Advances in Heat Transfer, Volume 26

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Author: James P. Hartnett † | Thomas F. Irvine | Young I. Cho | George A. Greene

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ADVANCES IN HEAT TRANSFER Volume 26

This Page Intentionally Left Blank

Advances in

HEAT TRANSFER Serial Editors James P. Hartnett

Thomas F. Irvine

Energy Resources Center University of Illinois at Chicago Chicago, Illinois

Department of Mechanical Engineering State Unii,ersity of New York at Stony Brook Stony Brook, New York

Serial Associate Editors Young I. Cho

George A. Greene

Department of Mechanical Engineering Drexel Unii,ersity Philadelphia, Pennsylvania

Department of Adranced Technology Brookha Len National Laboratory Upton, New York

Volume 26

ACADEMIC PRESS San Diego New York Boston London Sydney Tokyo Toronto

This book is printed on acid-free paper.

@

Copyright 0 1995 by ACADEMIC PRESS. INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW 1 7DX

International Standard Serial Number: 0065-27 17 International Standard Book Number: 0- 12-O20026-0 PRINTED IN THE UNITED STATES OF AMERICA 95 96 97 98 99 00 B B 9 8 7 6 5

4

3 2

1

CONTENTS

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

ix

Direct-Contact Transfer Processes with Moving Liquid Droplets PORTONOVO S. AYYASWAMY

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Fluid Mechanics Related to Direct-Contact Transfer Studies . 111. Direct-Contact Heat and Mass Transfer Studies . . . . . . . . . . IV . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 19 90 91 92

Single-Phase Liquid Jet Impingement Heat Transfer B. W . WEBBAND C.-F. MA I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 107 I11. Submerged Jets: Experimental Studies . . . . . . . . . . . . . . . . . 123 IV . Free-Surface Jets: Experimental Studies . . . . . . . . . . . . . . . . 135 170 V . Liquid Jet Arrays ................................ VI . Other Factors Affecting Transport . . . . . . . . . . . . . . . . . . . . 182 VII . Conclusions and Recommendations for Further Research . . . 204 206 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I1. Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . .

Thermal Design Theory of Three-Fluid Heat Exchangers D . P . SEKULIC A N D R . K. SHAH I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219

and Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . IV . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . The Mathematical Model of a Three-Fluid Heat Exchanger Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 237

11. Classification of Three-Fluid Heat Exchangers 111. Generalized Form of the Model Formulation

V

. . . . . . . . . . . 222

256

vi

CONTENTS

VI . Solution of a Three-Fluid Heat Exchanger Problem . . . . . . . . VII . Three-Fluid Heat Exchanger Effectiveness . . . . . . . . . . . . . . VIII . Three-Fluid Heat Exchanger Thermal Design Methodology . . IX. Conclusions .................................... Nomenclature ................................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Subject Index

...................................

269 290 304 319 321 324 329

CONTRIBUTORS

Numbers in parentheses indicate the pages on which the authors' contributions begin.

PORTONOVO S. AYYASWAMY (11, Department of Mechanical Engineering and Applied Mechanics, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104 C.-F. MA (103, Department of Thermal Science and Engineering, Beijing Polytechnic University, Beijing 100022 China D. P. SEKULIC (2191, Department of Mechanical and Industrial Engineering, Marquette University, Milwaukee, Wisconsin 53233' R. K. SHAH(2191, Harrison Division, General Motors Corporation, Lockport, New York 14094 B. W. WEBB (109, Department of Mechanical Engineering, Brigharn Young University, Provo, Utah 84602

'Permanent address: Department of Mechanical Engineering, University of Novi Sad, 21121 Novi Sad, Yugoslavia.

vii


PREFACE

For over a quarter of a century this serial publication, Aduances in Heat Transfer, has filled the information gap between the regularly scheduled journals and university-level textbooks. The series presents review articles on special topics of current interest. Each contribution starts from widely understood principles and brings the reader up to the forefront of the topic being addressed. The favorable response by the international scientific and engineering community to the 25 volumes published to date is an indication of the success of our authors in fulfilling this purpose. The Senior Editors are pleased to announce the addition of two younger Associate Editors to the editorial board of Aduances, Young I. Cho of Drexel University and George A. Greene of Brookhaven National Laboratory. Their presence on the board brings fresh new ideas to the series and ensures its continuity. The editorial board expresses appreciation to the contributing authors of Volume 26. They have certainly maintained the high standards associated with Advances in Heat Transfer over the years. Last, but not least, the editors acknowledge the efforts of the professional staff at Academic Press, especially Ellen Caprio, Production Editor of the series. They are responsible for the attractive presentation of the published volumes.

ix


ADVANCES IN HtATTRANSFER, V O L U M t 26

Direct-Contact Transfer Processes with Moving Liquid Droplets

PORTONOVO S. AYYASWAMY Department of Mechanical Engineering and Applied Mechanics School of Engineering and Applied Science Unirlersity of Pennsylcania, Philadelphia, Pennsylrania 19104-6315

1. Introduction

The study of direct-contact heat and mass transfer to liquid droplets (dispersed phase) moving in a fluid medium (continuous phase) is of interest because the transfer rates with direct-contact systems are usually much higher than those with surface-type exchangers. This is due to the availability of a larger surface area for a given volume. The transfer can also be effected with a lower potential difference (temperature/concentration) and a lesser pressure drop. Corrosion and fouling are virtually absent. The possibility of stream “contamination,” however, could be a serious disadvantage [981. Direct-contact transfer processes with moving liquid droplets may or may not involve phase change at the drop surface. Furthermore, the process may involve single- or multicomponent systems. In this article, we are concerned primarily with analytical/numerical studies of directcontact transfer phenomena, rather than the experimental aspects or the specifics of the design of direct-contact transfer systems. The main emphasis throughout is on convective flow effects. The literature related to direct-contact transfer to droplets can be roughly classified into three groups: (a) Studies on the fluid-mechanical aspects of droplet motion (for example, flow solutions, drag and terminal velocity calculations, and shape deformation). The motivation for these studies has been to develop insights and results that can be incorporated in evaluating transport rates. (b) Evaluation of heat and mass transport to the droplet in the absence of phase change. (c) Evaluations of transport rates in situations that involve phase change at the droplet surface. The 1

Copyright 8 1995 hy Academic Press, Inc. All rights of reproduction in any form reserved.

2

PORTONOVO S. AWASWAMY

book by Clift et al. [55] and the review articles by Harper [82], Grace [741, Narasimhamurty et al. [140], and Yeung [2513 contain comprehensive discussions on droplet motion. The book [55] and the review articles by Godfrey and Hanson [71] and Steiner and Hartland [212] provide extensive discussions on items (a) and (b), and the book by Pruppacher and Klett [ 1661 addresses all direct-contact transfer issues related to atmospheric sciences. The book edited by Chhabra and De Kee [43] contains several articles on selected topics that involve moving drops. A recent book by Sadhal et al. [193] offers succinct and critical discussions on all three aspects, and the review articles by Jacobs [98] and the book by JSreith and Boehm [lo91 are entirely devoted to direct-contact heat transfer and cover a variety of topics. Many authoritative review articles on direct-contact transfer involving phase change with drops are available in published literature (see, for example, the articles by Law [119], Sideman and Moalem Maron [203], Faeth [68], Chigier [46], Avedisian [16], Ayyaswamy [19-211, Dwyer [60], Sirignano [206], Annamalai and Ryan [12], and Jones [1081). This article offers a review of recent studies related to direct-contact heat and mass transfer processes with moving liquid droplets. In some instances, earlier contributions are recalled and discussed for completeness. In a companion article [20], mathematical methods employed in direct-contact transfer studies have been discussed and it must be consulted for such details. The review is classified under various subdivisions. Contributions to fluid mechanics related to direct-contact transfer processes have been discussed under the following headings: inertialess translation, weakly inertial translation, low Reynolds number translation but with a strong radial field to address situations involving phase change at the interface, motion at intermediate and high Reynolds numbers, effect of surfactants, drop deformation, and surface viscous effects. The directcontact heat and mass transfer studies are discussed under studies that do not involve a phase change at the drop surface (the internal problem, the external problem, and the conjugate problem) and studies involving phase change at the drop surface (spherically symmetric condensation and evaporation in the presence of a pure vapor, spherically symmetric condensation in the presence of a noncondensable, condensation processes with a moving drop at low, intermediate, and high translational Reynolds numbers, effect of surfactants on condensation, effect of shape deformation, condensation on a spray of drops, transfer processes with a moving compound drop, spherically symmetric vaporization, evaporation processes with a moving drop at low, intermediate, and high translational Reynolds numbers, evaporation of a multicomponent droplet, vaporization of a

DIRECT -CONTACT TRANSFER TO LIQUID DROPLETS

3

slurry droplet, droplet interactions, droplet spray vaporization, transport in the presence of an electric field, and vaporization of a molten metal droplet). The most notable omissions in this review are discussions on low-gravity and thermocapillary effects, droplet ignition and extinction characteristics, and turbulence phenomena affecting transport to droplets. A conscious attempt has been made to highlight the thermophysical aspects of the contributions. Details of mathematics or numerical procedures are not the focus of this review (see [20]). In spite of the scope and size of this review, many important and useful contributions may have been inadvertently overlooked by the author, and it is requested that such information be brought to the notice of the author so that interested researchers may be served in a future article. We now discuss direct-contact transfer under various headings.

11. Fluid Mechanics Related to Direct-Contact Transfer Studies

Analytical and numerical treatments of direct-contact transfer have primarily employed the axisymmetric formulation. For three-dimensional axisymmetric incompressible flows, the velocity is usually expressed as u

=

*

v x (-i,,,)

r sin 9

in spherical coordinates, ( r , f l , 4 ) ,with $ for the stream function. To facilitate solution procedures, a suitable coordinate system is commonly adopted so that the interface can be identified by one of the coordinates. 41,, the stream In a generalized axially symmetric coordinate system, ( t , ~ function $ ( t , q ) is the flow description in the continuous phase and I&,T) in the drop. The interface is identified by 5 = 5, in a coordinate system h e d to the drop. At a fluid-fluid interface, the continuity of tangential and normal components of both velocities and stresses have to be satisfied. In the far field, the uniform stream conditions must be satisfied. A listing of these conditions in a generalized axially symmetric coordinate system and related discussions are available in Sadhal et al. [193] (also see Ayyaswamy [20]). It is generally difficult to satisfy all the interface conditions, particularly the normal stress condition, which is nonlinear, and linear approximations are usually employed. These are adequate for interfaces that are spherical or slightly deformed from that shape.

4

PORTONOVO S. AWASWAMV

A. INERTIALESSTRANSLATION OF A SPHERICAL DROP

The momentum equation for the inertialess system is &($)

=

0,

,

where L - is the Stokes operator

The drag force for inertialess motion is given by Payne and Pel1 [1621:

F

=

8 r r ~lim r-m

i*- 9 0 )

(4)

rsin28 ’

where $o = tUmr2sin2 e

is the stream function representing a uniform flow with velocity U,. For a spherical drop, 6 = r , 17 = 8, the metric coefficients are h , = 1, h , = r , and the radial distance from the axis of symmetry is Q = r sin 8. The creeping flow solution (Rybczynski-Hadamard solution) is

$(r,e)

=

1 1 -U,R2sin28-[(i)2 4 1 + 4&

-

(i)4],

(7)

where 4fi = @ / p is the viscosity ratio. The motion within the drop is the Hill’s spherical vortex. The drag force from Eq. (4) is

In a gravity field, the balance of the viscous drag with the buoyant force gives the following terminal velocity of the drop:

LIT

=

32 SR2(P - 6) F

(1 + 4&) (2 + 34J

(9)

Good agreement between theoretical predictions and experimental measurements with exceptionally pure fluids have been reported in the literature.

-

DIRECTCONTACT TRANSFER TO LIQUID DROPLETS

5

B. WEAKLYINERTIALTRANSLATION OF A DROP AND DEFORMATION

The formal singular perturbation solution for low Re(= U,R/v) flows and the first-order corrections have been given in Taylor and Acrivos [226]. A composite solution that combines the inner and the outer solutions is given by Sadhal et al. 11931:

x(l

+ cosO){1

-

exp[-;Re(r/R)(l

-

+ O(Re2). For a drop profile of the form

cosO)])

i

( 10)

r ( O ) = R[1 + l ( O ) l , (11) expressions for the deformation to O(We2/Re) and the drag force are given by Taylor and Acrivos [226]. This result shows that up to O(We) the shape of the drop could be either an oblate or a prolate spheroid. For most fluids of practical interest, the deformed drop is an oblate spheroid. Higher order corrections carried out by Brignell [311 show that, for Re -zx 1 and Ca 1 has been studied in Elzinga and Banchero [65] and Brounshtein et al. [33]. The analysis is similar to that of Kronig and Brink [110]. An interface condition takes external resistance into account. The external Nusslet number is assumed to be known and equal to its steady value. Boundary layer solutions for the unsteady transport problem at high Re and for Pe >> 1 have been attempted by Vorotilin et al. [237], Ruckenstein [188], Chao [41], and Chao and Chen [42]. Chao [411, for example, has considered the transient response of a drop to a sudden increase in the temperature of the ambient fluid at large Re and Pe conditions. Major assumptions in this analytical treatment (similarity transformation procedure) include inviscid flow fields for both phases, fully developed internal circulation, thin thermal boundary layers on both sides of the interface, and constant properties. The thermal core temperature has been inadequately described since the transport by diffusion in the tl direction has been ignored in the boundary layer energy equation. Due to the elliptic nature of the interior region, the solutions presented here are inaccurate except during the very early stages of the transient process. The conjugate problem with steady flow fields inside and outside the drop, in the region 0 < Re < 50, 0 < < 10, and 0.01 < +k < 3 , has been solved by Oliver and Chung [155]. The flow fields are determined by the series-truncation spectral method, and the AD1 technique has been employed for integrating the transient energy equations. At time t = 0 the temperature of the continuous phase is subjected to a step change from T,, to T,. To quantify the transport rates, a time-dependent Nusselt number is defined by:

+,,

(;/ Nu where 77

=

=

a@/dq) IT

=

0,

I

sin t9 dtl 7

(54)

l/r and 0 is the dimensionless temperature based on the

24


initial temperature difference,

and 0, is the bulk temperature in the drop given by:

0,=

0

sin 8 d r d e .

0

(56)

300, 0.333 and 3, respectively. As Re increases, the continuous phase velocity near the drop surface and the strength of internal circulation both increase. The combined effect greatly enhances the heat transport between the phases. Increased internal circulation results in shorter oscillation cycles during the transient period. By comparing Figs. 5 and 6 we see that, at a Figures 5 and 6 for the transient Nu variation with Fo, at Pe

4@= 1,and 4k= 3 show the effects of different viscosity ratios, +&

)

9

0.I

0

A

=

=

0.2

FOURIER NUMBER, Fo c ~ = 5 ~1, 300. Reprinted with permission from Oliver and Chung [155].

FIG.5. Transient Nusselt number versus Fourier number for the case of

bk = 3, +u

=

0.333, and Pe

=

-


25

c

6

i I

0

I

I

I

0. I

FOURIER NUMBER,

I

L

0.2 $0

FIG. 6. Transient Nusselt number versus Fourier number for the case of (6, = 1, = 300. Reprinted with permission from Oliver and Chung [155].

bk = 3, c $ ~ = 3, and Pe

fixed 4k, increasing 4& causes a reduction in the strength of internal circulation resulting in longer oscillation cycles and lower Nu. By similar calculations done with fixed $ p and various $ k values, it is shown that Nu increases with increasing 4kdue to a higher rate of heat diffusion from the surface to the interior of the drop. The amplitude of the fluctuations decreases with increasing C#I~ because diffusion across the internal circulation loops becomes increasingly dominant, thereby reducing the role of convective circulation, which is primarily responsible for fluctuations. Nguyen et al. [146] have described a numerical procedure based on the hybrid spectral scheme that is suitable for investigating fully transient conjugate heat transfer. The motivation for this study appears to be to ascertain the validity of quasisteady approximations that are commonly invoked for the flow field descriptions and for continuous phase transport. The flow variables and temperature are expanded in terms of Chebyshev and Legendre polynomials. The governing partial differential equations

26


are reduced to a nonlinear system of algebraic equations. The results have been presented for a drop in a gaseous environment (methanol droplet in air, Re = 20 and cpPc > 1800) and for a drop in a liquid medium (gasoline drop in water, Re = 50). Thermal diffusion in the drop is stronger for the methanol-air system. With the liquid-liquid system, the streamlines inside the drop are nonisothermal, and the drop surface cannot be considered to be at a uniform temperature. The computed drag coefficients agree well with the predictions of Eq. (17). The average Nusselt number variations as a function of dimensionless time obtained here for Pe = 300, 4” = 1, and 4a= 1 for three different sets of Re, 4p,and 4kdiffer somewhat from the quasisteady results reported by Oliver and Chung [ 1551, especially during the intermediate stages of transport. For this class of problems, fully transient analyses involving massive computational requirements may not be warranted to develop reasonably accurate transport results. 4. The Internal Problem in the Presence of an Electric Field Oliver et al. [1511 and Manohar and Iyengar [133] have numerically investigated the transient heat transfer in a droplet suspended in an electric field. The transient energy equation has been solved by the AD1 method for [based on the maximum velocity in the droplet produced by the electric field, Eq. (2911 in the range of 2000 or less. The stream as a function of function I$ is given by Eq. (28). The variations in for low to moderate values of are shown in Fig. 7,and the variations at are shown in Fig. 8. Here, moderate to high I

NU = (2RQ)/4aR2(fs -

a00

OD5

0.10

fb)k,

0.15

(57)

0.20

Fa FIG. 7. Transient Nusselt number versus Fourier number for low to moderate Pe numbers. Dispersed phase quantities. Reprinted with permission from Oliver et al. [1511.

DIRECT - CONTACT TRANSFER TO LIQUID DROPLETS

27

&! = 200 b ) ee = 500 c ) ce = 1000 d ) Pe = 2000

a)

N"

40

30 I

1

0.00

I

0.05 FO

FIG.8. Transient Nusselt number versus Fourier number for moderate to high Pe numbers. Dispersed phase quantities. Reprinted with permission from Oliver et al. [151].

where Q is the net rate of heat entering the drop, Tb is the droplet bulk temperature, and Fo= &/R2. At short times, conduction is the dominant mode of heat transfer to the droplet due to sharp gradients near the interface. At longer times, the convection component becomes significant for larger values of E.At large values of pe, Nu oscillates during the initial transient. This oscillation is due to the internal circulation, which brings cold fluid to the surface, resulting in high heat transfer rates. As time progresses, Nu approaches a steady-state value. It is interesting to note from Fig. 8 that there appears to be a limiting maximum steady-state Nusselt number of about 30 for the pure electrically driven flows. The asymptotic (steady-state) Nusslet number in the limit cc has, in fact, been accurately determined by Oliver and DeWitt [156] to be 29.8. This compares with the maximum steady-state Nusslet number of 17.9 for the interior problem associated with a drop translating due to gravity in the absence of an electric field [1101 (compare with the discussions in section III.A.1). Chung and Oliver [541 have extended their numerical study of the internal problem described in Oliver et al. [ M I to the situation where the droplet is undergoing slow translation. This requires consideration of Rybczynski-Hadamard flow in addition to the flow induced by the electric field [see Eq. (28)l. The AD1 numerical method is employed to develop the solution. In Fig. 9 the steady-state Nusselt numbers are noted to increase with an increase in and the dimensionless parameter W [Eq. (37)]. The top curve corresponds to a drop suspended in an electric field ( W w> and the bottom curve represents a drop undergoing slow translation under the influence of gravity ( W 0). Both curves converge to Nu = 6.58 as Pe + 0, the limit of a conducting solid sphere [1421 (compare with discusI

A

-

-

c+

-+

-+

28


Pe FIG. 9. Steady-state Nusselt number versus Peclet number for various values of W. Reprinted with permission from Chung and Oliver (541.

sions in section III.A.l). For very large E, the top curve asymptotically approaches 29.8, and the bottom asymptotically approaches 17.9. For Pe- 0(1000), the effect of electric field is negligible for W < 0.5, and the effect of creeping motion is unimportant when W > 10. For smaller G, however, say, around 50, the effect of the electric field is small for values of W < 2, and the effect of translation is minor when W > 50. Chung and Oliver point out that boundary layer approximations are inapplicable to resolve transport in the internal problem because the approximations are invalid beyond the immediate transient.

5 . The External Problem in the Presence of an Electric Field

In the external problem, the resistance to transport is significant in the continuous phase. Morrison [1391 has studied the high Pe number [Pe O(lOOO)] unsteady heat (mass) transfer to a dielectric drop suspended in another dielectric fluid in the presence of a uniform electric field. Because translation under the action of gravity is absent, the stream functions are given by Eqs. (27) and (28). The thin thermal boundary layer approximation is invoked and the solution has been developed by similarity transformation in a manner similar to that in Chao [41] (see discussions in section III.A.3). The analysis concerns itself with a thin region on either side of the interface. The results show that Nu is proportional to the

-

-

DIRECTCONTACT TRANSFER TO LIQUIDDROPLETS magnitude of the applied electric field. For short times,

NU =

2 (1

+ P)&

’

and for longer times, the quasisteady result is,

where r

=

at/R2 and

Chung and Oliver [54] have provided discussions regarding the limitations of this study. Steady-state heat (mass) transfer at low Peclet numbers, 0 I Pe I60, has been estimated by using a regular perturbation analysis to solve the continuous phase energy equation in Griffiths and Morrison [77]. The expansion parameter is Pe/2. The stream function is given by Eq. (27). Because the fluid velocity far from the drop is O(vR2/r2),the ratio of the convection to the conduction flux far from the drop is of O[(R/r)Pe]. Therefore, for sufficiently small Pe, the conduction solution is a uniformly valid approximation for the zeroth-order solution in this case. Solution to higher order terms has been sought by expansion in terms of Legendre polynomials (series-truncation method). A series expression for Nu is given in terms of the expansion parameter. Sharpe and Morrison 12021 have developed steady-state heat (mass) transfer results for the intermediate Pe regime, 1 I Pe 5 1000, by solving the energy equation using a finite-difference scheme with upwind differencing for approximating the convective terms. The stream function is given by Eq. (27). The numerical results for the overall Nu agree with the low Pe perturbation results of Griffiths and Morrison [77] within 4.5% for up to a Pe of 75, and match with the thermal boundary layer solutions of Morrison [I391 within 3% when the Pe is larger than 750. The overall Nu for the pole-to-equator flow is noted to be within a percent of the equator-to-pole flow solution. However, the temperature distributions and local Nu values for equator-to-pole flow differ greatly from their counterparts in pole-to-equator flow. The external problem in the limits of low and high Pe for a drop in an alternating electric field is given by Griffiths and Morrison [78]. The quasisteady creeping flow in the continuous phase induced by an alternating electric field has been analyzed by Torza et af. [2351, and these

30


solutions have been used in this study. The energy equation in the low Pe number limit has been investigated by a perturbation technique similar to the one employed by Griffiths and Morrison 2771 but considerably more complicated. Excellent discussions related to the limits of very large and very small thermal vibration number ( f = 2wR2/a, where o is the angular frequency of the applied field) and the rigorous approach presented in this paper are noteworthy. At high 5, the overall time-averaged Nu is found to be a weak function of 5, and the transport is almost entirely due to the steady part of the creeping motion induced by the electric field. For very small values of 5, the steady and oscillating parts of the fluid motion make nearly equal contributions to the overall heat transfer. The transfer rate at low Pe is always higher for very low f than for very high f . The periodic part of the fluid motion tends to enhance the transfer rate. The high Pe limit is treated in a manner similar to that in Morrison [139]. At high Pe, the periodic part of the fluid motion may either enhance or detract from the overall time-averaged Nu. Series expressions for the Nu have been provided in this study. Chang and Berg [38] have reported fluid flow and mass transfer results for a drop translating in an electric field at intermediate Reynolds numbers, 0 < Re < 100, and high Pe. The analysis considers a liquid-liquid system. On the basis that the mass diffusivities of such systems may be usually small, rendering the Peclet number to be large, thin diffusional boundary layers are assumed to exist on both sides of the drop interface. The mass transfer process is regarded as quasisteady. The results include a closed form expression for the Sherwood number, Sh. It is found that only when W , [Eq. (4311 is substantially greater than unity is the rate of transport enhanced significantly by the imposed electric field. 6. The Conjugate Problem in the Presence of an Electric Field

The conjugate problem formulation has to be solved when the transport resistance of the continuous phase is comparable to that of the dispersed phase. Chang et al. [39] have addressed this problem in the context of a drop slowly translating in the presence of a uniform electric field. The stream functions reflect the effects of gravity and electric field on either side of the interface. Unsteady and quasisteady transfer rates have been obtained for the limit of high Pe. The thin thermal boundary layer approximation and similarity transformation similar to those employed by Chao [41] are adopted in this study. The shortcomings of the developments described in Chao [41] also apply here (see discussions in section III.A.3), and the results obtained here are of limited validity (see Chung and Oliver [54]for a detailed discussion of Chang et al.’s study, and a

-


31

closed-form solution based on the method of characteristics to the boundary layer equation described in Chang et al.). The results illustrate the enhancement in heat transfer due to the presence of the electric field. The overall steady-state Nu is noted to be independent of the direction of flow. It is found that only when the absolute value of W [Eq. (37)] is larger than unity is the rate of transport enhanced by the imposed electric field. The low Pe limit of the conjugate problem has been investigated by Nguyen and Chung [1431 by treating the continuous phase as quasisteady and the transport in the drop as a transient process. The paper appears to have a different definition for W [see Eq. (37)]. The singular perturbation method is used to obtain the temperature profile for the continuous phase, whereas a regular perturbation procedure along with the method of weighted residuals is employed for the dispersed phase. The temperature has been computed up to and including the first order in Pe; however, higher order effects are also examined in order to ascertain the influence of an external field on the transport rates. In the first-order solution, the electric field presence is found to alter the temperature profiles but the net heat transfer rate remains unchanged. The role played by electroconvection has been delineated by examining the variation of the ratio of heat transfer rates with and without the electric field as a function of time for two different values of Pe.

B. STUDIES INVOLVING PHASE-CHANGE AT THE DROPSURFACE A change of phase brings in radial convective transport along with other convective fields, and in general, at an interface undergoing phase change, mass balance and energy and/or species balance are required. 1. Spherically Symmetric Condensation and Evaporation of a Drop in the Presence of Pure Vapor Plesset and Prosperetti [1631 have estimated the spherically symmetric flow rate of pure vapor in the region surrounding a drop experiencing evaporation or condensation from kinetic theory considerations as:

where p ' ( T ) represents the equilibrium vapor pressure at the specified pressure. For drops in the size range of 1 mm, a (radial) Peclet number of the order Pe 100 would be the theoretical upper limit. The temperature field for a drop in an infinite vapor medium is given in Sadhal et al. [193]

-

PORTONOVO S . AWASWAMY

32 as T(r)=

( T , - Tme-Pe) ( T , - T , ) e - P e ( R / r ) 1 - e-Pe 1- e-Pe ,

(62)

where Pe = U R / a . For detailed kinetic theory treatments, see Shankar [201] and Loyalka [132]. 2. Spherically Symmetric Condensation in the Presence of a Noncondensable For a stationary drop experiencing spherically symmetric condensation in the presence of a noncondensable, the distribution of the normalized noncondensable mass fraction, w1 = ( m , - m l , m / m , , s- m,,,), is shown by Sundararajan and Ayyaswamy [216] to be given by

w ,=

[1 - exp( -uc,o/2r)1 [1 - exp( - $4c,0>l

’

(63)

where u , , ~is the condensation velocity at the interface. The velocity u , , ~ is related to the condensation parameter W = 1 - (ml,m/rnl,s)as follows: u,,,

=

21n( 1 - W )

for W < 1.

(64)

The parameter W is a function of the thermodynamic conditions p,, T,, and T,. It varies from 0 to 1, with the limit zero corresponding to a noncondensing situation and W = 1 to a pure vapor environment. C. CONDENSATION PROCESSES WITH A MOVING DROP The presence of radial flow will influence the pressure profile at the surface, the shear stress, and the total drag experiencing by a droplet in motion. For drop sizes of, say, 0.5 to 5 mm in diameter, the external flow may separate behind the drop and this separation must be considered in estimating the transport. In a spray of drops, the interaction between drops might have to be considered in ascertaining transport rates. 1. Condensation at Low Translational Reynolds Number Chung et al. [52, 531, have investigated laminar condensation on a droplet of radius R translating slowly at a constant velocity U, in a large content of a binary mixture of its own vapor (condensable) and a gas (noncondensable). The droplet is assumed to have fully developed internal

-


33

FIG. 10. Geometry and coordinate system. Droplet undergoing slow translation and experiencing condensation.

motion (see Fig. 10). The translational Reynolds number is Re, = (LI,R/v,) = E -=z1, whereas the Reynolds number associated with the normal velocity due to condensation at the interface Re, = uc, R/u, may be of order unity. Here, u,,,,, is the maximum of the normal velocity at the interface u,. The drop is initially cold at a temperature To,whereas the binary mixture is taken to be saturated at a temperature T,(T, > 7'"). The mass fraction, m,, for the gas component is taken to be prescribed. In


34

Chung et al. [52], the transport (heat/mass) in the gaseous phase is treated as quasisteady, and heat transport in the drop is regarded as a transient process. Flow fields in both phases are treated as quasisteady. The energy and species equations for the continuous phase have been solved through a singular matched-asymptotic technique due to the existence of a region of nonuniformity in the neighborhood of the point at infinity [51. In the singular perturbation procedure, the leading-order description for the far-field temperature and mass fraction includes the perturbed velocity field. A transient solution of the dispersed phase has been developed using the semianalytical series truncation method to provide the surface temperature, and the continuous and dispersed phases are connected through the energy flux continuity equation:

The local heat flux at the interface has been expressed as:

The total heat transfer rate is defined by

Q

=

l T q 2 ~ Rsin 2 0 de 0

and an average based on the surface area of the drop has been defined as:

a

Q/47rR2. (68) To exhibit the effects of the presence of the noncondensable gas and the forced flow explicitly, the heat transfer results have been presented in terms of the variations of the dimensionless heat flux ratio ;/GNU. The is the average heat flux corresponding to a “Nusselt-type” heat flux iNu condensation heat transfer to an isothermal solid sphere situated in an atmosphere of quiescent, pure, saturated stream. The temperature of the solid sphere is set equal to the initial temperature of the drop. An expression for GNU has been given in Yang [247]: -

GNU = 0.803

[

=

g ( 6 -pao)hk3(T, -

2RD

(69)

To evaluate the transport quantities, the fall-velocity of the drop has been established through a balance between drag and gravity forces. The drag force has been evaluated using results provided in Sadhal and Ayyaswamy [192]. For illustration, the droplet initial temperature has been taken to be


0

10-3

I

35

lo-'

10-2

t* FIG. 11. The comparison between numerical and asymptotic results for dimensionless heat transfer as a function of dimensionless time; Re at equilibration = 0.5, To = 30°C T, = 100°C. Reprinted with permission from Chung et al. [531.

30°C. The droplet environment has been assumed to consist of air saturated with steam, with the air-mass fraction ranging from 0.01 to 0.5. The ambient temperatures range between 100 to 150°C. Figure 11 shows the variation in dimensionless heat transfer with dimensionless time t* for various values of noncondensable (air) mass fraction in the bulk, m,. In the high condensation regime, at t* = the dimensionless heat transfer drops from about 2.8 to almost half its value as the mass fraction of air increases from 0.01 to 0.1. In the low condensation regime, (4/jN,> is a = mere 0.6 for a m y = 0.5. For the parameters used here, iNu 250 W/cm2. The reason for the reduction in heat transfer due to the presence of air is that a buildup of air film at the interface causes a reduction in the partial pressure of the vapor at the interface. In turn, this reduces the saturation temperature at which condensation takes place. The net effect is to lower the effective thermal driving force, thereby reducing the heat transfer. It is also evident from Fig. 11 that the dimensionless heat transfer is significantly influenced by the surface temperature of the droplet. In Chung et al. [53],both phases are treated as fully transient and the system formulation is solved by the semianalytical series-truncation

36

PORTONOVO S. AWASWM

method. The flow field, fluid temperature, and mass function of the noncondensable gas are expanded as complete series of Legendre polynomials, and in the actual implementation, a six-term expansion series has been employed. The numerical results are shown to compare very well with those of Chung et al. [52] (see Fig. ll), particularly for the low rates of condensation. The presence of a large noncondensable mass fraction in the bulk causes the radial flow of vapor to be weakened. As a consequence, the internal circulation is weaker, and the corresponding liquidside Peclet number, E, is smaller. The asymptotic solution should be capable of representing this situation very well, and it is seen to do so.

2. Condensation at Intermediate Reynolds Number Laminar condensation on an isolated moving droplet in the range Re = O(100) and 0 < W < 1 has been investigated by Sundararajan and Ayyaswamy [214-2171 and by Huang and Ayyaswamy [91, 94, 951. The quasisteady assumption for transport in the continuous phase and for flow in the phases has been invoked in the first three publications, and the other papers address the fully transient formulation. The equations for the flow fields and the transport in the continuous phase have been solved by a hybrid finite-difference scheme (see Sundararajan and Ayyaswamy [2171 for details). The transport in the drop interior is solved by the CrankNicolson procedure. These studies have considered a cold water droplet of radius R , and initial bulk temperature To that is introduced into an environment consisting of a mixture of steam and air. The droplet is assumed to be projected with an initial velocity U, and at an angle Po with respect to the vertical direction as shown in Fig. 12. The total pressure p, and temperature T,(T, > To)of the saturated mixture in the drop environment are taken to be prescribed. The formulation considers a coordinate frame that coincides with the drop center and moves with the instantaneous translational velocity U,. The instantaneous value of Re is taken to be O(100) (up to 300). For Re > 300, flow instabilities such as drop oscillations and vortex shedding could render invalid the solutions obtained here. Drop deformation due both to inertial effects (Weber number We) and to hydrostatic pressure variation (Eotvos number Eo) are assumed to be small, and are neglected. For the range of Re considered here, based on Us (the maximum circulation velocity at the drop surface) is of U(102) [18, 1221. Because is of 0(1O), i% is of O(103). The droplet trajectory has been determined by a gravity-drag force balance. In addition to the initial conditions for velocity, temperature, and mass fraction, and the interface conditions requiring the continuity of tangential velocity and shear stress, the continuity of mass flux and the

-


37

Y

Fic. 12. Geometry and coordinate system for a droplet introduced at an arbitrary angle.

impermeability of the noncondensable at the interface have been considered. In Sundararajan and Ayyaswamy [216], it is assumed that the instantaneous surface temperature of the drop, T,, is uniform over the drop surface, and this assumption is useful in decoupling the quasisteady equations from the formulation for the drop heat-up [84]. In the liquid phase, the stream surfaces are taken to be isothermal. For the numerical scheme, the governing equations and boundary conditions have been transformed in terms of the dimensionless stream function $ (scaled with U,R2) and vorticity (scaled with UJR), and the far-stream conditions have been specified on a large but finite spherical surface of radius R,. The changes in the internal and external flow structure due to condensation are shown in Fig. 13. The flow patterns on the left and right halves (hatched) correspond to a noncondensing ( W = 0) and a typical condensing situation ( W = 0.7), respectively, at Re = 300. For W = 0, a detached recirculatory wake is present in the rear of the drop in the gaseous phase. Within this wake, fluid particles recirculate. In the drop interior, a primary liquid vortex generated by the positive shear

38


-


39

stress in the front portion of the drop and a secondary vortex generated by the negative shear stress in the rear are noted. The secondary vortex strength is very small compared to that of the primary and the sense of circulation in the secondary vortex is opposite to that of the primary vortex. In the presence of condensation, the wake length and volume are reduced. A dividing stream surface ( b ' ) exists, inside which fluid particles either condense or recirculate. In the drop interior, the strength of the primary vortex is higher. This is due to the increased shear stress at the interface caused by the bending of the gaseous phase stream surfaces toward the drop. The internal secondary vortex does not exist at high rates of condensation. As W is increased, the primary vortex center shifts toward the drop equatorial plane, and there is a reduction in the asymmetry in the circulation. The comparisons between numerical results and some experimental data from Kulif. and Rhodes [ 1111 for the dimensionless, instantaneous drop bulk temperature as a function of time, t , are shown in Fig. 14. The quantity 0,= (fk- fo)/(T, - fo),where the instantaneous bulk temperature f b = l/u/Tdu, and u is the instantaneous drop volume. The comparisons are for two situations reported in experimental studies: (I) R , = 1.45 mm, U, = 1.91 m/sec, T, = 8O"C, p , = 1 bar, and T, = 16°C (Re = 265) and (11) R , = 1.40 mm, U, = 1.68 m/sec, T, = 72.5"C, p, = 1 bar, and T, = 18°C (Re = 225). The drop bulk temperatures are predicted very well. Note also that, for a larger noncondensable fraction in the bulk, the drop heat-up rate is slower. This illustrates the effect of gas-phase resistance. The rate of drop growth is examined in Fig. 15 through a plot of dimensionless radius (scaled by R,) against time. The dimensionless growth rate is given by:

where ii, is the average condensation velocity. The growth rate increases linearly with the temperature differential A f = T, - f,. The approximate drop size at the end of condensation, Z?, is predicted in Sundararajan and Ayyaswamy [216] to be given by:

R,=

tpAf 1 + -. 3h

FIG. 13. Effect of condensation on flow pattern. For Re = 300 and W = 0, a = 0.1, 0.0, c = -0.02, d = -0.001. e = -0.003, and f = O.oooO3. For Re = 300 and W = 0.7, a' = 0.3, b' = 0.1688, c' = 0.166, d' = 0.15, e' = 0.072, f' = -0.0023, and g' = -0.007. Reprinted with permission from Sundararajan and Ayyaswamy 12171. b

=

40


v Experimental - Numerical

o

t (sec) FIG. 14. The variation of drop bulk temperature with time. Comparison of numerical results with experimental data for p , = 1 bar: (I) R , = 1.45 rnm, U, = 1.91 m/sec, T, = 80°C and T, = 16°C. (11) R , = 1.40 mm, Urn = 1.68 m/sec, T, = 72.5"C, and To = 18°C. Reprinted with permission from Huang and Ayyaswamy [94].

The numerical calculations are in agreement with this. Because the drop growth rate is intima!ely copnected with the drop heat-up rate, the variation of l? with AT and R , can be regarded as a direct consequence of the manner in which the temperature profile inside the drop evolves in time. The following correlations for C, and U c for a drop experiencing condensation in the parameter range 30 < Re < 300, 0 < W < 0.9, +p 40, and + p lo3 have been proposed in Sundararajan and Ayyaswamy [216] and Huang and Ayyaswamy [941:

-

-

c, = f l ( Y ) X 2 where y

=

Re-'/2, and x fl( y )

=

=

In(1

+fi(Y)X -

+f3(Y),

W ) , and

1 1 3 0 3 ~-~3 3 0 4 . 6 ~+~3 0 0 . 3 7 ~- 8.8029,

+ f3( y ) = 2 5 5 9 . 5 ~-~7 3 1 . 6 9 ~+~7 4 . 3 3 5 ~- 2.0658,

f2(

and

y)

C -U UC,O

=

8 8 9 0 . 1 ~-~2 5 9 8 . 2 ~ ~2 2 8 . 7 3 ~- 6.0811,

1 + 0.261 Re'/2 S C ' / ~ ( ~W)--1'5

for W < 1,


41

Iml 1.06

1.05 -

lo-'

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

I

I

0.9

1.0

t lsec)

FIG. 15. The drop growth rate for p,

=

300 kPa, To = 37°C. U,

=

10 m /sec, and

Po = 0 deg. Reprinted with permission from Huang and Ayyaswamy [94].

where u , , ~is the given by the Eq. (64). Improved correlations may be available in the future with greater computational accuracy, which could cause slight changes in the numerical coefficients in the correlations. But the structure is likely to stay because it has been developed with reasonable scaling arguments. The numerical calculations also show that the dimensionless average heat flux i j defined by

increases in proportion to ln(1 - W ) and Re'/*. Based on this, a new correlation to the dimensionless surface average heat flux for a moving drop experiencing condensation, taking into account the presence of a non-condensable, of the form -

uc ij= -

2 Le has been proposed for W < 1.

42

PORTONOVO S. AYYASWAMY

The significant conclusions that may be drawn from these numerical studies are as follows:

1. In the front portion of the drop, the condensation velocity has an approximate cosine variation with angle. 2. The maximum condensate rate that occurs at the front stagnation point is approximately twice the average condensation rate on the droplet. Thus, analyses developed for the stagnation region [49-5 11 can serve to obtain useful bounds for the transport to the droplet. 3. The transport to the drop in the rear region is enhanced due to recirculation in the wake, and this enhancement increases with Re. 4. The shear stress and the friction drag coefficient increase with condensation. The pressure drag coefficient, on the other hand, rapidly decreases with condensation due to a large pressure recovery in the rear portion of the drop. The relative importance of pressure drag over the friction drag increases with drop size or temperature differential. The contribution to the total drag from the linear momentum of the inward flow is significant only for very small drops and at very high rates of condensation. 5. At high levels of condensation, the drag on the drop is very much reduced, and if the drop size is very large, the drag force cannot balance the weight of the drop. Such an imbalance might lead to excessive drop acceleration and eventual breakup. 6. For Re = O(lOO), the numerical results predict an approximate overall relationship of the form ( i i C / u c , J- 1 a Re1/’. Because ti, depends directly on the average concentration gradient of the noncondensable at the interface, this square-root dependence may be attributed to an overall boundary layer type variation for the noncondensable concentration. At high Re, the regions of steep radial gradients of velocity, temperature, and noncondensable concentration in the gas-vapor phase near the drop surface can be represented by thin boundary layers. The boundary layer thicknesses are of O(Re-’/’) or O(Pe-’/’). With typical values such as R 1 mm and U, 1 m/sec, the boundary layers would be established within a few milliseconds after the introduction of the drop. This feature is discussed in Sundararajan and Ayyaswamy 12151.

-

-

3. High Reynolds Number Condensation: Solution by Boundary Layer Formulation Outside of the immediate transient period following the introduction of the drop, it is reasonable to consider established, quasisteady boundary

-


43

layers in the continuous phase for the entire duration of condensation, and to evaluate transport quantities on the basis of boundary layer theory. However, due to flow separation and wake recirculation, the theory would be inapplicable in the rear region. Because transport is maximum in the front region, bounds for transport rates and approximate results for the overall transport may be derived. In Sundararajan and Ayyaswamy [215], the quasisteady continuous phase equations are decoupled from those for the liquid phase by prescribing both a suitable surface circulation velocity and the surface temperature. In the front stagnation region of the drop, a similarity transformation has been introduced to obtain a system of ordinary differential equations and a similarity solution. For the entire boundary layer region, a series-truncation procedure with expansions in terms of Legendre polynomials has been employed to develop ordinary differential equations. These are solved by a numerical scheme. A Crank-Nicolson procedure is used for the drop interior. Tables summarizing results applicable for a wide range of condensation rates have been provided.

D. EFFECTOF SURFACTANTS ON CONDENSATION Droplet motion and transport are seriously altered by the presence of even trace amounts of surface-active impurities. A boundary layer analysis of the effects of an insoluble monolayer surfactant on high Re condensation on a drop is presented in Chang and Chung [40]. The study covers droplet radii in the range 100 to 1000 p m and variations in surface diffusivity of the surfactant from lowyto lop3 m*/sec. The strength of internal motion ranges from an order of magnitude smaller than the free stream velocity for slight surfactant contamination to almost a complete stop for high concentrations. It is found that surfactants with lower surface diffusion coefficients are more effective in weakening the strength of internal circulation in condensing drops. Huang and Ayyaswamy 1921 have carried out a numerical study of the effect of insoluble surfactants in condensation on a drop moving in the intermediate Re regime. The droplet environment is taken to consist of its own vapor and a noncondensable, and the droplet is assumed to be initially contaminated with an insoluble monolayer of surfactant. The ambient pressure is 1 atm. The surfactant induced force F,(8) has been evaluated on the basis of a steady convection-diffusion balance model [28, 1301. This force modifies the shear stress at the drop surface. Results have been provided for the interface velocity, drag, surface vorticity, external and internal flow structures, surfactant concentration along the droplet surface, and the Nusslet and Sherwood numbers. Some conclusions from the study of Huang and

44


Ayyaswamy [92] follow: 1. The gradient of the surface concentration increases with increasing Re. At low Re, the surfactant concentration is highest at 6 = T.With increasing Re, the angular location of the maximum surfactant concentration moves toward the front. 2. The surface mobility and the strength of internal circulation both decrease with increasing amounts of surfactant. 3. The covering angle of the surfactant increases with increasing difisivity of the surfactant. 4. During early stages of the condensation, Nu and Sh are relatively lower for a contaminated drop.

E. EFFECTOF DROPLETSHAPEDEFORMATION ON CONDENSATION Drop deformation may be generally assumed to be small for Eo < 0.4 and We < 0.3. At higher Re numbers, the contribution from the inertial forces to the normal stress balance increases, and a spherical water droplet translating in a stream-air mixture progressively deforms into an approximately prolate spheroid. Deformation of a droplet experiencing condensation would be smaller compared to that of a noncondensing drop under otherwise identical conditions as a consequence of the higher pressure gain in the rear. However, deformation may become sufficiently important at Re = O(100). The shapes of water droplets falling through air and the effect of the pressure profiles on droplet deformation are discussed in LeClair et al. [124] and Pruppacher and Pitter [167]. Hijikata et al. [881 have investigated direct contact condensation at high Re (Re lo4) on a refrigerant (R113) and a methanol droplet by experimentation and semiempirical modeling. Droplet oscillation and shape deformation are shown to enhance heat transfer to the droplet. The oscillation is shown to mix the droplet interior. The experimental results yield a heat transfer coefficient about 10 times higher than that for a solid sphere and about 4 times higher than the theoretical result for a spherical droplet. The predictions of the model with oscillation and deformation taken into account compare well with experimental observations.

-

F. CONDENSATION ON A SPRAY OF DROPS

Most analytical/numerical studies in published literature have dealt with an isolated droplet. At present, no systematic, careful, and rigorous analysis exists for reporting on this important area. Earlier experimental studies mostly related to nuclear reactor emergency core cooling spray

-

DIRECTCONTACTTRANSFER TO LIQUID DROPLETS

45

systems, and some numerical models for sprays largely based on ad hoc and questionable assumptions, have been discussed in Ayyaswamy [21]. Some of the earliest attempts to study condensing sprays include the contributions by Tanaka [223], Tanaka et al. [224], and Ohba et al. [l50]. Significant contributions have been made in the recent experimental studies on various aspects of direct-contact condensation as evident in the report by Cum0 [58]. In particular, these include a study by Celata et af. [37] in which experimental results for condensation of saturated steam on subcooled water droplets with diameters in the 0.3- to 2.8-mm range have been provided. The injection has been varied from 0.85 to 9.0 m/sec with pressures up to 0.6 MPa. The droplet Reynolds number is in the range 150 < Re < 2000, and photographic observations indicate droplet oscillation and nonsphericity in several cases. The following equation for the average nondimensional droplet temperature Om, (see Pasamehmetoglu and Nelson [159, 1601 and Carra and Morbidelli [36] for details) is stated to fit the experimental data:

(79) where C is an “empirical convective factor” given by Celata et al. [37]:

-

Huang and Ayyaswamy [96] have numerically investigated condensation on a spray of equal-sized drops moving in the Re O(100) regime by using a unit cylinder cell model [220, 2211 with a body fitted coordinate system [63, 2271. The distance between neighboring droplet centers is assumed to remain the same during the entire process of condensation, although this distance in the flow direction (front to back) is allowed to be different from that in the lateral direction (side to side). The drop environment in each cell is taken to consist of its own vapor and a noncondensable. The ambient pressure is taken to be 1 atm. The results for the interface velocities, drag, surface vorticity, external and internal flow structure, far-stream velocity, and the variations in Reynolds, Nusselt, and Sherwood numbers for various drop arrangements have been provided. The variations in surface shear stress with angular location on the drop surface ( 6 = 0 corresponds to front stagnation point) for the drops A , B , C , and D are shown in Fig. 16. Here A is the lead drop, and the follower drops, B , C, and D, are traveling in tandem; h l and h2, are half-distances between neighboring drops in the plane of motion of the drops, and w l is the half-distance between neighboring drops in the plane

PORTONOVO S . AWASWAMY

46

FIG.16. The variation in surface shear stress with 0 for drops A , B , C, and D . Aspect ratios h l : w l : h2 = 5 :5 : 5, W = 0.41, and Re = 100.

20 1

1

0

0

30

60

90 120 ANGULAR POSITION

150

1

180

FIG. 17. The variation in Nusselt number with 0 for the first, second, third, and fourth drops. Aspect ratios h l :wl : h2 = 5 :5 : 5 , W = 0.41, and Re = 100.

-


47

perpendicular to the direction of motion. At a given value of Re and for any particular drop, the shear stress increases to a maximum value on the front surface and then decreases. Beyond the equatorial plane, the shear stress actually becomes negative for 130" < 8 < 180". This region of negative shear stress corresponds to recirculation on the rear of the drop beyond the point of separation. For the follower drops, the shear stress profiles are similar, except that the magnitudes are lower in view of the weaker flow fields experienced by them. The local variation of the Nusselt number with angular position for a row of drops is shown in Fig. 17. The Nu attains its highest value at the front stagnation point of the lead drop where the temperature gradient is the highest. Away from the stagnation point, with increasing angle, Nu decreases up to the separation ring, following an approximate cos6 variation. For the follower drops, the steepest temperature gradient does not necessarily occur at the front stagnation points due to the wake effects of the leader drops. The dynamics and transport associated with a row of drops are also strongly influenced by the presence of drops on lateral boundaries. The numerical calculations also show that drag coefficients increase with the proximity of the side drops.

G. TRANSFER PROCESSES WITH

A

MOVING COMPOUND DROP

A compound drop is formed when a liquid drop undergoes evaporation or when a vapor bubble experiences condensation while passing through another immiscible liquid. With a compound drop, the liquid and its vapor are both present. 1. Condensation of a Bubble in an Immiscible Liquid

The articles by Sideman and Moalem-Maron [203], Johnson and Sadhal [1071, and Jacobs [98] contain critical reviews of related information and must be consulted for details. The review by Jacobs [98] provides discussions of the studies by Lerner and Letan [127] (vapor condensation with a thin condensate film) and Jacobs and Major [99] (bubble collapse taking into consideration the heat and mass transfer in the continuous phase). A Freon vapor-liquid water compound drop has been investigated in Lerner et al. [126]. Water is lighter than condensate Freon. The experimental aspect of the study is based on visualizations of temperature (shadowgraphy), flow pattern (color entrainment), and condensate shape (dye injection) together with screen tracing of the videotaped bubble shape and path. As the spherical bubble accelerates away from the nozzle, its shape progressively deforms into an ellipsoid, and hydrodynamic and

48


thermal boundary layers are seen to form over its surface. Viscous and thermal wakes form in the rear. This is followed by deceleration. Vortices in the wake advance forward and a thermal cloud covers the bubble. Vapor material inside the collapsing bubble is noted to be spherical, eccentrically positioned, and adhering to the top (front stagnation point) of the bubble. The condensate film is progressively thicker away from the front stagnation point. On completion of condensation, the droplet (which may contain noncondensables) moves away from the vortices. In the analytical model, the noncondensable gases in the bubble are assumed to be uniformly distributed in the vapor. The bubble velocity is taken to be time dependent, and the drag coefficient is allowed to vary with the size and shape. Heat transfer in the condensate is assumed to be quasisteady conduction along radial flow paths emanating from the center of the inner sphere. The continuous phase transport is assumed to be quasisteady, and the thermal resistance is evaluated using correlations described in Lee and Barrow [125]. The results show that the rate of collapse is high during acceleration and is significantly reduced during deceleration. The predictions of the model are shown to compare favorably with the experimental measurements. Motion of a gas bubble completely engulfed by another liquid and moving in a third immiscible fluid is analytically examined using the bipolar coordinate system for low Reynolds number flow in Sadhal and Oguz [191]. The surface tension at both the interfaces is assumed large enough to preserve sphericity of the bubble and the drop. The transient convective heat transfer associated with a collapsing bubble moving at low Reynolds number under the influence of buoyant forces is evaluated using finite-difference methods in Oguz and Sadhal [149]. The drag component induced by radial velocity contributes to the total drag on the bubble in eccentric configuration. The time-dependent Nusselt number depends on the compound drop configuration and on the conductivities of the participating liquids. When the heat transfer rate is high enough, the bubble away stays inside the drop due to rapid shrinkage until it finally disappears. The velocity of the drop and the relative velocity of the bubble decrease as the bubble gets smaller due to the changes in the buoyant forces. Radial convection decreases the Nusselt number.

2. Evaporation of a Drop in an Immiscible Liquid The review articles by Johnson and Sadhal [lo71 and Avedisian [16] contain related information about evaporation in an immiscible liquid and must be consulted for details.

-


49

Vaporization of liquid drops in an immiscible and low volatility liquid medium has been studied extensively in view of the numerous applications of this process. The classic contribution of Sideman and Taitel [204] has been followed by a very large number of studies with varying degrees of success. The evaporation of a liquid drop into a vapor bubble can occur in three different configurations: nonengulfing, completely engulfing, and partially engulfing based on the balance of surface tension forces [17]. The partially engulfed configuration is most common in direct-contact heat transfer. Note that in single drops, significant superheating can occur before evaporation commences. We discuss only situations where evaporation is occurring. Sideman and Taitel [204] have experimentally studied pentane and butane drops evaporating in distilled water and seawater. An analytical expression for Nu has been developed by solving the external heat transfer problem assuming a spherical shape for the drop (valid only to a limited type of fluid system) and potential flow in the continuous phase. The predictions match the experimental data only at the stage when the drop has almost completely evaporated and the residual liquid is small and thin. Evaporating compound drops may experience oscillation, causing the dispersed phase liquid to slosh from side to side, forming a thin film of liquid over the top of the inside bubble surface. This feature has been observed by Simpson et al. [205] in experimental studies with a butane drop rising in brine. The oscillation is ascribable to periodic vortex shedding of the wake at high Reynolds number. The drop changes shape from spherical through ellipsoidal to a cap-shaped bubble. A compound drop rising in low- and moderate-viscosity fluids follows a zigzag trajectory. A pentane drop evaporating in highly viscous glycerol has been studied by Tochitani et ai. [228, 2291. In the analytical model, the Stokes solution for a sphere has been assumed for the flow field, and the heat transport is treated as a quasisteady process. The latter assumption is questionable is O(100). because Tadrist et al. [218, Part i] have provided results for evaporation and bubble growth based on an approximate momentum balance that relates compound-drop acceleration to volume and surface forces. The surface forces are evaluated for the limits of potential flow and Stokes flow; and, for finite values of Re, “correction factors” are employed. The drag coefficients for air bubbles moving in water from Haberman and Morton [801 have been generally employed in the force balance. The RayleighPlesset equation is used to determine the growth rate of the bubble. A semiempirical expression for the instantaneous velocity of a compound drop is given by Raina et al. [172], and this may be useful in computations. The energy conservation equation in these studies is usually a heat balance

-

50


statement incorporating an overall heat transfer coefficient. This heat transfer coefficient is obtained from semiempirical expressions for the instantaneous Nusselt number such as the one developed by Battya et al. [24]:

Nu = 0.64 Pe0.5Ja-0.35,

(81)

or the one given in Raina and Grover [171], which incorporates sloshing effects, contact angles, and spreading coefficients. In Raina and Grover [170], a model for determining the liquid-liquid area taking into account the effect of viscous shear on the spreading of dispersed liquid over the bubble surface has been described. A theoretical model for evaporation that takes into account the effects of temperature difference between the phases and the bubble growth rate is developed in Battya et al. [25]. The predictions of this model agree with Eq. (81). The mechanical equilibrium of the bubble droplet has been further explored in the second part of Tadrist et al. [218] to determine the contact angles and the effective area available for heat transfer. The limitations of the Raina-Grover viscous shear model have been discussed in this study. Experimental results, correlation for volumetric heat transfer coefficient, and a model extending the single-drop study to multidroplet systems have all been included. A partially engulfed drop has been analyzed in Vuong and Sadhal[238, 2391. In the first of this two-part study [238], the motion is analyzed in the limit of Stokes flow, and the bubble growth rate has been prescribed arbitrarily. An exact analytical solution for the axisymmetric flow field has been developed in a toroidal coordinate system by combining the solutions separately obtained for a flow field resulting from drop translation and a flow field resulting from the moving boundaries of the drop due to the growth. The drag force on the compound drop is shown to depend on many parameters besides the viscosity of the continuous phase, the drop size, and the free stream velocity. Among the important parameters are 4p,the drop geometry, that is, the liquid-to-vapor volume ratio together with contact angles that depend on the nature of the fluid systems, and the ratio of the growth to translational velocities. During the vaporization process, for a compound drop, the drag force has been shown to even exceed the value for a solid sphere whose volume is equal to the total liquid-vapor dispersed phase volume. This drag force attain: a maximum value where the dispersed phase liquid-vapor volume ratio, V,/V,, is 1. In part two of the study [239], the heat transfer problem is solved under the assumption that the liquid-vapor interface and the vapor phase are at the equiIibrium temperature corresponding to the hydrostatic pressure. The vapor is effectively decoupled from the liquid [163]. The energy equations, for both the continuous phase and the liquid portion of the

-

-


51

16-1

-

0

r = - Uot

r

2

4

6

-

8 10 12 14 16 18 20

7 =

Uot r

FIG. 18. Time history of the Nusselt number: (a) AT = 3.1 K and (b) AT d o = 1.4 mm. Reproduced with permission from Vuong and Sadhal [239].

=

0.1 K;

dispersed phase with the convective velocity taken from the fluid flow solution, have been numerically solved using finite-difference techniques. The time history of evaporation of a pentane drop immersed in a bath of glycerol has been provided, and the predictions compare favorably with the experimental results of Tochitani et al. [228, 2291 except for the heat transfer coefficient. Note that the measured heat transfer coefficient in the Tochitani et af. study for the 1.4-mm-diameter drop is higher than for the 0.8-mm-diameter drop. This contradicts the well-established theory that the heat transfer coefficient is inversely proportional to the size of the drop. Figures 18 (a) and (b) show the time history of the average Nusselt number, defined as Nu = Q/127-rkR0(T, - T')] for two superheat temperatures of AT = 3.1, and 6.1 K, respectively, as a function of dimensionless time t' = U,t/R,. Here, U, is the initial velocity of the compound drop and R , is the initial radius of the vapor bubble. The results reveal that when the drop is first introduced into the high-temperature fluid, the average Nusselt number for the liquid-liquid interface (denoted by Nu,) is very high. Soon after, a very thin conduction layer in the region of the liquid-liquid interface begins to form resulting in a steep drop of Nu,. Simultaneously, the temperature at the liquid-vapor interface begins to increase due to the energy absorbed from the external fluid, and a rapid rise in the Nusselt number for the liquid-vapor interface (denoted by Nu;) is noted. Both Nu, and Nu, approach asymptotic limits at a large time since the transient effect is no longer important. It may be observed that,

52


although Nuj behavior is similar for various AT, Nu, variations depend on the particular level of AT. This is explained by the transient oscillations in Nusselt number and the role of internal circulation, both of which depend on the level of AT (compare with section III.A.1). An approximate analytical model applicable for preagglomerative and postagglomerative stages of compound drop motion in multidroplet evaporation situations is described in Smith et al. [2101. During the preagglomeration stage, it is assumed that the droplets are relatively small and do not interfere with one another. For the postagglomeration stage, it is assumed that the rate of coalescence is such that the dispersed phase volume fraction is constant. Experimental measurements related to the evaporation of cyclopentane droplets in stagnant water are reported. Results for volumetric heat transfer coefficients in terms of travel distance that compare favorably with experimental observations are provided in this study. Seetharamu and Battya [ 1971 have experimentally investigated the evaporation of R-113 and n-pentane droplets in a stagnant column of distilled water. The variation of volumetric heat transfer coefficient with column height and dispersed flow rate has been established. The volumetric heat transfer coefficient increases with (a) a decrease in column height and (b) with an increase in the flow rate of the dispersed phase. A lower column height is associated with larger temperature differences, greater acceleration of the drops, and increased turbulence. With distilled water for the continuous phase, the volumetric heat transfer coefficient is lower for R113 compared to n-pentane. Based on their experimental observations and a multiple-linear regression analysis, Seetharamu and Battya propose the following expression for calculating the initial drop diameter: 0.35 do.?? do = CVorifice orifice7 (82) where C is a constant that depends on the physical properties of the system (for example, 0.376 for R113-water, 0.307 for pentane-water), Vorifice is the orifice velocity (cm/sec), and dorifice is the orifice diameter (cm). Modifications to the theoretical model of Smith et al. [210] are suggested in this study. Although the compound droplet problem is among the more difficult class of problems involving phase change, considerable progress on the problem can be expected in the coming years due to the recent availability of advanced computational methods and supercomputers.

H. DROPLET VAPORIZATION IN A GASEOUS ENVIRONMENT Droplet vaporization has been extensively studied, most notably in the context of spray combustion and spray cooling processes.

-


53

For steady evaporation of a pure liquid drop at a low temperature (so that the effect of vapor concentration is negligible), and for Re < 2000, the following correlation based on the studies of Frossling [70]and Ranz and Marshall [179] is recommended in Yuen and Chen [252]:

Nu = 2.+ 0.6 Re'/2 Pr1I3. (83) lFor mass transfer Pr is replaced by Sc. The properties are evaluated at the film conditions. The Ranz and Marshall correlation is based on certain quasisteady, constant radius, porous wetted sphere experiments and is only valid for small values of transfer number. The other commonly used correlation is the Spalding's correlation [211]:

where B = c(T, - T,)/A is the transfer number. The correlation of Eq. (84) is based on experiments at 800 < Re < 4000 and is suggested for values of 0.6 < B < 5. Neither of the two correlations accounts for transient heating, regressing interface, and internal circulation. Experimental correlations for the evaporation of water droplets in air have also been reported in Beard and Pruppacher [27], Yao and Schrock [249], Yuen and Chen [252, 2531, and Pruppacher and Rasmussen [168]. The evaporation rate of small water droplets moving in air has been numerically predicted by Woo and Hamielec [2451. Recent studies of droplet vaporization, particularly as related to fuel droplets, have contributed to a deeper understanding of the mechanisms involved, and many new and comprehensive correlations are available. Fuel sprays consist of a spectrum of droplets of various sizes and velocities. The major objective of fuel drop vaporization studies has been to develop suficiently accurate yet simple enough models for use in a spray analysis. A vaporizing droplet can experience a range of Reynolds numbers during its lifetime. Typically, Re will decrease with time as the droplet diameter and the relative velocity decrease. (There are exceptions to the monotonic behavior, e.g., oscillatory ambient flow where large fluctuations of relative velocity, including change in direction, can occur [206]). The smaller droplets (diameter < 30 pm) essentially follow the ambient flow field, and will be in the low translational Reynolds number regime (Re < 1) during most of their lifetimes. The bigger droplets 100 pm), because of their large inertia and momentum, may (diameter be in the intermediate Reynolds number regime [Re = O(l00)l for significant portions of their lifetime. They penetrate deep into the spray core. The bigger droplets determine the overall behavior of the spray combustion process, because they constitute most of the spray mass. The smaller

-

54


droplets have a smaller impact, because they last over a comparatively shorter period of time. From an energy transport point of view, in typical combustion situations, the duration of the transient droplet heating is comparable with the droplet vaporization time. A complete investigation of the various flow regimes governing droplet motion and the evaluation of the associated transient effects are prohibitively difficult. This is true in spite of the availability of advanced experimental and numerical techniques and supercomputers. However, with intelligent modeling of combustion spray systems, a wealth of information can be obtained, and optima1 systems can be designed. In regard to combustion modeling, two major categories are distinguished by Faeth [68]. A spray model consisting of very small droplets is termed the locally homogeneous flow (LHF) model; here, the gas and liquid phases are assumed to be in dynamic and thermodynamic equilibrium at each point in the flow. Although this model cannot describe a real spray, it provides the lower bound for the size of spray process. The second category, called the separated flow (SF) model, considers the effect of finite rates of transport between the phases and, hence, can model practical sprays with finite size droplets. It is noted in Faeth 1681 that the behauior of individual drops in a spray must be examined in order to assess the validity of LHF models and to undertake SF models. This involues drop-life-history computations to yield the size, velocity, temperature, and composition of individual drops as a function of position in the pow. Most analytical/numerical studies of droplet vaporization have employed a number of simplifying assumptions. Small Mach number flow is almost always considered so that kinetic energy and viscous dissipation are negligible, Gravity effects, droplet deformation, radiation, Defour energy flux, and mass diffusion due to pressure and temperature gradients are all usually neglected. The multicomponent gas-phase mixture is assumed to behave as an ideal gas. Phase equilibrium (usuaIly described by the Clausius-Clapeyron equation) is assumed at the single-component dropletgas interface. Gas-phase density and thermophysical parameters are generally considered variable. Liquid-phase viscosity is generally taken as variable but density and other properties are typically taken to be constant. 1. Spherically Symmetric Vaporization of a Fuel Drop

For droplet evaporation in a stagnant medium, the spherically symmetric continuous phase motion is Stefan convection in the radial direction. The liquid phase motion is also spherically symmetric. The droplet surface regresses into the liquid. At moderate pressures, the gas phase may be

-


55

treated as quasisteady because the diffusion of heat and mass in the gas is relatively fast compared to that in the liquid. Hubbard et al. 1971 have presented a numerical study of the effects of transients and variable properties of the continuous phase on single-droplet evaporation into an infinite stagnant gas. The results are developed by solving transient one-dimensional problems for the continuous and dispersed phases. Sample calculations are reported for octane droplets inim evaporating into air tially at 300 K with R , = 0.1, 0.5, and 2.5 X at temperatures and pressures in the ranges 600 to 2000 K and 1 to 10 atm, respectively. The study recommends that, for purposes of engineering calculations, the most appropriate reference state is a simple rule wherein the reference temperature and species mass fraction are, respectively, T, = T, + +(T, - 7'J and an identical expression for the mass fraction. Spherically symmetric vaporization in the presence of an exothermic reaction in the continuous phase between the fuel vapor and a suitable reactant is discussed in a number of articles [123, 2061. These articles contain details in regard to the following. At the liquid-vapor interface, the energy balance is:

For very fast reaction rates (reaction zone in the limit of zero thickness), constant specific heat, and unit Le,the spherico-symmetric mass vaporization rate is: mss =

47

[jR3

dr ]-'log(l

+B),

where

where Q is energy per unit mass of fuel. The position of the flame zone for a thin flame may be determined from:

'r=log[(l

+ ~YiW)/(1 - YFJ]

R

log(1 + Yo,)

(88)

The solution to the transient energy equation for the liquid phase will provide the necessary relation between the interface temperature and the liquid heating rate. The droplet heating time r H is of O(Ri/cF).

56


For constant pD, the droplet lifetime T~ can be estimated as p^Rg/ + B ) ] , where the transfer number, B , is given by Eq. (87). A number of limiting cases in sphericavy symmetric vaporization are discussed in Sirignano [206]. When k 1). A strained = 6'1) are defined for the coordinate Q = E r and a stream function outer region. In the outer region, it is inappropriate to scale velocity by the

-


63

evaporation velocity A, but it may be scaled by U,. Letting

*

= &*(,

+ &’Yl + . .

,

(93) the inner and outer solutions up to and including O ( E ’ ) have been developed. The liquid phase solutions to O ( E ’ )are obtained in a manner similar to that in Gogos et al. [73]. The drag force on the liquid drop (nondimensionalized by puU,R) is calculated up to and including terms of ’

O(E).

b. Solution for Temperature Field: Gaseous Phase The temperature and the mass fractions are nondimensionalized as T * = Tc/AH, Y$ = WFYo/(u,W0) and Y: = YF/uF.With Shvab-Zeldovich variables,

T * - T,* g=

+ Y$

T,* - T,*

-

Yom

and

h

=

Y,*

-

Y$

-

Ygm

+ +

+ &2U2) . vg - v2g = 0, E U ~+ E’u’) . Vh - V 2 h = 0.

+ YOm,

(94)

the equations are Sc(A,u,

Sc(A,u,

EU,

(95)

(96)

In the outer region,

c. Solution for Temperature Field Liquid Phase The temperature inside the droplet is nondimensionalized as T* = Tc/A H and time is normalized as T = &/R2.The governing equations are written in terms of the transformed variable &:

Similar to the expansion for the temperature in the gaseous phase, a perturbation scheme in terms of E and Legendre polynomials is used:

i= io0 + E [ iol+ P,(COSe)&,,] + E ’ [ gO2+ P,(COSe)$,, + P,(COSe)g,,]

+ ... .

(99)

With this expansion, the governing differential equations are solved numerically using a finite-difference method. An implicit algorithm is used to solve for the transient temperature field inside the droplet. Calculations

64


for the drop exterior are carried out simultaneously. At each time step, several iterations are required to obtain consistent convergent solutions.

d. Evaluated Quantities The mass burning rate at the droplet surface is given by

m = 27Rp

1

1-u , ~ , = ~ ~ (6c)o. s

( 100)

1

Nondimensionalizing the mass burning rate by 47rRp Aoo,we get m=1

+ E-A,, + A00

E2-

A 02 A”,

+ O(E2).

The drop regression rate is given by

8ki0,

1

+ O(&’) .

(6 - P)C,D,2

(102)

The heat quantities are nondimensionalized by 4 a R o k A H / c . The dimensionless heat transfer from the gaseous phase to the droplet is

The heat required for fuel evaporation is 4e =

1 R Sc ----j’ 2 R, AH

~~,i,=~dcose -1

Sc R - --- [ A , Ja R o

+ &A,, + &?A,’ + O ( & ” ] .

(104)

The heat used for liquid heating is

( 105 1 An equation for the velocity of the droplet can be obtained by setting the net force acting on the droplet (weight, drag, and buoyancy) equal to B=4-qe.

-


65

The interfacial heat transfer variation with time is shown in Fig. 22. For a droplet introduced at its wet bulb temperature, Twh,the heat from the gaseous phase, q , is completely utilized for fuel evaporation, q,. For a droplet whose initial temperature is less than the wet bulb temperature, for about a third of the droplet lifetime, part of the heat input is used for fuel evaporation and a substantial portion is used for liquid heating. As the surface temperature of the droplet increases, q decreases while qe increases. At higher T,, the mass fraction of the fuel at the interface is higher. For droplet initial temperatures higher than the wet bulb value, the interface may receive heat both from the droplet interior and the gas phase. This will result in very high values for the initial evaporation rates. Eo = 0.1

0.0125 r ......

__

-.-

! \.,I11

;

:;

,('

320 K 359.2 K(= 361 K

1 9

Twb) II 111

qe

i

66


4. High Reynolds Number Vaporization: Solution by Boundary Layer Formulation At high Re, the evaporation rate can be sufficiently accurately determined by solving the coupled boundary layer equations of motion, energy, and species concentration in the gas and liquid phases. Harpole [84] has examined water droplet evaporation into a high-temperature dry air or pure steam environment by solving the laminar boundary layer equations at an axisymmetric stagnation point. It is claimed that transport to the whole droplet may be estimated with suitable corrections to the stagnation-point solution. A number of correlations are given to demonstrate and justify this claim, but the validity has been seriously questioned by Renksizbulut and Yuen “41. Prakash and Sirignano [164, 1651 have used boundary layer theory for a single-component drop. Law et al. [122] and Lara-Urbaneja and Sirignano [115] have developed it for a multicomponent drop. The droplet is assumed to vaporize in a hot inert environment. It is also assumed that the specific heats ci and binary diffusion coefficients Dj of all the gas phase components are equal, and Le is unity. In the high Re regime, gas flow is essentially unsteady due to the temporal change in the size of the droplet. However, the characteristic time for changes in the gas phase is the residence time in the neighborhood of the droplet, and is of O(d/Um= 10 p s ) for a droplet of 100-pm in diameter in a free stream velocity of 10 m/sec. This time is much smaller than the droplet lifetime, which is typically of O(5 msec) for a droplet of this size vaporizing in a convective field. In the time scale of the drop heating, the gas phase is, therefore, taken to consist of quasisteady momentum and thermal boundary layers near the surface. The velocity and pressure distribution outside the momentum boundary layer are assumed to be those for potential flow over a sphere. These assumptions preclude the determination of the correct total drag coefficient. In many circumstances, the drag force effects are such as to rapidly decrease the flow Reynolds number with time. In those situations, the boundary layer assumption is invalid beyond a small portion of the droplet lifetime. The gas-phase boundary layer formulation is solved by the von Karman-Pohlhausen integral method in which fourth degree polynomial profiles are assumed for velocity, temperature, and concentration at any location along the flow. For the liquid motion in the drop, due based on droplet diameter, maximum liquid velocity, and to large 6, liquid properties can be of the same order of magnitude as or higher than Re. This results in high % values. The flow field is considered to be quasisteady. (See discussions in the sections on fluid mechanical studies and on the internal problem.) In the liquid-phase analysis, the boundary


67

layer velocity field is expressed as a perturbation to the Hill’s vortex solution valid in the core. The droplet core heating problem is transformed from a two-dimensional unsteady liquid-phase diffusion problem to a one-dimensional unsteady problem by using the quasisteady stream surfaces (denoted by 4) as coordinates and by assuming that, for large E, the temperature is uniform (but time varying) over each of these surfaces. Then diffusion essentially occurs only in the normal direction to the stream surfaces. The solution of the core temperature equation is developed by matching with a liquid-phase thermal boundary layer solution at the droplet surface. The thermal boundary layer solution is, in turn, matched with a gas-phase boundary layer solution at the droplet surface. The gas-phase and liquid-phase solutions are iterated until the desired convergence is achieved. The heat and mass transfer in the wake region is considered negligible. The results of Prakash and Sirignano [164, 1651 for the vaporization of n-hexane, n-decane, and n-hexadecane droplets in air at 1000 K and 10 atm (wide range of volatility values) show that the heat and vaporizing mass fluxes continually decrease in the downstream direction, and are of the order of 20% of their front stagnation region values near the point of boundary layer separation. The variation of dimensionless radius to the with dimensionless time, t&/R;,is linear three-halves power, (R/R,)3/2, except in the early evaporation period when the transient liquid heating is substantial and the vaporization is small. The results are shown to compare well with the predictions of Eq. (83) during the initial part of the lifetime and with Eq. (84) during the later part of the lifetime. (In applying the correlations, A,*, which is smaller than A by an amount equal to the liquid phase heat flux, ha5 been used.) Toward the end of the droplet lifetime, the boundary layer theory predictions are lower than the values of the two correlations, This discrepancy is attributed to the neglect of wake effects. The vaporization models described in Prakash and Sirignano [1651 and in Lara-Urbaneja and Sirignano [ 1151 are too complex for incorporation into spray programs. Simplified models based on several ad hoc assumptions have been suggested in Sirignano 12091,Tong and Sirignano 1230-2341, and Abramzon and Sirignano [41. When employing these models, remember that the droplet drag coefficient cannot even be approximated by these simplified analyses. Empirical correlations have to be used for evaluating the drag force. Tong and Sirignano have analyzed the high Re, Le = 1, problem with one-step chemistry. For the gas-phase, two separate boundary layer analyses, one for the stagnation region ( r = x and U, = ax, where a is the constant of curvature in stagnation point flow) and another for the


68

“shoulder region” (0 = 7r/2, r = R , U, = ;Urn> where the pressure gradient is zero and the flow locally behaves like a flat plate flow, have been developed. The Blasius differential equation is solved, and the vaporization rate per unit area is expressed in terms of the Blasius function, f , as

-Af(O,B , U s / U e ) ,

( 107) for the stagnation point flow and is equal to pepLe ( ~ , / 2 / & 3 , p , d x ’ ) ~for / ~ the shoulder region. The model assumes that most of the transport occurs on the front or on the “shoulders” of the droplet. The transport on the rear of the droplet is considered to be relatively smaller and is neglected. For the liquid phase, it has been shown by Tong and Sirignano that the liquid-phase heat diffusion equation developed in Prakash and Sirignano [165] given by Pds=

where A

=

af

aV

af

- = ~ - arb2 + b - , a4

( 108)

where

with f l , f i , and f3 being known functions, and the corresponding mass diffusion equation can be simplified when the change in droplet radius due to vaporization occurs slowly compared to changes in liquid temperature. In that case, the nonlinearities introduced by coefficients in Eq. (108) can be modified to give an approximate piecewise linear behavior for the equation. A Green’s function analysis reduces the equation to an integral form whereby a quadrature gives the liquid temperature at any point as a function of the surface heat flux. This procedure results in the development of a Volterra-type integral equation that relates surface temperature to surface heat flu. The integral equation formulation is subsequently transformed into a system of first-order ordinary differential equations, which are solved with proper matching conditions at the interface by the Runge-Kutta scheme, thereby offering considerable computational simplicity. From Tong-Sirignano analyses, the following relations applicable for high Re vaporization can be developed:

where k is a nondimensional coefficient of order unity, which is determined by averaging the heat flux over the droplet surface in an approxi-


69

mate manner (based on the two local solutions for the stagnation and shoulder regions; see Sirignano [209]) and by a similar averaging process for the vaporization rate, m

=

p e R [-f(O,

B , u,/ue)] Re’I2.

( 111)

Abramzon and Sirignano have described a model for droplet vaporization that covers a wide range of Re, %, and Le. It is assumed that the gas phase is quasisteady and the pressure drop in the gas is negligible. The thermophysical properties are evaluated by the rule. The gas-phase calculations are based on the simple one-dimensional thin-film resistance theory of heat/mass transfer. The temperature and vapor concentrations along the droplet surface are assumed to be uniform (cause for underestimation of vaporization rate). The film thicknesses so obtained are then “corrected” (called “extended film model” theory) to account for the presence of Stefan flow (which in the case of vaporization will have a thickening effect) by considering a range of Falkner-Skan solutions for flow past a vaporizing wedge. These, in turn, are used to calculate the average transport rates across the gas boundary layer on the droplet surface. The transient liquid heating inside the droplet is calculated by using an “effective” thermal diffusivity model that approximately accounts for the characteristic heating time and liquid heat capacity [ 103, 2221. The effective thermal diffusivity is defined by Gee = x& where x depends on the instantaneous E.The value of x has been developed by fitting data in Johns and Beckmann [1061: y, = 1.86

+ 0.86 tan h[2.225 log(@e/30)].

(112)

With this CEefl, a spherically symmetric pseudo temperature field in the liquid is solved for by using a one-dimensional diffusion equation for the dispersed phase. Note that although the surface temperature can be reasonably predicted with these ad hoc assumptions, the details of the internal temperature field can be seriously in error. With BH = ( h e h,)/AeR, and B, = (YFs - Y,&(l - Y,,), the following relations for the Nusselt number and the vaporization rate, based on Abramzon and Sirignano [4], are given in Sirignano [2061: Nu

=

2

1.

k Pr1/3Re’/2 log(1 + B H ) BH [l+2 F(BH)

and a similar equation applies for Sh with B,

(113)

replacing BH and Sc

70


replacing Pr. Also, AR

m = 47r-

log(1

[

C

=

where, for Le

=

47rpDR log(1

1, B,

= B,,

k Pr'/3Re'/2

+ BH) 1 + + B,)

F(BH)

[

1

k SC'/~R~'/~ 1+ F(BM)

and otherwise

BH = (1

+ BJ

-

1,

where

k Re'/' 1+-'F a = _ c Le

F(BM)

k Re'/2 ' 1+-F(BH)

in which

F( B )

=

(1

+ B)'.'

In( 1

+ B)

B

for 0 I B,, B, I20,l I Pr, and Sc I 3. Results from these equations disagree strongly with the predictions of the Ranz-Marshall correlation, and the reason is ascribed to the narrow range of B values on which the Ranz-Marshall correlation is based. Figure 23 shows comparisons between the predictions of the extended film model (curve 1)' the infinite conductivity model (curve 21, the conduction limit model (curve 31, and the effective diffusivity model (curve 4) for the droplet surface temperature variation with time. The results refer to an n-decane droplet of initial radius ro = 50 k m and temperature To = 300 K, which is injected into an air stream at T, = 1500K and pm = 10 atm. The results for the extended film liquid heating model fall, in general, between those for the infinite conductivity and conduction limit models. The predictions of the effective diffusivity model almost coincide with those of the extended film model. Range1 and Fernandez-Pello [177] have used boundary layer analysis to investigate mixed convective (effects of natural convection included) vaporization and combustion of a hexadecane droplet in air at a prescribed Reynolds number using constant property assumptions except for density variations. For the liquid droplet, it is postulated that for the mixed


71

Time, msec FIG. 23. Surface temperature versus time: Various models. Reprinted with permission from Abrarnzon and Sirignano [4].

convection case, the strength of the vortex can be taken to be A

=

3Re cv,-(1 2

+ 42)”4/R3,

where the constant C is determined by matching vorticities between the liquid boundary layer and the Hill’s vortex. In Eq. (118), the mixed convection ratio 4 = Gr/(3Re/2I2, and the Grashof number Gr = R3g(T, - T,)/(T’v~). Thus, 4 = 0 corresponds to the forced convective limit, and 4 + CQ to the free convective limit. Under the various conditions applicable to this analysis, 1191, the droplet burning rate is given by FernandezPello [69] as:

dR2 dt

-

kmg 0.2,($Re)1’2(1

+ +2)1’sll,

PC

where B

[ YomQ/(Movo)

c(T - T,)]/A

( 120) and Q is the heat of combustion per gram of oxidizer. The dependence of the burning rate on R e and B given by Eq. (119) agrees qualitatively with the experimental observations of Faeth 1681, Eisenklam el al. 1641, and Natarajan and Brzustowski [141]. =

-


72

5. Vaporizing Droplet : Intermediate Reynolds Number Numerical Solutions In most numerical studies of droplet vaporization, the axisymmetric, unsteady form of the governing equations is solved with stiff upstream boundary conditions and zero-derivative downstream boundary conditions. The liquid and gas flows are coupled at the spherical droplet surface by conditions of continuity of temperature, normal mass fluxes and tangential shear force, and balance of normal momentum and normal heat flux. The stream-function-vorticity method and implicit finite-difference techniques are typically employed for generating the solutions [931. Renksizbulut and Yuen [1841 have reported experimental results for heat transfer rates to simulated and freely suspended droplets of water, methanol, and heptane. The ranges of experimentation are 25 < Re < 2000, and 0.07 < B < 2.79 where

where Q R is radiative heat flux to drop. The data show that at higher temperatures, evaporation reduces heat transfer rates by a factor of (1 B,-)O.’ (subscript f denotes film condition, that is, arithmetic average molar concentration and temperature). Numerical solutions for evaporation rates due to high-temperature air flow past droplets of water and methanol and solid spheres, and superheated steam flow past water droplets, have been provided in Renksizbulut and Yuen [ M I . The temperatures range from 600 to 1000 K and 10 < Re < 100. The gas phase is assumed to be a binary mixture of ideal gases of equal molecular weights and equal and constant specific heats. The equal specific heat assumption has been questioned by Abramzon and Sirignano [4]. A constant-temperature liquid phase has been assumed, and the liquid-phase motion and core heating are neglected. The liquid-phase assumptions are also difficult to justify. The numerical solutions show that blowing at the interface increases the drag coefficient, C,, in agreement with their own experimental observations. This conclusion, however, is at complete variance with the numerical results of Dwyer and Sanders [61, 621, which show that C , decreases significantly as the droplet vaporizes and the relative velocity between the droplet and gas decreases. The constant property assumption invoked in the investigations of Dwyer and Sanders is questionable. The transient evaporation of a spherical n-heptane fuel droplet in superheated n-heptane streams at 800 K and 1- and 10-bar pressures has been studied by Renksizbulut and Haywood [1821 using a finite-volumebased numerical method (see Ayyaswamy [201 for a brief discussion of the numerical method described by Renksizbulut and Haywood [1821 and Haywood et al. [85]). Abramzon and Sirignano [4] noted that the results

+

-


73

obtained in Renksizbulut and Haywood [182] are not applicable to the usual situation of a nonvapor environment where diffusion may be a controlling factor. Haywood et al. [85] have extended the earlier analysis to an n-heptane droplet evaporating in air at 800 K and 1 atm. The droplet is initially uniform in temperature at 298 K with no internal motion and has an initial Reynolds number of 100 based on free stream thermophysical properties. The effects of all transients and of variable properties in both gas and liquid phases are included. The effect of surface regression and the deceleration of the droplet under the influence of its own drag have been examined. The major conclusions from the studies of Renksizbulut and Yuen are that the following correlations (based on drop evaporation studies in air streams up to 1059 K, 10 < Re, < 2000, together with a simple diffusion model for liquid-phase heating) can be used in a quasisteady manner to predict accurately droplet behavior for system pressures up to 10 bar. For droplet drag (10 < Re, < 3001, cD(1

+ B~,-)’.’ = 24Re;’ + 4.8Re;0.37.

( 122)

For heat transfer (10 < Re, < 20001, Nuf(l

+

=

2

+ 0.57Re!,’zPr//3,

(123)

where CD is the drag coefficient, BHf is the transfer number = (T, T,)c/h, and its range is 0.07 < B , < 2.79. All the properties are evaluated at film conditions except for the density in the Reynolds number, which takes the free stream value, and such a Reynolds number is indicated by Re,,. The liquid heating model accounts for the effects of liquid motion through the use of an effective thermal conductivity. At pressures higher than 10 bar, the surface regression rates are higher, and unsteady gas-phase effects a r e t h o u g h t t o b e c o m e i m p o r t a n t . H a y w o o d el al. [85] have speculated that the effects of increased surface motion and second-order drag effects [68] could render the correlation of Eqs. (122) and (123) inapplicable at higher pressures. The transient drag coefficient and Nusselt and Sherwood number histories computed by the fully transient numerical model in Haywood et al. [85] are well predicted by the quasisteady correlations of Eqs. (122) and (123) with suitable corrections for the effect of variable properties and liquid-phase heating. The correlations recommended in Haywood et al. [851 for C , and Nu are the same as Eqs. (122) and (123), respectively, except now the transfer number has to be modified as

In the preceding equation, Q/Q is the fraction of heat transfer into the


74

liquid phase and Q R / Q is the ratio of thermal radiation heat transfer rate at the interface to the gas-phase heat transfer rate. The transient heating of the liquid phase must be modeled in order to obtain the surface temperature and liquid heating rate for input to the transfer number that appears in the correlations. For mass transfer, the following equation is recommended by Renksizbulut et al. [183]: Sh, (1

+ B,,,,f)O" = 2 + 0.87 Re!,/*

Sc)l3

(125)

for 10 5 Re,,, I 2000 where

The dimensionless parameter Sh, = 2RhM/(p,Df) is the Sherwood number and Sc, = pf/(pfDf) is the Schmidt number. The drag coefficient and transport results for quasisteady evaporation of an n-heptane droplet in air obtained by Huang and Ayyaswamy [93] using a numerical procedure that is based on hybrid difference scheme agree very well with the predictions of Haywood et al. [851 and Renksizbulut et al. [183]. Some other conclusions from the studies of Renksizbulut et al. are as follows: 1. The dimensionless droplet radius, (R/R,), is a linear function of dimensionless time, (tUJR,), except at the very beginning of the transient. 2. The droplet spends a significant portion of its lifetime at relatively high Reynolds numbers where convection effects are dominant. However, at later times the instantaneous Reynolds numbers are small and boundary layer approximations may be invalid. Liquid heating is significant for about the first half of the droplet 3. lifetime. At elevated pressures, liquid heating will persist for a greater portion of the lifetime. 4. Except for a rapid reduction in the early transient period, the total drag coefficient increases as the droplet Reynolds number decreases. The early reduction is ascribed to an immediate decrease in shear stress and mixture viscosity at the onset of evaporation. At later times, these effects lose significance and other mechanisms (for example, pressure drag dominance) become important. 5. The transient dynamics of the mass transfer and heat transfer are similar. The thickened boundary layers and modified properties (reduced thermal conductivity and fuel-rich region near the surface) that accompany the onset of evaporation inhibit heat and mass transfer to the surface. A rapid initial decline is noted for both Nusselt and Sherwood numbers. At later times, with decreasing

-

DIRECTCONTACT TRANSFER To LIQUID DROPLETS

75

convection effects, these numbers continue to decrease but at much reduced rates. 6. Ignoring contributions from the wake region (defined as 0 2 120" ) is likely to introduce errors of about 10% in the predictions of average Nusselt and Sherwood numbers. Chiang et a!. [44] have used finite difference procedures to analyze the vaporization of a fuel droplet suddenly injected into high-temperature, high-pressure air environments, allowing for variable properties and variable density with multicomponent gaseous mixtures. The results show by comparison that the variable density but otherwise constant property calculations of Patnaik et al. 11611 could overpredict drag coefficients by as much as 20%. Figure 24 shows the variations in the three components of the drag coefficient as a function of gas hydrodynamic diffusion time at three different ambient temperatures. The major role of friction drag components on total drag is evident. Friction drag is substantially reduced at higher values of surface blowing. The pressure drag coefficient increases steadily as a result of the reductions in upstream velocity. Thrust drag becomes important at high transfer numbers. It is estimated that the time

1. T =1800 K, Pressura Drag

,1

Pressure Drag - ___ 32.. TT =~ 18002 5K,0K,Pressure Drag 4. T d800 K, Frlcllon D n g

0.0 217 5.4

/

8.1 10.7 13.4 16.1 18.8 21.5 24.2 26.8

Gas Hydrodynamic Diffusion Time FIG. 24. Three drag coefficient components versus time for different ambient temperatures. Reprinted with permission from Chiang er nl. [44].

76

PORTONOVO S. AWASWAMY 7.54 6.88

______. 2.7 +1

6.22

- - -. 1.1 =

4. T = I 0 K. H-N-R Corrrlatlon =1250 K, H-N-RCorrrlallon S. 1 800 K, H-N-RConrlrllon 7 . 1 = 1 0 K, Modlflrd Corntailon

--- -.

5.56 4.90 4.23 3.57 2.91 2.25 1.59

0.92 0.0 2.7 5.4 8.1 10.7 13.4 16.1 18.8 21.5 24.2 26.8

Gas Hydrodynamic Diffusion Time FIG. 25. Nusselt number versus time for different ambient temperatures: numerical results and correlations. Reprinted with permission from Chiang er al. [44].

-

required for the flow field to relax from the initially impulsive motion is A t 2 R/U, 0.4 diffusion times, and during this period, the drag coefficient falls rapidly, with steeper gradients at higher ambient temperatures. Subsequently C, increases as a result of a reduction in the Reynolds number. The drag results reported by Chiang et al. differ by as much as 20% from the predictions of correlations described by Haywood et al. Figure 25 shows the average Nusselt numbers at three different ambient temperatures and comparisons with the predictions of Haywood et al. (H-N-R correlations). Again there is deviation between the two predictions although the discrepancy becomes smaller at later times when the surface temperature approaches the wet-bulb temperature. Similar trends hold for the variations in the Sherwood number. Correlations of the numerical results are expressed in Chiang and Sirignano [45]and are as follows:

-

CD(l + B,,f)0'27 = 24.432 Nu,( 1 + BHf)0'67x = 1.275 Re:438 Sh (1

+ B,

)".568 =

( 127) f

'

1.224 Re:385 Sc0.492 f '

DIRECT - CONTACT TRANSFER TO LIQUID DROPLETS where 0.4 I B , I 13,0.2 IB,, I 6.5,25 I Re,,, I 200,0.7 and 0.4 2 Sc, I 2.2.

77

I Prf I1.0,

6 . Evaporation of a Multicomponent Droplet

Multicomponent droplet vaporization is governed not only by component volatility, but also by the rate of species diffusion and droplet surface regression, as well as internal circulation. Liquid-phase mass diffusion is commonly much slower than the liquid-phase heat diffusion so that thin diffusion layers can occur near the surface especially at high ambient temperatures where the surface regression rate is large. This is the reason for the dominant role played by liquid-phase mass diffusion. As noted by Law and Law [1201, with a multicomponent droplet, after the gasification is initiated, the concentration of the volatile component in the surface layer is rapidly reduced, although its value in the inner core is hardly affected because of diffusional resistance. Diffusion becomes efficient only toward the end of the droplet lifetime, at which the droplet size and thereby diffusion distance are sufficiently reduced. This leads to a rapid depletion of the more volatile component from the droplet interior. Because ?e for mass transfer, which is a ratio of the gasification rate constant to the characteristic liquid mass diffusivity, governs the effectiveness of liquid-phase mass diffusional resistance, large and small values of this parameter, respectively, indicate diffusion- and volatility-dominated gasification. The presence of volatile components in the drop may also lead to a microexplosion or droplet fragmentation [116, 118, 240, 2501. The occurrence of a microexplosion is succinctly explained in Law and Law [ 1201 as follows: As the droplet surface concentration layer is being established after initiation of gasification, the droplet surface temperature also increases because of the increase in the surface concentration of the less volatile component. The extent of increase is mostly controlled by the boiling point of the less volatile component. Because liquid-phase thermal diffusion is much faster than mass diffusion, the entire droplet heats up fairly rapidly. However, because the droplet inner core retains a high concentration of the more volatile component because of diffusional resistance, the inner core can be heated to the state at which homogeneous nucleation is initiated. The subsequent rapid internal gasification leads to the violent rupturing. The formulation of a multicomponent droplet vaporization problem requires, in addition to flow field descriptions, the simultaneous solutions of liquid-phase species continuity equations, multicomponent phase equilibrium relations (typically Raoult’s law), and gas-phase multicomponent energy and species continuity equations. The spherically symmetric vaporization of a heptane-octane droplet in air ( p , = 1 atm, T, = 2300 K, and

78


Le = 18) has been numerically examined by Landis and Mills [1141. During the early transient, the more volatile heptane is preferentially vaporized. Beyond the initial period, evaporation becomes essentially quasisteady, and both components evaporate at a rate nearly proportional to their initial concentrations. The effect of liquid-phase on the sphericosymmetric evaporation of a decane-dodecane droplet has been studied by 1, concentrations within the droplet remained nearly Law [118]. For spatially uniform throughout the droplet lifetime, and vaporization approached the limit of batch distillation. For G=10, evaporation was a transient process throughout. The core concentration of decane slowly 30, decreased, and the surface concentration was varying with time. At results were similar to those of Landis and Mills. The variations in average liquid-phase species concentration in two-component droplets vaporizing at Re = 0 have been examined by Randolph et al. [174]. Fuel component concentrations were noted to vary throughout the lifetime of the droplet, demonstrating the influence of diffusional resistance. As noted in an earlier section, the convective evaporation of decanehexadecane and hexane-decane binary droplets at Re 200, 10 atm, and 1000 K has been studied through a boundary layer formulation by LaraUrbaneja and Sirignano [1151. Substantial temperature and mass fraction gradients are noted to exist in the droplet in spite of vigorous internal circulation. The mass fraction at the internal vortex center remains near its initial value for the entire lifetime. Mass transport from this location to the droplet surface occurs as diffusion across vortex streamlines of uniform concentration. Tong and Sirignano [232, 2331 have reexamined the problem using simplified models (a liquid-phase vortex model with a onedimensional gas-phase model) discussed in an earlier section. The results obtained lie between those predicted by the rapid-mixing and pure-diffusion models. Megaridis and Sirignano [137] have examined the effects of volatility differential by comparing the vaporization rates of a 50% decane-50% benzene (by mass) droplet and a 50% octane-50% benzene droplet. Preferential vaporization of benzene is noted to be more pronounced, and a microexplosion is indicated as a distinct possibility with the larger volatility differential decane-benzene droplet. The total drag coefficient C, for the droplet is predicted to first decrease as the Reynolds number decreases to about 65 from its initial value of 100, after which it begins to increase. The drag results also show very large reductions in drag (of more than 50%) as compared to a solid sphere at the same Re [136]. These drag characteristics differ significantly from the predictions of Renksizbulut and Bussmann [1801who have carried out a detailed numerical study of decane-hexadecane droplet vaporization in air at 1000 K and a

c=

c=

-


19

pressure of 10 atm. The results show that liquid heating persists for the entire lifetime. The total drag coefficient C , is predicted to increase with a decreasing Reynolds number. The numerical results show that the drag coefficient, Sherwood number, and Nusselt number correlations originally developed for single-component droplets by Renksizbulut and coworkers can be used for multicomponent droplets as well. The mass transfer correlation is more accurate in its predictions for the predominantly vaporizing fuel component. A semianalytical model of convective, multicomponent droplet vaporization suitable for spray programs is described by Renksizbulut et al. [181]. The liquid-phase heating is calculated from a Nusselt number model that takes into account heat transfer enhancement due to internal circulation. The gas-phase momentum and heat and mass transfer are calculated based on the drag coefficient and Nusselt and Sherwood number correlations. The model predictions are shown to be in close agreement with detailed fully numerical solutions. Lage et al. [112, 1131 have investigated multicomponent vaporization by means of the extended film concept introduced by Abramzon and Sirignano [4].In Lage et al. [113] a parametric analysis of the solution of constant-property boundary layer transport equations including the effects of blowing, surface tangential displacement, and the role played by the interdiffusion term in the energy equation has been carried out. The chief motivation has been to develop correlations for multicomponent droplet evaporation that span a wide range of transfer numbers. The limitations of boundary layer analysis, inability to accommodate wake effects, and the consequent error in the transport estimates are stated. In Lage et al. [112] a vaporization model that takes into account variable properties, nonideal phase equilibrium, and the interdiffusion term is described. The nonideal phase equilibrium feature is to accommodate nonideal liquid mixtures. The model does not take liquid circulation into account and, hence, is applicable for stagnant droplets. The numerical procedure is based on the control-volume method of lines. Results show that the neglect of interdiffusion terms leads to small errors in maximum surface droplet temperature and concentration, but considerable error in the prediction of droplet lifetime could occur. 1. Effect of Droplet Shape Deformation on Euaporation

Haywood et al. [87] point out, on the basis of usual estimates, that a typical droplet in a hydrocarbon spray could have an initial R e of 114 and an initial We of 2. Droplets at this We level are likely to be significantly deformed. Haywood et al. [86] have developed a finite-volume numerical method that uses a nonorthogonal adaptive grid to examine both steady

80


deformed and transient deforming droplet behavior. Computations of the steady-state evaporation of n-heptane droplets in high-temperature air (T, = 1000 K, 10 IRe I 100, We I10) show deformed oblate shapes with major axes perpendicular to the mean flow direction. In the numerical study, liquid-phase motion and heating have been ignored. This is difficult to justify, although the effects of surfactants in real situations are likely to lessen the role played by the liquid-phase motion. But then, the theory may have to be developed to account for the presence of surfaceactive agents [92]. Allowing the droplet to deform, and yet preventing liquid motion, has necessitated the re-setting of liquid-phase velocities to zero at the end of each numerical iteration sequence. Droplets are generally noted to become more oblate with increasing We and decreasing Re, although this trend may not hold at high We, say, 10. The results show that the transfer rates of evaporating droplets are insensitive to changes in shape due to the surface blowing effect of vaporization. Quasisteady correlations for Nu and Sh given in Eqs. (123) and (125) based on a volume equivalent diameter are shown to predict adequately transport rates for steady deformed droplets (Fig. 26) (contrast with discussions in section 1II.E). The quasisteady drag correlation, based on the projected frontal area, is shown to predict adequately the drag (Fig. 27). In Fig. 27, C D $ Ris from Eq. (1221, based on the frontal area of an equivalent sphere. Note that the development of Eqs. (1221, (1231, and (125) is

-

3

FIG. 26. Nusselt and Sherwood numbers. Numerical results: Squares, We, = 1; circles, We, = 2; triangles, We, = 5; pluses, We, = 1 0 solid line, quasisteady correlations. Reprinted with permission from Haywood er al. [86].


0

I

I

I

I

40

I

20

60

80

100

81

1: 0

FIG. 27. Drag coefficient. Numerical results: squares, We, = I ; circles, We, = 2; triangles, We, = 5; pluses, We, = 10; solid line, quasisteady correlation. Reprinted with permission from Haywood et af. [861.

8. Vaporization of a Slurry Fuel Droplet The review article [13] must be consulted for details about vaporization of a slurry fuel droplet. A slurry fuel is a mixture of solid fuel particles and one or more liquid fuels. Sakai and Saito [195] used a two-stage model to describe the combustion of a single slurry droplet. During the first stage, vaporization and combustion of the liquid phase and any volatiles that are present in the coal particles occur, resulting in a dispersed phase residue agglomerate, During the second stage, which constitutes the majority of the overall process time [13], combustion of the agglomerate occurs. (This model may not apply in many cases; see, for example, Antaki and Williams [15].) Addition of a small amount of water to a slurry droplet results in shattering of the droplet during the first stage, and the agglomerate fragments burn in lesser time (disruptive burning). The mechanism here is the rapid growth of water vapor bubbles that nucleate on the coal particle surface in the slurry droplet [121]. Disruption may also be induced during the period of water vaporization as noted by Yao and Manwani [248]. Antaki [141 analyzed the transient internal processes of heat conduction and liquid surface regression during the vaporization and combustion of a slurry droplet. The effects of internal circulation have not been included. In the model, the volume fraction of solid particles in the slurry droplet is taken to be large. During vaporization, the droplet is assumed to consist of

82


a porous shell with a constant outer radius equal to the initial droplet radius, which surrounds an inner sphere of solid particles and liquid. The surface of the inner sphere regresses with time due to liquid evaporation, causing the shell thickness to increase simultaneously. On complete vaporization of the liquid, agglomerate (porous sphere) remains, with a radius equal to the initial droplet radius. In the first part of the analysis, an exact solution for the transient temperature distribution in the droplet has been obtained subject to the applicability of an approximate expression for the increasing surface temperature. The second part of the analysis applies when the surface temperature has reached the liquid boiling temperature. In this part, singular perturbation-matched asymptotic techniques are used to obtain approximate solutions for the temperature profiles of the inner sphere and the regression velocity of its surface. The expansion parameter, E (called the “heating parameter”) is the ratio of energy required to raise the inner sphere to the liquid vaporization temperature, to that required for liquid vaporization and, in a sense, it is an approximate comparison of the two competing processes of internal heating and liquid vaporization. The expansion parameter depends on the liquid volume fraction and hence on the distribution of particles in the droplet. Here, the liquid volume fraction is assumed to be a constant (uniform distribution of particles). The results show that, for small values of the parameter, the decrease in the cube of the diameter of the inner sphere is approximately linear with time, and vaporization time is linearly proportional to both the liquid volume fraction and the square of the overall droplet diameter. For large values of the parameter, the vaporization time is proportional to the liquid volume fraction raised to an exponent less than unity and the square of the overall droplet diameter. Miyasaka and Law [138] and Chung [48] note that the internal circulation induced by shear at the surface can have a significant influence on the agglomeration of particles and disruptive burning. A model for the internal circulation of a nonvaporizing slurry droplet with a small volume fraction of solid particles is described by Chung [48]. The particles are assumed to be carried by the internal flow but not to modify it. The interaction between particles is modeled by the use of an effective viscosity that considers the volume occupied by each particle. Inclusion of vaporization and regression is likely to change significantly the residence times predicted by this model. As noted by Antaki and Williams [15], the theories for slurry droplet vaporization and burning that are available at present do not predict irregular burning or conditions for onset of gas eruption from the interior, mass ejection, or total droplet disruption. By providing interior temperature profiles as functions of time, however, they yield information needed for beginning investigations of the causes of these irregularities.


83

9. Evaporation of a Molten Metal Drop Evaporation of metal droplets is an important area of study because it has considerable application value for the industry. As noted by Bayazitoglu and Cerny [26], there is very little analytical work to report in this important area. Bayazitoglu and Cerny have formulated three nonequilibrium lumped parameter models to study the evaporation of molten metal drops moving through an inhomogeneous alternating magnetic field: (1) vacuum-like high velocity model (high velocity of ambient gas), (2) a diffusion model (no motion of ambient gas relative to the droplet), and (3) a quasisteady diffusion-convection model. Infinite thermal conductivity is assumed for the droplet. The results show significant differences in drop temperature predictions of the diffusion and vacuumlike high-velocity models ( 10oO K), but there is relatively small disagreement ( - 8%) in predicting the final size of the drop. The quasisteady diffusion-convection model provides results that essentially lie between the predictions of the first two models. It is stated that the efficiency of the evaporation process can be improved by ambient gas flow in a direction opposite to that of droplet translation. Considerable scope for further research exists in this important area of study.

-

I. DROPLET EVAPORATION AND CONDENSATION: EFFECTSOF AN ELECTRIC FIELD Nguyen and Chung [1441 have investigated evaporation/condensation of a spherical liquid drop slowly translating in an electric field by perturbation techniques. The paper appears to have a different definition for W [see Eq. (37)l. The total flow field is a combination of that generated by the slow translation of the drop, the flow induced by the presence of an electric field, and a strong nonuniform radial velocity (at least an order of magnitude larger than the other two fields). The formulation is similar to that of Sadhal and Ayyaswamy [192], but more complicated because of the presence of the additional force field. The outward radial velocity corresponds to evaporation and the inward represents condensation. The flow field formulation is solved by a regular perturbation technique in which the Reynolds number (based on the terminal velocity of the drop and the kinematic viscosity of the continuous phase) is used as the expansion parameter. The conservation equations of heat and mass transfer for the continuous and dispersed phases are solved by singular and regular perturbation procedures, respectively. The zeroth- and first-order governing equations are solved analytically. The overall solution is second-order accurate with respect to the perturbation variable. The results from sample calculations indicate that the electric field alters the local tempera-

84


ture and concentration profiles. The droplet internal flow is dominated by the electric field presence. The drag force remains unchanged by the uniform electric field. The flow outside the droplet is dominated by the interfacial mass flux and recirculation is noted only for an evaporating drop under the influence of a negative electric field. J. DROPLET INTERACTIONS The recent review article by Annamalai and Ryan [12] must be consulted for comprehensive and thorough discussions of droplet interactions. Sirignano [206] has suggested that droplet interactions may be viewed, for convenience in modeling, at three different levels. At the first level of interaction, the distance between the droplets is so large that the influence of neighbors is negligible. In this situation, the Nusselt number, Sherwood number, and drag coefficients may be obtained from isolated droplet studies. At the next level of interaction, droplets are closer, on average, to each other, such that the ambient conditions are modified, and the Nusselt number, Shenvood number, and drag coefficients are different from those for an isolated droplet. For example, the neighbor($ may be within the gas film or wake of the droplet. In a convective flow field, a droplet can influence a second droplet at substantial distances of many tens of droplet radii if the latter is in its wake. If the droplets are placed side by side in a convective situation, significant influence occurs only over short distances of a few droplet radii. The third level of interaction involves collisions whereby the liquids of the different droplets actually make contact with each other. Here, the droplets might coalesce into one droplet or emerge from the collision as two or more droplets. Sirignano [2081 and Annamalai and Ryan [121 classify interactive droplet studies into three categories: droplet arrays, droplet groups, and sprays. Arrays involve a few interacting droplets with ambient gaseous conditions specified. There are many droplets in a group but gaseous conditions far from the cloud are specified and are not coupled with the droplet calculations. The spray differs from the group in that the total gas field calculation in the domain is strongly coupled to the droplet calculation. In the spray, either the droplets penetrate to the boundaries of the gas or, while the droplets may not penetrate, the impact of the exchanges of mass, momentum, and energy extends throughout the gas. The early work on droplets that were interactive at the second level involved vaporizing droplet arrays without forced convection. However, results of recent evaporation experiments with binary drop arrays of dissimilar composition suspended from fine wires [246] indicate that d i f i sion analyses (neglecting convective effects) might overpredict the effects (intensity and persistency) of droplet interactions. Interactive buoyant

-


85

convection is noted to augment substantially, the droplet gasification rate over the isolated drop value. Choi and Lee [47] have described experimental studies on the dynamics and evaporation of tandem liquid droplets in a hot gas flow. The effects of droplet interaction in the streamwise direction and mass transfer on the drag coefficient have been studied. The experimental parameters tested include initial droplet size, spacing between the droplets, and liquid properties. When the initial droplet spacing ratio, S = ( L / d ) , is less than 4,where L is the distance between the droplets in the axial direction and d is the diameter, the following empirical equation is recommended for the drag coefficient: C,

=

2.3 Re-0.37,

( 130) for 20 5 Re I 110 and with the properties evaluated by the f rule. For closely spaced tandem droplets, that is, S I4, the vaporization rate or mass transfer from droplets was found to be relatively small, resulting in negligible effect of mass transfer on the drag coefficient and a lower value for the drag coefficient. Raju and Sirignano [173] have numerically studied two vaporizing droplets moving in tandem in an intermediate Reynolds number flow taking into account variable density effects. All other thermophysical properties are regarded as constants (this is a questionable assumption). The significant advancement made by this contribution beyond most other studies of tandem droplets lies in the analysis of transient behavior (including unequal regression rates of the two droplet surfaces and temporal variation in droplet spacing due to differences in droplet drag and mass), a fully coupled Navier-Stokes solution (allowing for complete coupling of internal liquid flow and gas flow and for complete elliptic behavior with upstream influence), and different initial sizes for the upstream and downstream droplets. The droplets are assumed to maintain spherical shapes. Results have been provided for initial Reynolds numbers of 50, 100, and 200, a limited range of droplet spacing, and droplet radii ratio. The most interesting result is that a critical ratio of the two initial droplet diameters exists below which droplet collision does not occur. If the ratio of the downstream droplet initial diameter to the upstream droplet initial diameter is larger than the critical ratio, the reduced drag coefficient of the downstream droplet causes less deceleration and greater relative velocity with the gas for the downstream droplet. Therefore, collision is likely since the spacing decreases with time. Below the initial ratio, the reduced inertia of the downstream droplet causes the spacing to increase. The critical ratio is found to be less than unity and very weakly dependent on the initial Reynolds number. The two-tandem-droplet calculation has been extended to account for variable thermophysical properties by

86


FIG.28. Surface temperatures, Nusselt number, and Sherwood number versus position on droplet surface for the two-tandem-droplet case. --- Surface temperature for the surface temperature for the lead droplet (2), downstream droplet (l), surface Shenvood number for the downstream droplet (31, --- surface Sherwood number for the lead droplet (4), surface Nusselt number for the downstream droplet (51, and surface Nusselt number for the lead droplet (6). Fuel = N-decane; ambient temperature = 1000 K, initial droplet temperature = 300 K, initial Reynolds number = 100, initial R1 = 1.00, R 2 = 1.00, and D = 8.0. Reprinted with permission from Chiang and Sirignano t451.

Chiang and Sirignano (4.51. Correlations of the numerical results for drag coefficient, Nusselt number, and Sherwood number with Reynolds number, transfer number, spacing, and radii ratio have been provided in this study (see Nguyen et al. [1471 for discussion). The local Nu and Sh variations at the droplet surfaces are shown in Fig. 28. The decrease of the Nusselt number with time at the rear part of the lead droplet and at the front stagnation region of the downstream droplet is ascribed to the approach of the downstream droplet. As the spacing is reduced, the heat exchange is lowered. The increase of Nu at the rear part of the downstream droplet is caused by the hot ambient gas entrained by the recirculating flow. The highly nonuniform distribution of the surface temperature of the downstream droplet is attributable to the slow liquid circulation.

DIRECT - CONTACTTRANSFER TO LIQUID DROPLETS

87

(3

dn

0.00

76.83

81.31

85.90

90.44

94.98

99.51

INSTANTANEOUS REYNOLDS NUMBER FIG.29. Drag coefficient versus instantaneous Reynolds number for two-tandem-droplet case. Reprinted with permission from Chiang and Sirignano [451.

The variations in the drag coefficients are shown in Fig. 29. The drag coefficient of the lead droplet drops by about 6% from its isolated drop value due to the interactions with the downstream droplet. The discrepancy increases as the droplet spacing decreases or with decreasing Re. The downstream droplet experiences lower drag because of the wake effect. The figure also shows that constant property calculations overestimate the drag coefficient, particularly for the lead droplet. Studies on group vaporization have been discussed in elaborate detail by Annamalai and Ryan [12]. A brief review of some spray studies is included here.

1. Droplet Spray Vaporization

In a spray the dispersed phase may be in the form of droplets or ligaments with a gaseous medium for the continuous phase. Here, we consider situations where the dispersed phase is in the form of droplets and almost entirely use numerical models. Turbulent spray analyses are

88


not included in the following discussion. A recent review article [1751 must be consulted for related heat transfer issues. In view of the enormous complexity, substantial simplifications are required to predict transport rates for spray droplets. The simplifications usually involve ignoring some of the processes or the coupling between them [198]. In this context, the use of LHF and SF models in spray combustion has been critically examined by Faeth [68]. Various types of spray calculations are of interest depending on the desired length scale of resolution, and the associated computational issues are very different [ 1781. Four different formulations of spray equations have been commonly used: 1. The discrete particle formulation [57,59, 2071 in which spray behavior is resolved on a scale finer than the spacing between neighboring droplets. The resolution is achieved by following each individual droplet and resolving the liquid field within and the gas field surrounding it. Computational considerations limit the use of this method to a small number of droplets and to a small volume of mixture. Because both the discrete and continuous phases are being resolved here, only liquid properties or only gas properties exist at a given point in space or time. 2. The two-continua or multicontinua formulation allows resolution only on a scale larger than the average spacing between neighboring droplets [57, 59, 2071. The equations are formulated so that each dependent variable at any spatial point is an instantaneous average value over a neighborhood of that point that includes both liquid and gas. Therefore, both liquid and gas properties exist at a point independent of whether that point is actually in a gas or in a liquid at that instant. When droplets are divided into many subclasses according to initial values of velocity, position, diameter, and/or composition, the two-continua approach is expanded to a multicontinua approach 12061. The gas-phase equations are typically solved by finite-difference methods on a Eulerian mesh that is either fixed or defined through some adaptive grid scheme. 3. The probabilistic or distribution function formulation can be used both for coarse and fine resolution and is especially useful in dealing with a spray containing a very large number of droplets [207, 2431. In this procedure, where coarse resolution is sufficient, both gas-phase and liquid-phase properties are averaged over a neighborhood containing many droplets. In the fine-resolution case, uncertainty in droplet position remains a factor, and the probability density function describes this uncertainty. There will also be an uncertainty in the


89

gas-phase values here because of the coupling between the phases. For very dilute sprays, the uncertainty in the gas-phase properties can be neglected. In denser sprays, these uncertainties in the gasphase properties will appear in high-resolution practical spray problems [S]. 4. The maximum entropy formalism [198, 1991. The required physical constraints come from basic conservation laws applied on an integral or global basis. In all of the preceding formulations, either Lagrangian or Eulerian or both methods have been used, depending on the desired length scale of resolution and computational accuracy (control of numerical errors). A Eulerian description of the two-continua formulation is provided in Seth et al. [2001 and by Aggarwal and Sirignano [9] in the context of one-dimensional analyses of flames propagating through fuel sprays. The axisymmetric two-continua problem using the Lagrangian method has been investigated by Aggarwal et al. [7]. A discrete particle Lagrangian methodology has been used to study one-dimensional, planar unsteady spray configurations by Aggarwal et al. [lo]. Various subgrid vaporization models have been examined and it is noted that substantial global differences in the two-phase flow can result from different vaporization models. In Aggarwal [6], a simplified spray environment is used to represent multicomponent vaporization. Liquid-phase models are the diffusion limit, the infinite diffusion, and the vortex. Gas-phase models are the Ranz-Marshall correlation, and the axisymmetric model. The physical situation considered is that of a dilute spray in a hot convective environment. Three subsets of equations are solved by a hybrid Eulerian-Lagrangian, explicit-implicit scheme. Results show that the internal circulation is less important for the multicomponent case compared to the single component. Based on results obtained here, for most dilute spray situations, the diffision-limit model is recommended. Multicomponent liquid sprays in one-dimensional, planar, and spherically symmetric configurations have been investigated by Continillo and Sirignano [56]. The general conclusions of the one-dimensional planar, two-dimensional planar, and spherically symmetric studies are that (a) resolution on the scale of the spacing between droplet is important in the determination of ignition phenomena and flame structure, (b) more than one flame zone can exist at any instant with spray combustion, (c) an inherent unsteadiness results for the flame structure, (d) diffusion-like or premixed flames can occur, (el ignition and flame propagation for sprays can sometimes be faster than for gaseous mixtures of identical stoichimetry, and (f) droplets tend to burn in a cloud or group rather than individually.

90


IV. Conclusions Contributions to literature that deal with direct-contact transfer processes have been discussed with a view to highlight advances and progress made in the recent past. Critical comments and personal views have been expressed where appropriate. The emphasis in the review has been on the physics, and deliberately not on the equally important mathematical, numerical, application, or design aspects. As stated earlier, several important topics have not been included in this review; this decision was made on the basis of the author’s own extent of understanding of many areas in this vast field. The review reveals that although significant progress has been made during the past decade, many new questions have been raised and some old ones remain unresolved. For example, with droplet evaporation, the many research groups that regularly contribute to the literature do not seem to agree on such a fundamental issue as the variation of drag force affecting droplet motion. Lack of experimental results for isolated droplet vaporization at high transfer numbers has been a definite hindrance. The situation with multicomponent droplets is of even greater concern. Detailed and comprehensive studies (analysis and experimentation) of electrical effects are lacking. Considerable scope for study exists with regard to phase-change problems associated with molten metal droplets. The topic of fuel droplet spray flow still requires more research. Droplet distortion, possible break-up into smaller droplets, and the accurate determination of the liquid-gas interface are important areas that have remained largely unexplored from analytical/numerical viewpoints. Droplet vaporization problems at near-critical and supercritical pressures (particularly the effects of surface tension and ambient gas dissolution) need extensive study. Condensation on a spectrum of droplets of various sizes or on a spray of droplets remains essentially unexplored. There are no systematic studies in this area to report. The effect of surfactants in condensation and evaporation needs thorough investigation. Analytical/numerical studies of compound drops are at the initial stages. Very extensive studies are needed in this important area. The state-of-the art with slurry droplets is not much different.

Acknowledgments The author is very grateful to Professors W. A. Sirignano (WCI), C. K. Law (Princeton), S. S. Sadhal (USC), M. Renksizbulut (U. Waterloo), J. N. Chung (Washington State UJ, D. L. R. Oliver (U. Toledo), G . Gogos (U. Nebraska), and M. A. Jog (U. Cincinnati) for very

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91

many helpful discussions, for providing original ink drawings, and for granting permission for inclusion in this article. Mrs. Julie McNamara is gratefully acknowledged for her assistance in the production of this manuscript. The sustained and continued support of the author’s research in the specific area of direct-contact heat /mass transfer and in closely related areas by the U.S. National Science Foundation during the past 16 years is gratefully acknowledged.

Nomenclature Pe a

A A0

A MI

constant of curvature in stagnation point flow strength of Hill’s vortex radial velocity in the absence of translation radial Reynolds number (= A,,R/V)

B c C CD

CII

Ca d

D 8

Eo

f

F Fo

g

Gr

transfer number specific heat at constant pressure drag coefficient Total drag coefficient Gegenbauer polynomial of order n capillary number ( = p U m / u ) droplet diameter mass diffusivity, droplet center-tocenter spacing Damkohler number Eotvos number (= g A p d 2 / o ) Blasius function drag force, electric field strength Fourier number ( = at / R 2 ) acceleration due to gravity, ShvabZeldovich variable Grashof number [ = R3g(T, - T=)/

Tk: 1 h

i

Ja k

P Le m m ml

M Nu P pn

heat transfer coefficient, enthalpy, Shvab-Zeldovich variable, distance between droplet centers unit vector Jakob number [ = c(T, - T , ) / A ] thermal conductivity differential operator Lewis number ( = (Y / D ) droplet mass, mass fraction droplet mass vaporization rate noncondensable mass fraction Morton number ( = g p 4 A p / p Z o 3 ) Nusselt number (= h d / k ) pressure Legendre polynomial of order n

Pr 9

Q r

‘r

R R,

a

Re S

sc Sh I I‘

T 11

u, V V W

Wl

W

Peclet number ( = Ud / a or Ud/D) Prandtl number (= v/a) heat flux energy per unit mass radial coordinate flame radius instantaneous radius of the drop radius of the outer boundary gas constant Reynolds number (= U , d / u ) strength of stokeslet, total amount of surfactant, source term Schmidt number ( = v / D ) Sherwood number ( = dh, / p D ) time dimensionless time (= tcu / R 2 ) temperature velocity component free stream velocity velocity, volume given by Eq. (29) in electric effects dimensionless mass fraction ( = m - m,), distance between drops normalized mass fraction condensation parameter ( = 1 m I, m / m

W W,

We x, Y, z

Y

I,

.J

given by Eq. (37) in electric effects given by Eq. (43) in electric effects Weber number ( = I I * d p / o ) spatial coordinates mass fraction of the evaporating species

GREEKSYMBOLS a

P

thermal diffusivity instantaneous angle of drop trajectory, Shvab-Zeldovich variables


92

Euler-Mascheroni constant boundary layer thickness, small parameter ( = T ~ / o ) density difference translational Re, dielectric constant, heating parameter Blasius coordinate polar angle dimensionless temperature, activation energy surface shear dilatation viscosity latent heat dynamic viscosity kinematic viscosity, stoichiometric coefficient spatial coordinates density radial distance from the axis of symmetry surface tension, electrical resistivity dimensionless time ( = t a / R 2 ) , stress droplet heating time droplet lifetime azimuthal angle, normalized stream function ratio of dynamic viscosities (= f i / / I )

ratio of electrical resistivities

X

ratio of effective thermal diffusivity to thermal diffusivity stream function

(=

4

au b C

e

f

F h H i m

M 0

0 P P r S

U

wb 0 1

k/k)

ratio of thermal diffusivities ( =

6/u)

SUBSCRIPTS

e

ratio of densities ( = p ^ / p ) ratio yf thermal conductivities (=

$,

W

average bulk condensation evaporation, edge of boundary layer friction fuel horizontal heat transfer index for species component mean mass transfer creeping flow condition oxygen pressure product radial direction at the droplet surface vertical wet bulb at angular position at initial time noncondensable far stream

&/a)

ratio of specific heats (= t / c ) ratio of dielectric constants (= ;/El

SUPERSCRIPTS

-

ratio of kinematic viscosities (=

S/v)

*

dispersed phase average dimensionless

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ADVANCES IN HEAT TRANSFER. VOLUME 26

Single-Phase Liquid Jet Impingement Heat Transfer

B. W. WEBB Department of Mechanical Engineering Brigham Young Unioersity Provo, Utah

C.-F. MA Department of Thermal Science and Engineering Beijitig Polytechnic University Beijing 100022, China

1. Introduction

A. RELEVANCE

Impinging liquid jets have been demonstrated to be an effective means

of providing high heat/mass transfer rates in industrial transport processes. When a liquid jet strikes a surface, thin hydrodynamic and thermal boundary layers form in the region directly beneath due to the jet deceleration and the resulting increase in pressure. The flow is then forced to accelerate in a direction parallel to the target surface in what is termed the wall jet or parallel flow zone. The thickness of the hydrodynamic and thermal boundary layers in the stagnation region may be of the order of tens of micrometers. Consequently, very high heat/mass transfer coefficients exist in the stagnation zone directly under the jet. Transport coefficients characteristic of parallel flow prevail in the wall jet region. The high heat transfer coefficients make liquid jet impingement an attractive cooling option where high heat fluxes are the norm. Some industrial applications include the thermal treatment of metals [ 1-31, cooling of internal combustion engines [4], and more recently, thermal control of high-heat-dissipation electronic devices [5]. Both circular and planar liquid jets have attracted research attention. 105

Copyright 0 1995 by Academic Press, Inc All rights of reproduction in any form reserved

106

B. W. WEBBAND C.- F. MA

A tremendous body of technical literature exists that deals with the transport characteristics of gas jets. Exhaustive reviews summarize the work [6-81. Only recently has the research community begun to assess the literature dealing with liquid jet impingement heat and mass transfer [9-121. Recent work has contributed to the fundamental understanding of heat, mass, and momentum transport in liquid jet impingement applications. The purpose of this review is to compile and summarize the available analytical and experimental work in the area with the objective of correlating the research findings. This work is confined to single-phase flow and heat transfer. The reader is referred to the recent review by Wolf et al. [9] for investigations treating liquid jet impingement boiling. Every effort has been made to acquire and summarize all relevant literature.

B. FLOWCONFIGURATIONS Liquid jets may be configured in a variety of ways. In a submerged jet configuration the fluid exits a nozzle or orifice into a body of surrounding fluid that is most often the same as the jet itself. Submerged gas jets have been reviewed extensively by Martin [7]. As the jet exits the nozzle and proceeds toward the target surface, a region forms, termed the potential core, where the jet velocity remains largely unaffected by the stagnant ambient fluid. The potentiaI core is typically five to eight nozzle diameters (or widths) long, and a region of high turbulence exists at its end. Beyond the end of the potential core the centerline velocity of the jet is reduced. Submerged jets thus entrain surrounding fluid, which may be at a different temperature. Vertical confinement of the submerged jet may also be important and influence the local transfer if the jet is formed by an orifice plate, which bounds the flow. Generally speaking, gravitational effects are unimportant in submerged jets. Free-surface jets result when a liquid issues from a nozzle or orifice into a gas environment. Entrainment of surrounding fluid is therefore negligible. The free surface forms immediately at the nozzle exit and prevails through the impingement region and into the wall jet region. The shape of the free surface is governed by a balance of gravity, surface tension, and pressure forces. The jet speed, size, and orientation determine the relative strengths of these forces. The influence of gravity may be far more important in free-surface liquid jets than in their submerged jet counterparts. As the free-surface jet strikes the target surface, the flow stagnates and flows outward in a thin liquid layer. At some downstream location the liquid layer may experience a hydraulic jump, with associated sudden deceleration of the fluid and consequent degradation in transport charac-

-

SINGLE PHASE LIQUID JETIMPINGEMENT

107

teristics. Preimpingement jet destabilization and breakup and/or jet splattering may also be present in free-surface liquid jets. Liquid jets find use in both axisymmetric and planar configurations. Both configurations share the common feature of a very small stagnation zone at the impingement surface whose size is of the order of the jet dimension, with the subsequent formation of a wall jet region. However, axisymmetric and planar jets differ structurally in the wall jet region. In free-surface jet configurations, the axisymmetric jet experiences a deceleration of fluid velocity in the parallel flow zone, whereas the planar jet does not. Both are affected by viscous shear in submerged jet applications. Liquid jets may be oriented normal or oblique to the target surface. Oblique impingement obviously affects the hydrodynamic distribution of flow and consequently the heat/mass transfer. Both axisymmetric and planar jets may be configured in arrays in an attempt to achieve the high transport characteristic of the stagnation zone over a larger area. These arrays are operated with either submerged or free-surface jets. Axisymmetric jets are usually arranged in either repeating square or triangular modules, while the planar jet arrays are most typically configured in rows. Prior studies of gas jet arrays have documented the critical nature of the exhaust flow in such arrays; lateral confinement of the spent fluid with resulting crossflow effects results in diminished momentum of the impinging jets downstream with associated degradation in heat transfer. This review summarizes the work in both axisymmetric and planar jets operated in both submerged and free-surface configurations. Theoretical and experimental approaches are treated. 11. Theoretical Considerations This section surveys the analytical research in the area of liquid jet impingement. The summary will be confined to free-surface liquid jets, since the theory for confined jets has been presented in some detail elsewhere [7]. The objective of these analytical studies has been the prediction of the local heat, mass, and momentum transport characteristics in both the stagnation and wall jet zones. Consider a free-surface liquid jet illustrated schematically for both axisymmetric and planar jets in Fig. 1. The flow impinges either normally or at an angle of incidence to the impingement surface. The preimpingement jet velocity can be either uniform or nonuniform (e.g., parabolic) at average jet velocity oj. The velocity and temperature distributions in the jet cross section are known upstream of the stagnation point. At the

B. W. WEBBAND C.- F. %i a Ti

J vi radial flow zone

free

stagnation zone

surface

rad:gow

(\\\\\I\\,,,,,,\\\\\\\,\,\\\\\,

ZO

Y

4sta;gn;ion X

\\\\\\\\\\\\\\\

parallel flow zone

parallel flow zone

FIG. 1. Schematic of free-surface liquid jet: (a) axisymmetric jets and (b) planar jets.

stagnation point a laminar boundary layer will form, growing in the wall jet region, and ultimately reaching the free surface. Outside the hydrodynamic boundary layer lies the inviscid region wherein the effects of viscosity are negligible. Generally speaking, the hydrodynamic and thermal boundary layers are unlikely to be of equal thickness, and will therefore encompass the entire liquid layer in the wall jet region at different locations. Ultimately the boundary layer will experience a transition to turbulent flow and transport. Both circular and planar jets may experience a hydraulic jump, depending on drainage conditions. Transport in the stagnation zone is one of a class of classical stagnation flow problems, and predictive methodologies draw heavily from prior stagnation flow analyses. The free-surface character of the liquid jet studied, .however, presents a complication: The free-surface profile itself is unknown and must be determined as part of the overall solution. Transport in the wall jet region has traditionally been treated using conventional

SINGLE -PHASE LIQUIDJET IMPINGEMENT

109

Khrmhn-Polhausen integral techniques with good accuracy (when compared to experimental data). A. LAMINARAXISYMMETRIC JETS 1. Stagnation Zone

The analytical treatment of the flow and heat transfer in the stagnation zone of axisymmetric jets is a subset of classical stagnation flow where the viscous boundary layer flow is matched to the inviscid flow at the edge of the boundary layer. Early works include those of Homann [13], Schach [14, 151, Sibulkin [16], Shen [17], and Strand 1181. More recent treatments include those of Yonehara and Ito [191, Ma et al. [20], Wang et al. [21], Liu et al. [22, 231, and Nakoryakov et al. 1241. The preimpingement jet condition for these studies was one of uniform inlet velocity. Of particular importance in the solution of the inviscid region is the magnitude of the dimensionless radial velocity gradient at the stagnation point, defined here for axisymmetric jets as

The velocity gradient will be shown hereafter to affect directly the magnitude of the stagnation point heat/mass transfer. The velocity gradient is often determined experimentally from static pressure measurements beneath the liquid jet from the Bernoulli relation P 2

P ( r ) = q + -[u;

-

U'(r)],

where P ( r ) is measured and U r ) is the velocity in the inviscid free stream, determined from Eq.(2). One of the more noteworthy of the recent analytical treatments of the inviscid region is that of Liu ef al. [23], wherein the velocity potential was expanded in a series of Legendre polynomials including the influence of surface tension. The Legendre coefficients were determined numerically. The predicted variation of the radial velocity gradient and the velocity component parallel to the jet axis are shown in Fig. 2 for the case of negligible surface tension effects, We, + m. The axial velocity drops from the jet magnitude near y/d = 0.5, stagnating at the impingement surface. The radial velocity gradient in the free stream is a maximum at the surface and vanishes for y/d + 0.5. The predicted variation of the free stream velocity gradient at the impingement plate is shown in Fig. 3 as a function of jet Weber number. Note that the result of surface tension is to increase

B. W. WEBBAND C.-F. l l h

110

I .6

I .2

0.8 0.4

0

FIG. 2. Predicted profiles of the vertical component of velocity and radial velocity gradient for the case of negligible surface tension forces, We, + m. Reprinted with permission of ASME from Liu et al. [231.

the velocity gradient in the inviscid outer flow, thereby increasing the transport in the stagnation region. However, the result is small. The CQ (negligible surface tension effects) is G = asymptotic result for We, 0.916. The inviscid outer flow velocity distribution is necessary for matching velocity conditions at the edge of the viscous boundary layer. The prediction of Liu et al. for the inviscid region also showed that the velocity distribution was well approximated by a linear distribution --f

U

Gr d

_ = _

uj

(3)

Equation (3) has been written for negligible jet contraction effects. Otherwise, uj must be corrected according to Bernoulli's formula. In contrast to the uniform velocity jet of Liu et al., Scholtz and Trass [25] reported predictions for a preimpingement jet with parabolic velocity distribution. Such would be the character of jets issuing from long nozzles at jet Reynolds numbers in the laminar range of Re, < 2500. Inviscid solutions were presented for nozzle-to-plate spacings of 0.05 < zo/d < 0.5. The inviscid flow radial velocity gradient for the parabolic preimpingement jet profile was found to be G = 4.644. This result was found to be rather insensitive to nozzle-to-plate spacing in the range 0.05 < zo/d < 0.5. Note that the value of the velocity gradient for the parabolic velocity jet is more than fourfold higher than the velocity gradient of the uniform velocity jet. As will be shown, this has the effect of a rather dramatic increase in stagnation heat transfer. Also of note was the fact that the parabolic inlet

SINGLE -PHASE LIQUIDJET IMPINGEMENT

111

2G

FIG. 3. Predicted variation of the inviscid velocity gradient with jet Weber number. Reprinted with permission of ASME from Liu et al. [23].

velocity jet yielded a considerably smaller stagnation zone than that resulting from a uniform jet. Once the inviscid flow region has been characterized, attention may be turned to the boundary layer flow in the stagnation zone. The flow and heat transfer in this region are governed by the continuity, momentum, and energy equations: 1 a(ru) au -+--0, r ar ay au au u- + uar ay

=

(4)

dU dr

u-

+ u a2u 2 , ay

dT aT a2T + u- =a?. ar ay ay The stagnation region of axisymmetric jets is a special case of a Falkneru-

Skan flow [261 after application of Mangler's transformation [27]. The solution proceeds with the definition of a stream function I) and similarity coordinate 7 r JI = - t y v r U ) " 2 F ( q ) (7)

\/z

112


and

v=y

c 2-

where F ( 7 ) is the solution of the transformed momentum equation, Eq.

(51,

F”‘ + FF” = p( F f 2 - 1) (9) for the special case of p = 1/2. The energy equation displays similarity for either isoflw or isothermal wall boundary conditions. In this case the temperature distribution is expressed as

(T’ - T ) = (L- T,)%). ( 10) On substitution of Eq. (10) into the energy equation, Eq. (61, the ordinary differential equation results: Off + Pr FB’ - y Pr(2 - p ) F ‘ e = 0, (11) where the parameter y describes the variation in wall temperature, (T’ - I;.> = Kry. For uniform wall temperature the solution to Eq. (11) is

The dimensionless heat transfer coefficient, the Nusselt number, is determined directly from the solution of Eq. (12) based on Fourier’s law:

Here, the local Reynolds number is defined as Re, = Ur/u. Combining Eqs. (3) and (13) reveals that the Nusselt number is approximately independent of r in the stagnation zone. In other words, the thermal boundary layer is of uniform thickness there. The preceding results for the stagnation zone therefore hold for either uniform wall temperature or uniform heat flux thermal boundary conditions imposed at the impingement plate. The local Nusselt number in the stagnation zone, which exhibits no dependence on r, may thus be expressed as

The dependence of the Nusselt number on fluid Prandtl number is embodied in the function 8’(0)lp,1/2, which can be integrated very accurately. For most engineering applications, however, simplified expressions

-

SINGLEPHASE LIQUIDJET IMPINGEMENT

113

for the dependence of Nud,o on Pr over restricted ranges are sufficient. The stagnation Nusselt number depends approximately on Pr" where n varies between 1/2 at small Pr to 1/3 at large Pr. The result of Eq. (14) indicates the strong dependence of the stagnation Nusselt number on both jet Reynolds number and free stream velocity gradient. Note that the two effects are independent. That is, the velocity gradient may be altered for the same jet Reynolds number and vice versa. The intimate dependence of Nusselt number on velocity gradient in the inviscid region has prompted considerable effort in its experimental measurement. This is explored in a later section. Based on the classical stagnation flow solution, Liu el al. [22] summarize theoretical results for the stagnation zone Nusselt number (for a uniform velocity preimpingement jet) as

NUd,o =

(

0.15 < Pr < 3

0.715 0.797

Pr'l3

Pr > 3

(15)

This result is very close to that presented by Ma et al. [201 for radially variable heat flux imposed at the heating plate: 0.7212~-".~ Re;/*

0.7 < Pr < 3

0 . 7 2 1 2 ~ - " Re'/' .~~ d

3 < Pr < 10

0.8597&-'13 Re:/* Pr1l3

Pr > 10.

(16)

The factor E in Eq. (16) is determined from the radial variation in wall heat flux q O ( r )as

Note that for a uniform heat flux boundary condition E = 1. The stagnation zone result of Yonehara and Ito [19] is higher than that of Eq. (15) by approximately lo%, but is close to the result of Eq. (16) for Pr > 10. Wang et al. [21] investigated analytically the effect of non-power-law variations in wall temperature using an asymptotic technique. The solution revealed a considerable influence of even small wall temperature or wall heat flux variations on the stagnation zone heat transfer coefficient, attributed largely to the significant effect of radial temperature gradients there. This analysis was extended for use with conjugate heat transfer in the impingement plate as well [28].

114

B. W. WEBBAND C.-F. MA

2. Radial Flow Region The radial flow region exists for r/d > 0.4 to 0.8, depending on the velocity distribution of the preimpingement jet. The KhrmBn-Polhausen integral technique is generally employed for determining the heat/mass transfer characteristics in this region for axisymmetric jets. The hydrodynamic and thermal fields are divided into four laminar flow zones. Including the stagnation zone, the regions may be described as follows (for Pr > 1):

Regwn Z ( r < 0.4 to 0.8d): The stagnation zone. Hydrodynamic and thermal boundary layer thicknesses are independent of radial location. Region ZZ (0.4 to 0.8d < r < rJ: In this region neither the hydrodynamic nor thermal boundary layer has reached the free surface. Region Zii (rt, < r < r,): The hydrodynamic boundary layer has reached the free surface, but the thermal boundary layer has not. Region ZV ( r > rt): Both hydrodynamic and thermal boundary layers have grown to encompass the entire liquid film. Note that Regions I11 and IV can be interchanged for fluids with Pr < 1. In this case the thermal boundary layer reaches the free surface of the liquid layer before the hydrodynamic boundary layer. In practical systems transition to turbulence may occur, invalidating the laminar flow analyses presented in this section. Transition to turbulence is treated in Sec. IV. Watson investigated the flow in the radial layer of an axisymmetric free-surface liquid jet [29]. His analysis extended radially to include the hydraulic jump. Results were presented for both laminar and turbulent flow. Chaudhury analyzed the heat transfer for the same problem [30]. More recently, integral solutions for the flow and heat transfer in the radial flow region have been presented by Carper [311, Liu et al. [22], and Liu and Lienhard [32]. Ma et al. [20] and Wang et al. [33] solved the problem for arbitrary heat flux variation with radius. The integral solution for uniform temperature and uniform heat flux impingement surfaces was presented by Yonehara and Ito [19] and Ito and Yonehara [34], respectively. Buyevich and Ustinov [35] formulated a model for predicting the location of the hydraulic jump as well as transport characteristics upstream and downstream. Results of the integral analyses for specified variation of wall heat flux or wall temperature are now summarized for the three radial flow zones prior to the transition to turbulence. a. Specified Variation of Wall Heat Flux, Pr > 1 Both Ma et al. [20] and Liu and Lienhard [32] present integral solutions for the local heat transfer

TABLE I COMPARISON OF ANALYSES FOR THE LOCAL NUSSELT NUMBER UNDER LAMINAR, FREE-SURFACE LIQUID JETS FOR Pr > 1

i

0 . 7 2 1 2 ~ - "Re:/' ~

1

Nu,

=

Pr".4

0.7212~-'-" Re:/*

Nu,

=

=

r/d
10

2[qo(r*)r'd'r' E =

NU^(

0.715 Re'/2 Pr".' 0.797R~z;/~Pr'13

< r,,/d

rt./d

=

[

Red

Pr

IPr

s3

>3

r/d

< 0.787

Nud = 0.632Re:/2 P r ' ' 3 ( r / d ) - ' / 2 0.787 < r / d < r , / d

0.1773ReY3

25.735 1 . 5 8 7 4 ~ - ' / ~ R e ;P / ~r ' / 3 -( r / d ) 3

0.15

el a/. [22, 321

q,(r)r'

0 . 6 6 8 ~ - ' Re:/* /~ Pr'/3(r/d)-'/2

1 25001, and 8.0 for transformer oil and ethylene glycol [61] at lower Reynolds number ( < 1000). The trend toward shortening of the potential core length with a decreasing Reynolds number was also reported for air jets by Hrycak ef al. [741. Figure 6 illustrates the


128 "I"

water jet

Red

0

300

+ 0 0

0

5

10

15

~

6000 8100 10,000 11,900 15,600 19.600

20

z, Id FIG. 5. Variation of the stagnation Nusselt number with nozzle-to-plate spacing for submerged axisymmetric water jets. Replotted with permission of ASME from Sun et al. [57].

t

water jet

Red

A 6000 0 8100 A 10,000 0 11,900 0

15,600

o

/

4

0.1 1

100

FIG. 6. Correlation of stagnation point Nusselt number with nozzle-to-plate spacing for submerged axisymmetric water jets. Replotted with permission of ASME from Sun et al. [57].

SINGLE -PHASE LIQUID JET IMPINGEMENT

129

good agreement between Eq. (40)and experimental data for axisymmetric submerged water jets 1571. At jet Reynolds numbers below Re, = 800, Elison and Webb observed that the Nusselt number may be independent of zo/d in the range 0I tJd s 80, due to destabilization of the initially laminar jet issuing into the stagnant environment [%I. The reduction in jet centerline velocity is compensated by an increase in turbulence, and the net effect is a near-constant stagnation Nusselt number with variations in nozzle-to-plate spacing. b. Effect of Reynolds Number The Reynolds number dependence on the Nusselt number has been experimentally studied with various working fluids. The power-law function of the Reynolds number with exponent 0.5 suggested by laminar stagnation flow theory [37] has been determined in experiments of local heat transfer with water [57], R-113 [%I, kerosene [60], and transformer oil and ethylene glycol [61]. An exception is the result of Rao and Trass [?ill, who observed an unusually high dependence on Reynolds number for water jets. The reason for this discrepancy was pointed out by Martin [7]: The mass transfer surface might be eroded by the water jets in the stagnation zone. The square root dependence of the Reynolds number displays the laminar characteristic of heat transfer in the stagnation zone where the strong favorable pressure gradient tends to laminarize the impinging jet, resulting in the formation of a laminar boundary layer in the stagnation region. For Re, < 800 the dependence of the stagnation heat transfer on the Reynolds number is higher, with exponents for Eq. (39) in the range 0.70 Im I 0.83 [55]. Again, this is due to the destabilization of the initially laminar jet issuing into a stagnant ambient fluid.

c. Effect of Prandtl Number Because the Prandtl number of liquids may be several orders of magnitude higher than that of air, a precise determination of the Prandtl number dependence is critical in the prediction of the impingement heat transfer with liquid jets. A power-law dependence on the Prandtl number with exponent 0.42 was proposed by Martin [7] based on a comparison of the mass transfer measurements with the data of heat transfer to air and to water. This value has been accepted for water [53-571, R-113 [%I, and FC-77 [53, 541. Examination of the Prandtl number dependence has been extended to high Prandtl number liquids using transformer oil and ethylene glycol as the test liquids [61, 751. It was reported that for large Prandtl number liquids an exponent value of 1/3 should be used in the Prandtl number dependence, consistent with the laminar stagnation flow theory of Sec. II.A.l. For nozzle-to-plate spacings

130


corresponding to the target surface held within the potential core, the data of five different liquids can be well correlated by Eq. (39) with C = 1.29, m = 0.5, and n = 0.4 for gases and n = 1/3 for liquids [57, 58, 60-61, 751. The range of validity of the correlation is Pr = 0.7 to 351, and Re, = 120 to 37,000. For liquids of Prandtl number not far from unity, the choice of exponent values of 0.4 or 1/3 makes little difference in the calculation of stagnation heat transfer. However, for larger Prandtl number liquids, differences in predicted stagnation zone heat transfer of 30% or more might result. Although validating experimental work is apparently nonexis1, the theoretical dependence of NU^,^ on Pr'/* tent for liquids of Pr as illustrated in Eq. (21) should hold. Using a high Prandtl number liquid as the working fluid, the effect of viscous dissipation in the heat transfer process could become significant. For air flows, the practical importance of the viscous dissipation effect arises only in the case of high velocities. However, for large Prandtl number liquids, this effect could be significant even at moderate velocities. In this case for calculation of local or average impingement heat transfer, the static jet temperature q should be replaced by the adiabatic wall temperature Taw,where u;

Taw= ?;. + r - . 2%

(41)

The parameter r is the recovery factor of the jet flow [61, 761. Ma et al. [611 measured the recovery factor at the stagnation point for submerged round jets of transformer oil, recommending a correlation of r = Values of the average recovery factor were experimentally determined by Metzger et al. 1761 with free-surface circular jets of lubricating oil, and a correlation of r = was proposed. Velocities of oil jets slightly higher than 20 m/sec were reached in the two investigations [61, 761, resulting in values of u;/2c, close to 0.1 K. However, the recovery factors for oil jets were determined to be relatively large, greater than 10. As a consequence, the difference between the static jet temperature and the adiabatic wall temperature may be greater than 1 K at velocities above 20 m/sec. Significant errors in the heat transfer calculation may therefore result from neglect of viscous dissipation effects for high Pr liquids. 2. Radial Flow Region The radial distribution of the local heat transfer coefficient has been experimentally measured and correlated for submerged circular jets of water [57], R-113 [%I, kerosene [60], and transformer oil and ethylene

SINGLE -PHASELIQUID JET IMPINGEMENT

131

glycol [61]. Empirical correlations have been developed for the variation of the local Nusselt number with dimensionless radial location. These correlations are now summarized. a. Correlation of Local Heat Transfer Based on a semiempirical model, two correlations have been presented for the near-stagnation zone ( r / d

2) for submerged liquid jets by Ma and Bergles [58,721 for R-113 submerged circular jets:

I1.69(r / d ) -'"'

lUud,o

for r/d > 2.

(43)

Both transformer oil and ethylene glycol can be correlated quite well by the two equations in the stagnation zone and the wall jet region, respectively. Using the correlation technique proposed by Churchill and Usagi [77], Sun et al. [57] presented a general correlation to cover water jet data both in the stagnation and wall jet zones: ;)-'.07]'] 1/p tanh0.'(0.88r/d)

1'

( r/d )

'"

+

[

(

1.69 -

(44)

The average scatter in the water jet data relative to Eq. (44) with P = - 17 is only 5% as shown in Fig. 7 [57]. Note that the Nusselt number 1.21

5.9x1103

1.o

U

0

0.8

0

3

9.9 x 1.6 x 2.1 x 2.4 x

103

I04 104 104

0.6 U

z3

0.4.

0.2. -2 0 2

4

6

8 10 12 14 16 18

:0

r/d FIG. 7. Correlation of the variation of the local Nusselt number for submerged axisymmetric water jets. Reprinted with permission of ASME from Sun et al. [57].

B. W.WEBBAND C.- F. MA

132

falls to 40% of its maximum value by r/d 10% beyond r/d c- 12.

c-

4.Further, Nu,/Nu,,,

is only

b. Transition to Turbulence Transition from laminar to turbulent flow has been observed with submerged water jets at z o / d = 2 [571. Examination of the local heat transfer distribution for Re, = 24,000 in Fig. 8 indicates that an inflection point appears in the profile at approximately r/d = 1.9. These data are for submerged turbulent jets issuing from fully developed pipe-type nozzles. For lower Reynolds numbers, similar but less conspicuous changes in the slope of the Nusselt number versus dimensionless radial distance profiles are observed in the same location. It is interesting to note that Gardon and Akfirat [73] observed similar peaks in local heat transfer profiles at exactly the same radial position of r/d = 1.9 for submerged circular turbulent air jets with a Reynolds number greater than 25,000. This phenomenon was also reported by Baughn and Shimizu [78], Hrycak [79], and Obot et al. [80] for circular air jets. Rice and Garimella [62] report that the effect of vertical confinement of submerged jets may result in increased magnitude of the secondary peaks. The peaks and the sharp knees in the profiles can be attributed to the transition from laminar to turbulent boundary layer flow. Despite the observance of transition in experimental work, there appears to be no quantitative

30C

-

I

1w

9.9 x 1.6 x 104 2.1 x 104 2.4 x 104

200

100

0 -2 0

2

4

6

8 10 12 14 16 18

:

FIG.8.- Radial distributions of the local Nusselt number for submerged axisymmetric water jets. Reprinted with permission of ASME from Sun et al. [57].

-

SINGLEPHASELIQUIDJET IMPINGEMENT

-2

0

2

4

6

8

10

12

133

14

rld FIG. 9. Radial profiles of the local Nusselt number for submerged axisymmetric jets of kerosene with impingement surfaces held beyond the end of the potential core. Replotted with permission from Ma et al. 1601.

generalization in the literature of the associated secondary peak locations or magnitudes.

c. Local Heat Transfer Beyond the Potential Core The foregoing discussion is related to target surfaces within the potential core (z,/d < 5 to 8). At nozzle-to-plate spacings beyond the potential core, the radial variation in local Nusselt number is reduced while the maximum heat transfer coefficients decrease as described by Eq. (40). This trend is illustrated in Fig. 9 with submerged circular kerosene jets [61]. Interestingly, the local Nusselt number merges for all nozzle-to-plate spacings at large r/d. d. Average Heat Transfer Measurements have been reported for the average heat transfer coefficient with submerged axisymmetric jets of various fluids. Based on their water jet data, Sitharamayya and Raju [561 developed a correlation by separating the target area into the impingement region (r/d < 4) and wall jet region (r/d > 4):

Nu,

=

[32.4Re:523

+ 0.266(D/d - 8)Reis28](D/d)-2

(45)

B. W. WEBBAND C.. F. MA

134

This correlation is valid for water jets in the range of Re, = 2000 to 40,000, dimensionless heated target diameters D/d > 8, and nozzle-toplate spacings zo/d < 7. Womac et al. [54] developed a correlation of average heat transfer for square chip-size heaters of side length L:

Nu, where A, given by

=

=

0.785RegS( ;L) A r

+ 0.0257Re:t

(1 - A r ) ,

(46)

dl.9d)*/L2 and the average length of the wall jet region is L*

=

0 . 5 [ ( 0 . 5 f i L - 1.9d)

+ (OSL - 1.9d)I.

(47)

Both Eqs. (45) and (46) are based on an area-weighted combination of correlations for the impingement and wall jet regions, with a stagnation flow dependence of Nu, in the impingement zone, and an Re: in the assumed turbulent boundary layer dependence of Nu,. area occupied by the wall jet region. The good agreement between Eq. (46) and experimental data shown in Fig. 10 attests to the validity of the correlation approach. All water and FC-77 jet data are represented by Eq. (46) with a standard deviation of 14% 1541. For wide ranges of parameters,

-

-

1 o3 Wafer

0.978

4 3

1.65 1.65

3

Water

3

3.11 3.11

4

FC-77 Water FC-77

o m

3

Fc-rr

3 PP.4 1oz

10' 0.785R~~~(L/d)A, +0.0257 RB?'(&

I( )(l-Ar)

FIG. 10. Correlation of average heat transfer data for submerged axisymmetric water and FC-77 jets impinging on a square heat source [see Eq. (4611. Reprinted with permission of ASME from Womac et al. [54].

SINGLE -PHASE LIQUID JETIMPINGEMENT

135

Martin [7] proposed an empirical correlation:

NU, =

2 - 4.4( d / D ) 1 + 0.2[ ( ~ , / d ) 61 ( d / D ) (d)F(Re,)Pro.4‘, D

(48)

where

The range of validity of Eq. (48) is Re, = 2000 to 400,000, z,/d = 2 to 12, and D/d = 5 to 15. Womac et al. [54]found that both their water and FC-77 data could be correlated by Eq. (48) in the range of validity, but the use of Martin’s correlation beyond its range of applicability resulted in large errors.

B. PLANAR JETS Average heat transfer measurements have been reported for submerged and confined planar jet cooling of discrete heat sources with application to electronic cooling using water [81] and FC-72 [82] as the working fluid. Wadsworth and Mudawar [82] correlated the average Nusselt number based on discrete heater length L as

-

-Nu, -

pr1I3

- 4.33 Re:”

+ 0.157 R e t W

(50)

to within k 10%. This correlation was found to underpredict the experimental data of Schafer et al. [81] by as much as 35%, the difference being attributed to variations in nozzle design and drainage configuration. An interesting feature of confined laminar planar jet impingement was the prediction of a secondary recirculation zone on the impingement surface just downstream of the stagnation region [83]. The location of the recirculation zone was seen to affect the local heat transfer downstream.

lV. Free-Surface Jets: Experimental Studies The theoretical aspects of jet impingement heat transfer presented in the foregoing illustrate the intimate coupling between the flow field and the transport at the impingement plate. It is therefore critical that the flow structure be well understood. The available theoretical results are based almost exclusively on laminar flow. The importance of turbulent transport in liquid jet systems is self-evident. Recourse is generally taken to experi-


136

mental methods to characterize the turbulent flow structure and heat/mass transfer. This section first summarizes investigations treating the flow structure of axisymmetric and planar liquid jets. No attempt is made to survey the body of literature dealing with the flow structure of submerged jet systems, which are quite well understood [6-81. Rather, emphasis is on the reported free-surface jet work. Characterization of these free-surface systems is difficult due to the existence of the free surface and its incompatibility with much of the flow structure instrumentation available. Experimental studies on the heat/mass transfer at the target surface under liquid jets have also been conducted to verify the laminar jet theoretical results presented previously and to characterize phenomena that are intractable analytically. These phenomena could include, for example, the influence of free stream turbulence on the transport and the transition to turbulent flow in the parallel flow region. The problem of characterizing heat/mass transfer under liquid jets is a challenging one owing to the extremely thin boundary layers. Experimental apparatus must be carefully instrumented so as not to disturb mechanically the viscous and thermal/solutal boundary layers. The heat transfer under an impinging liquid jet could be a function not only of the nozzle but also the supply tubing, etc., which can influence the levels of jet turbulence at the target surface. In some cases the fluid supply systems and even the nozzle configurations themselves have not been completely described in the published studies. The latter part of this section focuses on the experimental investigations of heat and mass transfer under free-surface liquid jets. A.

~ I S Y M M E T R I CJET

FLOWSTRUCTURE

Relatively few attempts have been made at determining experimentally the hydrodynamic characteristics of axisymmetric free-surface liquid jets. What studies are available focus on flow development in the preimpingement jet, turbulent flow structure measurements in the stagnation and radial flow regions, characterization of the radial variation of liquid film thickness, transition to turbulence, etc. Relevant experimental studies are summarized in what follows according to classification in three flow zones: the preimpingement jet, the stagnation zone, and the radial flow region. 1. Preimpingement Jet

For axisymmetricjets at low exit speed, gravitational acceleration results in jet contraction. If one neglects the effect of surface tension, the jet diameter can be predicted as a function of distance from the jet exit with

-

SINGLEPHASE LIQUIDJETIMPINGEMENT

137

reasonable accuracy as [84]

where d, is the diameter of the liquid jet at a distance z from the jet exit and Frz, is the Froude number based on the jet length z,. Surface tension effects in the preimpingement jet were shown to be negligible relative to gravitational forces for 1 70 [90]. Obviously this jet breakup results in deterioration of the heat transfer under the jet and is to be avoided. The development of a fully turbulent jet issuing from a pipe-type nozzle was characterized by Stevens and Webb by measuring the fluctuating free-surface velocity of the preimpingement jet using laser-Doppler veIocimetry [91]. The results show that the free-surface velocity reaches a value of 90% of the average jet exit velocity u j just three to four jet diameters from the nozzle exit. The development of the free-surface velocity for smaller jet diameter was apparently slower for the same jet exit Reynolds number. The rms fluctuations of the free-surface velocity fell almost immediately from a value near 20% of the jet exit velocity to a value of approximately 8%, and were nearly constant thereafter. These relatively high free-surface fluctuations may be the result of measurement in a fully turbulent jet where the surface structure is unsteady. 2. Stagnation Zone Measurements of the radial component of fluid velocity have been made by Stevens et a f . [921 using laser-Doppler velocimetry (LDV) in the stagnation zone of free-surface liquid jets. The liquid jet was configured to


138

I

1 .a

o

0.037

o

0,119 0.193

rn

A

0

0.75

0.307 0.514 0.606

0.50

u/v,

J

0.25

1.o Yfd

0.751

55

A

Fully developed

nozzle

0.293

A

0.0

0.25

n.n -.-

0

0.1

0.3

0.2

0.4

0.5

r/d FIG. 11. Measured mean radial velocity distributions in the stagnation zone of an axisyrnmetric, free-surface liquid jet issuing from a contoured orifice and pipe-type nozzle, respectively. Reprinted with permission of ASME from Stevens et al. [92].

strike a transparent plate through which the beams of the LDV system were directed. The LDV diagnostic volume was positioned within the stagnation zone using a multidirectional traversing table. Due to the extremely thin boundary layer, measurements were made almost exclusively in the inviscid free stream region. Measurements were made at a nozzle-to-plate spacing of z,/d = 1 using jets generated with sharp-edged orifice, contoured orifice, and fully developed pipe-type nozzles in the fully turbulent flow regime. Turbulence-modifying screens were also placed in the plenum upstream of the sharp-edged orifice to assess the impact of turbulence generating devices on the velocity field and level of turbulence in the exiting jet. Figure 11 illustrates profiles of the mean radial velocity component as a function of distance from the impingement plate for the contoured orifice and fully developed nozzle. Results for the sharp-edged orifice were qualitatively similar. The maximum radial velocity is found near the impingement surface for all nozzles. With the exception of regions near the edge of the jet (r/d = 0.51, the velocity was seen to vary linearly with r/d for all values of y / d . The gradient of mean radial velocity at the jet centerline was determined for positions relative to the impingement plate ( y / d > using a least-squares fit of the data. This gradient was seen to be approximately independent of Reynolds number for these

-

139

SINGLEPHASELIQUIDJETIMPINGEMENT 3.02.5

Red: 33,200 39,800 53,100

-

Sharp-edged orifice (no screens) Sharp-edged orifice

0

Fully developed

A

8

o

2.0

d(dv i )

d (r / d )

1.5

Contoured

0

1.o

0.5

0.0 -0.5 0.0

0.1

0.2

0.4

0.3

0.5

0.6

0.7

Y/d

FIG. 12. Experimentally determined mean radial velocity gradient as a function of distance from the impingement plate for various nozzle configurations. Reprinted with permission of ASME from Stevens et al. [92].

nozzles studied. The variation of the radial velocity gradient with distance from the impingement surface is seen in Fig. 12. Note the qualitative similarity between the experimentally measured dependence of the velocity gradient on y/dfor these turbulent jets and the laminar flow prediction of Liu et al. [23] seen in Fig. 2. The experimental data show clearly the influence of nozzle configuration on the free stream velocity gradient at the wall, G = d(u/~~)/d(r/d)l,,~,~.The gradient G varies by nearly a factor of 2 between the extreme cases, the lowest gradient exhibited by the contoured nozzle and the highest seen in the sharp-edged orifice data. Corresponding differences in heat transfer were also observed, and will be summarized later. The rms turbulence fluctuations of the radial velocity component corresponding to the mean flow data just described showed that the levels of turbulence were a strong function of distance from the impingement plate. The different nozzle configurations and turbulence modification techniques yielded only minor changes in the levels of turbulence in the stagnation zone. In general, levels of radial component rms turbulence normalized by the jet exit velocity varied from 2 to 3% for the contoured orifice, to 7 to 12% for the sharp-edged orifice without turbu-

140


d(mm) 2.1 4.6 7.6 A 10.9 0 14.0 0 23.0 0 14.0 ( ~ ~ l d - 2 ) A

14.0 (zO/d=4)

8000 5 ReA5 62,000

1

Y/d FIG. 13. Experimentally measured variation of mean radial velocity gradient in the stagnation zone with distance from the impingement plate for fully developed turbulent jets issuing from pipe-type nozzles. Reprinted with permission from Stevens et al. [93].

lence-modifying screens. A related study focused on the flow structure in the stagnation zone under jets issuing from a fully developed pipe-type nozzle configuration [93]. That study revealed that the mean radial velocity gradient is insensitive to jet diameter and nozzle-to-plate spacing for 2.1 I d I 23.0 mm, 8000 I Re, I 62,000, and z , / d I 4, as shown in Fig. 13. Further, the velocity gradient at the stagnation line (r/d = 0) varies linearly with y / d . The mean radial velocity in the stagnation zone (0 I r/d < 0.5 and 0 s y / d 5 0.5) is well represented by the function

f)](f )

U = [1.83 - 3 . ~ (

(53)

*j

from which the dimensionless velocity gradient may be expressed as a function of height from the target surface as

These data illustrate the vertical extent to which the impingement surface influences the preimpingement flow. Equation (54) indicates that the radial velocity gradient vanishes near y / d = 0.5, suggesting that the im-

-

SINGLEPHASELIQUID JET IMPINGEMENT

141

pingement surface begins to cause deceleration of the approaching jet flow approximately half the nozzle diameter away. This is consistent with the laminar flow predictions of Liu ef al. [23], illustrated in Fig. 2. Low Reynolds number jets displayed contraction due to gravitational acceleration, manifested by negative velocity gradients just downstream of the nozzle exit. The low Reynolds number jet data were observed to coincide with the nonaccelerating jet data if the jet velocity and diameter were corrected for gravity assuming jet acceleration to a point one jet diameter above the impingement plate.

3. Radial Flow Region

a. Liquid Layer Thickness Olsson and Turkdogan measured the liquid film thickness downstream of the stagnation zone of a free-surface axisymmetric liquid jet using a vernier height gauge and a needle probe [941. The free-surface velocity of the jet was also measured crudely with floating tracer particles and high-speed photography. Comparison with Watson's theoretical results [291 was inconclusive, owing largely to the relatively imprecise nature of the experimentation. The variation in free-surface velocity in the radial flow region of a fully turbulent jet issuing from a pipe-type nozzle was determined using LDV [91]. The liquid (water) was opacified with a very dilute mixture of milk to prevent contamination of the free-surface velocity data by optical penetration into the interior of the liquid layer. The measurements revealed a local maximum in the free-surface velocity near r/d = 2.5 that was found to exceed the average jet exit velocity by as much as 20% in some cases. The normalized mean free-surface velocity, Uo/uj, was correlated according to the form -0.125(r/d)* -0.0936(r/d)

+ 0 . 6 2 5 ( r / d ) + 0.303 + 1.33

0.5 I r / d 5 2.86 2.86 Ir / d I 14.

(553) (55b)

This expression was developed for Reynolds numbers in the range 17,000 < Re, < 47,000 and for nozzle diameters in the range 2.1 < d < 9.3 mm. The measured free-surface velocities compared rather poorly with the laminar and turbulent flow analyses of Watson [29]. A velocity profile in the liquid layer was assumed of the form u ( y / h ) . From this profile assumption and the measured free-surface velocity, the radial variation of the liquid layer thickness was determined from global continuity considerations at each radial

142

B. W. WEBBAND C.. F. MA

location. The expression for the local liquid layer thickness is

h = d

(‘i L)( i -) 1

8C

r/d

Uo/vj ’

where the constant C depends on the assumed form of the velocity profile through the liquid layer thickness. The value of the constant C was found to be relatively insensitive to the profile assumption for the laminar or the turbulent flow regimes. The variation of the liquid layer thickness with radial location calculated from Eqs. (55) and (56) under the assumption of the quadratic profile for which C = 0.667 compared moderately well with the predictions from Watson 1291 in the range r / d > 4. b. Radial Layer Flow Structure A comprehensive set of experimental data has been reported for laminar flow in the radial liquid layer issuing from the gap formed by placement of a tube very near a solid surface [95-100]. LDV measurements inside the layer, which was made deliberately thick to facilitate optical probing, reveal that Watson’s analysis [29] represents quite well the velocity profile in the laminar sheet. The measurements of mean velocity profiles across the film and liquid film thickness both compared favorably with Watson’s analysis [97, 991. Some deviation was observed in the comparison of the layer free-surface velocity, but determination of this velocity was subject to somewhat higher uncertainty in the study. Probability density functions of the local velocity within the liquid layer were reported as a function of radial distance from the radial jet exit. The data show qualitatively that higher turbulence intensities occur near the solid surface, but no quantitative data are reported for the local radial turbulence intensity. It should be stressed that the liquid layer of Azuma and Hoshino was laminar. Measurements for a turbulent jet are discussed next. Measurements of the flow structure in the radial layer of an impinging, turbulent free-surface liquid jet were made using LDV [loll. Measurements were confined to the region r/d < 10 since the thinning layer made resolution of velocities difficult despite the 0.16-mm length of the optical diagnostic volume of the LDV system as configured. As with the technique of Azuma and Hoshino [99], reflection of the LDV probe volume off the free-surface created some ambiguity in the thickness of the liquid layer, which was circumvented by integrating the measured local mean radial velocity profiles over the layer thickness until global continuity was satisfied. This procedure defined the mean free-surface location. Mean radial velocity measurements are shown in Fig. 14 for two nozzle diameters. Watson made the assumption that the maximum velocity is at the free

-


143

a 0 0.50 0.75

free surface

+ A

1.0

0 0 0

2.0 2.5

3.0

1.5

-

0

UlVJ

1

-

0

1

UIV.

I

FIG. 14. Measured profiles of the mean radial velocity in the liquid layer of the radial flow region, free-surface jets issuing from fully developed pipe-type nozzles: (a) d = 10.9 m m and (b) d = 23.0 mm. Reprinted with permission from Stevens and Webb [loll.

surface, which travels at the jet exit velocity until the viscous boundary layer reaches the free surface. The data of Fig. 14 show that for a turbulent jet the maximum velocity occurs internal to the liquid layer for 0.5 < r/d < 2.5, and the y/d location corresponding to this maximum rises until it reaches the free surface. The radial location where this occurs coincides with the position of maximum free-surface velocity measured by Stevens and Webb [91]. Figure 15 illustrates the rms turbulent fluctuations of the radial velocity component at a given distance from the target

144

B. W. WEBBAND C.- F. 1MA

y/d = 0.018

2

4

6

8

53*100

10

12

rld FIG. 15. Radial variation in the rms fluctuations of the radial velocity component in the liquid layer of a free-surface jet. Reprinted with permission from Stevens and Webb [loll.

surface as a function of r/d. The turbulence data display a sharp rise near r/d = 1.0 with a local maximum in u'/uj near r/d = 2.5. This is presumably a transition from the laminar boundary layer, whose origin is at the stagnation point, to turbulent flow and transport. The radial location of transition is also apparently independent of jet diameter and Reynolds number. Note that this occurs considerably earlier than for a layer formed under a free-surface liquid jet with low free stream turbulence [22, 32, 95, 1001.

B. PLANAR JET FLOWSTRUCTURE As with the axisymmetricjet, the existing literature dealing with the flow structure under planar free-surface liquid jets is categorized into studies of the preimpingement jet, the stagnation zone, and the parallel flow region.

1. Preimpingement Jet Wolf et al. made pitot tube measurements within a jet issuing from a parallel plate channel for Reynolds numbers corresponding to turbulent flow [102]. Measurements were made no less than five jet widths down-

-


145

stream of the nozzle exit. The results showed that the mean axial velocity in a preimpingement jet dropped to nearly 85% of the centerline value near the air-liquid interface. The result was a considerably higher free stream velocity gradient at the impingement plate. Static pressure measurements at the impingement plate showed that the free stream velocity distribution deviated somewhat from the potential flow solution for a uniform preimpingement jet, with a higher free stream velocity gradient in the inviscid region than what is predicted for a uniform velocity jet. Fitting the local pressure data to a spline, the velocity gradient was evaluated as G = 0.968, which was compared to the uniform flow theoretical value G = 7r/4 = 0.785. Turbulence dissipation in a free-surface planar jet was investigated by Wolf et al. [46]. Channels of parallel plate and converging design were used to form the jet. Turbulence levels in the jet were altered by the use of wire screen or grid turbulence manipulators in the nozzle. The turbulent flow structure of the preimpinging jet was characterized using a conical hot-film anemometer. Figures 16 and 17 illustrate the development of the mean axial velocity and turbulence intensity distributions at several locations downstream of the nozzle exit. These measurements were made in the absence of the impingement plate. The coordinate z is measured from the nozzle exit. Included in the mean velocity data are results from

0.0 0.1

0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9

x/(w/2) FIG. 16. Axial mean velocity profiles across the jet width as a function of streamwise distance from the nozzle exit for a turbulent planar jet issuing from a fully developed parallel plate channel. Reprinted with permission of ASME from Wolf er nl. [a].


146

0.10

A 20

V

25 0 3n

0.08

->

0.06 0.04

0 0 0 0.02

0.0

0.1

0.2 0.3 0.4

0.5

0.6 0.7

0.8

0.9

x/( w/2) FIG. 17. Axial turbulence intensity profiles across the jet width as a function of streamwise distance from the nozzle exit for a turbulent planar jet issuing from a fully developed parallel plate channel. Reprinted with permission of ASME from Wolf et al. [46].

Hussain and Reynolds [lo31who reported measurements of turbulent flow structure in parallel plate channels. The mean velocity data of Fig. 16 illustrate the development of the flow downstream of the nozzle exit. The mean axial velocity u near the air-liquid interface has reached 96% of the mean centerline velocity ( u X = J 10 nozzle widths from the exit. This free-surface planar jet development appears to be somewhat slower than the corresponding axisymmetric jet [911. The velocity profile is uniform with no cross-jet gradients present 20 nozzle widths downstream. The corresponding turbulence intensity data of Fig. 17 show initially high, nonhomogeneous turbulence at the nozzle exit with values of u'/v as high as 11% near the air-liquid interface. The levels of turbulence dissipate, however, reaching levels of about 2% and exhibiting almost no variation across the width of the jet by a location 20 nozzle widths downstream. Data for configurations where turbulence manipulators were added to the nozzle exit revealed that despite high initial levels of turbulence, the turbulence intensity dissipated rapidly to levels near 2% within about 5 to 10 nozzle widths of the nozzle exit. Results showed that with the exception of regions very near the nozzle exit ( z / w < 3, the parallel plate nozzle with no turbulence manipulators yielded the highest levels of turbulence. Static pressure measurements along the impingement plate revealed that the velocity gradient was independent of Reynolds number (as with the axisymmetric jet results of Stevens and Webb [93]). This gradient was seen

SINGLE -PHASELIQUIDJET IMPINGEMENT

147

to be nearly independent of nozzle-to-plate spacing for the converging nozzle (which would be expected to yield the most uniform nozzle exit velocity profile). The free stream velocity gradient was observed to decrease with nozzle-to-plate spacing for the parallel plate nozzle, varying nearly linearly with z o / w in the range 2 < z o / w < 30. Interestingly, the free stream velocity distribution (and corresponding velocity gradient) for the fully turbulent parallel plate nozzle was found to differ from the potential flow solution for uniform velocity jets at low nozzle-to-plate spacing ( z , / w = 21, but agreed very well for higher values, z , / w = 30. 2. Stagnation Zone Several investigations have used static pressure measurements in the near-stagnation zone to determine the variation in free stream velocity. This variation is then used to calculate the free stream velocity gradient. Static pressure taps at the impingement surface reveal the local pressure just outside the viscous boundary layer, because the cross-stream pressure gradients are small there. Hence, the velocities inferred from these pressure measurements correspond to those at the edge of the viscous boundary layer. Such measurements have been reported by McMurray et al. [104], Zumbrunnen et al. [401, and Wolf et al. 146, 1021. McMurray et al. report that pressure measurements made under a planar jet issuing from an orifice plate yield results that compare very well with theoretical distributions [104]. This agreement was also found by Zumbrunnen et al. [40] for a convergent nozzle. Deviations from potential flow theory were found to be less than 4% for x / w < 2.

3. Parallel Flow Region

To the authors’ knowledge, no data exist that describe the turbulent flow structure of the parallel flow region (downstream of the stagnation zone) for planar free-surface liquid jets. Quantification of the point of transition from laminar to turbulent flow in the parallel flow region has generally been based on heat transfer measurements, and is summarized in Sec. IV.D.2.b. Unlike the radial flow region under axisymmetric freesurface liquid jets, the parallel flow region under planar jets does not experience a decreasing velocity with downstream distance. Hence, measurements of liquid layer thickness are not critical. C. HEATTRANSFER UNDER AXISYMMETRIC JETS Experimental research on the heat and mass transfer characteristics of axisymmetric free-surface liquid jets is summarized in Table 111. These

TABLE I11 EXPERIMENTAL INVESTIGATION OF HEAT/MASS TRANSFER UNDER FREE-SURFACE AXISYMMETRIC LIQUID JETS Fluid

Nozzle type ( d )

Red

Water

Pipe-type (10-20 mm)

52,000-210,000

2.5-10

Local and average heat transfer, isoflux surface

Electrolyte' (Sc = 1070)

Pipe-type (3-15 mm)

2000-30,000

1-4

Local and average mass transfer, isoconcentration surface

Di Marco et al. [MI

Water

Pipe-type (10,17 mm)

12,000-47,000

6-20

Stagnation heat transfer, isoflux surface

Elison and Webb [55]

Water

Pipe-type (0.25-0.58 mm)

300-7000

0.1-40

Local heat transfer, isoflux surface

Faggiani and Grassi [lo61

Water

Pipe-type (17 mm)

20,OOO-150,OOO

0.5-5


Grassi and Magrini [90]

Water

Pipe-type (10, 17 mm)

16,000-110,000

5-35

Stagnation heat transfer, isoflux surface

P A0 (Pr = 72 @ 30°C)

Convergent (0.76 mm)

1000-3500

0.7-21

Local heat transfer, isoflux surface, comparison with spray cooling

Authors Barsanti et al. 11051 Bensmaili and Coeuret [69]

M

2,

/d

Measurement type

P W

Gu

et

al. [I071

Jiji and Dagan [lo81

Water FC-77 (Pr = 25)

Contraction with straight section l / d = 6.4-12.8 (0.5, 1.0 mm)

Lienhard et al. [36]

Water


Liu and Lienhard [ 109, 1101

Water

Liu er a/. [221; Liu and Lienhard 1321

Water

Orifice-type (1.9 mm) Orifice-type (3.2-9.5 mm)

Ma and Bergles 1581

R-113

Metzger et al. [761 Nakoryakov ef al. 1241 Pan et al. [711

2800-12,600 FC-77; 10,000 and 20,000 water 17,000-62,000

3-20

Average heat transfer, variable temperature surface L = 12.7 m m

1.2-29

Local heat transfer, isoflux surface, jet splattering effects

- 50 30,600-85,500

-

Stagnation heat transfer, ultrahigh heat flux Local heat transfer, isoflux surface

Pipe-type (1.1 mm)

2500-29,OOO

1.5-21


R-113 (Pr = 8 ) Kerosene (Pr = 20) Oil (Pr = 260)

Pipe-type (1.0 mm)

50-23,000

2-20


Water Oil (Pr = 85-151)

Pipe-type (3.8,82 mm)

3-24

Average heat transfer, isothermal surface, D / d = 1.7-25.1

Electrolyte (SC = 1070)

Orifice-type (10 mm)

-

Local mass transfer, isoconcentration surface

Water

Pipe-type, converging, orifice-type with turbulence manipulator (10.9 mm)

1.o


2200-12,000 oil 6400-140.000 water

16,500-43,500

-

continues

TABLE 111-conrinued Measurement type

Fluid

Nozzle type ( d )

Stevens and Webb [ill]

Water


4ooO-50,000

0.6-18.5

Local and average heat transfer, isoflw surface

Stevens and Webb ill21

Water

Pipe-type (4.6, 9.3 mm)

6600-52,000

1.6,4.6

Local heat transfer, isoflw surface, normal and oblique jet impingement

Water FC-77 (Pr = 25)

Contraction with straight section I / d = 39-2.7 (0.46-6.6 mm)

2oO-5O,OoO

0.25-20

Average heat transfer, isothermal surface, square heater L = 12.7

Authors

Womac er al. [53,54]

“NaOH-Fe(CN2- / Fe(CN):-

solution.

Red

20

/d

mm

SINGLE - PHASELIQUIDJETIMPINGEMENT

151

studies include several measurement techniques, and describe both heat and mass transfer results for a variety of nozzle configurations. Working fluids investigated cover a broad range of Prandtl (Schmidt) numbers: water [36,53-55, 71,76,84,90, 105, 106, 108, 110-1121, viscous oil [20, 761, electrolyte solution [24, 691, R-113 [20, 581, P A 0 [1071, FC-77 [53-54, 1083, and kerosene [201. Information regarding working fluid, nozzle type and diameter, range of Reynolds numbers, range of nozzle-to-plate spacings, and other relevant information for each investigation is summarized in the table. Entries are left blank (-1 where information has not been specified in the published report. Reference will be made to these studies in the section to follow. 1. Srugnation Zone Experimental work reveals that the transport in the stagnation zone is a strong function of the jet speed. Low-speed jets are subject to gravitational and surface tension effects. At higher speeds the influence of free stream turbulence is likely to become important. The stagnation zone heat/mass transfer is discussed here in terms of low, moderate, and high Reynolds number studies. The heat transfer coefficient is uniform in the stagnation zone, which extends to r/d = 0.7 to 0.8 for preimpingement jets of near-uniform velocity profile. The zone is smaller for jets of nonuniform velocity. At low Reynolds number (Re, < 2000), the jet is likely to be laminar, and jet contraction due to gravitational acceleration and surface tension effects become important. Elison and Webb report experimental data for microjet applications using pipe-type nozzles for 0.25 5 d I 0.58 mm [55]. The Reynolds number range investigated spanned the laminar, transitional, and turbulent regimes in the jet nozzle. Below Re, = 2000, the transport was influenced by surface tension. The physical structure of the jet was observed to be changed resulting in surface-tension-induced broadening of the jet at the exit of the nozzle tube; the jet was found to attach to the nozzle tube at a diameter larger than the tube internal diameter due to the strong action of surface tension. At higher jet Reynolds numbers, the momentum and turbulence of the jet were dominant. The stagnation Nusselt numbers in both laminar (Re, < 2000) and turbulent (Re, > 2000) regimes were described by a relation of the form of Eq. (39) where C and m were determined from least-squares regressions of the experimental data. The value of n = 0.4 was used arbitrarily, because only water was used in the study. The data reveal notably that a dependence of NU,,, Re:85 was observed in this low Reynolds number regime owing to the influence of surface tension. Further, the correlations predicted

-

152

B. W.WEBBAND C.- F. IMA

stagnation Nusselt numbers (based on jet exit velocity and diameter) that were lower than the theoretical laminar limit as a result of surfacetension-induced broadening of the liquid jet at the nozzle exit. The study of Di Marco et al. is also relevant to free-surface liquids at low speed, where gravitational forces are dominant [84]. By neglecting surface tension effects in comparison to the gravity-induced jet contraction, a simplified theoretical model was developed relating the stagnation Nusselt number to the Froude number based on nozzle-to-plate spacing. The Froessling number Fd, a dimensionless grouping that arises in stagnation flows, is defined by Fd

o

Pr" '

(57)

where the exponent n describes the dependence of the stagnation heat transfer on fluid Prandtl number. Di Marco et af. adopted the value n = 0.4. Over a wide range of experimental conditions, experimental data for gravitationally contracting jets were correlated as a function of Froude number based on jet length 2 , with

where a is a constant reported to depend on the velocity gradient (and, presumably, the level of free stream turbulence) in the preimpingement jet. The value a = 0.8 represented the experimental data quite well for FrZ, < 2 for a broad range of experimental conditions. Beyond Fr," = 2 the data were constant at Fd = 1 with an error of f 10%. Thus correlated, the Froessling number was found to be independent of the dimensionless nozzle-to-plate spacing zJd; the influence of nozzle distance from the target surface was represented entirely by the Froude number FrZo. At intermediate Reynolds numbers, the jet may be laminar or turbulent, depending on the liquid supply and nozzle configuration. The generation of laminar jets requires special care in the design and fabrication of the nozzle and upstream plenum. For laminar jets theoretical considerations, verified by the experimental work of Liu et af. [22], should be quite adequate for the prediction of stagnation heat transfer under uniform velocity liquid jets of Pr > 3: Pr'i3. (59) Nud,, = 0.797 This relationship is in close agreement with the analytical result of Liu et af. [23] for the stagnation zone in the limit as We, -+ 01. This expression is compared to the experimental data of Liu et al. in Fig. 18 [22]. The

-

SINGLEPHASELIQUIDJETIMPINGEMENT

I

l0I 2x104

0

d=1/8inch

0

d=1/4 inch

o

d=3/8 inch

153

- Equmion(59) 3x104

5x104

1x105

Red FIG. 18. Comparison of the theoretical laminar stagnation zone result, Eq. (59), with experimental data for a laminar jet. Reprinted with permission of ASME from Liu et al. [22].

experimental data are observed to be somewhat low, particularly at lower Reynolds numbers, for laminar jets in the range 25,000 < Re, < 80,000. For laminar axisymmetric jets of nonuniform velocity profile, the difficulty becomes one of determining the radial velocity gradient in the free stream. A logical limiting case for laminar tube flow is the parabolic profile, for which Scholtz and Trass [251 predicted a velocity gradient of G = 4.644, as opposed to the G = 0.916 value predicted by Liu et al. [23] for uniform velocity jets. The enhancement in heat transfer for the parabolic velocity profile jet relative to the uniform jet may be estimated from Eq. (14) by scaling the lead constant in Eq. (59) by the square root of the ratio of radial velocity gradient for the two jet velocity distributions,

JqXz.

In most engineering applications the nozzle configurations used will produce free stream turbulence, enhancing the transport at the stagnation point. Most of the experimental studies listed in Table 111 used nozzles under turbulent flow conditions. Fully developed turbulent pipe flow, produced by long nozzle tubes ( l / d > 10 to 20) for Re, > 5000 to 7000, has been used as a standard condition for study. Turbulence intensities in this case are approximately 3 to 5% at the nozzle exit. Further, the velocity gradient can no longer be estimated from the laminar flow relationship for uniform preimpingement jets. The difficulty is that the velocity gradient and free stream turbulence intensity are not easily predicted, and will not be known a priori for arbitrary nozzle and liquid supply systems. As summarized in Sec. IV.A.2, Stevens et al. measured the radial velocity gradient for a variety of nozzle configurations for z o / d = 1 [92, 931.

154


However, the reported values of G will depend on the nozzle-to-plate spacing with z,/d > 1for all but a uniform velocity jet because the jet will experience viscosity-induced flow development prior to impingement. In most cases the dependence of velocity gradient and free stream turbulence has been described by the lead constant in an empirical correlation of the form of Eq. (39). The study of Pan et al. [711 has sought to address the simultaneous dependence of stagnation heat transfer coefficient on free stream turbulence and velocity gradient. Making heat transfer measurements for experimental conditions identical to those employed in the flow structure study of Stevens et al. [92], and using hydrodynamic data from that study, an empirical correlation was developed that was valid for the sharp-edged orifice, pipe-type, and converging nozzle configuration investigated. The correlation of experimental heat transfer and flow structure data was =

0.69ReL/2G'/2

(60)

which is valid for 15,000 < Re, < 48,000. Values of the dimensionless radial velocity gradient were in the range 1.2 I G < 2.2, bounded on the low and high end by the convergent nozzle and sharp-edged orifices, respectively. For fully turbulent pipe-type orifices, G = 1.83 and is independent of nozzle diameter, Reynolds number, and nozzle-to-plate spacing in the range z,/d < 4 [93]. This yields the following recommended correlation for the stagnation heat transfer for turbulent jets issuing from fully developed pipe-type nozzles

Nud,,

=

0.93

This agrees to within approximately 5 to 7% with the correlations of previous work employing pipe-type nozzles [84, 106, 1111 in the same range of Reynolds number and nozzle-to-plate spacing. The correlation (when interpreted in terms of the heat/mass transfer analogy) yields stagnation Nusselt numbers somewhat higher than the mass transfer results of Bensmaili and Coeuret [69]. Equation (59) agrees moderately well with the work of Gu et al. [ 1071 given the uncertainty in nozzle design and corresponding velocity gradient and turbulence intensity for that study. Note that the work of Stevens and Webb [ l l l ] observed a dependence of nozzle diameter on stagnation Nusselt number that was not described by an empirical correlation of the form of Eq. (61). This has not yet been resolved in the literature. The lead constant in Eq. (61) is expected to be somewhat smaller for higher nozzle-to-plate spacings, because development of the preimpingement flow will yield a slightly more uniform jet at the target surface with correspondingly low velocity gradi-


155

ent. The Prandtl number exponent adopted in Eq. (61) was based on the work of Ma et al. [61]. As originally proposed, the correlation of Eq. (60) included a term attempting to describe the heat transfer dependence on turbulence. However, the results showed no discernible dependence on the measured rms fluctuations in radial turbulence. Note that the turbulent fluctuations in radial velocity might not be the appropriate figure of merit for describing turbulence effects. Further, as pointed out by Wolf et al. [1131, predictions of turbulent jet impingement heat transfer reveal that the rms fluctuating velocity parallel to the target surface (i.e., the radial velocity component) changes only slightly as the flow approaches the surface, while that component normal to the target experiences significant increases. Note that for typical jet impingement heat transfer, Eq. (61) predicts stagnation heat transfer about 30% higher than the laminar, uniform velocity jet correlation of Eq. (591, presumably due to the estimated 3 to 5% turbulence intensity in the pipe-type nozzle. The converging nozzle is expected to minimize the free stream turbulence owing to the favorable pressure gradient. This type of nozzle, for which G = 1.1 to 1.2 [92], yields stagnation Nusselt numbers according to Eq. (60) that are only 4 to 8% higher than the laminar flow result. The study of Lienhard et al. [361 reports heat transfer coefficients for a fully developed pipe-type nozzle in the turbulent Reynolds number range that are 16% higher than the result of Eq. (61), and nearly 60% higher than the laminar jet result, presumably due to the high turbulence generated in the jets designed to study splattering phenomena. In the high and moderate Reynolds number range (and for sufficiently high-speed jets) the transport under free-surface liquid jets is relatively insensitive to nozzle-to-plate spacing (in the r,/d range prior to jet destabilization and breakup) [58, 69, 106, 108, 111, 114, 1153. Stevens and Webb [111] and Bensmaili and Coeuret [69] reported a slight decrease in stagnation Nusselt number for free-surface jets issuing from fuily developed pipe-type nozzles, experimentally correlating a dependence of Nu , (z,/d)" where n lies between -0.0336 and -0.11 over the range 1 < z,/d < 35. This is probably due to the combined effect of (a) development of a uniform velocity profile at large z,/d from the fully turbulent profile at the exit of pipe-type nozzles with the associated lower velocity gradient and (b) dissipation of turbulence generated in the nozzle and plenum. At a higher Reynolds number, the dependence on the jet Reynolds number increases, perhaps due to increased generation of free stream turbulence for increasing Re,. Gabour and Lienhard [115] find, for

-


156

FIG. 19. Comparison of stagnation Nusselt number correlations for free-surface axisymmetric jet impingement.

25,000 < Red < 85,000, NUd,o =

0.278 Re:.”’

Pr’’’,

(62)

which was reported accurate to within f3%. The higher dependence on Reynolds number is in qualitative agreement with the correlation of Faggiani and Grassi [lo61 for z,/d = 5 , who represented the exponent on the Reynolds number as a function of the Reynolds number itself NU,,c>

-1 -

1.10~ejl..47’ 0.229 ReyLS ~

~ 0 . 4

Red < 76,900 Re, > 76,900.

(63a) (63b)

Equations (62) and (63) agree to within 30% for Red < 90,000 at Pr = 8. A graphical comparison of stagnation Nusselt number correlations for axisymmetric jets summarized in this section is found in Fig. 19 for a liquid of arbitrary Prandtl number. 2. Radial Flow Region a. Laminar Flow Correlations For laminar jets, the integral theoretical analyses of Ma et al. [20] and Liu et al. 1221 summarized in Table I can be

SINGLE -PHASE LIQUIDJETIMPINGEMENT

157

used to evaluate the local heat transfer under laminar jets. These expressions have been validated by comparison with experimental data for liquid jets impinging against isoflux walls. The expression of Yonehara and Ito [19], Eqs. (18)-(20), can be used for isothermal impingement surfaces. b. Transition to Turbulent Flow Unless the preimpingement jet is char-

acterized by extremely high turbulence, a laminar boundary layer will begin at the stagnation point and proceed outward into the radial flow zone. At some location downstream, the flow will experience a transition to turbulence. Knowledge of the location of transition to turbulent transport is important because of the associated higher heat/mass transfer coefficient and higher skin friction there. Because transition to turbulence is a stochastic process, characterization of this transition phenomenon has been largely experimental in nature. Studies treating the transition to turbulent flow are now summarized. The hydrodynamic study of Azuma and Hoshino [95, 1001 is noteworthy in this regard. They investigated the onset of turbulent flow in the radial liquid layer formed by a nozzle placed in close proximity to the impingement plane, zo/d < 0.5. Such a configuration was chosen to investigate the onset of turbulence because it minimizes the turbulence at the origin of the radial jet. They show that a radial liquid layer generated in this manner represents that originating from an impinging free-surface liquid is used in place of the jet radius, where C , is the nozzle jet if discharge coefficient. Transition to turbulent flow was observed for Re, > 47,000, and the critical radial location corresponding to the onset of turbulence, r,, was correlated as

-4

r,/d

=

380

(64)

A subsequent study [1001 concluded that, based on temporally resolved pressure measurements, the onset of turbulence was the result of amplifications inside the liquid layer, rather than the shearing stress on the surface of the liquid. Liu et al. characterized the onset of turbulence in the radial layer for a laminar, free-surface liquid jet [22]. Based on visual observations of the layer stability, the location of transition was described empirically as

r,/d

=

1200 Re;".422.

(65)

Transition to turbulence was observed for jet Reynolds numbers as low as Re, = 13,000. The difference in correlated r,/d given by Eqs. (64) and (65) from the two independent studies is less that 15%, perhaps due to

158


experimental differences in the generation of the liquid layer. Liu et al. further observed a ring of turbulence development bounded by the onset of turbulence and the achievement of fully turbulent flow. They correlated the location of the achievement of fully turbulent transport, r f r ,based on the secondary (off-stagnation) peak in heat transfer coefficient, describing it as r f t / d = 2.86 X lo4 (66) Equations (64) and (65) indicate that transition to turbulent flow and heat transfer occurs in the range 9 < r,/d < 25 for laminar flow nozzles in the Reynolds number range 10,000 < Re, < 100,000. As summarized in Sec. IV.A.3, measurements of the mean and turbulent flow structure in the radial layer of free-surface liquid jets have been reported by Stevens and Webb [loll. Measurements were made for jets issuing from fully developed nozzle tubes, a condition for which the turbulence would be much higher than that reported in the laminar nozzles of Azuma and Hoshino [95, 1001 and Liu et al. [22]. LDV was used to make measurements of the mean and rms velocity fluctuations in the thin liquid layer through a transparent impingement plate. Despite the very small diagnostic volume employed, only limited measurements were possible across the thin layer depth. The measurements revealed that the normalized rms turbulence fluctuations, u'/uj, inside the liquid layer rose dramatically at r/d = 1, reached a peak near r/d = 2.5, and decayed shortly thereafter. The location of the critical radius was observed to be nearly independent of the jet Reynolds number in the range investigated, 13,000 < Re, < 53,000. The radial variation of the local rms free-surface velocity fluctuation normalized by the local mean free-surface velocity, U:/U,, reported by Stevens and Webb [911 reveals a local minimum in U;/U, near r/d = 2, followed by a dramatic rise thereafter. These data were for fully developed nozzles with Reynolds numbers in'the range 8500 < Re, < 48,000 for nozzle diameters in the range 2.1 < d < 9.3 mm. The radial location corresponding to the rise in free-surface turbulence coincides with the location of peak turbulence on the interior of the layer. These experimental turbulence measurements suggest that transition to turbulent flow may occur significantly sooner (radially) for fully developed nozzles in which the levels of free stream turbulence are substantially higher than the laminar jet studies cited previously. The measurements of the flow structure inside the liquid layer are supportive of heat transfer measurements made for fully developed nozzles yielding turbulent freesurface exit flow, where secondary maxima in the Nusselt number were observed for r/d < 5 [ill]. The magnitude of the secondary peak in heat


159

transfer coefficient was observed to rise with increases in the jet Reynolds number.

c. Local Heat Transfer Beyond Transition Beyond the point of transition, Liu et al. [22] developed an expression for the local Nusselt number based on the thermal law of the wall. For fluids of Pr B 1 the expression becomes: NU,

0.0052

Pr

=

1.07 + 12.7(Pr2I3- 1 ) m

where the local film thickness and skin friction coefficient are d

(68) and

Cf= 0.073 Re,

'I4( r/d)'l4.

(69) The radius at which the thermal boundary layer encompasses the entire liquid layer, r,/d, is given in Table I. Good agreement was found between Eq. (67) and experimental data at radial locations beyond transition to turbulent flow. For fully turbulent jets, Sec. IV.A.3.b indicated that transition to turbulence occurs much sooner (radially) than the laminar jets just treated. The empirical correlations of Stevens and Webb [ill] can be used to estimate the local variation in Nusselt number (for isoflux surfaces). For a wide range of experimental conditions, that study revealed that if the local Nusselt number is normalized by its value at the stagnation point, the local as some function of data collapse, suggesting a correlation of Nu,/Nu,,. r/d. This ratio was correlated according to the superposition of dual asymptote technique of Churchill and Usagi [77] for impingement heat TABLE IV CORRELATION COEFFICIENTS FOR THE LOCALNUSSELT NUMBER UNDER AXISYMMETRIC, TURBULENT JETS ISSUING FROM PIPE-TYPE NOZZLES [Eo. (71)l [ill]

d (mm)

2.2

4.1

5.8

8.9

a

1.15 - 0.23

1.34 - 0.41

1.48 - 0.56

- 0.70

b

1.57


160 1.2

0 0

A 0

Nud

Rd 6800 10,600 21,200 31,800 40,800

-Eq. (70)

NUd.o

0.4

-

0.2

0

2

4

6

8

rld FIG. 20. Comparison of radial profiles of the local Nusselt number under a turbulent, free-surface axisymmetric jet with the empirical correlation of Q. (70). Reprinted with permission of ASME from Stevens and Webb [lll].

transfer under free-surface liquid jets issuing from fully turbulent pipe-type nozzles. The correlation takes the form -= o

where f(r/d) is the asymptotic function describing the normalized Nusselt number in the large r/d limit. Any appropriate correlation for the stagnation Nusselt number may be used in evaluating Nud,,. Note that the asymptotic value of Nu,/Nu,,, in the stagnation zone is unity. The form of f ( r / d ) selected was

f(r/d) = a exp[b(r/d)l. (71) The parameters a , b, and P were chosen to best represent the experimental data. The value P = 9 was found to be representative of all experimental conditions with good accuracy, whereas a and b were observed to depend on nozzle diameter as summarized in Table IV. The correlation is illustrated in Fig. 20 for a 4.1-mm-diameter nozzle. As can be seen, Eq. (70) is valid only to the radial location where a rise in Nusselt number indicated a transition to turbulence, r/d = 3 to 4. Beyond that the expression serves as a lower bound for local heat transfer behavior.

SINGLE -PHASELIQUIDJETIMPINGEMENT

161

3. Aiierage Heat Transfer Often only the average heat transfer characteristics of liquid jets are required. Average heat transfer may be determined by integration of local heat transfer correlations presented in the foregoing sections. Expressed generally, this integration can be stated

A local average heat transfer coefficient may be determined and correlated as a function of the relevant spatial coordinate by varying the area over which the integrations in Eq. (72) are performed. Alternatively, several experimental investigations report only an average Nusselt number for heater surfaces of differing size. These investigations are reflective of combined transport behavior in both the stagnation and radial flow zones. Womac et al. [541 report on the average heat transfer characteristics of free-surface jets impinging on a square heater of side length L. A correlation form was proposed that reflects the differing transport characteristics of the stagnation and radial flow zones, with the two dependencies super-imposed by an area-weighted average: Nu, -

Pr 0.4

L

- 0.516 Re:5( d ) A r + 0.491 Ret232

(1 - A r ) . (73)

The Reynolds number and jet diameter should be corrected for gravitational contraction of the jet and associated increase in impingement velocity. The term Re, is the Reynolds number based on the average length of the wall jet region, L* = 0.25[(\/2L - d ) + (L - d)]. The term A , = 7rd2/4L2 is the ratio of impingement zone area to total heated surface area. Metzger et al. present average heat transfer data for a broad range of fluid Prandtl numbers and several heater diameters [76]. Their correlation takes the form of an average Stanton number

where p, and paware the fluid dynamic viscosity evaluated at the heated wall and adiabatic wall temperature, respectively. The adiabatic wall temperature is evaluated from the recovery factor r according to Eq. (41). The recovery factor was presented graphically as a function of fluid

162

B. W. WEBBAND C.- F. M A

Prandtl number and dimensionless heater diameter by Metzger et al. As a first approximation this may be estimated by r = Pr".6. Based on local Nusselt number data, Stevens and Webb [ l l l ] developed an expression similar to Eq. (70) for the local average Nusselt number at a given radial location, %,

s,]

Nu, = [l + F ( Nu*

- 1/ P '

(75)

with the same asymptotic limits. The function g ( r / d ) was expressed as

where a and b are drawn from Table IV and f(r/d) is given in Eq. (71). A value P' = 7 correlates the average heat transfer data well. This correlation has the same limitations as Eq. (70). D. HEATTRANSFER UNDER PLANARJETS

Experimental research on the transport characteristics of planar liquid jets is summarized in Table V [116-1181. The limited submerged planar jet studies treated in Sec. 1II.B are also included. Information regarding working fluid, nozzle type and diameter, range of Reynolds numbers, range of nozzle-to-plate spacings, and other relevant information for each investigation has been summarized in the table. Again, where information was not included in the original reference, the information has been indicated by a blank entry (-) in the table. Reference will be made to these studies in the section to follow. 1. Stagnation Zone

The stagnation zone for uniform velocity planar free-surface liquid jets is confined generally to the region x / w < 0.8. For laminar uniform velocity planar jets, the classical stagnation flow result expressed in Eq. (35) is recommended. This has been verified by comparison with experimental data at Re,,, = 940 for low-turbulence nozzles designed to produce a near-uniform velocity profile [391. Earlier work at significantly higher jet Reynolds numbers (Re, > 10,500) yielded Nusselt numbers in the stagnation zone that were 70 to 80% higher than this value, resulting from higher levels of turbulence in the free stream jet. This is in keeping 471. with experimental results for other turbulent planar jets as well [a,

EXPERIMENTAL INVESTIGATIONS Authors

OF

TABLE V HEATTRANSFER UNDER

PLANAR LIQUID JETS

/w

Fluid

Nozzle type ( w )

Re,

G u et al. [116]

P A 0 (Pr = 72Q 30°C)

Convergent (0.13-0.36 mm)

100-800

4-55

Free-surface jet, local and average heat transfer, isoflux surface

Inada ei af. [39]

Water

Orifice-type (1.1 mm)

940

16

Free-surface jet, local heat transfer, isoflux surface

McMurray ef a!. [lo41

Water

Orifice-type (6.4 mm)

50,000- 100,000

-

Free-surface jet, local heat transfer, isoflux surface. normal and oblique jet impingement

Miyasaka and Inada [1171

Water

Convergent (10 mm)

10,500- 157,000

-

Free-surface jet, local heat transfer, isoflux surface

Schafer et af. [81]

Water

Converging with / / w = 16 straight section (3.2 mm)

125-9300

1.5. 3.0

Submerged (confined) jet, average heat transfer, single and multiple square isothermal discrete heaters L = 12.7 mm

Vader er al. [47, 118)

Water

Convergent (10.2 m m )

20

Measurement type

Free-surface jet, local heat transfer, near-isoflux surface continues

TABLE V-continued

Authors

Fluid

Nozzle type ( w )

Re,

z,/w

Measurement type

FC-72

Convergent, contractionwith I / w = 5 straight section (0.13-0.51 mm)

500- 15,000

1-20

Submerged (confined) jet, average heat transfer, multiple square near-isothermal discrete heaters L = 12.7 mm

Parallel plate

15,500-46,500*

8.8

Free-surface jet, local heat transfer, near-isoflux surface

Wadsworth and Mudawar [82]

(Pr = 10)

Wolf er al. [lo21

Water

(10.2 mm)

Wolf et al. 11131

Water

Parallel plate, converging with turbulence manipulators (10.2 mm)

23,000 and 4 6 , M b

2-30

Free-surface jet, local heat transfer, near-isoflux surface

Zumbrunnen el al. [401

Water

Convergent (10.2,20.3nun)

17,700-86,000‘

-

Free-surface jet, local heat transfer, transient quenching technique

“Calculated from dimensional data. bData presented in terms of jet velocity and width corrected for gravitational acceleration. ‘Re, based on jet velocity and width corrected for gravitational acceleration.

SINGLE-PHASELIQUIDJETIMPINGEMENT

165

Nonuniform velocity profiles in the preimpingement jet result generally in a higher stagnation heat transfer coefficient than those of uniform jets. In addition, the extent of the stagnation zone is substantially reduced. The analysis of Sparrow and Lee [43]revealed that a parabolic velocity distribution in the preimpingement jet resulted in a nearly fourfold increase in the radial velocity gradient in the inviscid outer flow region. Again, the increase in heat transfer coefficient at the stagnation zone can be estimated by scaling Eq. (34) by the ratio -./, The velocity gradient for the uniform and parabolic jet velocity profiles may serve as the lower and upper bounds for G in evaluating the laminar flow heat transfer. Experiments with fully developed turbulent flow exiting a parallel plate channel revealed a velocity gradient that was 23% higher than the theoretical stagnation flow result for a uniform jet velocity distribution [102]. This would suggest heat transfer enhancement of the order of 11%. However, heat transfer enhancement was observed to be as high as 79% relative to the convergent nozzle results of Vader et al. [47]. This was explained by higher turbulence in the parallel plate channel nozzle configuration. McMurray et al. [lo41 also reported stagnation Nusselt numbers approximately 50% higher than the laminar jet result due to turbulence. The combined influence of velocity gradient and free stream turbulence on stagnation heat transfer under planar free-surface liquid jets was investigated by Wolf et al. [46, 1131. Hot-film anemometry measurements inside the preimpingement jet and static pressure measurements at the impingement surface (summarized in Sec. IV.B.1) were accompanied by local heat transfer measurements for the same experimental conditions. The jet velocities were relatively low, and all data were corrected for the effects of gravity. The stagnation Nusselt number was discovered to depend approximately on for the range of experimental conditions 0.79 IG I 1.13. Recall that the uniform velocity jet is characterized by a dimensionless gradient G = n/4 = 0.785. Employing the same form of empirical correlation as the circular jet experiments of Pan et ~ l 1711 . and Stevens et al. [92], the stagnation heat transfer data were expressed as

where ul/v is the axial turbulence intensity measured at a spacing from the nozzle exit corresponding to the z,/w used in the heat transfer experiments (in the absence of the impingement plate) and averaged over the jet width. The jet velocity and width in this expression were corrected for gravity. As pointed out by Wolf et al., Eq. (77) has the drawback that it 0. Additionally, the does not reduce to the laminar jet result for u'/u--, investigators concluded that the range of velocity gradient investigated was


166

too small to identify conclusively the dependence of Nu,,, on G; forcing the GI/* dependence as revealed by laminar flow theory resulted in little change in the accuracy of the correlations. However, the dependence of the stagnation heat transfer on free stream turbulence intensity is important. Equation (77) illustrates that a doubling in the jet turbulence intensity results in an enhancement of approximately 20% in the stagnation Nusselt number. Drawing from previous work seeking to identify the effect of free stream turbulence on heat transfer in stagnation flows [119, 1201, an expression that reduces properly to the correct limit for u'/D+ 0 was derived: Nuw,0 = 1 + lO.Z[' Nuw,o,lam

u'/u) Re!,," 100

]-

30.3['

2 u'/u) Re!,,I2 100

]

, (78)

where NU,,^,,^^ is the laminar limit for the stagnation Nusselt number taken from theory. A comparison with similar data for submerged axisymmetric (air) jets [1203 reveals a higher dependence on free stream turbulence with free-surface liquid jets. These results are shown graphically in Fig. 21.

....... 0

2

Hoogendoom [ 1201

4

6

FIG.21. Correlation of the dependence of stagnation Nusselt number on free stream turbulence for free-surface planar liquid jets. Reprinted with permission of ASME from Wolf et al. [113].


167

Note that for ul/v= 2 to 3% and G = 0.8 (as might be expected for convergent channel planar jets), Eq. (77) is in very close agreement with prior experimental results for previous planar jet investigations [47, 1181. The dependence of the stagnation Nusselt number on nozzle-to-plate spacing has generally been treated successfully for gravitationally contracting jets by scaling the jet width and velocity at the impingement plate using Bernoulli's equation. It might also be expected that the correlation of DiMarco et al. [84] for gravitationally contracting axisymmetric jets could be extended to planar jets. For higher speed jets, the influence of nozzleto-plate spacing is expected to be minimal. Inada et al. found little dependence of the stagnation heat transfer coefficient for z,/w > 0.5 [39].

2. Parallel Flow Region

a. Local Heat Transfer Distributions There appears to be little experimental data verifymg the local heat transfer in the parallel flow region for laminar planar jets. The laminar jet data of Inada et al. [39] extend only to x/w = 2.5. Further, the local heat transfer data reported in other investigations are for jets whose free stream turbulence is finite as reflected in higher Nu,,,, in comparison to the theoretical laminar limit. The limited local data of Inada et al., however, agree quite well with laminar flow theory. The theory for the parallel flow region resulting from laminar jet impingement may be confidently applied out to the point of transition to turbulent flow. For turbulent planar jets issuing from convergent nozzles Vader et al. [47] recommend the following correlation for isoflux surfaces prior to transition to turbulence: Nux* = 0.89 Re$?'

(79)

where Re,* is a local Reynolds number based on the free stream velocity outside the boundary layer Re,* = Ux/u, and Nu,* = hx/k is the local Nusselt number at this location. They recommend that the stagnation zone correlation be used for Re,* < 100. The inviscid velocity distribution U ( x ) is estimated from Eq. (38). This agrees well with the correlation of Zumbrunnen et al. [a]for a similar nozzle design and range of jet Reynolds number:

Nu,,, = g ( x / w ) Re; where for w

=

20.3 mm, m

=

(80)

0.666, and the function governing the

B. W.WEBBAND C.-F. 1MA

168

spatial variation in heat transfer, g ( x / w ) is given by (D.149 - (~/w)~[0.01303( x / w ) * - 0.06517( x / w ) g(x/w) =

I

+ 0.093331

0 <x/w < 1.5 (8la) 0.1208 - ( x / w ) [0.00233( x / w ) - 0.023831

I

1.5 < x / w < 5 (81b)

for 17,700 < Re,,, < 86,800. The correlation is illustrated in Fig. 22. The Reynolds number exponent in Eq. (80) was observed to be slightly lower for w = 10.2 mm, and a different spatial function g(x/w) was fit to the local data for the smaller jet width. Wolf el al. [lo21 used a correlation of the same form as Eq. (80) for local heat transfer under a free-surface planar jet issuing from a parallel plate channel under fully turbulent conditions and striking an isoflux surface. Their experimental data revealed an exponent of m = 0.71, and the local data were fit empirically

TL

o*20

0

0.15

0.00 O.O5L-++-0

1

Re, v, (W 17.700 1.26

0 23.600 0 36;lOO A 53.100 V 62,700 69.800 86.800

1.38 1.85 2.51 2.86 3.23 3.94

2

3

X/W

D

0

4

5

Fro. 22. Variation of local Nusselt number in the parallel flow region under a freesurface planar liquid jet of moderate turbulence. Reprinted with permission of ASME from Zurnbrunnen et al. [401.

SINGLE -PHASELIQUIDJET IMPINGEMENT

169

with (0.116 g(x/w)

=

I

+ (x/~)~[0.00404( x/w)’

-

O.O0187(x/w) - 0.01991 0 < x/w < 1.6 (82a)

0.111 - 0.020( X / W )

+ 0.00193( x / w ) *

I

1.6 < x / w < 6 (82b)

for 17,000 < Re, < 79,000 to within 9.6%. Beyond transition to turbulence the correlation of McMurray er al. [lo41 reduces to Nu,

=

0.037 Re:.’ Pr1I3,

(83)

where Re, is u . x / v . Equation (83) is valid to a local Reynolds number Re, = 2.5 x 10d . b. Transition to Turbulence Experimental observations of transition to turbulent flow have also been made for planar jets. McMurray er al. reported the onset of turbulence based on measurements of local heat transfer under planar free-surface liquid slot jets [104]. No information was provided relative to the nozzle geometry; the level of free stream turbulence with origin in the nozzle itself cannot be inferred. Their measurements reported a critical local Reynolds number corresponding to the onset of turbulence of Re,= = 3.6 X lo’, where the critical local Reynolds number was defined as Rexc= ujxC/v. Secondary (off-stagnation-line) maxima reported by Inada et al. [39] for low nozzle-to-plate spacings ( z , / w = 0.1 and 0.25) near the edge of the stagnation zone ( x / w = 0.3, sometimes attributed to transition, are most likely the result of acceleration of the flow exiting the nozzle parallel to the impingement plate at such low z,/w. This configuration yields acceleration of the flow exiting the nozzle-plate gap relative to the nozzle mean velocity for z,/w < 0.5. Such secondary maxima near the edge of the stagnation zone have been documented for axisymmetric air jets at nozzle-to-plate spacings that yield fluid acceleration through the gap between the nozzle exit and the impingement plate [ 1211. Characterization of transition inferred from local heat transfer under planar jets by Vader et al. [47] report Re,: = 3.6 X lo5. The nozzle used in the Vader et al. study was strongly convergent, suggesting some initial damping of the turbulence in the preimpingement jet. The local film Reynolds number in that study was based on the local free stream velocity in the inviscid region, Re,* = Ux/v, where the free stream velocity is determined from the approximation to the potential flow solution, Eq. (38). Note that outside the impingement zone ( x / w > l), the local film velocity rapidly approaches the jet exit velocity. Thus, in

170

B. W. WEBBAND C.- F. 1MA

most cases involving transition to turbulent flow Re,; = Re,< since tanh(x/w) = 1 for x / w > 3. An independent study of planar jet impingement heat transfer employing an experimental technique based on quenching yielded a correlation for the critical local Reynolds number at transition of Rexc= 1.9 X 10’ [40]. Differences in jet free stream turbulence might account for the broad range in reported critical Reynolds numbers describing the onset of turbulence in planar jets. Indeed, Wolf et al. [113] reported that for a given jet velocity, the critical Reynolds number decreases with preimpingement jet turbulence intensity, falling from Rexc= 4.5 X lo6 to Rexc = 1.5 x lo6 when the axial turbulence intensity in the jet increases from 1.2 to 5%. 3. Average Heat Transfer

Average heat transfer coefficients over a finite heater area may be determined by integration of the appropriate expression for the local heat transfer coefficient for isoflux surfaces. Additionally, Gu et al. report experimentally determined average heat transfer results for free-surface planar jet impingement of a finite-sized heater for a range of L/w and z,/w [1161.

V. Liquid Jet Arrays The extremely high heat transfer characteristics of impinging liquid jets has been demonstrated in the foregoing sections. The peak heat transfer, however, is confined to the stagnation zone, and falls to a fraction of its maximum value just a few jet diameters or widths from ;he stagnation point. Applications with larger heated areas may be cooled with arrays of liquid jets [122, 1231. In this case the proximity of the adjacent jets results in a higher average heat transfer coefficient and greater uniformity of cooling over the heater surface area. Drainage of the spent fluid is, however, a critical issue in jet array applications. Studies with arrays of air jets have shown that lateral confinefnent of the jet exhaust results in degradation of the overall heat transfer [124-1271, Arrays of liquid jets may be arranged either in submerged or free-surface configurations. For submerged jets, then, the same entrainment and core velocity considerations evident for single submerged jets may apply to jet arrays. Of course, these effects are superimposed with the aforementioned cross-flow effects in submerged arrays. Variations in nozzle-to-plate spacing may present additional unusual behavior, and must be considered with other phenomena already mentioned. Submerged air jet array characteris-

-


171

tics have been summarized and quantified in some detail previously [7]. The reader is referred to this exhaustive review for the heat transfer characteristics of submerged air jet arrays. This section deals exclusively with investigations of liquid jet array heat/mass transfer. Sample nozzle configurations for both axisymmetric and planar jet arrays are illustrated schematically in Fig. 23. The two common configurations for axisymmetric jets are square and triangular arrangements of the jet orifices. Planar jets are usually arranged in parallel rows. The parallel flow from neighboring jets will interact at a location approximately midway between. If the array is designed to provide good drainage of the spent liquid, the flow and heat transfer will exhibit symmetry, with repeating modules as shown in the figure. Module-average transport characteristics can be determined in these modules and subsequently applied to the entire array. Of course, cross-flow effects will destroy the symmetry due to the influence of drainage from upstream jets in the confinement. Published studies dealing with liquid jet array heat transfer have approached the problem largely from an experimental viewpoint. Table VI summarizes the published work in the area. Of these, only the work by Yonehara and Ito [19] draws from approximate theory to predict the heat

FIG. B. Example of (a) axisymmetric and (b) planar jet array nozzle configurations.

TABLE VI EWERIMENTAL INVESTIGATIONS OF HEAT/ MASSTRANSFER UNDER LIQUID JETARRAYS

Authors Bensmaili and buret [ 130,1351

Chang et UL 1591 I . N

Copeland [132]

Fluid Electrolytea (SC = 1070)

Array configuration Square, triangular

N

( d , mm) 4-37 (2.5-6.9)

R-113 (Pr = 8)

Square

25 (1.0,2.0)

FC-72

Square

4-100 (0.25-2.5)

Red

s/d

z,/d

1oO0-11,oO0

3-8

1-10

Free-surface jets, local and average mass transfer, circular target D = 1-70 mm, isowncentration surface, radial drainage from array center

5,lO

1.5-6

Submerged jets, average heat transfer, drainage through target plate at 2-mm holes geometrically centered between array nozzles

-

2, =

Measurement type

Submerged jets,

average heat

0.25-4mm transfer, square target L = 10 mm, radial drainage from array center


Water FC-77 (Pr -- 25)

Square

4,9 (0.5, 1.0)

2800-20,000

5.1, 10.2 mm

3-15

Free-surface jets, average heat transfer, square heater L = 12.7 mm, radial drainage from array center

Nanzer et al. [133, 1361

Panand Webb

Electrolyte" (Sc = 1070)

Square

4-36 (0.15- 1.O)

450-7000

2.5-10

1-15

Submerged jets, local and average mass transfer, circular target D = 1-70 mm, isoconcentration surface, radial drainage from array center

Water

Square, triangular

7,9 (1.0-3.0)

5000-20,000

2-8

2.5

Free-surface jets, local and module-average heat transfer, isoflux surface, radial drainage from array center

Water

Planar jets

12,500-27,0OOb

10.4, 16.5

-

Free-surface jets, local heat transfer, near-isoflux surface, lateral drainage between planar jets

12,000-26,000'

10.4, 16.5d

18.3

Free-surface jets, local heat transfer, near-isoflux surface, adjoining rows of circular jets, lateral drainage

6.2-20

5-10

Free-surface and submerged jets, average heat transfer, isothermal surface, square heater L = 12.7 mm, radial drainage from array center

5

Free-surface jets, module-average heat transfer, isothermal, radial drainage from array center

[128,129]

Slayzak et al. [I371

Slayzak et al.

5

Rows of adjacent circular jets

2 rows

Water FC-77(Pr = 24)

Square

4,9 (0.51, 1.0)

500-20,000

Water

Square

9 (1.0-8.0)

7100-48,100

Water

~381

w

Womac et al. I1311

Yonehara and It0 [19]

2

( w = 5.1)

(4.9)

"NaOH - Fe(CN)z-/ Fe(CN):- solution. bReynolds number based on planar jet width. 'Reynolds number based on individual circular jet diameter. dSpacing between nadjacent rows of circular jets.

13.8-330

(1 10-330 mm)

174

B. W. WEBBAND C.- F. l k b

transfer characteristics of free-surface liquid jet arrays. This section first outlines available theory for free-surface liquid jet arrays, then focuses on available experimental information relating to the effect of relevant array variables on the local and average transport in such systems.

A. THEORETICAL APPROACH Yonehara and Ito [19] appear to be the only investigators to have considered analytically the heat transfer under a square array of impinging free-surface liquid jets. A repeating module for each jet may be defined by the symmetry; a square of side length s is centered over each jet in the array. As an approximation, the jets were considered to act independently out to a radial location r,, where the heat transfer coefficient from the laminar flow model falls to that equal to natural convection in a stagnant film. (Note that the local jet Nusselt number will fall to this level only if s/d is large enough.) Thereafter, the heat transfer coefficient was determined to be equal to that projected by a suitable correlation for natural convection in a stagnant horizontal film, designated h,. To determine the average heat transfer coefficient in a region defined by a single module of repeating jets in the array, a cooling effectiveness is defined as

*

=

/ r 2 T h ( 5 ) 5 d6-9

(84)

where h(5) is an expression for the local heat transfer coefficient taken from a suitable laminar flow and heat transfer model (such as those presented in Sec. 11). This cooling effectiveness can be determined over the area defined by the square module in the array, and is termed +sq. The integration of Eq. (84) was carried out by Yonehara and Ito [191 for an isothermal target surface, considering three geometric conditions related to the square module. Special care was taken to consider the spatial regions beyond a circle of radius s/2 that lie in the corners of the square modules. The intermediate expressions are rather cumbersome and are not reproduced here. The module-averaged heat transfer coefficient, %mod, is evaluated from

-

hmod= +l,q/d2 s*~, (85) where s* = (s/d>/Refi/3. For the conditions of the experimental study of Yonehara and Ito- [19], the resulting expression for the module-averaged Nusselt number is Pr1I3(s/d) -4’3, NUd,,,,,d = 2.38 where the subscript 6 ‘ m ~ drefers ” to averaging over a single jet symmetry

-


175

module. A similar expression can be derived for an isoflux surface, and is likely to yield an average Nusselt number somewhat higher than the result of Eq. (86).

B. EXPERIMENTAL STUDIES 1. Module-Averaged Heat Transfer In support of the theoretical study summarized in Sec. V.A, Yonehara and Ito [ 191investigated heat transport under a square array of free-surface jets striking an isothermal surface. Nine jets of varying interjet spacing and jet diameter were employed in the study over a range of Reynolds s/d I 330 and numbers. The configurations studied correspond to 13.8 I 7100 I Re, I 48,000. The experimental results reveal that the theoretical expression for the module-averaged Nusselt number of Eq. (86) yields good agreement with experimental data for the region Re,/(s/dI2 2 5. At lower values of this parameter, the model overpredicts the experimental data by as much as 30 to 40%. It should be emphasized also that the experiments performed in validation of this model were designed to permit good drainage of the spent liquid from the array. Thus, cross-flow effects in the experiments were negligible, and were obviously not accounted for in the model development. Recognizing the discrepancy for Re,/(s/d2 < 5, Eq. (86) may be used confidently to predict the heat transfer under free-surface liquid jet arrays striking isothermal targets with good drainage. The result can also be used to approximate the heat transfer character for submerged liquid jet arrays within the potential core, z,/d < 5 to 8, although the module-averaged Nusselt number for submerged jets may be somewhat lower due to shear-induced deceleration in the radial flow region not present in the model development. One rationale for employing liquid jet arrays is to increase the uniformity of heat transfer under the array. To that end, arrays featuring smaller interjet spacings than those utilized by Yonehara and Ito [19] are of interest. Most of the available experimental work reported in the literature on liquid jet arrays has focused on smaller s/d. Only the work of Pan and Webb [128, 1291, however, measured and correlated module-averaged heat transfer. The local temperature of an isoflux surface was measured radiometrically under seven-jet triangular and nine-jet square free-surface array configurations, and the module-averaged Nusselt number (under the central jet) was determined. It is again stressed that this configuration was also designed to produce good drainage of the spent liquid, minimizing cross-flow effects in the central jet. The experimental results revealed that the heat transfer behavior was influenced by the nozile-to-plate spacing in

B. W. WEBBAND C.- F. IMA

176

TABLE VII MODULEAVERAGE NUSSELTNUMBERCORRELATION O F EO. (87)FOR SQUARE AND TRIANGULAR FREE-SURFACE JET ARRAYS IMPINGING AGAINST ISOFLUX SURFACES [I281

COEFFICIENTS IN THE

z,/d

C

m

a

2

0.386 0.129

0.59

0.09

0.71

0.1

5

the range studied, 2 I zo/d I 5; flooding of the central jet at lower nozzle-to-plate spacings resulted in the formation of a region of submerged jet heat transfer surrounding the central jet, with an interface defining the transition from submerged to free-surface jet behavior at some radial distance between the central and neighboring jets. The module-averaged Nusselt number was correlated empirically as

-

c Re’;

~/d)]. (87) The correlation coefficients are summarized for z,/d = 2 and 5 in Table VII, valid for 2 I s/d I 8 and 5000 I Re, I 20,000. No significant difNU,,

mod =

eXp[ -a(

ferences in module-averaged heat transfer were noted between the sevenjet triangular and nine-jet square array configurations. Note that nearly the same Reynolds number dependence as predicted theoretically by Indeed, without Yonehara and Ito is observed here, G d , m o d significant loss of accuracy, the correlation of Eq. (87) may be recast as

-

NU,, mod = 0.225 Re:/3 eXp[ - o m ( S / d ) ] , (88) which is good to within approximately &lo%for both nwle-to-plate spacings. The correlation of Pan and Webb [Eq. (8711 and the experimentally verified theoretical expression of Yonehara and Ito [Eq. (8611 are compared in Fig. 24 for two Reynolds numbers over the s/d range of validity of the experiments. The analysis of Yonehara and Ito yields a Nusselt number dependence on (s/d)-4I3 at larger s/d, and is verified experimentally for Red/(s/dl2 2 5. Despite the absolute discrepancy with theory for Red/(s/d)2 < 5, the functional power-law dependence on s/d is valid. At lower s/d, the empirical module-averaged Nusselt number correlation of Pan and Webb illustrates a somewhat different dependence on s/d. It may be conjectured that the two would merge in the s/d range of nonoverlap (keeping in mind the difference in thermal boundary conditions), permitting prediction of the module-averaged Nusselt number for well-drained free-surface liquid jet arrays. Also plotted in Fig. 24 is the single jet laminar value of the stagnation Nusselt number found in Sec.

SINGLE -PHASELIQUIDJET IMPINGEMENT Single laminar jet. stagnation zone

103F,(‘

- .

..

- . . m . m I ’

177

[a]: Nud = 0.797ReyzPr1’3

.

a

.v-m-7

Pan and Webb [128]

hd’20.000

Experimental data I191 13.8 s sld i 330

Pr = 8

1 0” 1 on

I

I

I

1

10’

.

1

1

1

1

1

10’

3

1 o3

sld FIG. 24. Variation of the module-averaged Nusselt number with jet-to-jet spacing in the array: experimental data of Pan and Webb [128] and analysis of Yonehara and Ito 1191.

II.A.l. This - serves as the limiting value for the module-averaged Nusselt number NU^,,,^^ as s/d + 0, because the jets in the array merge to form a for Re, = 20,000 single jet at vanishing s/d. The values of Gd,mod apparently higher than the single-jet laminar limit in the Pan and Webb jet array correlation can be attributed to the characteristics of the individual nozzles in the orifice plate, which result in a somewhat higher stagnation zone velocity gradient and jet turbulence intensity with a correspondingly higher heat transfer coefficient. 2. Other Correlations Other experimental studies have investigated the transport behavior under liquid jet arrays striking heated targets of arbitrary dimension [130-1331. The resulting average heat transfer correlations might not be valid for estimating module-averaged Nusselt numbers, but could be used in calculating overall heat transfer if the physical dimensions of the application exhibit similarity with the range of validity of the correlations. The jet array average heat transfer correlations are found in Table VIII, and the experimental conditions for these investigations were summarized in Table VI. Those studies employing the electrodiffusion mass transfer experimental technique can be extended to heat transfer situations by

TABLE VIII SUMMARY OF AVERAGE HEAT/ a s s TRANSFER EMPIRICAL CORRELATIONS FOR LIQUID JETARRAYS Authors

Correlation

-

0.66Re$5744Pr0.4(zo/d)-o.1m5[ 1 + 0.1147(r/d)1.81]

r / d 5 1.25

0.7017 Re:5744

r / d > 1.25

~ ~ / d ) - ~ r~/ d' )~- 0". 6(2

h = ~ ~ ~ , ~n , 0 . 5 8 ~ 0 . 2 ~ 0 . 0where 9 A, = a d 2 / 4 shd = 0.38 Re:54 S C ' / ~ ( S / ~ ) - ~forz,/d .~ =4

Copeland [132]' Nanzer el al. [133, 1361'

-


Nu,

=

3.84 R e a 2 Pr 'I3

Womac et 01. [131] Free-surface jets: 0.516 0.344 0.579 OS[(fis/2 - d / 2 ) + (s/2 - d/211 N?rd2/4L2 Submerged jets: 0.509 0.0363 0.8 0.5[(&~/2 - 1.9d) + (s/2 - 1.9d)l Nd1.9d)*/L2 Jet velocity and width are corrected for gravity ~~

~~

~~~

~~

" D / d range of correlation validity not specified. 'Correlation was presented in terms of dimensional quantities, the lead constant 7100 was estimated from graphical data.

-

SINGLE PHASELIQUID JET IMPINGEMENT

179

invoking the heat and mass transfer analogy. Generally speaking, the correlations of Table VIII yield predictions both above and below the module-averaged Nusselt numbers illustrated in Fig. 24, presumably due to the fact that the nozzle array configuration varies in relation to the target area. Thus, the average Nusselt (or Sherwood) number reported for a particular target geometry and area may be more or less heavily weighted by stagnation or radial zone heat transfer. Further, nozzle configurations are probably different, yielding varied levels in velocity gradient and turbulence intensity.

3. Local Effects The distribution of local heat transfer coefficient under liquid jet arrays is often of interest, because this is critical in assessing the uniformity of cooling in applications where the high transport coefficients of liquid jets are required. The experimental measurement of these distributions is difficult due to limitations in spatial resolution of most instrumentation. As a consequence, few experimental works report the distribution of local heat transfer. Pan and Webb report two-dimensional distributions of local heat transfer coefficient for triangular and square free-surface liquid jet arrays impinging against an isoflw surface E128, 1291. Local results reveal that the transport in the stagnation zone is not influenced by interjet spacing; each jet acts independently in the immediate vicinity of its impingement. However, the variation of local Nusselt number shows a strong dependence on s/d. This is illustrated in Fig. 25, where the local Nusselt number normalized by the value at the stagnation point is plotted as a function of r/s for three values of the interjet spacing, s/d, under a square nine-jet array. The results show that the spatial uniformity in Nusselt number increases with decreasing s/d. A secondary maximum in the Nusselt number is found for intermediate values of s/d at the location where the radial wall jets originating from adjacent jets interact, r/s I:1/2. However, the magnitude of the secondary peak Nusselt number, NU,,^,^ is diminished with further increases in s/d. The secondary peak in Nusselt number in the interference zone between adjacent jets is illustrated from two-dimensional distributions of Nu, in Fig. 26 for both triangular and square axisymmetric jet arrays. Figure 26 shows infrared images of the two-dimensional temperature distribution for an isoflux wall cooled by free-surface liquid jet arrays. Increases in the level of gray scale indicate higher Nu,; the darkest regions are in the stagnation zone, where Nu, is highest, and Nu, decreases radially until the interference zone is reached. The dark hexago-

B. W. WEBBAND C.- F. I%

180

0.7

-

-

0.0

s/d = 2 -s/d=4 -t ~ / d =6

0.1

0.3

0.2

0.4

0.5

r/s FIG. 25. Variation of the local Nusselt number in a square array of axisymmetric free-surface liquid jets. Data are shown along a radial line joining closest adjacent jets. Reprinted with permission from Pan and Webb [129].

FIG.26. Two-dimensional infrared images of the local wall temperature distribution under free-surface liquid jet arrays striking an isoflux surface, Re, = 15,ooO, s / d = 4, z , / d = 2: (a) triangular jet array and (b) square jet array. Darker regions correspond to cooler temperatures. Adapted with permission from Pan and Webb [1291.

-


181

nal and near-square bands in the interference zones between adjacent jets indicate secondary maxima in Nu, for the triangular and square array configurations, respectively. The magnitude of Nu,, o, increases with the jet Reynolds number. Pan and Webb quantified the magnitude of the secondary maxima as a ratio of relative differences between stagnation values and the minimum value of Nu, in the array [128, 1291. They proposed the parameter

Figure 27 illustrates the variation of A N u , , ~ , ~ / A N u ,with , ~ Re, for the seven-jet triangular array. The figure illustrates that the secondary maximum increases almost linearly with Re,, and A N U ~ , ~ , ~ / A N can U , , reach ~ values of 60%. A similar trend was also found for square arrays. However, the peak in Nu, in the interference zone is a strong function of s/d and zo/d in the range of these parameters studied. Nevertheless, variations in Nu, could be significant, and should be considered in a liquid jet array application. Similar secondary maxima in the local transport coefficient have been discovered in other round jets arranged in arrays [134-1361. Ishigai et al. characterized the local heat transfer behavior near the wall

0

s/d = 6

s/d = 6

0.6

5000

10000

15000

20000

25000

FIG. 27. Variation in the magnitude of the secondary maximum Nusselt number with the jet Reynolds number for triangular arrays of free-surface liquid jets. Reprinted with permission of ASME from Pan and Webb [128].


182

jet interference region, classifying the structure qualitatively in terms of the jet array parameters [134]. Studies for planar liquid jet arrays [137] and rows of closely spaced round jets [138] also reveal secondary maxima in the wall jet interference zone between adjacent jets. Maxima in Nu, were found to be as high as that at the stagnation line, depending on the jet exit velocity. Flow visualization revealed a fountain, which results from the interacting wall jets and is responsible for the secondary peaks in heat transfer there. Unequal jet velocities among neighboring planar jets results in a spatial shift of the interference zone to a location nearer the weaker (lower velocity) jet.

VI. Other Factors Afl'ecting Transport A. JET INCLINATION Some industrial applications of liquid jet impingement feature non-norma1 orientation of the jet with respect to the impingement plane. Such a configuration will affect the hydrodynamics of the jet and, hence, the heat transfer. As a point of nomenclature, the minor and major axis directions are those associated with flow toward and away from the direction of jet inclination, respectively. Those studies treating this problem for turbulent submerged and free-surface liquid jets are summarized. 1. Axisymmetric Jets The effect of inclination on local heat transfer under axisymmetric free-surface liquid jets was first investigated by Stevens and Webb [112]. Experiments were conducted with water jets to determine the variation of maximum heat transfer and the profiles of local heat transfer coefficients in the range of the jet Reynolds number from 6600 to 52,000. It was found that the point of the maximum heat transfer was displaced upstream (along the minor axis) of the geometric stagnation point a distance rd, as has been observed for circular air jets [139, 1401. However, the magnitude of the displacement was found to be significantly less than that of the submerged air jets, being confined to rd/d < 0.5. It was attributed to the relative unimportance of entrainment and the preimpingement jet spreading for the free-surface liquid jets. This trend of oblique liquid jets was also reported both with axisymmetric free-surface [1411 and submerged jets of transformer oil [142] for a jet Reynolds number of less than 1000. The displacement of the maximum heat transfer point was measured and

-


183

correlated for the two jet configurations [141, 1421. The observed effect of the Reynolds number on the displacement of peak heat transfer was not significant, being a function primarily of impingement angle:

rd/d

=

(M

+ Ne)cos 8,

(90)

where the empirical coefficients take values M = 0.119 for free-surface jets or M = 0.0176 for submerged jets; N = 0.00454 for free-surface jets or N = 0.00734 for submerged jets; and B is the jet inclination angle relative to the impingement surface, expressed in radians. The displacement with submerged jets is slightly higher than that with free-surface jets. This difference can be attributed to stronger entrainment and consequent preimpingement spreading for submerged jets. Although the Prandtl number of transformer oil is much higher than that of water, the results with the circular jets of the two liquids exhibit the same trend in terms of displacement variation. The values of the maximum displacement reported in references [112, 141, 1421 are very close: rd/d = 0.25 for water jets and rd/d = 0.27 for oil jets. These values are significantly lower than the measured maximum displacement values of rd/d = 1 to 2 with air jets [139, 1401. The displacement coincides more closely with the point of maximum pressure predicted and measured in oblique air jet investigations [143-1451. Besides its shift, the magnitude of the maximum heat transfer was also measured with circular liquid jets. Based on the data of transformer oil jets, a correlation for the maximum Nusselt number was developed for the two jet configurations [141, 1421:

NU^,,,^^

=

C Re7 Pr",

(91)

where the empirical constants C, m, and n are presented in Table IX. Figure 28 illustrates the effect of inclination on the magnitude of the maximum heat transfer with oblique free-surface axisymmetric jets of TABLE IX EMPIRICAL CONSTANTS I N Eo. (91) FOR THE DETERMINATION OF THE MAXIMUM NUSSELTNUMBER UNDER OBLIQUELY IMPINGING JETS[141, 1421 Free-surface jet B(deg)

90

80

70

60

Submerged jet 50

40

90

75

60

45

C 0.388 0.383 0.372 0.349 0.333 0.298 0.354 0.366 0.345 0.305 m 0.605 0.620 n 1/3 1/3


184 100

I -

-

N"d,max

8 (dw) 0 0

90

0

60

80 70

50 0 40

Pr"3 10

-

Nud,,,= = 0.388Re:605 (correlation,

t 1 10'

e

P$ '3

= 90 deg)

transformer oil I

I

I

I,lI,l

I

I

I

I I 1 1 1 1

1 o3

102

I

I

I

I III,

1d

Red

FIG.28. Effect of jet inclination on the magnitude of the maximum Nusselt number for axisymmetric free-surface transformer oil jets. Replotted with permission from Ma et al. [141].

transformer oil [142]. The Reynolds number dependence for the oil jets agrees well with the results of air jets [139, 1401, noting that the Reynolds number exponents (0.620 or 0.605) for the oil jets are identical with the value of 0.6 reported by Sparrow and Love11 [139]. Over a small range of jet angles from 90 to 60 deg, the decrease in the maximum Nusselt number was approximately 5% which is of the order of data uncertainty for most investigations. The insensitivity of maximum heat transfer with inclination in this small range precludes exact determination of the variation of peak heat transfer; sometimes even slight increases were observed [ 112, 139, 140, 1421. With further increases in jet inclination, moderate decreases were measured with oil jets [141, 1421. The greatest decrease encountered in the two investigations was about 20%. From the viewpoint of engineering practice, the significant imbalance of thermal capability caused by jet inclination could be of great importance. This imbalance was investigated with liquid jets by studying the local heat transfer behavior 1112, 141, 1421. Measurements were made to determine the local distributions of heat transfer coefficients with water 11121 or oil

-


185

[141, 1421 jets. Similar to oblique air jets, the results of liquid jets also exhibit a significant asymmetry in the local heat transfer profiles about the maximum heat transfer point. The oblique liquid jets provide a higher heat transfer rate on the downstream side and lower rate on the upstream side. The asymmetry was accentuated with increasing jet inclination. For freesurface water jets issuing from fully developed pipe-type nozzles, Stevens and Webb 11121 correlated experimental data for the local Nusselt number profiles along the major and minor axes for free-surface jets at higher jet Reynolds numbers. The form of the correlation is similar to that proposed by Goldstein and Franchett for circular air jets [140]; Nu,

=A

Re: exp[ ( PO2

+ me + a ) ( r / d ) ] ,

(92)

where 19 is the jet inclination angle expressed in radians. The coefficients appearing in the correlation A , a, p , rn, and n are given in Table X. Equation (92) fits 92% of the data within 10% for all measured Reynolds numbers and jet angles and for r/d < 3. Local heat transfer profiles were also correlated in the major and minor axes [141, 1421 for inclined low Reynolds number transformer oil jets. For the submerged oil jets, the local distribution of heat transfer coefficients can be expressed by Nu d 1 -= (93) NU^,^^^ 1 + AIr/dIP’ where A and P are functions of 8, calculated from A

= U,

+ ale + a,e2

P

=p o

+ ple + p2e2.

(94)

and (95) In Eqs. (94) and (99, 8 is expressed in radians. The coefficients aO,a,, a2 and p o , p , , p 2 are presented in Table XI, and NU^.,,^ is given by Eq. (91). TABLE X COEFFICIENTS FOR EQ. (92) CORRELATING THE LOCAL NUSSELT NUMBER ALONG THE MAJOR AND MINORAXES FOR INCLINED FREE-SURFACE AXISYMMETRIC LIQUIDJETS11121

d (mm) 4.6, r / d 4.6, r / d 9.3, r / d 9.3, r / d

>0 0 0.26. Zumbrunnen and Aziz investigated the phenomenon experimentally. A thin, constant heat flux plate was instrumented with thermocouples on its underside for instantaneous temperature measurement. The planar, free-surface jet was generated with a low-turbulence converging nozzle. A multibladed wheel was rotated about an axis parallel with the preimpingement jet, periodically interrupting the jet flow against the heated surface. The resulting impingement flow was termed intermittent,because the flow was periodically arrested completely during the time interval for which the flow was obstructed by the rotating blade. A combination of three- and six-blade wheels permitted the variation of pulse frequency and flow interruption time. Comparison was made of the pulsing jet results with the steady jet counterpart for the same nozzle Reynolds number. Experimental results are shown in Fig. 34 for a single Reynolds number over a range of pulse frequencies, 30 s f I130 Hz. The data confirm that heat transfer enhancement occurred for S > 0.26, and was explained in terms of complete boundary layer surface renewal due to the jet interruption. The parameter r is the dimensionless time for boundary layer renewal, defined as r = &,tb& where f b , is the time period for boundary layer growth during the intermittent jet cycle. The magnitude of the maximum enhancement in a time-averaged Nusselt number was seen to be as high as a factor of 5 relative to the steady jet value. Further, enhancement was observed in both stagnation and parallel flow regions, over the full range of instrumented heater apparatus, x / w I 6. The time-averaged spatial distribution of local heat transfer coefficient exhibited some asymmetry about the stagnation line for the pulsing planar jets. The technique has the added advantage that the rotating blade jet pulse generator technique results in no added pressure drop in the liquid supply system. Further work investigating the effect of jet pulse temporal shape revealed that enhancement is not assured for S > 0.26 [159]. A sinusoidal pulse shape was generated using a rotating ball valve upstream of the nozzle. The data were compared with results for the rotating blade


200

0 3.939 0,165

A

0 x

-

2.462 0.204 2.226 0.29) 1.994 0.320

1

I

-6

-4

59 94

I04

I17

00 0

-t

Lb''on

n

I

I

I

I

I

-2

0

2

4

6

x/w

Pr

FIG. 34. Local time-averaged Nusselt numbers for intermittent jets, Re, = 9450 and 5.6. Reprinted with permission of ASME from Zumbrunnen and Aziz 11571.

=

intermittent jet, which generated a square-wave temporal jet velocity variation. The pulse amplitude, defined as the ratio of periodic jet velocity amplitude to average jet velocity was less than 0.85 for the sinusoidal jet and, by definition, was 100% for the intermittent jet, The flow characteristics were quantified using hot-film anemometry. To separate the effects of flow periodicity and induced turbulence, the instantaneous velocity was characterized as a superposition of average, periodic, and turbulent velocity contributions. A spectral analysis of the temporal velocity data revealed that the turbulence intensity never exceeded 1% for both the stlady jet baseline and the pulsing jet cases. Thus, any observed differences in the transport characteristics were attributed to flow periodicity. The experimental results revealed that the sinusoidal flow pulsation yielded a degraded time-averaged Nusselt number of about 20% relative to the steady jet value. This is corroborated by submerged (air) jets [160, 1611. By contrast, the intermittent (square-wave) jet yielded enhancements of up to 33%, generally for jet Strouhal numbers (dimensionless pulse frequencies) exceeding the 0.26 criterion of the previous work [157]. The reduction in heat transfer for the sinusoidal jet velocity waveform was explained in terms of a combination of observed jet bulging and nonlinear dynamics in the jet boundary layer [162]. It appears that enhancement in time-averaged transport of liquid jets may be achieved if fully intermittent jets are employed, that is, jets that

SINGLE -PHASELIQUID JETIMPINGEMENT

20 1

are fully obstructed briefly in time. This allows complete surface renewal and boundary layer regeneration. By contrast, degradation of the heat transfer may result for pulsing jets of other impingement velocity waveforms, depending on the pulse magnitude, shape, and frequency. F. HYDRAULIC JUMP

A phenomenon of interest in free-surface jets under certain conditions is the formation of a hydraulic jump as the flow in the parallel flow film region makes a transition from a supercritical to a subcritical regime. This transition is accompanied by a significant reduction in film average velocity with a correspondingly sharp increase in the liquid film thickness. The hydraulic jump for axisymmetric jets is rather complex, and has been characterized for different regimes delineated by variations in the layer depths upstream and downstream of the jump [ 1631. Significant backflow may be present at the jump location. In general, the region of increased film thickness corresponds to a region of substantially increased turbulence [24]. The deceleration of the liquid layer appears to dominate over the increased turbulence, and the transport in the subcritical region experiences a dramatic reduction. Predictions for the flow and heat transfer using a one-dimensional transport assumption revealed significant decreases in the heat transfer coefficient due to transition from supercritical to subcritical flow regimes [ 1641. These approximate predictions were verified by numerical solution of the full two-dimensional transport equations including the prediction of the hydraulic jump [165]. The predicted reduction in heat transfer coefficient is supported by experimental measurements of the local heat transfer behavior near the jump region [134]. The ratio of thickness before and after the hydraulic jump may be determined from a mass and momentum balance. The result for uniform flow assumed across the layer depth is h

1

h,

2

2= -

( d m - l),

(115)

where h , and h , are the upstream and downstream film thicknesses, respectively, and Fr, is the upstream Froude number, U , / &. The constant preceding the Froude number becomes 9.6 if a parabolic velocity variation is assumed across the liquid layer [166]. Although the simplified theory of Eq. (115) appears to be adequate for most hydraulic jumps encountered in planar flows where the supercritical Froude number is typically less than 30, Liu and Lienhard point out that the prevailing conditions in axisymmetric situations may yield Froude numbers of several

202

B. W.WEBBAND C.-F. l h

hundred [163]. Further, the supercritical Froude number is not known a priori; it is intimately tied to the prediction of the jump location itself. The first analytical attempt to predict the hydrodynamics of the hydraulic jump was presented by Watson [29]. As summarized by Liu and Lienhard [ 1631, subsequent investigations compared experimental results with the Watson theory [24, 94, 134, 1671. Agreement with the theory has been mixed. Liu and Lienhard speculate that the Watson theory experiences breakdown when the downstream flow is too deep or when the upstream Froude number is too high. They conclude that a primary weakness of the Watson model is the assumption of uniform velocity across the layer depth in both the supercritical and subcritical flow regimes. From a thermal standpoint, the location of the hydraulic jump is critical because of the degraded transport characteristics there. Unfortunately, the location of the hydraulic jump is unsteady and depends heavily on the drainage configuration downstream; the jump location and height may be artificially manipulated by positioning and adjusting a raised lip on the flow plate downstream. Further, visualization of the jump in rotating systems reveals that plate rotation may eliminate the hydraulic jump completely [165]. Stevens and Webb presented a correlation of experimentally determined jump location data for fully developed axisymmetric nozzles of diameters 2.2 I d 5 8.9 mm impinging against a large, stationary plate and for the Reynolds number range 2000 I Re, I50,000 [lll]. The data were represented crudely as

rj/d = 0.0061 Rets2. (116) The average error in Eq. (116) was determined to be 15% over the full range of jet Reynolds number and jet diameter investigated. As with previous experimental studies, the correlation of Eq. (116) shows rather poor agreement with the Watson model. However, the study revealed that the jump location’s dependence on Reynolds number agreed quite well with the analytical result of Watson for Re, > 4000.Despite the relatively high uncertainty, the expression of Eq. (116) yields a crude estimation of the jump location for axisymmetric jets. Liu and Lienhard conclude that a rigorous prediction of the axisymmetric jump radius may be governed by surface tension considerations, and will depend on the jet Reynolds, Froude, and Weber numbers, the ratio of subcritical-to-supercritical layer depth, and the subcritical layer radius [163]. Buyevich and Ustinov propose that the jump phenomenon is due to the development of the adverse pressure gradient coupled with gravity effects [35]. They present a rather simplified analysis for the location of the hydraulic jump based on an integral momentum approach. Their analysis agrees quite well with the

-

SINGLEPHASE LIQUID JETIMPINGEMENT

203

experimental hydraulic jump data of Watson [29] and Nakoryakov et al. [24]. However, no comparison was made with the extensive jump data of Liu and Lienhard [163]. Attempts to predict the hydrodynamics and heat transfer numerically by solving the multidimensional conservation equations are computationally intensive, and compare only reasonably well with available experimental data [168-1751. A novel approximate analytical technique that recognizes the similarity between the hydraulic jump phenomenon and the transition from supersonic to subsonic flow in gas dynamics allows a unified treatment of the supercritical and subcritical regimes [166].Predictions for the flow and heat transfer were found to agree quite well with more computationally intensive two-dimensional numerical solutions. G . MOTIONOF THE IMPINGEMENT SURFACE

1. Axisymmetric Jet Impingement of Rotating Surfaces Several studies have investigated axisymmetric jets impinging on rotating surfaces. This finds application in simultaneous lubrication and cooling of power transmission gearing systems. This problem has been studied experimentally [176, 1771. The investigation dealt with the axis of the circular jet located coincident with the axis of rotation of an isothermal target surface (axisymmetric impingement), as well as at radial locations off the centerline (asymmetric impingement). Experiments were conducted for a range of target surface-to-jet diameter ratios, D/d. The average Nusselt number data (based on target surface diameter) were correlated in terms of the conventional jet Reynolds number Re, and the target rotational Reynolds number Re, = w D 2 / u . For the axisymmetric impingement configuration the data follow

which is valid for 16,000 I Re, I545,000, 180 IRe, I1300, and 87 I Pr I400. For the asymmetric impingement studies, an additional term describing the radial location of the circular jet relative to the axis of the rotating disk is necessary, and the correlation takes the form

where - rjet is the radial location of the jet. A nonmonotonic variation of Nu, with rjet/D observed in the experimental data has been neglected in this expression. Equation (118) is reported accurate for 180 I Red I 1300 and 0.2 I2rj,,/D I 0.8 for the same range of rotational Reynolds and

204


Prandtl numbers as Eq. (117). The empirical correlations reflect the increased average Nusselt number with target surface rotation. 2. Planar Jet Impingement of Moving Surfaces The impingement of planar liquid jets on moving surfaces arises in thermal treatment of metals. This has been treated experimentally [178] and analytically [179, 1801. In general, the results show that the use of heat/mass transfer correlations for the stationary plate configuration to predict the transport from moving surfaces may be inappropriate except for low surface velocities. In many cases, the impingement surface velocity may exceed the jet impingement velocity. The graphical data of Zumbrunnen [179] can be used to assess the importance of surface motion on heat transfer and the applicability of empirical correlations to movingsurface situations. VII. Conclusions and Recommendations for Further Research

This review summarized the body of literature related to transport under liquid jets. Significant progress has been made in understanding the fundamentals of heat, mass, and momentum transport in these systems. In addition, the following have been identified as areas where further research is needed. This list is not meant to be exhaustive, but may spawn interest in further research in these and other related areas. The analysis of impinging liquid metal jets has yet to be verified experimentally. Liquid metals will extend the characteristically high heat transfer coefficients due to their high thermal conductivity. The influence of free stream turbulence in liquids with such high molecular transport is uncertain. Such low Pr jets may find industrial application in ultrahigh heat flux situations. The local heat transport characteristics of turbulent axisymmetric jets have been quantified in the stagnation zone. However, despite the very thin thermal boundary layers there, this region accounts for only a small fraction of the total heat transfer under an axisymmetric jet; the majority of the total heat transfer takes place in the radial flow region. Heat transfer in the radial flow region for turbulent axisymmetric jets has yet to be adequately generalized. While the enhancing effect of free stream turbulence has been demonstrated for liquid jets, there is currently no generalized relationship between nozzle design and turbulence for a range of jet

SINGLE - PHASELIQUIDJET IMPINGEMENT

205

Reynolds numbers. Although it is true that the laminar flow correlations can safely be used as a lower bound on the transport, and that empirical correlations exist for idealized turbulent jets (e.g., fully developed turbulent pipe flow), these configurations are not likely to be encountered in most engineering applications owing to practical fluid delivery design considerations. The seriously degraded transport characteristics beyond the hydraulic jump may result in unacceptable temperature excursions there. Although there has been considerable experimental and analytical study of the jump phenomenon, there is as yet no acceptable predictive tool for determining the location of the hydraulic jump in a design setting. This is an area for further work. Liquid jet impingement cooling of moving surfaces has been briefly summarized here. However, the surface motion has been only in the direction normal to the axis of the jet. There has been no research investigating liquid jet cooling with surface motion parallel to the axis of the jet. This arises in the cooling of pistons in internal combustion engines. Air jet cooling of surfaces with curvature has been studied, and the results may be extended to submerged liquid jets. Free-surface liquid jet impingement heat transfer of curved surfaces might present some unusual hydrodynamic phenomena, because the liquid film thickness will be intimately coupled to the target curvature. This could influence dramatically the transport characteristics. Submerged and free-surface jet configurations are actually limiting cases. Situations could arise where partially submerged jets are encountered due to flooding of the impingement surface. To date, there appears to have been no work investigating the influence of a stagnant liquid layer of finite depth, which a free-surface jet must penetrate to reach the thermally active surface. Impingement heat transfer with gas jets has shown enhancement by the addition of a dilute dispersed second phase. This has not been explored for liquid jets. The dispersed phase in liquid jet systems may take the form of very small bubbles in an aerated jet or small solid particles suspended in the jet. There is considerable need for further research in liquid jet array applications, both in submerged and free-surface jet configurations. Cross-flow effects in these systems, which have been quite well characterized for submerged jets, have received only superficial treatment for free-surface jets. The physical phenomena here will be highly complex, requiring careful experimental investigation.

B. W.WEBBAND C.-F. MA

206

Acknowledgments The authors are grateful for the timely responses of researchers in the area, copies of whose published work was requested in the compiling phase of this review. The authors are also grateful to Prof. B. G. Pokusaev of the Institute of Thermophysics, Russian Academy of Sciences, who surveyed the Russian literature. The support of the U.S. National Science Foundation and the National Natural Science Foundation of China is also gratefully acknowledged.

Nomenclature ROMANL E ~ E R S

k

skin friction coefficient specific heat jet diameter local contracting jet diameter diameter of impingement surface mass diffusion coefficient frequency of jet pulsations function in transformed momentum, Eq. (9) Froessling number, Eq. (57) Froude number based on jet diameter L;. / @ Froude number based on nozzle-to-plate spacing uj/

k* I L

N

Kid Nu,, Nu,

NUd,o,NUw,o

NUd,o,2

&C

free stream velocity gradient at the impingement plate liquid film thickness; heat transfer coefficient local mass transfer coefficient stagnation heat transfer coefficient liquid layer thickness before and after hydraulic jump, respectively

- Nu,, Nu,

NUd,rnin

NUd,rnax

P

pl

thermal conductivity; rms surface roughness dimensionless surface roughness ( k / d ) nozzle hydrodynamic development length length of heated surface total number of jets in array average Nusselt number (%d/k) local Nusselt number (hr/k,h / k ) local Nusselt number (hd/k,hw/k) stagnation Nussdt number (h,d/k, h,w/k) secondary maximum local Nusselt number Nusselt number based on heater dimension (hD / k , hL / k ) average Nusselt number based on heater dimension ( X D / ~ , ~ Z L / ~ ) minimum local Nusselt number under jet array maximum Nusselt number under inclined jets local static pressure jet pressure

SINGLE -PHASELIQUIDJETIMPINGEMENT fluid Prandtl number (PCp/k)

Re,, Re,

total heat transfer from modified surfaces, Eq. (99) splattered jet flow total jet flow radial coordinate, axisymmetric jets; recovery factor, Eq. (41) dimensionless radial coordinate [ ( r / d ) / critical location for onset of turbulence radial displacement of peak Nusselt number from geometric stagnation point for axisymmetric jets radial location of hydraulic jump splattering radius radial location where thermal boundary layer reaches free surface radial location where viscous boundary layer reaches free surface jet Reynolds number

Re,, Re,

local Reynolds number

i'

207

local temperature adiabatic wall temperature jet temperature local wall temperature mean radial and axial velocity components, respectively rms fluctuations in u , L', respectively velocity in the inviscid free stream free-surface velocity rms fluctuations in U0 jet average velocity planar nozzle width Weber number ( p i f d / o ) coordinate parallel to impingement plate, planar jets coordinate normal to impingement plate, axisymmetric and planar jets coordinate measured from jet exit nozzle-to-plate spacing

( U J d / U ,f I J W / V ) I

( L ; r / u ,I ; X / U )

Re, Rex*

S

S

sc St Shd

rotational Reynolds number (oD2/u) local Reynolds number based on local free stream velocity, planar jets (Ux / u ) critical Reynolds number for onset of turbulence, planar jets ( L I ~ X/, v ) critical Reynolds number for onset of turbulence based on local free stream velocity, planar jets ( U x c / v ) center-to-center spacing of jets in jet array Strouhal number for pulsatile jets (fi/uJ) fluid Schmidt number ( u / ( B ) Stanton number (Nu, / Re, Pr) Sherwood number ( h , d / (Q)

GREEKL E ~ E R S thermal diffusivity wedge angle parameter hydrodynamic boundary layer thickness thermal boundary layer thickness variable wall flux parameter, Eq. (17) similarity coordinate, Eqs. (8) and (32) dimensionless temperature, Eq. (10); jet inclination angle relative to the impingement surface dynamic viscosity evaluated at adiabatic wall temperature dynamic viscosity evaluated at wall temperature

B. W. WEBBAND C.- F. M.4

P U

* w

fluid kinematic viscosity fraction of liquid mass splattered fluid density coefficient of surface tension similarity coordinate, Eqs. (7) and (31); cooling effectiveness, Eq. (84) splattering parameter, Eq. (102); or rotational speed of impingement surface

SUBSCRIPTS

lam P

par

unif mod

laminar potential core parabolic jet velocity distribution uniform jet velocity distribution jet array symmetry module

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21 1

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76. Metzger, D. E., Cummings, K. N., and Ruby, W. A. (1974). Effects of Prandtl number on heat transfer characteristics of impinging liquid jets. Heat Transfer, Proc. Int. Heat Transfer Conf. Sth, 1974, Vol. 2, pp- 20-24. 77. Churchill, S. W.,and Usagi, R. (1972). A general expression for the correlation of rates of transfer and other phenomena. AIChE J . 18, 1121-1128. 78. Baughn, J. W., and Shimizu, S. S. (1989). Heat transfer measurements from a surface with uniform heat flux and a fully developed impinging jet. J . Heat Transfer 111, 1096-1098. 79. Hrycak, P. (1983). Heat transfer from impinging jets to a flat plate. Int. J. Heat Mass Transfer 26, 1857-1865. 80. Obot, N. T., Majumdar, A. S., and Douglas, W. J. M. (1979). The effect of nozzle geometry on impingement heat transfer under a round turbulent jet. Am. Soc. Mech. Eng. [Pap.]19-WA/ HT-53. 81. Schafer, D., Incropera, F. P., and Ramadhyani, S. (1991). Planar liquid jet impingement cooling of multiple discrete heat sources. J . Electron. Packag. 113, 359-366. 82. Wadsworth, D. C., and Mudawar, I. (1990). Cooling of a multichip electronic module by means of confined two-dimensional jets of dielectric liquid. J. Hear Transfer 112, 891-898. 83. Schafer, D. M., Ramadhyani, S., and Incropera, F. P. (1992). Numerical simulation of laminar convection heat transfer from an in-line array of discrete sources to a confined rectangular jet. Numer. Heat Transfer, Part A 22, 121-141. 84. Di Marco, P., Grassi, W., and Magrini, A. (1994). Unsubmerged jet impingement heat transfer at low liquid speed. Proc. Int. Heat Transfer Conf., ZOth, Vol. 3, pp. 59-64. Hemisphere, New York. 85. Errico, M. (1986). A study of the interaction of liquid jets with solid surfaces. Ph.D. Dissertation, University of California at San Diego. 86. Hoyt, J. W., Taylor, J. J., and Runge, C. D. (1974). The structure of jets of water and polymer solution in air. J . Fluid Mech. 63, 635-640. 87. Chen, T.-F., and Davis, J. R. (1964). Disintegration of a turbulent water jet. J . Hydraul. Diu., Am. SOC.Ciu. Eng. HY-1, 175-206. 88. Nakoryakov, V. E., Pokusaev, B. G., and Shreiber, I. R. (1993). Waue Propagation in Gas-LiquidMedia. 2nd ed., CRC Press, Boca Raton, FL. 89. Bhunia, S. K., and Lienhard, J. H. V. (1993). Surface disturbance evolution and the splattering of turbulent liquid jets. J . Fluids Eng. 116, 721-727. 90. Grassi, W., and Magrini, A. (1991). Effect of the free jet fluid dynamics on liquid jet impingement heat transfer. Proc. Natl. Heat Transfer Conf., Italy, 9th, 1991, pp. 410-433. 91. Stevens, J., and Webb, B. W. (1992). Measurements of the free surface flow structure under an impinging free liquid jet. J. Heat Transfer 114, 79-84. 92. Stevens, J., Pan, Y.,and Webb, B. W. (1992). Effect'of nozzle configuration on transport in the stagnation zone of axisymrnetric, impinging free-surface liquid jets. Part 1. Turbulent flow structure. J . Heat Transfer 114, 874-879. 93. Stevens, J., and Webb, B. W. (1993). Measurements of flow structure in the stagnation zone of impinging free-surface liquid jets. Int. J . Heat Mass Transfer 36, 4283-4286. 94. Olsson, R. G., and Turkdogan, E. T. (1966). Radial spread of a liquid stream on a horizontal plate. Nature (London) 211, 813-816. 95. Azuma, T., and Hoshino, T. (1984). The radial flow of a thin liquid film. First report. Laminar turbulent transition. Bull. JSME 27,2739-2746. 96. Azuma, T., and Hoshino, T.(1984). The radial flow of a thin liquid film. Second report. Liquid film thickness. Bull. JSME 27, 2747-2754.


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136. Nanzer, J. O., and Coeuret, F. (1984). Distribution of local mass transfer coefficients over one electrode bombarded by submerged multijets of electrolyte. J. Appl. Electrmhem. 14, 627-638. 137. Slayzak, S. J., Viskanta, R., and Incropera, F. P. (1994). Effects of interaction between adjacent free surface planar jets on local heat transfer from the impingement surface. Int. J. Heat Mass Transfer 37, 269-282. 138. Slayzak, S. J., Viskanta, R., and Incropera, F. P. (1994). Effects of interactions between adjoining rows of circular, free-surface jets on local heat transfer from the impingement surface. J. Heat Transfer 116, 88-95. 139. Sparrow, E. M., and Lovell, B. J. (1980). Heat transfer characteristics of an obliquely impinging circular jet. J . Heat Transfer 102, 202-209. 140. Goldstein, R. J., and Franchett, M. E. (1988). Heat transfer from a flat surface to an oblique impinging jet. J. Hear Transfer 110, 84-90. 141. Ma, C.-F. ef al. (1994). Local characteristics of impingement heat transfer with oblique round free-surface jets of large Prandtl number liquid. To be published. 142. Sun, H. et al. (1994). Impingement heat transfer from a vertical flat surface to oblique submerged jets of large Prandtl number liquid. To be published. 143. Beltaos, S. (1976). Oblique impingement of circular turbulent jets. J . Hydraul. Res. 14, 17-36. 144. Donaldson, C. D., and Snedeker, R. S. (1971). A study of free jet impingement. Part 1. Mean properties of free and impinging jets. J. Fluid Mech. 45, 281-319. 145. Rubel, A. (1982). Oblique impingement of a round jet on a plane surface. A M J. 20, 1756-1758. 146. Sullivan, P. F., Ramadhyani, S., and Incropera, F. P. (1992). Use of smooth and roughened spreader plates to enhance impingement cooling of small heat sources with single circular liquid jets. Am. Soc. Mech. Eng., Heat Transfer Dio., HTD-206(2), 103-110. 147. Schlichting, J. (1960). Boundav Layer Theory, 4th ed. McGraw-Hill, New York. 148. Heindel, T. J., Ramadhyani, S., Incropera, F. P., and Campo, A. (1992). Surface enhancement of a heat source exposed to a circular liquid jet with annular collection of the spent fluid. Am. Soc. Mech. Eng., Heat Transfer DiL1. HTD-206(2), 111-118. 149. Sullivan, P. F., Ramadhyani, S., and Incropera, F. P. (1992). Extended surfaces to enhance impingement cooling with single circular jets. Pruc. ASME/JSME J. Conf. Electron. Packag. EEP-Vol. 1-1, pp. 207-216. 150. Priedeman, D., Callahan, V., and Webb, B. W. (1993). Enhanced surface liquid jet impingement heat transfer. Am. Soc. Mech. Eng., Heat Transfer Di". HTD-263, 43-48. 151. Priedeman, D., Callahan, V., and Webb, B. W. (1994). Enhancement of liquid jet impingement heat transfer with surface modifications. J. Hear Transfer 116, 486-489. 152. Wadsworth, D. C., and Mudawar, I. (1992). Enhancement of single-phase heat transfer and critical heat flux from an ultra-high-flux simulated microelectronic heat source to a rectangular impinging jet of dielectric liquid. In Adcances in Electronic Packaging 1992, Vol. 1, pp. 143-151. ASME, New York. 153. Wadsworth, D. C., and Mudawar, I. (1992). Enhancement of single-phase heat transfer and critical heat flux from an ultra-high-flux simulated microelectronic heat source to a rectangular impinging jet of dielectric liquid. J. Heat Transfer 114, 764-768. 154. Teuscher, K. L., Ramadhyani, S., and Incropera, F. P. (1993). Jet impingement cooling of an array of discrete heat sources with extended surfaces. A m . Soc. Mech. Eng., Heat Transfer Dic. HTD-263, 1-10. 155. Bhunia, S. K., and Lienhard, J. H. V. (1994). Splattering during turbulent liquid jet impingement on solid targets. J. Fluids Eng. 116, 338-344.

B. W. WEBBAND C.- F. RIA 156. Kestin, J. (1966). The influence of freestream turbulence on heat transfer rates. Adu. Heat Transfer 3, 1-32. Academic Press, New York. 157. Zumbrunnen, D. A., and Aziz, M. (1993). Convective heat transfer enhancement due to intermittency in an impinging jet. J. Heat Transfer 115, 91-98. 158. Zumbrunnen, D. A. (1992). Transient convective heat transfer in planar stagnation flows with time-varying surface heat flux and temperature. J. Heat Transfer 114, 85-93. 159. Sheriff, H. S., and Zumbrunnen, D. A. (1993). Effect of flow pulsations on the cooling effectiveness of an impinging planar water jet. Am. Soc. Mech. Eng., Heat Transfer Diu. HTD-249, 11-21. 160. Azevedo, L. F. A., Webb, B. W., and Queiroz, M. (1994). Pulsed air jet impingement heat transfer. Exp. Therm. Fluid Sci. 8, 206-213. 161. Disimile, P. J. (1994). Effect of impinging jet excitation on curved surface heat transfer. J. Propuls. 10, 293-294. 162. Mladin, E. C., and Zumbrunnen, D. A. (1993). Nonlinear dynamics of laminar boundary layers in stagnation flows due to temporal changes in flow velocity and surface heat flux. Am. Soc. Mech. Eng., Heat Transfer Diu. HTD-249, 23-34. 163. Liu, X., and Lienhard, J. H. V. (1993). The hydraulic jump in circular jet impingement and in other thin liquid films. Exp. Fluids 15, 108-116. 164. Thomas, S., Hankey, W., Faghri, A,, and Swanson, T. (1990). One-dimensional analysis of the hydrodynamic and thermal characteristics of thin film flows including the hydraulic jump and rotation. J. Heat Transfer 112, 728-735. 165. Thomas, S., Faghri, A., and Hankey, W. (1991). Experimental analysis and flow visualization of a thin liquid film on a stationary and rotating disk. J. Fluids Eng. 113, 73-80. 166. Rahman, M. M., Hankey, W. L., and Faghri, A. (1991). Analysis of the fluid flow and heat transfer in a thin liquid film in the presence of gravity. Int. J. Heat Mass Transfer 34, 103-114. 167. Craik, A. D. D., Lathan, R. C., Fawkes, M. J., and Gribbon, P. W. F. (1981). The circular hydraulic jump. J. Fluid Mech. 112, 347-362. 168. Rahman, M. M., Faghri, A., and Hankey, W. L. (1992). Fluid flow and heat transfer in a radially spreading thin liquid film. Numer. Heat Transfer, Part A 21, 101-120. 169. Faghri, A., Thomas, S., and Rahrnan, M. M. (1993). Conjugate heat transfer from a heated disk to a thin liquid film formed by a controlled impinging jet. J. Heat Transfer 115, 116-123. 170. Rahman, M . M., and Faghri, A. (1992). Analysis of heating and evaporation from a liquid film adjacent to a horizontal rotating disk. Int. J. Heat Mass Transfer 35, 2655-2664. 171. Rahman, M. M., and Faghri, A. (1992). Numerical simulation of fluid flow and heat transfer in a thin liquid film over a rotating disk. Int. J. Heat Mass Transfer 35, 1441-1453. 172. Rahman, M. M., Fdghri, A., and Hankey, W. L. (1990). New methodology for the computation of heat transfer in free surface flows using a permeable wall. Numer. Heat Transfer, Part B 18, 23-41. 173. Rahman, M. M., Faghri, A., Hankey, W. L., and Swanson, T. D. (1990). Prediction of heat transfer to a thin liquid film in plane and radially spreading flows. J. Heat Transfer 112, 822-825. 174. Rahman, M. M., Faghri, A., Hankey, W. L., and Swanson, T. D. (1990). Computation of the free surface flow of a thin liquid film at zero and normal gravity. Numer. Heat Transfer, Part A 17, 53-71.

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175. Rahman, M. M., Faghri, A., and Hankey, W. L. (1994). Computation of turbulent flow in a thin liquid layer of fluid involving a hydraulic jump. J . Fluids Eng. (in press). 176. Carper, H. J. Jr., Saavedra, J. J., and Suwanprateep, T. (1986). Liquid jet impingement cooling of a rotating disk. J . Heat Transfer 108, 540-546. 177. Carper, H. J., and Defenbaugh, D. M. (1978). Heat transfer from a rotating disk with liquid jet impingement. Heat Transfer, Roc. Znt. Heat Transfer Conf., 6th, 1978, Vol. 4, pp. 113-118. 178. Zumbrunnen, D. A,, Incropera, F. P., and Viskanta, R. (1990). Method and apparatus for measuring heat transfer distributions on moving and stationary plates cooled by a planar liquid jet. Exp. Therm. Fluid Sci. 3, 202-213. 179. Zumbrunnen, D. A. (1991). Convective heat and mass transfer in the stagnation region of a laminar planar jet impinging on a moving surface. J. Heat Transfer 113, 563-570. 180. Zumbrunnen, D. A., Incropera, F. P., and Viskanta, R. (1992). A laminar boundary layer model of heat transfer due to a nonuniform planar jet impinging on a moving surface. Warme-und Stoffuberfragung.27, 311-319.


ADVANCES IN HEAT TRANSFER, VOLUME 26

Thermal Design Theory of Three-Fluid Heat Exchangers D. P. SEKULIC* Department of Mechanical and Industrial Engineering Marquette University, Milwaukee, Wisconsin

R. K. SHAH Harrison Dicision General Motors Corporation Lockport, New York

I. Introduction

A. MOTIVATION FOR REVIEW The thermal and hydraulic design theory of a direct-transfer-type twofluid heat exchanger (tubular, plate-type, and extended-surface exchangers) is well developed and is available in standard heat transfer literature [ 1-31. However, the complete heat exchanger design problem is complex because in addition to the thermal and hydraulic design, one needs to consider the mechanical/structural design, manufacturing and cost considerations, trade-offs, and system-based optimization as discussed by Shah [4].In short, heat exchanger design involves a number of considerations in addition to heat transfer that are equally important in the engineering decision procedure. The well-established algorithm for thermal design of a two-fluid heat exchanger, however, has no adequate equivalent when the physical situation implies more than one thermal communication. We define thermal communication as heat transfer from one fluid to another, such as one thermal communication in a conventional (adiabatic) two-fluid heat ex-

* Permanent address: Department of Mechanical Engineering, University of Novi Sad, 21 121 Novi Sad, Yugoslavia. 219

Copyright 0 1995 hy Acddemlc Pres5. Inc All rightc of reproduction in m y form re5erved

D. P. SEKULIC AND R. K. SHAH

changer. One of the first practical problems of this type, noticed by engineers long ago, is a conventional two-fluid heat exchanger with heat losses to ambient. In this nonadiabatic exchanger, an additional thermal communication with (i.e., heat losses to) the environment has to be included [5-81. An even more complex situation arises when the second and/or the third thermal communication is introduced by the third fluid stream in the so-called three-fluid heat exchanger [9-131. Finally, a very complex situation corresponds to the interrelation of more than three fluids, that is, in the case of multistream and/or multifluid heat exchangers when more than two simultaneous thermal communications exist [14-171. Most heat exchanger applications in the process, power, transportation, thermal energy recovery, electronics, and aerospace industries involve transfer of thermal energy between two fluids through one thermal communication in two-fluid heat exchangers. In the production of cryogenic temperatures in the gas processing and petrochemical industries, however, transfer of thermal energy often takes place among three or more fluids or fluid streams. Three-fluid and multifluid heat exchangers are widely used in cryogenics and some chemical processes, for example, air separation systems, helium-air separation units, purification and liquefaction of hydrogen, and ammonia gas synthesis. The reasons for bringing more than two fluids into thermal contact might be different in different applications. For example, chemical processes carried out at low temperature are dominated by requirements for very small temperature differences between streams exchanging heat because of the very high cost associated with compressor power to achieve the desired cryogenic temperatures. This led to the development of heat exchangers that are very high compact, and this is associated with a large number of flow passages for streams [18]. The number of flowing streams in the exchanger is set by the process as well as by the flow rate and terminal conditions. Compact multifluid heat exchangers can result in significant savings in overall costs and space [19]. A three-fluid (or three-fluid-stream) heat exchanger may be desirable or even necessary due to space constraints (as in the aerospace industry; see, for example, Schubel [20]) and/or overall system thermal balance considerations (as in cryogenics and low-temperature refrigeration processes; Abadzic and Scholz [21]). Schubel suggested an application in which the hot fluid is engine lubricating oil. The two cold fluids are (a) fuel circulating from the aircraft fuel tank through the oil cooler and then back to the tank and (b) fuel from the outlet of the engine control flowing through the cooler and then to the engine fuel nozzles. In the aerospace industry, multifluid heat exchangers are used also in applications in which

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THREE FLUIDHEATEXCHANGER THERMAL DESIGN

22 1

a redundant cooling or heating fluid circuit is required to improve overall system reliability [19]. Finally, we note that in a number of modern applications that involve a multifunctional unit, more than two fluids are in thermal contact. For example, the design and construction of micromechanical components and devices might involve heat exchange between more than two fluids as in the case of a micro compact heat exchanger discussed by Friedrich et al. [22]. Another example is related to the oil-bearing strata exploitation at great depths when the problem solution requires the interaction of three fluid flows between the earth’s surface and a deep underground oil-bearing strata level [23]. The thermal design procedure in most such applications should follow an approach similar to that for a three-fluid (or multifluid) heat exchanger. It is noteworthy that, whereas a considerable number of papers have been published in the literature on thermal design theory of three-fluid and multifluid heat exchangers, unified design approach exists. Published analyses are not systematic, and a clear and relatively simple design procedure is not available. In contrast, interest in three-fluid and multifluid heat exchanger applications is continually increasing. Consequently, the primary objective of the present work is to outline the thermal design theory of a single-pass three-jluid heat exchanger with two thermal communica tions .

B. SCOPEOF REVIEW The complexity of the design and analysis of two-fluid heat exchangers has led to an impressive list of information published in a vast number of relevant references during the past 80 years or so. It is quite clear that the involvement of an additional fluid stream, a third one, will increase the complexity of the design of this new class of heat exchangers. Thus, the number of important issues concerning the design and the scope of the analysis will be much broader. The relevant literature, however, is not as abundant as in the case of two-fluid heat exchangers. Consequently, a compilation of available data and a thorough analysis of three-fluid heat exchangers should include, in addition to a complete review of existing knowledge, inquiry regarding a number of still unanswered questions. An effort to present comprehensive insight into the subject in an article like this is, therefore, too ambitious a task. The questions for which this article might provide some answers, therefore, are rather restricted in scope. Major attention is devoted to the most important aspect of heat exchanger analysis: the fluid temperature distribution for a linear problem formulation. Two major types of three-fluid heat exchangers are considered in detail: (a> the parallel stream and (b)

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the cross-flow three-fluid heat exchangers with two thermal communications. Related topics include temperature distributions, overall and temperature effectivenesses, temperature cross phenomena and thermal design procedures for both rating and sizing problems. It is worth noting that the term thermal design, in the sense in which it will be used in this article, has a somewhat restricted meaning. We outline only the effectivenessNTU (number of heat transfer units) approach and corresponding rating and sizing problems for the determination of the effectiveness or NTU for a three-fluid heat exchanger. Pressure drop analysis and determination of the physical size of a three-fluid heat exchanger are not covered here. Finally, the thermal design theory of multifluid heat exchangers (having more than three-fluid streams) and multistream plate-fin heat exchangers and the study of nonlinear problems are not considered here due to space and time constraints.

11. Classification of Three-Fluid Heat Exchangers

A. DEFINITION OF A THREE-FLUID HEATEXCHANGERS

In a three-fluid heat exchanger, heat transfer takes place among the three fluid streams. Therefore, an adiabatic three-fluid heat exchanger can be defined as a thermal device that provides for change of mutual thermal energy levels’ among three fluid streams in thermal contact, without external heat and work interactions [24]. The term “the stream” instead of “the fluid” is used here to emphasize the fact that a three-fluid heat exchanger can have one or more fluid streams of the same fluid at different temperatures with the remaining fluid streams being a different fluid, or all three streams of different fluids. In a three-fluid heat exchanger, three fluid streams enter and three streams leave the exchanger. Throughout the discussion of three-fluid heat exchangers in the literature, it is most often assumed that one hot fluid is transferring heat to two colder fluids or one cold fluid is receiving heat from two hotter fluids [25]. There may or may not be any heat transfer between two cold (or hot). ‘The wording “change of mutual thermal energy levels” is used as a generalization for the more appropriate expression “enthalpy change” instead of the conventional expression “heat transfer.” Namely, heat transfer is only a consequence of the fulfillment of the actual engineering goal, that is, to change the mutually thermal energy levels of the fluid streams involved.

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FIG. 1. Three-fluid heat exchangers: (a)-(c) two thermal communications; (d)-(f) three thermal communications.

fluids. In this article, we distinguish the fluids as fluids 1, 2, and 3 regardless of which fluids are hot and cold. Depending on how heat transfer takes place, the following are the categories (see Fig. 1): Only one fluid stream transfers heat to the other two fluid streams [two thermal communications among the fluid streams, Figs. l(a), (b), and (c)]. All three fluid streams transfer heat among each other [three thermal communications among the fluid streams, Figs. l(d), (e), and (f)]. The three-fluid heat exchangers are to be classified according to the heat transfer process as indirect-contact direct-transfer-type heat exchangers. Classification of three-fluid heat exchangers according to their construction and flow arrangements is discussed next.

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B. CLASSIFICATIONS OF THREE-FLUID HEATEXCHANGERS Three-fluid heat exchangers can be classified in many different ways. We classify them here according to the construction and flow arrangement.

1. Classification according to Construction

a. Tubular Three-Fluid Heat Exchangers The most common type of construction is a tubular three-fluid heat exchanger with one fluid transferring heat to the other two, or all three fluids transferring heat to each other. Two main categories of tubular exchangers are a triple-pipe tubular heat exchanger and a shell-and-tube-type heat exchanger. Schematics of two triple-pipe heat exchangers are presented in Fig. 2.

Fro. 2. Tubular triple-pipe type three-fluid heat exchangers: (a) three thermal communications; (b) two thermal communications.

a

Fluid 2 in

Fluid 2 out

t

Out

out

, Fluid1

out

+

Fluid 4

Fluid

out

out

-

Fluid 1 in Fluid 3 in Fluid in

Fluid 2 out

Fluic out

-

Fluid 2 in

Flud 3in

Fluid 3 out

-Fluid

C

Fluid1 in

Fluid3 in

1 out

Fluid1 in

r

Fluid 2 in

\

'J

PFluid 1 out

'Fluid 3 out

Fluid 2 out

FIG. 3. Shell-and-tube rnultifluid heat exchangers: (a) three-fluid heat exchanger, from Schneller [261; (b) from Gersh [271; (c) from Schubel "201; and (d) four-fluid heat exchanger D71.

D. P. S E K U LAND I ~ R. K. SHAH

226

a

b

1

Fluid 1 out

FIG. 4. Hampson heat exchanger: (a)'two-fluid paired tube multistrearn heat exchanger, from Kao [29]; (b) three-fluid multistream heat exchanger, from Gersh [271; (c) four-stream wound coil heat exchanger, from Mollekopf and Ringer [301.

-

THREEFLUIDHEATEXCHANGER THERMAL DESIGN

227

A triple-pipe three-fluid heat exchanger is the simplest possible construction for a multifluid heat exchanger; fluid flow passages are formed by three pipes (or pipe-like flow passages) to allow necessary thermal communications among three fluid streams. A shell-and-tube multifluid heat exchanger can be constructed in several different ways. Some examples are shown in Fig. 3 [26, 271. In contrast to the triple-pipe three-fluid heat exchanger, the shell-and-tube three-fluid heat exchanger has more complex fluid flow arrangements, which in general cannot be classified as parallel stream or cross-flow three-fluid exchangers. A special type of tubular two-, three-, or multifluid multistream heat exchanger is the so-called Hampson heat exchanger [18, 21, 281 as shown in Fig. 4 [29, 301. This is a coiled-tube exchanger that consists of a large number of equal lengths of tubes wound in helics around a central core tube. b. Extended Surface Plate-Fin Three-Fluid Heat Exchangers Extended surface (plate-fin) three-fluid (and multifluid) heat exchangers are generally compact heat exchangers used mostly in the process and cryogenic industries. An example is shown in Fig. 5 [31]. This heat exchanger is constructed by stacking alternate layers of corrugated surfaces (fins) between flat separator plates and then brazing the whole assembly. The entire heat exchanger might consist of more than one hundred layers.

2, Classification according to Flow Arrangement Two major types of flow arrangements of a three-fluid heat exchanger are the parallel stream arrangement (characterized by a parallel flow of fluid streams with respect to each other) and the cross-pow arrangement (characterized by the perpendicular direction of one fluid stream in relation to the other two). These two types are divided into several subtypes as follows depending on the specific orientation of the fluid streams. a. Parallel Stream Arrangements Three different parallel stream flow arrangements, as shown in Fig. 6, are (a) cocurrent (corresponding to Pl), (b) countercurrent (P2), and (c) countercurrent-cocurrent (P3) or cocurrent-countercurrent (P4). In the cocurrent parallel flow arrangement (Pl), all three streams flow in the same direction. In countercurrent flow arrangements (stream couplings P2-P4), one of the streams is flowing in a direction opposite to the other two.* 'A detailed discussion regarding the flow arrangement versus the number of stream couplings is given in Sec. VI.B.1.

D. P. S E K U W AND ~ R K. SHAH

228 a

1 Separatorplates 4

Side bars

7 Nozzles

v"out

in

1

b

Stream A out m

____

~

Repeating $pattern

-- _ - ------

Repeating pattern

6

Three-Passage Pattern

Four-Passage Pattern

FIG. 5. A plate-fin three-fluid heat exchanger: (a) schematic, from Diery [31]; (b) three-passage (ABC) and four-passage (ABCB) repeating pattern of passages.

If each stream transfers heat with the other two streams (three thermal communications), two different flow arrangements are possible: cocurrent and countercurrent (countercurrent-cocurrent or cocurrent-countercurrent couplings became the same as countercurrent). However, if only two streams are in mutual thermal communication, all three flow arrangements are possible. The P4 arrangement shown in Fig. 6 is identical to the P3 arrangement from the analysis point of view (identical effectiveness-NTU, formula) as long as we rename fluids 1 and 3 of the P4 arrangement as new fluids 3 and 1, and use the temperature distribution or effectiveness-NTU, formula for P3. In other words, one should identify two thermal communications by not using respective pairs of fluid numbers, but the

THREE- FLUIDHEATEXCHANGER THERMAL DESIGN

229

FIG. 6. Parallel stream couplings of a three-fluid heat exchanger: P1, cocurrent parallel flow; P2, countercurrent parallel flow; P3, countercurrent-cocurrent parallel flow; and P4, cocurrent-countercurrent parallel flow.

cocurrent-countercurrent attributes of the adjacent fluids. Hence, we do not need to identify P4 as an independent (distinct) flow arrangement from the analysis point of view. However, for a triple-pipe exchanger, for example, the P3 and P4 arrangements could represent different physical solutions and hence different performances as further discussed in Sec. VI. As a result, the P4 arrangement is included in this article. b. Cross-flow Arrangements Three-fluid single-pass heat exchanger arrangements with two thermal communications are shown in Fig. 7(a) and consider the mutual direction of the streams involved. Because each of the three fluids can be oriented in cross-flow with respect to the other two, many more different arrangements are possible than for the case of a parallel stream three-fluid heat exchanger. The number of possible crossflow arrangements depends not only on the mutual direction of the streams involved but also on the mixing condition of each fluid within its

D. P. S E K U LAND I ~ R K. SHAH

230

a C1

c3

c2

b

cTP1

CTP3

FIG. 7. (a) The cross-flow arrangements of three-fluid heat exchangers (two thermal communications): C1, cocurrent cross-flow; (2,countercurrent cross-flow; C3, cross-countercurrent flow; C4, cross-cocurrent flow. (b) Two-pass cross-flow arrangements of three-fluid heat exchangers (two thermal communications): CTP1, cross-cocurrent, overall cocurrent flow of the central fluid; W 2 , cross-cocurrent, overall countercurrent flow of the central fluid; and CTP3, cross-countercurrent flow.

own flow passage. Each of the three streams can be either completely mixed or unmixed or various combinations thereof. In addition, the number of configurations increases even more if either three or two thermal communications among the streams have to be included. In this paper, we focus our attention mainly on cross-flow arrangements with two thermal communications among the fluids and without lateral mixing in any fluid stream. Of particular interest is a cross-cocurrent flow arrangement [Fig. 7(a), C41 in which the two lateraI fluid flow streams are in a mutual cocurrent

-

THREE FLUID HEATEXCHANGER THERMAL DESIGN

23 1

arrangement [32-341. Of the several geometrical arrangements possible with three fluids in cross-flow, the one most likely to be of practical interest is the cross-cocurrent flow arrangement (C4) according to Willis and Chapman [32]. This is because in many cases this flow arrangement has a higher overall heat exchanger effectiveness than that for the crosscountercurrent arrangement (C3). The proof for this statement was given by Ellis [35](see the detailed discussion in the Sec. IV.A.2). The multipass design is considered in two-fluid heat exchanger applications when the design of a heat exchanger results in either an extreme length, significantly low fluid velocities, a low effectiveness, or other design criteria. Figure 7(b) presents three two-pass three-fluid cross-flow heat exchanger arrangements. For two-pass configurations, there are three possibilities for the behavior of the fluids in the elbow sections of multipass exchangers: (a) mixed, (b) unmixed, identical order, and (c) unmixed, inverted order. The terminology used is the same as in the two-fluid exchanger design theory (see also Sec. IV.A.2). 111. Generalized Form of the Model Formulation

and Dimensionless Groups A. GENERALIZED FORM OF

THE

MODELFORMULATION

The detailed formulation of the mathematical model is elaborated on later on in Sec. V. However, a generalized form of the analytical model is discussed here in order to present a general overview of the problem formulation and to discuss clearly the literature information. We use this model to determine temperature fields within a three-fluid heat exchanger. Basic equations, as derived from the energy balance and rate equations, are outlined in Sec. V for parallel stream as well as cross-flow three-fluid heat exchangers. As shown in Sec. V.C.l, generalized governing equations for a three-fluid heat exchanger in a parallel stream arrangement (for determining one-dimensional temperature fields in the x direction) can be represented in the form:

The summation on the right side of Eq. (1) depends on the number of thermal communications between the fluids (i.e., two terms for three thermal communications, and one or two terms for two thermal communications among the streams). Quantities Cj represent flow stream heat

232


capacity rates, and Ti and Tk denote, respectively, temperatures of fluid streams j and k at a cross section x along the heat exchanger of length L. Note that in the case of a two-fluid heat exchanger, the parameter q k Ajk/Cj is designated as NTU [361 if C j represents a minimum of C, and C,. In order for Eq. (1) to be valid in all cases, the equation must be directionalized; that is, i j is positive if the fluid flow rate is in the positive direction of the corresponding coordinate x (or y or z ) , and ij is negative if the flow is in the opposite direction. The positive direction is arbitrary. The quantity A j k represents the heat transfer surface area for the j’th stream exchanging heat with the k’th stream. For example, in a plate-fin heat exchanger for the flow in a single channel, A , represents one-half of the total channel surface area. When there is no heat transfer between the j’th and k’th channels, q k (which represents the overall heat transfer coefficient between the j’th and k’th channels based on the heat transfer area A j k )is taken as zero. Note that qj = 0 and qkAjk = UkiAkj. To solve Eq. (1) for a design problem, boundary conditions must be specified. The most simple problem is the case for which the temperature of each fluid at one end of the exchanger is known. If temperatures are specified for some fluids at one end and for the rest at the other end, the problem becomes more complicated. In both cases, it is assumed that all qkA,k/c, are specified (rating problem). However, when all inlet and outlet temperatures are prescribed and the exchanger area is to be determined, as in sizing a heat exchanger, the problem becomes even more complicated. As is shown later in Secs. V.E.l and V.E.2, the set of governing equations for both parallel stream and cross-flow three-fluid heat exchanger arrangements has the same generalized dimensionless form as follows:

where 0 is the nondimensional temperature and 6 is the nondimensional coordinate. Coefficients ujk have the following properties: ajk = 0 for j = k or ajk = 0 for j # k but for no thermal communication between the fluid streams j and k. The coefficients a,k represent dimensionless heat conductances between fluids j and k [see, for example, Eqs. (201, (21), and (22)l. The corresponding idealizations are discussed in Sec. V.3 The number of terms on the right side of Eq. (2) accounts for the number of thermal interactions for the corresponding fluid stream in a 3Note that the equations are linear if the thermophysical properties of fluids and heat transfer coefficients for each stream can be assumed constant.

-


233

FIG. 8. Three-fluid cross-flow heat exchanger arrangement with three-dimensional temperature distributions for all three fluids: (a) A possible design solution; (b) stream orientation.

concrete situation. For a heat exchanger with three thermal communications, the right side of Eq. (2) has two terms for each j because each fluid communicates thermally with the other two fluids. In contrast, for a heat exchanger with two thermal communications, one of the fluids is in thermal interaction with the other two fluids (for example, fluid j = 2), but each of the other two fluids has only one thermal interaction (fluid j = 1 and fluid j = 3). Let us consider a few distinct situations. For a three-fluid cross-cocurrent flow heat exchanger with two thermal communications [arrangement C4,Fig. 7(a>l,the set of equations given by Eq. (2) consists of three partial differential equations. Fluid 2 has a partial derivative of the fluid stream temperature with respect to tj = 6 = x / X , , j = 2 on the left side of Eq. (2) with the sum of two terms on the right side. Two remaining partial differential equations for fluids 1 and 3 have partial derivatives of the respective fluid stream temperature with respect to 6, = 77 = y/Y,, j = 1 or 3. In both cases, the derivatives are equal to the single term on the right side of the corresponding Eq. (2). In the case of the three-fluid heat exchanger of Fig. 8 (two thermal communications among the streams), three equations [Eq. (2)] contain partial differentials of stream temperatures with respect to = 6 , t2= 7, and t3= f , respectively. Equation (2), for j = 2 (in the x , i.e., 7) direction), has only one term on the right side. The same situation holds for j = 3. Finally, the equation for the first fluid ( j = 1) has two terms on the right side.

D.P. S E K U LAND I ~ R. K.SHAH

234

In conclusion, if all temperature derivatives [see Eq. (2)] are related to the same independent variable (i.e., tj = 6 for all j ) , the corresponding heat exchanger has a parallel stream arrangement (i.e., Oj = O j ( O for j = 1,2,3). If at least one derivative is with respect to another independent variable (for example, ti= 5 for j = 2 and ti = r) for j = 1,3), temperature fields are two dimensional, that is, the cross-flow arrangement [Oj = @,(&, 17)] holds. Finally, if all three derivatives are with respect to different independent variables (for example, t1= (, t2= r), and t3= 61, temperature fields are three dimensional [cross-flow, with Oj = @ j ( t , 17, 5)1. A similar model [Eq. (211 also holds for more than three fluids (streams) and, in such a case, the model represents the set of n equations for a multifluid heat exchanger [14, 15, 37, 381. For example, a system of equations for a parallel stream multifluid heat exchanger (n fluids all in mutual thermal interactions) can be written in matrix form as:

n @,

k= 1 n

0 2

k-1

an1

...

...

'

(3)

n

* * *

-

c

ank

@n

k=l

Because Eqs. (11, (21, and (3) represent first-order linear ordinary (for parallel streams) or partial (for cross-flow) differential equations, the solution will be of an exponential form (for parallel stream arrangements, Wolf [14]) or expressed by a class of special functions (for cross-flow three-fluid arrangements, BaEliC et af. [34]). Details of the model formulation and solution procedure for three-fluid heat exchangers are given in Secs. V. and VI. The general solution for a multifluid parallel stream exchanger [Eq. (3)] is given by Wolf [14], Settari [391, and Zaleski and Jarzebski [38]. B. DIMENSIONLESS GROUPS From Eq. (1) or (2), one can formulate independent dimensionless groups (parameters) for the temperature distributions; ajk as well as additional parameters from the boundary conditions (see, for example, 0, in in Table I or Atin in Table 11). The total number of independent

THREE- FLUIDHEATEXCHANGER THERMAL DESIGN

235

TABLE I DIMENSIONLESS GROUPSFOR THREEFLUID PARALLEL STREAMHUT EXCHANGERS WITH TWO THERMAL COMMUNICATIONS Dimensionless independent parameters Symbol

Definition

Title

Meaning

-

NTU,

(UAh.2

Number of transfer units Thermal size measure

(%)I

CT.2

R*

(~")' (%)*

(uA)3.2

Thermal balance Measure

Heat capacity rate ratio

Thermal balance measure

Conductance ratio

Heat conductance balance

Inlet temperature ratio

Inlet relative temperature levels

(UA h . 2 T3.in

@ 3 , in

Heat capacity rate ratio

- Tl.in

T2.in -

Dimensionless dependent variables (effectivenesses) 91

TI.,,^

-

T1,in Temperature

T ~in, - T,.in

E

Q Q,,

effectiveness of fluid 1

Three-fluid heat exchanger effectiveness

The degree to which the temperatures of outer fluids have approached the rature of the nuld Measure of the exchanger performance

groups is five for two thermal communications and six for three thermal communications in a three-fluid exchanger. To specify a three-fluid heat exchanger design problem in terms of design-oriented quantities, the parameters a . can be regrouped into a new set, which also has physical Jk (i.e., engineering) interpretation. In Table I, these dimensionless groups are summarized for a three-fluid heat exchanger with two thermal communications. A detailed analysis of how these parameters were arrived at, their physical meaning, and an order of magnitude and ranges of values are given after the detailed mathematical model. In the literature, there is

TABLE I1 COMPARISON OF LITERATURE VERSUS PROPOSED NOMENCLATURE FOR DIMENSIONLESS GROUPSFOR HEATEXCHANGER WITH Two OR THREE THERMAL COMMUNICATIONS Present work" Comm.

Sorlie [111

Willis [40] Ellis [35]

Fluid 1 Fluid 2 Fluid 3

2 cl h c2

m 1

Ntu

NTUl

c:

Kl

c : 2

THREE-FLUID

BaEliC

Aulds and Barron 1121

Rabinovich [46]

et al. (341

2 3 1

2

"I; I;

'I;

Kl

2

R31

K3

c;2

R*

Demetry and Platt [19]

A

_ Rl U2A2.1 -~

R*

R2

UIA, '2

=

h

=i

-

c

-c

e;

9 3 , in

Additional parameter 91

93

E l . cl

@I

01

ei

0 3 ~

0,=

Tz.out - Tz.in TI. in - 7'2, in

~~~

'See Table I. Symbols for all dimensionless groups are kept as in the original references. However, all definitions (the right side of the equality sign) are presented in terms of the nomenclature in this article. When definitions are not presented, they are identical to the present definitions. bDirnensionless outlet temperatures.

THREE- FLUIDHEAT EXCHANGER THERMAL DESIGN

237

no unified approach to the selection of governing independent parameters or a consistent nomenclature. Table I1 provides a comparison of nomenclature used in the literature to the present terminology for heat exchanger dimensionless groups. For a three-fluid heat exchanger with all inlet temperatures specified, the fluid temperatures within the exchanger (the dependent variables) can be nondimensionalized as shown later by Eq. (23). When these dimensionless temperatures are evaluated at the fluid outlet, they are related to the temperature effectiveness [ l l , 12, 40, 411. In addition, we can define overall heat exchanger effectiveness for a three-fluid heat exchanger [ l l ] similar to the two-fluid heat exchanger effectiveness [36]. Thus, for a three-fluid heat exchanger, two temperature effectiuenesses and one overall heat exchanger effectiveness are defined in Tables I and 11. The detailed discussion of the definition of the three-fluid heat exchanger effectiveness and the concerns related to the use of this definition are elaborated later on in Sec. VII. As can be seen from Table I1 for a three-fluid heat exchanger with three thermal communications, temperature effectiveness depend on six dimensionless independent parameters instead of five for the case with two thermal communications. The six parameters are as follows: one NTU parameter, two heat capacity rate ratios, two conductance ratios and one inlet temperature ratio. In general, for an n-fluid heat exchanger with all fluids (none at constant temperature) transferring heat among each other, there will be [ n ( n + 1) - 2]/2 independent parameters arising from differential equations (regardless of the flow arrangement). In contrast, out of n , if p fluids are not transferring heat among each other and r fluids are at constant temperature, there will be [n(n + 1) - 2]/2 - p ( p - 1)/2 - r independent parameters associated with differential equations. In addition, there will be several parameters associated with the temperature boundary conditions on which the solution will depend. For an n-fluid heat exchanger, there will be ( n - 2) inlet temperature ratios arising from the boundary conditions.

IV. Literature Review A. THREE-FLUID HEATEXCHANGERS 1. Parallel Stream Heat Exchangers The first analysis of a three-fluid heat exchanger like that shown on Fig. l(a) in steady state was performed by Morley [9]. He determined the

238

D. P. S E K U LAND I ~ R. K.SHAH

temperature distribution of all three fluid streams within the heat exchanger. Based on the energy balances and rate equations, he arrived at a third-order ordinary differential equation for the temperature distribution for one of the fluids involved. Using this result in conjunction with the energy balance equations for the other two fluids, it is possible to present formulas for all three fluid temperature distributions throughout the heat exchanger, although they would be in dimensional form. Furthermore, he obtained the solution in terms of unknown coefficients of integration, which he suggested should be evaluated in terms of boundary conditions specified for a given problem, because the general solution would be very complicated. Since this effort, note that a number of other authors have solved the same problem repeatedly, using a more or less similar procedure or expressing the set of equations in a slightly different form. As a matter of fact, Morley neither presented the final explicit temperature distribution as a function of relevant dimensionless groups, nor did he define an explicit design procedure. However, for the sizing problem, he suggested a trial-and-error method along with the “laborious calculations.” He presented some sample calculations for a supermiser (a combined feedwater heater and air heater) in which flue gas, air, and water are the three fluids. The flue gas [fluid 2 in Fig. l(a), flows through the annular spaces between pairs of concentric tubes] heats both the air (fluid 1 in the outermost flow passage) and the water (fluid 3 in the innermost tube). Air and water flow in the opposite direction to the gas. Correspondingly, he considered only the counterflow arrangement (P2 of Fig. 6). At any rate, his contribution to the theory of three-fluid heat exchangers could be treated in a manner similar to Nusselt’s [42] analytical solution of an unmixed-unmixed cross-flow two-fluid heat exchanger problem, which has been reinvented by more than a dozen investigators. Hausen [lo] was the first one to address Morley’s work and define an explicit form of temperature distributions for a three-fluid heat exchanger like that of Fig. l(a) (the countercurrent flow arrangement, P2 of Fig. 6). Hausen’s solution is algebraically well organized, but he did not recognize that one can determine in some cases the size of a heat exchanger without a trial-and-error method. It is important to note that Hausen suggested how to perform the calculation procedure in the general case of variable heat capacities of the fluids involved and/or variable heat transfer coefficients. The proposed procedure involves the division of a heat exchanger into sections, and afterward a successive repetition of the calculation using the same analytical formulas, but with a successive change in thermophysical properties and/or heat transfer coefficients. Paschkis and Heisler [43] used an electric analogue method for the design of a parallel stream extended-surface three-fluid heat exchanger

-

THREEFLUIDHEAT EXCHANGER THERMAL DESIGN

239

with three thermal communications [Fig. l(d) or (f) for both cocurrent and countercurrent flow arrangements]. The problem was defined in such a manner that some inlet and some outlet temperatures were specified; the length of the heat exchanger and unknown temperatures were to be determined. By changing the appropriate resistors and capacitors of an electric analogue model of a heat exchanger, they simulated variable fluid properties across the exchanger length. The whole analysis was verified by comparing some of the results with experiments carried out on a small heat exchanger mode1 involving heat exchange among nitrogen, oxygen, and air at low temperature, Excellent agreement was found between analogue and physical experiments (i.e., approximately 2% error in the determination of heat exchanger length). Based on their analog results, Paschkis and Heisler concluded that the idealization of temperature-independent properties leads, for the conditions studied, to serious errors in the calculation of the required exchanger length (i.e., leads to severely undersized exchangers). The general validity of this conclusion has been correctly questioned by Krishnamurty and Venkata Rao [441 for applications involving ordinary temperature range (Lea, neither cryogenic nor high-temperature range), because the average properties and average heat transfer coefficients are reasonable approximations for these applications. Okoio-Kutak [45] has represented a comprehensive analysis of a parallel stream three-fluid heat exchanger with both two and three thermal communications [Figs. l(a), (b), (d), (e), and (f)]. His interest was directed particularly to the problem of designing the Field-tube-type heat exchanger (i.e., a bayonet-tube two-fluid three-stream heat exchanger). In the study, a number of explicit analytical solutions for different parallel stream flow arrangements (couplings Pl-P4, see Fig. 6) are given and corresponding numerical examples (most of them for the rating problem) are discussed. In addition, some particular cases are elaborated on in detail with respect to the quantitative values of fluid heat capacity rates k e . , the sum of directionalized heat capacity rates is equal to zero or some of heat capacity rates are equal to infinity). Most of the results are formulated for the case of three thermal communications among the fluids, although the results for the case of two thermal communications are also presented. The study concludes with a detailed study of a Field-type heat exchanger (including both sizing and rating problems). The results were not presented in dimensionless form, nor were the dimensionless parameters important for thermal design of a heat exchanger explicitly recognized. Rabinovich [46] obtained an analytical solution for the system of equations describing the three-fluid parallel stream heat exchanger with two thermal communications [Fig. l(a)l. Rabinovich stated that the mathemati-

240


caI model for a multifluid heat exchanger with n fluids represents either a system of n differential equations of the first order or one differential equation of n’th order. The mathematical model that Rabinovich used is the same as that discussed by Morley [9], but Rabinovich formulated a generalized governing differential equation applicable to all parallel stream flow arrangements. He introduced important dimensionless parameters: two heat capacity rate ratios, NTUs defined for one of two thermal communications, and the ratio of UA’s for both thermal communications (i.e., the thermal resistance ratio). The list of parameters Rabinovich used is similar to that presented in Table I though with a different combination of relevant quantities (see Table 11). Rabinovich recognized the existence of four different parallel stream flow arrangements if only two thermal interactions are involved. He also showed that some two-fluid heat exchangers are special cases of corresponding three-fluid heat exchangers. He solved one third-order linear differential equation for one of the fluids, and obtained temperature distributions for the other two streams using energy balances. He presented temperatures for each stream in terms of five dimensionless groups for the parallel stream exchangers of Fig. 6, the cocurrent and countercurrent flow arrangements (P1 and P2). The particular case of heat transfer between three streams was considered, in which one of the streams has an infinitely large heat capacity rate. Finally, an example was given for the design of a two-fluid heat exchanger that lose heat losses to the surroundings. Luck [47] proposed an approximate explicit analytical design method (for solving the sizing problem) for the parallel stream three-fluid heat exchanger of Fig. l(a). He approximated the temperature differences between both the lateral fluids and the central fluid by linear functions with respect to the longitudinal coordinate. This is done by separation of the necessary differential equations and formulation of a coefficient for the temperature ratio. The procedure was elaborated on in detail for the four parallel stream arrangements of Fig. 6 and included an additional two cases in which the coldest fluid has uniform temperature across the exchanger while the two other fluids are mutually in either cocurrent or countercurrent flow. The procedure leads to approximate explicit analytical expressions for the determination of heat transfer areas for both thermal communications. However, the radical approximation was not further assessed, particularly in relation to the exact solutions; thus could result in serious errors. Sorlie [ll] developed a design theory for two parallel stream arrangements of a three-fluid heat exchanger with two thermal communications [Fig. l(a)]. The first one is cocurrent flow in which one hot and two cold fluids flow in the same direction (coupling P1, Fig. 61, and the second one

-

THREE FLUIDHEATEXCHANGER THERMALDESIGN

24 1

is countercurrent flow in which hot fluid flows in the opposite direction of two cold fluids (coupling P2, Fig. 6 ) . Sorlie derived closed-form formulas for the temperatures of each stream by solving a set of three first-order linear ordinary differential equations. Following the effectiveness-NTU approach for a two-fluid heat exchanger, the results were presented in terms of two temperature effectiveness (one for each cold fluid, see Table IIX4 Each temperature effectiveness was expressed as a function of five independent dimensionless groups: Ntu,, Cr, C,*, R*, and A t ; as defined in Table 11. All results for temperature effectivenesses were presented both in graphical and tabular forms for the selected range of practical interest of five nondimensional groups: Ntu,, I 5.0, Cy = 0.5 and 2.0, C,* = 0.5 and 2.0, R* = 0.5 and 2.0, and At: = 0.25, 0.5, and 1.0. Some of the theoretical results were compared with experimental results, and excellent agreement was found between the predicted and measured temperature effectivenesses (the largest discrepancy was k 3.9% within the uncertainty interval of &4%). Sorlie defined the overall heat exchanger effectiveness for a three-fluid exchanger similar to that for a two-fluid heat exchanger effectiveness. We should point out that this overall three-fluid heat exchanger effectiveness definition is rather questionable in the case for which the sum of heat capacity rates of two fluids exceeds the heat capacity rate of the hot fluid [41] (see also Sec. VII for a detailed discussion). Krishnamurty and Venkata Rao [44] analyzed the single-pass four-stream and three-stream heat exchangers of Figs. l(a), (b), and (f) and with any of the parallel stream arrangements of Fig. 6. The authors had stated that the set of simultaneous differential equations for an n-fluid single-pass heat exchanger can be reduced to a set of ( n - 1)’th-order differential equations, each applying to one of the fluids (i,e., three second-order equations for a three-fluid heat exchanger). These differential equations become linear equations with constant coefficients if the UA’s are independent of the temperature. By first integrating the energy balance equation for total heat transfer and then combining the energy balance and rate equations for each fluid, second-order linear differential equations were formulated for a three-fluid parallel stream heat exchanger. The subsequent solutions resulted in exact analytical expressions for the determination of the required heat transfer area. From this solution, an expression was devised for the heat exchanger length. Two important cases 4The symbolism and notation used when referring to different literature sources, if different from the symbolism adopted here, correspond as a rule to the symbolism and notation used in the original references. The comparison of major symbols and definitions is provided in Table 11.

D.P. SEKULI~ AND R. K. SHAH

242

were considered: (a) a general case, in which all three fluids were considered to be exchanging heat with each other [i.e., the case with three thermal communications, Fig. l(f)] and (b) one fluid exchanged heat with the other two fluids, but the other two fluids did not exchange heat between themselves [i.e., the situation with only two thermal communications, Fig. l(a)]. The design procedure proposed is explicit regarding the size (length) of a heat exchanger if the outlet temperatures are known (as in the sizing problem). The solution was formulated in a general form in which any of the flow streams can have any desired direction. If it is necessary to consider heat losses to the surroundings, the three-fluid heat exchanger problem becomes a four-fluid heat exchanger problem with the fourth fluid as surroundings with constant temperature. In another paper devoted to the same problem, Krishnamurty [481 solved the system of equations using a matrix method. The temperatures at one end of the exchanger were obtained in terms of a complicated function of eigenvalues of the matrix and the temperature at the other end of the exchanger. Aulds [25] and Aulds and Barron [12] presented analytical relationships between the design variables (effectiveness-NTU relationships) for a countercurrent parallel stream three-fluid heat exchanger (Fig. 6, P2) with three thermal communications between the fluids [Fig. Me)]. This investigation was thus an extension of Sorlie’s work [ l l ] by treating the case with three thermal communications. Note that this problem was already analyzed by Okdo-Kulak [451 in terms of temperature distributions (and not in terms of effectiveness, as was done by Aulds and Barron). The ranges of parameters considered were as follows: N,, s 5 , CJC,,, C,/C, 0.5 to 1.0, R,/R,, R , / R , 0.25 to 2.0, and X - 0.25 to 1.0. The parameters were defined in a manner similar to Sorlie’s work (see Table 11). According to Aulds and Barron, when all three streams are in thermal communication (i.e., three thermal communications), the two temperature effectivenesses are functions of six dimensionless groups (the additional dimensionless group is the second thermal resistance ratio). Analytically obtained heat exchanger effectiveness values were compared with the experimental results and, according to the authors, acceptable agreement was found for most of the cases. The difference between the predicted and measured values of overall three-fluid heat exchanger effectiveness, however, is within - 10% and +90% according to data published by Aulds and Barron [121. The differences in temperature effectivenesseswere even substantially larger, according to data reported by Aulds [25]. It was concluded that the experimental determination of thermal resistances was responsible for the major differences between the experimental and analytical results. Aulds and Barron [12] used the same definitions of overall and temperature effectivenesses proposed by Sorlie [l 11.

-

-

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243

Schneller [26] studied the same problem as Rabinovich [46] [Fig. l(a)], but obtained explicit formulas for fluid temperature and heat transfer rates for all four parallel stream three-fluid heat exchanger arrangements (Fig. 6). Some special arrangements were also discussed (such as Fieldtube-type heat exchangers) and several examples of rating problems were elaborated on in detail. However, the solutions obtained were not presented using the dimensionless parameters convenient for a design procedure. Demetri and Platt [191 used an analytical electrical analogue method to examine the effect of direct heat transfer between nonadjacent fluids due to conduction through the fins in a three-fluid plate-fin heat exchanger? In other words, this investigation was directed to the examination of the influence of a third thermal communication (due to conduction through the fins) on the performance of a three-fluid heat exchanger in which only one fluid exchanges heat with the other two directly. The method is based on representing heat transfer by means of the analogy between heat transfer paths and electrical circuits. Both parallel stream and cross-flow arrangements have been investigated. In this subsection, only the information regarding parallel stream arrangements [plate-fin configuration of Fig. l(d), countercurrent arrangement, P2, of Fig. 61 are discussed, with the cross-flow arrangements presented in the following subsection. The results, two temperature effectivenesses (the same as given by Sorlie [ll]), were presented parametrically in terms of six independent nondimensional parameters (the ratio of convective to fin conduction resistance in a passage, R R , 2 , as an additional parameter over those of Sorlie; see Table 11). The ranges of the parameters are as follows: NtU,,5 6, K,, K , 0.5 to 2.0, R , 0.5 to 1.0, A t i 0.25 to 0.75, and R R , z 0.0 to 10.0 (see Table 11). Note that R R . 2 = 0 represents a limiting case of negligible heat transfer between nonadjacent streams, that is, two thermal communications among the fluid streams. A comparison of the curves for temperature effectiveness for R R , 2= 0 with the corresponding curves presented by Sorlie [ l l ] shows good agreement. The results obtained demonstrate that the effect of fin conduction on performance is significant. The net effect of increasing the relative amount of conduction through the fins is to equalize the temperature levels of the fluids in the exchanger. In virtually all cases, the data obtained by Demetri and Platt show that increasing the heat transfer between nonadjacent fluids due to conduction through the

-

5

-

-

-

The indirect heat transfer between nonadjacent fluids is through intervening passage containing the third fluid. The three fluid streams are arranged in the four-passage repeating pattern with fluids 1 and 3 separated by an intervening passage containing fluid 2; that is, the pattern is as follows: 1-2-3-2 [see Fig. 5(b), pattern ABCB].

244

D. P. SEKULI~ AND R K. SHAH

fins tends to decrease the temperature effectiveness of fluid 1 and increase the temperature effectiveness of fluid 3. Barron and Yeh [49] obtained a numerical solution for the temperature distribution and overall heat exchanger effectiveness (according to a definition similar to that of Sorlie [ 111) of a three-fluid heat exchanger with the effect of longitudinal conduction included within separating surfaces. The three fluid streams were arranged to have two thermal communications: heat exchange occurs between the hot ( h ) and two lateral cold fluids -intermediate-temperature fluid (i) and cold (boiling) fluid ( c ) with constant temperature. A tubular triple-pipe countercurrent heat exchanger was analyzed [Fig. l(a) and countercurrent arrangement, P2 of Fig. 61. The central tube fluid had a constant temperature throughout the heat exchanger (boiling). The mathematical model consisted of a set of four differential equations (two of the first order for two fluid streams with the third stream at a constant temperature, and two of the second order for two walls). The standard idealizations from the two-fluid heat exchanger analysis [36] were used (except for relaxing the idealization of nonzero thermal conduction in walls), and corresponding boundary conditions associated with these linear differential equations were adopted. The general exponential form of temperature distributions was adopted in order to define the characteristic equation. Newton’s method was used to obtain six roots of the characteristic equation and the Gauss-Jordan method was utilized to solve numerically six simultaneous equations for unknown constants. The effect of longitudinal conduction was illustrated by a comparison of the numerical solution with the analytical one, obtained by Aulds and Barron [12] for a three-fluid heat exchanger without longitudinal conduction. It was concluded that longitudinal conduction reduces heat exchanger effectiveness. The magnitude of this reduction was found to depend on the value of longitudinal conduction parameters Ai = kWAwi/LCi, A , = k w A w , / L C jwhere k , is the thermal conductivity of the wall, AWi and A,, are conduction areas, L is the length in the longitudinal direction, and Ci is the capacity rate of the intermediatetemperature fluid. The influence of longitudinal conduction was illustrated by the comparison of temperature distributions as well as heat exchanger effectivenesses in both situations (with and without the conduction influences) for the following set of parameters: R,, = hh,Ah,/hiAi = 2.0, Rhi = hhiAhi/hjAi = 2.0, R , = h,A,/hiAi = SO, CR = 0.50, Oil = 0.5, Ai = A, = 0.5, and p = hhiAhi/Ci= 1.6; here the subscripts h, i, and c denote hot, intermediate, and cold fluid streams, respectively. Note that the preceding R parameters defined by Barron and Yeh are hA ratios (i.e., defined differently than in most of the other studies; see Table 11) where h is the convective heat transfer coefficient and A is the heat transfer surface area. The last two parameters are CR = Ci/ch, heat

THREE- FLUIDHEAT EXCHANGER THERMAL DESIGN

245

capacity rate ratio, and Oil = (q,- Tc)/(Thl - T',), the temperature ratio of the intermediate-temperature fluid at the inlet. The intermediate fluid exit temperature was lower with longitudinal conduction, whereas the hot fluid exit temperature was generally higher with conduction than without as expected. A complete set of data for a wide range of variables is given by Yeh [50]. The general conclusion reached was that the effect of longitudinal conduction did not influence the overall effectiveness very much, but its influence was not negligible. For example, for the set of parameters just given, the decrease in overall effectiveness due to longitudinal conduction ranged between 1.5 and 5% for p changing from 0.6 to 1.6. Note that A, = A, = 0.5 is unrealistically high. Rao [51] analyzed cocurrent and countercurrent (Fig. 6, P1 and P2) parallel stream three-fluid heat exchangers in terms of temperature differences between the fluids as dependent variables. The heat exchanger had three simultaneous thermal communications [as in Fig. l(f)l. The general solution for the temperature differences along the heat exchanger under steady-state operation and for constant properties was presented in dimensional form without defining relevant design parameters and without acknowledging the previous analytical solutions from literature. Neither qualitative nor quantitative evaluation of the obtained solutions was performed to compare other existing solutions. Schubel [20] investigated three-fluid heat exchangers like those of Fig. 3(c) under the assumption that the shell fluid is sufficiently baffled, that is, the heat exchanger can be considered as having a parallel stream flow arrangement. Three different fluid flow arrangements were analyzed: (a) cocurrent (Pl, Fig. 6), (b) countercurrent (P2, Fig. 6), and (c) countercurrent-cocurrent (P3, Fig. 6). The author used a combined analyticalnumerical approach to solve the mathematical model instead of providing an exact analytical solution. This is, indeed, a serious shortcoming of that analysis. The heat exchanger effectivenesses were defined (in the same manner as used by Sorlie [11] and temperature profiles were calculated for a set of relevant parameters. The set of parameters included two NTUs: one as defined by Sorlie [ll], that is, N , = NTU,, and an additional one, N2,which represents the ratio of the UA product for another thermal communication and the heat capacity rate of the hot fluid instead of the conductance ratio. Also included were two capacity rate ratios (Rl, R 2 ) , defined in the same manner as by Sorlie [ll], and the dimensionless inlet temperature of the intermediate-temperature fluid, TE = (Tit - T Z j ) / ( q - TZi), where T., T I i ,and T,, are the inlet temperatures of the hot fluid, the intermediate-temperature fluid, and the cold fluid, respectively.6 'Note that the symbolism and notation used by Schubel differs substantially from the symbolism used in this review (see footnote 4).

246

D. P. SEKULII~ AND R K. SHAH

The other two (cold and hot fluid) dimensionless inlet temperatures are normalized to be 0 and 1. The selection of parameters was as follows: N,,N2 I 10; R , , R , (0.1,l.Ok and T; (0,l.O). The final conclusion confirms the already known fact that the countercurrent arrangement (P2) has the best overall efficiency, while the cocurrent arrangement (PO has the worst. The author did not cite any literature references who had analyzed the three-fluid heat exchanger problem. Shpil'rain and Yakimovich [23] developed an analytical solution for heat exchange among three fluids (countercurrent flow arrangement, P2, of Fig. 6). The study is related to the problem of supplying heat to oil-bearing strata at great depths. The solution was obtained using the conventional analytical approach of solving the corresponding set of differential equations. The three fluids involved were water, air, and combustion products. BaEliC et al. [52] used the Laplace transform technique and obtained solutions for three-fluid parallel stream heat exchangers with two thermal communications [Fig. l(a>l. The temperature distributions were presented as implicit functions of the number of transfer units, the resistance ratio, the two heat capacity rate ratios, and dimensionless inlet and outlet temperatures. The results of the calculation were demonstrated for all four possible couplings (P1 through P4 of Fig. 6 ) using the same sizing design problem example of BaEliC et al. [34] for the cross-flow heat exchanger. Similar to Krishnamurty [48],BaEliC et al. [52] emphasized that the sizing problem can be solved explicitly if the inlet and outlet temperatures are already determined. The three distinct expressions for predicting three possible direct or indirect temperature crosses7among the fluids had been derived. Without acknowledging previous exact analytical solutions except for those of Morley [91 and Hausen [lo], Kancir [53] presented temperature distributions for the parallel stream heat exchanger of Fig. l(a) in countercurrent flow (Fig. 6, P2). He used a matrix algebra method. The form of the solution is similar to the existing solutions but neither detailed quantitative nor qualitative comparisons were made. All papers reviewed so far are devoted to the steady-state operation of three-fluid parallel stream heat exchangers. However, knowledge of the dynamic behavior of a three-fluid heat exchanger is necessary for startups and shutdowns, thermal stresses, fatigue behavior, erratic operations, and accidents. According to the best knowledge of the authors, only two papers [54,551 have been published on transient performance; the parallel stream three-fluid heat exchanger with two thermal communications [Fig. l(a); couplings P1 through P4 of Fig. 61 was analyzed both numerically and

-

-

'For a detailed discussion of temperature cross phenomena, see Sec. VI.

-


247

experimentally. From the preliminary comparison between the numerical and experimental results, it can be concluded that the transient behavior (in contrast to the steady-state operation) is strongly influenced by heat exchanger wall thermal capacities, similar to that for a two-fluid heat exchanger. In the two most recent studies [13,411, all four parallel stream three-fluid heat exchanger couplings were analyzed under steady-state operation. A compact form of temperature distributions obtained by the Laplace transform technique was given for the situation when two thermal communications are present. A single analytical expression is given for determining the temperature cross for any combination of fluids. involved, and all four couplings [ 131. In another paper [41], the study of overall three-fluid heat exchanger effectiveness was performed on the same class of three-fluid heat exchangers. It was demonstrated that the overall effectiveness provides limited insight into exchanger performance. A detailed study of the interrelation among temperature distributions, temperature effectiveness indicators, and overall effectiveness was elaborated. The results of these studies are integrated in concise form in this work in Secs. VI and VII. 2. Cross-flow Heat Exchangers Rabinovich [56] was the first investigator to analyze a simple three-stream cross-flow arrangement. The study performed was, in fact, related to a two-fluid cross-flow heat exchanger, but one of the fluids, having left the exchanger, enters it again in the opposite direction. Thus, this two-fluid heat exchanger may be considered to be a three-fluid heat exchanger. Rabinovich solved the corresponding mathematical model using the Laplace transform technique. Willis [40] and Willis and Chapman [32] analyzed a single-pass [Fig. l(c); arrangement C4, Fig. 7(a)] as well as a two-pass cross-flow three-fluid heat exchanger with two thermal communications and no lateral mixing in any of the fluids [i.e., unmixed fluids, Fig. 7(b), arrangements CTPl and CTP21. The authors have analyzed three cases of two-pass arrangements: (a) cross-cocurrent C"1, fluids mixed between passes; (b) cross-cocurrent CTP2, unmixed in identical order'; and (c) cross-cocurrent CTP2, unmixed in inverted order. Using the same idealizations as those of Kays and "For both the cocurrent and countercurrent flow arrangements, there are two possibilities for coupling two passes in the header sections of a two-pass exchanger when the fluid is unmixed in the header. The fluid may be completely unmixed in the header and approach the second pass from the same side as in the previous pass (unmixed-identical order). Alternatively, the fluid may be completely unmixed in the header but with a flow arrangement to invert the fluid prior to entering the second pass (unmixed-inverted order).


248

London [36], they formulated the first-order linear partial differential equations and solved them numerically (using a first order predictorcorrector integration scheme) after selecting parameters similar to those of Sorlie [ l l ] (see Table 11). Two temperature effectivenesses and the overall effectiveness were obtained for the following range of parameters: 0.25 to 1.0; R, 0.5 to 2.0; and A t i 0.25 to 1.0). NTU, I8; K,,K, The results were presented graphically. For example, the temperature effectiveness of the hotter of the two outer hot fluids generally increases with increasing NTU, for a single-pass three-fluid heat exchanger when the central fluid has to be used to cool both lateral fluids. The effectiveness of another lateral fluid, however, may decrease, or even become negative, for larger values of NTU,. (For more details about the definition of temperature effectiveness, see Sec. V1I.B; for a thorough discussion of the meaning of the temperature effectiveness indicators, refer to SekuliC and Kmekko [41].) Willis and Chapman [32] compare also the performance of two-pass cross-cocurrent three-fluid heat exchangers with a single-pass cross-flow three-fluid exchanger operating under the same conditions. For 0.5 and R , 2.0, both effectiveness indicators of a example, K,, K, two-pass exchanger with the overall countercurrent flow of fluid 2 [arrangement CTP2, Fig. 7(b)], are higher compared to the single-pass exchanger operating at the same conditions, The important conclusion is that the most significant increase occurs in the effectiveness of fluid 1 for small values of Ati ( - 0.25 to 1.0). Ellis [35] extended the work of Willis and Chapman [32, 401 by analyzing cross-flow arrangement C3 [Fig. 7(a)] for a one-pass three-fluid heat exchanger with two thermal communications (i.e., there was no heat transfer between the two lateral cold fluids). The temperature effectivenesses (see Table 11) where determined numerically by a first-order iterative, predictor-corrector integration scheme for the differential equations. The results were presented graphically for the range of parameters 0.5 to 2.0; and Ati as follows: NTU, I 5 ; K,, K, 0.25 to 1.0; R, 0.25 to 1.0 (see Table 11). Ellis compared his results for the C3 arrangement [Fig. 7(a)] with Willis and Chapman’s results for the C4 arrangement [Fig. 7(a)] and found the following. He concluded that, in general, the cross-countercurrent flow configuration [C3, Fig. 7(a)] is inferior to the cross-cocurrent flow configuration [C4, Fig. 7(a)l. For NTU, I2, both temperature effectivenesses for the cross-countercurrent flow configuration are lower or slightly higher (up to about 5%) than the corresponding values for the cross-cocurrent configuration for the same set of relevant parameters. The temperature effectivenesses for the same two flow arrangements become almost equal as Ati approaches unity when R , is high ( R , > 2). The cross-countercurrent configuration becomes much less

-

-

-

-

N

-

N

-

-


249

effective (the effectiveness for the cross-countercurrent configuration is 10% or more lower than that for the cross-cocurrent flow configuration) as R , becomes smaller and K , >> K,. For NTU, > 2, when R , is high and A l l approaches unity, the cross-countercurrent flow configuration effectiveness does not significantly differ from that for the cross-cocurrent arrangement. The cross-countercurrent flow arrangements is significantly inferior for all other cases when comparing temperature effectivenesses simultaneously for NTU, > 2 and the analyzed range of parameters K , , K , , and R,. Ellis [35] speculated that the preceding results make sense thermodynamically because the effect of counterflowing fluid 3 tends to cause fluid 2 temperature gradient transverse to the flow such that its average outlet temperature is actually less than that for the cocurrent flow case. As a quite general conclusion, for small heat exchangers (NTU, < 2) where R,, is high, the two arrangements are comparable. For large heat exchangers (NTU,> 21, the cross-countercurrent flow configuration is not desirable. Ellis also analyzed two-pass arrangements with both passes in cross-counterflow arrangements [arrangement CTP3, Fig. 7(b)] and with (a) mixed fluids between passes, (b) unmixed, identical order, and (c) unmixed, inverted order, but only for selected operating conditions. The comparison of CTP2 and CTP3 [see Fig. 7(b)] showed that the CTP3 arrangement is inferior to the CTP2 arrangement, that is, it is a similar result to the one reached for single-pass arrangements C3 and C4 [see Fig. 7(a)]. Demetri and Platt [19] analyzed the conduction effects between nonadjacent fluids due to conduction through the fins of intervening passages in a crossflow three-fluid exchanger [Fig. l(c)]. The configuration considered consisted of an exchanger involving three fluid streams arranged in the four-passage repeating pattern [see Fig. 5(b)]. The central fluid passages [B, Fig. 5(b)] are assumed to contain simple rectangular fins; the passages for both lateral fluids are unfinned. With this arrangement, direct heat exchange between lateral fluids occurs only through the fins of the intervening passage of the central fluid. This indirect transfer through the fins from the hottest fluid to the intermediate fluid is possible if the central fluid is the coldest fluid. Both situations, with lateral fluids cocurrent and countercurrent to each other, were analyzed [arrangements C4 and C3, respectively, of Fig. 7(a)]. A comparison of the results obtained for zero fin conduction resistance in a passage ( R R , 2= 0; see Table I1 for definitions of parameters) with the corresponding results presented by Willis [401 and Ellis [35] shows good agreement. With the increase in parameter RR, (i.e., with the decrease of the conduction resistance compared to the convective resistance), for a given N,u,l, the effectiveness of one lateral fluid will decrease, while at the same time the effectiveness of another fluid will

250

D. P. S E K U LAND I ~ R K. SHAH

increase. The net effect is to equalize the fluid temperatures in the exchanger. This effect of fin conduction is, according to Demetri and Platt [19], most pronounced for cases in which lateral fluids flow countercurrent to each other. The quantitative data are given graphically 1191 (see also Sec. VII.C.2). The reason is that the difference in temperature between fluids 1 and 3 remains relatively high throughout the exchanger rather than decreasing steadily as would be the case for cocurrent flow. Horvith [33] generalized the mathematical model of a three-fluid heat exchanger in order to include both parallel and cross-flow arrangements at the same time in the model. In particular, he analyzed a one-pass cross-flow three-fluid heat exchanger like that shown in Fig. l(c) with (a) two thermal communications and unmixed fluids with the flow arrangements of type C2, Fig. 7(a) and (b) three thermal communications and unmixed fluids with the flow arrangements of type C2. Horvith used the method employed by Jakob [57] for cross-flow two-fluid heat exchangers for the calculation of two-dimensional temperature distributions of three-fluid cross-flow heat exchangers. The solutions were obtained by solving numerically the corresponding Volterra integral equations obtained from the three governing differential equations. The mean outlet temperatures were obtained by numerical integration. The solutions for the temperature fields were presented as functions of relevant independent parameters (five parameters for two thermal communications, and six parameters for three thermal communications). Besides one inlet temperature ratio, four or five other parameters were defined in the form of different NTUs (i.e., the UA/C ratios). As a measure of the heat transfer efficiency of a heat exchanger, Horviith defined the effectiveness for each fluid in the form of the ratio of overall temperature change to the greatest temperature difference in the heat exchanger. No quantitative data regarding the effectivenesses were given. An analysis of all fluid arrangements of Figs. 6 and 7(a) was given by Horvith [%I. Shen [59, 601 analyzed the cross-flow three-fluid heat exchanger of Fig. 7(a) arrangement C4. The analytical solution was obtained by using the generalized Nusselt linear integral method. The solutions for the temperature effectivenesses (defined like those of Willis [40]) are given in the form of an infinite series as functions of five dimensionless parameters. The series convergence is very fast. For almost all cases, four and five terms are sufficient for results accurate to within 1%.The outlet mean temperatures of all three fluids are given. The reduction of the three-fluid heat exchanger solution to the one obtained for the two-fluid heat exchanger case was performed numerically and shows excellent agreement. His results compare with Willis’s numerical results to within 2%. The exact analytical treatment of an unmixed cross-flow three-fluid heat exchanger with two thermal communications [Fig. l(c), arrangement C4,

THREE- FLUID HEATEXCHANGER THERMAL DESIGN

25 1

Fig. 7(a)] was performed by BaEliC et nl. [34]. The governing three partial differential equations were solved by the Laplace transform technique and exact analytical solutions were obtained for temperature distributions within the heat exchanger and for mixed mean outlet temperatures. The relevant parameters used in the analysis are shown in Table 11. Besides the general analytical solutions, the following particular cases were discussed: (a) a symmetric heat transfer three-fluid heat exchanger (LIA’s for both thermal communications are equal to each other) as well as balanced (equal) fluid capacity rates for lateral fluid streams; (b) the same as case (a) but with unbalanced (unequal) fluid capacity rates for lateral fluid streams; and (c) an asymmetric (unequal U’s) heat transfer three-fluid heat exchanger. The dimensionless fluid outlet temperatures versus NTUs for a symmetrical and balanced heat exchanger were presented graphically. In two companion papers, Skladziefi [61, 621 investigated cross-flow three-fluid heat exchanger [Fig. l(c)] flow arrangements C1 through C4, Fig. 7(a)]. The standard idealizations (see Sec. V.B) were adopted. Starting from the energy balances, he derived governing differential equations in dimensionless form and obtained solutions for temperature distributions. Skladziefi mentioned two different analytical approaches to the solution procedure: the Laplace transform method and the Volterra integral equation method. According to Skladzieh, the Laplace transform technique cannot give a general solution and the corresponding solutions are inconvenient for numerical calculations. Note that this statement is not always correct as shown by BaEliC et al. [34]. The method that uses Volterra equations cannot give a general solution and the solutions have the form of an irregular (with respect to one of the coordinates) infinite series that converges rather slowly. Therefore, Skladziefi proposed a method in which the original set of governing equations is converted into an infinite sequence of equation sets that are easy to solve according to him. Every set of equations consists of one algebraic and two differential equations. The series in solutions converges sufficiently rapidly. When the mean temperatures are calculated, no more than six terms of the infinite series will result in an accuracy of ().I%.’ ’After this article was completed, the monograph of Sktadzien [62al came to our attention. He has compiled the work of Okdo-Kutak [45] and Skiadzien [61, 621 as reported here and in Sec. IV.A.l; he has provided a comprehensive review of efforts devoted to three-fluid and three-stream heat exchangers at “Silesian School of Thermodynamics,” Silesian University, Gliwice, Poland. In addition, he has provided the thermal analysis of two U-type spiral heat exchangers: three-fluid four stream U-type spiral parallel-flow exchanger and two-fluid, three-stream U-type cross-flow exchanger. The analysis was performed numerically using a finite difference method iteratively. He concludes based on the results that the thermal efficiency of the U-type spiral cross-flow heat exchanger is almost the same as that for the U-type tubular cross-flow exchanger for the same dimensionless parameters.

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D.P. SEKULIC AND R. K. SHAH

Nosach and Filipchuk [63] analyzed a cross-flow three-fluid heat exchanger in which all three fluids have three-dimensional temperature fields within the heat exchanger but only two thermal communications. This heat exchanger had all three streams flowing in a mutually perpendicular manner as shown in Fig. 8. They formulated governing differential equations and determined the mean mixed outlet temperatures and heat transfer rates. The design methodology was discussed, and relevant correction factors and parameters introduced into the design procedure were detailed. However, the details of the solution procedure and results are not presented in the published technical note. In another paper, Nosach and Filipchuk [64] defined two correlation factors for design of a three-fluid cross-countercurrent flow configuration [C3, Fig. 7(a)]. They solved numerically the corresponding set of differential equations and boundary conditions. The thermal design is based on correction factors for estimating the average temperature differences for both lateral fluids (for more detail, see Sec. VIII.B.3).

3. Summary of Research Efforts Related to Parallel Stream and Cross-Flow Three-Fluid Heat Exchangers

As a concluding remark to the literature survey in the preceding two sections, a number of three-fluid heat exchangers have been analyzed as summarized in Table 111. Often the same problem has been solved with more or less similar methods. Varied amounts of detail are presented in the investigations and the presentations are not systematic. We will unify the approach and systematize the solutions in Secs. V, VI, and VIII. B. OTHERTYPES OF MULTISTREAM HEATEXCHANGERS In addition to the list of investigations reviewed, a number of other research efforts have been devoted to the general problem of three-fluid and multifluid heat exchangers. The family of heat exchangers with more than two fluid streams involved is much larger than can be concluded from the previous reference list. Without attempting to discuss every relevant source, it is worth noting some of them, especially those related to multifluid applications. The field of cryogenics has many applications for multifluid heat exchangers [18, 28, 651. One particular design is the so-called Hampson (Giauque-Hampson) wound-coil or coiled-tube shell-and-tube heat exchanger (Hampson, British patent #10156) as shown in Fig. 4. A multifluid heat exchanger of the Hampson type [shell and paired tube exchangers; see Fig. 4(a)] was anaIyzed by Kao [291. Differential equations were

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253

TABLE I11 SUMMARY OF THREE-FLUID HEATEXCHANGER STUDIES Subject Flow arrangements

Number of thermal communications

P2 P2

2

P1, P2

3

Okdo-Kulak I451 Rabinovich [461 Luck [47] Sorlie [ 111

P1, P2, P3, P4 P1, P2 P1, P2, P3, P4 P1, P2

2 and 3 2 2 2

Krishnamurty and Rao; Krishnamarty [44,48] Adds [25] Aulds and Barron [ 121 Willis [40] Willis and Chapman [321

P1, P2, P3, P4

2 and 3

P2

3

c4

2

c3

2

P1, P2, P3, P4 P2, c3, c 4

2 2 and 3

c4

2 2

Author(s) Morley [9] Hausen [lo] Paschkis and Heisler [431

Ellis [35]

Schneller [26] Demetri and Platt [19] Shen [60] Barron and Yeh [49] Rao [51] Horvith [58] Schubel[20] BaEliC et al. [34] Shil’rain and Yakimovich [23] Skladzien [61] BaEliC et al. [521 Nosach and Filipchuk [631 Kancir [53] Nosach and Filipchuk (641 SekuliC [13]

2

Comments Analytical solution Analytical solution, variable properties Electrical /analog simulation experiment, variable properties Analytical solution Analytical solution Approximate solution Analytical solution / experiment Analytical solution

c4 P2

2 2

Analytical solutions / experiment Numerical solution / one- and two-pass arrangements Numerical solution / one- and two-pass arrangements Analytical solution Electrical analog method, transverse conduction Analytical solution Numerical solution, longitudinal conduction Analytical solution Seminumerical solution Analytical /numerical solution Analytical solution Analytical solution

c1, c2, c3, c 4 PI, P2, P3, P4 Cross-flow, Fig. 8 P2 c3 P1, P2, P3, P4

2 2 2 2 2 2

Analytical solution Analytical solution Analytical solution 3-D field Analytical solution Numerical solution Analytical solution

P2 P1, P2 Pl-P4,Cl-C4 P1, P2, P3

3 2 and 3 2

254


formulated for 2n-fluid streams. One shell stream is assumed to flow in parallel with 2n tube streams in n sets of paired tubing. The model was obtained in a matrix form. The temperature variations can be evaluated by a numerical finite difference procedure. For constant heat transfer coefficients and heat capacity rates, a direct analytical solution (eigenvalue approach for four streams or less) or the matrix solution can be obtained. Examples were worked out by Kao [29] for a three-fluid Hampson exchanger. A large number of references have been devoted to the analysis of parallel flow multistream heat exchangers, particularly plate-fin multifluid heat exchangers (Fig. 5). Kao [66] conducted a mathematical analysis of heat conduction along the interconnecting fins across all passages of a multistream plate-type heat exchanger. A computation procedure in terms of matrices on a digital computer was proposed that is applicable to heat exchangers with an arbitrary number of streams and variable heat transfer coefficients and heat capacity rates. Wolf [14, 67, 681, Settari [391, and Zaleski and Jarzebski [38] analyzed the parallel stream multichannel heat exchanger in detail. The set of n first-order homogeneous linear differential equations was put into matrix form and solved using the techniques of linear algebra. The solution was obtained for the general case of a parallel stream multichannel heat exchanger in the form of a bundle of channels. Settari and Venart [37] obtained approximate solutions for the steady-state lumped linear formulation of heat transfer in multichannel parallel stream and so-called mixed-flow heat exchangers (particularly characteristic for plate heat exchangers). Solutions were obtained using polynomial approximations for the temperatures in each stream. The method is applicable to both the linear and nonlinear problems with any boundary condition and allows nonuniform geometry along the streams. A comparison with the exact method was performed by Mennicke 169, 701 and it was concluded that polynomials of the third order yield practically exact solutions for the range of variables investigated. The method can also be utilized to consider the nonlinear and transient problems. Chato et al. [15] also analyzed a parallel multistream heat exchanger. Because the general theory outlined by Wolf [14] was difficult to apply numerically, Chato et al. [15] presented a computer-oriented analytical method to predict the temperature distributions and temperatures at one end of the multifluid heat exchanger when the temperatures at the other end are known. It was also assumed that heat transfer coefficients, heat capacity rates, and surface areas for each fluid stream were known. The method was applied to the plate-fin exchanger. The heat exchanger is assumed to operate in a steady state, The authors stated that well-balanced heat exchangers and good mixing of hot and cold streams (with a minimum of identical streams in

-


255

thermal contact) provided the best performance. Bentwich [ 161 formulated an idealized multistream countercurrent heat exchanger model. The solution is obtained by expressing the temperature field in terms of inlet conditions and heat flux across the partitions. Haseler [71] analyzed multistream plate-fin exchangers and showed that plate-fin matrix conduction heat transfer can be expressed in terms of a bypass efficiency in each layer. Methods of integrating the resulting heat transfer equations are suggested for cocurrent and countercurrent multistream heat exchangers. Prasad and Gurukul [72, 731 presented methods for sizing and rating a multistream plate-fin heat exchanger. A computer code was developed for the sizing method outlined. Differential methods of rating, based on dividing the heat exchanger into several sections, and a stepwise integration of heat transfer and pressure loss functions were developed for counterflow, cross-flow, and cross-counterflow heat exchangers. Prasad [74] discussed in detail the effect of various stacking arrangements [such as those given in Fig. 5(b)l on the exchanger performance of a multistream plate-fin exchanger. Prasad [75] has also presented the concept and formulas for the fin efficiency of a plate-fin multistream heat exchanger. Paffenbarger [17] proposed a general computer-oriented method of analysis for a plate-fin counterflow single-phase multistream heat exchanger. The method includes a number of important effects (variable physical properties, layer order, cross-layer conduction through fins, and axial conduction). An important field of interest relevant to the analysis of a multistream heat exchange process is the design of so-called plate heat exchangers. Although plate heat exchangers are two-fluid exchangers in most applications, an end effect occurs when the number of thermal plates in the exchanger is less than about 40 [761. Heat transfer in this exchanger can be more accurately computed considering multistream exchangers. One of the first efforts to analyze n parallel stream two-fluid heat exchanger was made by Mennicke in two companion papers [69, 701. He solved the mathematical model of n (for n up to 10) parallel, counter, and mixed-flow heat exchangers. For large n , his analysis becomes very complicated and a numerical approach in essential. The extensive list or related references involving different aspects of heat transfer phenomena and design theory for plate heat exchangers is given by Shah and Focke [77]. ON THE THREE-FLUID HEATEXCHANGER C. CONCLUDING REMARKS PROBLEM

To conclude the literature review, we should emphasize that the linearized problem of a parallel stream three-fluid heat exchanger has been

256


solved for all flow arrangements. The analytical solutions for the cross-flow heat exchanger problem, however, are not known for all of the flow arrangements. Generally speaking, for parallel stream and cross-flow exchangers, the design procedure itself (and the sizing problem in particular) is not outlined well in the literature, particularly in terms of the selection of parameters, the algebraic complexity of the analytical solutions, the evaluation of approximate approaches, the sound definition of a heat exchanger figure of merit (or effectiveness), the temperature cross phenomena, and nonlinear problems.

V. The Mathematical Model of a Three-Fluid Heat Exchanger Problem

A. SYMBOLISM AND NOTATION The effectiveness-NTU approach, well known for a two-fluid heat exchanger thermal design, is extended here to a three-fluid heat exchanger design. The nomenclature scheme adopted in this text follows as much as possible that proposed by Kays and London [36], which is frequently used in the United States and elsewhere in the world. The alternative nomenclature scheme often used in most of Europe is compared in Table IV for completeness (only the most characteristic symbois are compared). The complete set of symbols used in this article is given at the end in the Nomenclature section.

TABLE IV NOMENCLATURE SCHEME FOR HEATEXCHANGER ANALYSIS Entity Heat capacity rate, mcp Heat capacity rate ratio Overall heat transfer coefficient Number of heat transfer units Heat transfer area Heat transfer coefficient Heat exchanger effectiveness

USA

C C*

U

Europe W

R,6J k

NTU or Ntu

NTU,kF/ W

A h

A, F

E

E

a

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257

B. IDEALIZATIONS To define an adequate mathematical model for the analysis, one needs to adopt a set of rational idealizations. In this analysis, we adopt the following list of idealizations and approximations: 1. The three-fluid heat exchanger operates under steady-state conditions. Mass flow rates are constant, and fluid temperatures at the inlet and within the exchanger are also independent of time. 2. The heat exchanger is adiabatic; that is, heat losses to the surroundings are negligible. 3. The specific heats (as well as other fluid properties, implicitly used in NTU,) of each fluid are constant. 4. There are no internal thermal sources (or sinks) in the walls or fluids such as thermal energy generation by nuclear processes, chemical reaction, or electrical heating. 5. In parallel stream exchangers arrangements, perfect transverse mixing occurs in each flow passage; that is, there is no temperature gradient normal to the flow direction within the respective fluid streams. In cross-flow exchangers, each fluid is considered mixed or unmixed at every cross section depending on the specifications. 6. Each fluid transfers heat with the other two fluid streams or only one of the fluids transfers heat with other two depending on the specifications. 7. In the fluid streams flowing through the exchanger, either there is no phase change or phase change occurs at a constant temperature. 8. Zero heat conduction is assumed in fluids or in walls parallel to the fluid flow direction. 9. Heat transfer coefficients are independent of temperature, time, and position. (The overall conductances between the fluids are constant.) 10. The fluid flow rate is uniformly distributed through the exchanger on each fluid side. The velocity and temperature at the entrance of the heat exchanger on each fluid side are uniform. 11. The heat transfer area is distributed uniformly on each fluid side. The overall extended surface temperature effectiveness is considered uniform and constant on each fluid side of the exchanger if fins are employed. Because of space limitations, the mathematical model is elaborated on only for the case of a three-fluid heat exchanger with two thermal communications (the second choice in idealization 6 in the list just given).

258


Corresponding solutions are presented in detail without any derivations. A similar procedure can be used for the case with three thermal communications [25].

C. ENERGY EQUATIONS The governing energy equations for describing the temperature fields within a heat exchanger are obtained from the energy balance. With the idealizations invoked earlier, the term energy balance is identical to the enthalpy rate balance for respective fluid flow streams. The enthalpy balance can be twofold: (a) microbalance-when balancing the enthaipy rates on a differential fluid flow passage element or on a differential element of a heat exchanger-and (b) macrobalance-when balancing the enthalpy rates on specific fluid flow passages or on the exchanger as a whole. Note that the microbalances lead to the differential equations describing the temperature fields within the heat exchanger; the macrobalance (i.e., a black box approach) provides the information about global energy (enthalpy) changes on the fluid flows involved. One additional feature of both balances is that they are interrelated and mutually consistent. This is because, when utilizing either the microbalance or macrobalance, one should have in the final instance the same global (overall) energy changes of relevant entities within the heat exchanger. All of these facts are included in the following analysis. 1. Microbalance Equations The microbalance analysis is divided into two separate parts depending on the characteristics of the mutual fluid flow orientation: (a) parallel stream arrangements and (b) cross-flow arrangements. The basic difference is that in the former case the temperature distribution is one dimensional, whereas in the later case the temperature fields within a heat exchanger are two or even three dimensional. a. Parallel Stream Arrangements In general, the procedure for deriving energy equations for any parallel stream heat exchanger is the same regardless of the number of thermal communications and the flow arrangement. A schematic of a parallel stream three-fluid heat exchanger is shown in Fig. 9 with the countercurrent fluid flow arrangement and two thermal communications between the fluids. Three microbalances on differential elements of the corresponding fluid flow passages for fluids 1, 2, and 3 [see Figs. 9(a), (b), and (c)] are as

-


259

FIG.9. Energy balances on differential elements of a parallel stream three-fluid heat exchanger: (a)-(c) microbalances; (d) microbalance versus macrobalance.

follows:

+

Here ij = 1 or - 1 for j = 1,2,3 depending on the flow direction, and the values are given in Table V for the four-stream couplings of Fig. 6.

D, P. SEKULI~ AND R. K. SHAH

260

TABLE V FLUID FLOW INDICATOR i j FOR EQS.(4), (5), A N D ( 6 ) i , for three-fluid heat exchanger parallel stream coupling

Stream

P1

P2

P3

P4

1 2 3

+1

+1 -1 +1

+1

+1

-1 -1

+1 +1 -1

+1

Note that only fluid 2 is in thermal contact with the other two fluids. In the case of three thermal communications ke., thermal communication also exist between fluids 1 and 31, equations for fluids 1 and 3 [Eqs. (4)and (611 will have an additional term similar to that for fluid 2 in Eq. (5). Furthermore, in writing microbalances, the enthalpy rate change is expressed by the mcpT product, following idealizations 3, 4,5, and 10 listed earlier. However, thermal communications between fluid streams are formulated by using the concept of overall heat transfer coefficient for the corresponding separating heat transfer surface along with idealization 9 [i.e., d Q , = (uuY),,,(T, - T ~ )dQ3,2 , = ( u ~ A ) , , , (T T3>1. , 10 Equations (41, (5), and (6) can be modified by replacing heat transfer area as an independent variable by the pertinent axial distance because of idealization 11 [i.e., A ( x ) = Px]. After algebraic simplification, Eqs. (41, (3,and (6) reduce to:

Note that i, = + 1 always and, hence, it is omitted from differential equations from now onward. Note that the UP term in Eqs. (71, (8), and (9) involves pertinent fin efficiencies for extended surfaces [see also Eqs. (10) through (1411. It is easy to see that Eqs. (7), (8), and (9) can be further rearranged to result in the generalized form given earlier by Eq. (1). ''Note that Eqs. (41, (51, and (6) are valid regardless of the fact that fluid 2 may have either the highest or the lowest inlet temperature.

-


26 1

b. Cross-flow Arrangements In these arrangements, at least one fluid is flowing cross-flow to the other two. In such a situation, the temperature fields are two dimensional. This is the most frequent situation in practical applications of cross-flow three-fluid heat exchangers. It is possible to have a heat exchanger in which all three fluids are mutually in cross-flow (either with two or three thermal communications between them) as shown in Fig. 8. The corresponding temperature fields are three dimensional [63]. Let us start the analysis for the most frequent situation in which two lateral fluid streams are in parallel flow and the third fluid between them is in cross-flow orientation. The lateral fluid streams can be either mutually in cocurrent or countercurrent orientations. In Fig. 10(a), a schematic of a cross-cocurrent three-fluid heat exchanger with two thermal communications is presented [see also Fig. 7(a), arrangement C41. Using energy balances for a differential element [Figs. 10(a) and (b)], one can show that the differential equations for the three fluids involved are as follows:

To emphasize the fact that cross-flow heat exchangers frequently employ extended surfaces, the extended surface effectiveness 77, is included in Eqs. (lo), (111, and (12)." In Fig. 1Mb) and Eqs. (lo), (ll), and (12), notice that extended surface effectivenesses are taken as different for all four heat transfer surfaces on the two separating walls between the fluids, i.e., 770,1, % , 2 , and 77,,3.12 For a plate-fin surface (see Fig. 5), if the heat source (or sink) is symmetric for a fin, the fin efficiency is calculated based

7 7 L

11 To be formally consistent with all existing solutions for a three-fluid cross-flow heat exchanger from the literature [32-35, 40, 58, 591, the extended surface effectiveness [36] is defined with respect to the surface area A, = X,Y, [see Fig. 10(a)]. For design purposes, the extended surface effectiveness has to be redefined with respect to the total heat transfer area of the respective side. Regardless of the approach, one should consistently define the heat transfer surface on which U is based, that is, (UA) = ibidem. '*In compact heat exchangers, the extended surface effectivenesses for both finned walls of the same flow passage are considered to be the same. Here all four effectivenesses are taken as different to include a more general case.

262

D.P. SEKULI~ AND R K. SHAH

c

FIG. 10. Energy balance on a control volume of a cross-cocurrent three-fluid heat exchanger [arrangement C4 of Fig. 7(a)l: (a) general view; (b) differential element dudy.

on the fin length as one-half the distance between the plates, as is generally the case for a two-fluid heat exchanger. In a three-fluid heat exchanger, however, the fluids on each side of the fin passage would be at different temperatures, resuIting in asymmetric heating on the plates. As a result, half the distance between plates may not represent the correct fin length for the fin efficiency. Prasad 1751 presents a method to compute the fin efficiency in such a case. Ordinarily, in a proper design, the heating and cooling on both sides of a fin surface is not drastically asymmetric and the conventional concept of fin efficiency may not introduce a serious error in design. However, for certain designs, heating and cooling may be very asymmetric and the concept of a conventional definition of fin efficiency may not be useful as discussed by Kao [66] and Prasad [751.

THREE- ~

U I HEAT D EXCHANGER THERMAL DESIGN

263

The elimination of the wall temperatures can be performed by writing the rate equations for both sides of the walls:

%,2h2(T2

-

Tw.2) = %,*hl(Tw,1 -

Tl)?

(13)

77Z.2WT2 - Tw.3) = %,3h(Tw,3 - T3)' ( 14) With Eqs. (13) and (141, it is easy to show that balance equations (101, ( l l ) , and (12) will be expressed in terms of the fluid temperature differences and overall heat transfer coefficients between the fluids involved:

where X o and Yo are heat exchanger flow lengths in the x and y directions. The overall heat transfer coefficients are defined according to Qj,, = U,,2 A q , 2 , 4 0 ,for j = 1 and 3.13 Here ij = + 1 or - 1 for j = 1,2,3 depending on the flow direction and the coordinate system adopted. Again, the set of equations, Eqs. (19, (16), and (17), may be rearranged to result in the generalized form as presented earlier by Eq. (2).

2. Macrobalance Equations The macrobalance or overall energy balance of a three-fluid heat exchanger [see Figs. 9(d) and 10(a)] can be written in the following form: 3

C (q,out

-

q-in)(kcp)j =

0,

(18)

j= 1

where q,,,,, and 7;.,in are the mixed mean outlet/inlet temperatures. Note that the internal consistency between the microbalances [Eqs. (4146) or (15)-(17)] and macrobalances [Eq. (lS)] can be formally expressed with the relation: 3

/ c ij(ritc,)jdq j= 1

=

0.

(19)

I3Note that for design purposes !he overall heat transfer coefficients may be defined according to an alternative relation Q,,, = q2A q , , A where I!& is based on the total heat transfer area A of the respective side [36] (see footnote 11).


264

TABLE VI BOUNDARY CONDITIONS FOR PARALLEL STREAM COUPLINGS Parallel stream coupling j

P1

P2

P3

P4

D. BOUNDARY CONDITIONS

A list of possible boundary conditions (i.e., the corresponding inlet temperatures and the locations of inlets) for a parallel stream three-fluid heat exchanger (see Fig. 6) is given in Table VI. The corresponding boundary conditions for the cross-flow three-fluid heat exchanger of Fig. 10 are given in Table VII [see Fig. 7(a)]. Note that the list of parallel stream three-fluid heat exchangers with two thermal communications is complete in Table VI, whereas the list of possible cross-flow arrangements with two thermal communications given in Table VII is not complete, as mentioned in Sec. II.B.2.b.

E. THEMATHEMATICAL MODELIN DIMENSIONLESS FORM

The governing equations are now nondimensionalized to understand the complexity of the three-fluid heat exchanger problem and to determine the relevant parameters controlling the heat transfer process and the shape of temperature fields. Note that a number of investigations reported in the literature were performed without nondimensionalizing the governing equations.

TABLE VII BOUNDARY CONDITIONS FOR CROSS-FLOW ARRANGEMENTS

Fluid flow arrangements

i

c1

c2

c3

c4

-


265

The nondimensionalizations of governing equations for parallel stream and cross-flow arrangements are done separately. 1. Parallel Stream Arrangements

The set of governing equations, Eqs. (7), (81, and (9), can be nondimensionalized as follows:

i2-

i,-

dO,

d5 d@3

d5

=

Cf.2 NTU,(O, - 0,)+ C t 2 R * N T U I ( 0 3- O , ) ,

=

-R*NTU,(O,

C?,2

-

(21)

0,),

CT92

where

0,= I;. -

T1,in

T2,iti - T l , i n

for j

=

1 , 2 , 3 and

X

6 = -; L

(23)

It is worth noting that the preceding set of nondimensional groups chosen is just one of several possible combinations (see Table I for titles and Table I1 for a number of other possible selection^).'^ The corresponding boundary conditions of Table VI are given in Table VIII in nondimensional form. The macrobalance of Eq. (19) reduces to:

Id -(Cf,,0,

d5

+ i20,+ i3C?,,0,) d(

=

0.

(25a)

I4We should emphasize that in the formulation just presented, fluid 2 is the central fluid (with two thermal communications) and fluids 1 and 3 are lateral fluids. As pointed out in Sec. ILA, we will distinguish fluids as fluids 1, 2, and 3 regardless of which fluids are hot and cold (for the problem solution in a general case, the temperature levels of the fluids involved are irrelevant). For the definition of the heat exchanger effectiveness, though, it is important to distinguish which of the fluids is the hottest (coldest) one (see Sec. VI1.A).

D. P. SEKULIC AND R K. SHAH

266

TABLE VIII DIMENSIONLESS BOUNDARY CONDITIONS FOR PARALLEL STREAM COUPLINGS Parallel stream coupling

P1

P2

P3

P4

2. Cross-flowArrangements The set of governing equations, Eqs. (151, (16), and (171, can be generalized and nondimensionalized as follows:

The independent variables here are defined differently compared to the parallel stream arrangement in order to present concised closed-form solutions [see Eqs. (44) through (5011:

6'

X

= -NTU,

xo

C t 2 = 6 NTU, Ct2;

77'

=:

Y -NTU,

YO

=7

NTU,. (29)

The nondimensional parameters are defined in the same manner as for parallel stream arrangements (see Table I). Note that heat transfer areas in NTU, and R* are equal to A , = X,Y, (see footnote 11). Consequently, the ratio R* reduces to the ratio of overall heat transfer coefficients. The boundary conditions of Table VII are given in Table IX.

THREE - FLUIDHEATEXCHANGER THERMAL DESIGN

267

TABLE IX DIMENSIONLESS BOUNDARY CONDITIONS FOR CROSS-FLOW ARRANGEMENTS Fluid flow arrangements

c1

c2

j

5

v

@

1 2 3

1 1

0 -

0

-

5 -

s 0

1

0 e3j" - 1

c3 @

0 1 e3.i"

5 0

-

c4

s

0

1 0

0 1 @3,in

5

0

0 0 - 0

@

0

1 a3.i"

F. GOVERNING PARAMETERS In Sec. III.B, a list of dimensionless groups was given (see Tables I and 11). Let us consider more closely the physical meaning of the groups that

govern the mode! of a three-fluid heat exchanger with two thermal communications. As is well known from the analysis of a two-fluid heat exchanger (the effectiveness-NTU approach), two dimensionless parameters deduced from differential equations are NTU and capacity rate ratio C*. However, for a three-fluid heat exchanger with three thermal communications, five independent dimensionless groups should be formulated based on the dimensionless governing equations. For example, those five can be two capacity rate ratios, two thermal resistance ratios, and one NTU (as defined by Aulds and Barron [12]; see Table 11). For a three-fluid heat exchanger with two thermal communications, the number of independent parameters is four [as in Eqs. (241, i.e., as defined in Table I; instead of two thermal resistance ratios just one is needed, i.e., R*]. Additionally, if one fluid is at a constant temperature (as in condensation or evaporation), its thermal capacity rate is treated as infinite, and hence one heat capacity rate ratio drops from the parameter list. In addition, several parameters are associated with the temperature boundary conditions on which the solution of the differential equations will depend. In the case of a three-fluid heat exchanger with two or three thermal communications, three independent inlet temperatures exist. Two of three dimensionless inlet temperatures can be specified in a manner to have the values 1 or 0 [as shown in Tables VIII and IX; see also Eq. (2311. Therefore, the nondimensional inlet temperature e3, in of the third ffuid can be defined as a parameter. All parameters can be divided into two groups: operating condition parameters and design parameters. Operating condition parameters are related to the fluid flow rates and inlet temperatures of streams involved:

268

D. P. SEKULI~ AND R. K. SHAH

the heat capacity rates of fluid streams and the inlet temperatures. They are not dependent on the heat exchanger overall thermal conductances or physical size, fluid flow arrangement, or number of thermal communications. The heat capacity rate ratios of fluid streams are defined as follows (see, for example Tables I and 11):

The heat capacity rate ratios Cjlc2 represent the measure of thermal balance between the fluid streams j = 1 or 3 and fluid stream 2. For some applications, in a well-designed three-fluid heat exchanger, the combined heat capacity rates of the two outer fluids should not be significantly different from the capacity rate of the central fluid [32] (see also Sec. VII). With the foregoing definitions of CT2 when CT2 C z , = 1 (i.e., the heat capacity rate of the central fluid is equal to the sum of the heat capacity rates of lateral fluids) the heat exchanger is considered to be “balanced.” In general, however, the range of values of capacity rate ratios can be between zero and infinity (or zero and one, depending on the definition). For a heat capacity rate ratio equal to unity ( C : , or C z , = l), the relevant fluid streams are “balanced.” The dimensionless inlet temperature of the third fluid is defined as follows using Eq. (23):

+

m

m

According to the definition of dimensionless temperatures of Eq. (231, two other dimensionless inlet temperatures for fluid 1 and fluid 2 have fixed values 0 and 1, respectively (for example, fluid 1 has the lowest inlet temperature and fluid 2 the highest). Therefore, the range of values of the inlet temperature of the third fluid is between 0 and 1. Design parameters depend both on the size of a heat exchanger (a combination of the physical size and overall heat transfer coefficient) and/or on the heat capacity rates of fluids involved. Similar to the conventional two-fluid heat exchanger analysis, it is convenient to define NTU, (see Table I) as:

The value of NTU, (the thermal size of a heat exchanger) represents the ability of the heat exchanger to change the temperature of fluid 1 because

-


269

of the thermal communication between streams 1 and 2.” As just defined and in Table I for a three-fluid heat exchanger with two thermal communications, NTU, is related to fluids 1 and 2; however it can be defined differently. NTU, varies between 0 and infinity with a range of 0.5 to 3.0 for most industrial heat exchangers. Note, however, that in some applications the heat exchanger thermal size frequently has a much broader range of values (for example, in cryogenics). Finally, the ratio between the overall thermal resistances of heat transfer surfaces is also a design parameter [see Eqs. (21) and (22) or Eqs. (27) and (281, as well as Tables 1 and 111:

The ratio of the overall thermal resistances, the ( l / U A ) ’ s , indicates the relative ability of the separating surfaces to transfer heat. This parameter has the meaning of thermal resistance balance among the heat transfer surfaces involved. This ratio can be either larger, smaller than, or equal to 1. If a three-fluid heat exchanger has two thermal communications, only one parameter of this type has to be defined. If three thermal communications exist, two parameters of this type should be defined.

VI. Solution of a Three-Fluid Heat Exchanger Problem A. SOLUTION METHODS

Solution methods used in literature for solving the mathematical model of a multifluid heat exchanger can be divided into several different approaches: exact (closed-form) procedures [lo, 11, 14, 25, 26, 34, 461, approximate analytical [37, 471, numerical [7, 32, 35, 401, seminumerical methods [15, 16, 19, 331, and electric analogue [43]. The selection of a particular method depended on the investigator as well as the complexity of the physical configuration under consideration (three or more fluid streams; two, three, or more thermal communications; parallel stream or cross-flow arrangements, one-, two-, or three-dimensional temperature distributions, etc.) and the corresponding model adopted (constant or variable thermophysical properties, etc.). According to the well-known analytical procedure for solving a system of linear first-order differential equations [78], the set of three first-order ’’This is the reason why the symbol for NTU has the subscript 1. Subscript 1 denotes one of the two lateral fluids.

270

D. P. SEKULIC: AND R. K. SHAH

equations describing the three temperature distributions in a parallel stream three-fluid heat exchanger [Eqs. (71, (81, and (9) or Eqs. (20), (21), and (2211 can be transformed to one third-order equation in one unknown temperature distribution. This equation has to be integrated in order to obtain the general solution of the equation. Constants of the integration have to be determined by applying the boundary conditions (see Table VI or VIII). Subsequently, the other two temperature distributions are found by the substitution of the known temperature distribution without performing integration, Alternately, the original set of equations can be rearranged in order to be presented in matrix form, and afterward the determinant of the system of equations has to be defined. To obtain nontrivial solutions for the system, the determinant must be equal to zero, which leads to a cubic characteristic equation. Because derivatives of variables in the original set of equations are linearly dependent on variables, the solution of equations will be of an exponential form. Note that the complexity of algebra associated with the solution results in a cumbersome but straightforward procedure. The explicit analytical solution of the set of linear first-order differential equations can also be obtained very efficiently using the Laplace transform technique. The same technique can be utilized to obtain the explicit analytical solution for both parallel flow and cross-flow arrangements, as demonstrated by BaEli6 et al. [34, 521. In the case of one-dimensional temperature distributions, the solution procedure leads to a set of algebraic equations. The set of equations can be solved to express explicitly one of the unknown temperature distributions as a Laplace transform of the same variable as a function of other relevant parameters. Furthermore, the inverse Laplace transform of that variable has to be determined. Subsequently, the other two temperature distributions can be obtained by exploiting the algebraic relations among the Laplace transforms of the variables and by exercising the inverse Laplace transform procedures. In the case of a two-dimensional temperature field (e.g., for a cross-flow three-fluid heat exchanger), a similar analytical procedure [applied to Eqs. (13, (161, and (17) or Eqs. (261, (271, and (28) and corresponding boundary conditions; see Table VII or 1x1 leads to a differential equation with Laplace transforms of temperatures involved and to a set of algebraic relations among the corresponding Laplace transforms. These algebraic relations can be mutually combined in order to obtain one differential equation for one of the Laplace transform variables. Afterward, performing the inverse Laplace transform procedures and using the relationships between the variables, all three temperature fields can be determined. The solutions are algebraically very complex for both parallel flow and crossflow arrangements, In the case of a cross-flow heat exchanger, the solu-

THREE- FLUID HEATEXCHANGER THERMAL DESIGN

27 1

tions are expressed by a class of special functions in the form of an infinite series of modified Bessel functions of n’th order [341. According to the review of research work on the three-fluid and/or multifluid heat exchanger problems (see Sec. IV), some other approaches (modifications of the analytical procedures discussed, approximate methods, numerical methods, and seminumerical methods) are available. Particularly when the number of fluids is greater than 3, the temperature fields are multidimensional and some of the idealizations have to be ignored. The solution to the characteristics equation and determination of unknown coefficients can be carried out numerically using standard numerical analysis techniques. More advanced numerical procedures are unavoidable if nonlinearities are to be included into the model [171. All analytical results presented in this work were obtained by a Laplace transform method.

B. PARALLEL STREAM ARRANGEMENTS 1. Temperature Distributions

Without elaborating on the analytical procedure, only the final solutions are discussed. Refer to Sec. IV.A.l and V1.A for the list of pertinent references. The set of Eqs. (201, (21), and (22) is solved along with the boundary conditions (according to Table VIII) for four parallel stream couplings (see Table V and Fig. 6). The dimensionless temperature distributions for three fluid streams are given by [13]:

@,(S)

=

@2,+0@j(S) + @3,5,0y(S) f o r j = 1,2,3,

(34)

where the subscript j denotes the fluid flow stream. The functions @,(S> and q j ( S > are given in Table X, and the coefficients 02.5=o and [the dimensionless temperatures of fluids 2 and 3 collocated at 6 = 0; see Fig. 9(d)] are given in Table XI. Here, the dimensionless parameters NTU,, Ct2, C ; , , R*, and 0,in are defined in Table I (see also Sec. V.F). The fluid flow sign indicators 1, and i, in Eq. (34) (see Table XI should be used as given in Table V. (Note that i , = + 1 always holds as adopted by the convention; therefore, this indicator is not used explicitly in the solutions.) Note that the four stream direction combinations in a parallel stream three-fluid heat exchanger, P1 through P4, correspond to the four distinct heat exchanger physical situations concerning the coupling of stream inlets and outlets (as an example, we will consider a triple pipe heat exchanger). Therefore, even though effectiveness and NTU, for P3 and P4 (see Fig. 6 ) can be calculated using the same formula as noted in Sec. II.B.2.a, P3 and

TABLE X FUNCTIONS q ( S ) AND Yj(6) OF EQ.(34)

1

2a

P4 arrangements with the same inlet temperatures and flow rates may have different performances as discussed next with an example. In the triple-pipe heat exchanger of Fig. l(a), if the heat capacity rates of the two lateral (colder) fluids (for example, fluids 1 and 3) are not the same, the values of the heat capacity rate ratios for the three-fluid heat exchanger will be different for the P3 and P4 arrangements and, hence, the effectiveness, outlet temperatures, and heat transfer rates will be different in this heat exchanger. For example, consider C, = lOO,C, = 20,C3 = 25 W/K and ( U A ) , , = 100 and (UA)1,2= 50 W/K for the P3 arrangement, resulting in C ? , = C,/C, = 0.5, Cz, = C3/C2 = 0.25, and

@2,5=o

Coupling

P1

TABLE XI 03,c-o OF EQ.(34)

AND

@2,C-O

1

P2

P4

@3,6=n

@3, in @3, in

1


273

TABLE XI1 EXITTEMPERATURES FOR PARALLEL STREAM HEATEXCHANGES'

"Explicit expressions for @,(,$) and 'P,([) for j

= 1,

2, and 3 are given in Table X.

R*

= ( U A ) 3 , 2 / ( b ! ! ) ,= , 22.0. Now if we switch fluids 1 and 3 to the innermost and outermost pipes of Fig. l(a>, it will correspond to the P4 arrangement of Fig. 6. If we refer to switched fluid 3 as 1' and fluid 1 as 3', then the new values of the parameters are C t 2 , = C1,/C2 = 0.25, C f , 2= C , / C 2 = 0.5, and R*' = (UA)3b,2/(b!A),,,2 = 0.5. The temperature distributions, effectiveness, and heat transfer rates for the switched fluid case can now be determined using the formulas for the P3 arrangement, and the results will be different from the original case because the values of nondimensional parameters are different. Note that if we had not considered the switched case, the values of nondimensional groups (C?,, C z , , and R * ) for the P4 arrangement would be the same as that for the P3 arrangement, but the effectiveness would be different from the P3 case as found from the results using the P4 formula of Tables X, XI, and XI1 or results presented later in Fig. 17.

2. Exit Temperatures Explicit formulas for the temperature distributions of fluid streams within the parallel stream three-fluid heat exchanger provide an easy determination of exit temperatures regardless of the flow arrangement

274

D. P. SEKULIE AND R. K. SHAH

and/or coupling of streams. In Table XII,exit temperatures are summarized for all four stream couplings (P1 through P4). The exit temperature formulas were obtained collocating Eq. (34) at the corresponding outlets of respective fluid streams (see also Fig. 6). The functions Q j ( [ )and qj(,f) for j = 1, 2, and 3 should be calculated at 6 = 1 according to Table X. Note that fluid flow indicators i , and i, take values according to Table V. 3. Explicit Formulas for NTU,

Because of a very complex algebraic structure of the interrelation between outlet temperatures and pertinent parameters (see Tables XI and XII), a number of authors believe that a trial-and-error procedure is always needed to calculate NTU, even if terminal temperatures are given [9-111. Krishnamurty and Venkata Rao [44, 481 have derived explicit formulas for determining the length (Le., the corresponding heat transfer areas) of a parallel stream three-fluid heat exchanger valid for any flow arrangement and in terms of terminal temperatures. The approach is still iterative if some of the terminal temperatures are not known. Similar expressions, but in a more general form (as functions of independent parameters) have been obtained by BaElik et al. [52]. From the solution of Eq. (34), after cumbersome but straightforward algebraic manipulation, one can also obtain an explicit expression for NTU, similar to the previous ones from the literature [44, 48, 521. The solution valid for any parallel stream arrangement and/or stream couplings is presented in a compact form in Tables XI11 and XIV.16 4. Temperature Cross

Similar to some two-fluid exchangers, the temperature cross phenomenon can appear in a three-fluid heat exchanger. For a two-fluid heat exchanger, a temperature cross is defined to exist when the hot fluid outlet temperature is lower than the cold fluid outlet temperature [79].” In a IhNote that the values of different parameters given in Tables XI1 and XIV depend on the flow arrangement. The fluid flow indicators should be determined according to Table V and dimensionless temperatures according to Table XI. ”We should emphasize that the temperature cross phenomenon is evident for some values of parameters in a 1-2 shell-and-tube exchanger when the shell fluid inlet is at the U-tube bend end. For the identical values of parameters, one cannot see a temperature cross between the shell fluid and the second-pass tube fluid temperatures when the shell fluid inlet is at the tube fluid inlet end. In both cases, the effectiveness and outlet temperatures will be the same. Hence, now a more general definition of a temperature cross as defined here is used.

-


TABLE XI11 NUMBER OF TRANSFER UNITS FOR PARALLEL STREAMTHREE-FLUID HEATEXCHANGER

N,

V

1-4

5 6

2yi2i3-

a

1

c::2

1

b 2Y

C

1

TABLE XIV REOUIRED IN TABLE XI11

COEFFICIENTS nj,,,

275


276

P2

Pi

e

F

P4

P3

c FIG. 11. Temperature distributions and temperature cross in a parallel stream three-fluid heat exchanger with stream couplings P1 through P4, C t 2 = 0.8, C?, = 0.25, R* = 2.0, 03.in = 0.0, and NTU, = 1.25, according to Sekulif [13].

single-pass three-fluid heat exchanger, the temperature equalization of adjacent fluid streams (direct cross) and/or nonadjacent fluid streams (indirect cross) is theoretically possible (see Fig. 11). Heat transfer takes place simultaneously between all three fluid streams (directly or indirectly) and there is no violation of the second law of thermodynamics, Thus, in such a situation a temperature cross could exist between temperature distributions of the hot fluid and a cold fluid(s); the hot fluid is being heated by the cold fluid in the region beyond temperature cross. The heat transfer area in this region is then wasted. Therefore, there is a need to

-


277

predict the existence of the local temperature equalization even for a single-pass three-fluid heat exchanger in order to optimize the design. The indirect temperature cross mentioned earlier between two colder fluids (having no thermal communication) is of academic interest only; it does not affect heat transfer surface area requirements. Let us consider more carefully corresponding situations. The set of inlet fluid data has been defined. A parallel stream heat exchanger of a tubular type (three concentric tubes) is now analyzed for all four stream couplings. Lateral fluid 1 is flowing through the innermost tube, and lateral fluid 3 is flowing through the outermost annulus, both with the same inlet temperatures but different mass flow rates. The central fluid (fluid 2) has the highest inlet temperature and a given flow rate. If all three fluids enter the heat exchanger at the same side (Fig. 11, P1) for a given set of inlet temperatures and mass flow rates, fluid 2 will heat fluid 1 throughout the heat exchanger. However, fluid 2 will heat fluid 3 only until the point where the equalization of respective temperatures at 6 = (* = 0.301 occurs (i.e., the temperature pinch occurs at &*). The local heat transfer rate between fluids 2 and 3 is going to be reversed throughout the rest of the exchanger. In the P2 case (fluid 2 enters the heat exchanger at the outlet side of lateral fluids; all fluids have the same inlet conditions as in Pl), there is no direct cross between the central and any of the lateral fluid temperature distributions. Local temperature differences along the heat exchanger length never approach zero. The outlet temperature differences between the central and both lateral fluids, however, do change sign compared to the differences of the respective inlet values (i.e., 02,0ut < 03,0utr and > 03,in, In that sense, the outlet temperature crosses exist.” If the fluid that is flowing through the innermost tube (fluid 1) enters the heat exchanger at the heat exchanger side where fluids 2 and 3 are leaving it (Fig. 11, P3), all three aspects of the temperature cross phenomenon are present: (a) the direct cross between fluids 2 and 3 at 6* = 0.664, (b) the indirect cross between fluids 1 and 3 at 6* = 0.914, and (c) the change in sign of the exit temperature differences of both fluids 2 and 3, and fluids 2 and 1 in comparison to the inlet temperature differences. Finally, if the fluid flowing through the outermost annulus enters the heat exchanger where fluids 2 and 1 are leaving it (Fig. 11, P4), 18 From this discussion, it is clear that the term “fernperatwe cross” can be misleading if used when no actual cross of fluid stream temperature distributions exists in a three-fluid heat exchanger. The meaning of the term then is the same as in the case of a two-fluid heat exfhanger; that is, the outlet temperature of the hot fluid and the inlet temperature of a lateral cold fluid interchange their respective magnitudes. However, the local temperature difference between the respective streams is not approaching to zero at any place within the heat exchanger.

.‘ .

278

D.P.SEKULI~ AND R. K. SHAH TABLE XV PARAMETERS L i , AND D,,, OF EQ.(35)

the direct cross between fluids 1 and 2 is at (* = 0.981, the indirect cross between fluids 1 and 3 is at [* = 0.777, and the change in outlet temperature differences between both fluids 2 and 3 and fluids 2 and 1 exists. Note that in the last two cases, P3 and P4, regardless of the fact that the both cases can be analyzed using the P3 formulas for temperature distributions, effectiveness, and NTU,, the actual physical situations are different. It can be shown that the location of both direct and indirect temperatures cross in a three-fluid parallel stream heat exchanger with two thermal communications is given by SZj as follows [13]: 1 Di,j + Li,j =--In (35) y NTU, Dj,- L j , *

':J

Parameters Di,and L j , are given in Table XV, and the values of O,,f=o and @ 3 , 5 = o are given in Table XI for all four possible parallel flow arrangements. Subscripts i and j denote the temperature cross between the following specific fluid streams: {1,2}, {2,3}, and {1,31. Note that streams 1 and 3 (in contrast to the other two temperature cross situations) are not mutually in thermal contact (i.e., fluid 1 is separated from fluid 3 by fluid 2) and there is no practical implication about whether or not there is an indirect temperature cross.

5 . Reduction of the General Solution for Some Important Particular Cases Careful consideration of the analytical structure of the exact solution given by Eq. (34) for a parallel stream three-fluid heat exchanger can provide several useful results. The first important particular case is the

THREE- FLUID HEATEXCHANGER THERMAL DEStGN

279

reduction of the given three-fluid heat exchanger problem to a two-fluid heat exchanger case with the thermal communication between fluids 2 and 3 nonexistent. To reduce the three-fluid heat exchanger model [Eqs. (20), (211, and (2211 to the model for a conventional two-fluid heat exchanger, the conductance ratio R* should be set equal to zero. The R* = 0 condition means that the overall heat transfer coefficient for the thermal communication between the central fluid (fluid 2) and one of the lateral fluids (fluid 3) has to be equal to zero. In that case, the second term in Eq. (21) vanishes and the right side of Eq. (22) is equal to zero. Consequently, fluid 3 does not change temperature (no thermal interaction between fluid 2 and fluid 3). This is formally equivalent to the situation when i , = 0. Therefore, in the limit R* = 0 and i, = 0, the system of three governing equations [Eqs. (201, (211, and (2211 reduces to two differential equations that correspond to the two-fluid heat exchanger model:

along with the corresponding boundary conditions, depending on the flow arrangement. The solution of the set of equations given by Eqs. (36) and (37) should be exactly the same as the one obtained from the general three-fluid heat exchanger problem solution [Eq. (3411 in the limit R* -+ 0, i , = 0. For the sake of clarity, let us consider only the reduction of the dimensionless temperature of fluid 1. It is easy to conclude that the dimensionless outlet temperature of fluid 1 [see Eq. (23) for j = 1 at the outlet] in the two-fluid heat exchanger case corresponds to the definition of two-fluid heat exchanger effectiveness [361 for C , < C 2 . Therefore, the reduction of the dimensionless outlet temperature of fluid 1 given by Eq. (34) should lead to the definition of a two-fluid heat exchanger effectiveness. The reduction was first performed by Sorlie [ll],utilizing the definition of temperature effectiveness of a three-fluid heat exchanger, but only for cocurrent and countercurrent fluid flow arrangements. Later BaEliE et al. [52] demonstrated that the same holds true for all four flow arrangements, as anticipated. In our case [Eq. (3411, the reduction gives:


280

The two basic flow arrangements exist for a parallel stream two-fluid heat exchanger (is., i, = +1, cocurrent; and i, = -1, countercurrent). One can show, after performing the necessary algebraic manipulation, that Eq. (38) reduces to Cocurrent flow arrangement:

Countercurrent flow arrangement:

- 1 - exp[ - ( 1 - C;,,)NTU,] Rlim * - O [01(t)]6=1,i2=-1 - 1 - C;,z exp[ - (1 - C;,,)NTU1] r,=O

The second important limiting case is when a phase change occurs in a heat exchanger (condensation or evaporation of a fluid at constant temperature). Let us consider the situation when the central hot fluid stream has constant temperature throughout the heat exchanger. In such a situation, the two heat capacity rate ratios C t 2 and C?, should be equal to zero (i,e., the heat capacity rate of the central fluid IS considered to be infinitely large). The mathematical model of a three-fluid heat exchanger [Eqs. (20), (20, and (2211 reduces in this situation to the set of only two decoupled differential equations as follows:

dO,

-d5 -,i

-

NTUl( 1 - e l ) ,

d0,

= C;,,R*NTU,(l - 0,). d5 Note that 0,= 1 = fixed. Integrating Eq. (40, one can show that the dimensionless outlet temperature of fluid 1 in a three-fluid heat exchanger with the central fluid remaining at the constant temperature is equal to the two-fluid exchanger effectiveness (which corresponds to the class of heat exchangers known as condensers or evaporators), i.e.,

The same results [but with a corresponding ( U ! ) , ,JC3 value] hold for the dimensionless outlet temperature of fluid 3. The result defined by Eq. (43) can be obtained reducing the general solution [Eq. (3411 to the particular case that corresponds to the limit

THREE- FLUIDHEATEXCHANGER THERMAL.DESIGN

28 1

C z 2 = CZ2= 0. In that case, a = m and p = y = 1; therefore, V&l)= 0 and @,(1) = 1 - exp(-NTU,), as expected. Finally, for the special case when the sum of the heat capacity rates of both lateral fluids tends to be equal to the heat capacity rate of the central fluid k e . , C t 2 + C ; , = 1, the so-called balanced three-fluid heat exchanger), analytical expressions obtained in the literature as well as the general solution given by Eq. (34) are singular. The nonsingular general solution can be obtained in a straightforward manner by using the same Laplace transform solution procedure, but taking into account that the relevant parameters are given as follows: (Y = 0; that is, p = y. Notice that the general solution of the problem given by Eqs. (201, (211, and (22) can be readily reduced to this particular case. The procedure assumes determination of adequate limeses of the solution in that case. Some particular cases are discussed in the literature (explicit solutions for cocurrent and countercurrent flow arrangements are given by Sorlie [ll] and Krane and SekuliC [801, but in a different format). C. CROSS-FLOW ARRANGEMENTS 1. Temperature Distributions

The analytical procedure to obtain an explicit two-dimensional temperature distribution within a three-fluid cross-flow heat exchanger is far more complex than the corresponding one for a parallel stream arrangement. It requires the solution of partial differential equations (261, (271, and (28) along with the corresponding boundary conditions (Table IX). Because of the complexity of the problem, one can find very few attempts in the literature devoted to the analytical treatment of cross-flow heat exchangers. An exact closed-form analytical solution is available only for the cross-cocurrent flow arrangement C4 of Fig. 7(a); Table IX [34, 601. As stated earlier in Secs. IV.A.2 and VLA, the other solution approaches are numerical, seminumerical, or approximate analytical [19, 32, 33, 35, 40, 58, 61, 62, 641. The lack of closed-form solutions causes a considerable problem in presenting the numerical results in a manner that would be useful to a designer. Namely, a three-fluid cross-flow heat exchanger has five independent parameters (in the case of two thermal communications among the fluid streams) with one or more dependent variables, the same as for parallel stream arrangements. In addition, the temperature distributions are now at least two dimensional. Therefore, the effort to present comprehensive numerical results that are important for design purposes, and to cover at the same time all important ranges of parameters values, is very

282

D. P.SEKULI~ AND R. K. SHAH 1.0

0.4 0.2

0

FIG. 12. Temperature fields in a three-fluid cross-flow heat exchanger (arrangement C4) where C t 2 = C.Fz = 1.0, R* = 1.0, NTU, = 2.0, and a,,,, = 0.5, from BaElii ef al. [34]. The dashed line fluid temperature distributions denote values lower than the solid line temperature distributions for the other fluid in the same region. Thus one can see a temperature cross between fluid 2 and fluid 3 in this figure.

demanding. As an example, Fig. 12 shows the complexity of temperature distributions in a three-fluid cross-flow heat exchanger for a particular set of governing parameters. From a designer’s point of view, temperature effectivenesses are presented graphically as functions of relevant parameters for a selected set of parameter values (see Sec. VII). However, temperature effectivenesses (in fact, the dimensionless mixed-mean outlet temperatures of the lateral fluids or their combinations) provide no information on the existence of a temperature cross (see Fig. 12), hot and cold spots, etc. In addition, multiple interpolation is needed for the determination of the temperature effectiveness. Therefore, in this section, major attention is paid to the spatial distribution of temperature fields. Willis [401, Willis and Chapman [32], and Ellis [35]concluded that the cross-cocurrent flow arrangement C4 of Fig. 7(a) has the best overall performance among the different one-pass cross-flow arrangements. Let us consider in detail the analytical solution for this arrangement.

-


283

Equations (261, (271, and (28) (including boundary conditions from Table IX for the C4 arrangement) were solved using the Laplace transform technique 1341. The spatial temperature distributions of each of the three fluids are given by the following equations:

284


where

f o r m 2 1 . (50) The I,, function in the special function denoted by V,, Eq. (50), is a modified Bessel function of n’th order. To better understand the temperature conditions in a cross-flow arrangement, it is useful to plot the so-called “temperature gradient map” [35], which gives a spatial two-dimensional temperature distributions in the ( x , y ) or (6,771 plane of a fluid flow passage. The numerical solution obtained for the C4 arrangement is presented graphically by Ellis [35] for the most simple situation of the same dimensionless inlet temperatures = 0) for both lateral fluids. Also, the lateral fluids have equal (Le., 03,in heat capacity rate ratios such that their sum equals that of fluid 2 0 4-

9 $

0.4

g

0.2

fiE

0

- 0.2

0

1.o

2.0

NTUl

3.0

4.0

FIG. 20. Temperature effectiveness (a,,dashed lines; a,, solid lines) for countercurrent parallel stream three-fluid heat exchangers (P2, Fig. 6). Direct heat transfer beween nonadjacent fluids due to conduction through the fins (a third thermal communication) is not considered negligible. The set of parameters (Cz,, and C;*, R', and At$ = 0.25) is the same as in Fig. 19, from Demetri and Platt [19].

results, which correspond to the same set of parameters as in Fig. 19, are presented in Fig. 20. Inspection of these diagrams shows that the third thermal communication (situations when parameters RR, is not equal to 0; see Table I1 for the meaning of parameters) can substantially influence both temperature effectivenesses.

2. Cross-flowArrangements

In Fig. 21, the temperature effectivenesses for two cross-flow arrangements are presented. The cross-countercurrent flow arrangement tempera-

looa'

--

'

'

~

2

0

'

'

' -At, = 0.25 '

8

6

4 m 1

100

Ib

T

I

I

'

1 80

l9 60

40

20

i

FIG. 21. Temperature effectivenesses in percent (IY, , dashed lines; a,, solid lines) for cross-flow arrangements: (a) cross-countercurrent arrangement C3 of Fig. 7(a), from Ellis [351; (b) cross-cocurrent arrangement C4 of Fig. 7(a), from Willis [40].The set of parameters is the same as in Fig. 19.

D. P. SEKULI~ AND R K. SHAH ture effectivenesses are depicted in Fig. 21(a) [35],and Fig. 21(b) shows a similar set of curves for the cross-cocurrent flow arrangement [401. The set of independent parameters is the same as in Fig. 19 for parallel stream arrangements. It is obvious that temperature effectiveness reaches higher values in cross-cocurrent arrangements when compared to the cross-countercurrent arrangement (see also comments given in Sec. IV.A.2 concerning the work of [35,401). Again, as in the case of parallel stream arrangements, the effectiveness of the colder of the two lateral fluids may become negative at larger NTUs. The reason is related to the inversion of the heat transfer rate following the temperature cross phenomenon.

VIII. Three-Fluid Heat Exchanger Thermal Design Methodology

In this section, we discuss thermal design theory for a three-fluid heat exchanger with two thermal communications. The other important aspects, such as the hydraulic design, mechanical design, manufacturing considerations, and thermoeconomical optimization, are beyond the scope of this article, but the reader can refer to Shah [4] for a qualitative description. Because of significant differences in the thermal design procedure, approximate and exact methodologies are reviewed with detailed step-by-step procedures. We should emphasize that the design procedures to be outlined here use the solutions (i.e., the linear theory) presented in Sec. VI, which are valid only when the idealizations of Sec. V.B are met. If those idealizations are not met, a numerical approach should be used; for example, if the fluid properties are not constant over the temperature range of interest, or if the axial conduction has to be included in the analysis (as in many cryogenic applications). Another option is to apply the linear theory after dividing the exchanger into a certain number of segments (the first one who suggested such an approach was Hausen [ 101; also see Prasad and Gurukul [72,731, and Paffenbarger [17]). The design of multifluid heat exchangers (including the three-fluid heat exchanger), in which a numerical approach has been used, was discussed recently by Paffenbarger [17]. That effort was devoted to the development of a general computer code for the solution of a rating problem for multistream, countercurrent plate-fin heat exchangers, and includes the effects of variable physical properties, axial and cross-layer conduction within the core, and the stacking order. The program developed uses the finite-element differential equation solver COLSYS [82-841 for solving equations governing temperature and pressure distributions within the

-


305

heat exchanger. The temperature prediction for a three-fluid parallel stream heat exchanger with constant fluid properties agrees within ‘r 0.01% with the exact analytical solution (see Sec. VI). Paffenbarger’s approach has been developed to overcome the computational limitations of the first-order finite difference scheme (roundoff error and stability problems) used in previous efforts [15, 71, 721. For some other important aspects of design procedures, the reader should consult the literature. (See the review given in Sec. IV and, in particular, the papers of Kao [29, 661 and Prasad and Gurukul [73].) As pointed out by Haseler [71] and Paffenbarger [17], methods for the design and performance analysis of multistream heat exchangers have been developed primarily for the cryogenic gas processing industry.

OF A THREE-FLUID HEATEXCHANGER THERMAL A. DEFINITION DESIGNPROBLEM

Two characteristic design problems need to be addressed: rating (performance) and sizing (design). In both problems, it is assumed that all heat transfer coefficients are given or determined beforehand, the same as in the case of a two-fluid heat exchanger [3, 36].*’

I. Rating Problem The determination of outlet temperatures for all three fluid streams and/or heat transfer rates constitutes the main task in the rating problem. Inputs to the rating problem of a three-fluid heat exchanger are as follows: The three-fluid heat exchanger type, configuration, and overall dimensions Flow arrangement and number of thermal communications Complete details on the materials and heat transfer surface geometries Fluid mass flow rates Fluid thermal properties Inlet temperatures of all three streams Fouling factors. 25Becau~eof space limitations, only the major steps for the determination of the outlet temperatures, or NTU, values, are outlined for rating and sizing problems. It is assumed that one needs to calculate appropriate overall heat transfer coefficients in a step-by-step methodology; one also needs to determine the physical size from known NTU, and R* in a sizing problem.

D. P. SEKULIC? AND R. K. SHAH

306

Therefore, the determination of outlet temperatures for all three streams and subsequent computations of the heat transfer rates for each fluid stream will constitute the desired output for a rating problem, The pressure drop analysis is also important in thermal design of a three-fluid heat exchanger. The basic procedure is similar to the wellknown procedure for a two-fluid heat exchanger design [36, 851, hence the details are omitted. 2. Sizing Problem The sizing problem in general refers to the determination of construction type, fluid flow arrangement, physical size, etc., needed to meet the specified heat transfer (and allowed pressure drop) requirements. For an already selected construction type and flow arrangement (including the way of coupling the fluid streams), the sizing problem reduces to the determination of a physical size (length, width, and height of the exchanger). However, because we are not going to include the pressure drop considerations, we will determine NTU, and not the physical size in the sizing problem. In that case, the inputs to the problem are as follows: Surface geometries for both thermal communications Fluid flow rates of all three streams, as well as pertinent fluid thermophysical properties Inlet and outlet temperatures of all three streams.

B. DESIGN PROCEDURES 1. Approximate, Log-Mean Temperature Difference Approach 26 This method for three-fluid heat exchangers parallels the log-mean temperature difference (LMTD) method for the two-fluid heat exchanger design. It is well known that the log-mean temperature difference for two-fluid countercurrent and cocurrent flow arrangements corresponds to the true mean temperature difference. For all other flow arrangements, it is customary to define a correction factor as a ratio of the true mean temperature difference to the log-mean temperature difference. The extension of this method to the three-fluid heat exchanger case, even for ''Sorlie 1111, who analyzed only countercurrent and cocurrent parallel stream arrangements, stressed that this method has been used frequently in industry. The methodology outlined in this section follows the exposition given by Sorlie.

-


307

parallel stream heat exchangers, requires the use of true mean temperature differences; the log-mean temperature difference leads to an inaccurate design as demonstrated later, Sec. VIII.B.2.c. The step-by-step procedure we outline here allows for the development of LMTD correction factors. However, the exact approach proposed later in Sec. VIII.B.2 is quite simple and noniterative regarding NTU, for a number of cases. Hence, there is no need to present an approximate method, but since it is referred to in the literature, it is included here for completeness. a. Rating Problem The input data are as follows: A , , , , A o , z ,lJ,,2, U3.,, ( h c p ) , , ( h c J 2 , ( t i ~ c ~T,,in, ) ~ , T2,in,and T3,in.The output data are: T2,0ut,and T,,,,,. The LMTD method is iterative. It presumes the two-fluid relationship between the heat transfer rate, heat transfer area, and LMTD to be valid; that is, Eqs. (94) and (95) presented later are accurate for the three-fluid heat exchanger. The iterative procedure requires the following steps in which we assume that the central fluid is hot fluid, and that both lateral fluids are cold fluids, but a similar procedure also holds for the case in which the central fluid cools both lateral fluids: 1. Initially assume the magnitudes of heat transfer rates across two existing thermal communications: Q , , , = Q, and Q 3 , 2 = Q3. 2. Determine outlet temperatures of three fluids using energy balances for fluid 1:

Q , = ( h c p ) l ( ~ l , o u t- '],in)?

for fluid 2:

and for fluid 3:

(82)


308

3. Compute LMTDs for heat transfer between the central (hot) and both lateral (cold) streams using the conventional two-fluid exchanger definitions. We assume that the mean temperature differences are functions of terminal temperatures of the streams for a given flow arrangement:

where and for j = 1 and 3 are temperature differences between the appropriate lateral fluids (fluid j for j = 1 or 3) and fluid 2 at the fluid 2 inlet or outlet section: AT1,zin = (T2,in AT1,2out = (T2,out AT3,2in AT3,20ut

= (T2,in

- T1,2in)' - T1,20"t)9 - T3,2in),

= (T2,0ut - T 3 . 2 ~ ~ t ) .

(90) (91) (92)

(93)

In Eqs. (90) through (931, T,2in or q,Zout, for j = 1,3, are the temperatures of fluid j for j = 1,3 at the three-fluid heat exchanger section where fluid 2 has the inlet or outlet, respectively. Depending on the fluid flow arrangement, temperature differences given by Eqs. (90) through (93) can take different values for the same set of inlet/outlet temperatures because of different combinations of the temperatures from the set. 4. Compute heat transfer rates: Q1.2 = ( U W 1 . 2 A ~ ( 1 . 2 W Q3,2

=

AT(3,2)lm'

(94)

(95)

If the Q's of Eqs. (94) and (95) do not agree with those assumed in step 1, iterate steps 1 through 4 until the Q's are converged within the desired accuracy. The desired outlet temperatures are then those from Eqs. (83), (85), and (87) from the last iteration. The input data are as follows: V , , J ,U3,2,( h c p l 1 , Tz,~,,,T3,in,TI,,,,, T2,0ut,and T3,0ut.The output data and A3,2.The step-by-step procedure for the sizing problem is as

b. Sizing Problem

(rja~,,)2~ (thcp)3,Tl,in,

are

-


309

follows: 1. Calculate LMTDs using Eqs. (88) through (93). 2. Determine heat transfer rates among the fluid streams across the two thermal communications:

3. Subsequently, compute two heat transfer areas:

2. Exact Design Method for Three-Fluid Parallel Stream Heat Exchangers The approximate method presented in Sec. VII.B.l has been used frequently, probably because of its simplicity. In some cases, when conventional idealizations €or the heat exchanger design are justified, this approach has been acceptable [ll]. However, as demonstrated later, the approximate approach can lead to unacceptably large errors. Therefore, only an approach based on the exact (analytical or numerical) solutions for temperature distributions within the heat exchanger has to be recommended for the thermal design procedure. The procedure for determining NTU, outlined in this section is rather simple and in some applications noniterative. However, if one needs to include into the consideration the variable fluid properties, the only choice is a numerical approach. The design procedure for a parallel stream three-fluid heat exchanger is based on the analytical solution of the three-fluid heat exchanger problem given in Sec. VI. Both the rating and sizing problems can be solved explicitly and noniteratively in some cases. While the rating problem utilizes a straightforward application of explicit solutions for temperature distributions, the sizing problem needs an explicit expression for NTU, of a three-fluid heat exchanger as a function of other parameters (see Sec. VI.B.3). a. Rating Problem The input data for the rating problem are the same as those given in Sec. VII.B.1.a. The task is to calculate all three outlet

D. P. SEKULIC AND R K. SHAH

310

TABLE XVI LOCATIONS FOR DIMENSIONLESS EXITTEMPERATURES FOR PARALLEL STREAM HEATEXCHANGERS Dimensionless outlet temperatures Stream

P1

P2

P3

P4

temperatures. The procedure is explicit, exact, and noniterative:

1. Determine the dimensionless groups: Cz2 [ j = 1,3; Eq. (3011, 03,in [Eq. (31); note that = 0 and 02,in = 1 in all cases according to the definition of the dimensionless temperatures, Eq. (2311, NTU, [Eq. (3211, and R* [Eq. (33)l. 2. Define the fluid flow indicators ij following the data in Table V in accordance with the specified flow arrangement and/or stream coupling. 3. Calculate the dimensionless outlet temperatures collocating Eq. (34) at either 6 = 0 or 6 = 1, depending on the flow arrangement and/or stream coupling as given in Table XVI. The explicit formulas are given in Table XII. Using the definition of the dimensionless temperatures, determine outlet temperatures. 4. Check for the existence of temperature crosses, using Eq. (35) and Tables X, XI, and XV,and decide on a plan of action if necessary. Temperature effectiveness charts were also utilized in the literature for the solution of the rating problem [ll].However, the limited set of existing effectiveness-NTU, diagrams available (for which an interpolation is usually needed) results in an unacceptable level of accuracy for an engineer and thus makes that approach outdated.' The programming of Eq. (34) is straightforward even though the formula is quite complicated due to the expressions of Tables X and XI. Step 4 should be adopted as a standard step in the design of a three-fluid heat exchanger. A designer should formulate a plan of action if the temperature cross does exist-and a temperature cross might be unavoidable, particularly for large NTU,. b. Sizing Problem The key step in the sizing problem of a three-fluid heat exchanger is to calculate the thermal size (NTU,), having known inlet


311

and outlet temperatures and an estimate for R*. Fortunately, an explicit relationship between NTU, and other parameters is available (see Table XII). The steps for the solution of a sizing problem are as follows: 1. Define the dimensionless governing parameters Czz [ j = 1,3; Eq. (3011 and terminal dimensionless temperatures Oj,in(out), using the definition of dimensionless temperature of Eq. (23) and the inlet data of Table XVI. 2. Assume an initial value or accept the given value of parameter R*. 3. Define the fluid flow indicators ij to be consistent with the fluid flow arrangement and/or stream coupling adopted (Table V), and determine from the given set of data all other dimensionless inlet and outlet temperatures. 4. Calculate the thermal size NTU,using the equation from Table XIII. 5 . Calculate heat transfer areas:

6. Check for the temperature cross using Eqs. (35) and Table XV and decide the plan of action if the temperature cross exists and needs to be avoided. 7. Repeat steps 2 to 6 with a new value of R* if necessary according to the plan of action adopted in step 6.

Note that the sizing problem procedure is not necessarily iterative. If the outlet temperatures are given as in the classical sizing problem (see the list of given data in Sec. VIII.B.l.b), and if overall heat conductances in a three-fluid heat exchanger are given beforehand or assumed (i.e., the parameter R* should be known),27 the calculation of both heat transfer surface areas is straightforward as outlined in steps 1 through 7. Whether 21

The overall heat conductance ratio R* is in general not known. The magnitude of this ratio depends on the heat transfer and fluid flow conditions related to both thermal communications. As pointed out at the beginning of this section, the complete step-by-step procedure should include both heat transfer and fluid flow considerations. We assume that the R* ratio is given and, if needed, it can be changed in a repeated and/or iterative procedure (step 2) to satisfy additional requirements (for example, the actual heat transfer surface area distribution imposed by the construction type of a heat exchanger). If this is not the case, the procedure is inevitably iterative.

312

D. P. SEKULIC: AND R. K. SHAH

or not the repetition of the procedure is necessary, if a temperature cross exists, will depend on the engineering judgment for the desired task.28 Finally, if the design problem is not a pure sizing or rating problem, the procedure might inevitably be iterative. For example, a typical design problem can be formulated in such a way that we can determine the size of a heat exchanger and the mass flow rate of the centraI fluid required to provide the specified outlet temperatures of both lateral fluids with given inlet temperatures of all fluids and mass flow rates of the two lateral fluids. This problem is equivalent to the problem of knowing both temperature effectivenesses but not the heat capacity rate ratios, the thermal size NTU,, and the heat conductance ratio R*. For every problem of this or similar type, an iterative procedure based on exact solutions that are discussed in relation to the rating and sizing problems can be utilized to build a combined approach in order to obtain the solution to a design problem. The precise sequence of steps for designing a heat exchanger in such situations varies and depends on the particular set of inlet data given. This general problem is beyond the scope of this article. c. A Comparison of an Approximate and an Exact Methodology for Parallel Stream Three-Fluid Heat Exchangers To demonstrate the use of both the approximate and exact methodologies and to assess the feasibility of using the approximate log-mean temperature as a true mean temperature in the LMDT method, let us consider the following design problem (adapted from Sorlie [ll]) We want to design a three-fluid parallel stream cocurrent heat exchanger with two thermal communications (stream coupling Pl). The heat exchanger has to operate under the conditions given in Table XVII with the heat conductance ratio R* assumed as 0.25. In the sizing problem, the heat conductance (UA) has to be determined for both heat transfer surfaces (i.e., between fluids 1 and 2, the first thermaI communication, and between fluids 3 and 2, the second thermal communication). Following the exact sizing procedure (Sec. VIII.B.2.b), the first step is to calculate dimensionless parameters CTz ( j = 1,3) and the dimensionless inlet and outlet temperatures needed for the calculation (see step 1, Sec. VIII.B.2.b). Using Eqs. (30) and (23), one can determine numerical **Forexample, if the purpose of a heat exchanger is to heat both lateral streams using the central stream, the large NTU, and the large dimensionless inlet temperature of the third fluid may lead to the presence of the temperature cross between the central and the third fluid for the adopted flow arrangement (for example, stream coupling P3). If any wastage of the heat transfer surface is not acceptable, the calculation may be repeated for another flow arrangement (for example, P2) and/or with an altered set of input data.

-


313

TABLE XVII INPUT DATAFOR A SIZINGPROBLEM' Fluid streams Cold fluids Fluid data mc,

(W/K)

Tin(K) To,, (K)

Hot fluids

Fluid 1

Fluid 3

Fluid 2

553.5 274.8 372.0

329.5 322.0 352.6

975.3 519.3 453.7

' From Sorlie [ll], the decimal places shown are truncations.

values of these parameters as follows: CT,, = 0.568, C;, = 0.338, and 03,in = 0.193, = 0.398, 02,0ut = 0.732, and = 0.318. (Note that = 0 and 02.in = 1 according to the definition.) In addition, R* = 0.25 is given. According to Table V, the fluid flow indicators are i, = + 1, and i, = + 1. Therefore, fluids 1, 2, and 3 have inlets at 6 = 0. Now, NTU, is calculated as 0.395 using the expressions from Tables XI11 and XIV. Finally, heat conductances are determined using Eqs. (32) and (33) as (UA),,, = 218.65 W/K and ( U A ) , , = 54.66 W/K. Exercising the approximate procedure (Sec. VIII.B.1 .b) using the same input data one = can determine LMTDs using Eqs. (88) and (89) as follows: A7&,,, 148.47 K and = 143.86 K. Heat transfer rates from Eqs. (96) and (97) are Q,,, = 53.8 kW and Q,,,= 10.1 kW,and finally from Eqs. (98) and (991, (UA),,, = 362.47 W/K and (UA),,, = 69.97 W/K. Table XVIII contains corresponding dimensionless parameters. In the same table, results of the calculations based on the approximate method (Sec.

TABLE XVIII COMPARISON OF APPROXIMATE AND EXACTSOI.UTIONS FOR THE SIZING PROBLEM

NTU, R*

Approximate

Effectiveness charts

Exact

0.655 0.193 (calculated)

0.750 (a first estimate) 0.25 (given)

0.395 0.25 (given)

0.899

0

0

0

- 0.228

314


VIII.B.1.b) and using the temperature effectiveness charts of Sorlie [ l l ] are also given. The later data, however, are only estimates, made by utilizing a multiple interpolation. It is clear that both the approximate method and chart utilization approach give either an underestimation or overestimation, respectively. Correspondingly, an exact approach has to be used, indeed, it is the only acceptable design methodology. A check of the existence of the temperature cross [Eq. (391 shows that only an indirect temperature cross between nonadjacent streams (fluid 1 and fluid 3) exists. It is important to note that a reliable prediction of the discrepancy between the approximate method based on the LMTD and exact calculation does not exist. In the case of the temperature cross present within the heat exchanger, the erroneous prediction by the approximate method may or may not be serious, depending on the set of inlet parameters. In addition, the change of fluid flow arrangement can cause both an increase or decrease in the discrepancy between the results obtained by exact and approximate procedures. To illustrate this, let us compare the results of a single sizing problem solution (for thermal size NTU, only) by both the approximate LMTD method and the exact approach for all four stream couplings (P1 through P4). Specific numerical data are taken from the examples discussed in the literature (adapted from Man'kovskij et al. [86] and BaEliC et al. [34, 521. Two cryogenic fluid streams (fluids 1 and 3) with heat capacity rates of 400 and 40 kW/K, respectively, are to be heated from 198 K by using the third fluid stream (fluid 2) with the heat capacity rate of 400 kW/K at the inlet temperature of 213 K. The heat conductance ratio is 0.8. The desired outlet temperatures for two cold streams are 201 and 209 K, and the outlet temperature of fluid 2 is 208 K. We want to determine the NTUs required for all four parallel stream couplings. The results of the calculations are presented in Table XIX along with the corresponding temperature distribution trends. In a cocurrent flow arrangement, the temperature cross between fluids 3 and 2 does indeed exist. Note that there is no temperature cross between fluids 1 and 2. The temperature cross at t*= 0 between fluid 1 and fluid 3 is a direct consequence of inlet data. A similar analysis of the countercurrent design shows that temperature crosses do not exist. Again, the cross at 5" = 0 is imposed by the set of inlet data. In both situations, the use of the approximate method results in a substantial error in the estimation of the NTU, values. In the countercurrent-cocurrent flow arrangement (stream coupling P3), a direct temperature cross (between fluids 3 and 2) and an indirect temperature cross (between fluids 1 and 3) exist. It is interesting to note that in this particular case, the simple approximate calculation gives a magnitude for the NTU, that is

-


315

TABLE XIX COMPARISON OF NUMBER OF TRANSFER UNITS, CALCULATED BY APPROXIMATE AND EXACTDESIGN PROCEDURES FOR FOURPARALLEL STREAM COUPLINGS P2

P1

P4

P3

Exact

Appr.

Exact

Appr.

Exact

Appr.

Exact

Appr.

0.391

0.322

0.181

0.308

0.315

0.308

0.312

0.322

-

17%

+ 70%

- 2%

+ 3%

very close to the exact value. Finally, for the P4 case, the direct temperature cross does not exist, and again the approximate and exact calculations give NTU, values close to each other. In conclusion, it is clear that the approximate approach could lead to design solutions that either overestimate or underestimate the exact heat exchanger size. It is important, though, to keep in mind an additional important hint. The exact approach, as defined in this section, assumes a set of idealizations that are frequently challenged in engineering practice (see the comment at the beginning of this section), such as the constant thermophysical properties of fluids. In such a situation, only a numerical analysis of the modified initial set of governing differential equations or the iterative (zonal) approach has to be utilized [lo, 17, 251.

3. Design Procedure for a Three-Fluid Cross-flow Heat Exchanger

The design problems of cross-flow three-fluid heat exchangers are much more complex than their counterparts involving parallel streams. In the open literature, only a few papers deal with explicit design methodologies. Nevertheless, the basic approach to both the rating and sizing problems is similar to the procedures elaborated on in previous sections.

316


The rating problem can be solved using numerical, seminumerical, or analytical procedures for determining the temperature distributions and/or mixed-mean outlet temperatures of all three fluid streams [32, 34, 35,58, 601. For example, the exact analytical solutions for the temperature distributions for a cross-cocurrent flow arrangement [C4, Fig. 7(a)], is given by the set of equations outlined in Sec. V1.C [Eqs. (44) through (6531. The first step in the rating problem then should be the determination of dimensionless groups, and subsequently the use of Eqs. (44) through (65) to find the temperature distributions, outlet temperatures, and heat transfer rates. For only a few other cross-flow arrangements, analytical and/or numerical solutions exist, but not in a closed form (arrangements C1 through C4 [35,581). A check of the existence of temperature crosses can be performed a posteriori. The solution of a sizing problem is iterative, because no explicit closedform formula for NTU, is available, even when the analytical solutions are available for temperature distributions such as for C4. A standardized methodology does not exist. BaEli6 et al. [52] made a specific comparison of the solution of a sizing problem for a cross-cocurrent heat exchanger (arrangement C4 [341) with solutions similar to those from Table XIX for heat exchangers with parallel stream arrangements Ge., stream couplings P1 through P4) for the same set of input data. These comparisons showed that the cross-cocurrent heat exchanger (C4) has a 25% smaller heat transfer area A,,2 compared to the heat exchanger with cocurrent parallel stream arrangement (Pl), but it has an approximately 40% larger heat transfer area A,,2 than a countercurrent heat exchanger (P2). However, no general assessment on the trend of surface area requirements can be made for crossflow versus parallel stream three-fluid heat exchangers. As an example of the approximate procedure, let us consider the methodology proposed by Nosach and Filipchuk [64] for sizing the threefluid heat exchanger with a cross-countercurrent flow arrangement (C3). The procedure is a modification of the well-known LMTD approach for two-fluid heat exchangers, and it uses correction factors for estimating the true mean temperature differences. The procedure is outlined as follows:

1. CaIculate the log-mean temperature differences:

-


317

where ~ T 1 , 2 ) i n= ( F 2 , i n

- Fl,in),

AT(1,2)0ut = (T2,m

AT(3, 2)in = (2' ,

in

-

( 104)

T,"m)?

(105)

in)

( 106)

- 3'.

9

AT3,2)our = ( c , o " t

-

(107)

7,3,0ut).

Note that the notation is not the same as that used in Eqs. (88) through (93) for the solution of a sizing problem for a parallel stream for j = 1, 2, three-fluid heat exchanger. Temperatures and and 3 are constant inlet or integral mean outlet temperatures for fluid streams. Fluid 2 is the central fluid, and fluids 1 and 3 are lateral fluids. Calculate the following parameters:

q,out

for j

for j

=

1 or 3,

=

1 or 3.

(110)

Determine the correction factors2':

The relation given by Eq. (111) was obtained numerically from the original set of governing partial differential equations and overall energy balances, and presented graphically. A set of results is given in Fig. 22 [64]. 29Thecalculation of the correction factors for this cross-flow arrangement (C4) is similar to the standard procedure for determining the correction factor as a ratio of the true mean temperature difference to the LMTD in a two-fluid heat exchanger. Consequently, the same methodology can be established for the cross-cocurrent flow arrangement (C4) using the exact solution given in Sec. V1.C.


318

a

.

__

0.6

Xl 0.8

0.4 3. 2.01.0

$I3 = 4.0

h

@1

U

Xl

0.8

0.4

C Xl

0.8

0.4

0

0.8

0.4

0

01

0.4

0.8 03

FIG. 22. Correction factors xj [arrangement C3, Fig. 7(a)l for j = 1 or 3: (a) 5 = 3; (b) 5=2; (c) 9= 1.5, from Nosach and Filipchuk [a].

4. Determine the heat transfer rates: Qj

=

( I j t ~ ~ ) ~ -( ?T,in) ; l , ~ ~ for ~j

=

1, or 3.

(112)

5. Calculate the heat transfer surface areas:

Note that the xi correction factors are presented graphically for only limited values of pertinent parameters [631. Either detailed tabular values

-


319

or closed-form equations are desirable for design purposes. This is the major drawback of the proposed procedure. In addition, an assessment regarding the accuracy of the methodology does not exist in the literature.

IX. Conclusions The present review was done because the use of three-fluid heat exchangers in a number of important applications is not accompanied by a sound design theory and because the thermal design theory of three-fluid and multifluid heat exchangers has not yet been adequately standardized. The major goal of the present effort has been threefold: (a) to present existing comprehensive information on the thermal design theory of three-fluid heat exchangers in a uniform format, (b) to fill in some gaps in that body of knowledge in order to construct a unified approach to the thermal design of a three-fluid heat exchanger with two thermal communications among the streams, and (c) to present step-by-step design procedures for simplified heat exchanger rating and sizing problems. When one fluid stream transfers heat to the other two fluid streams, it is referred to as two thermal communications in the exchanger; in contrast, when all three fluids transfer heat among each other, it is referred to as three thermal communications in the three-fluid heat exchanger. The two most common construction types of three-fluid heat exchangers in practice are the tubular and extended surface plate-fin types. Both construction types may have parallel, cross-flow, or combined fluid flow arrangements. Two or three thermal communications among the three fluid streams are possible. The set of parallel stream arrangements includes in general three distinct situations: cocurrent, countercurrent, and countercurrent-cocurrent or cocurrent-countercurrent parallel flow with three independent arrangements (Pl, P2, and P3) and four possible stream couplings P1 through P4 (see Fig. 6). There are four basic single-pass cross-flow arrangements as shown in Fig. 7(a), all of which will have many specific solutions depending on whether one or more fluids are mixed or unmixed in respective flow passages. Two arrangements have been analyzed: cross-countercurrent flow and cross-cocurrent flow [C3 and C4 in Fig. 7(a)], both with unmixed fluid streams at every cross section. An extensive review of the literature clearly demonstrates the lack of a unified approach for thermal design even for a single class of three-fluid heat exchangers. The existing analytical solutions are most frequently valid only for a particular design and/or flow arrangement and not suited for use in general-purpose computer codes. There is no unified design

320


methodology and no assessment is made of the existing approximate design procedures. In this article, we have presented a unified theory for all flow arrangements of parallel stream three-fluid heat exchangers with two thermal communications. A compact form of temperature distributions for all three fluid streams for each flow arrangement (see Fig. 6) is given in an explicit form in Eq. (34) and NTU, is given in Table XITI. It has been demonstrated that an explicit relation [Eq. (391 can be used to predict the occurrence of the temperature cross among fluid temperature distributions within a heat exchanger regardless of the flow arrangement. The temperature cross phenomena can cause substantial deterioration in heat exchanger performance. A closed-form solution of the temperature distributions and exit temperatures of all three fluids is available for crosscocurrent flow (arrangement C4, BaEliE et al. [34]) as given by Eqs. (44) through (65). The seminumerical and numerical solutions for single-pass cross-flow arrangements are also available in the literature (C1 through C4, [581; C4, [40, 601; C3, [351). We have not made an assessment of the idealizations incorporated in heat exchanger basic design theory outlined in Sec. V.B. If some of the idealizations incorporated in the mathematical analysis have to be relaxed, explicit analytical solutions cannot be obtained. A numerical procedure though can usually overcome these difficulties. However, a number of idealizations (in particular, constant and uniform heat transfer coefficients, uniform distribution of heat transfer surface area, negligible longitudinal conduction in fluids and walls, and no flow maldistribution in the heat exchanger) are valid for an actual heat exchanger problem for a moderate temperature range of operation and well-designed inlets and outlets. For cryogenic and high-temperature applications, the only practical and accurate tool is numerical discretization of the corresponding physical and/or mathematical models. The concept of two-fluid heat exchanger effectiveness can be extended to three-fluid heat exchangers regardless of the flow arrangement and operating conditions, but the pragmatic use of the three-fluid heat exchanger effectiveness is of limited value. It is restricted to an assessment of overall performance. The concept of temperature effectiveness as discussed here is more useful practically. Finally, step-by-step procedures have been presented for rating and sizing problem solutions for three-fluid heat exchangers. It has been also demonstrated that the solution to the sizing problem is in some cases noniterative for parallel stream three-fluid heat exchangers since NTU, can be explicitly determined. The feasibility of the use of the approximate LMTD method in three-fluid heat exchanger design is not validated. It has

-


321

been shown that the use of a direct analogy in defining overall heat exchanger effectiveness as well as the use of the LMTD approach in the thermal design of three-fluid heat exchangers can produce significant errors; therefore, the temperature effectiveness approach is recommended in favor of the conventional overall exchanger effectiveness approach. Note that this exposition of the theory and design of one class of three-fluid heat exchangers is far from complete. Further studies and theory unification are needed, in particular with respect to the relaxation of standard idealizations. In addition, further studies of the influence of temperature cross phenomena on overall heat exchanger performance should be conducted, as well as reconsideration of the overall three-fluid heat exchanger effectiveness definition. A similar theoretical/analytical approach should be extended to other classes of three-fluid heat exchangers, for example, three-fluid heat exchangers with all three fluids in thermal contact or cross-flow three-fluid heat exchangers. We hope the readers will be exposed to the many existing gaps in this specialized heat exchanger design problem and will be encouraged to fill these gaps to arrive at improved designs for three-fluid heat exchangers.

Nomenclature A

An

a

c

CP

heat transfer surface area on one fluid side of a threefluid heat exchanger, m2 cross-flow heat exchanger core size area defined as Xd',,, m2 parameter defined in Table XIII, dimensionless coefficient in Eq. (2), dimensionless parameter defined in Table XIII, dimensionless fluid heat capacity rate, mc, for j = 1,2,3, W/K heat capacity rate ratio C;f2 = C J / C , for j = 1.3, dimensionless parameter defined in Table XIII, dimensionless specific heat of fluid at constant pressure, J/kgK

.([)

parameter defined in Table XV, Eq. (35), dimensionless functions defined in Table X, dimensionless function defined by Eq. (49), dimensionless temperature effectiveness defined by A d d s and Barron 1121, Table 11, dimensionless heat transfer surface area, Table IV only, m2 heat capacity rate ratio defined by Eq. (108). d = C , / C , , dimensionless parameters defined by Eqs. ( 5 5 ) and (591, dimensionless functions defined by Eqs. (47) and (481, for i = 2,3, dimensionless

D.P. SEKULIC AND R. K. SHAH heat transfer coefficient, W/m*K modified Bessel function of n'th order, dimensionless fluid flow indicator as defined in Table V for parallel stream exchanger, dimensionless heat capacity rate ratio deTable fined by Willis [a], If, dimensionless thermal conductivity, W/mK overall heat transfer coefficient in Table IV only, W/m2K heat exchanger length, m parameter defined in Table XV, Eq. (351, dimensionless mass flow rate, kg/sec number of heat transfer units defined by BaEli6 er al. [34], Table 11, dimensionless parameter defined in Table XIII, dimensionless number of heat transfer units NTU, = ( U A ) l , 2 / ( h c p ) ldimen, sionless number of heat transfer units defined by Sorlie 1111, Table 11, dimensionless number of heat transfer units defined by Aulds and Barron 1121, Table 11, dimensionless number of fluid streams, dimensionless parameters defined in Table XIV, dimensionless wetted perimeter of heat exchanger passages for any given fluid, m number of fluids not in direct thermal contact in a multifluid heat exchanger, dimensionless

dimensionless temperature as defined by Eq. (109), j = 1 or 3, dimensionless heat transfer rate, W conductance ratio defined in Table II; heat capacity rate ratio, Table IV, dimensionless conductance ratio R* = (UA),,2/(UA),,2 , dimensionless number of fluids at constant temperature in a multifluid heat exchanger, dimensionless dimensionless temperature defined by Eq. (110), j = 1 or 3 parameters defined in Table X, dimensionless temperature, K temperature difference defined in Eqs. (88) through (93) and Eqs. (104) through (107), K inlet temperature difference ratio, A t ; = 1 - @3*i,,, dimensionless inlet temperature difference ratio defined by Willis [40], Table 11, dimensionless overall heat transfer coefficient [36], W/m'K parameter defined by Rabinovich [46], Table 11, dimensionless special function, defined by Eq. (SO), dimensionless heat capacity rate, Table IV only, W/K inlet temperature difference ratio defined by Adds and Barron [12], Table 11, dimensionless cross-flow heat exchanger length in x direction, m

-

THREEFLUIDHEAT EXCHANGER THERMAL DESIGN X

Yn Y

Z

Cartesian axial coordinate, m cross-flow heat exchanger length in y direction, m Cartesian coordinate, m parameter defined by Eq. (571, dimensionless

K

A

GREEKLEITERS (Y

Y E

5

parameter defined in Table X, dimensionless heat transfer coefficient, Table IV, W/m2K parameter defined in Table X, dimensionless conductance ratio defined by Rabinovich [46], Table 11, dimensionless parameter defined in-Table X, dimensionless three-fluid heat exchanger effectiveness, Table I, dimensionless dimensionless coordinate

d,

XI

0 , w*

(= Z/Z")

11.17'

Ilu

@j

B3.i"

dimensionless coordinates defined by Eq. (291, dimensionless extended surface effectiveness [ = 1 - A,(1 ? / ) / A , , where Af and A , are total fin area on one side and surface area on one side on which overall heat transfer coefficient is based, respectively, and 9, is fin temperature effectiveness [36]), dimensionless dimensionless temperature defined by Eq. (231, j = I, 2,3, dimensionless dimensionless inlet tempera= ture of fluid 3, 03.,n ( T ~ i n- T~in)/(T~,inTl.in)

323

overall heat transfer coefficients ratio defined by Barlif el al. [34], Table 11, dimensionless longitudinal conduction parameter defined by Barron and Yeh [49], dimensionless dimensionless coordinates defined by Eqs. (23) and (291, dimensionless location of the temperature cross defined by Eq. (39, dimensionless function defined in Table X; (Dj(l) = aj(S)evaluated at 8 = 1, for j = 1,2,3, dimensionless correction factor defined by Eq. (111) and Fig. 22, j = 1 or 3, dimensionless function defined in Table X, Yj(l) = Y,(.$) evaluated at $. = 1 for j = 1,2,3, dimensionless heat capacity rate ratios defined by BaElif er af. [341, Table 11, dimensionless

SUBSCRIFTS

actual c

exact

f h i

in

i i,k

actual value cold fluid (stream) exact value fin hot fluid (stream) fluid (stream) with the intermediate temperature inlet fluid stream, j = 1,2,3 thermal communication between fluids (streams) j and k ( ( j , k ) = {1,2), (3,2), and {1,3)h j = 1 and 3 corresponds to lateral streams, j = 2 corresponds to the central stream

324


j , in

j, out

j,5

=

0

( j , 2Nm

j, 2in

j . 2out

max out

at the fluid (stream) j inlet, j = 1,2,3 at the fluid (stream) j outlet, j = 1,2,3 fluid (stream) j at 5 = 0, j = 1,3 log-mean temperature between fluids (streams) j = 1 or 3, and fluid 2 corresponds to the heat exchanger section where fluid (stream) 2 has the inlet, j = 1 or 3 corresponds to the heat exchanger section where fluid (stream) 2 has the outlet, j = 1 or 3 maximum value outlet

U W

x=o x=L x = X" y=o Y = yo 2FHE 3FHE 1,2,3

refers to overall heat transfer coefficient at the wall atx=O atx=L at x = X, aty=O at y = Yo two-fluid heat exchanger three-fluid heat exchanger fluid stream 1, 2, or 3

SUPERSCRIPTS 00

-

at NTU = 00 integral-mean value

References 1. Taborek, J. (1983). Shell-and-tube heat exchangers: Single-phase flow. In Heat Exchanger Design Handbook (E. U. Schliinder, ed.), Vol. 3, Sec. 3.3. Hemisphere, Washington, DC. 2. Shah, R. K., and Wanniarachchi, A. S. (1991). Plate heat exchanger design theory. In Zndustrial Heat Exchangers (J. M. Buchlin, ed.), Lect. Ser. No. 1991-04. Von Kirmin Institute for Fluid Dynamics, Belgium. 3. Shah, R. K. (1988). 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.), pp. 255-266. Hemisphere, Washington, DC. 4. Shah, R. K. (1991). Multidisciplinary approach to heat exchanger design. In Industrial Heat Exchangers (J. M. Buchlin, ed.), Lect. Ser. No. 1991-04. Von KBrmin Institute for Fluid Dynamics, Belgium. 5. Nesselman, K. (1927). Der Einfluss der Warmeverluste auf Doppelrohrwarmeaustauscher (The influence of the heat loss on a double-pipe heat exchanger) (in German). Veroeff. Siemens-Konzern 6 , 174-183. 6. Krishnamurty, V. V. G., Ramanjam, T. K., Sagar, D. V. and Rao, C. V. (1964). Liquid-liquid heat transfer: Development of calculation techniques. Indian J . Technol. 2, 244-246. 7. Barron, R. F. (1983). Effect of heat transfer from ambient on cryogenic heat exchanger performance. Adv. Cryog. Eng. 29, 265-212. 8. Chowdhury, K., and Sarangi, S. (1983). Performance of cryogenic heat exchangers with heat leak from the surroundings. Adu. Cryog. Ens. 29, 273-280. 9. Morley, T. B. (1933). Exchange of heat between three fluids. Engineer 155, 134. 10. Hausen, H. (1950). Warmeiibertragung im Gegenstrom, Gleichsirom und Kreutzstrom. Springer-Verlag, Berlin.

-


325

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SUBJECT INDEX

A

Activation energy asymptotics, 61 AD1 technique, 22, 23, 27 Aerospace industry, heat exchangers, 220-221 Air jet cooling, 205 Alternating direction implicit technique, see AD1 technique Average heat transfer free-surface jet, 161-162, 170 submerged jet, 133-135 hisymmetric jet, 107 array, 171 flow structure, 136-144 free-surface liquid jet, 109-119 heat transfer, 124-126, 131-135, 137-162, 189-193, 204 hydraulic jump, 114, 201-203 jet inclination, 182-186 motion of impingement surface, 203-204 stagnation region, Falkner-Skan flow, 11 1, 120, 122 submerged jet, 123-135 surface liquid jet, 107-108

B Benzene, compound droplet, vaporization, 78 Binary droplet, vaporization, 78-79 Boundary layer, droplet vaporization, 66-71 Bubble, in immiscible liquid, 47-50

C Cocurrent-countercurrent parallel flow heat exchanger, 228, 229 Cocurrent cross-flow heat exchanger, 230-231 Cocurrent parallel flow heat exchanger, 228-229, 245 temperature effectiveness, 300-302 COLSYS, 304-305 Combustion spray system, 54, 59 slurry fuel droplet, 81-82

Compound drop, transfer processes, 47-52 Condensation gas bubble in immiscible liquid, 47-48 moving liquid droplet, 32-43 compound drop, 47-48 droplet shape deformation, 44 electric field, 17, 83-84 gaseous environment, 6, 31-32 spherically symmetric, 31-32 spray of drops, 44-47 surfactants, 43-44 Condensation parameter, 32 Condensation velocity, 32 Conjugate problem, moving liquid droplet, 22-26,30-31 Convective droplet, vaporization, 58-59, 70-71 Cooling air jet, 205 liquid jets, 105, 170, 205 Countercurrent-cocurrent parallel flow heat exchanger, 228, 229 Countercurrent cross-flow heat exchanger, 230 Countercurrent parallel flow heat exchanger, 228, 229, 242, 245 temperature effectiveness, 300-302 Creeping flow solution, moving liquid droplets, 4 Cross-cocurrent flow heat exchanger, 230-231,282, 284, 286 temperature effectiveness, 302-304 Cross-countercurrent flow heat exchanger, 230, 231, 285-286 temperature effectiveness, 302-304 Cross-flow heat exchanger, 229-231 boundary conditions, 264 design, 315-319 energy equations, 261-263, 266-267 exit temperature, 290 literature review, 247-252 mathematical model, solution, 281-290 temperature effectiveness, 302-304 Cryogenics, heat exchangers, 220, 222, 252, 305

329

330

SUBJECT INDEX D

Damkohler number, 60-61 n-Decane, vaporization, 67, 78-79 Dimensionless groups, three-fluid heat exchanger, 234-237 Dimensionless inlet temperature, three-fluid heat exchanger, 268 Direct-contact transfer fluid mechanics, 3-19 drop deformation, 12-13, 18-19 electric field effects, 13-19 inertialess translation, 4 Reynolds number translations, 5-1 1, 16-18 surface-viscous effects, 13-16 surfactants, effect on drop motion, 11-12 weakly inertial translation, 5 heat and mass transfer, 19-89 Disruptive burning, 81 Drag force compound drop, vaporization, 50 moving liquid droplet electric field, 17 gaseous environment, 7-9 inertialess system, 4 intermediate Reynolds number, 9, 10 low Reynolds number, 34 multicomponent drop, 79 Drop bulk temperature, 39, 40 Drop deformation n-heptane, 80 moving liquid droplet, 12-13 condensation, 44 electric field, 18-19 evaporation, 79-81 low Reynolds number translation, 6 weakly inertial translation, 5 Drop geometry, 50 Droplet, see also Moving liquid droplet binary, vaporization, 78-79 compound, 47-52 convective, vaporization, 58-59, 70-71 direct-contact transfer, 1-3 fluid mechanics, 3-19 heat and mass transport, 19-89 electrohydrodynamics, 14- 19 interactions, 84-89 lifetime, 56-57

molten metal, evaporation, 83 motion, 11-19, 43-44 multicomponent, vaporization, 66, 77-79 oscillation, 44-49 slurry fuel, vaporization, 81-82 spherical, inertialess terminal velocity, 4 spray, 44-47, 53-57, 81-82 deformation, 79-80 interaction, 84 transport, 87-89 transport, 43-44 Droplet array, moving liquid droplet, 84 Droplet group, moving liquid droplet, 84 Droplet lifetime, 56-57 Droplet oscillation, 44-49 Drop motion electric field effect, 13-19 surface-viscous effect, 13 surfactants, 11-12, 43-44 Drop transport, surfactants, 43-44

E

Effectiveness-NTU approach, 222, 256, 267 Electric field effect, moving liquid droplet, 13-19, 26-31, 83-84 Electrohydrodynamics, droplet, 14-19 Empirical convective factor, 45 Energy equations, three-fluid heat exchanger, 258-264 Energy flux continuity equation, 34 Eotvos number, 36 Euler-Mascheroni constant, 21 Evaporation compound drop, 48-52 convective droplet, 58-59, 70-71 molten metal drop, 83 moving liquid droplet electric field, 83-84 gaseous environment, 6, 31-32, 52-83 Reynolds number, 59-77 shape deformation, 79-81 multicornponent droplet, 66, 77-79 slurry fuel droplet, 81-82 stagnant medium, 54-57 Exit temperature cross-flow heat exchanger, 290 parallel-stream heat exchanger, 273-274

SUBJECT INDEX Extended film model theory, 69 Extended surface plate fin heat exchanger, 227, 228

33 1

Gas bubble, condensation in immiscible liquid, 47-48 Gas jet, 106, 205

F Falkner-Skan flow, axisymmetric jet stagnation region, 111, 120, 122 Flow structure, liquid jet impingement, 136-147 Fluid mechanics direct-contact transfer studies drop deformation, 12-13, 18-19 electric field effect, 13-19 inertialess translation, 4 Reynolds number translation, 5-1 1, 16-18 surface-viscous effect, 13-16 surfactants, effect on drop motion, 11-12 weakly inertial translation, 5 Four-stream heat exchanger, 226 Free-surface liquid jet, 106-107, 205 axisymmetric jet, 109-119 stagnation zone, 137-141 turbulent flow, 153-154, 157 experimental studies, 135-170 heat /mass transfer, 147-162 hydraulic jump, 114, 201-203, 205 local heat transfer, 159-160, 167-169 planar jet, 107-108, 119-123 flow regions, 146-147, 167-170 stagnation zone, 147, 162-167 radial flow region, 141-144, 156-160 splattering, 195-198 stagnation zone, 137-141, 147, 162-167 theory, 107-123 Froessling number, 152, 206 Froude number, 152, 201-202, 206 Fuel spray slurry fuel droplet, vaporization, 81-82 stagnant medium. evaporation, 54-57 vaporization, 53-54, 83-84

G Galerkin coefficients, 18 Galerkin method, 17

H Harnpson heat exchanger, 226, 227, 252, 254 Heat capacity rate ratio, three-fluid heat exchanger, 268 Heat exchangers, see also specific hear exchangers cryogenics, 220, 222, 252, 305 effectiveness, 290-304 heat transfer, 219 Heat flux local heat flux, 34 wall jet zone, 114-118 Heating parameter, 82 Heat transfer heat exchanger, 219 liquid jet, 189-193 array, 171-179, 175-177 free-surface jet, 122, 147-162, 162-170 jet inclination, 182-189 jet splattering, 196-198 planar jet, 122, 162-170, 192-195 stagnation zone, 110-113, 128, 129 submerged jet, 124-126, 131-135 wall roughness, 193-195 moving liquid droplet condensation, 32-47, 83-84 droplet interaction, 84-89 droplet oscillation, 44 phase change at drop surface, 19-32 vaporization, 48-84 thermal communication, 219 turbulent flow, 162-170 n-Heptane droplet deformation, 80 droplet vaporization, 72-73 n-Hexadecane, vaporization, 67, 70-71, 78-79 n-Hexane, vaporization, 56-57, 59, 67, 70 Hill’s spherical vortex, moving liquid droplet, 4, 7, 9, 10 Hydraulic jump, free-surface liquid jet, 114, 201-203, 205

SUBJECT INDEX

332

Hydrocarbons moving liquid droplet drop deformation, 79-80 vaporization, 56-57, 59, 67, 70, 72, 78-79 Hydrodynamics, liquid jet impingement, 136-147. 157

I Ignition, droplet, 60-61 Inlet temperature, three-fluid heat exchanger, 267, 268 Inviscid theory strength, 10

J Jet impingement, see Liquid jet impingement Jet inclination, liquid jet impingement, 182-189 Jet pulsation, liquid jet impingement, 198-201 Jet splattering, liquid jet impingement, 195- 198

K Kbrmin-Polhausen integral technique, 114, 118 Kinetic theory, moving liquid droplet, 31-32

L Laminar jet, 152-153 axisymmetric jet heat flow, 156-157 radical flow region, 114-119 stagnation zone, 109-113, 123, 127-130 planar jet parallel flow region, 122-123 stagnation zone, 119-122 Laser-Doppler velocimetry (LDV), liquid jet impingement, 137, 142, 158 Liquid jet, 106-107; see also specijic liquid jets array, 170-182, 205

cooling, 105, 170, 205 heat transfer, 189-193 hydraulic jump, 114, 201-203, 205 jet inclination, 182-189 jet pulsation, 198-201 mass transfer, 124-126, 147-162, 171-174 modified impingement surface, 189-193 motion of impingement surface, 203-204 stagnation zone, velocity gradient, 109-110, 120 wall roughness, 193-195 Liquid jet impingement arrays, 170-182 flow structure, 136-147 free-surface jet, 109-123, 135-170 heat transfer arrays, 171-179 jet inclination, 182-189 jet pulsation, 198-201 liquid jets, 124-127, 131-135, 147-162 modified impingement surface, 189-193 motion of impingement surface, 203-204 wall roughness, 193-195 hydraulic jump, 114, 201-203, 205 hydrodynamics, 136-147, 157 jet pulsation, 198-201 jet splattering, 195-198 laser-Doppler velocimetry (LDV), 137, 142, 156 metals, 204 Nusselt number, 206 submerged jet, 123-135 theory, 107-123 transport, factors affecting, 182-204 LMTD method, see Log-mean temperature difference method Local heat f l u , 34 Local heat transfer array, 179 jet inclination, 184-186 jet pulsation, 198-201 liquid jets axisymmetric jet, 182 free-surface jet, 159-160, 167-169 submerged jet, 130-133 Locally homogeneous flow (LHF) model, 54 Log-mean temperature difference method, three-fluid heat exchanger design, 306-309, 320-321

SUBJECT INDEX M

333

heat and mass transport, 19-89 condensation, 44 Macrobalance equations, three-fluid heat exevaporation, 79-81 changer, 263 drag force Mass burning rate, droplet, 64 electric field, 17 Mass transfer gaseous environment, 7-9 liquid jet inertialess system, 4 array, 171-174 Reynolds number, 9, 10, 34 free-surface jet, 147-162 droplet array, 84 submerged jet, 124-126 droplet group, 84 moving liquid droplet droplet oscillation, 44-49 condensation, 32-48, 83-84 drop motion, 11-19, 43-44 droplet interaction, 84-89 electric field effect, 13-19, 26-31, 83-84 no phase change at drop surface, 19-31 evaporation phase change at drop surface, 31-32 electric field, 83-84 vaporization, 48-84 gaseous environment, 6, 31-32, 52-83 Mathematical model Reynolds number, 59-77 multifluid heat exchanger, 234 shape deformation, 79-81 three-fluid heat exchanger, 231-237, heat /mass transfer, 19-89 256-269 Hill’s spherical vortex, 4,7, 9, 10 cross-flow solution, 281-290 hydrocarbons parallel-stream solution, 271-281 drop deformation, 79-80 Metals vaporization, 56-57, 59, 67, 70, 72, liquid jet impingement, 204 78-79 moving liquid droplet, evaporation, 83 kinetic theory, 31-32 Microbalance equations, three-fluid heat exmetals, evaporation, 83 changer, 258-263 Nusselt number, 21-23 Module-averaged heat transfer, 175-177 surfactants, 11-12, 43-44 Molten metal droplet, evaporation, 83 vortex, 10, 11 Momentum equation, moving liquid droplet, Multicomponent droplet, vaporization, 66, inertialess system, 4 77-79 Moving liquid droplet Multifluid heat exchanger, 220-222,225-227 condensation, 32-43 design, 304 compound drop, 47-48 dimensionless groups, 237 droplet shape deformation, 44 literature review, 252, 254-255 electric field, 17, 83-84 mathematical model, 234 gaseous environment, 6, 31-32 spherically symmetric, 31-32 spray of drops, 44-47 surfactants, 43-44 N direct-contact transfer, 1-3 fluid mechanics, 3-19 NTU, 222, 256, 267, 274 drop deformation, 12-13, 18-19 Nusselt number electric field effects, 13-19 liquid jet impingement, 206 inertialess translation, 4 arrays, 174-175, 177, 179-181 Reynolds number translation, 5-11, free-surface jet, 115-1 17 16-18 jet configuration, 183-187 surface-viscous effects, 13-16 stagnation zone, 112-113, 121, 151-152, surfactants, effect on drop motion, 154-156, 162, 166-167, 194-195 11-12 submerged jet, 123, 127-129, 131-133 weakly inertial translation, 5 module-average, 174, 175, 177

SUBJECT INDEX

334 moving liquid droplet, 21-23 compound drop, 49-52 deformed droplets, 80 droplet interactions, 84, 86 multicomponent drop, 79 transient, 25-26

0

Octane, compound droplet, vaporization, 78

P Parallel flow region, 105 free-surface planar jet, 147, 167-170 laminar planar jet, 122-123 Parallel-stream heat exchanger, 227-229 boundary conditions, 264 design, 309-315 energy equations, 258-260, 265-266 exit temperature, 273-274 literature review, 237-247 mathematical model, solution, 271-281 multifluid exchanger, 254 temperature effectiveness, 300-302 Planar jet, 107 array, 171, 182 flow structure, 144-147 free-surface liquid jet, 107-108, 119-123 heat transfer, 122, 162-170, 192-195 jet inclination, 186-189 submerged jet, 135 Plate-fin heat exchanger, 227, 228 Potential core, 106 Prandtl number, submerged jet, 129-130 Preimpingement jet, 136-137, 144-147, 165

R Radial flow region free-surface jet, 141-144, 156-160 submerged jet, 129-134 Radical flow region, laminar axisymmetric jet, 114-119

Ranz-Marshall correlation, 53, 70 Reynolds number moving-liquid droplet condensation processes, 32 direct-contact transfer fluid dynamic studies, 5-11. 16-18 evaporation, 59-77 Rybczynski-Hadamard solution, 4, 11, 27 S

Shell-and-tube multifluid heat exchanger, 225, 227 Sherwood number, 22,79,80,84 Slurry fuel droplet, vaporization, 81-82 Spalding’s correlation, 53 Spherical droplet, inertialess terminal velocity, 4 Spray condensation, 44-47 droplet deformation, 79-80 droplet interaction, 84 droplet transport, 87-89 evaporation, 53-54 Square array, 171, 181 heat transfer, 174, 175 Stagnant-cap model, 12 Stagnation zone, 105, 108, 205 liquid jets free-surface jet, 137-141, 147, 162-167 laminar jet, 109-113,119-122.123, 127-130 submerged axisymmetric jet, 123, 127- 130 velocity gradient, 109-110, 120 transport, 108 Strouhal number, jet pulsation, 199 Submerged liquid jet, 106, 205 axisymmetric jet, 123-135 turbulent flow, 132-133 cooling, 205 experimental studies, 123-135 heat transfer, 124-126, 131-135 mass transfer, 124-126 planar jet, 135 Prandtl number, 129-130 radial flow region, 129-134 Surface-viscous effect, drop motion, 13 Surfactants, moving liquid droplet, 11-12, 43-44

SUBJECT INDEX T Temperature cross, parallel-stream exchanger, 274, 276-278 Temperature distribution cross-flow heat exchanger, 281-286 parallel-stream heat exchanger, 271-273 Temperature effectiveness, three-fluid heat exchanger, 297-304 Terminal velocity, intertialess, spherical drop, 4 Thermal communciation, 219 Three-fluid heat exchanger, 219-223, 319-321 design, 304-321 log-mean temperature difference method, 306--309,320-321 dimensionless groups, 234-237 dimensionless inlet temperature, 268 effectiveness, 290-304 energy equations, 258-264 extended surface plate fin exchanger, 227, 228 heat capacity rate ratio, 268 inlet temperature, 267,268 literature review, 237-253. 255-256 macrobalance equations, 263 mathematical model, 231-237, 256-269 solution, 269-290 microbalance equations, 258-263 temperature cross, 274, 276-278 Transport equations, conjugate problem, 22 Triple-pipe three-fluid heat exchanger, 227, 229 Tubular three-fluid heat exchanger, 224-227 Turbulent flow free-surface jet, 204-205 axisymmetric, 153-154, 157 planar, 146, 162-170 heat transfer, 162-170 submerged axisymmetric jet, 132-133 Two-fluid heat exchanger, 219-220, 231 effectiveness, 290-291

335 U

Uniform retardation model, 12

V Vaporization benzene, 78 compound drop, 48-52 convective droplet, 58-59, 70-71 n-decane, 67, 78-79 drag force, 50 droplet spray, 87-89 fuel spray, 53-54, 83-84 n-heptane droplet, 72-73 n-hexadecane, 67, 70-71, 78-79 n-hexane, 56-57, 59,67, 70 moving liquid droplet electric field, 83-84 gaseous environment, 52-54 Reynolds number, 59-77 shape deformation, 79-81 multicomponent droplet, 66, 77-79 octane, 78 slurry fuel droplet, 81-82 stagnant medium, 54-57 Velocity gradient, liquid jet stagnation zone, 109-110, 120 Vortex, liquid droplet, 10, 11

W Wall jet zone, 105, 107 heat flux, 114-118 temperature, 118 transport, 108-109 Weber number, 36,202


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