS

11.6.4

Analysis Based on an Exergy Rate Balance

In previous sections, we demonstrated the utilization of both energy and cost rate balances. These analyses, though, have not considered the quality of energy rate ﬂows. Under the term energy rate ﬂow, we consider either the enthalpy rate or the heat transfer rate. Consequently, we should examine more closely the role of the irreversibility analysis for heat exchanger thermal design and optimization. In this section, we formulate the exergy rate balance. In the following two sections, we use this concept to deﬁne a thermodynamic ﬁgure of merit (thermodynamic eﬃciency of a heat exchanger) and to introduce the cost of irreversibility. Let us assume that a heat exchanger represents a component of a process or power plant. The plant can be optimized for the total annual cost of operation. However, to ﬁnd the cost of the related irreversibilities (see Section 11.6.6), we should deﬁne the irreversibility on an energy basis in a quantitative manner. This cost can be calculated as a sum of the cost of compensation for the irreversibility and the capital investment. Irreversibility may be measured by its energy measure—either the entropy generation multiplied by the appropriate weighting temperature factor (Ahrendts, 1980), or exergy (Kotas, 1995). So far in our analysis we measured irreversibilities in terms of entropy generation. As is known from thermodynamics (Moran and Shapiro, 1995), irreversibility can conveniently be deﬁned in terms of energy rate units using the exergy rate as follows (excluding chemical exergy): e_j;k ¼ m_ j ½ðhj;k href Þ Tref ðsj;k sref Þ ð11:53Þ The exergy rate represents the rate of the available energy of a given ﬂuid stream with respect to the conveniently selected reference state. Each ﬂuid stream (entering or leaving the heat exchanger; see Fig. 11.13) carries certain exergy, deﬁned by Eq. (11.53). In this equation, the subscript j denotes either ﬂuid 1 or ﬂuid 2, or alternatively, the hot or cold ﬂuid, k denotes inlet or outlet, and ‘‘ref ’’ denotes a state of the respective ﬂuid at the thermodynamic condition deﬁned by a selected reference state (often, but not always, the state of the environment). Exergy does not obey a conservation principle because it includes not only the properties of a thermodynamic system at an exchanger terminal port, but also the reference thermodynamic state. In our case, the term thermodynamic system refers to whichever ﬂuid stream exposed to heat transfer in a heat exchanger. In many applications, that state can simply be a state of the system in thermodynamic equilibrium with the environment. The usefulness of the concept of exergy is in providing a reference level regarding the maximum possible useful energy potential that would be available from a particular energy source with respect to the surroundings. The quality of energy ﬂow rate can be interpreted by a simple example as follows. The quality of the same amount of available energy from a ﬂuid at 5008C is higher than for the same ﬂuid at 508C if both have to be used (of course at diﬀerent ﬂow rates) to heat another ﬂuid entering at 08C in a heat exchanger. The ﬁrst ﬂuid stream would have a larger exergy rate and would therefore be able to transfer more heat over a wider temperature span. In other words, its potential to do work is high and its use at the low temperature level would be wasteful if there is a second stream at a lower temperature (< 5008C) that is available to do the same job. Applied to a situation in a heat exchanger, the exergy balance can be closed only by introducing into the balance the exergy destruction d. Consequently, the exergy rate balance for a heat exchanger (Fig. 11.13) may be written as follows: d ¼ ðe_1;i e_1;o Þ þ ðe_2;i e_2;o Þ

ð11:54Þ

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

Hot fluid . with inlet temperature Th,i , pressure ph,i , enthalpy rate Ih,i = f(Th,i , ph,i ), and exergy rate Eh,i = f(Th,i , ph,i, Tref , pref )

787

T

E h,,i Tref > Th,i Heat exchange, q

E h,o = f(Th,o , ph,o, Tref , pref) Exergy destruction, D

Th,o

Ec,o = f(Tc,o, pc,o, Tref, pref) Ec,i

Th,i

Tref < Tc,i

Tc,o Tc,i

Hot fluid . with inlet temperature Tc,i, pressure pc,i , enthalpy rate Ic,i = f(Tc,i, pc,i) and exergy rate Ec,i = f(Tc,i, pc,i, Tref , pref) FIGURE 11.13 Exergy rate ﬂows through a heat exchanger.

where the two expressions in parentheses on the right-hand side of the equality represent individual exergy destruction rates for each of the two ﬂuids. The total exergy destruction rate given by Eq. (11.54), d in watts (Btu/hr), represents the quantitative energy measure of irreversibility. As demonstrated by Bosnjakovic´ (1972), this irreversibility can also be calculated from entropy generation using the Gouy–Stodola theorem as follows: d ¼ Tref S_ irr

ð11:55Þ

Equation (11.55) obviously implies a need for insight into the heat transfer thermodynamic intricacies (expressed in terms of entropy generation). We started the analysis in this chapter with the argument that insight into temperature distributions must be gained and that the heat exchanger designer should understand how the exchanger performance depends on the most important sources of irreversibility, S_ irr . Now we can clearly see that this understanding may contribute to a reduction of exergy destruction in a system. A system engineer knows how to relate that destruction directly to the monetary value of the capital investment and operating costs and to optimize the system by providing appropriate changes in heat exchanger design. By writing exergy rate balances (or calculating entropy generation), energy rate measures of irreversibilities can be determined and subsequently their monetary value calculated (see Section 11.6.6). 11.6.5

Thermodynamic Figure of Merit for Assessing Heat Exchanger Performance

Let us now introduce a thermodynamic ﬁgure of merit{ in the form of exergy eﬃciency for the performance of a heat exchanger in a system. The performance level may be {

The second law eﬃciency is deﬁned in many diﬀerent ways and they are in general referred to as the thermodynamic ﬁgure of merit.

788

THERMODYNAMIC MODELING AND ANALYSIS

deﬁned using the concept of exergy. The exergy balance for a heat exchanger, based on Eq. (11.54), states simply that exergy entering a heat exchanger (carried in by both ﬂuid streams) must be equal to the sum of exergy leaving the exchanger and the destruction of exergy caused by heat exchanger operation (as a consequence of irreversibilities), that is e_i ¼ e_o þ d where

e_i ¼ e_1;i þ e_2;i

and e_o ¼ e_1;o þ e_2;o

ð11:56Þ ð11:57Þ

The exergy balance given by Eq. (11.56) can be rearranged as follows: ðe_1;o e_1;i Þ þ d ¼ e_2;i e_2;o

ð11:58Þ

In a particular case, when ﬂuids 1 and 2 are identiﬁed as cold and hot ﬂuids, respectively, Eq. (11.58) indicates that the exergy increase in the cold ﬂuid plus the exergy destruction must be equal to the exergy decrease in the hot ﬂuid utilized in this process. Equation (11.58) in such a case can be divided by the diﬀerence in the exergy rates on the righthand side of the equality, to obtain e_c;o e_c;i d þ ¼1 e_h;i e_h;o e_h;i e_h;o

ð11:59Þ

Now deﬁne an exergy eﬃciency of a heat exchanger as ¼

e_c;o e_c;i d ¼1 _ _ _ eh;i e_h;o eh;i eh;o

ð11:60Þ

In general, the exergy eﬃciency of a heat exchanger is deﬁned as follows:

¼

8 e_c;o e_c;i > > > > < e_h;i e_h;o

for Tc;i Tref

> > e_h;o e_h;i > > : _ ec;i e_c;o

for Tc;i Tref

ð11:61Þ

The eﬃciency deﬁned by Eq. (11.61), 1, takes care of diﬀerent objectives of a heat exchanger used in a system. Namely, if the purpose of a heat exchanger is to increase the exergy rate (i.e., energy availability or energy quality) of a cold-ﬂuid stream (at the expense of a decrease of exergy rate of the hot-ﬂuid stream), is calculated using the ﬁrst expression of Eq. (11.61). An objective of heating a ﬂuid by cooling another ﬂuid is an example for the ﬁrst expression of of Eq. (11.61) (the heat source may be waste thermal energy, a heat exchanger in a heat pump, etc.). On the contrary, if the purpose of the exchanger is to cool the hot stream at or below the reference temperature (refrigeration for temperatures below the environment temperature), the second expression of Eq. (11.61) should be used. The change of design parameters may lead to diﬀerent behaviors of the heat exchanger eﬀectiveness and thermodynamic eﬃciency for the same heat exchanger. Let us demonstrate this fact by an example.

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

789

Example 11.8 Assess the following two designs for a counterﬂow heat exchanger. The heat exchanger is to heat 1 kg/s of air from 2278C (500 K) and 3 bar to 3078C (580 K). The hot-ﬂuid stream is also air at the inlet temperature of 3278C (600 K) and 2 bar. The two options available are as follows: The heat exchanger is designed for a hot-ﬂuid mass ﬂow rate of either 1.24 kg/s or 4.94 kg/s. Decide which design will yield better exergy eﬃciency in the system. The surroundings is at 300 K and 1 bar. SOLUTION Problem Data: Operating conditions: (a) cold ﬂuid is air with Tc;i ¼ 500 K, Tc;o ¼ 580 K, m_ c ¼ 1 kg=s, pc;i ¼ 3 bar; (b) hot ﬂuid is air with Th;i ¼ 600 K, ph;i ¼ 2 bar, m_ h ¼ 1:24 or 4.94 kg/s. The heat exchanger is a counterﬂow unit. The surroundings is at Tref ¼ 300 K, and pref ¼ 1 bar. Determine: Which of the two ﬂuid ﬂow rates will lead to a design with higher exergy eﬃciency? Assumptions: The assumptions are as listed in Section 3.2.1. Air is assumed to behave as an ideal gas. Pressure drops are negligible, i.e., pc;i ¼ pc;o and ph;i ¼ ph;o . Idealize the same overall heat transfer coeﬃcient in both designs. Assume surroundings to be relevant for exergy deﬁnition (i.e., at reference conditions as follows: Tref ¼ 300 K and pref ¼ 1 bar). Analysis: A study of input data reveals the following. The heat capacity rates of the two ﬂuids are not known, but we do know that they are not the same because the mass ﬂow rates are diﬀerent. The speciﬁc heats of both ﬂuids are not known. Note that the speciﬁc heat of an ideal gas is a function of the temperature only. So, to determine the heat capacity ratio, we have to know the exit temperature of the hot ﬂuid. However, the exit temperature of the hot ﬂuid cannot be determined a priori because the speciﬁc heat is not known. However, both ﬂuids are of the same type (i.e., air) and we can idealize that the speciﬁc heats at constant pressure of both ﬂuids will not diﬀer signiﬁcantly from each other. The mass ﬂow rate of the hot ﬂuid is speciﬁed as larger than that for the cold ﬂuid (i.e., m_ h ¼ 1:24 kg/s or 4:94 kg=s > m_ c ¼ 1 kg/s). Therefore, in both cases, the heat capacity rate of the hot ﬂuid will be larger than the corresponding value for the cold ﬂuid. Hence, cold air is the Cmin ﬂuid, and " ¼ "c . The heat exchanger eﬀectiveness represents in this case a dimensionless outlet temperature of the cold ﬂuid (" ¼ "c , i.e., the cold ﬂuid has a smaller heat capacity outlet rate, Cc ¼ Cmin ; see Section 11.2.2). Since both terminal temperatures of the cold ﬂuid are known and ﬁxed, the heat exchanger eﬀectiveness "c and heat transfer rate q for both designs are the same (a subject of Review Question 3.26). However, because the heat capacity rate ratios are not the same, the two designs must have diﬀerent NTUs. Diﬀerent NTUs for the same U and Cmin must correspond to diﬀerent heat transfer areas. One can arrive at the same conclusion by noting that the heat transfer rate, the inlet conditions for both ﬂuids, and the smaller heat capacity rate all are predetermined. From the deﬁnition of heat exchanger eﬀectiveness based on its thermodynamic interpretation (see Example 11.2), it becomes clear that the heat exchanger eﬀectiveness must be identical in both cases. Therefore, the change in NTU is due to diﬀerent heat transfer surface areas. Thus, the two designs considered would have the same " and q but diﬀerent A’s. So the two exchangers would extract the same heat transfer rate from the hot ﬂuid but at diﬀerent temperature levels and with diﬀerent local temperature distributions. Hence,

790

THERMODYNAMIC MODELING AND ANALYSIS

the irreversibility level (responsible for thermodynamic performance) would not be the same. The question is: Which of the two designs (the one with the larger or the one with the smaller mass ﬂow rate of the hot ﬂuid) will provide higher exergy eﬃciency? To be able to use one of the two deﬁnitions of the exergy eﬃciency of Eq. (11.61), one should recognize the purpose of the device. According to the problem formulation, the goal is to heat the cold ﬂuid at the expense of the hot ﬂuid. Therefore, the exergy eﬃciency is deﬁned by the ﬁrst of the two equations given by Eq. (11.61). In other words, the increase in the exergy rate of the cold ﬂuid will be accomplished at the expense of a decrease in the exergy rate of the hot ﬂuid: ¼ where

e_c;o e_c;i e_h;i e_h;o

ð1Þ

e_j;k ¼ m_ j ½ðhj;k href Þ Tref ðsj;k sref Þ

ð2Þ

with hj;k ¼ j;k ðair; T ¼ Tj;k Þ

0 sj;k ¼ j;k ðair; T ¼ Tj;k ; p ¼ pj;k Þ

ð3Þ

In Eqs. (2) and (3), j ¼ h or c and k ¼ i or o. The condition of the reference state (and values of stream properties at the condition of the surroundings) is denoted by ‘‘ref.’’ The design problem formulated in this example is a sizing problem in which both NTUs and one outlet temperature are unknown. The outlet temperature of the hot ﬂuid in terms of Tc;i can be determined from the deﬁnition of " and C* as follows: Th;o ¼ Tc;i ð1 C *"ÞðTc;i Th;i Þ

ð4Þ

The calculation of exergy eﬃciency must be performed numerically using Eqs. (1)–(4) because thermophysical properties depend on the temperature or on both temperature and pressure [see Eqs. (3) and (4)]. The results of the calculation of exergies and are listed below. These results are obtained using EES software (2000). Air is treated as an ideal gas. C* and " are calculated from input data and subsequently, NTU is calculated for these values of " and C* for a counterﬂow exchanger. m_ h (kg/s)

Th;o (K)

C*

"

NTU

e_c;i (kW)

e_c;o (kW)

e_h;i (kW)

e_h;o (kW)

1.24 4.94

536 584

0.8 0.2

0.8 0.8

2.94 1.79

142.1 142.1

178.9 178.9

191.7 762.5

152.6 721.6

0.941 0.899

A review of the results above indicates that the exergy eﬃciency is larger for the design that requires larger NTU (smaller mass ﬂow rate of the hot ﬂuid). Both designs are characterized by the identical heat exchanger eﬀectiveness. Discussion and Comments: The results obtained conﬁrm a statement made at the beginning of this chapter that heat exchanger (or temperature) eﬀectiveness does not necessarily provide suﬃcient information about exchanger performance. The diﬀerences between the terminal temperatures on the hot ﬂuid inlet side are the same for both exchangers but the temperature diﬀerence between the ﬂuids at the cold ﬂuid inlet side increases from 36 K (for a smaller mass ﬂow rate or a larger NTU) to 84 K (for a larger mass ﬂow rate or a smaller NTU). The results clearly demonstrate that smaller local

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

791

temperature diﬀerences along the respective ﬂow paths lead to larger exergy eﬃciency (as a consequence of the smaller entropy generation for the smaller m_ h case). Moreover, in the case of the smaller exergy eﬃciency, the exit temperature of the hot ﬂuid (584 K) is quite close to the cold-ﬂuid outlet temperature (580 K). To understand the implication, review the S* vs. NTU curve for C * ¼ 1 for a counterﬂow exchanger in Fig. 11.5. The trend for the S* vs. NTU curve for other values of C* is identical except for C * ¼ 0 and S*max is of the same order of magnitude for all C* values (Sekulic´, 1990) for a given #. Hence, for this problem, one can see that the operating point approaches from the left to S*max at a high airﬂow rate, and the operating point for a low airﬂow rate will be considerably to the right of S*max . Thus, S* and S_ irr ð¼ S *Cmax Þ will be higher for the high-airﬂowrate case. So this high-airﬂow-rate case is an unfavorable design solution from the exergy point of view. An engineer’s ultimate decision, though, will also depend on a number of additional considerations (pressure drop considerations, cost analysis, etc.). These considerations may include a trade-oﬀ between the physical size of the heat exchanger (cost of exchanger) and the mass ﬂow rate of the hot ﬂuid (operating cost), or alternately, the minimum cost of thermodynamic irreversibility, as discussed in the next subsection. 11.6.6

Accounting for the Costs of Exergy Losses in a Heat Exchanger

With the thermodynamic background developed so far, we can now point out how to relate an exergy ﬂow rate to the cost rate. The heat exchanger transforms the total exergy rate input, carried in by both streams, into an exergy rate output that is always smaller than the one at the input due to the inevitable presence of irreversibilities (caused by heat transfer rate at ﬁnite temperature diﬀerences, ﬂuid mixing, friction phenomena, etc). Each exergy rate ( j streams into and out of the heat exchanger) has an associated cost rate C_ j as C_ j ¼ Cj e_j

ð11:62Þ

where Cj denotes the cost per unit of exergy (c=J). The cost per unit of exergy of ﬂuid ﬂows that enter a heat exchanger has to be determined before an exergy costing can be performed. For example, if a particular ﬂuid stream has to be utilized in a system, a designer must know the cost of the utilization of that stream. This cost has to be determined from an analysis located ‘‘upstream’’ from the location of the stream utilization (Bejan et al., 1995). Now we can write a cost rate balance for a heat exchanger: X X ð11:63Þ C_ j;i C_ j;o ¼ Z_ cap þ Z_ op j¼h;c

j¼h;c

In Eq. (11.63), Zcap and Zop denote the capital investment cost and operating expenses{ of a heat exchanger that must be balanced by a diﬀerence between the sum of the cost rates of all inlet ﬂuid ﬂows and the cost rates of all outlet ﬂuid ﬂows. It should be noted that cost balancing as presented by Eq. (11.63) has to be performed for a heat exchanger as a component in a system. Note also that cost rates do not obey the conservation principle. {

For simplicity, we ignore here the installation cost, which can be comparable to capital and operating costs for some shell-and-tube heat exchangers and PHEs, and can be added as a third term on the right-hand side or can be included in the capital cost..

792

THERMODYNAMIC MODELING AND ANALYSIS

Before one attempts to perform an optimization, a careful utilization of a commonsense second law of thermodynamics approach to the design of the entire system has to be conducted (Sama et al., 1989; Sama, 1995b). This approach is based on a selection of thermodynamic rules such as the ones given in the summary of this chapter based on the results of Section 11.3 and this section for a heat exchanger design. If input cost rates of the form of Eq. (11.62) are known, and if both the capital investment and operating expenses are known, Eq. (11.63) can be used directly to determine the output cost rate of a stream. Once determined, a cost rate balance can be combined with the cost rate balances of other components and used to formulate an objective function (Bejan et al., 1995) for a system. Such an objective function combines thermodynamics and economics, exploiting the concepts of exergy and/or entropy generation for a system; that is the reason this branch of engineering is referred to as thermoeconomics (Bejan et al., 1995). Equations (11.62) and (11.63) clearly show how the cost rate balance can be related to the exergy rate balances (assigning a dollar sign to exergy ﬂows) for a heat exchanger as a component of a system. Determination of cost per unit exergy ﬂow is diﬃcult and a single cost model could not be applied. This complex thermoeconomic topic is beyond the scope of our presentation. A practical methodology to account for the cost of irreversibilities in a heat exchanger as a stand-alone unit has been developed by London (1982). This methodology has at its core an identiﬁcation of the individual costs of irreversibilities incurred during heat exchanger operation, which is in fact related to both the capital (initial design) and operating costs. This is because some small irreversibilities may be very expensive (such as airﬂow friction irreversibility for an automotive radiator), and some large irreversibilities may be less expensive (such as liquid-side friction irreversibility in a gas–liquid heat exchanger). London’s approach does not use the exergy rates explicitly but does use the exergy destruction rate, referred to as the energy measure of the irreversibilities. Individual energy measures of irreversibility are determined by multiplying individual entropy generation rates with a temperature-weighting factor, Tref . Total irreversibility destruction is related to total entropy generation through the relationship given by Eq. (11.55). Details of London’s procedure are also presented by London and Shah (1983). With the thermodynamic background developed so far, we can outline a procedure to evaluate in monetary terms the various irreversibilities in a heat exchanger as shown in Fig. 11.14. For example, ﬁve irreversibilities of signiﬁcance in a heat exchanger may be the ﬁnite temperature diﬀerence between hot and cold ﬂuids, ﬂuid mixing at the exchanger ports (if applicable), pressure drops on the hot and cold sides, and irreversibility associated with heat leakage between the exchanger and the environment. The total energy measure of irreversibility normalized by the heat exchanger duty is then given by d I_irr I_ I_ I_ I_ þ irr þ irr þ irr ð11:64Þ ¼ þ irr q T q mixing q p;h q p;c q leak q where I_irr;i ¼ Tref S_ irr;i , with the subscript i representing individual irreversibilities of Eq. (11.64). These are summarized in Table 11.3 for an ideal gas or an incompressible liquid, all in measurable system operation quantities (temperatures, pressures, mass ﬂow rates, etc.). These irreversibilities have an energy monetary value that is dependent on the system in which a particular heat exchanger is used. Assigning a monetary value to irreversibility is analogous to assigning a cost rate to exergy rates as in Eq. (11.62). Once the monetary values or costs of various irreversibilities are determined, the analyst

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

793

FIGURE 11.14 Thermodynamic optimization of a heat exchanger, including the development of trade-oﬀ factors. (From London, 1982.)

is in the position of deciding which particular irreversibilities are most costly and should ﬁrst be reduced for a cost-eﬀective heat exchanger. Hence, the industrial approach to the design of a heat exchanger is to reduce the most costly irreversibilities rather than reducing all irreversibilities in a heat exchanger [i.e., not to minimize d of Eq. (11.64), but to minimize only those irreversibilities on each ﬂuid side that are the most expensive]. Note that reducing one irreversibility may increase or require the addition of another, or involve an increase in capital investment. These considerations lead to the development of trade-oﬀ factors, useful criteria for arriving at an optimum heat exchanger design as a component. Thus, coming back to Fig. 11.14, the design and optimization of a heat exchanger involves the process of evaluating various irreversibilities on an entropy basis and then converting them into an energy basis (either in the form of exergy ﬂows or energy measures of irreversibility), assigning monetary values, minimizing the most costly irreversibilities and as a result developing trade-oﬀ factors, and continuing this process until the optimum heat exchanger is developed. To understand this process, London (1982) and London and Shah (1983) have provided a detailed example of a condenser in a thermal power plant with a clear demonstration of the accounting procedure above, including how to develop the trade-oﬀ factors. Problems 11.17 and 11.18 are based on this example. Even more intricate problems of second-law-based thermoeconomic optimization of exchangers (including both single- and two-phase heat exchangers) are proposed by Zubair et al. (1987), estimating the economic value of entropy generation in the heat exchanger caused by ﬁnite temperature diﬀerences and pressure drop. That method permits an engineer to trade the cost of entropy generation on each ﬂuid side of the heat exchanger against its capital expenditure. Now let us relate the cost rate analysis for a system [Eqs. (11.62) and (11.63)] with a similar analysis based on minimizing the most costly irreversibilities using the approach of Fig. 11.14 and the associated Eq. (11.64). The exergy cost analysis of a system, based on Eq. (11.63), includes all irreversibilities present individually within all components [say, a compressor, a condenser, an expansion device, an evaporator, and an accumulator/dehydrator in an automotive air-conditioning system, for which cost rates are calculated using Eq. (11.62)] if the ﬂuid variables are determined at the boundaries of each component and each component is considered to be a black box. Hence, optimization of the system is performed by taking into account the irreversibility losses of all components without distinguishing various components of the total irreversibility in

794

Source: Data from London and Shah (1983).

Heat leak

Pressure drops on hot and cold ﬂuid sides

Fluid mixing (exchanger outlet into environment)

Finite temperature diﬀerence

j ¼ c or h

Hot fluid:

Tref 1 m_ p Tref 1 m_ p I_irr I_irr ¼ ; cold fluid: ¼ q p;h Th;lm q q p;c Tc;lm q h c qleak Tref qleak Tref I_irr I_irr ; from environment: To environment: ¼ ¼ 1 1 q leak q Th q leak q Tc

Tj;i Tj;o 1 1 I_irr where Tj;lm ¼ ¼ Tref q T lnðTj;i =Tj;o Þ Tc;lm Th;lm Tref ðm_ cp Þ Tc;o Tc;o I_irr ¼ 1 ln q mixing Tref Tref q

TABLE 11.3 Normalized Energy Measure of Various Irreversibilities in a Heat Exchanger

ENERGY, EXERGY, AND COST BALANCES OF HEAT EXCHANGERS

795

each component. Equation (11.64) is used to minimize the most costly irreversibilities to produce a cost-eﬀective heat exchanger, and it does not necessarily lead to an optimum design of the system as a whole. So a system-based optimization would use an objective function based on Eq. (11.63), while a heat exchanger component optimization would be based on an analysis of individual irreversibilities. It should be noted that the cost rates in Eq. (11.63) can be calculated using Eq. (11.62) in such a way as to determine exergy rates [using Eq. (11.53)] by excluding from the entropy change all less costly irreversibility contributions. In such a way, the system-based optimization will include only the most costly irreversibilities in the same manner as would be done by using an individual irreversibility calculation for each component. However, the opposite is not possible; an optimization based on Eq. (11.64) only cannot provide an optimum system as a whole. Example 11.9 Determine the annual cost of exergy destruction in a heat exchanger caused by ﬁnite temperature diﬀerences. The heat exchanger heats a cold ﬂuid stream having a heat capacity rate of 5:8 106 W=K from 198C (292 K) to 278C (300 K). The hot ﬂuid reduces its temperature from 308C (303 K) to 258C (198 K). The reference temperature (surroundings) is 178C (290 K). The annual capital cost of the equipment involved is 103 c=kW of energy at 10% annual interest rate. The average yearly energy cost is 150 c=kW yr. SOLUTION Problem Data and Schematic: The terminal temperatures of the ﬂuids entering and leaving the heat exchanger and the cold-ﬂuid heat capacity rate are as follows: Tc;i ¼ 292 K, Tc;o ¼ 300 K, Th;i ¼ 303 K, Th;o ¼ 298 K, and Cc ¼ 5:8 106 W=K. The reference surrounding temperature is To ¼ 290 K (178C). The annual capital cost of the equipment is Ceqp ¼ 1000 c=kW and the annual interest rate r ¼ 0:1/yr. The average annual cost of energy is Cq ¼ 150 c=kW yr. See Fig. 11.13 as a representative sketch. Determine: The annual cost of exergy destruction in the exchanger caused by ﬁnite temperature diﬀerences. Assumptions: All appropriate assumptions of Section 3.2.1 are invoked here. Analysis: The exergy destruction is given by Eq. (11.55): d ¼ Tref S_ irr

ð1Þ

Also, according to Eq. (11.64), taking into account only the ﬁnite temperature diﬀerence contribution d I_irr ð2Þ ¼ q q T

In other words, the energy measure of irreversibility in the heat exchanger caused by ﬁnite temperature diﬀerences is equal to corresponding exergy destruction since no other irreversibility contributions are included. The entropy generation caused by ﬁnite temperature diﬀerences is given by Eq. (11.30): Tlm;h Tlm;c S_ irr ¼ q Tlm;h Tlm;c

ð3Þ

796

THERMODYNAMIC MODELING AND ANALYSIS

where Tlm; j ¼

Tj;o Tj;i lnðTj;o =Tj;i Þ

ð4Þ

Equation (4) provides the magnitudes of the log–mean temperatures of the two ﬂuids (equal to the logarithmic mean between the inlet and outlet temperatures of each ﬂuid) with j ¼ h and c. Inserting problem data into Eq. (4), we get Tlm;h ¼

Th;o Th;i ð298 303Þ K ¼ ¼ 300:5 K ln Th;o =Th;i lnð298=303Þ

Tlm;c ¼

Tc;o Tc;i ð300 292Þ K

¼ ¼ 296:0 K lnð300=292Þ ln Tc;o =Tc;i

Entropy generation is, from Eq. (3), 1 1 S_ irr ¼ Cc ðTc;o Tc;i Þ Tlm;c Tlm;h ¼ ð5:8 106 W=KÞ ð300 292Þ K

1 1 296:0 K 300:5 K

¼ 2347 W=K

The exergy destruction caused by ﬁnite temperature diﬀerences, from Eq. (1), is d ¼ Tref S_ irr ¼ 290 K 2347 W=K ¼ 0:681 106 W Now, keeping Eq. (2) in mind and taking the cost data into account, we can calculate the cost of this exergy destruction: C ¼ dðCeqp r þ Cq Þ ¼ 0:681 106 W ð1 0:1 þ 0:15Þðc=W yrÞ ¼ 0:17 106 c=yr Discussion and Comments: This example illustrates how one can determine in a simpliﬁed way a cost value of the lost work (exergy destruction) caused by irreversibilities in a heat exchanger. In this example, only the ﬁnite temperature diﬀerence irreversibility contribution is considered. Problems 11.17 and 11.18 include all other relevant irreversibility contributions for a representative heat exchanger application. 11.7 PERFORMANCE EVALUATION CRITERIA BASED ON THE SECOND LAW OF THERMODYNAMICS{ All performance evaluation criteria for heat transfer surfaces presented in Section 10.3.2 are based on the ﬁrst law of thermodynamics (i.e., energy and mass balances). In a diﬀerent approach to a performance evaluation criterion, we introduce the thermodynamic quality of heat transfer and ﬂuid ﬂow processes to evaluate the heat transfer surface performance. Such an evaluation requires an assessment of the irreversibility level of heat transfer and ﬂuid ﬂow phenomena. Hence, a combined ﬁrst and second law of { The body of knowledge usually called the second law of thermodynamics analysis always involves both the ﬁrst and second laws of thermodynamics. However, it is customary to name the product of such an analysis by indicating the second law of thermodynamics only.

EVALUATION CRITERIA BASED ON THE SECOND LAW OF THERMODYNAMICS

797

thermodynamics analysis becomes necessary. Here, we emphasize only the possible formulation of a PEC that reﬂects these considerations. For details of this approach and a review of the literature, refer to Bejan (1988). The second law of thermodynamics performance evaluation criteria are based on objective functions that include both heat transfer and pressure drop (ﬂuid friction) irreversibilities and hence gauge the combined eﬀect of these irreversibilities. Separation of the two irreversibilities can subsequently be performed if necessary. Note that heat transfer and pressure drop irreversibilities should be translated into costs separately because the unit costs of these irreversibilities are in general not equal. The entropy rate balance for a duct/channel control volume (i.e., a ﬂow passage of a heat exchanger) deﬁnes total entropy generation between the ﬂuid inlet and outlet as follows: S_ irr ¼ S_ irr;T þ S_ irr;p ¼ S_ irr;T ð1 þ Þ

ð11:65Þ

where the S_ irr;T and S_ irr;p terms denote the contributions to overall entropy generation incurred by either ﬁnite temperature diﬀerence between the ﬂuid and the wall (T) or pressure drop (p) (see Problem 11.19). The irreversibility distribution ratio, ð¼ S_ irr;p =S_ irr;T Þ expresses, by deﬁnition, the trade-oﬀ between the two contributions. An explicit form of Eq. (11.65) depends on heat transfer and ﬂuid ﬂow conditions and idealizations (e.g., boundary conditions, free-ﬂow area geometry, ﬂow regime, selection of dimensionless parameters). One such expression in terms of dimensionless parameters for a constant-cross-section duct and constant-property ﬂuid can be written as S* ¼

Nq2 ðDh =LÞ 4j Pr2=3

þ

2f Ec Dh =L

ð11:66Þ

where the dimensionless heat transfer rate Nq and Eckert number Ec are deﬁned as Nq ¼ q=ðm_ cp Tm Þ and Ec ¼ u2 =cp Tm Jgc , respectively. Equation (11.66) represents a dimensionless form of the corresponding entropy generation expression derived by Bejan (1988). These dimensionless groups are conveniently deﬁned using ﬂuid bulk mean temperature Tm [physical modeling involving the derivation of Eq. (11.66) is the subject of Problem 11.20]. Note that the dimensionless group Ec is usually deﬁned based on a temperature diﬀerence, not the bulk temperature. Such a deﬁnition can easily be introduced in Eq. (11.66), leading to the introduction of an additional temperature ratio parameter. For simplicity, the Ec number is deﬁned as given above. Equation (11.66) assumes constant and known heat transfer and mass ﬂow rates. For a speciﬁed constant mass ﬂow rate m_ and given duct length L, a change in the hydraulic diameter Dh causes a change in S*. From the algebraic structure of Eq. (11.66) and Reynolds analogy (see Section 7.4.5), it becomes clear that the two terms on the right-hand side of Eq. (11.66) have opposite trends with respect to a change in Re (which aﬀects j and f factors) or the hydraulic diameter. Thus, this objective function [i.e., Eq. (11.66)] may have an extremum (a minimum.) It has been recognized that what is good for the reduction of friction irreversibility (by decreasing surface area) is apparently bad for the reduction of ﬁnite temperature irreversibility (i.e., for an increase in exchanger eﬀectiveness), and vice versa. An optimum trade-oﬀ between these two inﬂuences may exist. Consequently, such an objective function may be used as a basis for deﬁning a new thermodynamic performance evaluation criterion. An optimum geometry (or ﬂow regime) can ultimately be deﬁned for a given selection of characteristic parameters. We can illustrate this approach by the following example.

798

THERMODYNAMIC MODELING AND ANALYSIS

Example 11.10 Thermoeconomic optimization of a large energy system requires minimization of irreversibility costs of a plant. Within the scope of that analysis, a heat exchanger designer must decide which geometry of a heat exchanger passage cross section would contribute the least to the overall irreversibility. The length and freeﬂow area of the duct representing the passage are known and ﬁxed, as well as the ﬂuid (air) inlet thermal state and mass ﬂow rate. Temperature of the heating ﬂuid is constant and equal to 1008C (373 K). The wall thermal resistance can be neglected. Duct geometry options include (1) square, (2) rectangular (aspect ratio * ¼ 18), and (3) circular cross sections. Determine which cross section would be best from the point of view of irreversibility minimization. Compare the ﬁndings with an analysis of the magnitude of heat transfer and pressure drop for each of the duct shapes. The following data are available. Free-ﬂow area of the cross section is 5 103 m2 , the duct length 5 m, the mass ﬂow rate of air 5 102 kg=s, and the ﬂuid inlet temperature 300 K. The thermophysical properties of the air are as follows: density, 1.046 kg/m3; speciﬁc heat at constant pressure, 1.008 kJ/kg K; dynamic viscosity, 2:025 105 Pa s; Prandtl number, 0.702; and thermal conductivity, 2:91 102 W=m K. SOLUTION Problem Data and Schematic: The duct geometries for this problem are shown below along with the input data for the problem.

.

Ao = 5×10–3 m2, L = 5 m, m = 5×10–2 kg/s, Ta,i =300 K, Th,i = Th,o = 373 K ρ = 1.046 kg/m3, cp = 1,008 kJ/kg K, µ = 2.025×10–5 Pa.s, Pr = 0.702, k = 2.91x10–2 W/m .K

FIGURE E11.10 Square, rectangular and circular duct geometries.

Determine: Which of the three cross-sectional shapes [square, rectangular ( * ¼ 18), or circular] will lead to minimum heat transfer and ﬂow friction entropy generation? Assumptions: The ﬂow regime is assumed fully developed and the thermophysical properties are constant. Heat transfer from the duct to the ﬂuid is accomplished at a uniform rate across the assumed temperature diﬀerence between the wall and bulk mean temperature (determined as an arithmetic mean of the inlet and outlet temperatures). The temperature of the wall is uniform, constant, and equal to the temperature of the heating ﬂuid (assume that the ﬂuid is changing its phase at a constant temperature of 373 K). Consequently, the wall thermal resistance is neglected. Analysis: We determine the dimensionless entropy generation associated with heat transfer and ﬂuid friction, Eq. (11.66), for each duct and then compare the results to determine which geometry is most favorable. The calculation of heat transfer and pressure drop characteristics is summarized in Table E11.10. For details of some calculations for the square and rectangular ducts see Example 7.5. Entropy generation can subsequently be calculated in dimensionless form by Eq. (11.66), or in the corresponding dimensional

EVALUATION CRITERIA BASED ON THE SECOND LAW OF THERMODYNAMICS

799

TABLE E11.10 Thermal and Hydraulic Characteristics and Entropy Generation of Various Ducts Variable or Parameter

Equation

Square

4Ao =P m_ =Ao GDh = Nusq ¼ Nurect ¼ 0:024Re0:8 Pr0:4 Nucirc ¼ 0:023Re0:8 Pr0:4 j ¼ St Pr2=3 ¼ Nu Pr1=3 =Re kNu=Dh PL hA=m_ cp 1 eNTU q ¼ "m_ cp ðTw Ta;i Þ Ta;i þ q=m_ cp ðTa;i þ Ta;o Þ=2 frect ¼ 0:0791Re0:25 ð1:0875 0:1125 *Þ fcirc ¼ 0:00128 þ 0:1143Re1=3:2154 p ¼ 4fLG2 =2gc Dh q=m_ cp Tm u2 =cp Tm Jgc Nq2 ðDh =LÞ=4j Pr2=3 2f Ec=ðDh =LÞ * þ Sp * ST

Dh (m) G (kg/m2 s) Re Nu j h (W/m2 K) A (m2) NTU " q (W) Ta;o ð8CÞ Tm ð8CÞ f pðPaÞ Nq Ec * ST * Sp S*

Rectangular 2

Circular

7:071 10 10 34,919 89.78

4:444 10 10 21,946 61.92

2

7:979 102 10 39,402 94.77

2:893 103 36.94 1.4142 1.038 0.6458 2,376 74.1 50.6 5:642 103

3:174 103 40.55 2.25 1.83 0.8396 3,089 88.3 57.7 6:976 103

2:706 103 34.56 1.25 0.87 0.5810 2,138 69.4 48.2 5:534 103

76.3 0.1456 2:799 104 2:043 102 2:240 104 2:065 102

150.0 0.1852 2:738 104 1:899 102 4:292 104 1:942 102

66.3 0.1320 2:820 104 2:021 102 1:963 104 2:041 102

form (per unit of the duct length) as follows: S_ irr ðq=LÞ2 Dh 2m_ 2 f þ 2 2 ¼ 2 L 4Tm m_ cp St Ao cp Tm Jgc Dh From the last line of Table E11.10 it is obvious that the rectangular duct generates the minimum entropy. Therefore, the rectangular duct would be considered as most favorable from an entropy-generation point of view. Discussion and Comments: To understand the inﬂuence of various sources of irreversibility on duct performance, the results of the analysis are summarized below. For the three duct geometries, the following quantities are compared: heat transfer rate, pressure drop, dimensionless entropy generation caused by temperature diﬀerence, dimensionless entropy generation caused by pressure drop, and total dimensionless entropy generation. The results are presented qualitatively in terms of the highest heat transfer rate, the lowest pressure drop, and the lowest entropy generations, all marked with the symbol :: ^ . The lowest heat transfer rate, the highest pressure drop, and the highest irreversibility :: :: _ . Finally, the medium values are marked . are each marked with the symbol Variable or Parameter q p S*T * Sp S*

Square

Rectangular

Circular

:: :: :: _ :: : : _

:: ^ :: _ :: ^ :: _ :: ^

:: _ :: ^ :: : : ^ : :

800

THERMODYNAMIC MODELING AND ANALYSIS

It is obvious that the rectangular duct has the best performance in terms of both heat transfer rate and irreversibility level. The penalty for high heat transfer performance is paid by a very high pressure drop. It is interesting to note that the worst geometry from the heat-transfer-rate point of view is the circular duct, but from the entropy-generation point of view, it is a square duct. The reason for that is related to the diﬀerent orders of magnitude of the entropy generation caused by temperature diﬀerence and pressure drop. If the ultimate goal is to reach the minimum entropy generation for a given geometry, say a circular duct, an optimization based on the objective function given by Eq. (11.66) has to be performed. Solving ð@S*Þ=@Re will lead to the determination of an optimum Re, or the optimum duct hydraulic diameter for a given mass ﬂow rate. For the circular duct of this problem, this optimum diameter of the duct (the hydraulic diameter considered as the only degree of freedom, the other variables ﬁxed) is found as very large: 135 mm. The optimum trade-oﬀ between heat transfer and friction (pressure drop) irreversibilities may or may not exist. Also, the existence of a minimum of entropy generation is not always present for the range of parameters selected. It must also be emphasized that optimization based on a thermodynamic criterion cannot be a goal per se in a design eﬀort related to an isolated heat exchanger. Usually, the ultimate goal is the minimization of cost within the framework of a system analysis. SUMMARY In this chapter, several interdisciplinary issues are discussed. These include (1) the use of energy and mass balances (ﬁrst law of thermodynamics only) and mathematical modeling to obtain temperature distributions and temperature diﬀerence distributions in various ﬂow arrangements, (2) the application of the ﬁrst and second laws of thermodynamics (combined) to identify irreversibility sources that have a detrimental eﬀect on heat exchanger performance, (3) a heuristic approach to the assessment of heat exchanger eﬀectiveness, (4) thermodynamic analysis, including exergy and thermoeconomic accounting for heat exchanger optimization, and (5) performance evaluation criteria based on the minimization of entropy generation. Our goal has been to understand why certain heat exchanger designs would lead to a higher or lower eﬀectiveness and/ or thermodynamic (exergy) eﬃciency than would a similar one, and to develop the skills needed for an approach to optimization of a heat exchanger as part of a system. Finite temperature diﬀerences, ﬂuid mixing, and ﬂuid friction are important irreversible phenomena associated with exchanger performance. Details on these irreversibilities are presented in the text. The most important guidelines for the design of a heat exchanger as a component in a system are as follows. An approach to optimum design must be based on sound engineering judgment, along with utilization of a commercial or proprietary software (if any) with understanding. Such an approach should be performed utilizing not only the energy balances (implied by the ﬁrst law of thermodynamics), but also entropy-generation balances (implied by the combined ﬁrst and second laws of thermodynamics). An exergy balance or an entropy-generation calculation has to be accompanied by economic evaluations. The components of exergy balances or total entropy-generation rates should have assigned monetary values. Exergy cost balances may be used to deﬁne an objective function in a search for the optimum design of the system in which the analyzed heat exchanger is a component. The methodology outlined in Fig. 11.14 should be adopted for optimization of a heat exchanger in a system if the minimization of the most costly

801

irreversibilities is considered. Trade-oﬀ factors are also developed in this methodology as part of the optimization procedure. Based on the analysis presented throughout this chapter, a set of guidelines important for an assessment of a heat exchanger as a component in a system can be deﬁned. These include the following: . The thermodynamic driving potential (local temperature diﬀerences between the ﬂuids) should be reduced as much as possible. . Fluid mixing within the exchanger or at exchanger terminal ports (e.g., tanks, headers, etc.) has to be avoided whenever possible. . Fluid streams in a heat exchanger network exchanging heat have to be matched beyond the temperature pinch (i.e., the point in the temperature enthalpy rate diagram where the composite curves are closest to each other). That means that one should not transfer heat across the pinch (cold and hot utilities should be used only below and above the pinch, respectively). . Fluid streams have to be balanced as much as possible (i.e., C* should have a value close to unity in an exchanger) for minimum irreversibilities. . Fluid friction, throttling, and all the other inherently irreversible phenomena should be minimized. . High temperatures (compared to the reference thermodynamic state) and large mass ﬂow rates of the ﬂuid streams should be avoided if a ﬂuid stream having lower operational variables can be utilized (to minimize the exergy losses in the system). In other words, do not use the large thermal potentials if not needed. However, if heat recovery is desired from an available high-temperature stream, avoid dilution of that stream with a colder stream if possible. If the high-temperature stream can be used without adding cost for high-temperature materials, it is better to use the undiluted stream, since that will preserve more of the exergy in the high-exergy stream. In conclusion, it should be noted that thermodynamic irreversibilities cause substantial deterioration of the performance level of a heat exchanger. They can never be eliminated, but should always be assessed, and if cost eﬀective, should be minimized.

REFERENCES Ahrendts, J., 1980, Reference states, Energy, Vol. 5, pp. 667–677. Alt-Ali, M. A., and D. J. Wilde, 1980, Optimal area allocation in multistage heat exchanger systems, ASME J. Heat Transfer, Vol. 107, pp. 199–201. Bejan, A., 1988, Advanced Engineering Thermodynamics, Wiley, New York. Bejan, A., G. Tsatsaronis, and M. Moran, 1995, Thermal Design and Optimization, Wiley, New York. Bosnjakovic, F., 1965, Technical Thermodynamics, Holt, Rinehart and Winston, New York (6th German ed., Technische Thermodynamik, Steinkopf, Dresden, 1972). EES, 2000, F_Chart Software, Engineering Equation Solver, Middleton, WI. Gregorig, R., 1965, Exergieverluste der Wa¨rmeaustauscher (Exergy Losses in a Heat Exchanger), Teil 1, Reibung (Friction), Chem. Ing. Techn. Vol. 37, pp. 108–116; Teil 2, Endlichen Temperaturunterschiedes (Finite Temperature Diﬀerences), pp. 524–527.

802

THERMODYNAMIC MODELING AND ANALYSIS

Gunderson, T., and L. Naess, 1988, The synthesis of cost optimal heat exchanger networks: an industrial review of the state of the art, Comput. Chem. Eng., Vol. 12, pp. 503–530. Kays, W. M., and A. L. London, 1998, Compact Heat Exchangers, reprint 3rd ed., Krieger Publishing, Malabar, FL. Kmecko, I., 1998, Paradoxical Irreversibility of Enthalpy Exchange in Some Heat Exchangers, M.S. thesis, University of Novi Sad, Novi Sad, Yugoslavia. Kotas, T. J., 1995, The Energy Method of Thermal Plant Analysis, Krieger Publishing, Melbourne, FL. Linnhoﬀ, B., D. W. Townsend, D. Balard, G. F. Hewitt, B. E. A. Thomas, A. R. Guy, and R. H. Marshand, 1982, User Guide on Process Integration for Eﬃcient Use of Energy. Institution of Chemical Engineers and Pergamon Press, Oxford. London, A. L., 1982, Economics and the second law: an engineering view and methodology, Int. J. Heat Mass Transfer, Vol. 25, pp. 743–751. London, A. L., and R. K. Shah, 1983, Costs of irreversibilities in heat exchanger design, Heat Transfer Eng., Vol. 4, No. 2, pp. 59–73; discussion by W. Roetzel, in Vol. 5, No. 3–4, 1984, pp. 15, 17, and Vol. 6, No. 2, 1985, p. 73. Moran, M. J., and H. N. Shapiro, 1995, Fundamentals of Engineering Thermodynamics, Wiley, New York. Peters, M. S., and K. D. Timmerhaus, 1980, Plant Design and Economics for Chemical Engineers, McGraw-Hill, New York. Sama, D. A., 1995a, Diﬀerences between second law analysis and pinch technology, J. Energy Resour. Technol., Vol. 117, pp. 186–191. Sama, D. A., 1995b, The use of the second law of thermodynamics in process design, J. Energy Resour. Technol., Vol. 117, pp. 179–185. Sama, D. A., S. Qian, and R. Gaggioli, 1989, A common-sense second law approach for improving process eﬃciencies, Proc. Int. Symp. Thermodynamic Analysis and Improvement of Energy Systems, Beijing, International Academic Publishing, Pergamon Press, New York, pp. 520–532. Shah, R. K., and Skiepko, T., 2002, Entropy generation extrema and their relationship with heat exchanger eﬀectiveness – number of transfer units behavior for complex ﬂow arrangements, Heat and Mass Transfer 2002, Proc. 5th ISHMT-ASME Heat Mass Transfer Conf., Tata McGrawHill., New Delhi, India, pp. 910–919. Sekulic´, D. P., 1990a, A reconsideration of the deﬁnition of a heat exchanger, Int. J. Heat Mass Transfer, Vol. 33, pp. 2748–2750. Sekulic´, D. P., 1990b, The second law quality of energy transformation in a heat exchanger, ASME J. Heat Transfer, Vol. 112, pp. 295–300. Sekulic´, D. P., 2000, A uniﬁed approach to the analysis of unidirectional and bi-directional parallel ﬂow heat exchangers, Int. J. Mech. Eng. Educ., Vol. 28, pp. 307–320. Sekulic´, D. P., 2003, A heuristic approach to an assessment of heat exchanger eﬀectiveness, to be published in Int. J. Mech. Eng. Education, Vol. 31. Shah, R. K., and A. Pignotti, 1989, Basic Thermal Design of Heat Exchangers, Report Int-8513531, National Science Foundation, Washington, DC. Sontag, R. E., and G. J. Van Wylen, 1982, Introduction to Thermodynamics, Wiley, New York. Zubair, S. M., P. V. Kadaba, and R. B. Evans, 1987, Second-law-based thermoeconomic optimization of two-phase heat exchangers, ASME J. Heat Transfer, Vol. 109, pp. 287–294.

REVIEW QUESTIONS Where multiple choices are given, circle one or more correct answers. Explain your answers brieﬂy.

REVIEW QUESTIONS

803

11.1

Circle the following statements as true or false and provide detailed reasons. (a) T F A workable solution of a heat exchanger design problem requires neither irreversibility analysis nor economic analysis. (b) T F The true meaning of the concept of temperature eﬀectiveness cannot be derived without invoking explicitly the second law of thermodynamics.

11.2

Optimization of a heat exchanger as an isolated component always makes sense. (a) true (b) false (c) The answer depends on the system application.

11.3

An ideal heat exchanger operates under the following conditions: (a) " ¼ 1 (b) S_ irr ¼ 0 and C * ¼ 1 _ * (d) # ¼ 0 (c) Sirr ¼ 0 and C ¼ 0

11.4

Temperature distributions of the two ﬂuids in a mixed–mixed crossﬂow arrangement are: (a) both two-dimensional (b) both one-dimensional (c) one two-dimensional and the other one-dimensional

11.5

The existence of an internal temperature cross implies the existence of an external temperature cross. (a) always true (b) always false (c) true only for some ﬂow arrangements

11.6

A heat exchanger with equal exit temperatures of the two ﬂuids is characterized with: (a) minimum entropy generation (b) maximum entropy generation (c) minimum heat exchanger eﬀectiveness (d) maximum heat exchanger eﬀectiveness

11.7

The concept of exergy is based on: (a) application of the ﬁrst law of thermodynamics only (b) utilization of energy balances only (c) application of both the ﬁrst and second laws of thermodynamics (d) application of the second law of thermodynamics only

11.8

Circle the following statements as true or false. Provide detailed reasons. (a) T F Reduction of local temperature diﬀerences between the ﬂuids in a heat exchanger always contributes positively to the decrease in the system total entropy generation. (b) T F The presence of mixing in a heat exchanger causes deterioration of heat exchanger performance compared to that of an exchanger without mixing (all other design parameters remaining the same). (c) T F An initially hot ﬂuid in thermal contact with an initially cold ﬂuid in a heat exchanger can become (locally, anywhere within the heat exchanger) colder than the other ﬂuid.

804

11.9

THERMODYNAMIC MODELING AND ANALYSIS

(d) T

F

(e) T

F

(f) T

F

(g) T

F

(h) T

F

(i) T

F

(j) T

F

For identical overall NTU and C*, two identical exchangers in series coupling will yield an overall eﬀectiveness higher than that for a similar pair of identical exchangers in parallel coupling. In a parallel-coupled arrangement of two exchangers, the overall exchanger eﬀectiveness will be higher if the Cmax ﬂuid is in the series (and Cmin ﬂuid in parallel) compared to the Cmin ﬂuid being in series. An external temperature cross can occur for any ﬂow arrangement only if there is an internal temperature cross. The lower the mean temperature diﬀerence (MTD), the lower is the counterﬂow exchanger eﬀectiveness. Fluid mixing decreases irreversibility and results in a decrease in exchanger eﬀectiveness. As NTU increases, the irreversibility increases and hence the exchanger eﬀectiveness increases in a counterﬂow exchanger. For all exchanger ﬂow arrangements and C* > 0, as NTU increases, the exchanger eﬀectiveness monotonically increases.

The total entropy on hot plus cold ﬂuid sides in an industrial exchanger increases due to: (a) heat transfer in the exchanger (b) pressure drops in the exchanger (c) leakage of hot ﬂuid to cold ﬂuid (d) fouling (e) all of these (f) none of these

PROBLEMS 11.1

Consider a 1–2 TEMA J shell-and-tube heat exchanger. Deﬁne the analytical model (the set of diﬀerential equations and boundary conditions) that fully describes the temperature distributions for both tube and shell ﬂuids. Assume that all ﬂuid properties, process parameters, and heat exchanger dimensions are known.

11.2

The problem of predicting local temperatures and/or temperature diﬀerences along ﬂuid stream paths in a plate-ﬁn crossﬂow heat exchanger requires modeling of ﬂuid temperature ﬁelds. Temperature ﬁelds for such a crossﬂow heat exchanger with both ﬂuids unmixed throughout the exchanger core obey a model described with the set of equations as follows: @1 þ 1 ¼ 2 @

@2 þ 2 ¼ 1 @

with the boundary conditions 1 ð0; Þ ¼ 1

2 ð; 0Þ ¼ 0

where dimensionless variables and the coordinates are as follows: j ¼ ðTj T2;i Þ= ðT1;i T2;i Þ, ¼ ðx=L1 ÞNTU, and ¼ ðy=L2 ÞC* NTU (see Section 11.2.4 for

PROBLEMS

805

details). Determine temperature distributions j as explicit functions of dimensionless coordinates and . 11.3

Temperature distributions of both ﬂuids in a crossﬂow heat exchanger with ﬂuid 1 mixed throughout and ﬂuid 2 unmixed are deﬁned by the following relationships [see Example 11.3, Eqs. (8) and (9)]: 1 ðÞ ¼ ek

2 ð; Þ ¼ ð1 e Þek

where k ¼ ½1 expðC* NTUÞ=ðC* NTUÞ. Show that the heat exchanger eﬀectiveness of this heat exchanger is given by " ¼ 1 eK=C

K ¼ 1 expðC * NTUÞ

11.4

Consider a crossﬂow heat exchanger with the smaller heat capacity rate ﬂuid unmixed and the other ﬂuid mixed throughout. Show that the heat exchanger eﬀectiveness of this ﬂow arrangement is given by " ¼ ½1 expðMC*Þ=C*, where M ¼ 1 expðNTUÞ. Determine the numerical values for NTU that correspond to the maximum entropy generation of this exchanger for C * ¼ 1 and 0.1.

11.5

Using mass and energy balances, formulate a mathematical model of a general case of a crossﬂow heat exchanger. The inlet temperatures of either of the two ﬂuids may be nonuniform at respective inlets. The formulation must be presented in dimensionless form. The model should consist of diﬀerential equations for determining temperature distributions and corresponding boundary conditions. Assume steady-state operation. Invoke traditional assumptions for heat exchanger analysis, except for nonuniformity of the inlet temperatures.

11.6

Formulate corresponding reduced mathematical models for determining temperature distributions of both ﬂuids of a crossﬂow heat exchanger using the general model obtained in Problem 11.5. Consider the following particular cases: (a) The ﬂuid with the larger heat capacity rate is mixed; the other ﬂuid is unmixed, (b) The ﬂuid with the smaller heat capacity rate is mixed; the other ﬂuid is unmixed, (c) Both ﬂuids are mixed.

11.7

Determine the exact analytical solutions for temperature distributions of both ﬂuid streams in a particular case of a mixed–mixed crossﬂow arrangement. The ﬂuid inlet temperatures are uniform, and the heat exchanger operation is steady. All other traditional assumptions for heat exchanger theory are invoked.

11.8

Determine the exact analytical solutions for temperature distributions of both ﬂuid streams in a particular case of an unmixed–mixed crossﬂow arrangement. The mixed ﬂuid is the ﬂuid with larger heat capacity rate. The unmixed ﬂuid inlet temperature is nonuniform, and the heat exchanger operation is steady. All other traditional assumptions for heat exchanger theory are invoked. Reduce the

806

THERMODYNAMIC MODELING AND ANALYSIS

general solution to a simpliﬁed one by assuming the uniformity of both inlet temperatures. 11.9

Determine the exact analytical solutions for temperature distributions of both ﬂuid streams in a particular case of an unmixed–mixed crossﬂow arrangement. The mixed ﬂuid is the ﬂuid with smaller heat capacity rate. The unmixed ﬂuid inlet temperature is nonuniform, and the heat exchanger operation is steady. All other traditional assumptions for heat exchanger theory are invoked. Reduce the general solution to a simpliﬁed one by assuming the uniformity of both inlet temperatures.

11.10 Derive the analytical expression for determining dimensionless temperature diﬀerences in a parallelﬂow/counterﬂow heat exchanger. Highlight important reasoning in the derivation with an explanation. 11.11 A two-ﬂuid heat exchanger of an arbitrary ﬂow arrangement has a speciﬁed number of transfer units NTU, and the heat capacity rate ratio C*. The inlet temperatures of both ﬂuids are known and their ratio is # ¼ T1;i =T2;i . Derive the relationship between the entropy generated in this exchanger and its design parameters as given by Eq. (11.33), in which the pressure drops are neglected. Subsequently, calculate the entropy generation for a counterﬂow heat exchanger having C * ¼ 1 and an inlet temperature ratio of 0.5. Perform the calculations for NTU ¼ 1, 5, and 10 and discuss the change in entropy generation with increased heat exchanger size. Finally, show how Eq. (11.28) would change if there is ﬁnite pressure drop. 11.12 Determine the entropy generation caused by ﬂuid friction for ﬂuid ﬂow in a heat exchanger passage. Fluid is assumed to be an incompressible liquid described by mass density and mass ﬂow rate m_ . The pressure drop along the ﬂow length is p ¼ pi po . The inlet and outlet temperatures are Ti and To , respectively. 11.13 The relationship between design parameters and entropy generated in a heat exchanger is deﬁned by Eq. (11.33). This relationship is the subject of Problem 11.11. Show that the operating point corresponding to equal outlet temperatures must correspond to maximum entropy generation. Assume that the temperature eﬀectiveness increases monotonically with NTU. Calculate the number of transfer units that correspond to the maximum entropy generation operating point for a counterﬂow exchanger with C* ¼ 1. 11.14 Compare the magnitude of the eﬀectiveness of the two-pass cross-parallelﬂow and cross-counterﬂow arrangements presented in Fig. 11.10. Formulate a thermodynamic argument for comparison, and verify conclusions calculating the eﬀectiveness for an arbitrarily selected set of design parameters, including NTU ¼ 1 and C* ¼ 1: 11.15 Establish the rank of single-pass crossﬂow heat exchangers shown in Table P11.15. The rank is to be established by comparing the values of heat exchanger eﬀectiveness using a heuristic approach in assessing irreversibility levels of the pairs of ﬂow arrangements. 11.16 Establish the rank of two-pass crossﬂow heat exchangers shown in Table P11.16. The rank is to be established by comparing the values of heat exchanger eﬀective-

PROBLEMS

807

TABLE P11.15 Single-Pass Crossﬂow Heat Exchanger Schematic

Flow Arrangement

Single-pass crossﬂow, ﬂuid 1 unmixed and coupled in inverted order, ﬂuid 2 split into two streams with equal mass ﬂow rates, individually mixed.

Single-pass crossﬂow, ﬂuid 1 unmixed and connected in inverted order, ﬂuid 2 split into three streams with equal mass ﬂow rates, individually mixed

Single-pass crossﬂow, ﬂuid 1 unmixed and connected in inverted order, ﬂuid 2 split into four streams with equal mass ﬂow rates, individually mixed

ness using a heuristic approach in assessing irreversibility levels of the pairs of ﬂow arrangements. Verify the conclusions by performing a numerical evaluation of heat exchanger eﬀectiveness for each heat exchanger with C* ¼ 0:8 and NTU ¼ 1, 2, and 3. 11.17 The condenser of a power plant operates under conditions as follows. The condensing water vapor enters the condenser at 308C (303 K) with the quality (vapor fraction) x < 1. The pressure on the condensing side is 4 kPa. The pressure drop on the condensing side is 0.6 kPa. The cold-ﬂuid (river water) inlet temperature is TABLE P11.16 Two-Pass Crossﬂow Heat Exchanger Schematic

Flow Arrangement

Two-pass cross-counterﬂow, ﬂuid 1 unmixed and coupled in inverted order ﬂuid 2 mixed throughout

Two-pass cross-counterﬂow, ﬂuid 1 unmixed and coupled in identical order, ﬂuid 2 mixed throughout

Two-pass cross-parallelﬂow, ﬂuid 1 unmixed and coupled in inverted order, ﬂuid 2 mixed throughout

Two-pass cross-parallelﬂow, ﬂuid 1 unmixed and coupled in identical order, ﬂuid 2 mixed throughout

808

THERMODYNAMIC MODELING AND ANALYSIS

178C (290 K). The temperature of the water at the heat exchanger outlet is 278C (300 K). The pressure drop on the coolant side of the heat exchanger (including manifolds and connecting pipes) is 50 kPa. It is assumed that the water pump operates with negligible losses. In addition, the exchanger (including the connecting piping) heat losses to the environment constitute approximately 2% of the condenser heat transfer rate. Determine the magnitude and relative importance of all irreversibilities associated with the operation of the heat exchanger. 11.18 Reconsider the analysis of Problem 11.17 (a steam electric power plant condenser), but now include the economic aspect of the entropy generation estimation. Assign a monetary value to the entropy-generation contributions in terms of busbar energy delivery costs. The relevant additional data are as follows: Variable

Value

Rate of fuel consumption (MW) Boiler eﬃciency (%) Net electric power delivered (MW) Combined turbine/generator eﬃciency (%) Combined motor/pump eﬃciency (%) Capital cost of the equipment with 12% annual interest rate ($/kW) Average energy cost ($/109 J) Operation time (h/yr)

1600 80 700 85 80 700 3 4000

11.19 Show that entropy generation caused by heat transfer and ﬂuid friction associated with ﬂow through a duct can be described by Eq. (11.65). Elaborate explicitly all the assumptions needed for derivation of this relationship. 11.20 A heat exchanger passage carries a constant property ﬂuid. The cross-sectional area is Ao and the wetted wall perimeter is P. The mass ﬂow rate of the ﬂuid is ﬁxed. The heat transfer rate between the ﬂuid and the wall is across a mean temperature diﬀerence T, and it is considered to be constant along the ﬂow direction of a short passage under consideration. Show that dimensionless entropy generation can be written in the form of Eq. (11.66) if the ﬂuid represents a simple compressible substance with constant thermophysical properties. No phase change is present. 11.21 It can be shown that dimensionless entropy generation caused by heat transfer at ﬁnite temperature diﬀerences and ﬂuid friction takes the following form for ﬂow through an isothermal duct: 2=3

2=3 ð# 1Þe4jPr S_ irr ¼ ð# 1Þð1 e4jPr L=Dh Þ þ ln # m_ cp

L=Dh

þ1

2=3

1 f ð# 1Þe4jPr L=Dh þ 1 Pr2=3 Ec ln þ 2=3 2 j #e4jPr L=Dh Using mass, energy, and entropy rate balances, show that this result is correct if the ﬂow is assumed to be fully developed. Demonstrate that entropy generation may have a global minimum for a selected set of operating parameters. Perform the analysis for both air and water as working ﬂuids.

12

Flow Maldistribution and Header Design

One of the common assumptions in basic heat exchanger design theory is that ﬂuid be distributed uniformly at the inlet of the exchanger on each ﬂuid side and throughout the core. However, in practice, ﬂow maldistribution{ is more common and can signiﬁcantly reduce the desired heat exchanger performance. Still, as we discuss in this chapter, this inﬂuence may be negligible in many cases, and the goal of uniform ﬂow through the exchanger is met reasonably well for performance analysis and design purposes. Flow maldistribution can be induced by (1) heat exchanger geometry (mechanical design features such as the basic geometry, manufacturing imperfections, and tolerances), and (2) heat exchanger operating conditions (e.g., viscosity- or density-induced maldistribution, multiphase ﬂow, and fouling phenomena). Geometry-induced ﬂow maldistribution can be classiﬁed into (1) gross ﬂow maldistribution, (2) passage-topassage ﬂow maldistribution, and (3) manifold-induced ﬂow maldistribution. The most important ﬂow maldistribution caused by operating conditions is viscosityinduced maldistribution and associated ﬂow instability. In this chapter, we consider geometry-induced ﬂow maldistribution in Section 12.1 and operating condition–induced ﬂow maldistribution in Section 12.2. Next, mitigation of ﬂow maldistribution is discussed in Section 12.3. Finally, header design for compact heat exchangers is summarized in Section 12.4.

12.1 GEOMETRY-INDUCED FLOW MALDISTRIBUTION One class of ﬂow maldistribution, which is a result of geometrically nonideal ﬂuid ﬂow passages or nonideal exchanger inlet/outlet header/tank/manifold/nozzle design, is referred to as geometry-induced ﬂow maldistribution. This type of maldistribution is closely related to heat exchanger construction and fabrication (e.g., header design, heat exchanger core fabrication including brazing in compact heat exchangers). This maldistribution is peculiar to a particular heat exchanger in question and cannot be inﬂuenced signiﬁcantly by modifying operating conditions. Geometry-induced ﬂow maldistribution is related to mechanical design-induced ﬂow nonuniformities such as (1) entry conditions, (2) bypass and leakage streams, (3) fabrication tolerances, {

Flow maldistribution is deﬁned as nonuniform distribution of the mass ﬂow rate on one or both ﬂuid sides in any of the heat exchanger ports and/or in the heat exchanger core. The term ideal ﬂuid ﬂow passage/header/heat exchanger would, as a rule, denote conditions of uniform mass ﬂow distribution through an exchanger core.

Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

809

810

FLOW MALDISTRIBUTION AND HEADER DESIGN

(4) shallow bundle eﬀects,{ and (5) general equipment and exchanger system eﬀects (Kitto and Robertson, 1989). The most important causes of ﬂow nonuniformities can be divided roughly into three main groups of maldistribution eﬀects: (1) gross ﬂow maldistribution (at the inlet face of the exchanger), (2) passage-to-passage ﬂow maldistribution (nonuniform ﬂow in neighboring ﬂow passages), and (3) manifold-induced ﬂow maldistribution (due to inlet/outlet manifold/header design). First, we discuss gross ﬂow maldistribution. Subsequently, the passage-to-passage ﬂow maldistribution is addressed, followed by a few comments related to manifold-induced ﬂow maldistribution.

12.1.1

Gross Flow Maldistribution

The major feature of gross ﬂow maldistribution is that nonuniform ﬂow occurs at the macroscopic level (due to poor header design or blockage of some ﬂow passages during manufacturing, including brazing or operation). The gross ﬂow maldistribution does not depend on the local heat transfer surface geometry. This class of ﬂow maldistribution may cause (1) a signiﬁcant increase in the exchanger pressure drop, and (2) some reduction in heat transfer rate. To predict the magnitude of these eﬀects for some simple exchanger ﬂow arrangements, the nonuniformity will be modeled as one- or two-dimensional as follows, with some speciﬁc results. Gross ﬂow maldistribution can occur in one dimension across the free-ﬂow area (perpendicular to the ﬂow direction) as in single-pass counterﬂow and parallelﬂow exchangers, or it can occur in two or three dimensions as in single- and multipass crossﬂow and other exchangers. Let us ﬁrst model a one-dimensional gross ﬂow maldistribution with an N-step inlet velocity distribution function. The heat exchanger will be represented by an array of N subunits, called subexchangers, having uniform ﬂow throughout each unit but with diﬀerent mass ﬂow rates from unit to unit. The number of subexchangers is arbitrary, but it will be determined to be in agreement with the imposed ﬂow maldistribution. The set of standard assumptions of Section 3.2.1 is applicable to each subexchanger. The following additional idealizations are introduced to quantify the inﬂuence of ﬂow nonuniformity caused by gross ﬂow maldistribution on each subexchanger and the exchanger as a whole. 1. Total heat transfer rate in a real heat exchanger is equal to the sum of the heat transfer rates that would be exchanged in N subexchangers connected in parallel for an idealized N-step inlet velocity distribution function. 2. The sum of the heat capacity rates of the respective ﬂuid streams for all subexchangers is equal to the total heat capacity rates of the ﬂuids for the actual maldistributed heat exchanger. With these auxiliary assumptions, the temperature eﬀectiveness of a counterﬂow/ parallelﬂow heat exchanger can be calculated by modeling the heat exchanger as a parallel coupling of N subexchangers for a maldistributed ﬂuid stream having N indivi{ Use of an axial tube-side nozzle (if the depth of the head is insuﬃcient to allow the jet from the nozzle to expand to the tubesheet diameter) may result in larger ﬂow in central tubes and lead to ﬂow maldistribution (Mueller, 1987).

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

811

dual uniform ﬂuid streams (i.e., having an N-step function velocity distribution). The other ﬂuid side is considered as having uniform ﬂow distribution for such an analysis. If ﬂow nonuniformity occurs on both ﬂuid sides of a counterﬂow or parallelﬂow exchanger, the exchanger is divided into a suﬃcient number of subexchangers such that the ﬂow distributions at the inlet on both ﬂuid sides are uniform for each subexchanger. For all other exchangers, the solution can only be determined numerically, and the solutions of Sections 12.1.1.1 and 12.1.1.2 are not valid in that case. 12.1.1.1 Counterﬂow and Parallelﬂow Exchangers. In this section, we derive an expression for the exchanger eﬀectiveness and hence heat transfer performance for counterﬂow and parallelﬂow exchangers having an N-step velocity distribution function on the ﬂuid 1 side and perfectly uniform ﬂow distribution on the ﬂuid 2 side, as shown in Fig. 12.1a. Here ﬂuid 1 can be either the hot or cold ﬂuid, and in that case, ﬂuid 2 will be the cold or hot ﬂuid. Subsequently, we apply this analysis to a heat exchanger having a two-step velocity distribution function at the inlet. Heat Transfer Analysis. Let us consider a counterﬂow exchanger with an N-step inlet distribution function of ﬂuid 1, shown in Fig. 12.1a. The same analysis would be valid for a parallelﬂow heat exchanger. Fluid 2 is considered uniform. We may model this exchanger as an array of N subexchangers, each obeying the standard assumptions of Section 3.2.1. Hence, q¼

N X

ð12:1Þ

qj

j¼A

where q represents the total heat transfer rate, and qj , j ¼ A, B, . . . , N, the fractions of heat transfer rate in N hypothetical subexchangers, each having uniform mass ﬂow rates on both sides, as shown in Fig. 12.1b. The assumptions invoked, including the auxiliary ones introduced above, lead to the following results: q ¼ C1 T1;i T1;o P1 ¼

T1;i T1;o T1;i T2;i

and qj ¼ C1; j ðT1;i T1;o Þj and

P1; j ¼

ðT1;i T1;o Þj T1;i T2;i

j ¼ A; B; . . . ; N

ð12:2Þ

j ¼ A; B; . . . ; N

ð12:3Þ

FIGURE 12.1 Idealized two-step function ﬂow nonuniformity on ﬂuid 1 side and uniform ﬂow on ﬂuid 2 side of a counterﬂow exchanger.

812

FLOW MALDISTRIBUTION AND HEADER DESIGN

Substituting Eqs. (12.2) and (12.3) into Eq. (12.1) and rearranging, the expression for ﬂuid 1 temperature eﬀectiveness becomes P1 ¼

N 1 X C P C1 j¼A 1; j 1; j

ð12:4Þ

where C1 ¼

N X

ð12:5Þ

C1; j

j¼A

Note that Eqs. (12.4) and (12.5) are valid for a maldistributed ﬂuid regardless of whether it is hot or cold, Cmin or Cmax , or denoted as 1 or 2. The subscript 1 in these equations may be replaced by a designator of the maldistributed stream, say as in Pms and Cms . The temperature eﬀectiveness of ﬂuid 1 for each of the subexchangers of Eq. (12.4) is computed knowing individual NTU and heat capacity rate ratio: C1; j P1; j ¼ P1; j NTU1; j ; C2; j

j ¼ A; B; . . . ; N

ð12:6Þ

where P1; j ðÞ on the right-hand side of Eq. (12.6) is computed for each exchanger using the expression provided in Table 3.6 as follows:

P1; j ¼

8 1 exp½NTU1; j ð1 R1; j Þ > > > < 1 R exp½NTU ð1 R Þ

counterflow

> 1 exp½NTU1 ð1 þ R1; j Þ > > : 1 þ R1; j

parallelflow

1; j

1; j

1; j

ð12:7Þ

Application of Eq. (12.6) requires the values of NTU1; j , C1; j , and C2; j for each subexchanger. To determine these variables, we should invoke the standard assumptions of Section 3.2.1 to get the free-ﬂow area and heat capacity rate ratio as Ao ¼

N X

ð12:8Þ

Ao; j

j¼A

C1; j m_ 1; j u1; j Ao; j ¼ ¼ C1 u1 Ao 1 m_ 1

j ¼ A; B; . . . ; N

ð12:9Þ

Similarly, for ﬂuid 2, we get C2; j u2; j Ao; j ¼ C2 u1 Ao 2

j ¼ A; B; . . . ; N

ð12:10Þ

Note that the sets of relations given by Eq. (12.9) and (12.10) may be reduced by one equation each by utilizing Eq. (12.5) for ﬂuid 1 (and similarly for ﬂuid 2). The number of heat transfer units and the capacity rate ratios for Eq. (12.7) can be determined from their deﬁnitions as follows:

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

NTU1; j ¼ R1; j ¼

UA1; j UA1 A1; j C1 ¼ C1; j C1 A1 C1; j

C1; j C2; j

j ¼ A; B; . . . ; N

813

ð12:11Þ ð12:12Þ

Note also that Dh ¼ 4Ao L=A with Dh and L identical for all subexchangers j ¼ A; B; . . . ; N for the counterﬂow/parallelﬂow heat exchanger. A1; j =A1 of Eq. (12.11) is obtained then from the deﬁnitions of Dh as Ao; j Aj ¼ j ¼ A; B; . . . ; N ð12:13Þ Ao 1 A 1 In addition,

Ao; j Ao

¼ 1

Ao; j Ao 2

j ¼ A; B; . . . ; N

ð12:14Þ

The reduction in the temperature eﬀectiveness on the maldistributed side can be presented by the performance (eﬀectiveness) deterioration factor as P*1 ¼

P1;ideal P1 P1;ideal

or "* ¼

"ideal " "ideal

ð12:15Þ

where P1;ideal represents the temperature eﬀectiveness for the case of having no ﬂow maldistribution. The inﬂuence of gross ﬂow maldistribution is shown in Fig. 12.2 for a balanced (C * ¼ 1) counterﬂow heat exchanger in terms of "* for a two-step inlet velocity distribution function (i.e., for two subexchangers). For a particular value of umax =um

FIGURE 12.2 Performance deterioration factor "* for a balanced heat exchanger, C* ¼ 1, N ¼ 2. (From Shah, 1981.)

814

FLOW MALDISTRIBUTION AND HEADER DESIGN

and given NTU, we ﬁnd the greatest reduction in the heat exchanger eﬀectiveness occurring when two-step function ﬂow maldistribution occurs in equal ﬂow areas (50 : 50%). The eﬀect of ﬂow maldistribution increases with NTU for a counterﬂow exchanger. Note that the reduction in the temperature eﬀectiveness P1 of Eq. (12.4), obtained using Eq. (12.15), is valid regardless of whether the maldistributed ﬂuid is the hot, cold, Cmax , or Cmin ﬂuid. We can idealize an N-step function velocity distribution into an equivalent two-step function velocity distribution. Based on the analysis of passage-to-passage ﬂow maldistribution presented in Section 12.1.2, it is conjectured that the deterioration in the exchanger eﬀectiveness is worse for the two-step function velocity distribution. Hence, conservatively, any ﬂow maldistribution can be reduced to a two-step function, and its eﬀect can readily be evaluated on the exchanger eﬀectiveness, which will represent the highest deterioration. As indicated above, the discussion of the eﬀect of the gross ﬂow maldistribution in this section refers to a heat exchanger with counterﬂow arrangement and balanced ﬂow (C* ¼ 1). Hence, the increased eﬀect of ﬂow maldistribution with increasing NTU is valid only for this special situation. If either C * 6¼ 1 or if the ﬂow arrangement is parallelﬂow, the inﬂuence of ﬂow maldistribution may decrease with increasing NTU. This can be determined from Eq. (12.15), assuming the validity of the appropriate eﬀectiveness– NTU relationships for each subexchanger and for the heat exchanger as a whole. Also, the actual ﬂow rate conditions in most practical cases would not correspond to a balanced heat exchanger case. Pressure Drop Analysis. There is no rigorous theory available for predicting a change in the pressure drop due to ﬂow maldistribution in the exchanger. This is because for nonuniform ﬂow distribution, the static pressures at the core inlet and outlet faces will not be uniform, and hence, constant pressure drop across the core is not a valid assumption. The following is a suggested approximate procedure. This approach is not based on a rigorous modeling of the actual ﬂow conditions and must be used very cautiously. Consider a two-step function velocity distribution at the core inlet on ﬂuid 1 side as shown in Fig. 12.1a for N ¼ 2. Subexchangers in Fig. 12.1b for N ¼ 2 are in parallel. Using Eq. (6.28), evaluate the pressure drop pj for a speciﬁc subexchanger which has the highest ﬂuid velocity in the ﬂow passages. Also compute puniform for ﬂuid 1 considering the ﬂow as uniform at the core inlet in Fig. 12.1. Therefore, as a conservative approach, this largest pj (i.e., pmax ) will be the pressure drop on the ﬂuid 1 side having imposed ﬂow nonuniformity. The increase in pressure drop due to ﬂow nonuniformity is then ðpÞincrease ¼ pmax puniform

ð12:16Þ

It should be emphasized that the entrance and exit losses in addition to the core friction contribution will be higher (in the evaluation of pmax ) than those for uniform ﬂow. If ﬂow nonuniformity occurs on both sides of an exchanger, the procedure outlined above is applied to both sides, since the pressure drops on both sides of a two-ﬂuid exchanger are relatively independent of each other, except for the changes in ﬂuid density due to heat transfer in the core. Hence, the analysis above is applicable to any ﬂow arrangement. Example 12.1 A counterﬂow heat exchanger has a severe ﬂow maldistribution due to poor header design. On the ﬂuid 1 side, 25% of the total free-ﬂow area has the ﬂow

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

815

velocity 50% larger than the mean ﬂow velocity through the core as a whole. The number of heat transfer units of the heat exchanger is NTU1 ¼ 3. The total heat capacity rates through the exchanger are nearly the same (i.e., the heat exchanger is balanced). Determine the reduction in the temperature eﬀectiveness of ﬂuid 1 and an approximate increase in the pressure drop due to ﬂow maldistribution. Assume fully developed laminar ﬂow on both ﬂuid sides (i.e., U remains constant). SOLUTION Problem Data and Schematic: A schematic of the heat exchanger under consideration is similar to that in Fig. 12.1 with only A and B subexchangers. The following data are known. NTU1 ¼ 3

u1;A ¼ 1:5u1

C1 ¼ C2

Ao;A ¼ 0:25Ao;1

Determine: The temperature eﬀectiveness P1 of the maldistributed ﬂuid. Assumptions: All the assumptions listed in Section 3.2.1 are valid with the exception of nonuniformity of the mass ﬂow rate of ﬂuid 1, which is assumed to have a two-step function velocity distribution. Even with this ﬂow maldistribution, U based on A1 or A2 is assumed constant. Analysis: The eﬀectiveness of the maldistributed heat exchanger is given by Eq. (12.4) with j ¼ A and B as C1;A C1;B P1 ¼ P þ P C1 1;A C1 1;B Let us determine all the parameters in this equation after determining the area ratios using Eqs. (12.13) and (12.14). Ao;A A1;A ¼ ¼ 0:25 Ao 1 A1 Ao;A Ao;A ¼ ¼ 0:25 Ao 1 Ao 2 Also,

Ao;B A1;B ¼ ¼ 0:75 Ao 1 A1 Ao;B Ao;B ¼ ¼ 0:75 Ao 1 Ao 2

u1;A ¼ 1:5 u1

where u1 is the mean ﬂuid velocity on the ﬂuid 1 side. The ratios of the heat capacity rates in the maldistributed subexchangers to the total capacity rate of ﬂuid 1 are then given by Eq. (12.9) as C1;A u1;A Ao;A C1;B C1;A ¼ ¼ 1:5 0:25 ¼ 0:375 ¼1 ¼ 1 0:375 ¼ 0:625 C1 u1 Ao 1 C1 C1 Similarly, the heat capacity rate ratios of subexchangers to the total exchanger on ﬂuid 2 side are given by Eq. (12.10) as C2;A Ao;A Ao;A C2;B C2;A ¼ ¼ ¼ 0:25 ¼1 ¼ 1 0:25 ¼ 0:75 C2 Ao 2 Ao 1 C2 C2

816

FLOW MALDISTRIBUTION AND HEADER DESIGN

We will determine temperature eﬀectivenesses, using Eq. (12.7), after calculating respective NTUs and R’s. The NTUs from Eq. (12.11) are NTU1;A ¼

UA1;A UA1 A1;A C1 1 ¼ ¼ 3 0:25 ¼ 2:00 C1;A C1 A1 C1;A 0:375

NTU1;B ¼

UA1;B UA1 A1;B C1 1 ¼ ¼ 3 0:75 ¼ 3:60 C1;B C1 A1 C1;B 0:625

The heat capacity rate ratios, required for eﬀectiveness calculations, are computed using Eq. (12.12) as follows: R1;A ¼

C1;A C1;A C2 C1 1 ¼ ¼ 0:375 1 ¼ 1:50 C2;A C1 C2;A C2 0:25

R1;B ¼

C1;B C1;B C2 C1 1 ¼ ¼ 0:625 1 ¼ 0:8333 0:75 C2;B C1 C2;B C2

Therefore, the temperature eﬀectivenesses of subexchangers are given by Eq. (12.7) as P1;A ¼

1 eNTU1;A ð1R1;A Þ 1 exp½2:00ð1 1:50Þ ¼ ¼ 0:5584 NTU ð1R Þ 1;A 1;A 1 1:5 exp½2:00ð1 1:50Þ 1 R1;A e

P1;B ¼

1 eNTU1;B ð1R1;B Þ 1 exp½3:60ð1 0:8333Þ ¼ ¼ 0:8315 1 R1;B eNTU1;B ð1R1;B Þ 1 0:8333 exp½3:60ð1 0:8333Þ

The temperature eﬀectiveness from Eq. (12.4) is P1 ¼

C1;A C1;B P1;A þ P ¼ 0:375 0:5584 þ 0:625 0:8315 ¼ 0:7291 C1 C1 1;B

The heat exchanger eﬀectiveness of a balanced counterﬂow heat exchanger without any ﬂow maldistribution on either ﬂuid side would be P1;ideal ¼

NTU1 3 ¼ ¼ 0:750 1 þ NTU1 1 þ 3

Finally, the quantitative measure of the reduction in the eﬀectiveness due to maldistribution is [see Eq. (12.15)] P*1 ¼

P1;ideal P1 0:750 0:7291 ¼ ¼ 0:0279 P1;ideal 0:750

Ans:

Discussion and Comments: From the results, it becomes clear that a relatively large ﬂow maldistribution on the ﬂuid 1 side in this particular case causes a deterioration of the temperature eﬀectiveness of approximately 2.8%. With all other parameters ﬁxed, a heat exchanger with high NTU will suﬀer more pronounced eﬀectiveness deterioration (see Fig. 12.2).

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

817

12.1.1.2 Crossﬂow Exchangers. A direct extension of the approach used for a counterﬂow/parallelﬂow exchanger to that for a crossﬂow exchanger with diﬀerent combinations of ﬂuid mixing/unmixing on each ﬂuid side is not necessarily straightforward. Only when ﬂow nonuniformity is present on the unmixed ﬂuid side with the other ﬂuid side as mixed can a simple closed-form solution be obtained, as outlined next. Mixed–Unmixed Crossﬂow Exchanger with Nonuniform Flow on the Unmixed Side. Let us consider a single-pass crossﬂow exchanger having the unmixed ﬂuid (ﬂuid 1) maldistributed. The inlet velocity distribution is represented with an N-step function (Fig. 12.3). Fluids 1 and 2 can be arbitrarily hot and cold, or vice versa. The total heat transfer rate in the exchanger is given by q¼

N X

qj

ð12:17Þ

j¼A

where the qj represent individual heat transfer rates/enthalpy rate changes as follows{: qA ¼ P1;A C1;A ðT1;i T2;i Þ ¼ C2 ðT2M;A T2;i Þ qB ¼ P1;B C1;B ðT1;i T2M;A Þ ¼ C2 ðT2M;B T2M;A Þ .. .

ð12:18Þ

qN ¼ P1;N C1;N ðT1;i T2M;N1 Þ ¼ C2 ðT2M;N T2;o Þ In Eq. (12.18), T2M; j ( j ¼ A; B; . . . ; N) represent the mixed mean temperatures of ﬂuid 2 between the subexchangers. Note that the left-hand side of Eq. (12.17) can also be presented in the form q ¼ P1 C1 ðT1;i T2;i Þ

ð12:19Þ

FIGURE 12.3 Idealized two-step function ﬂow nonuniformity on the unmixed ﬂuid 1 side and uniform ﬂow on the mixed ﬂuid 2 side of a crossﬂow exchanger. {

For the sake of simpliﬁed notation, the absolute value designators are omitted.

818

FLOW MALDISTRIBUTION AND HEADER DESIGN

Our objective is to determine the relationship between the ﬂuid 1 temperature eﬀectiveness P1 and the temperature eﬀectiveness and heat capacity rates of the subexchangers of Fig. 12.3b. From Eq. (12.19), we get P1 ¼

q C1 ðT1;i T2;i Þ

ð12:20Þ

Replacing q in Eq. (12.20) with q from Eq. (12.17), and utilizing the relationships provided by Eq. (12.18), we get P1 ¼

T1;i T2M;A T1;i T2M;N1 1 ð12:21Þ P1;A C1;A þ P1;B C1;B þ þ P1;N C1;N T1;i T2;i T1;i T2;i C1

Temperature diﬀerence ratios in Eq. (12.21) can be eliminated by manipulating relationships from Eq. (12.18) as follows: T1;i T2M;A P1;A C1;A ¼1 T1;i T2;i C2 T1;i T2M;B P1;A C1;A P1;B C1;B ¼ 1 1 T1;i T2;i C2 C2 ... Y T1;i T2M;N1 N1 P1;k C1;k ¼ 1 T1;i T2;i C2 k¼A

ð12:22Þ

Combining Eqs. (12.22) and (12.21) and after rearrangement, we get P1 ¼

j1 N Y X P1;k C1;k 1 P1;A C1;A þ 1 P1; j C1; j C2 C1 j¼B k¼1

ð12:23Þ

To reemphasize, the ﬂuid 1 side is unmixed and the ﬂuid 2 side is mixed for the expression of a crossﬂow exchanger above. The temperature eﬀectiveness of the ideal heat exchanger of Fig. 12.3 as a whole, and those of the subexchangers, can be expressed in terms of corresponding heat capacity rate ratios and numbers of transfer units as follows (see Table 3.6): P1;ideal ¼

P1; j

C2 C 1 exp 1 1 eNTU1 C1 C2

T1;i T1;o j C1; j C ¼ ¼ 2 1 exp 1 eNTU1; j T1;i T2;i C1; j C2

ð12:24Þ

j ¼ A; B; . . . ; N (12.25)

where NTU1 ¼

UA C1

NTU1; j ¼

UAj C1; j

j ¼ A; B; . . . ; N

ð12:26Þ

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

819

When the ﬂow maldistribution on the ﬂuid 1 side is assumed to be a two-step velocity distribution function, Eq. (12.23) can be simpliﬁed to P1 ¼

P1;A C1;A 1 P1;A C1;A þ P1;B C1;B 1 C2 C1

ð12:27Þ

Calculation of the temperature eﬀectiveness of a maldistributed ﬂuid stream using Eq. (12.23) or (12.27) is valid only if the maldistributed ﬂuid side is the unmixed ﬂuid side and the mixed side has uniform ﬂow. However, if the maldistributed ﬂuid is a mixed ﬂuid and the unmixed ﬂuid side has uniform ﬂow, a closed-form solution cannot be obtained using this simpliﬁed approach. This is because T2M temperatures for this case will not be uniform, and hence we cannot determine the eﬀectiveness of this exchanger using the formula of Table 3.6 when the ﬂow at the inlet of a subsequent subexchanger is not uniform. The problem becomes inevitably nonlinear, and no closed-form solution is available for this case; only a numerical solution is the option. Note that Eqs. (12.6), (12.9), and (12.11) are also valid for this mixed–unmixed crossﬂow exchanger, while Eqs. (12.10), (12.12), and (12.14) are not valid and we don’t need them to determine the temperature eﬀectiveness of this ﬂow maldistributed case. Unmixed–Unmixed Crossﬂow Exchangers. The two-dimensional ﬂow maldistribution has been analyzed numerically only for an unmixed–unmixed crossﬂow exchanger. In a series of publications as summarized by Chiou (1980) and Mueller and Chiou (1988), Chiou has studied the eﬀects of ﬂow maldistribution on an unmixed–unmixed crossﬂow single-pass heat exchanger with ﬂow maldistribution on one and both ﬂuid sides. When ﬂow maldistribution is present on only one ﬂuid side, the following general conclusions have been obtained. . For ﬂow maldistribution on the Cmax ﬂuid side, the exchanger thermal performance deterioration factor "* approaches a single value of 0.06 for all C* < 1 when NTU approaches zero. The performance deterioration factor decreases as NTU increases. For a balanced heat exchanger (C* ¼ 1), the exchanger thermal performance deterioration factor increases continually with NTU. . For ﬂow maldistribution on the Cmin ﬂuid side, the thermal performance deterioration factor ﬁrst increases and then decreases as NTU increases. . If ﬂow nonuniformities are present on both sides, the performance deterioration factor can be either larger or smaller than that for the case where ﬂow nonuniformity is present on only one side, and there are no general guidelines about the expected trends. A study of the inﬂuence of two-dimensional nonuniformities in inlet ﬂuid temperatures (Chiou, 1982) indicates that there is a smaller reduction in exchanger eﬀectiveness for the nonuniform inlet temperature than that for the nonuniform inlet mass ﬂow rate. For various nonuniform ﬂow models studied, the inlet nonuniform ﬂow case showed a decrease in eﬀectiveness of up to 20%; whereas for the nonuniform inlet temperature case, a decrease in eﬀectiveness of up to 12% occurred, with even an increase in eﬀectiveness for some cases of nonuniform inlet temperature. This occurs when the hotter portion of the inlet temperature is near the exit end of the cold ﬂuid, whose inlet temperature is uniform. In a recent study, Ranganayakulu et al. (1996) obtained numerical

820

FLOW MALDISTRIBUTION AND HEADER DESIGN

solutions for the eﬀects of two-dimensional ﬂow nonuniformities on thermal performance and pressure drop in crossﬂow plate-ﬁn compact heat exchangers. Example 12.2 Analyze a crossﬂow heat exchanger with ﬂuid 1 unmixed and ﬂuid 2 mixed having pronounced maldistribution on ﬂuid 1 side and NTU1 ¼ 3. The total heat capacity rates of the two ﬂuids are nearly the same. Determine the temperature eﬀectiveness of the maldistributed ﬂuid 1 if 25% of the total free-ﬂow area has the ﬂow velocity 50% larger than the mean velocity through the core on the ﬂuid 1 side corresponding to the uniform ﬂow case. SOLUTION Problem Data and Schematic: A schematic of the heat exchanger under consideration is similar to that in Fig. 12.3 with only A and B subexchangers. The following data are known: NTU1 ¼ 3

u1;A ¼ 1:5u1

C1 ¼ C2

Ao;A ¼ 0:25Ao;1

Determine: The temperature eﬀectiveness of ﬂuid 1. Assumptions: All the assumptions of Section 3.2.1 are valid here except for ﬂow maldistribution on the ﬂuid 1 side. Analysis: Let us calculate the temperature eﬀectiveness of ﬂuid 1 under the idealized conditions of uniform mass ﬂow rates on both ﬂuid sides for a balanced unmixed–mixed crossﬂow heat exchanger using Eq. (12.24) as follows: P1;ideal ¼ 1 exp½ð1 eNTU1 Þ ¼ 1 exp½ð1 e3 Þ ¼ 0:6133 However, under given ﬂow maldistribution conditions, this ideal eﬀectiveness cannot be achieved. Thus, the temperature eﬀectiveness should be calculated using Eq. (12.27). For this example, similar to Example 12.1, we can determine C1;A ¼ 0:375 C1

C1;B ¼ 0:625 C1

NTU1;A ¼ 2:00

NTU1;B ¼ 3:60

Now P1;A and P1;B are computed using Eq. (12.25) as follows after incorporating C2 ¼ C1 : C1;A C P1;A ¼ 1 1 exp 1 eNTU1;A C1;A C1

1 ¼ 0:7385 1 exp 0:375 1 e2:00 0:375 C1;B C ¼ 1 1 exp 1 eNTU1;B C1;B C1

¼ P1;B

¼

1 1 exp 0:625 1 e3:60 ¼ 0:7288 0:625

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

821

Now, compute the temperature eﬀectiveness of ﬂuid 1 given by Eq. (12.27) as (with C1 ¼ C2 Þ P1 ¼

P1;A C1;A C1;A C1;B P1;A C1;A 1 ¼ P1;A P1;A C1;A þ P1;B C1;B 1 þ P1;B 1 C1 C1 C1 C1 C1

¼ 0:7385 0:375 þ 0:7288 0:625 ð1 0:7385 0:375Þ ¼ 0:6063

Ans:

This actual eﬀectiveness is, indeed, smaller than the one calculated for an idealized situation, 0.6063 vs. 0.6133. Finally, the fractional deterioration in the temperature eﬀectiveness is given by Eq. (12.15) as P* ¼

P1;ideal P1 0:6133 0:6063 ¼ ¼ 0:0114 P1;ideal 0:6133

Discussion and Comments: Deterioration in the temperature eﬀectiveness caused by a relatively large ﬂow maldistribution for this crossﬂow exchanger is 0.0114, much smaller than 0.0279 for the counterﬂow exchanger (see Example 12.1), for the same operating conditions. The results of Examples 12.1 and 12.2 emphasize the fact that ﬂow maldistribution has the highest eﬀect on a counterﬂow exchanger (since it has the highest " for given NTU and C*) compared to exchangers with other ﬂow arrangements for similar operating parameters. 12.1.1.3 Tube-Side Maldistribution and Other Heat Exchanger Types. Tube-side maldistribution in a 1–1 TEMA E shell-and-tube counterﬂow heat exchanger studied by Cichelli and Boucher (1956) led to the following major conclusions: . For Cs =Ct small, say Cs =Ct ¼ 0:1, the performance loss is negligible for large ﬂow nonuniformities for NTUs < 2. . For Cs =Ct large, say Cs =Ct > 1, a loss can be noticed but diminishes for NTUs > 2. . Cs =Ct ¼ 1 is the worst case at large NTUs as can be found from Fig. 12.2. Fleming (1966) and Chowdhury and Sarangi (1985) have studied various models of ﬂow maldistribution on the tube side of a counterﬂow shell-and-tube heat exchanger. It is concluded that high-NTU heat exchangers are more susceptible to maldistribution eﬀects. According to Mueller (1977), the well-baﬄed 1–1 counterﬂow shell-and-tube heat exchanger (tube side nonuniform, shell side mixed) is aﬀected the least by ﬂow maldistribution. Shell-and-tube heat exchangers, which do not have mixing of the uniform ﬂuid [(1) tube side nonuniform, shell side unmixed; or (2) tube side uniform, shell side nonuniform in crossﬂow], are aﬀected more by ﬂow maldistribution. According to Kutchey and Julien (1974), the radial ﬂow variations of the mismatched air side and gas side reduce the regenerator eﬀectiveness signiﬁcantly. 12.1.2

Passage-to-Passage Flow Maldistribution

Compact heat exchangers with uninterrupted (continuous) ﬂow passages, while designed for nonfouling applications, are highly susceptible to passage-to-passage ﬂow maldistribution. That is because the neighboring passages are geometrically never identical, due to imperfect manufacturing processes. It is especially diﬃcult to control the passage size

822

FLOW MALDISTRIBUTION AND HEADER DESIGN

precisely when small dimensions are involved [e.g., a rotary regenerator with Dh ¼ 0:5 mm (0.020 in.)]. Since diﬀerently sized and shaped passages exhibit diﬀerent ﬂow resistances and the ﬂow seeks a path of least resistance, a nonuniform ﬂow through the matrix results. This phenomenon usually causes a slight reduction in pressure drop, while the reduction in heat transfer rate may be signiﬁcant compared to that for nominal (average) size passages. The inﬂuence is of particular importance for continuous-ﬂow passages at low Re (i.e., laminar ﬂow) as found in compact rotary regenerators. For a theoretical analysis for passage-to-passage ﬂow maldistribution, the actual nonuniform surface is idealized as containing large, small, and/or in-between size passages (in parallel) relative to the nominal passage dimensions. The models include (1) a two-passage model (London, 1970), (2) a three-passage model, and (3) an N-passage model (Shah and London, 1980). Although triangular and rectangular passage cross sections have been studied, similar analysis can be applied to any cross-sectional shapes of ﬂow passages. The analysis to follow can also be utilized for analyzing ﬂow maldistribution in viscous oil cooler with constant-wall-temperature boundary conditions (i.e., condensation or vaporization taking place on the other ﬂuid side). See Section 12.2.1 for further details. Let us ﬁrst deﬁne the two-passage-model ﬂow nonuniformity. From the methodological point of view, this approach is the most transparent and oﬀers a clear idea of how the modeling of ﬂow nonuniformity can be conducted. Also, the two-passage model predicts a more detrimental eﬀect on heat transfer and pressure drop than that of an Npassage (N > 2) model. 12.1.2.1

Models of Flow Nonuniformity

Two-Passage Model. Let us consider that a heat exchanger core characterizes ﬂow nonuniformity due to two diﬀerent ﬂow cross sections diﬀering in either (1) cross-section size of the same passage type, (2) diﬀerent cross-sectional shapes of ﬂow passages, or (3) a combination of both. The two most common types of idealized passage-to-passage nonuniformities are plate spacing and ﬁn spacing, shown in Fig. 12.4a and b, respectively. For the analysis, the actual heat exchanger core will be assumed to be a collection of two (or more) distinct sets of uniform ﬂow passages, passages 1 and passages 2

FIGURE 12.4 Two-passage nonuniformity model: (a) plate-spacing nonuniformity; (b) ﬁnspacing nonuniformity. Note that passages diﬀer in size. The nominal size of the passage may be large, small, or in between, depending on how it is deﬁned. (From London, 1968.)

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

823

(or N passages). Our objective here is to determine the reduction in heat transfer and pressure drop due to this passage-to-passage ﬂow nonuniformity. The following assumptions are invoked for setting up the model. . Flow is hydrodynamically and thermally fully developed (Nu ¼ constant, f Re ¼ constant). . Thermophysical properties of the ﬂuids are constant and uniform. . Entrance and exit pressure losses are negligible (the core friction component is dominant). . Static pressures are constant and uniform across the cross section at the entrance and exit of this multipassage exchanger. . The total ﬂow rate through all nonuniform ﬂow passages is identical to that going through all nominal ﬂow passages. . The lengths of all ﬂow passages are the same. The pressure drop for all ﬂow passages (regardless of the size, shape, and distribution of ﬂow passages) will be the same in the core based on the fourth assumption above: ðpÞj ¼ ðpi po Þj

ð12:28Þ

where pj ¼ fj

4L Dh

! j

m u2m 2gc

! j ¼ 1; 2; n

ð12:29Þ

j

where j denotes the ﬂow passage type. Invoking the deﬁnitions of the Reynolds number and mass ﬂow rate, Eq. (12.29) is regrouped as pj ¼

2L f Re m_ gc Ao D2h j j

ð12:30Þ

For a two-passage model, applying Eq. (12.30) for j ¼ 1 and 2 and taking the ratio and rearranging, we get m_ 1 ð f ReÞ2 Dh;1 2 Ao;1 ¼ m_ 2 ð f ReÞ1 Dh;2 Ao;2

ð12:31Þ

since p1 ¼ p2 from Eq. (12.28). Equation (12.31) provides the ﬂow fraction distribution in the two types of ﬂow passages. Normalizing ﬂow rates with m_ n , hydraulic diameters with Dh;n , and free-ﬂow areas with Ao;n , Eq. (12.31) becomes m_ 1 =m_ n ð f ReÞ2 Dh;1 =Dh;n 2 Ao;1 =Ao;n ¼ m_ 2 =m_ n ð f ReÞ1 Dh;2 =Dh;n Ao;2 =Ao;n

ð12:32Þ

where m_ n ¼ m_ 1 þ m_ 2 and all variables with a subscript n denote nominal values (selected by the choice of an analyst), either the passage geometry 1, the passage geometry 2, or some nominal passage geometry in between (for normalization of Dh and Ao used in the equation) for a two-passage nonuniformity.

824

FLOW MALDISTRIBUTION AND HEADER DESIGN

To compute the ﬂow area ratios in Eq. (12.32), we maintain approximately the same frontal area of the heat exchanger core with actual and nominal ﬂow passages. There are two choices for selecting the nominal passage geometry, and accordingly, the values of Ao; j =Ao;n ð j ¼ 1; 2Þ will be diﬀerent. They are as follows. 1. The number of ﬂow passages for the nominal geometry is the sum of the number of ﬂow passages for passage types 1 and 2 and the frontal area is the same. In this case, Ao;1 Ao;2 A^o;1 A^o;2 ¼ 1 ¼ 2 ð12:33Þ ^ Ao;n Ao;n Ao;n A^o;n where A^o;1 , A^o;2 , and A^o;n are the ﬂow area for one passage of passage types 1, 2, and n, respectively, and 1 and 2 are the corresponding fractions of the number of passages of types 1 and 2.{ This case applies when comparing sharp and rounded triangular (or any two similar passages), where the frontal area remains constant for the same total number of ﬂow passages, regardless of which is the nominal ﬂow passage. However, the free-ﬂow area will be diﬀerent for the nominal ﬂow passages since the ﬂow areas of sharp and rounded corner passages are diﬀerent (see Example 12.3). 2. In the alternative case, the total number of ﬂow passages for the nominal passages could be different from the actual number of ﬂow passages for the same frontal area. This case applies when we compare the two-passage model (e.g., large and small rectangular or triangular passages with 50% : 50% or any other percent distribution) with the nominal passage geometry having approximately the same frontal area.{ In this case, the number of ﬂow passages for the nominal passage geometry will be different from the sum of the number of ﬂow passages of passage types 1 and 2. The ﬂow area ratios Ao;1 =Ao;n and Ao;2 =Ao;n are given by Ao;1 1 A^o;1 Ao;2 2 A^o;2 ¼ ¼ ð12:34Þ Ao;n 1 A^o;1 þ 2 A^o;2 Ao;n 1 A^o;1 þ 2 A^o;2 where the deﬁnitions of A^o;1 , A^o;2 , 1 , and 2 are the same as deﬁned above after Eq. (12.33). Note that we may use A^o; j =A^o;n in Eq. (12.34) instead of A^o; j , j ¼ 1 or 2, since the fraction is primarily known. The pressure drop ratio (the ratio of the pressure drop for either of the two passage types, either 1 or 2, to the nominal passage pressure drop) can be calculated, using Eq. (12.30), as p1 ð f ReÞ1 m_ 1 Dh;n 2 Ao;n ¼ pn ð f ReÞn m_ n Dh;1 Ao;1

ð12:35Þ

{ Note that 1 ¼ 1 2 . The parameter , by deﬁnition, represents a ratio of the number of the ith shaped passage P to the total number of passages. If more than two passages are involved, the following relation holds: i ¼ 1. Also note that ðAo;1 þ Ao;2 Þ=Ao;n 6¼ 1 in general. This is because we have presumed the same frontal area, and as a result, the wall thickness is diﬀerent for diﬀerently shaped passages (see Example 12.3). { The number of passages must be an integer, so the frontal area for a particular selection of passages may not necessarily be the same when compared to a two-passage model with a nominal passage model. However, in a compact heat exchanger with a very large number of ﬂow passages, the diﬀerence will be negligible.

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

825

Note that the ﬂow area of two nominal passages is the same as the two (large and small) passages of the nonuniform core (see Fig. 12.4 for two examples). As we know, since the ﬂuid seeks the path of least ﬂow resistance, if we replace some nominal passages with diﬀerent ﬂow passages having larger and smaller ﬂow areas, a larger fraction of the ﬂow will go through the larger ﬂow area passages. Then for a constant ﬂow rate, the pressure drop (and hence heat transfer) will reduce for this exchanger with mixed passages.{ This means that p1 =pn ð¼ p2 =pn Þ will be less than unity. This gain (reduction) in the pressure drop due to passage-to-passage nonuniformity is pgain ¼ 1

p1 pn

ð12:36Þ

Let us now determine a change in heat exchanger eﬀectiveness due to passage-topassage nonuniformity. Heat transfer through diﬀerently shaped passages would be diﬀerent, which in turn would produce diﬀerent temperature diﬀerences between ﬂuids 1 and 2. Hence, one cannot consider diﬀerent passages, let us say two passages A and B, in parallel to arrive at an eﬀective h as the average of conductances hA and hB . To arrive properly at an eﬀective value of h for a two-passage geometry heat exchanger, the passage geometrical properties, ﬂuid physical properties, exchanger ﬂow arrangement, and "-NTU relationship must be considered. A procedure is outlined in the following subsections for the two most important cases of a two-passage geometry for a counterﬂow exchanger with C* ¼ 1 (a rotary regenerator case) and an exchanger with C* ¼ 0 (an oil cooler case with constant wall temperature). Refer to Shah and London (1980) for an analysis of other ﬂow arrangements. For both these cases, the heat transfer results are presented in terms of the number of transfer units ntuj for each type of passage on the maldistributed ﬂuid side as follows: hA Nu 4L 4kL NuAo ntuj ¼ ¼ ¼ j ¼ 1; 2 ð12:37Þ m_ cp j RePr Dh j cp m_ D2h j For the nominal passage, deﬁne ntun by Eq. (12.37) with j ¼ n. The normalized ntuj =ntun , based on Eq. (12.37), is ntuj Nuj m_ n Dh;n 2 Ao; j ¼ ð12:38Þ ntun Nun m_ j Dh;j Ao;n Nuj and Nun in this equation should be obtained from the results of Table 7.3 for the appropriate thermal boundary conditions for fully developed laminar ﬂow. Note that for H1 or * H2 , while the a counterﬂow exchanger with C * ¼ 1, the boundary conditions are * T for the C * ¼ 0 case. boundary condition is * counterflow heat exchanger with C* ¼ 1. In this case, "j and ntuj are related as follows using Eq. (3.85): "j ¼

{

ntuj 1 þ ntuj

j ¼ 1; 2; n

ð12:39Þ

This is what we have shown through Eqs. (12.28)–(12.36) that the performance (q and p) of a continuous-ﬂowpassage regenerator matrix will be lower when there are small and large passages in parallel compared to the performance of nominal (average)-size passages of the same shape.

826

FLOW MALDISTRIBUTION AND HEADER DESIGN

where j depends on whether the heat exchanger unit considered has all uniform (nominal) passages, "n ð j ¼ nÞ, or it refers to a maldistributed heat exchanger that consists of two subexchangers (with the eﬀectivenesses "1 or "2 for the passage geometries j ¼ 1 or j ¼ 2, respectively). Note that since ntuj is deﬁned using a heat transfer coeﬃcient (not the overall heat transfer coeﬃcient U as in NTU), the heat exchanger eﬀectiveness of Eq. (12.39) must be deﬁned based on the passage wall temperature: To; j Ti ntuj ¼ "j ¼ 1 þ ntuj Tw Ti

j ¼ 1; 2; n

ð12:40Þ

where Tw represents the mean wall temperature of the heat transfer surface, Ti is the inlet temperature of ﬂuids in both subexchangers and nominal exchanger, and the distribution of Tw; j vs. x are parallel to the distribution of Tj vs. x as shown in Fig. 1 of Shah and London (1980). The temperature Tw is assumed to be the same for both passage geometries at the inlet (thus leading to the same inlet temperature diﬀerence for both passage types). The ntuj for Eq. (12.40) are computed from Eq. (12.38) for a speciﬁed value of ntun and known ﬂow fraction distribution from Eq. (12.32). The average eﬀectiveness of the maldistributed heat exchanger can be calculated from the eﬀectiveness of two subexchangers using a simple energy balance and assuming constant speciﬁc heat of the ﬂuids as follows: m_ "ave ¼ m_ 1 "1 þ m_ 2 "2

ð12:41Þ

It must be emphasized that the analysis presented here is for one ﬂuid side of the exchanger (either the hot- or cold-ﬂuid side of a rotary regenerator). To ﬁnd the resultant eﬀect on the exchanger performance, the eﬀect of the other ﬂuid side needs to be taken into account, as will be shown in Example 12.3. The eﬀective ntu on one ﬂuid side is then given by ntueff ¼

"ave 1 "ave

ð12:42Þ

The ‘‘cost’’ of the inﬂuence of passage-to-passage nonuniformity on ntu is deﬁned as ntucost ¼ 1

ntueff ntun

ð12:43Þ

We need to compute ntueff for both ﬂuid sides and subsequently calculate NTUeff for the exchanger to determine a reduction in the exchanger eﬀectiveness due to passage-topassage nonuniformity, as shown for a speciﬁc exchanger in Example 12.3. London (1970) determined ntucost and pgain for plate-spacing and ﬁn-spacing type nonuniformities and concluded that the deviation in passage size causes a more severe reduction in the number of transfer units than does the pressure drop gain. Speciﬁc results from the two-passage model for the passage-to-passage nonuniformity are presented in Fig. 12.5 for rectangular passages. This two-passage model consists of 50% of the ﬂow passages large (c2 > cn ) and 50% being small (c1 < cn ) compared to the nominal passages, and the nominal aspect ratios n* ¼ 1, 0.5, 0.25, and 0.125. In Fig. H1 and * T boundary conditions and for a 12.5a, a reduction in ntu is presented for the *

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

827

(a)

(b) FIGURE 12.5 Deterioration factors for two-passage nonuniformities in rectangular passages: (a) percentage loss in ntu as a function of c , n*, and thermal boundary conditions; (b) percentage reduction in p as a function of c and n*. (From Shah, 1981.)

828

FLOW MALDISTRIBUTION AND HEADER DESIGN

nominal (design) ntun of 5.0. Here, ntucost , a percentage loss in ntu, and the channel deviation parameter c are deﬁned as ntucost ¼ 1

ntueff ntun

ntu*cost ¼

ntueff 100 1 ntun

c ¼ 1

c1 cn

ð12:44Þ

where ntueff is the eﬀective ntu for the two-passage model passage-to-passage nonuniformity, and ntun is the ntu for nominal (or reference) passages. It can be seen from Fig. 12.5a that a 10% channel deviation (c ¼ 0:10, which is common for a highly compact surface) results in 10 and 21% reduction in ntuH1 and ntuT , respectively, for n* ¼ 0:125 and ntun ¼ 5:0: In contrast, a gain in the pressure drop due to the passage-topassage nonuniformity is only 2.5% for c ¼ 0:10 and n* ¼ 0:125, as found from Fig. 12.5b. Here p*gain is deﬁned as p*gain ¼

pactual 100 1 pnominal

ð12:45Þ

The following observation may be made from Fig. 12.5a and additional results preT sented by Shah and London (1980): (1) the loss in ntu is more signiﬁcant for the * H1 boundary condition; (2) the loss in ntu increases with boundary condition than for the * higher values of nominal ntu; and (3) the loss in ntu is much more signiﬁcant than the gain in p at a given c . N-Passage Model. The previous analysis was extended for an N-passage model by Shah and London (1980). In the N-passage model, there are N diﬀerent-size passages of the same basic shape, either rectangular or triangular. The results of Fig. 12.5a and b for rectangular passages are also applicable to an N-passage model in which there are N diﬀerent-size passages in a normal distribution about the nominal passage size with a proper deﬁnition of the channel deviation parameter c as follows: c ¼

" N X

i

i¼1

c 1 i cn

!2 #1=2 ð12:46Þ

Here i is the fractional distribution of the ith shaped passage. For N ¼ 2 and i ¼ 0:5, Eq. (12.46) reduces to Eq. (12.44) for c . Similar results are summarized in Fig. 12.6 for the N-passage nonuniformity model associated with equilateral triangular passages. In this case, the deﬁnition of the channel deviation parameter c is modiﬁed to c ¼

N X i¼1

" i

rh;i 1 rh;n

!2 #1=2 ð12:47Þ

where rh;n is the hydraulic radius of the nominal passages, rh;i is the hydraulic radius of the ith passage, and they are related for a two-passage model as follows: 2r2h;n ¼ r2h;1 þ r2h;2 , but this particular case corresponds to an equilateral triangular passage. Qualitative trends of the results in Fig. 12.6 are similar to those in Fig. 12.5 for rectangular ﬂow passages.

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

829

FIGURE 12.6 Percentage loss in ntu and percentage reduction in p as functions of c for Npassage nonuniformities in equilateral triangular passages. (From Shah, 1985.)

Note that the percentage reduction in ntu and p vs. c curves for N ¼ 2 and N > 2 are identical (as shown in Figs. 12.5 and 12.6), except that the value of c is higher for a two-passage model compared to the N-passage model for the same value of cmax =cn . Hence, the two-passage model provides the highest deterioration in performance. heat exchanger with C* ¼ 0. In this case, "j and ntuj are related as follows using Eq. (3.84): "j ¼ 1 entuj

j ¼ 1; 2; n

ð12:48Þ

830

FLOW MALDISTRIBUTION AND HEADER DESIGN

where j depends on whether the heat exchanger unit considered has all nominal passages, "n ð j ¼ nÞ, or it refers to a maldistributed heat exchanger that consists of two subexchangers (with the eﬀectivenesses "1 or "2 for the passage geometries j ¼ 1 or j ¼ 2, respectively). The average eﬀectiveness of the passage-to-passage maldistributed heat exchanger can be calculated using Eq. (12.41). Similar to the previous case, the cost of the inﬂuence of passage-to-passage nonuniformity on ntu is deﬁned as follows: ntueff ntun

ð12:49Þ

1 1 "ave

ð12:50Þ

ntucost ¼ 1 where ntueff ¼ ln

Refer to Shah and London (1980) for further details. Example 12.3 A vehicular gas turbine counterﬂow rotary regenerator is made up of triangular ﬂow passages. Due to brazing of the core, some of the ﬂow passages became triangles with rounded corners. Hence, idealize that the matrix is made up of 50% passages having all three corners rounded and 50% passages having all three sharp corners. Determine: (a) The ﬂow fraction distribution in the two types of passages (b) The change in pressure drop due to passage-to-passage nonuniformity. Does it represent a loss or a gain in comparison to all passages of ideal sharp corners? (c) The change in the exchanger effectiveness due to the nonuniformity. Does it represent a loss or a gain? (d) The subsequent change in ntu The additional data are as follows:

Characteristic

Sharp Corner Passage

Rounded Corner Passage

f Re

13.333

15.993

NuH1 A^o =A^o;

3.111

4.205

1

0.868

Dh =Dh;

1

1.125

For this regenerator, the ﬂow split gas : air ¼ 50% : 50%. NTUn ¼ 2:5. Idealize negligible wall resistance, C * ¼ 1 and Cr* ! 1. SOLUTION Problem Data and Schematic: The passage-to-passage ﬂow nonuniformity is caused by diﬀering ﬂow resistance of sharp and rounded corner ﬂow passages shown in Fig. E12.3. All data for thermal and hydraulic characteristics of the ﬂow passages are provided in the table above, and 1 ¼ 2 ¼ 0:5, NTUn ¼ 2:5, and C * ¼ 1.

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

831

FIGURE E12.3

Determine: The inﬂuence of passage-to-passage ﬂow nonuniformity on heat transfer and pressure drop. Assumptions: The pressure drop is uniform across all heat exchanger ﬂow passages. All the assumptions of Section 3.2.1 are valid except for the nonuniform ﬂow through the regenerator due to diﬀerently shaped ﬂow passages. Analysis: The ﬂow fraction distribution can be determined from the assumption of uniform pressure drop distribution across the heat exchanger core. We consider the ideal sharp corner ﬂow passage as a nominal passage for this example. Therefore, using Eq. (12.33), Ao;1 ¼ 1 Ao;n

A^o;sharp ¼ 0:5 1 ¼ 0:5 A^o;sharp

Ao;2 A^o;round ¼ 2 ¼ 0:5 0:868 ¼ 0:434 Ao;n A^o;sharp

The ratio of mass ﬂow rates through these passages, using Eq. (12.31), is given by m_ 1 ð f ReÞ2 ¼ m_ 2 ð f ReÞ1

Dh;1 Dh;2

2

2 Ao;1 15:993 1 0:5 ¼ ¼ 1:092 Ao;2 13:333 1:125 0:434

ð1Þ

From Eq. (1) we get 1:092 m_ 1 m_ 1 =m_ 2 m_ 1 ¼ ¼ ¼ 0:522 and ) m_ n 1 þ m_ 1 =m_ 2 1 þ 1:092 m_ n

m_ 2 ¼ 0:478 m_ n

Ans:

The ratio of the pressure drop for the sharp corner triangular passages to the nominal pressure drop can be determined as follows using Eq. (12.35): psharp p1 ð f ReÞ1 m_ 1 Dh;n 2 Ao;n 13:333 1 ¼ ¼ ¼ 0:522 12 ¼ 1:044 pn pn ð f ReÞn m_ n Dh;1 Ao;1 13:333 0:5 Ans: Similarly, ratio of the pressure drop within the rounded triangular passages to the pressure drop through the nominal passages should, ideally, be 1.044. We could have calculated this ratio using an analogous relationship to the one given by Eq. (12.35) by replacing the subscript 1 with 2.

832

FLOW MALDISTRIBUTION AND HEADER DESIGN

pround p2 ð f ReÞ2 m_ 2 Dh;n 2 Ao;n ¼ ¼ pn pn ð f ReÞn m_ n Dh;2 Ao;2 2 15:993 1 1 ¼ 0:478 ¼ 1:044 13:333 1:125 0:434 The heat exchanger eﬀectiveness due to the nonuniformity depends on the number of transfer units for the respective passages. The number of transfer units for sharp and rounded corner passages, normalized with respect to nominal passages, are determined using Eq. (12.38) as ntuj Nuj m_ n Dh;n 2 Ao; j ¼ ntun Nun m_ j Dh;j Ao;n where j ¼ 1 or 2 (i.e., sharp-corner triangular passages or rounded-corner triangular passages, respectively). Utilizing the given and calculated data, we get ntusharp ntu1 Nu1 m_ n Dh;n 2 Ao;1 3:111 1 ¼ ¼ ¼ 12 0:5 ¼ 0:9579 ntun ntun Nun m_ 1 Dh;1 Ao;n 3:111 0:522 2 nturound ntu2 Nu2 m_ n Dh;n 2 Ao;2 4:205 1 1 ¼ ¼ ¼ 0:434 ¼ 0:9697 ntun ntun Nun m_ 2 Dh;2 Ao;n 3:111 0:478 1:125 For the given NTUn ¼ 2:5, we obtain ntun ¼ ntuh ¼ ntuc ¼ 5:0 for C* ¼ 1 [see, for example, Eq. (9.23)]. With ntun ¼ 5, we get ntu1 ¼ 5 0:9579 ¼ 4:7893

ntu2 ¼ 5 0:9697 ¼ 4:8483

Consequently, the heat exchanger eﬀectivenesses for the two types of passages for C * ¼ 1 would be "j ¼

ntuj 1 þ ntuj

j ¼ 1; 2

Therefore, "1 for sharp-corner triangular passages and "2 for rounded-corner triangular passages are "1 ¼

4:7893 ¼ 0:8277 1 þ 4:7893

"2 ¼

4:8483 ¼ 0:8290 1 þ 4:8483

The average heat exchanger eﬀectiveness can be calculated from a simple energy balance, using Eq. (12.41), as "ave ¼

m_ 1 m_ " þ 2 " ¼ 0:522 0:8277 þ 0:478 0:8290 ¼ 0:8283 m_ n 1 m_ n 2

Therefore, ntueﬀ from Eq. (12.43) is ntueff ¼

"ave 0:8283 ¼ ¼ 4:8241 ¼ ntueff;h ¼ ntueff;c 1 "ave 1 0:8283

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

833

The eﬀective NTU for this regenerator, using Eq. (5.54), is given by NTUeff ¼

1 1 ¼ ¼ 2:412 1=ntueff;h þ 1=ntueff;c 1=4:8241 þ 1=4:8241

Subsequently, the regenerator eﬀectiveness with the passage-to-passage ﬂow nonuniformity is given by "eff ¼

NTUeff 2:412 ¼ ¼ 0:7069 1 þ NTUeff 1 þ 2:412

In contrast, the eﬀectiveness of the heat exchanger with nominal uniform passages is "n ¼

NTUn 2:5 ¼ ¼ 0:7143 1 þ NTUn 1 þ 2:5

Thus, the loss in the regenerator eﬀectiveness is "loss ¼

"n "ave 0:7143 0:7069 100 ¼ 100 ¼ 1:0% "n 0:7069

Ans:

Discussion and Comments: It should be noted that in pressure drop analysis, sharp-corner passages are considered as nominal passages. Thus, the pressure drop of the matrix with nonuniform ﬂow passages is 4.4% larger than that for the ideal matrix with uniform sharp corners. This is because the sharp-corner triangular passage (used for the comparison) has a lower f Re than that for the rounded-corner triangular passage. In contrast, there is a reduction (1.0%) in regenerator eﬀectiveness due to nonuniform ﬂow because of the poor performance of the rounded-corner ﬂow passages. The comparison was performed by comparing the performance of nonuniform passages with nominal sharp-corner passages which have a lower heat transfer coeﬃcient and a lower friction factor. If we would have considered the rounded-corner triangular passages as nominal passages, there would have been a reduction in the pressure drop and a slight gain in heat exchanger eﬀectiveness. 12.1.2.2 Passage-to-Passage Flow Nonuniformity Due to Other Eﬀects. Finally, passage-to-passage ﬂow nonuniformity for very compact surfaces may be induced by brazing and/or fouling in addition to manufacturing imperfection. Both controlled atmosphere brazing and vacuum brazing have a negligible eﬀect on j and f data if the plates/tubes/primary surface is clad and ﬁns are unclad, and the ratio of the joint area to free-ﬂow area is less than 10%. For ultracompact surfaces/ﬂow passages, this ratio may not be small (i.e., ﬂow area blockage and brazing-induced surface roughness may not be negligible, and accurate experimental j and f data are essential in this case). Gross blockage due to brazing may increase the pressure drop substantially. The inﬂuence of surface roughness induced by salt dip brazing (currently an outdated technology due to environmental concerns) is generally nonnegligible (i.e., can increase p considerably with only a slight increase in h or j ) in highly compact surfaces (Shah and London, 1971). Controlled atmosphere brazing, a state-of-the-art manufacturing process for compact heat exchangers (Sekulic´, 1999), provides a very uniform ﬂow passage

834

FLOW MALDISTRIBUTION AND HEADER DESIGN

distribution due to uniform distribution of a re-solidiﬁed microlayer of cladding residue on a heat transfer surface and uniform ﬁn area distribution (Sekulic et al., 2001). 12.1.3

Manifold-Induced Flow Maldistribution

Whereas manifolds are integral in plate heat exchangers due to construction features, manifolds are common and attached separately in many other applications. In the PHEs, the ﬂuids enter and exit the manifolds laterally and ﬂow within the core axially; here the axial direction is deﬁned as the main direction of ﬂuid ﬂow within the PHE passages (see Fig. 12.7a and b). In other applications, the ﬂuids enter and exit the core also axially, or a combination of axial and lateral entry and exit. In the PHEs, the manifolds are of two basic types: dividing ﬂow and combining ﬂow. In dividing-ﬂow manifolds, ﬂuid enters laterally and exits the manifold axially. The velocity within the manifold, parallel to the manifold axis, varies from the inlet velocity to zero value. Conversely, in combining-ﬂow

FIGURE 12.7 Manifold conﬁgurations: (a) U-ﬂow or parallelﬂow conﬁguration; (b) Z-ﬂow, Sﬂow, or reverse-ﬂow conﬁguration. Pressure proﬁle in (c) U-ﬂow conﬁguration, (d) Z-ﬂow conﬁguration. In these conﬁgurations, (e) typical ﬂow distribution, and ( f ) typical temperature distribution. [Parts (e) and (f ) from Thonon, 2002.]

GEOMETRY-INDUCED FLOW MALDISTRIBUTION

835

manifolds, ﬂuid enters axially from the PHE core and exits at the end of the manifold laterally, with the velocity within the exit manifold varying from zero to the outlet velocity. When interconnected by lateral branches, these manifolds result in paralleland reverse-ﬂow systems, or U- and Z-ﬂow (or S-ﬂow) arrangements, as shown in Fig. 12.7a and b. Because the inlet and outlet manifolds have the same eﬀective-diameter pipes connected by the lateral branches, this construction has built-in inherent ﬂow nonuniformity, as evidenced by the resultant typical pressure proﬁles as shown in Fig. 12.7c and d, and the mass ﬂow rate distribution as shown in Fig. 12.7e. Modeling of a manifold requires determination of both axial and lateral velocity and static pressure distributions. Available solutions of the manifold ﬂow models may be either analytical (in simpliﬁed cases, Bajura and Jones, 1976; Edwards et al., 1984; Shen, 1992) or numerical (Majumdar, 1980; Thonon et al., 1991; Heggs and Scheidt, 1992) by considering the inlet and outlet manifolds connected by the ﬂow channels in a PHE. The key problem in analytical modeling is the diﬃculty of identifying a relevant streamline on which to calculate energy and pressure losses and apply the Bernoulli equation. The state-of-the-art design procedures utilize commercial and/or proprietary CFD codes. The accurate modeling can be done only numerically and the reader is referred to the references noted in this section. Such manifold-induced ﬂow maldistribution has traditionally been analyzed by a simpliﬁed approach without explicitly including the ﬂow resistance in the lateral branches. Bajura and Jones (1976) deﬁned a set of generalized equations for manifold systems. The model consists of (1) pressure-ﬂow equations, and (2) ﬂow distribution equations, including the related boundary conditions. These equations are obtained by writing continuity and momentum equations for dividing and combining ﬂow branch control volumes near branching points, and the discharge equation, which provides the relationship of the pressure diﬀerences between the manifold and the lateral branch. So this model is based on an application of ﬁrst principles, using continuity and momentum equations for header ﬂows and discharge equation for lateral ﬂows. The experimental results for simple dividing and combining ﬂows were in very good agreement with this theory. Shen (1992) extended the work of Bajura and Jones (1976) numerically to include the eﬀect of ﬂow friction in the manifolds and the momentum losses associated with turning the ﬂow in the lateral branches. Datta and Majumdar (1980) conducted a numerical analysis of both U- and Z-ﬂow arrangements and found that the ﬂow distribution within the heat exchanger core is dependent on the following three dimensionless groups: (1) the branch-to-manifold ﬂow area ratio Ao*, (2) the manifold friction parameter F, and (3) the lateral branch pressure loss coeﬃcient Kb , all deﬁned as follows: Ao* ¼

Nc A2o;b A2o;m

F¼

fm Lm Dp Nc A2o;b

Kb ¼

p u2m =2gc

ð12:51Þ

Here Nc is the number of channels on one ﬂuid side in a PHE, Ao;b is the free-ﬂow area for a branch or an exchanger and Ao;m is the ﬂow area of the manifold, Dp is the port or manifold diameter, and the subscript m is for manifold. Note that the pressure loss coeﬃcient Kb of Eq. (12.51) is equal to the bracketed term of Eq. (6.28). The main conclusions of the inﬂuence of these parameters are as follows (Datta and Majumdar, 1980): . The relative variation in the lateral ﬂow distribution increases with increased Ao*. . A reverse-ﬂow manifold provides relatively more uniform ﬂow distribution than a parallel-ﬂow manifold for otherwise identical conditions.

836

FLOW MALDISTRIBUTION AND HEADER DESIGN

. In parallel- and reverse-ﬂow manifolds, maximum ﬂow occurs through the last port and ﬁrst port, respectively. . The eﬀect of friction parameter is in general less signiﬁcant than that of the area ratio Ao*. A few general conclusions from these studies for more uniform ﬂow through manifold systems and some design guidelines for manifolds are as follows: . Flow maldistribution is insigniﬁcant in PHEs with less than 20 ﬂow channels on a given ﬂuid pass. . Flow maldistribution in the PHEs due to the manifold system (U- or Z-ﬂow) increases with increasing ﬂow rate, increasing the number of plates in a given pass and decreasing the liquid viscosity. . In a U-ﬂow manifold system, the maximum ﬂow occurs through the ﬁrst port, and in the Z-ﬂow manifold system through the last port. Neither arrangement provides uniform ﬂow through the PHE or lateral branches. However, ﬂatter (relatively more uniform) ﬂow distribution is obtained with the U-ﬂow manifold system than with the Z-ﬂow system (see Fig. 12.7e). . To minimize ﬂow maldistribution in a PHE, the ﬂow area of the inlet manifold (area of the actual or simulated pipe before lateral branches) should be larger than the ﬂow area of the lateral branches (heat exchanger core). The larger the port diameter, the more uniform ﬂow through the heat exchanger core. Alternatively, ﬂow maldistribution in a PHE plate pack (core) increases as the fraction of the total pressure drop in the manifold becomes signiﬁcant. . The ﬂow area of a combining-ﬂow manifold in Fig. 12.12b (the outlet manifold/ pipe in Fig. 12.7a and b) should be larger than that for the dividing-ﬂow manifold in Fig. 12.12a (the inlet manifold/pipe in Fig. 12.7a and b) for a more uniform ﬂow distribution through the core in the absence of heat transfer within the core. If there is heat transfer in lateral branches (core), the ﬂow areas should be adjusted ﬁrst for the density change and then the ﬂow area of the combining manifold should be made larger than that calculated previously. . Flow reversal is more likely to occur in a Z-ﬂow system, which is subjected to poor ﬂow distribution. . Based on the limited tests, a 2-pass 2-pass Z-ﬂow arrangement can be treated as if each pass were in a separate exchanger. Thonon et al. (1991) and Heggs and Scheidt (1992), among others, have analyzed heat transfer in a PHE with U- and Z-ﬂow arrangements having 60 channels (30 channels on each ﬂuid side). They found that when both ﬂuids enter the same end of the PHE with either a U- or a Z-ﬂow arrangement, the reduction in the exchanger performance is small (ca. 2%) compared to the ideal uniform-ﬂow case and may be neglected for practical purposes. Typical temperature distributions are shown in Fig. 12.7f. However, two ﬂuids can enter at diﬀerent ends in Z- or U-ﬂow arrangements. Based on Fig. 12.7e, the Z-ﬂow has a smaller ﬂow rate/velocity at the entrance-end ﬂow channels and a large ﬂow rate/ velocity at the exit-end ﬂow channels. When both ﬂuids in the Z-ﬂow arrangement enter from the diﬀerent ends, there are two possibilities: (1) the ﬂow rate in the end channels can be the largest, or (2) the ﬂow rate in the end channels can be the smallest. A similar

OPERATING CONDITION–INDUCED FLOW MALDISTRIBUTION

837

situation exists for the U-ﬂow arrangement with ﬂuids entering from the diﬀerent ends. Heggs and Scheidt (1992) show that the eﬀect of ﬂow maldistribution on the PHE performance is severe (up to 15% for the case that they analyzed) when the two ﬂuids enter the exchangers from diﬀerent ends. From the foregoing results, we ﬁnd that if both ﬂuids enter from the same end in a PHE, the manifold system has a signiﬁcant negative impact on ﬂow maldistribution and pressure drop, and a less degrading eﬀect on overall heat transfer. However, if the ﬂuids enter from diﬀerent ends, both signiﬁcant ﬂow and temperature maldistribution can occur in a PHE. While in Section 1.5.2.1, we mentioned one of the advantages of PHEs as having the same residence time for all ﬂuid particles on any ﬂuid side, the foregoing results indicate that severe ﬂow and temperature maldistributions can occur due to inherent construction features of a PHE, resulting in diﬀerent residence times. This can have an impact on the use of a PHE in the chemical and food industries if the residence time or temperatures are to be controlled over the entire surface area.

12.2 OPERATING CONDITION–INDUCED FLOW MALDISTRIBUTION Operating conditions (temperature level, temperature diﬀerences, multiphase ﬂow conditions, etc.) inevitably inﬂuence thermophysical properties (viscosity, density, quality) and/or process characteristics (such as the onset of oscillations) of the exchanger ﬂuids, which in turn may cause various ﬂow maldistributions, both steady and transient in nature. We next summarize the inﬂuence of viscosity-induced ﬂow maldistribution, common with oil ﬂows, on exchanger performance. For ﬂow maldistribution with phase change, refer to Hewitt et al. (1994) for details. 12.2.1

Viscosity-Induced Flow Maldistribution

Viscosity-induced ﬂow instability and maldistribution are results of large changes in ﬂuid viscosity within the exchanger as a result of diﬀerent heat transfer rates in diﬀerent tubes (ﬂow passages). We discuss below two cases: (1) ﬂow instability and associated ﬂow maldistribution for liquid cooling when the wall temperature is kept constant (i.e., condensing or evaporating ﬂuid on the other ﬂuid side), and (2) a single-phase exchanger with viscous liquid on one ﬂuid side and gas or liquid on the other ﬂuid side in which an oil/viscous liquid is being heated or cooled. 12.2.1.1 Flow Instability with Liquid Coolers. A possibility for ﬂow instability is present whenever one or more ﬂuids are liquids in a heat exchanger and if the viscous liquid is being cooled. Flow maldistribution and ﬂow instability are more likely in laminar ﬂow (p / ) than in turbulent ﬂow (p / 0:2 ). Mueller (1974, 1987) has proposed a procedure for determining the pressure drop or mass ﬂow rate (in a singletube laminar ﬂow cooler) above which the ﬂow instability due to ﬂow maldistribution is eliminated within a multitubular heat exchanger. Putnam and Rohsenow (1985) have investigated the ﬂow instability phenomenon that occurs in noninterconnected parallel passages of laminar ﬂow heat exchangers. If a viscous liquid stream is cooled, depending on the liquid ﬂow rate and the length of the tube, the liquid local bulk temperature Tm may or may not reach the wall temperature Tw along the ﬂow length L (see Fig. 12.8a). If it reaches the constant wall temperature, the liquid temperature and hence its viscosity remains constant farther downstream,

838

FLOW MALDISTRIBUTION AND HEADER DESIGN

Liquid cooling

T

∆pmin

A

B

Tm C

∆p

Tw

xw

L

.

m

Liquid heating

.

mmin

FIGURE 12.8 (a) Temperature distribution of a viscous liquid in the entrance region of a tube, and (b) pressure drop vs. mass ﬂow rate for a single tube (ﬂow passage) in laminar liquid ﬂow cooling. (From Mueller, 1974.)

dependent on Tw [i.e., ¼ w ðTw Þ. In the preceding region, the viscosity will be a function of the local bulk temperature, ¼ ðTm Þ. The total pressure drop between the inlet and outlet of the ﬂow passage could be approximated as a sum of the two terms that are based on the two viscosity regions: (1) a region between tube inlet and an axial location xw below which is dependent on Tm , and (2) a region between the axial location xw and the tube end in which is dependent only on the wall temperature Tw (and it is constant) as shown in Fig. 12.8a. Assume that one can deﬁne an average viscosity ave for the region between the tube entrance and location xw such that when used in the standard pressure drop equation it gives the true pressure drop for that region. From xw to the tube exit (x ¼ L), the pressure drop is calculated using the viscosity w . The total pressure drop is the sum of the above-mentioned two pressure drops and behaves as shown by the solid line in Fig. 12.8b as a function of the ﬂow rate (Mueller, 1974); it can be explained as follows. At any low ﬂow rates m_ , the pressure drop increases rapidly and almost linearly with m_ since the entire tube has the viscosity w . As the ﬂow rate increases, more of the tube has the ﬂuid with viscosity ave and less at w . A maximum pressure drop for a tube is reached (at point A in Fig. 12.8b) when the point x ¼ xw in Fig. 12.8a reaches x ¼ L, and it depends on the ratio of the ave to w . The discharge temperature then starts to rise, the entire tube is then at a viscosity ave which continuously decreases, and the pressure drop continuously decreases up to point C in Fig. 12.8b. With further increase in the ﬂow rate m_ , the pressure drop will again increase since the decrease due to the inﬂuence of ave becomes smaller than the inﬂuence of the increase in m_ . Note that if the exchanger contained more than one ﬂow passage (i.e., for a multitubular exchanger), more than one ﬂow rate would be possible for a given pressure drop in the operating range between points A and B in Fig. 12.8b. It is in this region that the ﬂow instability is produced. There will be no ﬂow maldistribution–induced instability in a multitubular cooler if the mass ﬂow rate per tube m_ m (assuming uniform ﬂow distribution) is greater than m_ min in Fig. 12.8b. The foregoing analysis is based on (1) fully developed laminar ﬂow in the tube; (2) the viscosity is the only ﬂuid property that can vary along the ﬂow path; (3) the fact that only the frictional pressure drop contribution is signiﬁcant; and (4) the wall temperature is considered constant and lower than the ﬂuid inlet temperature. Note that if a viscous liquid is being heated in a tube with constant wall temperature, the liquid viscosity will decrease along the tube length with increasing ﬂow, however, at a

OPERATING CONDITION–INDUCED FLOW MALDISTRIBUTION

839

rate lower than the increase in the ﬂow rate. As a result, the pressure drop will increase monotonically with the liquid ﬂow rate, as shown by a dashed-line curve for the liquid heating case in Fig. 12.8b. In this case, there will not be any ﬂow instability as found for liquid cooling. For gases, the viscosity increases with the temperature. Hence, ﬂow maldistribution can occur with the constant-tube-wall-temperature case when heating the gas and not when cooling the gas, a phenomenon just opposite that for the liquids. With increasing temperature drop for liquids or the increasing temperature rise for gases, the ﬂow maldistribution becomes more pronounced for the constant-wall-temperature boundary condition.

Example 12.4 A viscous liquid ﬂows under steady, fully developed laminar ﬂow conditions through a tubular heat exchanger having two tube rows connected to inlet and outlet pipes through lateral headers. The pressure drops within the headers are negligible. The dynamic viscosity of the liquid decreases exponentially with an increase in temperature while the other properties may be considered as being nearly constant. The temperature of the wall of the channels is either lower or higher than the temperature of liquid due to evaporating/condensing ﬂuid stream on the tube outside. Thus, the wall temperature is uniform and constant along the tube length. The ﬂow rate in each tube corresponds to the ﬂow rate between that for points A and B in Fig. 12.8b. Determine which of the following conditions may exist in this heat exchanger: (1) diﬀerent mass ﬂow rates of the viscous liquid may be established in the two tube rows for the cases of both heating and cooling of the liquid, (2) a condition such as item (1) is possible only in the case of heating, or (3) such a condition is possible only in the case of cooling.

SOLUTION Problem Data and Schematic: A liquid has the following property: ¼ ðTÞ; if T1 < T2 , then 1 ¼ ðT1 Þ > 2 ¼ ðT2 Þ, and if T1 > T2 , then 1 ¼ ðT1 Þ < 2 ¼ ðT2 Þ. Here subscripts 1 and 2 denote any two temperatures T1 6¼ T2 of the liquid within the range of temperatures considered. The ﬂuid ﬂows through the two tube rows (A and B in Fig. E12.4A). The liquid temperature T1 is either lower or higher than the uniform wall temperature Tw . Determine: Whether or not two diﬀerent mass ﬂow rates may be established in the two tube rows connected to the same inlet and outlet headers under the conditions of either cooling and/or heating of the viscous liquid. Assumptions: The ﬂow is steady laminar. The pressure drops along the headers are negligible. Consequently, the pressure drops along the tube rows between the headers will be the same. The thermophysical properties of the liquid, except for the viscosity, are assumed to be constant. We assume that the liquid viscosity varies with the temperature as ¼ C1 expðC2 =TÞ where C1 and C2 are constants. Entrance and exit pressure drops are negligible. Thermal resistance on the tube side is controlling (i.e., the temperature of the wall is constant, that is, the heat transfer coeﬃcient outside the channels is very large).

840

FLOW MALDISTRIBUTION AND HEADER DESIGN

FIGURE E12.4A

Analysis: The pressure drop under fully developed laminar ﬂow through tubes can be obtained from Eq. (6.67a) or directly from Eq. (13.2) by treating P independent of Dh as p ¼

1 1 16L _ m ð f ReÞ D3h 2gc P

ð1Þ

Hence, we may conclude from Eq. (1) that the pressure drops depend on the products of viscosity and mass ﬂow rate for each of the two tube rows: pj / ðm_ Þj

ð2Þ

where j ð j ¼ A or BÞ represents an average viscosity between inlet and outlet of either tube row A or B.{ For liquids, the dynamic viscosity may be approximated by an exponential decreasing function in terms of the local temperature (Mueller, 1987): j ¼ C1 eC2 =Tj

ð3Þ

where C1 and C2 are constants, T is the local temperature, in K or 8R and j ¼ A or B. Hence, to determine the ﬂow distribution in two tube rows through Eq. (2), we need to know the corresponding temperature distributions for j evaluation. With constant wall temperature, temperature distributions along the channels are given by [see Eq. (11) in Example 11.1] j ¼

Tj Tj;i ¼ 1 eNTUj / 1 eC3 ðx=m_ j Þ Tw Tj;i

j ¼ A or B

ð4Þ

where ¼ x=L, NTUj ¼ UA=ðm_ cp Þj ¼ hA=ðm_ cp Þj ¼ ð3:66 kÞA=ðDh m_ cp Þj and C3 is a constant. In writing the last equality, assumptions that the fully developed laminar ﬂow and controlling liquid side thermal resistance are utilized. {

Note that letters A and B in Figs. 12.8b and E12.4A are not related and denote diﬀerent entities.

OPERATING CONDITION–INDUCED FLOW MALDISTRIBUTION

841

From Eq. (2) it is clear that pressure would diﬀer along the ﬂow direction at each location within tube rows A and B everywhere but at the terminal ports. Note that the pressure gradient along the ﬂow ðp=LÞ is proportional to viscosity [see Eq. (1) above], which changes along the ﬂuid ﬂow direction as indicated by Eqs. (3) and (4). So even though the pressure drop between the inlet and outlet for each tube row is going to be the same, the local pressure distributions along the ﬂow direction are not necessarily the same for the two tube rows [see Eq. (2)] in the ﬂow instability region AB of Fig. 12.8b (speciﬁed input). From Eq. (2) it must be clear that the same pressure drop for both tube rows, j ¼ A and B, may be reached by having diﬀering viscosities and diﬀering mass ﬂow rates in these two channels, at least in principle. We may say that if a larger mass ﬂow rate and corresponding lower viscosity are established in one tube row, the same product of these two entities may be obtained for a lower mass ﬂow rate and larger viscosity that may be established in the second tube row. The question is whether such conditions are possible for both heating and/or cooling. An answer would lead to a solution of the problem. So, based on the relationships of Eqs. (2)–(4), the temperature and pressure distributions for a case of liquid cooling (i.e., TA;i ¼ TB;i > Tw ) are presented in Fig. E12.4B. From these plots we can conclude that if a viscous liquid is cooled, diﬀering tube rows may have diﬀering mass ﬂow rates despite the same pressure drop in the ﬂow instability region of p vs. m_ relationship. This is because the liquid viscosity increases with a decrease in its temperature. This means that ﬂow misdistribution in diﬀerent tube rows will be present in such a situation. In the case of heating of the liquid, its viscosity decreases with an increase in temperature, and the situation is going to be diﬀerent. Namely, the liquid with an assumed larger mass ﬂow rate would have lower local temperatures than those of the liquid that would have the smaller mass ﬂow rate based on Eq. (4). If that is the case, the viscosity along the ﬂow direction for the ﬂuid with a higher mass ﬂow rate (lower temperatures) would have higher viscosities along the channel due to the relationship given by Eq. (3). The situation for the ﬂuid with smaller mass ﬂow rate (higher temperatures) would lead to the presence of lower viscosities along the ﬂow direction. So, in both cases, both the mass ﬂow rate and the viscosity are either increased or decreased for a considered tube row. However, to have an invariant pressure drop for both channels, the product of the mass ﬂow rate and viscosity must stay invariant as shown by Eq. (2). That would be quite opposite from what would actually happen based on the analysis above. Therefore, such conditions could not be satisﬁed: It is not possible to have heating of a viscous liquid that would lead to maldistribution if the pressure drop must stay the same between the headers. That

p

T TA,i = TB,i

•

mB •

∆pA = ∆pB

•

mA > mB •

mB Tw 0 x=0

•

•

mA > m B

TA,o TB,o x=L FIGURE E12.4B

0 x=0

x=L

842

FLOW MALDISTRIBUTION AND HEADER DESIGN

means there is no instability region for a p vs. m_ relationship for viscous liquid heating (see the dashed curve in Fig. 12.8b). Discussion and Comments: The analysis presented indicates that operating conditions may cause the ﬂow maldistribution in a certain operating range of liquid ﬂow rate. For example, a liquid with viscosity decreasing with an increase of temperature may feature diﬀering mass ﬂow rates in diﬀerent tube rows within the heat exchanger, if the ﬂuid is cooled and the pressure drops are kept the same for these channels at their respective terminal ports under certain operating conditions. If the ﬂuid is heated, the mass ﬂow rates in diﬀering tubes must be approximately the same to keep the same pressure drops. It should be noted that this analysis was conducted for a viscous liquid that is characterized with a decreasing viscosity with increasing temperature. Gases may feature the opposite behavior (although the changes of the properties would be even smaller). Such a situation is considered in Problem 12.8. Mueller (1974) proposed the following procedure to determine the minimum pressure drop, pmin , above which the ﬂow maldistribution–induced instability would not be possible. The case considered is for a viscous liquid of a known inlet temperature being cooled as it ﬂows through the length of a tube of known constant temperature Tw . 1. From viscosity data, determine the slope m of the curve ln vs. 1/T, where T is temperature on the absolute temperature scale. 2. With the known slope m and liquid viscosities at the inlet (i ) and wall (w ) temperatures, determine the average liquid viscosity (ave ) using Fig. 12.9. This ﬁgure is based on the assumption that the ﬂuid temperature reaches the wall temperature within the tube. If the ﬂuid exit temperature is still larger than the wall temperature, the average viscosity should be modiﬁed. The details are provided by Mueller (1974). 3. With these viscosities, determine xw of Fig. 12.8 from xw ¼

L 2ð1 ave =w Þ

ð12:52Þ

m_ cp 0:4 kL

ð12:53Þ

4. Calculate the mass ﬂow rate from Gz ¼

The calculated mass ﬂow rate corresponds to the breakthrough mass ﬂow rate (Mueller, 1974). The maximum point A in Fig. 12.8b occurs when the actual mass ﬂow rate through the tube approximately equals to the breakthrough value given by Eq. (12.53), and the ratio of viscosities must be less than 0.5. 5. Calculate the minimum pressure drop from pmin ¼

128 m_ ½xw ave þ ðL xw Þw gc D4h

ð12:54Þ

OPERATING CONDITION–INDUCED FLOW MALDISTRIBUTION

843

FIGURE 12.9 Viscosity ratio chart for various slopes of m of ¼ C1 eC2 =T : (From Mueller, 1974.)

If the pressure drop for the given design is found to be less than that calculated by Eq. (12.54), the ﬂuid ﬂow length should be increased (either increasing the duct length or considering a multipass design) to eliminate the ﬂow instability. If the ratio of the average and wall temperature viscosities is larger than 0.5, the maximum pressure drop can occur at a ﬂow rate larger than the breakthrough mass ﬂow rate. If the exchanger pressure drop exceeds pmin , ﬂow instability will not be a problem. However, ﬂow maldistribution is still possible, as considered in the next subsection. When an exchanger has multiple (N) tubes, and the total ﬂow rate on the tube side is less than N m_ min (where m_ min is deﬁned in Fig. 12.8 per tube), diﬀerent ﬂow rates will establish in diﬀerent tubes depending on their operating average temperature. Conceptually, the ﬂow rate in individual tubes could be calculated based on the preceding method. Hence, one can visualize the ﬂow maldistribution due to diﬀerent ﬂuid viscosity in diﬀerent tubes. While Mueller (1974) provides a method for computing pmin or m_ min per tube and recommends operation of the multitubular exchanger at ﬂow rates higher than m_ min , Putnam and Rohsenhow (1985) provide a method for operation when m_ < m_ min for a speciﬁc case. 12.2.1.2 Flow Maldistribution When No Flow Instability Present. When the ﬂow rate m_ > N m_ min (in reality, each tube or ﬂow passage should have the ﬂow rate greater than m_ min Þ, there is no ﬂow instability. However, there will be passage-to-passage ﬂow

844

FLOW MALDISTRIBUTION AND HEADER DESIGN

maldistribution, due to the viscosity change in diﬀerent ﬂow passages in parallel. In that case, the pressure drop in individual ﬂow passages can be determined from Eq. (12.29). In this equation, fj will be diﬀerent in diﬀerent ﬂow passages, due to the viscosity change; this is because the change in the viscosity will result in um; j and hence Rej . For the analysis, consider a two-tube-row design with fully developed laminar ﬂow in both tube rows. Now for the fully developed laminar ﬂow case, we can use Eq. (12.30) for the two passages having diﬀerent but the same f Re because the ﬂow passage geometry of both tube rows is identical in the present case. From Eq. (12.30), we get the mass ﬂow rate ratio for the two passages as m_ 1 2 ¼ m_ 2 1

ð12:55Þ

For turbulent ﬂow, use f / Re0:2 or a similar relationship in Eq. (12.29) to get the ratio m_ 1 =m_ 2 in terms of the 2 =1 ratio (see, e.g., an equation for this ratio in Problem 12.6). In a similar manner, we can analyze developing laminar ﬂow with the correct values for the f factors in Eq. (12.29). Once the ﬂow fraction distribution is found from Eq. (12.55), the pressure drop ratio for fully developed laminar ﬂow case can be determined by Eq. (12.35), where the subscript n represents the case corresponding to the mean temperature for both tube rows (i.e., for the exchanger on that ﬂuid side). For other ﬂow types, calculate the pressure drop ratio from the following equation, derived from Eq. (12.29): p1 f1 m;1 um;1 2 ¼ pn fn m;n um;n

ð12:56Þ

The heat transfer case can be analyzed for the constant-wall-temperature boundary condition using Eq. (12.38) to determine ntuj for the given ntun and the computed ﬂow fraction distribution from Eq. (12.55). Subsequently, follow Eqs. (12.48)–(12.50) to determine the impact on the heat transfer performance due to ﬂow maldistribution in a viscous cooler. Be sure to modify these equations as appropriate when the ﬂow is turbulent or developing laminar ﬂow on the ﬂow maldistributed side.

12.3 MITIGATION OF FLOW MALDISTRIBUTION Flow maldistribution in heat exchangers may have serious consequences on thermal and mechanical performance as elaborated in detail by Kitto and Robertson (1989). Flow maldistribution in a heat exchanger may be reduced through modiﬁcations in the existing design or taken into account by incorporating its eﬀect in the design methodology. Most gross ﬂow maldistributions may result in relatively minor heat transfer performance reduction, smaller than 5% for NTU < 4 for tube-side ﬂow maldistribution in a shelland-tube exchanger as reported by Mueller and Chiou (1988). At high NTUs (NTU > 10), the performance loss may be substantially larger. However, the increase in pressure drop is generally substantial with gross ﬂow maldistribution. Where large temperature diﬀerences exist, the eﬀect of gross ﬂow maldistribution can result in excessive thermal stress in the heat transfer surface wall (Mueller, 1987). Passage-to-passage maldistribution may result in a signiﬁcant reduction in heat transfer performance, particularly for laminar ﬂow exchangers.

HEADER AND MANIFOLD DESIGN

845

Any action to prevent ﬂow maldistribution must be preceded by an identiﬁcation of possible reasons that may cause the performance deterioration and/or may aﬀect the mechanical characteristics of the heat exchanger. Possible consequences from the performance viewpoint are (1) deterioration in heat exchanger eﬀectiveness and increase in pressure drop; (2) ﬂuid ‘‘freezing,’’ as in viscous ﬂow coolers; (3) ﬂuid deterioration; (4) enhanced fouling; and (5) mechanical and tube vibration problems due to ﬂow instabilities, wear, fretting, erosion, and corrosion and mechanical failure. No generalized recommendations can be made for preventing the negative consequences of ﬂow maldistribution. Most problems must be solved by intelligent designs and diagnosis on an individual basis. A few broad guidelines for shell-and-tube heat exchangers are: . Gross ﬂow maldistribution may be induced at inlet nozzles on the shell side. Placing an impingement perforated baﬄe about halfway to the tubesheet will break up the inlet jet stream (see Fig. 10.2). . The shell inlet and exit baﬄe spaces are the regions prone to gross ﬂow maldistribution. An appropriate design of the baﬄe geometry (e.g., the use of double segmental or disk-and-doughnut baﬄes) may reduce this maldistribution. . Passage-to-passage ﬂow maldistribution may be reduced by improved control of the manufacturing process (tolerances and gaps). For example, for brazed compact heat exchangers, use of the state-of-the-art controlled-atmosphere brazing, with improved temperature uniformity during brazing, good control of brazing parameters such as cladding ratio and ﬂux amount, and proper ﬁxturing, may signiﬁcantly reduce the presence of passage-to-passage nonuniformities. . Manifold-induced maldistribution may be controlled by careful control of the area ratio and lateral ﬂow resistance. These parameters may be ﬁxed in many systems by requirements other than these considerations. In such cases, the relative length of the manifold, the friction factors, and the orientation between the manifolds may be used as factors that may reduce ﬂow maldistribution. . Operating condition–induced ﬂow maldistribution is diﬃcult to control. For the laminar ﬂow maldistribution, a design must be such as to allow suﬃcient pressure drop to prevent maldistribution or to resort to multipassing. It should be noted that heat exchangers involving multiphase ﬂows might be most prone to ﬂow maldistribution. However, this type of heat exchanger is outside the scope of this book. For single-phase heat exchangers, good engineering judgment, involving the considerations discussed above, may greatly reduce any possible inﬂuence of ﬂow maldistribution. In many cases, even though the overall inﬂuence of maldistribution on the heat exchanger performance is small, local phenomena caused by ﬂow nonuniformity may be of great importance. These may cause increased corrosion, erosion, wear, fouling, and even material failure (Kitto and Robertson, 1989).

12.4 HEADER AND MANIFOLD DESIGN Headers and manifolds are ﬂuid distribution elements connecting the heat exchanger core and the inlet and outlet ﬂuid ﬂow lines. An inlet header is the transition duct joining the inlet face of the heat exchanger core or matrix to the inlet pipe for each ﬂuid. Similarly, an outlet header joins the outlet face of the exchanger core to the outlet

846

FLOW MALDISTRIBUTION AND HEADER DESIGN

FIGURE 12.10 Typical compact heat exchangers with (a) normal headers and (b) turning headers.

(return) pipe. The header is variously referred to as a tank, box, or distributor. Manifolds have a bit more complex function, especially for compact heat exchangers. An incoming stream must be distributed uniformly into a heat exchanger core for the lowest core p and the highest achievable q. Basically, a manifold is a ﬂow channel/duct with one (side or central) inlet and multiple sidewall outlets to the heat exchanger core, or vice versa. Compact heat exchangers having normal and turning headers are illustrated in Fig. 12.10 and oblique ﬂow headers in Fig. 12.11. Examples of manifolds with ﬂow distributions are shown in Fig. 12.12.

FIGURE 12.11 Oblique-ﬂow headers: (a) parallelﬂow; (b) counterﬂow; (c) free discharge header. (From London et al., 1966.)

HEADER AND MANIFOLD DESIGN

Dividing flow

Combining flow

(a)

(b)

Reverse flow

(c)

847

Parallel flow

(d)

FIGURE 12.12 Major types of manifolds: (a) dividing-ﬂow manifolds, (b) combining-ﬂow manifolds. The corresponding inlet/outlet conﬁgurations (see also Fig. 12.7); (c) reverse-ﬂow conﬁguration, (d) parallelﬂow conﬁguration.

Two important requirements may be identiﬁed for header and/or manifold design. They should be designed so that they result in (1) uniform distribution of the ﬂuid stream within a heat exchanger core, and (2) minimal pressure drop within the header/manifold, since in general we do not get any heat transfer for that pressure drop expenditure. The design of the inlet header is more critical. An area increase from the inlet pipe to the core face may be 5 to 50 times. It is impossible to maintain streamline ﬂow in headers for such a large area enlargement. Hence, the ﬂow is normally separated in an inlet header with either a completely detached or a singly attached jet on one wall. Flow separation results in increased pressure drop in the inlet header and nonuniform ﬂow distribution at the core face. In addition to ﬂow separation, the shape of inlet header could produce high-velocity regions, and this could lead to localized erosion at the core face (tube entrance from the tank), particularly for liquid ﬂows. The design of the outlet header should match that of the inlet header (or vice versa) so that the pressure drop across the core is uniform, resulting in the uniform ﬂow distribution. To minimize pressure losses due to ﬂow separation, area contraction in the outlet header should be smooth. Also, sharp turns in the outlet header should be avoided. Turns and bends create centrifugal forces, resulting in nonuniform pressure at the core face, which may lessen ﬂow uniformity in the core. From the foregoing viewpoints, the header design objective is to provide for acceptably uniform ﬂow through the core with an acceptable header geometry and acceptably low pressure drops since this spent p in the headers is not associated with heat transfer between two ﬂuids (the heat exchanger core is designed for this heat transfer). In baﬄed shell-and-tube exchangers and in some multipass crossﬂow heat exchangers, good ﬂuid mixing takes place on the shell side or within the core. Flow nonuniformity at the core inlet generally does not degrade the performance of the exchanger if there is good mixing within the core. The design of headers is important for those exchangers in which there is very little ﬂuid mixing in the core, and for the gas side of a gas-to-ﬂuid heat exchanger. In the highly compact gas-to-ﬂuid heat exchanger, the header design on the gas side is more diﬃcult because the exchanger core shape is characterized by a large ﬂow frontal area and a short ﬂow length for the gas ﬂow path. In such cases, either no header is used, as in an automobile radiator (on the air side), or an oblique ﬂow header of the type

848

FLOW MALDISTRIBUTION AND HEADER DESIGN

mentioned in Section 12.4.1 (or a conical/tapered manifold of round cross section) is used. Normal-ﬂow headers are also used. We now discuss these two types of headers (oblique ﬂow and normal ﬂow) for compact heat exchangers. 12.4.1

Oblique-Flow Headers

The design theory for oblique-ﬂow headers for heat exchangers has been derived by London et al. (1968), based on the work of Perlmutter (1961). This theory is based on the study of ﬂow conditions and corresponding header shape, assuming steady, constantdensity, inviscid ﬂow. Three single-pass header conﬁgurations are of particular interest, as shown in Fig. 12.11. In an oblique-ﬂow header, the ﬂuid inlet ﬂow direction with respect to the core face is at an angle diﬀerent from 908 (i.e., normal ﬂow, as in a normal-ﬂow header). A special class of oblique-ﬂow headers has an inlet ﬂow direction parallel to the core face area. The main feature of this type of header is the minimization of header volume and ﬂow separation. The three main types of oblique headers are: 1. Parallelﬂow headers (Fig. 12.11a). The ﬂuid inlet and outlet are on opposite sides of the core and the ﬂuid ﬂows in the same directions through both headers. 2. Counterﬂow headers (Fig. 12.11b). The ﬂuid inlet and outlet are on the same side of the core and the ﬂuid ﬂows in counterﬂow through the headers. 3. Free discharge headers (Fig. 12.11c). Only the inlet header exists and ﬂuid discharges freely at the outlet without ducting. In Table 12.1, the model predictions for pressure distributions, theoretical shapes, and pressure drops are compiled for the three types of oblique headers based on the results of London et al. (1968). The assumptions adopted are given in Example 12.5. A simple box conﬁguration is considered for the outlet header (if the outlet header exists) from the construction and cost points of view; then inlet headers for parallelﬂow and counterﬂow oblique headers require a special shape to achieve the uniform ﬂow distribution through the core; and the shapes are derived theoretically as summarized in Table 12.1. All geometrical characteristics and the notation are presented in Fig. 12.11. If a designer has freedom to select the header type, a counterﬂow header would be the best option for the lowest p in headers, followed by the free discharge header and parallelﬂow header. Reviewing the results in Table 12.1, note that p for the inlet header is higher than that for the outlet header in all three cases, and p for the outlet headers is largely associated with the nonuniform velocity distribution at the exit shown in Fig. 12.11a and b. In addition to the three conﬁgurations of oblique headers presented in Table 12.1 and Fig. 12.11, various other conﬁgurations are possible (such as headers at diﬀerent incoming angles, or with turning vanes or guide vanes to turn the ﬂow and minimize ﬂow nonuniformity). No systematic studies are reported in the literature for such headers, due to many geometrical variations and lack of any available theory. One of the best alternatives is to conduct three-dimensional CFD analysis when accurate ﬂow distribution and pressure drop predictions are required. Example 12.5 Determine the geometry of the inlet header, z=yo as a function of X* ¼ x=L, of the parallelﬂow oblique header of Fig. 12.11a (note that the outlet header

849

pi pðX *Þ 2 zi 2 ¼ ðX *Þ2 i 4 o yo Hi

Hi ¼

u2m 2gc

Inlet header proﬁle

Inlet header proﬁleb

Inlet header proﬁleb

i

Ho ¼

z ¼ 1 X* zi

u2m 2gc

o

X* ¼

x L

z 1 X* ¼ yo fðyo =zi Þ2 ð2 =4Þði =o Þ½1 ð1 X *Þ2 g1=2

z 1 X* ¼ yo ½ð2 =4Þði =o ÞðX *Þ2 þ ðyo =zi Þ2 1=2 pðX *Þ pi 2 i zi Inlet header pressure ¼ ½1 ð1 X *Þ2 Hi 4 o yo

Inlet header pressure

Theoretical Modela

Inlet:

pi ¼1 Hi pt ¼1 Total: Hi

pt H ¼ 1 þ 1:467 o Hi Hi pi 1 ¼ Inlet: Hi 3 H d po Outlet : ¼ 0:645 o Hi Hi pt Total: ¼ 0:595 Hi Total:

po H ¼ 0:645 o Hi Hi

pi 2 Ho ¼1þ Hi 12 Hi

Outlet:

Inlet:

Pressure Drop

Source: Data from London et al. (1968). a The geometry of header designs is presented in Fig. 12.11. Subscripts i and o denote the entrance of the inlet header and the exit of the outlet header. b The outlet header is a box header (required for a minimum inlet header loss for counterﬂow conﬁguration and imposed for parallelﬂow conﬁguration). pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c For the parallelﬂow header, the inlet header dimension zi can be either larger or smaller than the outlet dimension yo ; for counterﬂow, zi must be ð2=Þyo o =i to assure the matching pressure proﬁle needed for uniform ﬂow distribution and for minimum header loss. The outlet header is also considered to be a box header. d Ho =Hi ¼ 4=2 for a counterﬂow header.

Free discharge header

Counterﬂow headerc

Parallelﬂow header

Header Type

TABLE 12.1 Theoretical Shape, Pressure Proﬁles, and Pressure Drops of Oblique-Flow Headers

850

FLOW MALDISTRIBUTION AND HEADER DESIGN

is a box type) for the case when inlet and outlet velocity heads are equal ðHi ¼ Ho Þ and i =o ¼ 1:4. Use the inlet header proﬁle given in Table 12.1 in which z=yo is dependent on yo =zi , the ratio of outlet to inlet header height. SOLUTION Problem Data and Schematic: The inlet header is a parallelﬂow oblique header as shown in Fig. 12.11a with appropriate notations. The following data are given: Ho ¼

u2m 2gc

! ¼ Hi ¼ o

u2m 2gc

! i

i ¼ 1:4 o

Determine: The inlet header geometry [i.e., z=yo ¼ ðx=LÞ], shown in Fig. 12.11a. Assumptions: The following assumptions are made (London et al., 1968): (1) the inlet and outlet header ﬂuid mass densities are individually constant, (2) the inlet header velocity distribution is uniform, (3) the ﬂow velocity and pressure in the inlet header are functions of x only, (4) the inlet and outlet header ﬂuid ﬂows are inviscid, (5) the outlet header pressure is a function of x only, (6) the velocity is uniform through the core, and (7) the depth of the header (the third dimension) is unity. Analysis: From Table 12.1, the geometry of an inlet parallelﬂow oblique header is given by Z¼

z 1 X* ¼ 2 yo ½ð =4Þði =o ÞðX *Þ2 þðyo =zi Þ2 1=2

ð1Þ

Let us ﬁrst demonstrate brieﬂy how this equation may be derived (for details, see London et al., 1968). By applying the Bernoulli equation for a streamline through the inlet header, core, and outlet header (Fig. 12.11a), we can relate inlet pressure to a pressure at any x position as follows: pi pðxÞ ¼

i 2 ðu u2i Þ 2gc

ð2Þ

Invoking assumptions concerning constant ﬂuid density, uniform inlet velocity, and uniform velocity v through the core (in the negative z direction in Fig. 12.11a), the continuity equation will have the following form{: ui zi ¼ uzðxÞ þ vx ¼ vðL xÞ þ vx ¼ vL ¼ uo yo

{

o i

ð3Þ

Note that x-direction mass ﬂow rate through any cross section x plus the mass ﬂow rate through the core in the y direction from x ¼ 0 to x is equal to mass ﬂow rate entering the inlet header at x ¼ 0. Alternatively, x-direction mass ﬂow rate through any cross section x is the same as the mass ﬂow rate going through the core in the y direction from x ¼ x to L.

HEADER AND MANIFOLD DESIGN

851

Note v ¼ vm ¼ constant; also vm L ¼ ui zi . Now it can be shown, based on the analysis of London et al. (1968), that the pressure drop described by Eq. (2) for y ¼ constant for a box type outlet header must be equal to (2 =4ÞHo ðx=LÞ2 ; that is, pi pðxÞ ¼

2 i 2 u u2i ¼ Ho ðX *Þ2 2gc 4

ð4Þ

Invoking the deﬁnitions of Ho and Hi from Table 12.1, we get 2 uo H ¼ o i ui H i o

ð5Þ

Finally, combining Eqs. (3) and (4) to eliminate u from the second equality in Eq. (4) and keeping in mind the deﬁnition of velocity heads from Table 12.1 and as indicated above by Eq. (5), we obtain Eq. (1). A detailed derivation of Eq. (1) is the subject of Problem 12.9. We will now derive the ratio yo =zi as a function of the velocity head ratio Hi =Ho from the continuity equation considering mass ﬂow rates to be the same at the entrance of the inlet and outlet headers: ðAo um Þi ¼ ðAo um Þo Therefore, um;i o Ao;o o yo ¼ ¼ um:o i Ao;i i z i

or

um;i um;o

2 ¼

2 2 o yo i zi

ð6Þ

From the deﬁnition of the velocity heads (see Table 12.1), we get Hi ðu2m =2gc Þi ¼ Ho ðu2m =2gc Þo

or

Hi i um;i 2 ¼ H o o um;o

ð7Þ

Now, combining Eqs. (6) and (7), we get Hi o yo 2 ¼ Ho i zi

ð8Þ

Substituting ðyo =zi Þ2 from Eq. (8) into Eq. (1), we get the geometry of the inlet header as z 1 X* ¼ yo ði =o Þ½ð2 =4ÞðX *Þ2 þ Hi =Ho 1=2

ð9Þ

Since the inlet and outlet velocity heads must be equal (imposed by the problem formulation; i.e., Hi ¼ Ho ), Eq. (9) can be rewritten as follows using i =o ¼ 1:4 as given z 1 X* 1 X* ¼ ¼ yo ði =o Þ½ð2 =4ÞðX *Þ2 þ11=2 1:4½ð2 =4ÞðX *Þ2 þ11=2

ð10Þ

852

FLOW MALDISTRIBUTION AND HEADER DESIGN

We can now calculate the coordinates of the inlet header shape z=yo : X*

0

0.2

0.4

0.6

0.8

1.0

z=yo

0.714

0.545

0.363

0.208

0.089

0

Discussion and Comments: The inlet header proﬁle calculated demonstrates a need to design this header with a variable header cross section according to the functional relationship given by Eq. (10) for the case when the outlet header is box type and uniform ﬂow is through the core. In this case, the inlet and outlet velocity heads would be equal, and the header loss would be, from Table 12.1, pt =Hi ¼ 1 þ 1:467ðHo =Hi Þ ¼ 1 þ 1:467 1 ¼ 2:47 velocity heads. 12.4.2

Normal-Flow Headers

Normal-ﬂow headers are characterized as having the ﬂow direction perpendicular to the heat transfer core (see Fig. 12.10a). The design of a normal-ﬂow header follows the design of a diﬀuser with a large increase in the free-ﬂow area from the inlet pipe to the heat exchanger core face. This type of header design is qualitatively discussed by Wilson (1966). The pressure drop, ﬂow separation, and recirculation (if any) depend on the diﬀuser geometry, which includes the type (two dimensional vs. three dimensional, rectangular vs. conical, etc.), included angle, aspect ratio (diﬀuser throat to length ratio), and ﬂow type. For a heat exchanger, the diﬀuser (inlet normal header) is followed by the heat exchanger core having ﬁnite pressure drop. Hence, the design information for a diﬀuser having no downstream ﬂow resistance will be conservative for a heat exchanger. If the inlet header has a box conﬁguration, a jet will be formed from the inlet pipe and will increase in diameter before impinging on the core face. So, to minimize the header volume and pressure losses, it is desirable to make the inlet header as a conical section (to match the spreading jet diameter reasonably) followed by a plenum chamber, instead of making it one large plenum chamber. For liquid ﬂows, a plain or perforated baﬄe or a perforated plate is used in the inlet tank for distributing ﬂow to all tubes in the noﬂow direction (see Section 1.6.1.3 for the deﬁnition of noﬂow direction). Such a design reduces header/tank volume and maintains an acceptably uniform ﬂow distribution. 12.4.3

Manifolds

Two major types of manifolds, as noted in Section 12.1.3, are shown in Fig. 12.12a and b: dividing-ﬂow manifolds and combining-ﬂow manifolds. In dividing-ﬂow manifolds, ﬂuid enters either axially or laterally and exits the manifold laterally. The axial velocity within the manifold varies from the inlet velocity to the zero value. Conversely, in combining ﬂow manifolds, ﬂuid enters laterally and exits at the end of the manifold either axially or laterally. Inlet and outlet manifolds in a heat exchanger can be arranged as (1) reverse-ﬂow conﬁguration and (2) parallelﬂow conﬁguration, as shown in Fig. 12.12c and d, interconnected by lateral branches represented by ﬂow passages in the heat exchanger core (see also Fig. 12.7). Modeling a manifold requires determination of both axial and lateral velocity and static pressure distributions. Available solutions of the manifold ﬂow models may be either analytical (in simpliﬁed cases, Bajura and Jones, 1976) or numerical (Majumdar,

SUMMARY

853

1980). The key problem in analytical modeling is the diﬃculty in identifying a relevant streamline on which to calculate energy and pressure drop losses and apply the Bernoulli equation. The state-of-the-art design procedures utilize commercial and/or proprietary CFD codes. Still, simple analytical modeling has merit for assessment purposes (Shen, 1992). SUMMARY This chapter deals with several important ﬂow maldistributions (geometry and operating conditions induced) in heat exchangers. Design theory is provided to determine quantitatively the inﬂuence of two major geometry-induced ﬂow maldistributions: (1) gross ﬂow maldistribution and (2) passage-to-passage ﬂow maldistribution. . Simple modeling for gross ﬂow maldistribution is possible only for counterﬂow, parallelﬂow, and one crossﬂow unmixed–mixed arrangements. For other ﬂow arrangements, the inﬂuence of ﬂow maldistribution can be evaluated numerically. In many situations, gross ﬂow maldistributions do not reduce heat transfer signiﬁcantly but may cause a signiﬁcant increase in pressure drop. Some speciﬁc gross ﬂow maldistributions in an unmixed–unmixed crossﬂow exchanger can increase the exchanger eﬀectiveness. This is mentioned in the paragraph just before Example 12.2. . Compact heat exchangers with continuous-ﬂow passages are highly susceptible to passage-to-passage ﬂow maldistribution important in laminar ﬂows. This maldistribution reduces the pressure drop slightly but may reduce heat transfer signiﬁcantly. Usually, the pressure drop reduction is neglected in the design. The simplest model of passage-to-passage ﬂow maldistribution is the two-passage model, where two diﬀerent-size passages are in parallel. The two-passage model reduces exchanger eﬀectiveness and hence heat transfer performance more than that for an Npassage (N > 2) model of passage-to-passage nonuniformity. Hence, for a conservative design, a two-passage model is most appropriate to determine the eﬀect of passage-to-passage ﬂow maldistribution. Among operating condition–induced ﬂow maldistributions, the most important is viscosity-induced ﬂow maldistribution. It can induce ﬂow instability in a multitube or multicontinuous passage exchanger if the ﬂow rate is below some critical value (m_ min in Fig. 12.8). If m_ > m_ min , the viscosity-induced ﬂow maldistribution problem reverts to the problem of passage-to-passage nonuniformity. It should be mentioned that no generalized recommendations can be made for preventing negative inﬂuences of ﬂow maldistribution. Each case should be considered separately. Header and manifold design is very important in controlling the level of ﬂow maldistribution within the core and reducing undesired pressure drop in headers and manifolds, particularly for compact heat exchangers. Speciﬁc information for designing the header cross-section proﬁle is presented in Section 12.4 for oblique-ﬂow headers. REFERENCES Bajura, R. A., and E. H. Jones, Jr., 1976, Flow distribution manifolds, ASME J. Fluid Eng., Vol. 98, pp. 654–666.

854

FLOW MALDISTRIBUTION AND HEADER DESIGN

Chiou, J. P., 1980, The advancement of compact heat exchanger theory considering the eﬀects of longitudinal heat conduction and ﬂow nonuniformity, in Compact Heat Exchangers: History, Technological Advancement and Mechanical Design Problems, R. K. Shah, C. F. McDonald, and C. P. Howard, eds., Book G00183, HTD-Vol. 10, American Society of Mechanical Engineers, New York, pp. 101–121. Chiou, J. P., 1982, The eﬀect of nonuniformities of inlet temperatures of both ﬂuids on the thermal performance of crossﬂow heat exchanger, Heat Transfer 1982, Proc. 7th Int. Heat Transfer Conf., Vol. 6, pp. 179–184. Chowdhury, K., and S. Sarangi, 1985, The eﬀect of ﬂow maldistribution on multipassage heat exchanger performance, Heat Transfer Eng., Vol. 6, No. 4, pp. 45–54. Cichelli, M. T., and D. F. Boucher, 1956, Design of heat exchanger heads for low holdup, AIChE Chem. Eng. Prog., Vol. 52, No. 5, pp. 213–218. Datta, A. B., and A. K. Majumdar, 1980, Flow distribution in parallel and reverse ﬂow manifolds, Int. J. Heat Fluid Flow, Vol. 2, pp. 253–262. Fleming, R. B., 1966, The eﬀect of ﬂow distribution in parallel channels of counterﬂow heat exchangers, in Advances in Cryogenic Engineering, Vol. 12, K. D. Timmerhaus, ed., Plenum Press, New York, pp. 352–363. Heggs, P. J., and H.-J. Scheidt, 1992, Thermal performance of plate heat exchangers with ﬂow maldistribution, in Compact Heat Exchangers for Power and Process Industries, American Society of Mechanical Engineers, HTD-Vol. 201, pp. 87–93. Hewitt, G. H., G. L. Shires, and T. R. Bott, 1994, Process Heat Transfer, CRC Press, Boca Raton, FL. Kitto, J. B., and J. M. Robertson, 1989, Eﬀects of maldistribution of ﬂow on heat transfer equipment performance, Heat Transfer Eng., Vol. 10, No. 1, pp. 18–25. Kutchey, J. A., and H. L. Julien, 1974, The measured inﬂuence of ﬂow distribution on regenerator performance, SAE Trans., Vol. 83, SAE Paper 74013. London, A. L., 1968, Laminar ﬂow gas turbine regenerators – the inﬂuence of manufacturing tolerances, T.R. No. 69, Department of Mechanical Engineering, Stanford University, Stanford, California, 1968. London, A. L., 1970, Laminar ﬂow gas turbine regenerators: the inﬂuence of manufacturing tolerances, ASME J. Eng. Power, Vol. 92, Ser. A, pp. 45–56. London, A. L., G. Klopfer, and S. Wolf, 1968, Oblique ﬂow headers for heat exchangers – the ideal geometries and the evaluation of losses, T.R. No. 63, Department of Mechanical Engineering, Stanford University, Stanford, California. London, A. L., G. Klopfer, and S. Wolf, 1968, Oblique ﬂow headers for heat exchangers, ASME J. Eng. Power, Vol. 90, Ser. A, pp. 271–286. Majumdar, A. K., 1980, Mathematical modeling of ﬂows in dividing and combining manifolds, Appl. Math. Model., Vol. 4, pp. 424–434. Mueller, A. C., 1974, Criteria for maldistribution in viscous ﬂow coolers, Heat Transfer 1974, Proc. 5th Int. Heat Transfer Conf., Vol. 5, pp. 170–174. Mueller, A. C., 1977, An inquiry of selected topics on heat exchanger design, AIChE Symp. Ser. 164, Vol. 73, pp. 273–287. Mueller, A. C., 1987, Eﬀects of some types of maldistribution on the performance of heat exchangers, Heat Transfer Eng., Vol. 8, No. 2, pp. 75–86. Mueller, A. C., and J. P. Chiou, 1988, Review of various types of ﬂow maldistribution in heat exchangers, Heat Transfer Eng., Vol. 9, No. 2, pp. 36–50. Perlmutter, M., 1961, Inlet and exit header shapes for uniform ﬂow through a resistance parallel to the main stream, ASME J. Basic Eng., Vol. 83, pp. 361–370. Putnam, G. R., and W. M. Rohsenow, 1985, Viscosity induced nonuniform ﬂow in laminar ﬂow heat exchangers, Int. J. Heat Mass Transfer, Vol. 28, pp. 1031–1038.

REVIEW QUESTIONS

855

Ranganayakulu, C. H., K. N. Seetharamu, and K. V. Sreevatsan, 1996, The eﬀects of inlet ﬂuid ﬂow nonuniformities on thermal performance and pressure drops in crossﬂow plate-ﬁn compact heat exchanger, Int. J. Heat Mass Transfer, Vol. 40, pp. 27–38. Sekulic´, D. P., 1999, Behavior of aluminum alloy micro layer during brazing, in Recent Res. Dev. Heat Mass Momentum Transfer, Vol. 2, pp. 121–140. Sekulic´, D. P., C. Pan, F. Gao, and A. T. Male, 2001, Modeling of molten cladding ﬂow and diﬀusion of Si across a clad-core interface of an aluminum brazing sheet, DVS Berichte, Vol. 212, pp. 204–219. Shah, R. K., 1981, Compact heat exchangers, in Heat Exchangers: Thermal-Hydraulic Fundamentals and Design, S. Kakac¸, A. E. Bergles, and F. Mayinger, eds., Hemisphere Publishing, Washington, DC, pp. 111–151. Shah, R. K., 1985, Compact Heat Exchangers, in Handbook of Heat Transfer Applications, 2nd Ed., Eds. W. M. Rohsenow, J. P. Hartnett and E. N. Ganic´, Chapter 4, Part III, pp. 4–174 to 4–311, McGraw-Hill, New York. Shah, R. K., and A. L. London, 1971, Inﬂuence of Brazing on Very Compact Heat Exchanger Surfaces, Paper 71-HT-29, American Society of Mechanical Engineers, New York. Shah, R. K., and A. L. London, 1980, Eﬀects of nonuniform passages on compact heat exchanger performance, ASME J. Eng. Power, Vol. 102, Ser. A, pp. 653–659. Shen, P. I., 1992, The eﬀect of friction on ﬂow distribution in dividing and combining ﬂow manifolds, ASME J. Fluids Eng., Vol. 114, pp. 121–123. Thonon, B., 2002, Private communication, CAE-GRETh, Grenoble, France. Thonon, B., P. Mercier, and F. Feidt, 1991, Flow distribution in plate heat exchanger and consequences on thermal and hydraulic performances, Proc. 18th Eurotherm Conference, Springer Verlag, Hamburg, Germany. Wilson, D. G., 1966, A Method of Design for Heat-Exchanger Inlet Headers, Paper 66-WA/HT-41, Americal Society of Mechanical Engineers, New York.

REVIEW QUESTIONS Where multiple choices are given, circle one or more correct answers. Explain your answers brieﬂy. 12.1

The gross ﬂow maldistribution is independent of: (a) surface geometry (b) passage-to-passage nonuniformity (c) ﬂow rate (d) outlet header (e) heat exchanger ﬂow arrangement

12.2

The following are characteristic of gross ﬂow maldistribution: (a) increased total heat transfer (b) increased core pressure drop on the maldistributed side

12.3

The following is a method of computing pressure drop on the side of gross ﬂow maldistribution: (a) a weighted average pressure drop based on apportioned ﬂow rates m_ i (b) the maximum pressure drop based on the largest m_ i component (c) an arithmetic average of pi ’s (d) a sum of pi ’s

12.4

Flow nonuniformity in laminar ﬂow surfaces can be caused by: (a) large frontal areas and small core depths that keep ﬂow velocity low but present ﬂuid distribution problems

856

FLOW MALDISTRIBUTION AND HEADER DESIGN

(b) manufacturing tolerances that are a signiﬁcant fraction of the surface hydraulic diameter (c) fouling (d) deposition of condensable substances 12.5

The temperature eﬀectiveness of a maldistributed ﬂuid stream of a counterﬂow heat exchanger can be determined using the following relationship: 1 X 1 Y 1 X Ci; j Pi; j (b) Pi ¼ Ci; j Pi; j (c) Pi ¼ Ci; j Pi; j (a) Pi ¼ Ci j Ci j Ci i where the subscript i denotes a ﬂuid stream, and j denotes a subexchanger having the uniform ﬂuid ﬂow distribution.

12.6

Gross ﬂow maldistribution increases irreversibilities caused by the following phenomena in a counterﬂow exchanger for heat transfer performance: (a) temperature diﬀerence (b) ﬂuid mixing (c) ﬂow friction (d) none of these (e) all of these

12.7

The following phenomena that cause irreversibilities in a counterﬂow exchanger are important for ﬂuid pumping power requirement when gross ﬂow maldistribution exists on the ﬂuid side of interest: (a) temperature diﬀerence (b) ﬂuid mixing (c) ﬂow friction (d) none of these (e) all of these Explain whether the irreversibility will increase or decrease.

12.8

Moderate gross ﬂow maldistribution results in more total irreversibility in an exchanger than that for moderate temperature maldistribution. (a) true (b) false (c) It depends on the ﬂow arrangement. (d) can’t tell Explain your reasoning from the irreversibility point of view.

12.9

Using the knowledge of irreversibilities gained in Chapter 11, explain why the temperature maldistribution increases exchanger eﬀectiveness for an unmixed– unmixed crossﬂow exchanger when the hotter portion of the hot ﬂuid maldistributed inlet temperature is near the exit end of the cold ﬂuid, whose temperature is uniform.

12.10 The following are characteristics of passage-to-passage ﬂow maldistribution on one ﬂuid side of an exchanger: (a) heat transfer and pressure drop generally unaﬀected (b) a signiﬁcant decrease in total heat transfer and also a slight decrease in pressure drop (c) lower j and f factors for continuous-ﬂow-passage surfaces (d) lower j and f factors for interrupted surfaces (e) poor header design 12.11 Passage-to-passage ﬂow maldistribution: (a) can be corrected by careful placement of turning vanes in the inlet and outlet headers (b) results from core heat transfer surface nonuniformity for continuous-ﬂow passages

REVIEW QUESTIONS

857

(c) is a critical design concern for all types of exchangers (d) can improve overall heat exchanger performance by promoting local turbulence 12.12 Brazing can aﬀect exchanger performance: (a) very little at most because additional heat transfer surface created by the brazing roughness negates the eﬀect of increased pressure drop due to the roughness (b) adversely when the ﬂow passage geometry is aﬀected appreciably by brazing roughness (c) positively because it signiﬁcantly increases the j factor while slightly increasing the f factor 12.13 Gross blockage due to brazing may reduce: (a) the f factor greatly (b) the f factor slightly (c) the j factor greatly (d) the j factor slightly 12.14 Circle the following statements as true or false. (a) T F For a given deviation in ﬂow uniformity for a particular passage type of a compact heat exchanger, the gain in the pressure drop is greater than the reduction in NTU. (b) T F Roughness introduced by brazing does not have any appreciable eﬀect on ﬂow maldistribution due to ﬂow passage geometry if the passages are large. (c) T F The problem of ﬂow maldistribution due to fouling with the same ﬂuid is more severe in heat exchangers of large-ﬂow-passage geometry than those having small-ﬂow-passage geometry. 12.15 Passage-to-passage ﬂow maldistribution increases the following irreversibilities in a counterﬂow exchanger: (a) temperature diﬀerence (b) ﬂuid mixing (c) ﬂow friction (d) none of these (e) all of these 12.16 The following irreversibilities in a counterﬂow exchanger are important for ﬂuid pumping power requirements when passage-to-passage ﬂow maldistribution exists on the ﬂuid side of interest: (a) temperature diﬀerence (b) ﬂuid mixing (c) ﬂow friction (d) none of these (e) all of these Explain whether the irreversibility will increase or decrease. 12.17 Manifold-induced ﬂow maldistribution in a PHE increases with: (a) increasing relative pressure drop in the manifold compared to the exchanger core (b) increasing a large number of plates in a 1-pass 1-pass PHE (c) larger ﬂow area of the dividing-ﬂow manifold than for the combining-ﬂow manifold 12.18 Viscosity-induced ﬂow maldistribution (beyond the ﬂow instability region) on one ﬂuid side in an exchanger results in: (a) increase in heat transfer (b) decrease in heat transfer (c) increase in p (d) decrease in p

858

FLOW MALDISTRIBUTION AND HEADER DESIGN

12.19 The pressure drop is increased signiﬁcantly compared to that for the uniform ﬂow case if the following ﬂow maldistributions exist on one ﬂuid side of an exchanger: (a) gross ﬂow maldistribution (b) viscosity-induced maldistribution (c) manifold-induced maldistribution (d) passage-to-passage maldistribution 12.20 The primary function of the headers/manifolds is: (a) to avoid ﬂow separation within the header/manifold (b) to provide uniform ﬂow distribution over the core face (c) to result in the lowest possible pressure drop within the headers/manifolds (d) to yield uniform temperature distribution at the core inlet 12.21 The design of the inlet header/manifold is more/less critical than the design of the outlet header/manifold: (a) more (b) less (c) can’t tell 12.22 Flow separation in the headers, caused by an area change in the free-ﬂow area, leads to: (a) increased pressure drop (b) decreased pressure drop (c) high-velocity regions (d) localized erosion 12.23 The header/manifold design problem is more important in a: (a) shell-and-tube heat exchanger (b) compact heat exchanger (c) spiral plate heat exchanger 12.24 Oblique-ﬂow headers are characterized by: (a) The inﬂow is orthogonal to the heat transfer core face. (b) The inﬂow is parallel to the heat transfer core face. (c) The inﬂow is at an angle diﬀerent from 908 to the core face. (d) The inlet and outlet headers are on the same side of the exchanger with side inlets. (e) Only the inlet header is always present. 12.25 For speciﬁed heat transfer performance in an exchanger, important design considerations for normal and oblique ﬂow headers are to: (a) match inlet and outlet header designs (b) minimize the core pressure drop (c) allow nonuniform ﬂow through the core to reduce the core pressure drop (d) Any header design is acceptable as long as the heat exchanger core has perfectly manufactured ﬂow passages. 12.26 Circle the following statements as true or false. (a) T F More uniform ﬂow distribution through the core is achieved by a reverse-ﬂow manifold system than by a parallelﬂow manifold system (b) T F More uniform ﬂow distribution through the core is achieved if the ﬂow area of the combining-ﬂow header is smaller than that of the dividing-ﬂow header. (c) T F Maintaining a larger pressure drop in an exchanger core than in the headers is important to provide uniform ﬂow distribution through the core.

PROBLEMS

859

12.27 For a compact heat exchanger, header/manifold design is important because: (a) turbulence must be minimized at the inlet (b) of the awkward shape of the core, a relatively large frontal area, and a short ﬂow length (c) performance falls oﬀ sharply in a maldistributed situation 12.28 The header/tank on the coolant side in an automobile radiator is a: (a) counterﬂow header (b) oblique ﬂow header (c) normal ﬂow header (d) dividing/combining ﬂow manifold 12.29 Rank the following ﬂuids in order of least to most important for good header design: (a) low-pressure gas (b) water (c) air (d) oil 12.30 Which header conﬁguration has the lowest pressure drop for the same inlet velocity and same inlet ﬂow area? (a) counterﬂow (b) parallelﬂow (c) free discharge (d) none of these 12.31 For a free discharge header: (a) The inlet header conﬁguration is not important since the discharge can be considered as in an inﬁnitely large reservoir. (b) The outlet pressure varies linearly along the length of core discharge. (c) The inlet header ﬂow area decreases linearly along the length of the core inlet to ensure uniform ﬂow distribution through the exchanger core. (d) The total header pressure drop, exclusive of core pressure drop contribution, equals one velocity head. (e) The pressure loss for the inlet header is higher than that for the outlet header. 12.32 Major functions of turning vanes in headers are to: (a) promote turbulence and hence increase heat transfer (b) protect header external walls from wear (c) deﬂect any solid particles in the ﬂow stream away from the core to prevent plugging (d) improve ﬂow distribution through the core (e) stiﬀen the header duct to increase the natural frequency PROBLEMS 12.1

On the ﬂuid 1 side of a counterﬂow exchanger, 80% of the total free-ﬂow area is fouled such that the velocity through that portion of the ﬂow area constitutes only 60% of the mean velocity through the ﬂuid 1 side core as a whole. The ratio of the heat capacity rates of ﬂuids 1 and 2 for the heat exchanger as a whole is equal to 1. The number of transfer units of the heat exchanger is 3.5. Determine the temperature eﬀectiveness of ﬂuid 1 if ﬂow on the ﬂuid 2 side is uniform.

12.2

The crossﬂow unmixed (ﬂuid 1) – mixed heat exchanger of Example 12.2 has to be analyzed for the inﬂuence of ﬂow maldistribution in the entire range of possible nonuniformities of the mass ﬂow rate on the ﬂuid 1 side. Determine the change in the temperature eﬀectiveness of the maldistributed ﬂuid (ﬂuid 1) if the X fraction of the total free-ﬂow area on the ﬂuid 1 side is characterized by a ﬂow velocity

860

FLOW MALDISTRIBUTION AND HEADER DESIGN

larger by Y% than the mean velocity through the core. Consider the cases for which X ¼ 13, 0.5, 23, and Y ¼ 25, 50, and 75, respectively, for each of the X values. 12.3

Consider a gas turbine rotary regenerator made up of a deepfold surface (rectangular passages of * ¼ 0:125). Due to the manufacturing process, some of the passages are close to a trapezoidal ð ¼ 858Þ rather than a rectangular cross section. Assume that the matrix is made up of 50% rectangular and 50% trapezoidal passages, as shown in Fig. P12.3. Determine the mass ﬂow fraction distribution and the reduction in heat exchanger eﬀectiveness and p due to passage-

FIGURE P12.3

to-passage nonuniformity. Consider the design operating point as ntun ¼ 5, C*r > 5, and C* ¼ 1. Following are f Re and NuH1 for fully developed ﬂow and some pertinent geometry information:

12.4

Characteristic

Rectangular Passage

Trapezoidal Passage

Re NuH1 Ao; =Ao; Dh; =Dh;

20.585 6.490 1 1

15.659 3.256 1 0.9961

Consider a gas turbine rotary regenerator made up of rectangular passages of * ¼ 16. Due to manufacturing imperfections, 70% of the passages are rectangular with * ¼ 16, and 30% of the passages have * ¼ 14. Determine: (a) The free-ﬂow area distribution Ao; j =Ao;n with j ¼ 1, 2. (b) The ﬂow fraction distribution in the two types of passages. (c) The change in pressure drop due to passage-to-passage nonuniformity. Does it represent a loss or a gain in comparison to all passages of ideal * ¼ 16 shape? (d) The change in exchanger eﬀectiveness due to the nonuniformity. Does it represent a loss or a gain? (e) The change in ntu. Use Fig. P12.4 and the following data for analysis: ntun ¼ 5, C*r > 5, and C * ¼ 1.

FIGURE P12.4

PROBLEMS

Characteristic

* ¼ 14

* ¼ 16

f Re NuH1

18.233 5.331

19.702 6.049

861

Note:

Dh ¼

2b* 1 þ *

Ao; ¼1=4 ¼ 1:5 for one passage each only Ao; ¼1=6

12.5

A crossﬂow regenerator for a gas turbine plant is characterized by the following data. The heat exchanger eﬀectiveness is equal to 0.8, the air-side core relative pressure drop p=pi ¼ 0:0042 (for an inlet air pressure of 0.91 MPa), the inlet-tooutlet air density ratio is 1.5, and the mass ﬂow rate of air is 25 kg/s. The frontal area of the air-side core is 2 m2. The inlet header air velocity ui has to be 30 m/s and air density at the inlet i ¼ 7 kg=m3 . Consider both parallelﬂow and counterﬂow header designs. Assume that the exit header has to be designed so as to have Ho ¼ Hi for a parallelﬂow header design. Determine the inlet header shape and header pressure losses.

12.6

Channel-to-channel ﬂow maldistribution occurs in a plate heat exchanger (PHE) when two diﬀerent plate groups are used. For example, consider a PHE with 47 thermal plates (24 channels for each ﬂuid) having 8 channels with chevron plates of ¼ 308 and 16 channels with chevron plates of ¼ 39:88 for a 308608 mixed plate. Because of the turbulent ﬂow in the channels, use a friction factor correlation of f ¼ a Ren , where a and n are constants. The channel-to-channel ﬂow maldistribution for this exchanger can be derived as m_ I ¼ m_ II

1=ð2nÞ n=ð2nÞ De;I ð1þnÞ=ð2nÞ Ao;I aII II aI I De;II Ao;II

where aI ¼ 0:8, aII ¼ 3:44 and n ¼ 0:25 and the subscripts I and II denote plate groups I and II. Consider a water-to-water counterﬂow single-pass PHE with the aforementioned two groups of plates. Water ﬂow rates on the hot and cold sides are 18 and 10 kg/s, respectively, and the inlet temperatures are 40 and 208C, respectively. The total fouling resistance and wall resistance are given as 0.00004 and 0.000003 m2 K=W, respectively. The following is additional information. Plate

Fluid Properties

Plate width W ¼ 0:05 m Plate height L ¼ 1:1 m Channel spacing 2a ¼ 0:0035 m Equivalent diameter De ¼ 0:007 m Projected area per plate WL ¼ 0:55 m2

¼ 0:00081 Pa s ¼ 995:4 kg=m3 k ¼ 0:619 W=m K cp ¼ 4177 J=kg K Pr ¼ 5:47

Consider the same ﬂuid properties for water for each plate group and identical equivalent diameter for both plate groups.

862

FLOW MALDISTRIBUTION AND HEADER DESIGN

(a) Determine the ﬂow distribution of hot and cold ﬂuids in plate groups I and II. (b) Outline a step-by-step procedure to calculate the heat duty of this exchanger. (c) Compute the heat duty of this PHE. Use the following equation to calculate the heat transfer coeﬃcients: k 0:646 h ¼ 0:724 Re0:583 Pr1=3 Dh 30 12.7

Consider an air-cooled tubular exchanger with engine oil ﬂowing in the tubes. Because of the diﬀerent amount of heat transfer taking place in the tubes, assume that oil ﬂows at 300 K in 50% of the tubes and at 380 K in the remaining 50% of the tubes. The objective of this problem is to determine the inﬂuence of viscosityinduced ﬂow maldistribution considering laminar ﬂow in the tubes. Assume that C* ¼ 1 and ntum ¼ 1. Since no results are presented in Table 3.6 or 3.3 on the "NTU relationship for a crossﬂow heat exchanger with a ﬁnite number of tubes and diﬀerent NTUs associated with each tube, assume that the exchanger is counterﬂow. (a) Determine the ﬂow fraction distribution in the two types of passages. (b) Calculate the change in p due to viscosity-induced ﬂow maldistribution. Does it represent a loss or a gain in comparison to the ‘‘nominal’’ or base case with oil ﬂow in both tubes at 340 K? (c) Determine the change in exchanger eﬀectiveness due to the ﬂow nonuniformity. Does it represent a loss or a gain? Use the following viscosity data for the oil and assume its density to be constant. T ðKÞ

300

340

380

ðPa sÞ

0.486

0.053

0.014

12.8

A gas ﬂows through the circular tubes (lateral branches) under laminar condition through the U-ﬂow manifold conﬁguration of Fig. 12.7. The two lateral headers are large enough to feature a negligible pressure drop along the ﬂuid ﬂow direction within each of the headers. For this gas, the dynamic viscosity increases linearly with temperature. The temperature of the tube wall is either lower or higher than the temperature of the gas ﬂowing inside it, but it is uniform and constant along the ﬂow length, due to high heat transfer on its outside surface. Determine which conditions will exist in this heat exchanger: (1) diﬀerent mass ﬂow rates of gas may be established in diﬀerent tubes for both heating and cooling of the gas ﬂowing through the tubes; (2) such a condition is possible only in the case of heating; or (3) such a condition is possible only in the case of cooling.

12.9

Derive an analytical expression that describes the geometry of the inlet parallelﬂow oblique header given in Table 12.1. That expression should be written in the form of a relationship between the local dimensionless header wall coordinate Z ¼ z=yo and the dimensionless coordinate in the ﬂow direction X * ¼ x=L, as presented by Eq. (1) in Example 12.5. The main steps in the derivation are discussed in Example 12.5, but the complete derivation is to be performed for the solution of this problem.

13

Fouling and Corrosion

Fouling is an accumulation of undesirable material (deposits) on heat exchanger surfaces. Undesirable material may be crystals, sediments, polymers, coking products, inorganic salts, biological growth, corrosion products, and so on. This process inﬂuences heat transfer and ﬂow conditions in a heat exchanger. Fouling is a synergistic consequence of transient mass, momentum and heat transfer phenomena involved with exchanger ﬂuids and surfaces, and depends signiﬁcantly on heat exchanger operation conditions. However, most manifestations of these various phenomena lead to similar consequences. In general, fouling results in a reduction in thermal performance, an increase in pressure drop, may promote corrosion, and may result in eventual failures of some heat exchangers. Corrosion represents mechanical deterioration of construction materials of heat exchanger surfaces under the aggressive inﬂuence of ﬂowing ﬂuids and environment in contact. In addition to corrosion, other mechanically induced phenomena are important for heat exchanger design and operation, such as fretting (corrosion occurring at contact areas between metals under load subjected to vibration and slip). Fouling and corrosion represent heat exchanger operation-induced eﬀects and should be considered for both the design of a new heat exchanger and operation of an existing exchanger. In this chapter, we explain the impact of fouling and corrosion on heat transfer and pressure drop in Section 13.1. In Section 13.2, we present a detailed description of various fouling mechanisms and phenomenological considerations of fouling. The methodology to take into account the eﬀect of fouling on exchanger performance and design is outlined in Section 13.3. Various techniques of prevention and mitigation of detrimental eﬀects of fouling are summarized in Section 13.4. Finally, a brief account of the importance of corrosion, in particular its inﬂuence on heat exchanger operation and design practices, is provided in Section 13.5.

13.1 FOULING AND ITS EFFECT ON EXCHANGER HEAT TRANSFER AND PRESSURE DROP Thermal fouling (in the presence of a temperature gradient) means accumulation of any undesirable deposition of a thermally insulating material (which provides added thermal resistance to heat ﬂow) on a heat transfer surface occurring over a period of time.{ This solid layer adds an additional thermal resistance to heat ﬂow and also increases hydraulic resistance to ﬂuid ﬂow. Also, the thermal conductivity of fouling deposits is usually lower y There are other types of fouling phenomena in nature (e.g., clogging of arteries) that are not of importance in heat exchanger design.

Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

863

864

FOULING AND CORROSION

than that for the metals used for heat transfer surfaces. Fouling is an extremely complex phenomenon characterized by combined heat, mass, and momentum transfer under transient conditions. Liquid-side fouling occurs on the exchanger side where liquid is being heated, and gas-side fouling occurs on the gas cooling side; however, reverse examples can be found. Fouling is very costly since it (1) increases capital costs due to the need to oversurface the heat exchanger and for cleaning; (2) increases maintenance costs resulting from cleaning, chemical additives, or troubleshooting; (3) results in loss of production due to shutdown or reduced capacity; and (4) increases energy losses due to reduced heat transfer, increased pressure drop, and dumping of dirty streams present. Gas-side fouling can also be a potential ﬁre hazard in a fossil-ﬁred exhaust environment, resulting in catastrophic lost production and repair costs. In some applications, increased pressure drop due to fouling may reduce gas ﬂows aﬀecting adversely heat transfer and increasing solvent concentration (such as during waste heat recovery from paint oven exhausts) which is not acceptable environmentally. Fouling signiﬁcantly reduces heat transfer with a relatively small increase in ﬂuid pumping power in systems with liquid ﬂows and high heat transfer coeﬃcients. For systems having low heat transfer coeﬃcients, such as with gases, fouling increases the ﬂuid pumping power signiﬁcantly with some reduction in heat transfer. Note that plugging will also increase pressure drop substantially but doesn’t coat the surface and still may be considered as fouling in an application. Let us ﬁrst discuss only qualitatively the inﬂuence of a deposit on a heat transfer surface. We consider either fully developed laminar or turbulent ﬂows. Using the results/ correlations for laminar ﬂow (Nu ¼ constant; see Table 7.3) and turbulent ﬂow [the Dittus–Boelter correlation, Eq. (7.79) in Table 7.6], we express the heat transfer coeﬃcient as follows:

h¼

8 Nu k > > > < D

with Nu ¼ constant

for laminar flow

h

0:8 > k 4m_ > > : Pr0:4 0:023 Dh P

ð13:1Þ for turbulent flow

Note that in the turbulent ﬂow expression of Eq. (13.1), we substituted Re ¼ m_ Dh =Ao ¼ 4m_ =P using the deﬁnition of Dh . Here P is the wetted perimeter of all ﬂow passages in the exchanger. In general, we treat P as independent of Dh . For example, having a double number of 5 mm diameter tubes compared to a speciﬁed number of 10 mm diameter tubes in an exchanger will have the same total P but diﬀerent Dh . A similar situation can exist for an extended surface. Using Eqs. (6.29) and (6.67b), we express the following pressure drop relationships: 8 1 1 16L > > _ > m ð f ReÞ > < D3h 2gc P 4L G2 p ¼ f ¼ Dh 2gc > > 1 0:046 4L 0:2 4m_ 1:8 > > : 3 2gc P Dh

for laminar flow ð13:2Þ for turbulent flow

In Eq. (13.2), f Re is approximately constant for fully developed laminar ﬂow (a theoretical value for a circular tube is f Re ¼ 16), while the following relationship is

FOULING ON EXCHANGER HEAT TRANSFER AND PRESSURE DROP

865

applied for turbulent ﬂow: f ¼ 0:046Re0:2 . The most important physical outcome of fouling is the ﬂow cross section getting plugged and resulting in a reduced hydraulic diameter of ﬂow passages. Therefore, for a given mass ﬂow rate m_ , ﬂuid ﬂow length L, heat transfer area A ð¼ PLÞ, and known ﬂuid properties, one gets from Eqs. (13.1) and (13.2), h/

1 Dh

p /

1 D3h

ð13:3Þ

The functional relationships given by Eq. (13.3) are obtained assuming total wetted perimeter as constant regardless of the change in the hydraulic diameter. In practice, when Dh for an exchanger changes, A may change as well as for a circular tube. In that case, since P ¼ di Nt ¼ Dh Nt for a tubular exchanger (Nt ¼ total number of tubes), p / 1=D4h from Eq. (13.2) for laminar ﬂow and p / 1=D4:8 h for turbulent ﬂow instead of 1=D3h of Eq. (13.3) for a tubular exchanger. Alternatively, the p expression for turbulent ﬂow in a circular tube can be expressed as follows using Eq. (6.29) with the deﬁnitions G ¼ m_ =Ao , Ao ¼ ð=4Þdi2 and Dh ¼ di : p ¼ f

4L G2 4L 1 32Lm_ 2 f m_ 2 ¼f ¼ di 2gc di ð=4Þ2 di4 2gc 2 gc di5

ð13:4Þ

Substitution of f ¼ 0:046Re0:2 will change the exponent of m_ in Eq. (13.4) to 1.8 and the exponent of di to 4.8. Also, the surface roughness change due to fouling on the f factors should be included as an additional eﬀect (generally, we neglect the eﬀect of surface roughness on the heat transfer coeﬃcient for conservatism). Actual inﬂuences of fouling on the heat transfer coeﬃcient and pressure drop are substantially more complex than those presented by Eqs. (13.3) and (13.4), due to inherently transient nature of fouling processes. The pressure drop ratio pf =pc of a fouled and a clean exchanger for a constant mass ﬂow rate, from Eq. (13.4), is given by pf ff ¼ pc fc

Dh;c Dh; f

5 ð13:5Þ

If we consider that fouling does not aﬀect the friction factor (i.e., fc ff ) and also consider that reduction in the tube inside diameter due to fouling is only 10 to 20%, the resulting pressure drop increase will be approximately 69% and 205%, respectively, according to Eq. (13.5), regardless of whether the ﬂuid is liquid or gas [note that in contrast, h / 1=Dh , as shown in Eq. (13.1) or (13.3)]. This increased p can be translated into increased ﬂuid pumping power using Eq. (6.1); and for liquids, the density being signiﬁcantly higher than that for gases, a substantially higher p due to fouling can be allowed for liquids for a reasonable increase in liquid pumping power. Also, the equipment cost of ﬂuid pumping power is lower for liquids than for gases for a given amount of pumping power.{ Now let us review the impact of fouling on exchanger heat transfer. As fouling will reduce the free-ﬂow area and hence the passage Dh , it will increase the convection heat y For example, for a midsized automobile, the cost of a 300-W fan for the radiator airﬂow was $35 to 40, compared to $20 to 25 for an equivalent power radiator coolant water pump in 2001.

866

FOULING AND CORROSION

transfer coeﬃcient h [of Eq. (13.1)] between the ﬂuid and heat transfer surface (which may be covered with a fouling layer), for two reasons: increased ﬂow velocity with a reduction in the ﬂow area, and increased surface roughness due to the fouling layer. Both these eﬀects would increase the pressure drop substantially. Fouling layers (deposits) on one or both ﬂuid sides increase thermal resistance to heat ﬂow from the hot ﬂuid to cold ﬂuid by conduction through the fouling layers (see Fig. 3.4), which also have lower thermal conductivity. The added thermal resistances in general reduce the exchanger overall UA substantially compared to the increase in h due to fouling, as mentioned above.{ Due to a large uncertainty, transient nature, variations in the fouling resistance ^ f ¼ 1=hf ), and no accurate means of its measurement, the increase in h due to fouling is (R ignored or lumped into the reported values of fouling resistances. Hence, the heat transfer coeﬃcients hh and hc for hot and cold ﬂuids are determined for unfouled clean surfaces for the UA computation of fouled surfaces. From the overall thermal resistance Eq. (3.20) or (3.24), we ﬁnd that fouling deposits will reduce UA and hence q more signiﬁcantly in liquids than in gases. This is because liquids have h an order of magnitude higher than that for gases in general. To understand this, consider a process plant heat exchanger with clean U ¼ 1500 W/m2 K or the overall unit thermal resistance ^ f ;h þ R ^ f ;c ¼ 3 104 [a reasonable ^ o ¼ 6 104 m2 K=W. If the fouling resistances R R value from TEMA (1999)] are considered, 50% extra heat transfer ^ o;new ¼ ð6 þ 3Þ 104 m2 K=W and area is chargeable to fouling since R ^ o;new . In contrast, for a gas-to-gas clean compact heat exchanger, consider q ¼ A Tm =R ^ o ¼ 3 103 m2 K=W. For the same fouling resistances, Uc ¼ 300 W=m2 K or R 4 2 ^ ^ Rf ;h þ Rf ;c ¼ 3 10 m K=W, the heat transfer surface area chargeable to fouling is only 10%. Based on the foregoing discussion, fouling in liquids has a signiﬁcant detrimental eﬀect on heat transfer with some increase in ﬂuid pumping power. In contrast, fouling in gases reduces heat transfer somewhat (5 to 10% in general) but increases pressure drop and ﬂuid pumping power signiﬁcantly (up to several hundred percent) from the cost point of view. It should be emphasized that the same magnitude of a fouling factor (or fouling unit thermal resistance){ can have a diﬀerent impact on performance for the same or diﬀerent applications. For example, the same fouling factor may represent heavy fouling in a clean service (such as a closed-loop refrigerant system) or low fouling in a dirty service (such as a reﬁnery crude preheat train). As another example, the same fouling factor in two diﬀerent plants may have radically diﬀerent fouling rates because of diﬀerent feedstocks, preprocessing, or equipment design.

13.2 PHENOMENOLOGICAL CONSIDERATIONS OF FOULING As noted in Section 13.1, fouling is an extremely complex phenomenon characterized by a combined heat, mass, and momentum transfer under transient conditions. Fouling is aﬀected by a large number of variables related to heat exchanger surface, operating conditions, and involved ﬂuid streams. Despite the complexity of the fouling process, a general practice is to include the eﬀect of fouling on the exchanger thermal performance y

For example, see the added thermal resistance terms ð1=o hf AÞ for the hot and cold ﬂuid sides in Eq. (3.20) or (3.24), which may reduce 1=UA more than the increase in hh and/or hc due to fouling, depending on their relative magnitudes in the equation. z The concept of fouling resistance introduced in Section 3.2.4 is explained further in Section 13.2.6.

PHENOMENOLOGICAL CONSIDERATIONS OF FOULING

867

by adding thermal resistances of fouling layers in the thermal circuit using empirical data, as explained through Fig. 3.4 and discussed further in Section 13.3. The problem, though, is that this simpliﬁed modeling approach does not (and cannot) reﬂect a real transient nature of the fouling process. The current practice is to use fouling factors or fouling unit thermal resistances from TEMA Standards (1999) (see Section 13.3 and Table 9.4 for tubular and shell-and-tube heat exchangers). However, probably a better approach would be to perform cost analysis for cleaning frequency by taking into account any initial overdesign (by including fouling resistances). This overdesign may provide added heat transfer performance initially due to larger surface area and ﬂow area than required for a clean exchanger but will reduce the ﬂow velocity and hence may accelerate initial fouling in some applications. Let us now consider in detail diﬀerent types of fouling mechanisms, sequential events in fouling, and modeling of a fouling process as an example. 13.2.1

Fouling Mechanisms

There are six types of liquid-side fouling mechanisms: (1) precipitation or crystallization fouling, (2) particulate fouling, (3) chemical reaction fouling, (4) corrosion fouling, (5) biological fouling, and (6) freezing (solidiﬁcation) fouling. Only biological fouling does not occur in gas-side fouling since there are in principle no nutrients in the gas ﬂows. In reality, more than one fouling mechanisms is present in many applications and their synergistic eﬀect makes the fouling even worse than predicted/expected with a single fouling mechanism present. Note that there are additional examples of fouling that may not fall in the foregoing categories, such as accumulation of noncondensables in a condenser. In addition, plugging will also increase pressure drop substantially, but doesn’t coat the surface and still may be considered as fouling in applications. Refer to Melo et al. (1988) and Bott (1990) for a detailed study of fouling. In precipitation or crystallization fouling, the dominant mechanism is the precipitation of dissolved salts in the ﬂuid on the heat transfer surface when the surface concentration exceeds the solubility limit. Thus, a necessary prerequisite for an onset of precipitation is the presence of supersaturation. Precipitation of salts can occur within the process ﬂuid, in the thermal boundary layer, or at the ﬂuid–surface (fouling–ﬁlm) interface. It generally occurs with aqueous solutions and other liquids of soluble salts which are either being heated or cooled. When the solution contains normal solubility salts (the salt solubility and concentration decrease with decreasing temperature such as wax deposits, gas hydrates and freezing of water/water vapor), the precipitation fouling occurs on the cold surface (i.e., by cooling the solution). For inverse solubility salts (such as calcium and magnesium salts), the precipitation of salt occurs with heating the solution. Precipitation/crystallization fouling is common when untreated water, seawater, geothermal water, brine, aqueous solutions of caustic soda, and other salts are used in heat exchangers. This fouling is characterized by deposition of divalent salts in cooling water systems. Crystallization fouling may occur with some gas ﬂows that contain small quantities of organic compounds that would form crystals on the cold surface. If the deposited layer is hard and tenacious (as often found with inverse solubility salts such as cooling water containing hardness salts), it is often referred to as scaling. If it is porous and mushy, it is called sludge, softscale, or powdery deposit. The most important phenomena involved with precipitation or crystallization fouling include the following. Crystal growth during precipitation require formation of a primary nucleus. The mechanism controlling that process is nucleation, as a rule heterogeneous in the presence

868

FOULING AND CORROSION

of impurities and on the heat transfer surface. Transfer of particulate solids to the fouled surface is accomplished by diﬀusion. Simultaneously with deposition, removal phenomena caused by shear stress are always present. Deposit mechanical integrity changes over time either by strengthening or by weakening it due to crystalization/recrystalization, temperature change, and so on. All these phenomena are controlled by numerous factors, the most dominant being local temperature and temperature gradient levels, composition of the ﬂuid including concentration of soluble species. Particulate fouling refers to the deposition of solids suspended in a ﬂuid onto a heat transfer surface. If the settling occurs due to gravity, the resulting particulate fouling is called sedimentation fouling. Hence, particulate fouling may be deﬁned as the accumulation of particles from heat exchanger working ﬂuids (liquids and/or gaseous suspensions) on the heat transfer surface. Most often, this type of fouling involves deposition of corrosion products dispersed in ﬂuids, clay and mineral particles in river water, suspended solids in cooling water, soot particles of incomplete combustion, magnetic particles in economizers, deposition of salts in desalination systems, deposition of dust particles in air coolers, particulates partially present in ﬁre-side (gas-side) fouling of boilers, and so on. The particulate fouling caused by deposition of, for example, corrosion products is inﬂuenced by the following factors: metal corrosion process factors (at heat transfer surface), release and deposition of the corrosion products on the surface{; concentration of suspended particles, temperature conditions on the fouled surface (heated or nonheated), and heat ﬂux at the heat transfer surface. Chemical reaction fouling is referred to as the deposition of material (fouling precursors) produced by chemical reactions within the process ﬂuid, in the thermal boundary layer, or at the ﬂuid–surface (fouling–ﬁlm) interface in which the heat transfer surface material is not a reactant or participant. However, the heat transfer surface may act as a catalyst as in cracking, coking, polymerization, and autoxidation. Thermal instabilities of chemical species, such as asphaltenes and proteins, can also induce fouling precursors. Usually, this fouling occurs at local hot spots in a heat exchanger, although the deposits are formed all over the heat transfer surface in crude oil units and dairy plants. It can occur over a wide temperature range from ambient to over 10008C (18008F) but is more pronounced at higher temperatures. Foulant deposits are usually organic compounds, but inorganic materials may be needed to promote the chemical reaction. This fouling mechanism is a consequence of an unwanted chemical reaction that takes place during the heat transfer process. Examples of chemical fouling include deposition of coke in petrochemical industries in cracking furnaces where thermal cracking of hydrocarbons is realized. This fouling mechanism is found in many applications of process industry, such as oil reﬁning, vapor-phase pyrolysis, cooling of gas and oils, polymerization of process monomers, and so on. Furthermore, fouling of heat transfer surface by biological ﬂuids may involve complex heterogeneous chemical reactions and physicochemical processes. The deposits from chemical reaction fouling may promote corrosion at the surface if the formation of the protective oxide layer is inhibited. All fouling deposits may promote corrosion. In corrosion fouling (in situ), the heat transfer surface itself reacts with the process ﬂuid or chemicals present in the process ﬂuid. Its constituents or trace materials are carried by the ﬂuid in the exchanger, and it produces corrosion products that deposit on the surface. Hence, corrosion fouling could be considered as chemical reaction fouling y It should be borne in mind that corrosion products may be soluble in a working ﬂuid, and hence both precipitation and particulate fouling would usually occur concurrently.

PHENOMENOLOGICAL CONSIDERATIONS OF FOULING

869

in which heat transfer fouling aﬀects the exchanger mechanical integrity, and the corrosion products add thermal resistance to heat ﬂow from the hot ﬂuid to the cold ﬂuid. If corrosion products are formed upstream of the exchanger and then deposited on the heat transfer surface, the fouling mechanism refers to particulate or precipitation fouling, depending on whether the corrosion products are insoluble or soluble at the bulk ﬂuid conditions. The interaction of corrosion and other types of fouling is the major concern for many industrial applications. Corrosion fouling is dependent on the selection of exchanger surface material and can be avoided with the right choice of materials (such as expensive alloys) if the high cost is warranted. Corrosion fouling is prevalent in many applications where chemical reaction fouling takes place and the protective oxide layer is not formed on the surface. Corrosion fouling is of signiﬁcant importance in the design of the boiler and condenser of a fossil fuel–ﬁred power plant. The important factors for corrosion fouling are the chemical properties of the ﬂuids and heat transfer surface, oxidizing potential and alkalinity, local temperature and heat ﬂux magnitude, and mass ﬂow rate of the working ﬂuid. It should be noted that although growth of corrosion inﬂuenced deposit has a detrimental eﬀect on heat transfer, this inﬂuence is less important than fouling caused by particulate fouling of corrosion products formed elsewhere within the system. For example, fouling on the water side of boilers may be caused by corrosion products that originate in the condenser or feedtrain. Biological fouling or biofouling results from the deposition, attachment, and growth of macro- or microorganisms to the heat transfer surface; it is generally a problem in water streams. In general, biological fouling can be divided into two main subtypes of fouling: microbial and macrobial. Microbial fouling is accumulation of microorganisms such as algae, fungi, yeasts, bacteria, and molds, and macrobial fouling represents accumulation of macroorganisms such as clams, barnacles, mussels, and vegetation as found in seawater or estuarine cooling water. Microbial fouling precedes macrobial deposition as a rule and may be considered of primary interest. Biological fouling is generally in the form of a bioﬁlm or a slime layer on the surface that is uneven, ﬁlamentous, and deformable but diﬃcult to remove. Although biological fouling could occur in suitable liquid streams, it is generally associated with open recirculation or once-through systems with cooling water. Since this fouling is associated with living organisms, they can exist primarily in the temperature range 0 to 908C (32 to 1948F) and thrive in the temperature range 20 to 508C (68 to 1228F). Biological fouling may promote corrosion fouling under the slime layer. Transport of microbial nutrients, inorganic salts, and viable microorganisms from the bulk ﬂuid to the heat transfer surface is accomplished through molecular diﬀusion or turbulent eddy transport, including organic adsorption at the surface. Freezing or solidiﬁcation fouling is due to freezing of a liquid or some of its constituents, or deposition of solids on a subcooled heat transfer surface as a consequence of liquid–solid or vapor–solid phase change in a gas stream. Formation of ice on a heat transfer surface during chilled water production or cooling of moist air, deposits formed in phenol coolers, and deposits formed during cooling of mixtures of substances such as paraﬃn are some examples of solidiﬁcation fouling (Bott, 1981). This fouling mechanism occurs at low temperatures, usually ambient and below depending on local pressure conditions. The main factors aﬀecting solidiﬁcation fouling are mass ﬂow rate of the working ﬂuid, temperature and crystallization conditions, surface conditions, and concentration of the solid precursor in the ﬂuid. Combined fouling occurs in many applications, where more than one fouling mechanism is present and the fouling problem becomes very complex with their synergistic

870

FOULING AND CORROSION

eﬀects. Some combined fouling mechanisms found in industrial applications are (Panchal, 1999): . Particulate fouling combined with biofouling, crystallization, and chemicalreaction fouling . Crystallization fouling combined with chemical-reaction fouling . Condensation of organic/inorganic vapors combined with particulate fouling in gas streams . Crystallization fouling of mixed salts . Combined fouling by asphaltene precipitation, pyrolysis, polymerization, and/or inorganic deposition in crude oil . Corrosion fouling combined with biofouling, crystallization, or chemical-reaction fouling Some examples of the interactive eﬀects of corrosion and fouling are as follows (Panchal, 1999): . Microfouling-induced corrosion (MIC) (sustained-pitting corrosion) . Under-deposit corrosion in petroleum and black liquor processing (concentration buildup of corrosion-causing elements) . Simultaneous corrosion and biofouling in cooling water applications . Fouling induced by corrosion products It is obvious that one cannot talk about a single, uniﬁed theory to model the fouling process wherein not only the foregoing six types of fouling mechanisms are identiﬁed, but in many processes more than one fouling mechanism exists with synergistic eﬀects. However, it is possible to extract a few variables that would most probably control any fouling process: (1) ﬂuid velocity, (2) ﬂuid and heat transfer surface temperatures and temperature diﬀerences, (3) physical and chemical properties of the ﬂuid, (4) heat transfer surface properties, and (5) geometry of the ﬂuid ﬂow passage. The other important variables are concentration of foulant or precursor, impurities, heat transfer surface roughness, surface chemistry, ﬂuid chemistry (pH level, oxygen concentration, etc.), pressure, and so on. For a given ﬂuid–surface combination, the two most important design variables are the ﬂuid velocity and heat transfer surface temperature. In general, higher ﬂow velocities may cause less foulant deposition and/or more pronounced deposit erosion, but at the same time may accelerate corrosion of the surface by removing the heat transfer surface material. Higher surface temperatures promote chemical reaction, corrosion, crystal formation (with inverse solubility salts), and polymerization, but reduce biofouling and prevent freezing and precipitation of normal solubility salts. Consequently, it is frequently recommended that the surface temperature be maintained low.

13.2.2

Single-Phase Liquid-Side Fouling

Single-phase liquid-side fouling is most frequently caused by (1) precipitation of minerals from the ﬂowing liquid, (2) deposition of various particles, (3) biological fouling, and (4) corrosion fouling. Other fouling mechanisms are also present. More important, though,

PHENOMENOLOGICAL CONSIDERATIONS OF FOULING

871

TABLE 13.1 Inﬂuence of Operating Variables on Liquid-Side Foulinga Operating Variable Temperature Velocity Supersaturation pH Impurities Concentration Roughness Pressure Oxygen

Precipitation

Freezing

Particulate

Chemical

Corrosion

Biological

"# #$ " " — " " $ $

# "# " — # " " $ $

"#$ # — "# — " "$ — —

"# # — — — — — " "

"# "#$ — "# — — "$ " "

"#$ "# — "# — — " "# "#

Source: Data from Cannas (1986). a When the value of an operating variable is increased, it increases ("), decreases (#), or has no eﬀect ($) on the speciﬁc fouling mechanism listed. Dashes — indicate that no inﬂuence of these variables has been reported in the literature.

is the combined eﬀect of more than one fouling mechanism present. The qualitative eﬀects of some of the operating variables on these fouling mechanisms are shown in Table 13.1. The quantitative eﬀect of fouling on heat transfer can be estimated by utilizing the concept of fouling resistance and calculating the overall heat transfer coeﬃcient under both fouling and clean conditions (see Section 13.3). An additional parameter for determining this inﬂuence, used frequently in practice, is the cleanliness factor. It is deﬁned as a ratio of an overall heat transfer coeﬃcient determined for fouling conditions to that determined for clean (fouling-free) operating conditions. The eﬀect of fouling on the pressure drop can be determined by the reduced free-ﬂow area due to fouling and the change in the friction factor, if any, due to fouling. 13.2.3

Single-Phase Gas-Side Fouling

Gas-side fouling may be caused by precipitation (scaling), particulate deposition, corrosion, chemical reaction, and freezing. Formation of hard scale from the gas ﬂow occurs if a suﬃciently low temperature of the heat transfer surface forces salt compounds to solidiﬁcation. Acid vapors, high-temperature removal of an oxide layer by molten ash, or salty air at low temperatures may promote corrosion fouling. An example of particulate deposition is accumulation of plant residues. An excess of various chemical substances, such as sulfur, vanadium, and sodium, initiates various chemical reaction fouling problems. Formation of frost and various cryo-deposits are typical examples of freezing fouling on the gas side. An excellent overview of gas-side fouling of heat transfer surfaces is given by Marner (1990, 1996). Qualitative eﬀects of some of the operating variables on gas-side fouling mechanisms are presented in Table 13.2. 13.2.4

Fouling in Compact Exchangers

Small channels associated with compact heat exchangers have very high shear rates, perhaps three to four times higher in a plate heat exchanger than in a shell-and-tube

872

FOULING AND CORROSION

TABLE 13.2 Inﬂuence of Operating Variables on Gas-Side Foulinga Operating Variable Temperature Velocity Impurities Concentration Fuel-air ratio Roughness Oxygen Sulfur

Particulate

Freezing

Chemical

Corrosion

"# "#$ — " " "$ $ —

# # # " — — $ —

" "#$ — — " — " "

"#$ "$ — " — "$ — "

Source: Data from Cannas (1986). a When the value of an operating variable is increased, it increases ("), decreases (#), or has no eﬀect ($) on the speciﬁc fouling mechanism listed. Dashes — indicate that no inﬂuence of these variables has been reported in the literature.

exchanger. This reduces fouling signiﬁcantly. However, small channel size creates a problem of plugging the passages. To avoid plugging, the particle size must be restricted by ﬁltering or other means to less than one-third the smallest opening of heat exchanger passages. Even with this guideline, particulate fouling can occur and agglomerate, such as with waxy substances.

13.2.5

Sequential Events in Fouling

From the empirical evidence involving various fouling mechanisms discussed in Section 13.2.1, it is clear that virtually all these mechanisms are characterized by a similar sequence of events. The successive events occurring in most cases are the following: (1) initiation, (2) transport, (3) attachment, (4) removal, and (5) aging, as conceptualized by Epstein (1978). These events govern the overall fouling process and determine its ultimate impact on heat exchanger performance. In some cases, certain events dominate the fouling process, and they have a direct eﬀect on the type of fouling to be sustained. Let us summarize these events brieﬂy (Cannas, 1986). Initiation of the fouling, the ﬁrst event in the fouling process, is preceded by a delay period or induction period d as shown in Fig. 13.1. The basic mechanism involved during this period is heterogeneous nucleation, and d is shorter with a higher nucleation rate. The factors aﬀecting d are temperature, ﬂuid velocity, composition of the fouling stream, and nature and condition of the heat exchanger surface. Low-energy surfaces (unwettable) exhibit longer induction periods than those of high-energy surfaces (wettable). In crystallization fouling, d tends to decrease with increasing degree of supersaturation. In chemical reaction fouling, d appears to decrease with increasing surface temperature. In all fouling mechanisms, d decreases as the surface roughness increases due to available suitable sites for nucleation, adsorption, and adhesion. Transport of species means transfer of a key component (such as oxygen), a crucial reactant, or the fouling species itself from the bulk of the ﬂuid to the heat transfer surface. Transport of species is the best understood of all sequential events. Transport of species takes place through the action of one or more of the following mechanisms:

PHENOMENOLOGICAL CONSIDERATIONS OF FOULING

873

. Diﬀusion: involves mass transfer of the fouling constituents from the ﬂowing ﬂuid toward the heat transfer surface due to the concentration diﬀerence between the bulk of the ﬂuid and the ﬂuid adjacent to the surface. . Electrophoresis: under the action of electric forces, fouling particles carrying an electric charge may move toward or away from a charged surface depending on the polarity of the surface and the particles. Deposition due to electrophoresis increases with decreasing electrical conductivity of the ﬂuid, increasing ﬂuid temperature, and increasing ﬂuid velocity. It also depends on the pH of the solution. Surface forces such as London–van der Waals and electric double layer interaction forces are usually responsible for electrophoretic eﬀects. . Thermophoresis: a phenomenon whereby a ‘‘thermal force’’ moves ﬁne particles in the direction of negative temperature gradient, from a hot zone to a cold zone. Thus, a high-temperature gradient near a hot wall will prevent particles from depositing, but the same absolute value of the gradient near a cold wall will promote particle deposition. The thermophoretic eﬀect is larger for gases than for liquids. . Diﬀusiophoresis: involves condensation of gaseous streams onto a surface. . Sedimentation: involves the deposition of particulate matters such as rust particles, clay, and dust on the surface due to the action of gravity. For sedimentation to occur, the downward gravitational force must be greater than the upward drag force. Sedimentation is important for large particles and low ﬂuid velocities. It is frequently observed in cooling tower waters and other industrial processes where rust and dust particles may act as catalysts and/or enter complex reactions. . Inertial impaction: a phenomenon whereby ‘‘large’’ particles can have suﬃcient inertia that they are unable to follow ﬂuid streamlines and as a result, deposit on the surface. . Turbulent downsweeps: since the viscous sublayer in a turbulent boundary layer is not truly steady, the ﬂuid is being transported toward the surface by turbulent downsweeps. These may be thought of as suction areas of measurable strength distributed randomly all over the surface. Attachment of the fouling species to the surface involves both physical and chemical processes, and it is not well understood. Three interrelated factors play a crucial role in the attachment process: surface conditions, surface forces, and sticking probability. It is the combined and simultaneous action of these factors that largely accounts for the event of attachment. . The properties of surface conditions important for attachment are the surface free energy, wettability (contact angle, spreadability), and heat of immersion. Wettability and heat of immersion increase as the diﬀerence between the surface free energy of the wall and the adjacent ﬂuid layer increases. Unwettable or low-energy surfaces have longer induction periods than wettable or high-energy surfaces, and suﬀer less from deposition (such as polymer and ceramic coatings). Surface roughness increases the eﬀective contact area of a surface and provides suitable sites for nucleation and promotes initiation of fouling. Hence, roughness increases the wettability of wettable surfaces and decreases the unwettability of the unwettable ones. . There are several surface forces. The most important one is the London–van der Waals force, which describes the intermolecular attraction between nonpolar mole-

874

FOULING AND CORROSION

cules and is always attractive. The electric double layer interaction force can be attractive or repulsive. Viscous hydrodynamic force inﬂuences the attachment of a particle moving to the wall, which increases as it moves normal to the plain surface. . Sticking probability represents the fraction of particles that reach the wall and stay there before any reentrainment occurs. It is a useful statistical concept devised to analyze and explain the complicated event of attachment. Removal of the fouling deposits from the surface may or may not occur simultaneously with deposition. Removal occurs due to the single or simultaneous action of the following mechanisms: shear forces, turbulent bursts, re-solution, and erosion. . Shear forces result from the action of the shear stress exerted by the ﬂowing ﬂuid on the depositing layer. As the fouling deposit builds up, the cross-sectional area for ﬂow decreases, thus causing an increase in the average velocity of the ﬂuid for a constant mass ﬂow rate and increasing the shear stress. Fresh deposits will form only if the deposit bond resistance is greater than the prevailing shear forces at the solid–ﬂuid interface. . Randomly distributed (about less than 0.5% at any instant of time) periodic turbulent bursts act as miniature tornadoes lifting deposited material from the surface. By continuity, these ﬂuid bursts are compensated for by gentler ﬂuid back sweeps, which promote deposition. . The removal of the deposits by re-solution is related directly to the solubility of the material deposited. Since the fouling deposit is presumably insoluble at the time of its formation, dissolution will occur only if there is a change in the properties of the deposit, or in the ﬂowing ﬂuid, or in both, due to local changes in temperature, velocity, alkalinity, and other operational variables. For example, suﬃciently high or low temperatures could kill a biological deposit, thus weakening its attachment to a surface and causing sloughing or re-solution. The removal of corrosion deposits in power-generating systems is done by re-solution at low alkalinity. Re-solution is associated with the removal of material in ionic or molecular form. . Erosion is closely identiﬁed with the overall removal process. It is highly dependent on the shear strength of the foulant and on the steepness and length of the sloping heat exchanger surfaces, if any. Erosion is associated with the removal of material in particulate form. The removal mechanism becomes largely ineﬀective if the fouling layer is composed of well-crystallized pure material (strong formations); but it is very eﬀective if it is composed of a large variety of salts each having diﬀerent crystal properties. Aging of deposits begins with attachment on the heat transfer surface, and refers to any changes the fouling material undergoes as time elapses. The aging process includes both physical and chemical transformations, such as further degradation to a more carbonaceous material in organic fouling, and dehydration and/or crystal phase transformations in inorganic fouling. A direct consequence of aging is change in the thermal conductivity of the deposits.{ Aging may strengthen or weaken the fouling deposits. y A common nonfouling example of aging is the transformation of fresh, soft, ﬂuﬀy snow in an open ﬁeld into hard, crystalline, yellowish ice after a week or so of exposure to the sun resulting diﬀerences in its material properties.

PHENOMENOLOGICAL CONSIDERATIONS OF FOULING

13.2.6

875

Modeling of a Fouling Process

Regardless of the type of fouling process, the principal characteristic feature of any type of fouling is that the net mass fouling rate (i.e., the change of the mass m of foulant deposited on the heat transfer surface for a given time, dm=d, is a consequence of a net diﬀerence between the foulant deposit rate m_ d and the foulant reentrainment rate m_ r : @mðs; Þ ¼ m_ d ðs; Þ m_ r ðs; Þ @

ð13:6Þ

In Eq. (13.6), s denotes symbolically the spatial dependence (say, x, y, and z) of the mass of foulant. Note that the mass m of the foulant deposited uniformly is given as a simple equation: m ¼ f Af

ð13:7Þ

where f represents foulant mass density, A denotes heat transfer surface area covered with the foulant, and f is the thickness of the foulant layer. In general, all three terms of Eq. (13.6) are spatially nonuniform and dependent on time. Equation (13.6) can conveniently be reformulated in terms of mass per unit heat transfer surface area, MA ¼ m=A, and for a uniform spatial distribution of deposit, it is dMA ¼ M_ A;d M_ A;r d

ð13:8Þ

Equation (13.8) is a direct consequence of Eq. (13.6) after idealizing a uniform distribution of the fouling deposit over the surface A. Furthermore, mass per unit heat transfer surface (uniformly distributed along the heat transfer surface) can be written as ^f MA ¼ f f ¼ f kf R

ð13:9Þ

^ f ¼ f =kf , the fouling factor, represents fouling unit thermal resistance; it reprewhere R sents the thermal resistance of the layer of foulant deposited for a unit area of heat transfer surface. Concisely, we refer to this entity as fouling resistance. From the fouling ^ f . Consequently, factor deﬁnition, we obtain f ¼ kf R ^f df dR dMA ¼ f ¼ f kf d d d

ð13:10Þ

In Eq. (13.10), it is assumed that both mass density and thermal conductivity of the deposited layer are invariant with time. Combining Eqs. (13.8) and (13.10), we obtain ^f dR ^_ ^_ R ¼R d r d

ð13:11Þ

^_ ¼ M_ = k represents deposition ( j ¼ d) and removal ( j ¼ r) fouling resiswhere R j A; j f f tance rates. To solve either Eq. (13.8) or (13.11), one needs the explicit forms of either mass rates per unit heat transfer area (for both deposit or reentrainment process) or unit thermal resistances [the terms on the right-hand sides of Eqs. (13.8) and (13.11)]. A number of

876

FOULING AND CORROSION

models for determining these variables have been developed; some of them are summarized in Table 13.3. Let us consider the model of Taborek et al. enlisted in that table as an illustration. According to Taborek et al. [as reported by Epstein (1978)], the deposition and removal mass rates have the form E M_ A;d ¼ c1 }1 n exp Cc > CL

"¼

1 1 1 þ 1 "c "h

Cc ¼ Ch ¼ C ¼ CL

Cc > CL > Ch

"¼

Ch CL

Cc CL

1 Ch þ " h CL

1

"¼

1 Cc þ "c CL "¼

"¼

1

1 1 "h

CL =C 1 1 þ 1 "c "h

1 1 1 C þ "c "h CL

"¼

1 1 1 þ 1 "c "h

1 1 "c

Source: Data from Kays and London (1998).

The analysis of individual exchangers in the liquid-coupled system is straightforward using the "-NTU theory presented in Section 3.3. The individual eﬀectiveness values for the overall system, the hot- and cold-side exchangers in Fig. B.1, are designated as ", "h , and "c using the deﬁnition used in Section 3.3 (i.e., based on the Cmin values for individual changers). For example, if CL > Ch > Cc , "¼

Cc ðTc;o Tc;i Þ Cc ðTh;i Tc;i Þ

"h ¼

Ch ðTh;i Th;o Þ Ch ðTh;i TL;i Þ

"c ¼

Cc ðTc;o Tc;i Þ Cc ðTL;o Tc;i Þ

ðB:1Þ

Note that "h and "c here are the exchanger eﬀectiveness values of the hot and cold ﬂuids, and they are not the temperature eﬀectiveness values deﬁned by Eqs. (3.51) and (3.52). The "h and "c are related to the overall eﬀectiveness " of the liquid-coupled exchangers as shown in Table B.1, where diﬀerent formulas are presented depending on the relationships among the heat capacity rates of the hot ﬂuid, cold ﬂuid, and the circulating liquid. Note that in most applications, CL is larger than Ch and Cc .

REFERENCES Kays, W. M., and A. L. London, 1998, Compact Heat Exchangers, reprint 3rd ed., Krieger Publishing, Malabar, FL. Reay, D. A., 1979, Heat Recovery Systems, E.&F.N. Spon, London.

APPENDIX C Two-Phase Heat Transfer and Pressure Drop Correlations Although the focus in this book is on single-phase ﬂow heat exchanger design and analysis, there are situations when phase-change (condensation or vaporizing) ﬂuid having negligible thermal resistance is on one ﬂuid side of a two-ﬂuid heat exchanger; the design and analysis for such an exchanger can be done using the slightly modiﬁed single-phase theory outlined in this book. However, we need to compute the heat transfer coeﬃcient on the phase-change side even for this situation. Additionally, if one would like to estimate approximately the performance or size of the phase-change exchanger, it can be treated as a single-phase exchanger once the average heat transfer coeﬃcient on the phase-change side is determined. Hence, in this appendix we provide some correlations for condensation and convective boiling. For the detailed information on the phasechange correlations and related phenomena, a comprehensive source is the handbook by Kandilkar et al. (1999). For completeness, we also provide a method to compute the pressure drop on the phase-change side and present it before the heat transfer correlations. Of course, many important topics of phase-change exchangers, such as the phasechange side not having the negligible thermal resistance, rating and sizing of the exchanger when phase change occurs on both ﬂuid sides, ﬂow maldistribution, and so on, are beyond the scope of this appendix and the book. C.1 TWO-PHASE PRESSURE DROP CORRELATIONS Due to the phase change during condensation or vaporization, the pressure gradient within the ﬂuid changes along the ﬂow path or axial length. The pressure drop in the phase-change ﬂuid can then be computed by integrating the nonlinear pressure gradient along the ﬂow path. In contrast, the pressure gradient is linear along the ﬂow length (axial direction) in many single-phase ﬂow applications, and hence we generally work directly with the pressure drop since there is no need to compute the pressure gradient in singlephase ﬂow. The total local pressure gradient in two-phase ﬂow through a one-dimensional duct can be calculated as follows{: dp dpfr dpmo dpgr þ þ ðC:1Þ ¼ dz dz dz dz {

Additional symbols used in this appendix are all deﬁned here and are not included in the main nomenclature section.

Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

913

914

TWO-PHASE HEAT TRANSFER AND PRESSURE

where the three terms on the right-hand side correspond to the contributions by friction, momentum rate change, and gravity denoted by the subscripts fr, mo, and gr, respectively. The analysis that follows is based on a homogeneous model. The entrance and exit pressure loss terms of single-phase ﬂow [see Eq. (6.28)] are lumped into the pfr term since the information about these contributions is not available, due to the diﬃculty in measurements. The in-tube two-phase frictional pressure drop is computed from the corresponding pressure drop for single-phase ﬂow as follows using the two-phase friction multiplier denoted as ’2 :

dp dz

¼ flo fr

4 G2 ’2 Dh 2gc l lo

where ’2lo ¼

ðdp=dzÞfr ðdp=dzÞfr;lo

ðC:2Þ

where flo is the single-phase Fanning friction factor (see Tables 7.3 through 7.8) based on the total mass ﬂow rate as liquid and G is also based on the total mass ﬂow rate as liquid; this means that the subscript ‘‘lo’’ indicates the two-phase ﬂow considered as all liquid ﬂow. The subscripts l and g in Eqs. (C.2) and (C.3) denote liquid and gas/vapor phases, respectively, and the subscript lo stands for entire two-phase ﬂow as liquid ﬂow. Alternatively, ðdp=dzÞfr is determined using the liquid or vapor-phase pressure drop multiplier as follows.

dp dz

¼ fr

dp dz

’2l ¼ fr;l

dp dz

’2g

ðC:3Þ

fr;g

where ’2l

ðdp=dzÞfr ¼ ðdp=dzÞfr;l

’2g

ðdp=dzÞfr ¼ ðdp=dzÞfr;g

dp 4fl G2 ¼ dz fr;l 2gc l Dh

dp dz

¼ fr;g

4fg G2 ðC:4Þ 2gc g Dh

where the subscripts l and g denote liquid and gas/vapor phases. ’2lo and ’2l or ’2g are functions of the parameter X (Martinelli parameter). ’2go [deﬁned similar to ’2lo of Eq. (C.2), with the subscript lo replaced by go] is a function of Y (Chisholm parameter). The X and Y are deﬁned as follows: ðdp=dzÞfr;l ðdp=dzÞfr;g

X2 ¼

Y2 ¼

ðdp=dzÞfr;go ðdp=dzÞfr;lo

ðC:5Þ

Here the subscript go means the total two-phase ﬂow considered as all gas ﬂow. The correlations to determine the two-phase frictional pressure gradient are presented in Table C.1 for various ranges of G and l =g (Kandlikar et al., 1999, p. 228). The momentum pressure gradient can be calculated integrating the momentum balance equation (Collier and Thome, 1994), thus obtaining

dp dz

¼ mo

d G2 x2 ð1 xÞ2 þ dz gc g ð1 Þl

ðC:6Þ

where represents the void fraction of the gas (vapor) phase (a ratio of volumetric ﬂow rate of the gas/vapor phase divided by the total volumetric ﬂow rate of the two-phase mixture), and x is the mass quality (a ratio of the mass ﬂow rate of the vapor/gas phase

915

TWO-PHASE PRESSURE DROP CORRELATIONS

TABLE C.1 Frictional Multiplier Correlations Used for Determining the Two-Phase Frictional Pressure Gradient in Eq. (C.2) Correlation

Parameters

Friedel correlation (1979) for l =g > 1000 and all values of G: ’2lo ¼ E þ

3:24FH Fr0:045 We0:035

Accuracy for annular ﬂow: 21% (Ould Dide et al., 2002

fgo E ¼ ð1 xÞ þ x l g flo 2

2

F ¼ x0:78 ð1 xÞ0:24

0:91

H¼

l g

Fr ¼

G2 gdi 2hom

g l

0:19 1

We ¼

1 x 1x ¼ þ hom g l Chisholm correlation (1973) for l =g > 1000 and G > 100 kg/m2 s:

’2lo ¼ 1 þ ðY 2 1Þ½Bxn ð1 xÞn þ x1n n* ¼

2n 2

g l

0:7

G 2 di hom

¼ surface tension (N/m)

Y deﬁned in Eq. (C.4); n ¼ 14 (exponent in f ¼ C Ren Þ G ¼ total mass velocity, kg/m2 s 8 > < 4:8 B ¼ 2400=G > : 55=G1=2

Accuracy for annular ﬂow: 38% (Ould Didi et al., 2002)

( B¼

G < 500 500 G 1900 G 1900

520=ðYG1=2 Þ

G 600

21=G

G > 600

9 > = > ;

for 0 < Y 9:5

) for 9:5 < Y 28

B ¼ 15,000=ðY 2 G1=2 Þ for Y > 28 Lockhart-Martinelli correlation (1949) for l g > 1000 and G < 100 kg=m2 s: ’21

d=dzÞfr c 1 ¼ ¼1þ þ 2 ðdp=dzÞl X X

’2g ¼

Correlation constant by Chisholm (1967): c ¼ 20 c ¼ 10 c ¼ 12 c¼5

for liquid and vapor both turbulent for liquid-turbulent, vapor-laminar for liquid-laminar, vapor-turbulent for liquid and vapor both laminar

ðdp=dzÞfr ¼ 1 þ cX þ X 2 ðdp=dzÞg

Accuracy for annular ﬂow: 29% (Ould Didi et al., 2002)

divided by the total mass ﬂow rate of the two-phase mixture). Equation (C.6) is valid for constant cross-sectional (ﬂow) area along the ﬂow length. For the homogeneous model, the two-phase ﬂow behaves like a single phase and the vapor and liquid velocities are equal. A number of correlations for the void fraction are given by Carey (1992) and Kandlikar et al. (1999). An empirical correlation for the void fraction whose general form is valid for several frequently used models is given by Butterworth (Carey, 1992) as 1 x p g q l r 1 ¼ 1þA l g x

ðC:7Þ

916

TWO-PHASE HEAT TRANSFER AND PRESSURE

where the constants A, p, q, and r depend on the two-phase model and/or empirical data chosen. These constants for a nonhomogeneous model, based on steam–water data, are A ¼ 1, p ¼ 1, q ¼ 0:89, and r ¼ 0:18. For the homogeneous model, A ¼ p ¼ q ¼ 1 and r ¼ 0. For the Lockhart and Martinelli model, A ¼ 0:28, p ¼ 0:64, q ¼ 0:36, and r ¼ 0:07. For engineering design calculations, the homogeneous model yields the best results when the slip velocity between the gas and liquid phases is small (for bubbly or mist ﬂows). Finally, the pressure gradient due to the gravity (hydrostatic) eﬀect is

dp dz

¼ gr

g sin ½g þ ð1 Þl gc

ðC:8Þ

Note that the negative sign (i.e., the pressure recovery) stands for downward ﬂow in inclined or vertical tubes/channels, and the positive sign (i.e., pressure drop) represents upward ﬂow in inclined or vertical tubes/channels. And represents the angle of tube/ channel inclination measured from the horizontal axis.

C.2

HEAT TRANSFER CORRELATIONS FOR CONDENSATION

Condensation represents a vapor–liquid phase-change phenomenon that usually takes place when vapor is cooled below its saturation temperature at a given pressure. The heat transfer rate per unit heat transfer surface area from the pure condensing ﬂuid to the wall is given by q 00 ¼ hcon ðTsat Tw Þ

ðC:9Þ

where hcon is the condensation heat transfer coeﬃcient, Tsat is the saturation temperature of the condensing ﬂuid at a given pressure, and Tw is the wall temperature. We summarize here the correlations for ﬁlmwise in-tube condensation, a common condensation mode in TABLE C.2 Heat Transfer Correlations for Internal Condensation in Horizontal Tubes Stratiﬁcation Conditions Annular ﬂowa (ﬁlm condensation) (Shah, 1977), accuracy 14:4% (Kandlikar et al., 1999)

Stratiﬁed ﬂow (Carey, 1992), accuracy: 18% (Ould Didi et al., 2002) a

Correlation hloc ¼ 0:023 Rel ¼

Gdi ; l

kl 3:8x0:76 ð1 xÞ0:04 0:4 ð1 xÞ0:8 þ Re0:8 l Prl di ðpsat =pcr Þ0:38 G ¼ total mass velocity ðkg=m2 sÞ

11 G 1599 kg=m2 s 0:002 psat =pcr 0:44 Prl > 0:5 21 Tsat 3108C; 0 x 1; 3 uvap 300 m=s; no limit on q Rel > 350 for circular tubes 7 di 40 mm 1 x g 2=3 3=4 k3l l ðl g Þghlg0 1=4 hm ¼ 0:728 1 þ l 1 ðTsat Tw Þdi x where hlg0 ¼ hlg þ 0:68cp;l ðTsat Tw Þ

Valid for horizontal, vertical, or inclined tubes.

HEAT TRANSFER CORRELATIONS FOR BOILING

917

most industrial applications. The two most common ﬂow patterns for convective condensation are annular ﬁlm ﬂow in horizontal and vertical tubes and stratiﬁed ﬂow in horizontal tubes. For annular ﬁlm ﬂow, the correlation for the local heat transfer coeﬃcient hloc ½hcon ¼ hloc in Eq. (C.9)] is given in Table C.2; and also for stratiﬁed ﬂow, the correlation for mean condensation heat transfer coeﬃcient hcon ¼ hm is given in Table C.2. Shah et al. (1999) provide condensation correlations for a number of noncircular ﬂow passage geometries.

C.3 HEAT TRANSFER CORRELATIONS FOR BOILING Vaporization (boiling and evaporation) phenomena have been investigated and reported extensively in the literature. In this case, the heat transfer rate per unit heat transfer surface area from the wall to the pure vaporizing ﬂuid is given by q 00 ¼ htp ðTw Tsat Þ

ðC:10Þ

where htp is the two-phase heat transfer coeﬃcient during the vaporization process. We present here a most general intube forced convective boiling correlation proposed by Kandlikar (1991). It is based on empirical data for water, refrigerants and cryogens. The correlation consists of two parts, the convective and nucleate boiling terms, and utilizes a ﬂuid–surface parameter. The Kandlikar correlation for the two-phase heat transfer coeﬃcient is as follows: 8 < ½0:6683Co0:2 f2 ðFrlo Þ þ 1058 Bo0:7 Ffl ð1 xÞ0:8 htp ¼ larger of : hlo ½1:136Co0:9 f ðFr Þ þ 667:2Bo0:7 F ð1 xÞ0:8 2

ðC:11Þ

fl

lo

where

hlo ¼

( f2 ðFrlo Þ ¼

8 > > > < > > > :

Relo Prl ð f =2Þðkl =di Þ 1:07 þ 12:7ðPr2=3 1Þð f =2Þ0:5

104 Relo 5 106

Relo Prl ð f =2Þðkl =di Þ 1:07 þ 12:7ðPr2=3 1Þð f =2Þ0:5

2300 Relo 10

ð25Frlo Þ0:3 1

ðC:12Þ 4

for Frlo < 0:04 in horizontal tubes for vertical tubes and for Frlo 0:04 in horizontal tubes ðC:13Þ 1 f ¼ ½1:58 lnðRelo Þ 3:282

ðC:14Þ

Here hlo is the single-phase heat transfer coeﬃcient for the entire ﬂow as liquid ﬂow. Also, the convection number Co, the nucleate boiling number Bo, and the Froude number Fr for the entire ﬂow as liquid are deﬁned as follows: Co ¼

g l

0:5

1x x

0:8 Bo ¼

q 00 Gh‘g

Fr ¼

G2 2l gdi

ðC:15Þ

918

TWO-PHASE HEAT TRANSFER AND PRESSURE

TABLE C.3 Fﬂ Recommended by Kandlikar (1991) Fluid Water R-11 R-12 R-13B1 R-22 R-113

Ffl

Fluid

Ffl

1.00 1.30 1.50 1.31 2.20 1.30

R-114 R-134a R-152a R-32/R-132 (60%–40% wt.) Kerosene

1.24 1.63 1.10 3.30 0.488

Ffl is a ﬂuid–surface parameter and depends on the ﬂuid and the heat transfer surface. Ffl values for several ﬂuids in copper tubes are presented in Table C.3. Ffl should be taken as 1.0 for stainless tubes. This correlation is valid for either vertical (upward and downward) or horizontal intube ﬂow. A mean deviation of slightly less than 16% with water and 19% with refrigerants has been reported by Kandlikar (1991). Note that being ﬂuid speciﬁc, Ffl cannot be used for other ﬂuids (new refrigerants) and mixtures. It is also not accurate for stratiﬁed wavy ﬂows and at high vapor qualities since it is not based on the onset of dryout. The Thome model (Kattan et al., 1998; Zrcher et al., 1999), based on a ﬂow pattern map, is recommended for those cases.

REFERENCES Carey, V. P., 1992, Liquid-Vapor Phase Change Phenomena, Taylor & Francis, Bristol, PA. Chisholm, D., 1967, A theoretical basis for the Lockhart–Martinelli correlation for two-phase ﬂow, Int. J. Heat Mass Transfer, Vol. 10, pp. 1767–1778. Chisholm, D., 1973, Pressure gradients due to friction during the ﬂow of evaporating two-phase mixtures in smooth tubes and channels, Int. J. Heat Mass Transfer, Vol. 16, pp. 347–358. Collier, J. G., and J. R. Thome, 1994, Convective Boiling and Condensation, 3rd ed., McGraw-Hill, New York. Friedel, L., 1979, Improved friction pressure drop correlations for horizontal and vertical two-phase pipe ﬂow, European Two-Phase Flow Group Meeting, Ispra, Italy, Paper E2. Hewitt, G. F., 1998, Gas–liquid ﬂow, in Handbook of Heat Exchanger Design, G. F. Hewitt, ed., Begell House, New York, Sect. 2.3.2. Kandlikar, S. G., 1991, Development of a ﬂow boiling map for subcooled and saturated ﬂow boiling of diﬀerent ﬂuids in circular tubes, ASME J. Heat Transfer, Vol. 113, pp. 190–200. Kandlikar, S. G., M. Shoji, and V. K. Dhir, eds., 1999, Handbook of Phase Change: Boiling and Condensation, Taylor & Francis, New York. Kattan, N., J. R. Thome, and D. Favrat, 1998, Flow boiling in horizontal tubes, Part 1; Development of a diabatic two-phase ﬂow pattern map, ASME J. Heat Transfer, Vol. 120, pp. 140–147; Part 2; New heat transfer data for ﬁve refrigerants, ASME J. Heat Transfer, Vol. 120, pp. 148– 155; Part 3; Development of a new heat transfer model based on ﬂow patterns, ASME J. Heat Transfer, Vol. 120, pp. 156–165. Ould Didi, M. B., N. Kattan, and J. R. Thome, 2002, Prediction of two-phase pressure gradients of refrigerants in horizontal tubes, Int. J. Refrig., Vol. 25, pp. 935–947. Shah, M. M., 1977, A general correlation for heat transfer during subcooled boiling in pipes and annuli, ASHRAE Trans., Vol. 83, No. 1, pp. 205–215; also, M. M. Shah, 1982, Chart correlation

REFERENCES

919

for saturated boiling heat transfer: equations and further study, ASHRAE Trans., Vol. 88, No. 1, pp. 185–196. Shah, R. K., S. Q. Zhou, and K. Tagavi, 1999, The role of surface tension in ﬁlm condensation in extended surface passages, J. Enhanced Heat Transfer, Vol. 6, pp. 179–216. Zrcher, O., J. R. Thome, and D. Favrat, 1999, Evaporation of ammonia in a smooth horizontal tube: heat transfer measurements and predictions, ASME J. Heat Transfer, Vol. 121, pp. 89–101.

APPENDIX D U and CUA Values for Various Heat Exchangers

920

Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

921

5,000

1,000

q=T (W/K)

Boiling organic liquid Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water Low-viscosity organic liquid High-viscosity liquid

Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water Low-viscosity organic liquid High-viscosity liquid Boiling water

Cold-Side Fluid

LowPressure Gas (< 1 bar) 55 5.70 93 5.02 120 5.51 105 4.89 99 4.96 68 5.39 105 4.89 99 4.96 55 2.11 93 1.63 120 2.26 105 1.56 99 1.59 68 1.86

Parameter

UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)} UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] 93 5.02 300 4.18 350 4.81 484 3.98 375 4.09 138 4.61 467 3.99 375 4.09 93 1.63 300 1.11 350 1.89 484 1.00 375 1.05 138 1.43

MediumPressure Gas (20 bar) 120 5.51 350 4.81 400 6.25 600 4.56 450 4.38 200 5.50 550 4.91 450 4.38 120 2.26 350 1.89 400 2.25 600 1.10 450 1.46 200 1.93

HighPressure Gas (150 bar)

TABLE D.1 U and CUAð¼C Þ Values for Shell-and-Tube Heat Exchangersa

102 4.93 429 4.03 600 4.56 938 3.77 600 3.91 161 4.46 875 3.79 600 3.91 102 1.58 429 1.02 600 1.10 938 0.88 600 0.95 161 1.36

Process Water 99 4.96 375 4.09 450 4.38 714 3.85 500 3.97 153 4.51 677 3.87 500 3.97 99 1.59 375 1.05 450 1.46 720 0.91 500 0.99 153 1.38

LowViscosity Organic Liquid 63 5.50 120 4.76 200 5.50 142 4.59 130 4.67 82 5.16 140 4.60 130 4.67 63 1.95 120 1.49 200 1.93 142 1.41 130 1.46 82 1.71

HighViscosity Fluid

Hot-Side Fluid

107 4.87 530 3.95 600 4.56 1607 3.61 818 3.81 173 4.42 1432 3.64 818 3.81 107 1.55 530 0.98 600 1.10 1607 0.83 818 0.89 173 1.32

Condensing Steam 100 4.95 388 4.07 400 4.82 764 3.83 524 3.95 155 4.50 722 3.85 524 3.95 100 1.59 388 1.05 400 1.45 764 0.90 524 0.98 155 1.37

Condensing Hydrocarbon

86 5.11 240 4.28 300 4.81 345 4.12 286 4.20 214 4.33 336 4.13 286 4.20 86 1.68 240 1.18 300 1.45 345 1.07 286 1.13 124 1.48

Condensing Hydrocarbon with Inert Gas

922

100,000

30,000

q=T (W/K) 105 1.56 99 1.59 55 1.11 93 0.76 120 1.06 105 0.71 99 0.74 68 0.94 105 0.71 99 0.74 55 0.95 93 0.58 120 0.93 105 0.52

Parameter

UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)]

Cold-Side Fluid

Boiling water

Boiling organic liquid Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water

Boiling organic liquid Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water Low-viscosity organic liquid High-viscosity liquid Boiling water

LowPressure Gas (< 1 bar)

TABLE D.1 Continued

467 1.00 375 1.05 93 0.76 300 0.37 350 0.62 484 0.29 375 0.33 138 0.57 467 0.29 375 0.33 93 0.58 300 0.23 350 0.35 484 0.17

MediumPressure Gas (20 bar) 550 1.20 450 1.46 120 1.06 350 0.62 400 0.94 600 0.40 450 0.53 200 0.73 550 0.49 450 0.53 120 0.93 350 0.35 400 0.58 600 0.24

HighPressure Gas (150 bar) 875 0.88 600 0.95 102 0.73 500 0.28 600 0.40 938 0.23 600 0.27 161 0.52 875 0.23 600 0.27 102 0.54 429 0.18 600 0.24 938 0.116

Process Water 677 0.93 500 0.99 99 0.74 375 0.33 450 0.53 714 0.25 500 0.38 153 0.53 677 0.25 500 0.28 99 0.55 375 0.20 450 0.28 714 0.134

LowViscosity Organic Liquid 140 1.42 130 1.46 63 0.99 120 0.63 200 0.73 142 0.56 130 0.59 82 0.83 140 0.56 130 0.59 63 0.83 120 0.47 200 0.64 142 0.41

HighViscosity Fluid

Hot-Side Fluid

1432 0.84 818 0.89 107 0.71 530 0.28 600 0.40 1607 0.19 818 0.24 173 0.50 1432 0.20 818 0.24 107 0.52 530 0.16 600 0.24 1607 0.086

Condensing Steam

722 0.91 524 0.98 100 0.73 388 0.32 400 0.62 764 0.24 524 0.28 155 0.53 722 0.25 524 0.28 100 0.55 388 0.19 400 0.32 764 0.129

Condensing Hydrocarbon

336 1.08 286 1.13 86 0.80 240 0.42 300 0.62 345 0.34 286 0.38 124 0.60 336 0.35 286 0.38 86 0.62 240 0.27 300 0.39 345 0.21

Condensing Hydrocarbon with Inert Gas

923

UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] 99 0.55 68 0.77 105 0.52 99 0.55

375 0.20 138 0.42 467 0.168 375 0.20

450 0.28 200 0.64 550 0.26 450 0.28

609 0.145 161 0.37 875 0.121 600 0.146

500 0.162 153 0.38 677 0.137 500 0.162

130 0.44 82 0.65 140 0.41 130 0.44

a

Source: Selection and Costing of Heat Exchangers, ESDU Engineering Data 92013, ESDU International, London, 1994. A CUA value for a ðq=Tm Þ between values of ðq=Tm Þj , j ¼ 1, 2 should be calculated by logarithmic interpolation: lnðCUA;1 =CUA;2 Þ ln½ðq=Tm Þ=ðq=Tm Þ1 CUA ¼ exp ln CUA;1 þ ln½ðq=Tm Þ1 =ðq=Tm Þ2

Boiling organic liquid

Low-viscosity organic liquid High-viscosity liquid Boiling water

818 0.125 173 0.35 1432 0.091 818 0.125

524 0.158 155 0.38 722 0.133 524 0.158

286 0.24 124 0.45 336 0.22 286 0.24

924

10,000

Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water Low-viscosity hydrocarbon High-viscosity hydrocarbon Boiling water

5,000

Boiling hydrocarbon Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water Low-viscosity hydrocarbon High-viscosity

Cold-Side Fluid

q=T (W/K)

UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ

Parameter

491 1.55 NUS NUS

NUS

NA

NA

315 1.55 NUS

NUS

NUS

402 3.10 217 1.55 325 1.55 NA

491 3.10 NUS

315 3.10 NUS

270 3.10 163 1.57 217 1.55 NA

217 3.10 325 3.10 NA

MediumPressure Gas (20 bar)

163 3.10 217 3.10 NA

LowPressure Gas (< 1 bar)

NA

NUS

NA

NA

NA

NA

NA

NA

NA

NA

NA

NA

NA

NUS

NA

NA

NA

NA

NA

NA

NA

NA NA

NA

NA

NA

NA

Process Water

NA

HighPressure Gas (150 bar)

TABLE D.2 U and CUA (¼C) Values for Plate-Fin Heat Exchangersa

NUS

NUS

NUS

453 3.10 264 1.55 377 1.55 NA

NA

NUS

NUS

NUS

264 3.10 377 3.10 NA

LowViscosity Hydrocarbon Liquid

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

HighViscosity Hydrocarbon Liquid

Hot-Side Fluid

NUS

NUS

NUS

NA

NUS

NUS

NA

NA

NUS

NUS

NUS

NA

NUS

NUS

Condensing Steam

NUS

NUS

NUS

530 3.10 270 1.55 402 1.55 NA

NA

NUS

NUS

NUS

270 3.10 402 3.10 NA

Condensing Hydrocarbon

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

Condensing Hydrocarbon with Inert Gas

925

Boiling hydrocarbon

Boiling hydrocarbon Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water Low-viscosity hydrocarbon High-viscosity hydrocarbon Boiling water

C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] UðW=m2 KÞ C[£/(W/K)] NA

NA 402 0.227

NUS

NUS

270 0.273

491 0.210 NUS

NA

NA

315 0.250 NUS

NUS

NUS

402 0.532 217 0.301 325 0.245 NA

491 0.513 NUS

315 0.560 NUS

270 0.579 163 0.336 217 0.301 NA

402 1.55 217 0.607 325 0.551 NA

NA

270 1.55 163 0.677 217 0.607 NA

NA

NA

NA

NA

NA

NA

NUS

NA

NA

NA

NA NA

NA

NA

NA

NA

NA

NA

NUS

NA

NA

NA

NA

NA

NA

NA

NA NA

NA

NA

NA

NA

NA

NA

NA

NA

NA

453 0.216

NA

NUS

NUS

NUS

453 0.527 264 0.280 377 0.231 NA

NA

NUS

NUS

NUS

453 1.55 264 0.574 377 0.537 NA

NA

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

Source: Selection and Costing of Heat Exchangers, ESDU Engineering Data 92013, ESDU International, London, 1994. a A CUA value for a ðq=Tm Þ between values of ðq=Tm Þj , j ¼ 1, 2 should be calculated by logarithmic interpolation: lnðCUA;1 =CUA;2 Þ ln½ðq=Tm Þ=ðq=Tm Þ1 CUA ¼ exp ln CUA;1 þ ln½ðq=Tm Þ1 =ðq=Tm Þ2

100,000

30,000

Boiling hydrocarbon Low-pressure gas (< 1 bar) Medium-pressure gas (20 bar) High-pressure gas (150 bar) Treated cooling water Low-viscosity hydrocarbon High-viscosity hydrocarbon Boiling water

hydrocarbon Boiling water

NA

NA

NUS

NUS

NUS

NA

NUS

NUS

NA

NA

NUS

NUS

NUS

NA

NUS

NUS

NA

NA

530 0.205

NA

NUS

NUS

NUS

530 0.518 270 0.273 402 0.227 NA

NA

NUS

NUS

NUS

530 1.55 270 0.579 402 0.532 NA

NA

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

NUS

NUS

NUS

NA

GENERAL REFERENCES ON OR RELATED TO HEAT EXCHANGERS Ackerman, R. A., 1997, Cryogenic Regenerative Heat Exchangers, International Cryogenics Monograph Series, Plenum Publishing, New York. Afgan, N., M. Carvalho, A. Bar-Cohen, D. Butterworth, and W. Roetzel, eds., 1994, New Developments in Heat Exchangers, Gordon & Breach, New York. Afgan, N. H., and E. U. Schlu¨nder, eds., 1974, Heat Exchangers: Design and Theory Sourcebook, McGraw-Hill, New York. Andreone, C. F., and S. Yokell, 1997, Tubular Heat Exchanger: Inspection, Maintenance and Repair, McGraw-Hill, New York. Apblett, W. R., Jr., ed., 1982, Shell and Tube Heat Exchangers, American Society for Metals, Metals Park, OH. Au-Yang, M. K., 2001, Flow-Induced Vibrations of Power Plants Components: A Practical Workbook, ASME Press, New York. Azbel, D., 1984, Heat Transfer Applications in Process Engineering, Noyes Publications, Park Ridge, NJ. Bar-Cohen, A., M. Carvalho, and R. Berryman, eds., 1998, Heat exchangers for sustainable development, Proc. Heat Exchangers for Sustainable Development, Lisbon, Portugal. Beck, D. S., and D. G. Wilson, 1996, Gas Turbine Regenerators, Chapman & Hall, New York. Bhatia, M. V., and P. N. Cheremisinoﬀ, 1980, Heat Transfer Equipment, Process Equipment Series, Vol. 2, Technomic Publishing, Westport, CT. Bliem, C., et al., 1985, Ceramic Heat Exchanger Concepts and Materials Technology, Noyes Publications, Park Ridge, NJ. Blevins, R. D., 1990, Flow-Induced Vibration, 2nd ed., Von Nostrand Reinhold, New York. Bohnet, M., T. R. Bott, A. J. Karabelas, P. A. Pilavachi, R. Se´me´ria, and R. Vidil, eds., 1992, Fouling Mechanisms: Theoretical and Practical Aspects, Eurotherm Seminar 23, Editions Europe´ennes Thermique et Industrie, Paris. Bott, T. R., 1990, Fouling Notebook: A Practical Guide to Minimizing Fouling in Heat Exchangers, Institution of Chemical Engineers, London. Bott, T. R., 1995, Fouling of Heat Exchangers, Elsevier Science Publishers, Amsterdam, The Netherlands. Bott, T. R., L. F. Melo, C. B. Panchal, and E. F. C. Somerscales, 1999, Understanding Heat Exchanger Fouling and Its Mitigation, Begell House, New York. Bryers, R. W., ed., 1983, Fouling of Heat Exchanger Surfaces, Engineering Foundation, New York. Bryers, R. W., ed., 1983, Fouling and Slagging Resulting from Impurities in Combustion Gases, Engineering Foundation, New York. Buchlin, J. M., ed., 1991, Industrial Heat Exchangers, Lecture Series 1991–04, von Ka´rma´n Institute for Fluid Dynamics, Belgium. Chen, S. S., 1987, Flow-Induced Vibration of Circular Cylindrical Structures, Hemisphere Publishing, Washington, DC. Chisholm, D., ed., 1980, Developments in Heat Exchanger Technology, Vol. I, Applied Science Publishers, London. Chisholm, D., ed., 1988, Heat Exchanger Technology, Elsevier Applied Science, New York. Dragutinovic´, G. D., and B. S. Bacˇlic´, 1998, Operation of Counterﬂow Regenerators, Vol. 4, Computational Mechanics Publications, WIT Press, Southampton, UK. Dzyubenko, B. V., L.-V. Ashmantas, and M. D. Segal, 1999, Modeling and Design of Twisted Tube Heat Exchangers, Begell House, New York. 926

Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

GENERAL REFERENCES ON HEAT EXCHANGERS

927

Foster, B. D., and J. B. Patton, eds., 1985, Ceramic Heat Exchangers, American Ceramic Society, Columbus, OH. Foumeny, E. A., and P. J. Heggs, eds., 1991, Heat Exchange Engineering, Vol. 1; Design of Heat Exchangers, Ellis Horwood, London. Foumeny, E. A., and P. J. Heggs, eds., 1991, Heat Exchange Engineering, Vol. 2; Compact Heat Exchangers: Techniques for Size Reduction, Ellis Horwood, London. Fraas, A. P., and M. N. Ozisik, 1989, Heat Exchanger Design, 2nd ed., Wiley, New York. Ganapathy, V., 1982, Applied Heat Transfer, PennWell Publishing, Tulsa, OK. Ganapathy, V., 2002, Industrial Boilers and Heat Recovery Steam Generators – Design, Applications, and Calculations, Marcel Dekker, New York. Garrett-Price, B. A., S. A. Smith, R. L. Watts, J. G. Knudsen, W. J. Marner, and J. W. Suitor, 1985, Fouling of Heat Exchangers, Noyes Publications, Park Ridge, NJ. Gupta, J. P., 1986, Fundamentals of Heat Exchanger and Pressure Vessel Technology, Hemisphere Publishing, Washington, DC; also as Working with Heat Exchangers, in soft cover, Hemisphere Publishing, Washington, DC, 1990. Hausen, H., 1983, Heat Transfer in Counterﬂow, Parallel Flow and Cross Flow, McGraw-Hill, New York. Hayes, A. J., W. W. Liang, S. L. Richlen, and E. S. Tabb, eds., 1985, Industrial Heat Exchangers, American Society for Metals, Metals Park, OH. Hesselgreaves, J. E., 2001, Compact Heat Exchangers: Selection, Design, and Operation, Elsevier Science, Oxford. Hewitt, G. F., exec. ed., 1998, Heat Exchanger Design Handbook, three vols. (ﬁve parts), Begell House, New York; former publication: G. F. Hewitt, coord. ed., 1989, Hemisphere Handbook of Heat Exchanger Design, Hemisphere Publishing, New York. Hewitt, G. F., G. L. Shires, and T. R. Bott, 1994, Process Heat Transfer, CRC Press and Begell House, Boca Raton, FL. Hewitt, G. F., and P. B. Whalley, 1989, Handbook of Heat Exchanger Calculations, Hemisphere Publishing, Washington, DC. Hryniszak, W., 1958, Heat Exchangers: Applications to Gas Turbines, Butterworth Scientiﬁc Publications, London. Idelchik, I. E., 1994, Handbook of Hydraulic Resistance, 3rd ed., CRC Press, Boca Raton, FL. Ievlev, V. M., ed., 1990, Analysis and Design of Swirl-Augmented Heat Exchangers, Hemisphere Publishing, Washington, DC. Jakob, M., 1957, Heat Transfer, Vol. II, Wiley, New York. Kakac¸, S., ed., 1991, Boilers, Evaporators, and Condensers, Wiley, New York. Kakac¸, S., ed., 1999, Heat Transfer Enhancement of Heat Exchangers, Kluwer Academic Publishers, Dordrecht, The Netherlands. Kakac¸, S., A. E. Bergles, and E. O. Fernandes, eds., 1988, Two-Phase Flow Heat Exchangers: Thermal Hydraulic Fundamentals and Design, Kluwer Academic Publishers, Dordrecht, The Netherlands. Kakac¸, S., A. E. Bergles, and F. Mayinger, eds., 1981, Heat Exchangers: Thermal-Hydraulic Fundamentals and Design, Hemisphere Publishing, Washington, DC. Kakac¸, S., and H. Liu, 1998, Heat Exchangers: Selection, Rating, and Thermal Design, CRC Press, Boca Raton, FL. Kakac¸, S., R. K. Shah, and W. Aung, eds., 1987, Handbook of Single-Phase Convective Heat Transfer, Wiley, New York. Kakac¸, S., R. K. Shah, and A. E. Bergles, eds., 1983, Low Reynolds Number Flow Heat Exchangers, Hemisphere Publishing, Washington, DC.

928

GENERAL REFERENCES ON HEAT EXCHANGERS

Katinas, V., and A. Zˇukauskas, 1997, Vibrations of Tubes in Heat Exchangers, Begell House, New York. Kays, W. M., and A. L. London, 1998, Compact Heat Exchangers, reprint 3rd edn., Krieger Publishing, Malabar, FL; ﬁrst ed., National Press, Palo Alto, CA (1955); 2nd ed., (1964), 3rd ed., McGraw-Hill, New York (1984). Kern, D. Q., 1950, Process Heat Transfer, McGraw-Hill, New York. Kern, D. W., and A. D. Kraus, 1972, Extended Surface Heat Transfer, McGraw-Hill, Chaps. 9–12, pp. 439–641. King, R., ed., 1987, Flow Induced Vibrations, BHRA Publication, London. Kraus, A. D., 1982, Analysis and Evaluation of Extended Surface Thermal Systems, Hemisphere Publishing, Washington, DC. Kraus, A. D., A. Aziz, and J. R. Welty, 2001, Extended Surface Heat Transfer, Wiley, New York. Kro¨ger, D. G., 1998, Air-Cooled Heat Exchangers and Cooling Towers, Tecpress, Uniedal, South Africa; also, Begell House, New York. Kuppan, T., 2000, Heat Exchanger Design Handbook, Marcel Dekker, New York. Lokshin, V. A., D. F. Peterson, and A. L. Schwarz, 1988, Standard Handbook of Hydraulic Design for Power Boilers, Hemisphere Publishing, Washington, DC. Ludwig, E. E., 1965, Applied Process Design for Chemical and Petrochemical Plants, Vol. III, Gulf Publishing, Houston, TX, Chap. 10. Manzoor, M., 1984, Heat Flow through Extended Surface Heat Exchangers, Springer-Verlag, Berlin. Martin, M., 1992, Heat Exchangers, Hemisphere Publishing, Washington, DC. Marto, P. J., and R. H. Nunn, eds., 1981, Power Condenser Heat Transfer Technology, Hemisphere Publishing, Washington, DC. Marvillet, Ch., gen. ed., 1994, Recent Developments in Finned Tube Heat Exchangers: Theoretical and Practical Aspects, DTI Energy Technology, Danish Technological Institute, Taastrup, Denmark. Marvillet, C., and R. Vidil, eds., 1993, Heat Exchanger Technology: Recent Developments, Eurotherm Seminar 33, Editions Europe´ennes Thermique et Industrie, Paris. McNaughton, K. J., ed., 1986, The Chemical Engineering Guide to Heat Transfer; Vol. 1; Plant Principles, Vol. 2; Equipment, Hemisphere Publishing, Washington, DC. Melo, L. F., T. R. Bott, and C. A. Bernardo, eds., 1988, Advances in Fouling Science and Technology, Kluwer Academic Publishers, Dordrecht, The Netherlands. Miller, D. S., 1990, Internal Flow Systems, 2nd ed., BHRA Fluid Engineering Series, Vol. 5, BHRA, Cranﬁeld, UK. Minton, P. E., 1986, Handbook of Evaporator Technology, Noyes Publications, Park Ridge, NJ. Mori, Y., A. E. Sheindlin, and N. H. Afgan, eds., 1986, High Temperature Heat Exchangers, Hemisphere Publishing, Washington, DC. Mu¨ller-Steinhagen, H., ed., 2000, Heat Exchanger Fouling: Mitigation and Cleaning Technologies, Publico Publications, Essen, Germany. Palen, J. W., ed., 1987, Heat Exchanger Sourcebook, Hemisphere Publishing, Washington, DC. Panchal, C. B., T. R. Bott, E. F. C. Somerscales, and S. Toyama, 1997, Fouling Mitigation of Industrial Heat Exchange Equipment, Begell House, New York. Podhorsky, M., and H. Krips, 1998, Heat Exchangers: A Practical Approach to Mechanical Construction, Design and Calculations, Begell House, New York. Putman, R. E., 2001, Steam Surface Condensers: Basic Principles, Performance Monitoring and Maintenance, ASME Press, New York. Reay, D. A., 1979, Heat Recovery Systems, E&FN Spon, London.

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Reay, D. A., 1999, Learning from Experiences with Compact Heat Exchangers, CADDET Analyses Series 25, Centre for the Analysis and Dissemination of Demonstrated Energy Technologies, Sittard, The Netherlands. Rifert, V. G., 1998, Condensation Heat Transfer Enhancement, Computational Mechanics Publications, WIT Press, Southampton, UK. Roetzel, W., P. J. Heggs, and D. Butterworth, eds., 1991, Design and Operation of Heat Exchangers, Springer-Verlag, Berlin. Roetzel, W., and Y. Xuan, 1998, Dynamic Behaviour of Heat Exchangers, Vol. 3, Computational Mechanics Publications, WIT Press, Southampton, UK. Saunders, E. A. D., 1989, Heat Exchangers: Selection, Design and Construction, Wiley, New York. Schlu¨nder, E. U., ed.-in-chief, 1982, Heat Exchanger Design Handbook, 5 vols., Hemisphere Publishing, Washington, DC. Schmidt, F. W., and A. J. Willmott, 1981, Thermal Energy Storage and Regeneration, Hemisphere/ McGraw-Hill, Washington, DC. Shah, R. K., K. J. Bell, H. Honda, and B. Thonon, eds., 1999, Compact Heat Exchangers and Enhancement Technology for the Process Industries, Begell House, New York. Shah, R. K., K. J. Bell, S. Mochizuki, and V. V. Wadekar, eds., 1997, Compact Heat Exchangers for the Process Industries, Begell House, New York. Shah, R. K., A. W. Deakin, H. Honda, and T. M. Rudy, eds., 2001, Compact Heat Exchangers and Enhancement Technology for the Process Industries 2001, Begell House, New York. Shah, R. K., and A. Hashemi, eds., 1993, Aerospace Heat Exchanger Technology, 1993, Elsevier Science, Amsterdam, The Netherlands. Shah, R. K., A. D. Kraus, and D. Metzger, eds., 1990, Compact Heat Exchangers: A Festschrift for A.L. London, Hemisphere Publishing, Washington, DC. Shah, R. K., and A. L. London, 1978, Laminar Flow Forced Convection in Ducts, Supplement 1 to Advances in Heat Transfer Series, Academic Press, New York. Shah, R. K., and A. C. Mueller, 1985, Heat exchangers, in Handbook of Heat Transfer Applications, W. M. Rohsenow, J. P. Hartnett, and E. N. Ganic´, eds., McGraw-Hill, New York, Chap. 4, pp. 1–312. Shah, R. K., and A. C. Mueller, 1989, Heat exchange, in Ullman’s Encyclopedia of Industrial Chemistry, Unit Operations II, Vol. B3, Chap. 2, VCH Publishers, Weinheim, Germany. Shah, R. K., and D. P. Sekulic´, 1998, Heat exchangers, in Handbook of Heat Transfer, W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, eds., McGraw-Hill, New York, Chap. 17. Shah, R. K., E. C. Subbarao, and R. A. Mashelkar, eds., 1988, Heat Transfer Equipment Design, Hemisphere Publishing, Washington, DC. Sheindlin, A. E., ed., 1986, High Temperature Equipment, Hemisphere Publishing, Washington, DC. Singh, K. P., and A. I. Soler, 1984, Mechanical Design of Heat Exchangers and Pressure Vessel Components, Arcturus Publishers, Cherry Hill, NJ. Smith, E. M., 1997, Thermal Design of Heat Exchangers: A Numerical Approach: Direct Sizing and Stepwise Rating, Wiley, New York. Smith, R. A., 1987, Vaporisers: Selection, Design, Operation (Designing for Heat Transfer), Wiley, New York. Somerscales, E. F. C., and J. G. Knudsen, eds., 1981, Fouling of Heat Transfer Equipment, Hemisphere/McGraw-Hill, Washington, DC. Soumerai, H., 1987, Practical Thermodynamic Tools for Heat Exchanger Design Engineers, Wiley, New York. Stasiulevicius, J., and A. Skrinska, 1987, Heat Transfer of Finned Tube Bundles in Crossﬂow, Hemisphere Publishing, Washington, DC.

930

GENERAL REFERENCES ON HEAT EXCHANGERS

Sukhotin, A. M., and G. Tereshchenko, 1998, Corrosion Resistance of Equipment for Chemical Industry Handbook, Begell House, New York. Sunde´n, B., and M. Faghri, eds., 1998, Computer Simulation in Compact Heat Exchangers, Computational Mechanics Publications, WIT Press, Southampton, UK. Sunde´n, B., and P. J. Heggs, eds., 1998, Recent Advances in Analysis of Heat Transfer for Fin Type Surfaces, Computational Mechanics Publications, WIT Press, Southampton, UK. Sunde´n, B., and R. M. Manglik, eds., 2001, Plate and Frame Heat Exchangers, Computational Mechanics Publication, WIT Press, Southampton, UK. Taborek, J., G. F. Hewitt, and N. Afgan, eds., 1983, Heat Exchangers: Theory and Practice, Hemisphere/McGraw-Hill, Washington, DC. Taylor, M. A., 1987, Plate-Fin Heat Exchangers: Guide to Their Speciﬁcation and Use, HTFS, Harwell Laboratory, Oxon, UK. Walker, G., 1990, Industrial Heat Exchangers: A Basic Guide, 2nd edn., Hemisphere Publishing, Washington, DC. Webb, R. L., 1994, Principles of Enhanced Heat Transfer, Wiley, New York. Willmott, A. J., 2001, Dynamics of Regenerative Heat Transfer, Taylor & Francis, New York. Yokell, S., 1990, A Working Guide to Shell-and-Tube Heat Exchangers, McGraw-Hill, New York. Zˇukauskas, A. A., 1989, High Performance Single-Phase Heat Exchangers, Hemisphere Publishing, Washington, DC. [This book has a misleading title. It should be Forced Convection Heat Transfer.] Zˇukauskas, A., and R. Ulinskas, 1988, Heat Transfer in Tube Banks in Crossﬂow, Hemisphere Publishing, Washington, DC. Zˇukauskas, A. A., R. Ulinskas, and V. Katinas, 1988, Fluid Dynamics and Flow Induced Vibrations of Tube Banks, Hemisphere Publishing, Washington, DC.

Reference Books by Title Advances in Industrial Heat Exchangers, HEE96, Institution of Chemical Engineers, London, 1996. ASME Boiler and Pressure Vessel Code, Sec. VIII, Div. 1, Rules for Construction of Pressure Vessels, American Society of Mechanical Engineers, New York, 1998. Condensers: Theory and Practice, IChemE Symposium Series 75, Pergamon Press, Elmsford, NY, 1983. Eﬀectiveness Ntu Relationships for Design and Performance Evaluation of Multi-pass Crossﬂow Heat Exchangers, Engineering Sciences Data Unit Item 87020, ESDU International, McLean, VA, October 1987. Eﬀectiveness Ntu Relationships for Design and Performance Evaluation of Two-Stream Heat Exchangers, Engineering Sciences Data Unit Item 86018, ESDU International, McLean, VA, July 1986. Shell-and-Tube Heat Exchangers for General Reﬁnery Services, API Standard 660, 4th ed., American Petroleum Institute, Washington, DC, 1982. Standard of the Tubular Exchanger Manufacturers Association, 8th edn., TEMA, New York, 1999. Standard for Closed Feedwater Heaters, 4th ed., Heat Exchanger Institute, Cleveland, OH, 1984. Standard for Power Plant Heat Exchangers, Heat Exchanger Institute, Cleveland, OH, 1980. Standard for Steam Surface Condensers, 8th ed., Heat Exchanger Institute, Cleveland, OH, 1984. Standards of the Brazed Aluminum Plate-Fin Heat Exchanger Manufacturers’ Association (ALPEMA), 2nd ed., AEA Technology, Didcot, Oxon, UK, 2000.

Index Absorptivity, 539 gas, 542 Advection, 439 ASME code(s), 13 Analogy between ﬂuid ﬂow and electric entities, 98–99 Analytical correlations, 473. See also Correlations, and Heat transfer coeﬃcient fully developed ﬂows, 475 hydrodynamically developing ﬂows, 499 laminar ﬂow, 475 simultaneously developing ﬂow, 507 thermally developing ﬂows, 502 Annular ﬂow, 916 Arithmetic mean, 187 Baﬄe(s): disk-and-doughnut, 683 grid, 682 impingement, 684 plate, 682 rod, 684 segmental, 682, 683 strip, 683 Baﬄe geometry, 588 Balances, 776 cost, 783 energy, 779 exergy, 786 Balance equations, 102, 115, 260, 269, 314, 739, 750 Bavex welded-plate, 30 Bell-Delaware method, 294, 647. See also Heat exchanger design methodology correction factors, 648, 650 Bend losses: circular cross section, 405 miter bends, 409 rectangular cross section, 409 Bhatti-Shah correlation, 482 Biological fouling, 869

Borda-Carnot equation, 400 Boundary layers, 426, 432 inviscid region, 435 momentum, 426 temperature, 428 thermal, 428 thickness, 429 velocity region, 435 velocity, 426 Brazed plate heat exchanger, 30 Bulk temperature, 439 Capital investment cost, 791 Carryover leakages, 360 cross bypass, 360 pressure, 360 side bypass, 360 Chemical reaction fouling, 868, 892. See also Fouling Chen and Chiou correlation, 483 Chisholm correlation, 915 Chisholm parameter, 914 Classiﬁcation of heat exchanges, 3. See also Heat exchanger construction features, 12 ﬂow arrangements, 56 heat transfer mechanisms, 73 multiﬂuid, 8 three-ﬂuid, 8 transfer process, 3 two-ﬂuid, 8 Cleanliness coeﬃcient, 881 Cleanliness factor, 881 Cleaning strategies, 892. See also Fouling Circular ﬁns on circular tubes, 569 Colburn correlation, 483 Colburn factor, 447 uncertainty, 459 Cold-gas ﬂow period, 311 Combined entrance region, 436 Compact heat exchanger surfaces, 711 general relationships, 711

Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic Copyright © 2003 John Wiley & Sons, Inc.

931

932

INDEX

Composite curves, 779 Controlling resistance, 110 Convection, 439 forced, 439 natural or free, 439 Convection conductance ratio, 320 Convection heat transfer, 426, 438, 474 Core mass velocity, 379 Core mass velocity equation, 618, 632 Core rotation, 320 Core volume goodness factor comparisons, 705 Correction factor, 736 Correlations, 511. See also Heat transfer coeﬃcient corrugated ﬂat ﬁns, 521 crossed rod geometries, 524 individually ﬁnned tubes, 519 louver ﬁns, 516 mixed convection, 536 oﬀset strip ﬁns, 516 plain ﬂat ﬁns on a tube array, 520 plate heat exchanger surfaces, 514 plate-ﬁn extended surfaces, 515 regenerator surfaces, 523 tube bundles, 512 Corrosion, 893 factors, 894 Corrosion control, 897 Corrosion fouling, 868, 892 Corrosion locations, 895 crevice, 897 erosion, 897 galvanic, 895 pitting, 896 selective leaching, 897 stress, 896 uniform (general ), 895 Corrosion types, 895 crevice, 895 erosion, 895 galvanic, 895 pitting, 895 stress, 895 uniform, 895 Corrugated ﬁn, 39 multilouver, 39 oﬀset strip, 39 perforated, 39 plain rectangular, 39 plain triangular, 39 wavy, 39 Corrugated louver ﬁn exchanger, 580 Cost balance, 791

Cost rate balance, 776, 783 Counterﬂow exchanger, 122, 125, 126, 136, 190, 748 temperature distribution, 739, 741, 748 Coupling, 773 identical order, 773 inverted order, 773 Cross ﬂow exchanger, 61, 62, 129, 749 both ﬂuids unmixed, 62, 63 cross-counterﬂow cross-parallelﬂow, 66 energy balances, 750 face-U ﬂow arrangement, 65 identical order, 63 mixing, 61, 62 models, 751 multipass, 65 one ﬂuid unmixed, 62 overall counterﬂow, 65 over-and-under passes, 65 parallel coupling, 65 partically mixed, 63 side-by-side passes, 65 temperature diﬀerence ﬁelds, 753 temperature distributions, 749 Crystallization fouling, 892. See also Fouling Darcy friction factor, 413 Dealuminumiﬁcation, 897 Delay period, 872 Denickeliﬁcation, 897 Dezinciﬁcation, 897 DIM standards, 13 Dimensionless axial distance, 446, 448 Dimensionless groups, 441–443 table of, 442 Dittus-Boelter correlation, 482, 484 Divided-Flow exchanger, 64 Double-blow method, 468 Double-pipe heat exchangers, 21 Echelon tube arrangement, 566 Eckert number (Ec), 797 Eﬀectiveness factor, 736 Eﬀectiveness (") NTU formulas, 114, 128 comparison, 341 table of, 144 Emissivity, 539 carbon dioxide, 543 correction factor, 541, 543 water vapor, 541 Energy balance, 102, 736 Energy rate balance, 779, 783 analysis, 779

INDEX

Enthalpy rate change, 83, 735, 736, 783 Entrance and exit losses, 388 Entrance region, 435 Entropy generation, 756, 757, 759, 762, 763 ﬁnite temperature diﬀerences, 756 ﬂuid friction, 762 ﬂuid mixing, 759 maximum, 763 Entropy generation analysis, 776 Euler number, 394, 413 Exchanger arrays, 164 Exergy, 791. See also Irreversibility and Entropy generation analysis, 786 available energy, 756, 776 destruction, 788 losses, 791 rate balance, 776, 786 Exhaustion coeﬃcient, 320 Extended surface eﬃciency, 289 Extended surface exchangers, 36, 37, 258, 694. See also Fins extended surfaces, 258 ﬂat ﬁns on a tube array, 698 individually ﬁnned tubes, 698 louver ﬁns, 696 oﬀset strip ﬁns, 696 perforated ﬁns, 697 plain ﬁn surfaces, 695 primary surface, 258 surface area, 258 tube-ﬁn surfaces, 697 wavy ﬁn surfaces, 695 External ﬂows, 432 F factors, 190 Fanning friction factor, 379, 338, 413 circular tubes, 400 Film coeﬃcient, 429, 440 Film temperature, 530 Finned tube exchanger, 41 Fins: assumptions for the analysis, 259, 285 boundary conditions, 262, 265 energy balance, 260 ﬁn heat transfer, 278 heat transfer rates, 265 interrupted ﬁns, 38 multilouver, 38 plain ﬁns, 38, 277 plain triangular, 277 plate, 38 straight ﬁn of uniform thickness, 261 temperature distributions, 265, 266, 274

933

thin ﬁn thermal behavior, 259 total ﬁn heat transfer, 263 wavy, 38 Fin density, 37 Fin eﬃciency, 258 circular ﬁns, 276, 286 dimensionless groups, 279 plate-ﬁn, 283 plate-ﬁn surfaces, 280 rectangular straight ﬁn, 273 straight ﬁns, 276 tube-ﬁn, 283, 286 Fin eﬀectiveness, 258, 288 Fin frequency, 37 First law of thermodynamics, 735, 776 Fixed-matrix regenerator, 53 Flow arrangements, 56 1–2 TEMA E, 159 1–2 TEMA G, 160 1–2 TEMA H, 161 1–2 TEMA J, 161 bi-directional, 748 both ﬂuids unmixed, 62 counterﬂow, 57 crossﬂow, 60 cross-parallelﬂow, 66 mixing, 62 multipass cross-counterﬂow, 168 multipass cross-parallelﬂow, 170 multipass exchangers, 164 one ﬂuid unmixed, 62 paralleﬂow, 58 parallel coupling, 172 Plate heat exchanger, 72 P-NTU formulas, 144 P-NTU relationships, 141, series coupling. overall parallelﬂow, 168 single-pass, 57 tube-side multipass two-pass, 57 unidirectional, 748 Flow friction characteristics, 425 Flow instability with liquid coolers, 837 Flow lengths, 563 heat transfer and pressure drop calculations, 563 Flow maldistribution, 809, 834, 843, 844 geometry-induced, 809 manifold-induced, 834 mitigation, 844 no ﬂow instability present, 843 operating condition-induced, 809, 837 viscosity-induced, 837 Flow maldistribution-induced instability, 842

934

INDEX

Flow nonuniformity, see Flow maldistribution increase in pressure drop, 814 Flow regimes: horizontal circular tubes, 533 vertical circular tubes, 535 Flow reversal symmetry, 215, 429 Flow types, 429 external, 432 fully, developed, 435 hydrodynamically developing, 435 imposed, 429 internal, 432 laminar, 430, 434 laminarization, 431 periodic, 432 reattachment, 437 recirculation zone, 438 recirculation, 437 reverse transition, 431 self-sustained, 429 separation, 437 simultaneously developing ﬂow, 436 steady, 429 streamline, 430 thermally developing, 435 transition, 430, 431 turbulent, 430 unsteady, 429 viscous, 430 Fluid mean temperature(s), 601 approximate, 602 arithmetic mean, 604 counterﬂow and crossﬂow heat exchangers, 604 heat exchangers with C* ¼ 0, 603 multipass heat exchangers, 604 Fluid pumping devices, 380 blower, 380 compressor, 380 exhauster, 380 fan, 380 head, 380 Fluid pumping power, 379, 438 Form drag, 438 Fouling, 863 aging of, 874 cleaning strategies, 892, 893 combined maximums, 869 compact exchangers, 871 deposition and reentrainment models, 877 diﬀusion, 873 eﬀect on heat transfer and pressure drip, 863 electrophoresis, 873

empirical data, 886 factor, 107, 866, 875 gas-side, 871, 888 impact performance, 882 inertial impaction, 873 initiation of, 872 Ken-Seaton correlation, 880 liquid-side, 870 mechanisms, 867 mitigation of gas-side, 892 mitigation of water side, 891 modeling of, 875 operating variables, 871, 872 phenomenological considerations, 866 prevention and control of liquid-side, 890 prevention and mitagation of, 890 prevention and reduction of gas-side, 891 removal of, 874 removal resistance, 876 resistance values, 660 resistance, 875, 881, 886 sequential events in, 872 thermophoresis, 873 time dependence, 878 transport of, 872 turbulent downsweeps, 873 unit thermal resistance, 866 Free convection, 532 superimposed, 532 Freezing or solidiﬁcation fouling, 869 Friction factor, 444, 451 apparent Fanning, 444 Darcy, 445 Fanning, 444, 451 hot, 451 factor determination, 471 Friction velocity, 496 Friedel correlation, 915 Froude number (Fr), 915, 917 Fully developed laminar ﬂow correlations, 480 inﬂuence of speciﬁc variables, 480 Fully developed region, 435 Galvanic series, 896 Gasketed plate heat exchangers, 23 basic construction, 23 Gas-to-gas heat exchangers, 38 Geometrical characteristics, 563–598 chevron plate geometry, 597 circular ﬁns on circular tubes, 569 corrugated louver ﬁn exchanger, 580 inline arrangement, 563 oﬀset strip ﬁn exchanger, 574 plain ﬂat ﬁns on circular tubes, 572

INDEX

plate-ﬁn surfaces, 584 staggered arrangement, 566 triangular passage regenerator, 585 tube-ﬁn exchangers, 574 tubular heat exchangers, 563 Gnielinski correlation, 482, 484 Gouy-Stodola theorem, 787 Graetz number (Gz), 448 Grashof number (Gr), 532 Gross ﬂow maldistribution, 810 counterﬂow and parallelﬂow exchangers, 811 crossﬂow exchangers, 817 mixed-unmixed crossﬂow exchanger, 817 tube-side madldistribution, 821 unmixed-unmixed crossﬂow exchangers, 819 Guy-Stodola theorem, 756 Hagen number (Hg), 442, 445, 512 Harper-Brown approximation, 286 Headers, 846, 848, 849 counterﬂow, 848 design, 809, 845–852 free discharge, 848 normal, 846, 852 oblique-ﬂow, 848, 849 parallelﬂow, 848 turning, 846 Header and manifold design, 845 Heat capacitance, 310 Heat capacity rate ratio, 141 Heat capacity rate, 310 Heat exchanger, 1, 3, 216. See also Classiﬁcation of heat exchangers 1–2 TEMA E, 142 1–2 TEMA G, 160 1–2 TEMA H, 161 1–2 TEMA J, 161 as a black box, 736 as a component, 738, 801 as part of a system, 737 compact heat exchanger, 8, 9 comparison of the analysis methods, 207 control volumes, 739 counterﬂow, 57 cross counterﬂow, 65 crossﬂow, 60 design problems, 216 designer controlled parameters, 104 direct transfer type, 1,4 direct-contact, 7 energy balances, 739 extended-surface, 12 epsilon (") -NTU method, 207, 208 face-U ﬂow arrangement, 65

935

ﬂuidized-bed, 6 gas-liquid, 8 gas-to-ﬂuid, 11 heat transfer elements, 3 immiscible ﬂuid, 8 indirect transfer type, 1 indirect-contact, 3 irreversibilities, 755 laminar ﬂow, 9 liquid-coupled liquid-to-liquid, 12 liquid-vapor, 8 meso heat exchanger, 9 micro heat exchanger, 9 modeling, 738 MTD method, 209 multipass cross-counterﬂow, 168 multipass crossﬂow exchangers, 65 multipass cross-parallel ﬂow, 66, 170 multipass, 64 number of shells in series, 163 operating condition variables, 104 overall counterﬂow, 65 over-and-under passes, 65 P1 -P2 method, 211 paralleﬂow, 58 parallel coupling, 65 performance, 787 phase-change, 12 P-NTU method, 209 psi ( )-P method, 210 principal features, 676 recuperators, 1, 4 sensible, 1 series coupling, 65 side-by-side passes, 65 single-pass, 57, 122 storage type, 5 surface compactness, 8 surface geometrical characteristics, 563 surface heat exchanger, 3 train, 164 tubular, 13 two-pass, 57 Heat Exchanger Arrays, 201 Heat exchanger design methodology, 78. See also Heat exchanger costing, 90 exchanger speciﬁcation, 81 manufacturing considerations, 90 mechanical design, 87 optimum design, 93 overview, 78 problem speciﬁcations, 79

936

INDEX

Heat exchanger design methodology (continued) process and design speciﬁcation, 79 thermal and hydraulic design methods, 84 thermal and hydraulic design, 83 trade-oﬀ factors, 92 Heat exchanger design problems, 84 design solution, 85 performance problem, 84 simulation problem, 84 surface basic characteristics, 85 surface geometrical properties, 85 thermal design problems, 84 thermophysical properties, 85 Heat exchanger design procedures, 601 Heat exchanger eﬀectiveness, 114, 212, 745, 772 approximate methods, 213 chain rule methodology, 214 condenser, 125 counterﬂow exchanger, 125 epsilon (")-NTU formulas, 128 evaporator, 125 exact analytical methods, 213 ﬂow-reversal symmetry, 215 heuristic approach, 772 matrix formalism, 214 nondimensional groups, 117 numerical methods, 213 paralleﬂow exchanger, 129 solution methods, 212 traditional meaning, 745 true meaning, 745 unmixed-unmixed crossﬂow exchanger, 129, 130 vs. eﬃciency, 114 Heat exchanger ineﬀectiveness, 238 Heat exchanger optimization, 664, 776 as a component, 776 as part of a system, 776 Heat exchanger selection, 673 Heat exchanger selection criteria, 674, 723 cost evaluation basis, 675, 724 fouling and cleanability, 675 operating pressures and temperatures, 674 Heat exchanger surface selection quantitative considerations, 699 screening methods, 700 Heat pipe heat exchangers, 44 Heat transfer analysis, 100, 308 assumptions, 100 assumptions for regenerator, 308 Heat transfer characteristics, 425 basic concepts, 426

Heat transfer coeﬃcient, 105, 429, 440, 647 adiabatic, 441 correction factor for baﬄe conﬁguration, 647 correction factor for baﬄe leakage eﬀects, 647 correction factor for bypass , 647 correction factor for larger baﬄe spacing, 647 correction factor in laminar ﬂows, 647 correction factor streams, 647 mean, 105 shell-side, 647 Heat transfer correlations, 916, 917 condensation in horizontal tubes, 916 vaporization, 917 Heat transfer rate equation, 83, 103 Heat transfer surface, 3 extended, 3 indirect, 3 primary or direct, 3 secondary, 3 uniform distribution, 740 Heat transfer surface area density, 311 Heat wheel, 51 Hot-gas ﬂow period, 311 Hydraulic diameter, 9, 312, 384, 441 window section, 394 Hydraulic radius, 384 Hydrodynamic entrance length, 435, 499 Hydrodynamic entrance region, 435 Incremental pressure drop number, 445 Inlet temperature diﬀerence, 105 Inline array, 568 Irreversibility, 755, 756, 763, 796 cost of, 786 design parameter, 758 energy measure of, 792, 794 entropy measure, 757 Kandlikar correlation, 917 Kays and London technique, 451 experimental procedure, 451 theoretical analysis, 452 Lambda ()–Pi (II) method, 339 Lamella heat exchangers, 33 Laminar ﬂow, 427, 430 fully developed, 436 velocity proﬁle, 427 Laplace transforms method, 742 Length eﬀect, 244, 249 correction factor, 250 Leveque number (Lq), 443, 448, 514

INDEX

937

Limitations of j vs. Re plot, 510 Liquid cooling, 841 Liquid-coupled exchangers, 911 Liquid metal heat exchangers, 233 Ljungstrom, 51, 361 Lockhart-Martinelli correlation, 915. See also Two-phase pressure drop correlation Log-mean average temperature, 453 Log-means average temperature, 186 Log-mean temperature, 758 Log-Mean temperature diﬀerence correction factor F, 187 counterﬂow exchanger, 190 counterﬂow exchanger, 190 heat exchanger arrays, 201 parallelﬂow exchanger, 191 Longitudinal conduction parameter, 235 Longitudinal wall heat conduction , 232 crossﬂow exchanger, 239 exchangers with C* ¼ 0, 236 multipass exchangers, 239 single-pass counterﬂow exchanger, 236 single-pass parallelﬂow exchanger, 239 Louver pitch, 696 Louver with, 696 Low-Reynolds-number turbulent ﬂows, 432

Microbial fouling, 869 Microchannels, 698 Microﬁn heat exchanger, 37 Miter bends, 409 Mitigation of ﬂow maldistribution, 844 shell-and-tube heat exchangers, 845 Modeling of a heat exchange, 735 Molecular diﬀusion, 430 Moody diagram, 399 Multipass crossﬂow exchangers, 164 Multipass exchangers, 164 compound coupling, 181 parallel coupling, 172 plate exchangers, 185 series coupling: overall counterﬂow, 164, 168 Multipassing, 56 Munter wheel, 51

Macrobial fouling, 869 Manifold-induced ﬂow maldistribution, 834 Manifolds, 852 combining-ﬂow, 834, 847 design guidelines, 836 dividing-ﬂow, 834, 847 parallel-and reverse-ﬂow systems, 835 S-ﬂow, 835 U-ﬂow, 835 Z-ﬂow, 835 Martinelli parameter, 914 Mass velocity equation, 619 Material coeﬃcient, 881 Materials for noncorrosive and corrosive service, 679 Matrix heat exchanger, 38 Mean beam length, 540 Mean overall heat transfer coeﬃcient, 245, 247 area average, 245 temperature and length eﬀects, 247 Mean speciﬁc volume, 384 Mean temperature diﬀerence, 11, 97, 105, 187 Mean temperature diﬀerence method, 186 Mean temperatures, 439 Mean velocity, 439 dependence of heat transfer coeﬃcient, 509 dependence of pressure drop, 509

Oblique-ﬂow header, 848 Oﬀset strip ﬁn(s), 574 Operating cost, 785, 786 Operating expenses, 791 Overall energy balance, 115 Overall heat transfer coeﬃcient, 11, 244, 319 combined eﬀect, 251 length eﬀect, 249 modiﬁed, 319 nonuniform, 244 step-by-step procedure, 251 temperature eﬀect, 248

Newton’s law of cooling, 440 Newton’s second law of motion, 383 Noﬂow height, 61, 281 Nominal passage geometry, 824 Normal-ﬂow headers, 852 Number of transfer units, 119, 319 NTU vs. " and C* 131 Nusselt number (Nu), 442, 446

Packing density, 311 Panelcoil Heat Exchanger, 35 Parallelﬂow exchanger, 136, 748. See also Heat exchanger and Flow arrangements temperature distribution, 739, 741, 748 Participating media, 538 gases, 538 liquids, 538 Particulate fouling, 868, 892 Particulate or precipitation fouling, 869 Passage-to-passage ﬂow maldistribution, 821 assumptions, 823 counterﬂow heat exchanger, 825 N-passage model, 828

938

INDEX

Passage-to-passage ﬂow maldistribution (continued) Other eﬀects, 833 two-passage model, 822 Peclet number (Pe), 443, 448 Performance (eﬀectiveness) deterioration factor, 813 Performance evaluation criteria, 699, 713, 714 algebraic formulas, 717 direct comparisons of j and f, 700 ﬁxed ﬂow area, 714 ﬁxed geometry, 714 ﬂuid pumping power, 700 reference surface, 700 variable geometry, 714 Induction period, 872 Periodic ﬂow, 437 Periodic ﬂow regenerator, 47 Petukhov-Popov correlation, 482, 484 Pinch analysis, 776, 779 Pipe losses, 399 Plate-ﬁn heat exchanger, 37, 584, 605 Plate heat exchanger, 185, 597, 632. See also Heat exchanger heat transfer-limited design, 635 limiting cases for the design, 633 mixed channels, 635 multipass, 185 pressure drop-limited design, 635 rating a PHE, 637 rating and sizing, 635 sizing, 645 Plate pack, 23 rating problem, 605 sizing problem, 617 super elastically deformed diﬀusion bonded, 40 Plate-type heat exchangers, 22, 693 advantages and limitations, 28 channel, 25 ﬂow arrangements, 27 gasket materials, 26 geometrical and operating condition characteristics, 27 hard or soft plates, 25 looped patterns, 71 major applications, 29 multipass, 64, 71 pass, 25 series ﬂow, 71 thermal plates, 27 U-arrangement, 72 Z-arrangement, 72

P-NTU method, 139 P-NTU relationship, 141, 142 Porosity, 312, 586 Prandtl number (Pr), 430, 436, 442, 448 Precipitation or crystallization fouling, 867 Pressure drop , 378, 380, 412, 825 analysis, 378 assumptions, 381 bend, 404 core exit pressure rise, 387 core, 382 dependence properties, 418 dimensional presentation, 414 ﬂuid distribution elements, 399 gain, 825 geometry and ﬂuid properties, 418 importance, 378 loss coeﬃcient, 385, 386 major contributions, 380 nondimensional presentation, 413 plate heat exchanger, 397 plate-ﬁn heat exchangers, 382 presentation, 412 reduction, 825 regenerator, 392 shell-and-tube exchangers, 393 shell-side, 648 sudden contraction, 382 sudden expansion and contraction, 399 total core, 388 tube banks, 393 tube-ﬁn heat exchangers, 391 Pressure gradient, 432 adverse, 432 favorable, 432 Pressure loss coeﬃcient, 413 Property ratio method, 244, 530, 531 Printed-circuit heat exchangers, 34 Radiation, 537 gases, 538 liquids, 538 superimposed, 537 Radiation heat transfer coeﬃcient, 540 Rating problem, 84, 208. See also Heat exchangers design methodology Rayleigh number (Ra), 532 Rectangular Fin, 261 Recuperator, 450 Reduced length, 339 Reduced period, 339 Reference temperature method, 530 Regenerators, 47, 361, 585 advantages, 50

INDEX

assumptions for regenerator, 308 balanced and symmetric, 321 boundary conditions, 315 carryover leakage, 360 counterﬂow, 321, 344 cross bypass leakage, 360 designation of various types, 340 dimensionless groups, 316 disadvantages, 51 eﬀectiveness, 318 energy balance, 314 energy rate balance, 314 epsilon (")-NTU0 method, 316 ﬁxed matrix, 49, 338 gas ﬂow network, 362 governing equations, 312 heat transfer analysis, 308 important parameters, 310 lambda ()- pi () method, 337 Ljungstrom, 47 longitudinal wall heat conduction, 348 matrix material, 366 matrix utilization coeﬃcient, 340 modeling pressure and carrover leakages, 360 operating schedule, 53, 54 parallelﬂow, 326, 345 periodic-ﬂow, 53 porosity, 312 pressure leakage, 360 rotary, 47 rotary regenerator, 313, 343 Rothemuhle, 49, 50 Schumann dimensionless independent variables, 337 seals, 361 side bypass leakage, 360 stack conduction, 352 stationary, 53 transverse wall heat conduction valve, 53 variables and parameters, 315 Regenerator surfaces, 699 Residence time, 120 Reversal period, 311 Reynolds analogy, 508 Reynolds number, 379, 442 Rollover phenomenon, 458 Rotary regenerators, 47, 51 Roughness Reynolds number, 496 Rough surface ﬂow regimes, 497 fully rough, 497 hydraulically, 497 smooth, 497

939

transition, 497 Run-around coil system, 911 Sand-grain roughness, 497 Schmidt number (Sc), 509 Second law eﬃciency, 787 Second law of thermodynamics, 723, 735, 776, 796 evaluation, 723, 796 performance evaluation criteria, 796 Sedimentation fouling, 868 Selection guidelines for major exchanger types, 680 extended-surface exchangers, 694 plate heat exchangers, 693 plate-ﬁn exchanger surfaces, 694 regenerator surfaces, 699 shell-and-tube exchangers 680 Shell-and-tube exchangers, 13, 68, 183, 291, 646, 766. See also Flow arrangements additional considerations, 291 approximate design method, 658 baﬄes, 18, 682 bundle-to-shell bypass stream, 292 comparison of various types, 21 correction factor pressure drop, 649 crossﬂow section, 591 crossﬂow stream, 292 design features, 689 disk-and-doughnut baﬄe, 683 divided-ﬂow exchanger, 71 external low-ﬁnned tubes, 648 ﬁnite number of baﬄes, 297 front and rear end heads, 18, 688 grid baﬄes, 18, 682 heat transfer calculation, 646 helical baﬄe, 18 impingement baﬄes, 684 increase heat transfer, 693 leakage and bypass streams, 292 low ﬁns, 17 multipass, 183 no-tubes-in-window design, 648 nozzles, 17 parallel counterﬂow exchanger 68 plate baﬄes, 18, 682 preliminary design, 646 pressure drop calculation, 646 rating, 646 rear-end heads, 688 reduce pressure drop, 693 rigorous thermal design method, 663 rod baﬄes, 18, 684

940

INDEX

Shell-and-tube exchangers (continued) segmental baﬄe, 682, 683 shell ﬂuid bypassing and leakage, 291 shells, 17, 686 shell-side ﬂow patterns, 291–293, 295 shell-side pressure drop, 648 shell-to-baﬄe leakage stream, 292 split-ﬂow exchanger, 70 strip baﬄe, 683 support plate, 683 tube count, 587 tube pitch and layout, 681 tubes return end, 162 tubes, 16, 680 tubesheets, 18 tube-to-tubesheet joints, 21 unequal heat transfer area, 296 window section, 589 windows and crossﬂow sections geometry, 589 Single-blow technique(s), 467 Sizing problem, 84, 207. See also Heat exchangers design methodology counterﬂow exchanger, 619 crossﬂow exchanger, 622 Spiral plate heat exchangers, 31 Spiral tube heat exchangers, 22 Split-ﬂow exchanger, 63 Stack height, 61 Stacked plate heat exchanger, 30 Staggered array rotated square, 568 rotated triangular, 568 square, 568 triangular, 568 Staggered ﬁnned-tube arrangement, 571 unit cell, 571 Staggered parallel arrangement, 55 Staggered tube arrangement, 566 unit cell, 567 Standard types of pitches, 680 Stanton number, 442, 447 Steady-state technique, 451. See also Kays and London technique Stefan-Boltzmann constant, 538 Stratiﬁed ﬂow, 916 Stream analysis method, 294 Stream symmetry exchanger, 133 exchanger conﬁguration correction factor, 188 log-mean temperature diﬀerence correction factor, 188 Surface area density, 9 Surface characteristics, 449

Surface ﬂow area goodness factor comparison, 704 Swing regenerator, 47 TEMA E Shell, 68 TEMA G shell, 70 TEMA Standards, 13 Temperature approach, 105 Temperature cross, 107, 143, 765 external, 107, 765 ﬂuid mixing, 765 internal, 107, 765 Temperature-dependent ﬂuid properties, 529 correction schemes, 530 Temperature diﬀerence, 187, 294 distribution, 738, 741, 744 counterﬂow, 744 parallelﬂow, 744 eﬀectiveness, 140, 244, 248 enthalpy rate change diagram, 776 head, 105 proﬁles of shell-side streams, 297 range, 105 ratio, 120 span, 105 swing, 366 weighting factor, 756 Test core design, 457 Test technique, 450 The ligament, 681 Thermal boundary conditions, 474 Thermal circuit, 100, 107 Thermal conductance, 111 Overall thermal conductance, 111 Thermal Design, 97, 232, 308 additional considerations, 232 basic thermal design, 97 numerical analysis, 256 regenerators, 308 Thermal entrance length, 435, 502 Thermal inertia, 98 Thermal length, 119 Thermal resistance, 450 controlling, 450 noncontrolling, 450 Thermodynamic analysis, 766 Thermodynamic eﬃciency, 786 Thermodynamic ﬁgure of merit, 787 Thermodynamic irreversibility, 755 ﬁnite temperature diﬀerence, 755 ﬂuid friction, 755 ﬂuid mixing, 755 Thermodynamic modeling and analysis, 735

INDEX

Thermodynamic quality, 796 Thermodynamic system, 786 Thermoeconomics, 779, 792 Thermophysical properties, 906 Transient test techniques, 467 experimental procedure, 468 theoretical model, 469 Transition-ﬂow correlation, 481, 482 True mean temperature diﬀerence, 602 Tube-ﬁn heat exchangers, 41, 631 Tube layout arrangements, 681 conventional, 41 ﬂat ﬁns, 42 heat transfer calculations, 631 individually ﬁnned tube exchanger, 41 plate ﬁnned tube, 42 plate-ﬁn and tube, 42 pressure drop calculations, 632 rating and sizing problems, 631

941

surface geometries, 631 Turbulent boundary layer, 430 fully, 430 turbulent region, 430 viscous sublayer, 430 Turbulent ﬂow, 430, 436 Turbulent ﬂow correlations, 487 smooth circular tube, 484 Turbulent mixing, 430 Two-phase pressure drop correlations, 913 two-phase, 913 U-ﬂow arrangement, 835. See also Manifolds Unsteadiness, 429 Utilities, 776 cold, 777 hot, 777 Valve switching frequency, 320