Advances in Heat Transfer, Volume 23

ADVANCES IN HEAT TRANSFER Volume 23

Contributors to this Volume R. P. Chhabra Thomas E. Diller Frank P. Incropera Massoud Kaviany Michel Quintard B. P. Singh Raymond Viskanta Stephen Whitaker D. H. Wolf

Advances in

HEAT TRANSFER Serial Editors James P. Hartnett

Thomas F. Irvine, Jr.

Energy Resources Center University of Illinois Chicago, Illinois

Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, New York

Serial Associate Editor Young I. Cho Department of Mechanical Engineering Drexel University Philadelphia, Pennsylvania

Volume 23

ACADEMIC PRESS, INC. Harcourt Brace & Company, Publishers Boston San Diego New York London Sydney Tokyo Toronto

This book is printed on acid-free paper.

@

COPYRIGHT 0 1993 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.

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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 63-22329

ISBN 0-12-020023-6

Printed in the United States of America 93 94 95 96 BB 9 8 I 6 5 4 3 2

1

CONTENTS

Preface .

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vii

Jet Impingement Boiling

D. H . WOLF. FRANK P. INCROPERA. AND RAYMONDVISKANTA I . Introduction . . I1. Background . . I11. Nucleate Boiling . IV . Critical Heat Flux V . Transition Boiling VI . Film Boiling . . VII. Research Needs . Acknowledgments Nomenclature . . References . . .

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1 2 10 53 108 117 120 123 124 126

I . Introduction . . . . . . . . . . . . . . I1. Continuum Treatment . . . . . . . . . . . . . I11. Solution Methods for Equation of Radiative Transfer . . IV. Properties of a Single Particle . . . . . . . . . . V . Radiative Properties: Dependent and Independent . . . VI . Noncontinuum Treatment: Monte Carlo Simulation . . VII . Radiant Conductivity . . . . . . . . . . . . . VIII. Modeling Dependent Scattering . . . . . . . . . IX. Effect of Solid Conductivity . . . . . . . . . X . Conclusions . . . . . . . . . . . . . . . . X . Acknowledgments . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

133 134 136 141 152 161 166 168 179 182 183 183 184

Radiative Heat Transfer in Porous Media

MASSOUDKAV~ANY AND B . P. SlNGH

V

CONTENTS

vi

Fluid Flow, Heat, and Mass Transfer in Non-Newtonian Fluids: Multiphase Systems

R. P. CHHABRA I. Introduction . . . . . . . . . . . . . . . . IT. Rheological Considerations . . . . . . . . . . . 111. Non-Newtonian Effects in Packed Beds . . . . . . . IV. Non-Newtonian Effects in Fluidised Beds . . . . . . V. Sedimentation of Concentrated Suspensions . . . . . VI. Concluding Summary . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

187 189 192 232 26 1 263 265 267

Advances in Heat Flux Measurements

THOMAS E. DILLER I. Introduction . . . 11. Measurement Methods 111. Calibration . . . . IV. Applications . . . V. Conclusions . . . Acknowledgments . Nomenclature . . . References . . . .

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279 28 7 342 343 352 353 353 354

One- and Two-Equation Models for Transient Diffusion Processes in Two-Phase Systems

MICHELQUINTARD AND STEPHEN WHITAKER I. Introduction . . . . . . . . . . . . . . . . 11. Volume Averaging . . . . . . . . . . . . . . 111. Closure . . . . . . . . . . . . . . . . . . IV. Prediction of the Effective Transport Coefficients . . . V. Comparison of One- and Two-Equation Models . . . . VI. Conclusions . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . Index

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369 378 386 407 425 458 459 459 460 465

PREFACE

The serial publication Advances in Heat Transfer is designed to fill the information gap between the regularly scheduled journals and universitylevel textbooks. The general purpose of this publication is to present review articles or monographs on special topics of current interest. Each chapter starts from widely understood principles and brings the reader up to the forefront of the topic in a logical fashion. The favorable response by the international scientific and engineering community to the volumes published to date is an indication of how successful our authors have been in fulfilling this purpose. The Editors are pleased to announce the publication of Volume 23 and wish to express their appreciation to the current authors, who have so effectively maintained the spirit of this serial publication.

vii

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ADVANCES IN HEAT

TRANSFER, VOLUME

23

Jet Impingement Boiling

D. H. WOLF, F. P. INCROPERA, AND R. VISKANTA Heat TransJer Laboratory, School of Mechanical Engineering, Purdue University, West Lafayetre, Indiana

I. Introduction

Increasing needs for high-heat-flux convective cooling of solids have directed considerable effort toward the development of effective cooling schemes. In some industries experiencing rapid technological growth, such as high-speed computing and data processing, thermal engineering could foreseeably become the factor which limits further growth. This statement is certainly true for the field of microelectronics, where the seemingly inexorable trend of achieving ever larger scales of circuit integration is straining the capabilities of existing high-flux cooling technologies (Incropera [1,2]). In such cases, cooling requirements are exacerbated by restrictions on available space, choice of coolant, local environmental conditions, and maximum allowable surface temperatures. Likewise, the production of steel, aluminum, and other metals having desired mechanical and metallurgical properties requires accurate temperature control during processing (Viskanta and Incropera [3]). The surface temperature and heat flux are typically very large and acceptable cooling times are relatively short. One means of achieving very high rates of heat transfer is through the use of impinging liquid jets. Heat transfer coefficients for systems of this type typically exceed 10,000 W/m2-"C for single-phase convection and are much larger in the presence of boiling. Impinging liquid jets have found usage in many industrial applications, in both submerged (liquid-into-liquid) and freesurface (liquid-into-gas) arrangements. Because of the attractiveness of jet impingement cooling for high-heat-flux applications, numerous studies have been performed for both single- and 1

Copyright 0 1993 by Academic Press, Inc All rights of reproduction in any lorn reserved ISBN 0-12-020023-6

2

D. H.WOLF ET AL.

two-phase conditions. This statement is particularly true for jet impingement boiling, which is distinguished by its ability to dissipate heat fluxes at the high end of the cooling spectrum. However, although the related literature is extensive, ambiguities and contradictions do exist, and there is need for a comprehensive review to assess the state of current knowledge. Such a review has been performed in order to identify strengths and weaknesses in the existing knowledge base and to identify areas requiring additional research. In so doing, every attempt has been made to retrieve and review all of the archival literature on the topic of jet impingement boiling, regardless of source. 11. Background

A. JET IMPINGEMENT HYDRODYNAMICS This review addresses liquid jets with continuous cross sections, thereby excluding spray and droplet impingement studies. Throughout this review,jet configurations will be delineated into the five categories of free-surfacejets, plunging jets, submerged jets, confined jets, and wall jets. These configurations are shown schematically in Fig. 1. The free-surfacejet is injected into an immiscible atmosphere (liquid into gas), and the liquid travels relatively unimpeded to the impingement surface. The plungingjet differs only in that it impinges into a pool of liquid covering the surface, where the depth of the pool is less than the nozzle-to-surface spacing. The submerged jet is injected directly into a miscible atmospshere (liquid into liquid), and the confined jet is injected into a region bounded by the impingement surface and nozzleplate. The wall jet flows parallel to the surface and occurs in both free-surface and submerged configurations. The first four configurations induce flow fields on the impingement surface which are qualitatively similar, and Fig. 2 depicts representative conditions for a planar, free-surface jet. The inviscid pressure and streamwise velocity distributions for a uniform jet velocity profile (Milne-Thomson [4]) are also shown. The pressure is a maximum at the stagnation point due to the dynamic contribution of the impinging jet. With increasing streamwise distance, the pressure declines monotonically to the ambient value. Conversely, the streamwise velocity is zero at the stagnation point and increases to the velocity of the jet with increasing distance along the surface. To clarify the discussion of boiling at various locations on the impingement surface, the flow has been demarcated into stagnation, acceleration, and parallel-flow regions. The stagnation region coincides with that of the impinging jet, in both size and location (x/wj < OS), and contains a nearly linear increase in

JET IMPINGEMENT BOILING

A +

k?-

Liquid

4

HI

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

3

+ Liquid + \\\\\\\\\\\\\\\\\\\\\\rn

a. Free-surface

b. Plunging

Gas

Nozzle plate +

1

Liquid

\\\\\\\\\\\\\\\\\\\\\\\\T

c. Submerged

Nozzle

d. Confined

-,

-

Gas

Liquid

e. Wall (free-surface) FIG. 1. Schematic of the various jet configurations.

+

4

D. H. WOLF ET AL.

Free surface

-, 1-

\\\\\\\\\\\\\\\-n\\\\\\\\\\\\\\\T-X A. Stagnation Region

B. Acceleration Region C. Parallel-Flow Region

wj

FIG.2. Inviscid pressure and velocity distributions for a planar, free-surface jet with uniform velocity profile, along with the respective flow regions.

the streamwise velocity. Within the acceleration region (0.5 I x/wj 6 2), the fluid continues to accelerate and approaches the jet velocity to within a few percent. For x/wj 2 2 (parallel-flow region), the streamwise velocity is essentially that of the jet and the hydrodynamic effects of impingement are no longer realized within the flow. Exceptions to this scenario can occur for plunging and submerged configurations when the exchange of momentum between the jet and miscible fluid is large, causing the flow to decelerate and expand laterally prior to impingement. Such cases typically occur for low jet velocities and/or large nozzle-to-surface spacings (or large pool heights in the case of plunging jets), resulting in lower stagnation pressures and a spatially broader velocity and pressure distribution than that shown in Fig. 2 (see, for example, Gardon and

JET IMPINGEMENT BOILING

5

Akfirat [S]). Exceptions can also occur for confined arrangements, where the nozzle-to-surface spacing is so small that the fluid accelerates further due to a decrease in flow cross-sectional area (see, for example, Miyazaki and Silberman [6]). In addition to governing the hydrodynamics, the pressure distribution controls the local saturation conditions along the surface. For a saturated water jet with an ambient pressure of Pa = 1.013 bar and an impingement velocity of 5 = 10m/s, the resulting saturation temperature at the stagnation point would be 111°C (P = P , = 1.492 bar), compared to 100°C several jet dimensions downstream. Consequently, variations in T,,, cause attendant variations in the degree of subcooling AKub and wall superheat AT,,,. Mudawar and Wadsworth [7] have addressed this issue for a confined jet, where an additional decrease in pressure can be realized for flow between the impingement surface and confining wall. Since the pressure distribution along the surface for a confined jet is a function of the velocity and nozzle-tosurface spacing, the local subcooling will exhibit a similar dependence. For small nozzle-to-surface spacings and large velocities, Mudawar and Wadsworth have shown the streamwise variation in ATubto affect the critical heat flux. Most of the other publications cited herein have based saturation conditions on the ambient pressure. However, for nonconfined arrangements where the heater size greatly exceeds that of the nozzle, ambient conditions exist over nearly the entire surface (excluding the region within several jet dimensions of the stagnation point). In such cases, the use of ambient saturation conditions is justifiable. For the purpose of this review, quantities such as the wall superheat and subcooling will be based on the saturation temperature corresponding to the ambient pressure (Pa);special mention is made of the few cases in which this does not apply. In each of the jet configurations, the velocity can vary between the nozzle exit (V,) and impingement surface (q).For free-surface jets, gravity accelerates the flow for downward impingement and decelerates it for upward impingement. The nozzle and impingement velocities are related, to a good approximation, by the expression vj = ( V ; f 2gz)'lZ, where differences in V , and 5 become negligible for large V,or small z. Indirectly, gravity also causes the jet dimension to vary in order to satisfy continuity. For plunging, submerged, and confined arrangements with large nozzle-to-surface spacings (or large pool heights), momentum exchange, initially occurring at the perimeter of the jet, will ultimately move inward and retard the velocity at the jet's centerline with increasing distance from the nozzle. The axial distance over which the centerline velocity remains equal to that of the nozzle exit (the so-called potential core) typically ranges from 5 to 8 nozzle dimensions. Hence, spacings outside this range will cause the impingement velocity to be lower than the corresponding nozzle velocity. The majority of the impinge-

D. H. WOLFET AL.

6

ment boiling literature has made no distinction between the two values and has used the nozzle velocity (V,) to compare and correlate data. Several investigators, reporting results for free-surface jets, have accounted for gravity and presented their data in terms of the velocity (5) and jet dimension at the point of impingement. In this review, the term jet velocity will generally refer to conditions at the nozzle exit; special mention will be made of the few exceptions for which the impingement velocity 5 is intended. B. JET IMPINGEMENT BOILING The most descriptive representation of boiling data is obtained by plotting the surface heat flux, q”, as a function of the difference between the wall and saturation temperatures (the wall superheat, AT,,,), yielding the boiling curue shown schematically in Fig. 3 for a saturated liquid.

-;:

Single-Phase Nucleate Transition C Forced o n v e c t i o nBoiling TRegimeT

Film

-BoilingRegime

Regime

D

I

I B

Heat Flux

Nuclcatc Boiling

Wall Superheat

log AT,,,

FIG. 3. Schematic of the boiling curve for a saturated liquid.

JET IMPINGEMENT BOILING

7

1. Single-Phase Forced Convection

The single-phase forced-convection regime represents heat transfer in the absence of boiling, and the relationship between the heat flux and wall superheat is governed by Newton’s law of cooling, 4” = h(AT,,, + ATub). For jet impingement, the convection coefficient (h) varies over the surface due to hydrodynamic variations in the streamwise direction. In addition to effects caused by the inviscid flow field discussed in Section II.A, factors such as boundary layer development and transition also affect the distribution of the convection coefficient. As a result, either the wall temperature (heat flux constant) or heat flux (wall temperature constant) will also vary. Single-phase jet impingement heat transfer is extensively discussed in existing surveys of experimental and numerical investigations (Martin [S] ; Downs and James [9]; Polat et al. [lo]; Viskanta and Incropera [ S ] ) .

2. Nucleate Boiling The forced-convection regime extends to wall temperatures that exceed that of saturation, and the nucleate boiling regime exists in the temperature range between points A and B. Although the formation of vapor within surface cavities commences at T,,,, temperatures above this value are required for the vapor to emerge and form a thermally stable bubble. Point A marks the onset of nucleate boiling (ONB), where discrete bubbles begin to detach from the surface and enhance the local fluid motion, causing the convection coefficient to increase. With increasing heat flux or wall temperature, the generation of vapor progresses from a few relatively small bubbles at point A to many larger bubbles coalescing near point B. The points A‘ and B form the extremes of what this review refers to as the fully deoeloped nucleate boiling region. Point A‘ has been chosen because it marks the beginning of the linear nucleate boiling region in log-log coordinates (4” AT,”,,) and the end of the transition from single-phase convection. This definition has been chosen as a matter of convenience in order to convey the reported results in a clear, welldefined manner. Although a universal definition of fully developed nucleate boiling (FNB) does not appear to exist, the term is commonly associated with behavior that is insensitive to conditions in the bulk liquid, such as velocity or subcooling (Collier [ll]). Most of the data to be discussed embody this definition, but exceptions are shown to exist. The attractive feature of nucleate boiling is the large increase in heat transfer that accompanies only moderate changes in the surface temperature. Consequently, it is the desired region of operation for many high-heat-flux cooling applications. However, controlled cooling depends on accurate knowledge of the location of point B, commonly referred to as the maximum or critical heat flux (CHF). The term maximum heat flux will be used for

-

8

D. H. WOLF ET AL.

boiling curves obtained through a quench. The large degree of bubble coalescence ultimately prevents liquid from reaching the surface, and the vapor forms an insulative barrier to heat transfer. Depending on whether the surface boundary condition is heat flux-controlled or temperature-controlled, a large increase in AT,,, (B to B') or decrease in q" (B to C) will result, respectively. Due to the wall temperature (heat flux constant) or heat flux (wall temperature constant) distribution on the impingement plane, both singlephase forced convection and nucleate boiling can occur simultaneously at different locations on the surface. Vader et al. [lZ], for example, have shown through local temperature measurements and high-speed photography that finite regions of nucleate boiling can develop amidst surrounding regions of single-phase convection for a free-surface, planar jet. They showed boiling to initiate near the transition from a laminar to a turbulent boundary layer (a local maximum in temperature for a constant heat flux surface) and subsequently propagate upstream and downstream to envelop the entire surface with increased heating. Cho and Wu [131 similarly reported single-phase convection at the center of the heater with nucleate boiling around the perimeter for a free-surface, circular jet. With increased heating, the nucleate boiling region propagated inward toward the stagnation point. Observations of the heating surface near the critical heat flux (point B) have consistently reported blanketing to initiate at the perimeter of the heated section (Katto and Kunihiro [14]; Katto and Ishii [lS]; Monde and Katto [16]; Monde [17]; Ma and Bergles [18]; Cho and Wu [13]). Blanketing of the inner surface area was generally reported to occur either immediately thereafter, without any additional heating, or upon a marginal increase in the heat flux. In either case, however, the vapor blanket at the heater's edge causes a substantial increase in the local surface temperature, which eventually propagates inward toward the stagnation point, inducing additional blanketing (heat flux-controlled boundary condition). 3. Transition Boiling

The transition boiling regime represents conditions where unstable vapor blankets form and collapse accompanied by intermittent wetting of the surface, The regime is demarcated by point B, the maximum heat flux, and point C, the minimum heat flux and temperature (qkin, Tmin).The q"-AT,,, relationship within the regime depends on the surface boundary condition. The temperature-controlled condition follows the solid curve (B to C), and the heat flux-controlled condition follows the dashed lines (B to B or C to C),depending on whether the heat flux is increasing or decreasing. With one possible exception (Miyasaka et al. [19]), operation in the transition boiling

JETIMPINGEMENT BOILING

9

regime for impinging jets has been limited to temperature-controlled conditions obtained through transient quenches (here the term temperaturecontrolled is used loosely). The primary focus of these investigations has been the measurement and prediction of &,,, Tmi,,,and T.,, (the wetting temperature). In the quench of a specimen, the temperature and heat flux will decline from point D to point C . However, the initiation of liquid-surface contact (wetting) will occur at a temperature (TWet) that is somewhat larger than Tmi,, thereby inducing the local minimum. While qkinand Tminresult directly from measurement of the boiling curve, T,,, can be obtained from measurement of the surface temperature and simultaneous measurement or observation of the liquid-surface contact. Use of the variables Tminand T,,, is intended to differentiate between these experimental approaches.

4. Film Boiling

The film boiling regime (C to beyond point D) represents heat transfer from the surface to the liquid across a vapor film. The mode of heat transfer is primarily forced convection of the vapor, with radiation becoming dominant at higher surface temperatures. For impinging jets, film boiling can often accompany other regimes of boiling on the same surface. Observations of a transient quench with an impinging jet reveal that, at low subcoolings and high plate temperatures, the jet is isolated from the surface by the vapor layer. As the plate temperature declines, the jet penetrates the vapor and wets the surface surrounding the stagnation point while film boiling persists at locations farther downstream (Kokado et al. [20]). 5. System-SpeciJic Efects

Heat transfer associated with each of the modes of boiling is sensitive, in varying degrees, to the experimental conditions used in the measurement. Factors such as surface finish (Rohsenow [21]), surface contamination (Joudi and James [22]), noncondensible gases (Fisenko et al. [23]), heater thickness (Guglielmini and Nannei [24]), heater material (Klimenko and Snytin [25]), method of heating (ac or dc powered) (Houchin and Lienhard [26]), and the type of experiment conducted (steady state or transient) (Bergles and Thompson [27]) have all been shown to affect one or more of the modes of boiling. However, the foregoing results pertain mainly to pool boiling, and there are, in fact, few data for forced-convection boiling. Hence, no attempt has been made in the following review to interpret results in terms of such system-specific effects. Nevertheless, details of each experimental investigation have been provided in tabular form.

10

D. H. WOLF ET AL.

111. Nucleate Boiling This chapter attempts to provide a comprehensive review and analysis of the current knowledge base on nucleate boiling heat transfer in impinging jet systems. Because existing studies have concentrated on CHF phenomena, this knowledge base is somewhat sparse, with parametric investigations having typically been limited to the most basic variables (velocity and subcooling). Nevertheless, the nucleate boiling regime is important, and it is appropriate to address issues such as boiling incipience; fully developed nucleate boiling (see Section II.B.2 for working definition)for single, multiple, and wall jet configurations; local boiling; and other effects such as hysteresis and surface motion. The impingement nucleate boiling literature cited in this chapter is summarized in Tables I and 11, along with particulars concerning the respective experiments. Of the numerous investigations to be discussed in this section, many have employed experimental arrangements where a single surface temperature has been measured at the stagnation point, others have computed averages of several temperatures on the surface, and still others have reported local measurements. Although it will be shown that the wall temperature is uniform for fully developed nucleate boiling ouer the entire heated surface, there may exist regions at low heat fluxes where boiling is fully established over only a portion of the heated surface. A lucid description of the physical mechanisms surrounding this localized boiling has been provided by Vader et al. [12] for a planar, free-surfacejet. Figure 4 shows a series of schematics (based on actual local data and observations) depicting the evolving temperature distribution with increased surface heating. At low heat fluxes (A), the temperature is lowest at the stagnation point and increases downstream with the development of the laminar thermal boundary layer. With sufficient heater length, a local maximum in the surface temperature is observed as the boundary layer begins a transition to turbulence. Additional heating (B), however, is shown to accelerate transition (denoted by the critical Reynolds number Re,) due to the onset of nucleate boiling (ONB), while subsequent increases in the heat flux (C through E) further accelerate transition and produce localized patches of boiling. Despite the observation of vapor bubbles over the entire surface at higher heat tluxes (F and G), the wall temperature remains nonuniform until, finally, fully developed nucleate boiling is achieved over the whole heater (H). Hence, as shown in Fig. 4G, experiments that use a single thermocouple at the stagnation point could infer local fully developed nucleate boiling despite its absence elsewhere on the surface. Although it is not the intent to dispute the presence of global, fully developed nucleate boiling in any of the investigations to be cited, the type of

JETIMPINGEMENT BOILING

- Visible Boiling

11 OPB

FIG. 4. Schematic of typical surface temperature distributions for forced convection boiling with a planar, free-surface jet. Effect of increasing heat flux: (A) single-phase convection; (B) incipient boiling; (C-E) single-phase convection and partial nucleate boiling; (F, G) partial nucleate boiling; (H) fully developed nucleate boiling. (Vader et al. [lZ], used with permission of ASME.)

temperature measurement used in each investigation has been summarized in the interest of clarity (Table 11). All of the steady-state investigations have employed a near-uniform heat flux surface, generated by either direct or indirect means (Table 11). A.

ONSET OF

NUCLEATEBOILING

The onset of nucleate boiling, or boiling incipience, has received limited attention for jet impingement systems. Toda and Uchida [28] reported boiling incipience results for a free-surface, planar, wall jet of water. The onset of nucleation was recognized by both visual observations and a marked change in the slope of the q"-AKat data. Good agreement between the two methods was reported. The following correlation for boiling incipience derived for mist cooling (Toda [29]) was employed with little success: q&NB

= m(AT,at)&&

(1)

where q&NB and ( A T a t ) 0 N B have units of W/m' and "C,respectively. Equation (1) consistently underpredicted the incipient wall superheat. However, the effects of increased jet velocity and subcooling were clearly shown to elevate ( A L >ONE*

TABLE I NUCLEATE BOILING INVESTIGATIONS-OPERATWG PARAMETERS' Author

c. h,

Aihara et al. [61] Chen and Kothari [70] Chen et a/. 1711 Cho and Wu [13] Copeland [40] Ishigai eta/. [53]' Ishirnaru et a/. [62] Karnata et a/. [58j Karnata et al. [59] Katsuta and Kurose 1481 Katsuta and Kurose [48] Katsuta and Kurose [48] Katto and Ishii Cl5j Katto and Kunihiro [14] Katto and Kunihiro [14j Katto and Monde [44j Ma et d.[49] Ma and Bergles [18] Ma and Bergles [35] McGillis and Carey [74] Miyasaka and Inada [34j Miyasaka er al. [19] Monde [ l q Monde and Furukawa [45] Monde and Furukawa [45] Monde and Katto [16, 51, 571 Monde and Katto [16,51,57] Monde and Katto [57] Monde and Okuma 1471 Monde ef a/. [64]

Circ-sub Circ-free Circ-free Circ-free Circ-free Planar-free Circ-sub Ci-conf Circ-conf Circ-free CUC-free Circ-free Planar-free Circ-free Ckc-sub Circ-free Circ-free CUc-sub Circ-sub Cir c conf Planar-free Planar-free cic-free Circ-free Circ-plunge Circ-free Circ-free Circconf Circ-free Circ-free

-

Nitrogen Water Water R-113 Water Water Nitrogen Water Water Mixtured R-11 R-113 Water Water Water Water Water R-113 R-113 R-113 Water Water Water R-113 R-113 R-113 Water Water R-113 R-113

0 75 75 -

4-78 35-75 0 0

0 1 bar Pa > 1 bar 1-9 jets 1-9 jets 1-4 jets

'Each range of operating parameters applies to nucleate boiling only; the range for the overall investigation, including other types of boiling data. may have been broader. Degas refers to whether or not the test fluid was degassed. 'Velocity range is at the point of impingement, not the nozzle exit. Transient experiments (quenches). Water mixed with a surfactant: Rapisool 8-80 (0.02% by weight) (a= 35.5 x N/m at 25°C). Subcooling is based on the saturation temperature corresponding to the stagnation pressure (Pa f pr Subcooling is based on the saturation temperature corresponding to the outlet pressure (measured downstream of the confined region), and the liquid temperature measured at the nozzle inlet. hRanges listed for Nonn ef al. C37.461 are based on those cited for the entire boiling investigation. Ranges for the CHF data only were not provided. 50% volumetric mixture of FC-72 and FC-87 (T,,, = 41°C).

+

e).

TABLE V CRITICAL (MAXIMUM)HEATFLUXINVESTIGATIONS-EXPERIMENTAL APPARATUS"

Author Aihara et al. [61] Andrews and Rao [l00] Baines er a/. [lo81 Cho and Wu 1131 Copeland [40] Ishigai and Mizuno [83] Ishigai et al. [53] Ishimaru et af. [62] Kamata et a/. [58, 591 Katsuta and Kurose [48] Katto and Haramura [lo71 Katto and Ishii [lS] Katto and Kunihiro [14] Katto and Kurata [lo61 Katto and Monde [44] Katto and Shimizu 1761 Ma and Bergles [lS]

% Area coverage 6-12 -

0 0.14 0.021-0.042 32-280 1.8 16 1.2 0.91-7.5 0 0 0.50-2.6 0 4.9 4.0 3.6

Angle (deg)

__ 90 90 0 90

90 90 90 90

90 90 0

15-60 90 0 90

Orientation

Material

Vertical UP Vertical & inclined Vertical Down UP UP Vertical UP UP Down Down

Copperb Stainless steel foil Copperb Copper Copper stainless steel Stainless steel

UP Vertical UP

90

Down

90

Vertical

-

Copper Copper Stainless steel Copper Copper Copper Stainless steel foil Copper Constantan foil

Heating scheme

Sue (mm)

Surface finish

Indirect Direct-dc Indirect Indirect Indirect Directc? Transient Indirect Transient Indirect Direct-dc Indirect Indirect Indirect Direct-ac Indirect Direct-dc

D = 2.3 3.2 x ? 66 x 114 D = 20.5 D = 19.1 8 x 10 12 x 80; x 2 D = 1.5 D = 20 15 5 D 5 25 10ILC40 10 5 L I 20 D = 10 10 I L S 20 8x8 D = 10

No. 100 emery; acetone No. 3 0 0 emery; acetone No. 1500 emery; acetone No. 800-110. 016 emery; acetone

5 x 5

Acetone

16-32p mill finish Wire wool; acetone -

25-pm nickel plating

-

-

No. 0 emery; acetone -

Acetone -

Matsumara er a/. [94] McGillis and Carey 1741 Miyasaka and lnada [34] Miyasaka et al. [191 Monde [I7'J Monde and Furukawa [45] Monde and Katto [16, 51, 5 q Monde and Okuma [47] Monde et a/. [64] Monde et al. [773 Monde et al. [78] Mudawar and Wadsworth [q Nonn et a/. [46] Nonn et a/. [37] &hi ef nl. [93] Skema and SlanEiauskas [99] Wadsworth [38] Wadsworth and Mudawar [IW]

0.010 1.9

>I00 1100 0.075-4.0 0.034 0.91-3.2 0.031 - I. 1 0-2.6 1.o 0.25-4.0 1.0-4.0 0.49-4.4 0.12-4.4 0.22-3.5 1.6-400 1.0-4.0 2.0-4.0

90 90 90 90 90

90 90 90

90 90 90 90 90 90 90

90 90 90

UP Vertical UP Down UP UP Up & down Down UP Down Down Vertical Vertical Vertical UP UP Vertical Vertical

Copper Cppper Platrnum foil Pt foil on copper Copper Copper Copper Copper Copper Copper Copper Copperb Brass Brass Stainless steel Copper Copperb Copperb

Transient Indirect Direct-ac Indirect Indirect Indirect Indirect Indirect Indirect Indirect Indirect Indirect Indirect Indirect Transient Direct-dc Indirect Indirect

D=200; xZOO 6.4 x 6.4 4x8 D = 1.5 11.9 5 D S 25.5 D = 59.8 11.2 D I I21.0 40 85°C) jet of water in the velocity range from 1.5 to 15.3 m/s. Results were also presented for subcooled

80

D. H.WOLF ET AL.

(30 I;AT,,, I;85°C) pool boiling of water. Their experimental apparatus for impingement boiling is shown in Fig. 25a. Heat was generated ohmically by supplying alternating current to the molybdenum plate. Primarily by radiation, energy was transferred from the molybdenum heater to a carbon plate, which was in contact with a conical copper block. Heat was conducted through the carbon, copper, and thin platinum foil prior to reaching the impingement surface. The surface temperature was inferred by extrapolating temperatures measured in the cylindrical portion of the copper block to the impingement plane. The measured temperature gradient was used to determine the heat flux in accordance with Fourier's law. The heater-measurement assembly used for pool boiling was nearly the same as that shown in Fig. 25a; however, the heated surface was facing upward, and the dimensions of the copper block were somewhat larger. Figures 25b and c show the reported boiling curves for pool and impingement boiling, respectively. Both the pool and impingement data reveal a nearly constant heat flux region, despite substantial increases in the wall superheat (ATatas large as 800°C). The authors have delineated these data into first and second transition regions. For the case of pool boiling (AT,, = 85"C), they observed discrete bubbles (approximately 0.1 mm in diameter) leaving the surface and condensing rapidly for heat fluxes in the range from 1.2 x lo6 to 8.1 x lo6 W/m'. At heat fluxes above 8.1 x lo6

Jet n&le

All dimensions BR in mm.

(a)

FIG. 25. Boiling curves for pool and impingement boiling. Shown are (a) a schematic of the experimental apparatus used to obtain the jet impingement data, (b) pool boiling in water, and (c) impingement boiling for a planar, free-surfacejet of water. (Replotted from Miyasaka et at. [19], used with permission.)

JET IMPINGEMENTBOILING

81

ATsub =85K

5

lo1

lo3

5

lo2

ATsat

(b)

K

I

lo8

:

5 -

10'

r

E

s

5 :

m

ie,

D=ISmm Subcooling ATsub =8SK

%.(I91

lo6 5

r

i

; i

15.3 I

I

,,,,*I

I

I

I

I

, , I #

82

D. H. WOLF

ET AL.

W/m2 (AT,,, x 50"C), however, increases in both the population density of nucleation sites and bubble diameter (0.2 to 0.3 mm) were reported, along with a discernible change in the slope of the boiling curve (Fig. 25b). Coalescence of the bubbles was observed at a wall superheat of approximately l W C , and the slope of the boiling curve was shown to change a second time at a superheat of about 200°C. For AT,,, x 3WC, the customary boiling sound was reported to cease, with formation of a stable film over the heater. Although no observations were reported for the impinging jet, Fig. 25c shows that the overall behavior of the boiling curve is the same. Two striking features of Fig. 25b warrant comment. First, data are reported over the extended wall superheat range of 100 IAT,,, I800°C for moderate changes in the heat flux (similar results are presented in Fig. 25c for the impinging jet). Results of this type do not appear to have been reported by other authors in the boiling literature, pool or otherwise. Although suggestions have been made that this type of behavior is related to the unique heater design (Miyasaka [95]) and to the highly subcooled liquid (Inada [SS]), the lack of comparable data from other investigators precludes a definitive explanation of these unique results. Second, the magnitude of the heat flux in the first and second transition regions (8.1 x lo6 I q" I 3 x lo7 W/m2) is large compared to CHF values typical of saturated pool boiling with water at atmospheric pressure (q&pool = 1.4 x lo6 W/m2, Kutateladze [96]). Based on the analytical relation proposed by Zuber et al. [97] (validated with the subcooled water (0 I AT,,, 5 65°C) results of Kutateladze and Schneiderman [98]), the enhancement in CHF for pool boiling with subcooled water at AT,,, = 85°C (the subcooling employed in Fig. 25b) over that for saturated water is 5.4 (i.e., q&F,,,, x 7.5 x lo6 W/mZ for AT,,, = 85"C), which is close to the heat flux at the start of the first transition region in Fig. 25b. However, the trend of the data in the first and second transition regions makes identification of a critical heat flux somewhat arbitrary. Similarly, the impingement data shown in Fig. 25c reveal that, for V, = 15.3 m/s and = W C , q" approaches 60 x lo6 W/mz, which is the largest heat flux reported in the impingement literature (equaled only by Skema and SlanEiauskas [99]). Clearly, one of the major factors in achieving such a large flux was the sizable subcooling. Since the jet dimension (w, = 10 mm) greatly exceeded that of the heater (D = 1.5 mm), the boiling surface was subjected to the sum of the static and dynamic pressures (Pa + ipfV:). For V, = 15.3 m/s and Pa = 1.013 bar, the stagnation point pressure was P = 2.183 bar with a corresponding saturation temperature of T,,, = 123°C; hence, the subcooling was AT,,, = 108°C. The heat flux at the beginning of the first transition region shown in Fig. 25c was recognized as the departure from nucleate boiling (DNB) and was correlated by the following dimensional expression (Miyasaka et al. [191

JETIMPINGEMENT BOILING

83

routinely interchanged the terms DNB and CHF in their discussion. As a reminder of the somewhat unusual nature of their data, DNB is used below.) qhNB

= q~HF,oor[l

+ 0*86v~'38i(1 + &sub)

(54)

where q~HF,ool is the critical heat flux for pool boiling proposed by Kutateladze [96]

and E , , ~ is a correction factor for the effect of subcooling

The units of V, in Eq. (54) are m/s. Equation (56) was obtained from the pool boiling data of Miyasaka et al. [19]; however, although the term (1 + &,,b) correlated the pool boiling data (30 < ATubI 8 5 ) , the impingement data were restricted to a relatively narrow range of subcoolings (85 I ATub I 108°C).Although not stated explicitly, it is assumed that the authors evaluated all fluid properties at the saturation temperature corresponding to the stagnation pressure (P,). The data represented by Eq. (54) were obtained for a 1.5-mm-diameter heater cooled by a 10-mm-wide, planar jet; hence, the heated surface was completely immersed within the stagnation region of the jet. Based on extensive evidence for free-surface, circular jets, which indicates that CHF is clearly a function of the heater dimension (see Section 1V.A.l.d), the validity of Eq. (54) for heated surfaces much larger than the jet width is highly suspect. The dependence of CHF on heater size may be attributed to the relatively poor cooling characteristics in the parallel-flow region of the jet. Since the transition between nucleate and film boiling is typically very unstable, vapor blanketing initiated at a downstream location of the heater is likely to spread quickly and envelop the entire surface. Therefore, the critical heat flux is commonly determined by conditions at the surface location of poorest heat transfer. The literature for circular jets has shown, virtually without exception, that CHF decreases at the heater dimension increases, suggesting that Eq. (54) will overpredict DNB for surfaces larger than the jet width. The transition regions shown in Figs. 25b and 25c were correlated in terms of the wall superheat and jet velocity. The first transition region for impingement boiling was given as

84

D. H. WOLF ET AL.

where the heat flux in the first transition region for pool boiling at a subcooling of 85°C is q;pool = 0.690 x lo6 AT:;:’

(58)

The units for q”, AT,,,, and V, are W/mz, “C, and m/s, respectively. The dimensionless parameter x was included to account for the effects of stagnation pressures which significantly exceeded atmospheric pressure and for subcoolings other than 85°C. Equation (58) was obtained for atmospheric conditions, and to compensate for pressure in Eq. (57), the authors suggested that x be the ratio of Eq. (54), evaluated at the stagnation pressure and desired subcooling, to Eq. (54), evaluated at atmospheric pressure and a subcooling of 85°C. That is,

As previously indicated, the jet impingement results were obtained at a single subcooling of 85°C. Hence, without experimental confirmation, inclusion of the term (1 + &,,b) in Eq. (59), which is based on the assumption that different subcoolings would be well predicted by this formulation, is speculative. The second transition region for impingement boiling was similarly correlated

+

(60) where ql;pooIpertains to the second transition region for pool boiling at a subcooling of 85°C 41; = xq~pool(l 0.4V:.4)

4ZPO0l - 11.2 x lo6 -

AT:!;^

(61)

The units are the same as those for Eqs. (57) and (58), and x is given by Eq. (59). The only other investigation of CHF that could be identified was that of Ishigai et al. [53], who reported entire boiling curves obtained by transient measurements (T = lOOOOC) for a subcooled jet of water. The maximum heat flux was shown to increase from approximately 3.4 MW/mZ at a jet velocity of 1 m/s to about 5.5 MW/m2 at 3.17 m/s (Axub= 15°C). However, no correlation of the data was provided. b. Subcooling. Ishigai et al. [53] have also examined the effects of subcooling on the complete boiling curve. Increases in the maximum heat flux by over a factor of 4 were reported for attendant increases in the subcooling from 5 to 55°C at a jet velocity of 2.1 m/s. Miyasaka et al. [191 have proposed a CHF correlation which accounts for the effects of subcooling [Eqs. (54) through (56)]. However, the subcooling

JET IMPINGEMENT BOILING

85

parameter given by Eq. (56) was obtained from pool boiling data, while the impingement experiments were performed at a single jet temperature of 15°C. Favorable agreement was demonstrated between Eqs. (54) through (56) and the subcooled, circular jet data of Ishigai and Mizuno [83] and Monde and Katto [Sl]. 3. Effects of System Parameters on the Critical Heat Flux for Submerged, Confined, and Plunging Jets

a. Jet Velocity. Katto and Kunihiro [14] reported CHF results for both submerged and plunging, circular jets of saturated water. Regardless of whether the jet was submerged or plunging, a linear relationship existed between CHF and the jet velocity, which ranged from 1 to 3 m/s; however, the form of that linear relationship was shown to be a function of the nozzleto-surface spacing and pool height. Results for 5- and 30-mm-deep pools are presented in Figs. 23b and 23c, respectively. These data, despite scatter, show several salient features concerning the effects of the jet environment (i.e., free surface, submerged, plunging) on CHF and are delineated as follows: (1) the closest nozzle-to-surface spacings consistently yielded the highest values for CHF; (2) Fig. 23b indicates that the critical heat flux for the submerged arrangement (spacings of 1 to 3 mm) is larger than that for the plunging configuration (spacings of 10 to 30 mm), independent of jet velocity; (3) the data also suggest that the spacing of 1 to 3 mm yielded larger values of CHF for a pool height of 5 mm than for a height of 30 mm; (4)the CHF data for nozzle spacings of 1 to 3 mm and 5 mm pool height exceeded the free-surface results (no pool) for a fixed jet velocity (Fig. 23a); ( 5 ) the effects of velocity were negligible for a nozzle-to-surface spacing of 30 mm in a 30-mm-deep pool, since the initial momentum of the jet was insufficient to affect conditions on the heater surface, and the value of CHF was approximately that of pool boiling (q&F,,,, % 1.4 x lo6 W/mZ, Kutateladze [96]). The linear relationship between CHF and jet velocity, which has been proposed by Katto and Kunihiro, has not been substantiated by others; rather, a cuberoot dependence has been cited most often. It is possible that the inference of a linear relationship was driven by the existence of a limited data base encompassing a small range of velocities. In a similar investigation, Monde and Furukawa [45] examined the effects of velocity and pool height on CHF for a saturated, circular jet of R-113. While maiantaining the nozzle-to-surface spacing at 5 mm, the pool height was adjusted between 0 and 8 mm, creating free-surface, plunging, and submerged conditions. The results are shown in Fig. 26, along with Monde’s [80] correlation for a circular, free-surface jet [Eq. (31)] and the correlation for pool boiling proposed by Kutateladze [96] [Eq. (55)]. Although the data

D. H. WOLFET AL.

86

Vn (m/s) FIG. 26. Effect of velocity on CHF for a circular jet of saturated R-113 with free-surface ( H = 0, z = 5 mm), plunging (0 s H s 4 mm, z = 5 mm), and submerged ( H = 8 mm, z = 5 mm) arrangements. (Monde and Furukawa [45], used with permission.)

are not as definitive as those of Katto and Kunihiro [14], similar features are evident. For example, CHF data for the free-surfacejet and a shallow liquid pool (0 IH I1 mm) are consistently below those with more substantial fluid levels (2 s H I 8 mm). Moreoever, the highest value of CHF is obtained for the submerged arrangement (H= 8 mm), with the plunging configurations (2 IH I 4 mm) yielding somewhat lower values. Perhaps the most striking feature of Fig. 26 is the comparatively large critical heat flux predicted by the pool boiling correlation. The results would seem to suggest that pool boiling provides a more effective means of cooling than jet impingement. However, the low values of CHF for jet impingement, at least for the free-surfacejet (H= 0 mm), resulted from the large heater diameter (59.8 mm) relative to that of the nozzle (1.1 mm). In Section IV.A.l.d, the adverse influence of a large heater and a small nozzle dimension on CHF was described. Andrews and Rao [lo01 reported CHF data for a submerged, planar jet of subcooled water (0 IAT,,, 5 62°C). Impingement occurred at the top surface of a thin, horizontal foil heater, whose lower surface was also exposed to the coolant. Increases in the jet velocity (0.3-2.0 m/s) produced accompanying increases in the critical heat flux. However, the extent of the increases in CHF appeared to become more pronounced with increasing subcooling [i.e., q& , : ' , I where m = f(AT,,b)], a result not reported elsewhere. Ma and Bergles [l8] obtained a small amount of CHF data for a circular, submerged jet of R-113. The basic trends in the data revealed a cube-root velocity dependence (1.08 I V, 5 2.72 m/s) for CHF.

-

JETIMPINGEMENT BOILING

87

Kamata et al. [58, 591, in a two-part investigation, examined the effects of jet velocity and nozzle-to-surface spacing on CHF for a confined, circular jet of saturated water. Part one of the study [58] used an arrangement where a circular plate was attached to the nozzle exit and was parallel to the heater surface. Clearances between the nozzle-plate and heater were kept small (0.3-0.6 mm), and the diameter of the confinement area was the same as that of the heater, 20 mm. For a nozzle-to-surface spacing of 0.3 mm, increases in the maximum heat flux from about 3.7 to 5.3 MW/m2 (43%)were reported for increases in the jet velocity from 10 to 17 m/s (70%). Part two of the investigation [59] employed the same nozzle arrangement, but with the addition of a 0.2-mm brim around the circumference of the nozzle-plate (between the nozzle-plate and heated surface) to prevent stratification of the liquid and vapor at large heat fluxes. The effects of velocity on the maximum heat flux were smaller. For a nozzle-to-surface spacing of 0.3 mm (100 pm clearance at the brim), increases in the maximum heat flux from about 6.9 to 8.5 MW/mz (23%)were shown for increases in the jet velocity from 12 to 20 m/s (67%). However, the maximum heat flux for the brimmed nozzle-plate exceeded that of the brimless version by 50% (V, = 17 m/s and z = 0.3 mm). Information concerning the pressure loss across the nozzle was not provided; however, it is likely to have been large, considering the constraints imposed on the flow. McGillis and Carey [74] examined the effects of jet velocity on the critical heat flux for confined, circular jets of subcooled (1 S ATubI30°C) R-113 (10 jets impinging on 10 heaters). Data were reported for heaters which were both flush mounted and protruding with respect to the substrate (the nozzleto-surface spacing was held constant at 10 nozzle diameters). Increases in the jet velocity (0.95-3.08 m/s) produced attendant increases in the critical flux. The higher-velocity data were well correlated by Eqs. (26) and (27) (q& V,!,I3)>. proposed by Monde and Katto [16], but with the coefficient of the subcooling parameter reduced from 2.7 to 0.456, as suggested by Nonn et al. [46]. The lower-velocity data approached values of the critical heat flux that were typical of subcooled pool boiling. No differences in the critical heat flux could be detected between the protruding and flush-mounted heater arrangements for similar operating conditions. Skema and SlanEiauskas [99] investigated the effects of jet velocity on the critical heat flux for a submerged, circular jet of highly subcooled water (85 IAT,,, I. 151°C).For velocities ranging from 1 to 35 m/s, the CHF data were well correlated by the expression N

+ 0.92v:'44](1 + &sub)

(62) where q&Fp,,, and &,,b are given by Eqs. (55) and (56), respectively, and the properties are evaluated at the stagnation pressure (P,). As for the study by q&fF= q&IFpool[f

D. H. WOLFET AL.

88

Miyasaka et al. [19], it is assumed that the correlation requires evaluation of all of the fluid properties at the saturation temperature corresponding to the stagnation pressure. Comparison of the data of Skema and Slani5auskas with those of Miyasaka et al. [19] for a planar, free-surface jet was favorable. Similar to the experimental conditions of Miyasaka et al., the nozzle dimension (d = 18 mm) exceeded that of the heated surface (D = 9 mm). Mudawar and Wadsworth [7] investigated the effects of velocity (I I V, I13 m/s) on CHF for a confined, planar jet of subcooled FC-72. They identified medium- and high-velocity regimes for which the dependence on the jet velocity differs. In the medium-velocity regime, the critical heat flux increased with increasing velocity, whereas in the high-velocity regime, CHF was approximately independent of velocity (Fig. 27) and even decreased for small nozzle-to-surface spacings. The velocity at which this transition occurred decreased with decreasing nozzle-to-surface spacing and subcooling.

Width, w,= 0.508 mm Subcooling = 10 i 0.15 “c

100 h

N

1

E

5l

,o

5

e 0

Q

0

:

0

un(m/s) FIG. 27. Effect of velocity and nozzle-to-surfacespacing on CHF for a planar, confined jet 01991, Pergamon Press PLC.) of FC-72. (Reprinted with permission from Mudawar and Wadsworth [7]

JETIMPINGEMENT BOILING

89

Mudawar and Wadsworth suggest that the trend of decreasing CHF with increasing velocity is unique to conditions of small nozzle-to-surface spacings and high jet velocities, because of the lower degree of local subcooling. In such flows, the jet strikes the surface, relatively unimpaired by the ambient fluid, and flows as a wall jet along the surface. Because of the large velocity along the surface, vapor bubbles lack sufficient momentum to enter the bulk flow and induce mixing of the cooler, free-stream fluid with the warmer fluid near the surface. Hence, although all the data shown in Fig. 27 were obtained at a subcooling of lWC, which is based on the saturation temperature corresponding to the outlet pressure measured downstream of the confined region and the liquid temperature measured at the nozzle inlet, the authors contend that the local subcooling was lower for conditions of high jet velocity and small separation distance, thereby providing lower values of CHF. The medium-velocity data were correlated with a mean absolute deviation of 7.4%by the expression

where Asubis a subcooling parameter given by

The properties cprand pf were evaluated at the nozzle inlet temperature (q), and the remaining properties (p,, hfg, and o) were evaluated at the saturation temperature (T,,,) corresponding to the outlet pressure measured downstream of the confined region (Wadsworth, [38]). The exponent of the density ratio was chosen in accordance with other CHF correlations, since the density ratio was essentially constant (92.7 < pf/pe < 101.5). This correlation indicates a velocity dependence of the form 4& V11.702, which is much larger than that reported by others for impinging or wall (discussed in Section 1V.C) jets. The authors suggested that, consistent with the results reported by Grimley et al. [loll for a confined, falling film, the uncommonly high exponent may be the result of confining the flow. No other correlation or extensive data base for a highly confined jet is available for comparison. Aihara and co-workers (Aihara et al. [61]; Ishimaru et al. [62]) have reported CHF data for a submerged,circular jet of saturated liquid nitrogen. The jet impinged into a region with radial confinement at the perimeter of the heater (i.e., the liquid impinged on the heater, flowed radially to the confining wall, turned 90°,and flowed along the confining wall in a direction opposite that of the incoming jet). Ishimaru et al., providing results for impingement on a flat heater surface, showed the relationship between the critical heat flux

-

90

D. H. WOLFET AL.

and jet velocity (0.22 5 V, I1.34 m/s) to be of the form q& V:.6. Likewise, Aihara et al., providing results for impingement on a concave, hermispherical heater surface, revealed the same dependence between CHF and jet velocity (0.77 5 V, I1.64 m/s). In both investigations,jet velocities above 1 m/s were able to sustain critical heat fluxes in excess of 1 MW/m2 [referred to the wetted surface area A, (see Fig. 19a)l. N

b. Subcooling. Andrews and Rao [loo] examined the effects of subcooling (0 IATub 5 62°C) on CHF for a submerged, planar jet of water. Although trends of increasing CHF with increased subcooling were evident, the degree of subcooling also affected the dependence of CHF on velocity. For a saturated jet, the approximate dependence was qgHF V:.', while for a For both subcooling of 33"C, the approximate dependence was (I& subcoolings, the data were well correlated over the entire range of investigated velocities (0.3 5 V,, 2 2.0 m/s). This type of behavior has not been reported by others. Ma and Bergles [18] reported critical heat flux data for a submerged, circular jet of R-113 at subcoolings from 11.5 to 29.5"C. Although only three CHF data points with nonzero subcoolings were presented, augmentations ranging from approximately 30 to 80% over those for saturated conditions were evident. The authors commented that the enhancement was not as large as the predicted by the Monde and Katto [16] subcooling parameter [( 1 + &sub), Eq. (27)] and attributed the difference to operating at subcooling beyond the correlated range for Eq. (27) coefficients (@sub = cpfAT,&,,) [Monde and Katto [161: 0 I@sub I0.1 1; Ma and Bergles [181: 0 I @sub I 0.201. Similarly, Nonn et al. [46] reported that the Monde and Katto factor of (1 + &sub) greatly overestimated the effects of subcooling for a free-surface jet when Osubexceeded the correlated limits. Nonn et al. confirmed the functional form of Eq. (27) but achieved better quantitative agreement with their data by replacing the coefficient of 2.7 with 0.456. McGillis and Carey [74] applied Eqs. (26) and (27) to their data for a confined, circular jet of subcooled (1 IAT,,,,, 5 30°C) R-113 and found favorable agreement, provided the coefficient of 0.456 was used in lieu of 2.7. Mudawar and Wadsworth [7] reported subcooled, CHF data for a confined, planar jet of FC-72. For subcoolings from 0 to 40"C, the data were well correlated by Eqs. (63) and (64), where the two-term subcooling parameter (A,,b) was based on a CHF model proposed by Mudawar et al. [1021 for falling films. Mudawar and Wadsworth also identified a secondary role of subcooling related to the existence of two different regimes of CHF, each of which had a different dependence on the jet velocity (see Section IV.A.3.a). At velocities below some critical value, sizable increases in CHF were reported for increasing velocity, while a plateau in CHF (or decline for

--

JETIMPINGEMENT BOILING

91

some conditions) was observed for velocities beyond the critical value. The demarcating velocity between the regimes was found to depend on the nozzle-to-surface spacing, as well as the degree of subcooling. Higher levels of subcooling were shown to postpone the transition between regimes to higher velocities, thereby allowing more significant increases in CHF with jet velocity. c. NozzlelHeater Dimensions. SkCma and SlanEiauskas [99] reported results for a submerged jet of water with variable nozzle and heater diameters (3 I d I 18 mm and 9 I D I20 mm). Although the data for a diameter ratio (D/d) of 0.5 were well correlated by Eq. (62), which is independent of any characteristic dimension, data for D/d > 0.5 deviated from Eq. (62) by more than 25% (constant nozzle velocity). The maximum value of CHF was reported to occur at a ratio of D/d = 2.5. The authors attributed this peak to the maximum radial velocity of the jet being achieved at D/d z 2. Mudawar and Wadsworth [7] varied the nozzle width from 0.127 to 0.508 mm for a confined jet of FC-72. Higher levels of CHF were achieved with increasing nozzle width, consistent with results reported for free-surface, circular jets (Section 1V.A.l.d). The correlation of' their data, given by Eqs. (63) and (64), exhibits this trend. d. Nozzle-to-Surface Spacing. Katto and Kunihiro [14] varied the nozzleto-surface spacing from 1 to 30 mm (0.6 Iz/d I 42) for circular, submerged, and plunging jets of saturated water (Figs. 23b and c). Relatively significant effects of spacing are shown for both jet configurations, with the smallest separation distances yielding the largest values of CHF. Moreover, as the spacing increases, the effects of jet velocity diminish. Figure 23c indicates complete invariance when the separation distance reaches 30 mm (19 I z/d I 42), for which the critical heat flux is then governed by pool boiling parameters. Andrews and Rao [IOO] reported effects of nozzle-to-surface spacing on CHF for a planar, submerged jet of water. Although only a limited amount of data were presented, the critical heat flux was shown to decrease linearly with increasing separation distance (-30% decline in CHF for a factor of 4 increase in nozzle-to-surface spacing). Kamata et al. [58, 591, performing a two-part investigation, examined the effects of nozzle-to-surface spacing on the maximum heat flux for a confined, circular jet of saturated water. Part one of the study [58] reported little effect of the separation distance on the maximum heat flux for clearances between the nozzle-plate and heater from 0.4 to 0.6 mm (0.18 Iz/d I 0.27). However, for d spacing of 0.3 mm (z/d = 0.14), an increase in the maximum heat flux of 36% over the larger spacings was reported (V, = 17 m/s). Part two of the

92

D. H. WOLF ET AL.

investigation [59], which employed a 0.2-mm brim around the circumference of the nozzle-plate, showed monotonic increases in the maximum heat flux with decreased spacing over the entire range from 0.6 to 0.3 mm (0.14 I z/d 5 0.27). Decreasing the spacing from 0.6 to 0.3 mm enhanced the maximum heat flux by over 60% (V, = 20 m/s). By preventing stratification of the liquid and vapor, the brim postponed (to larger heat fluxes and wall superheats) the transition to film boiling. By inducing higher pressures along the impingement surface, the brim may also have enhanced heat transfer by increasing the amount of subcooling. Kamata et al. reported the jet temperature to be equal to the saturation temperature, which, to avoid flashing within the nozzle, must be based on the ambient pressure (Pa= 1.01 bar, T,,, = 100T). Although the pressure loss across the nozzle was not reported, it is likely to have been large, considering the constraints imposed on the flow, particularly for the brimmed arrangement. Due to the larger pressure loss incurred by the brimmed nozzle-plate, the static pressure (and T,,,) at the stagnation point exceeds that for the brimless version, all other parameters (V, and z) remaining fixed. Hence, the subcooling at the stagnation point and along the surface may have been larger for the brimmed arrangement, yielding higher values for the maximum heat flux. Similar findings were reported by Nonn et al. [46] for a circular jet of FC-72. The fluid was confined between a nozzle-plate (7.5 mm diameter) and the heater surface (12.7 mm square). Spacings down to 0.5 mm (z/d = 0.5) had no effect on CHF for a jet velocity of 6 m/s, but further decreases in the spacing produced attendant increases in CHF. A separation distance of 0.1 mm (z/d = 0.1) yielded approximately 30% enhancement in the critical heat flux. Mudawar and Wadsworth [7] reported CHF data for a confined, planar jet of FC-72 with nozzle-to-surface spacings from 0.508 to 5.08 mm (1 5 z/ w, 5 40). The separation distance was found to affect the transition between two CHF regimes, identified as having a different dependence on jet velocity (see Section IV.A.3.a). The transition velocity delineating these regimes was shown to decrease as the nozzle-to-surface spacing decreased. However, at jet speeds below this transition value, only a weak dependence on the spacing was evident, while more marked effects were realized at higher velocities (Fig. 27). Monotonic increases in CHF with decreased nozzle-to-surface spacing (0.6 I z 1 2 . 2 mm) were reported by Aihara et al. [61] for a submerged, circular jet (d = 0.8 mm) of saturated liquid nitrogen impinging onto a radially confined, concave, hermispherical surface. For a jet velocity of 1 m/s, CHF was enhanced approximately 120%by reducing the separation distance from 2.2 mm to 0.6 mm.

JETIMPINGEMENT BOILING

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4. Differences between Circular and Planar Jets Since only three publications on planar jets could be identified (Ishigai et al. [53]; Miyasaka et al. [19]; Mudawar and Wadsworth, [7]), a comparison between results for circular and planar geometries is likely to be incomplete. The only known comparison between the two configurations (all else being fixed) was reported by Miyasaka et al. [19], who compared their CHF data for a planar, free-surface jet with those of Ishigai and Mizuno [83] and Monde and Katto [Sl], for free-surface, circular jets. The agreement was favorable, tending to suggest little or no dependence on nozzle geometry. However, since the comparison was not made on the same apparatus with identical flow conditions, this evidence is considered weak and inconclusive.

5. Differences between Free-Surface and Submerged Jets Although few investigations have compared CHF data between free-surface and submerged configurations, the general finding has been that submerged jets provide critical heat fluxes which are larger than those for free-surface jets. Katto and Kunihiro [14] reported CHF data for free-surface, submerged, and plunging jets of saturated water (see Fig. 23). Their results indicated the maximum value of CHF to occur for a submerged jet with a nozzle-to-surface spacing between 1 and 3 mm, and differences between the two configurations increased with increasing velocity (CHF for the submerged jet was approximately 25%larger than that of the free-surface jet at 3 m/s). Monde and Furukawa [45] reported CHF results for free-surface, submerged, and plunging jets of R-113 (see Fig. 26). Maximum critical heat fluxes were shown to occur for the submerged configuration (H = 8 mm, z = 5 mm), exceeding results for the free-surface jet by as much as 50 to 60% at a velocity of approximately 3 m/s. No discussion of the mechanism responsible for these differences was provided in either of the aforementioned papers.

6. Approaches to the Correlation of Critical Heat Flux Data Many authors have successfully correlated their data through dimensional arguments (Buckingham pi theorem, for example), previously proposed correlations, and occasionally trial-and-error procedures. Several authors, however, have approached the problem by considering the fundamental mechanisms involved in the transition from nucleate to film boiling (CHF). In doing so, a rationale has been provided for the various dimensionless groups which appear in most of the correlations presented thus far. Lienhard and Eichhorn [89] proposed a correlation for free-surface, circular jets of saturated liquid that was based on a balance of mechanical

94

D. H. WOLF ET AL.

energy in the liquid/vapor flow. The concept had been applied earlier to cylinders in cross-flow (Lienhard and Eichhorn [88]). They suggest that, at the point of burnout, the vapor traveling normal to the surface has kinetic energy that is approximately equal to the energy expended in the formation of droplets at the vapor-liquid interface (surface energy). This energy balance is given by

where a is the fraction of liquid converted into droplets, upis the velocity of the vapor, and ddropis the droplet diameter (Sauter mean). The left-hand side (LHS) of the equation represents the rate of transport of the kinetic energy of the vapor, and the right-hand side (RHS) represents the rate at which work is done by surface tension forces to form droplets. The first term on the RHS is the number of droplets formed per unit time (the numerator is the mass rate of liquid flow for all droplets, and the denominator is the mass of a single droplet). The second term on the RHS is the surface energy required to form one droplet (product of surface tension and area). The vapor velocity at the critical heat flux is estimated from the thermodynamic consideration that the latent heat required to generate the vapor (hfnpgu,nD2/4)is approximately equal to the heat extracted from the surface (q&,FnD2/4). Hence, the vapor velocity is estimated to be u, = &F/Pghfg, and Eq. (65) may be rewritten as

The ratio of jet to droplet diameters (d/ddrop)was estimated from both dimensional analysis and the available literature. They invoked an empirical expression proposed by Nukiyama and Tanasawa [lo31 for the Sauter mean diameter and used u, as the characteristic velocity in that expression. The fraction of liquid convkrted to droplets (a) was assumed to be governed by the density ratio (pf/p,). The final expression is of the form

where both f and 5 were shown by Lienhard and Hasan [go] to be functions of the density ratio, y = pf/pg[see Eqs. (48) and (4911. Sharan and Lienhard [91] modified Eq. (67) slightly by assumsing V, to be the characteristic velocity used in the correlation for the drop diameter and by arguing that the fraction of liquid converted to droplets (a) was governed by both the density ratio and Weber number. Despite a somewhat different form to the correlation [see Eqs. (50) and (5111, the essence of the model was maintained.

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Kandula [86] extended the analysis of Lienhard et al. by suggesting that the physical mechanisms associated with CHF differ, depending on the relative velocity between the radially flowing liquid film and vapor bubbles leaving the surface. When the liquid and vapor velocities are comparable, bubbles are able to penetrate the liquid film and induce droplet formation at the free surface. This flow condition was identified as the surface tension-controlled regime, and the mechanical energy stability criterion of Lienhard and Eichhorn [89] was used to predict the critical heat flux. When the liquid velocity greatly exceeds that of the vapor, the bubbles lack sufficient momentum to penetrate the liquid film and are dragged radially along the surface. This flow condition was identified as the liquid viscosity-controlled regime, because burnout was associated with liquid boundary layer separation from the surface due to vapor blanketing (shown to depend on the viscosity of the liquid). The critical heat flux in this high-velocity regime was based on a similarity solution of the momentum equation for blowing near a two-dimensional stagnation point

where b is the radial velocity gradient and C is a constant. Introducing the approximation that u$V, x q&/(PghfgI/n), it follows that

which indicates that CHF is independent of jet velocity and is a function of the liquid viscosity. Conditions where CHF is independent of the jet velocity were classified earlier as corresponding to the I-regime and were observed experimentally at elevated pressures [low density ratios (pf/pg)]. Kandula argues that the surface tension-controlled regime (q& P‘,!,’~)exists at larger density ratios (lower pressures), while the viscosity-controlled regime (q& independent of V,) dominates at smaller density ratios (higher pressures). Hence, the author suggests that Eq. (69) be utilized to correlate Iregime data. Comparing Eq. (69) with the I-regime correlation of Katto and Shimizu [76] [Eq. (28)], Kandula suggested that Cb”’ = 0.127(q/d)’/’, where q results from a corresponding-states analysis [Eq. (46)]. Replacement of Cb1I2with this expression yields the correlation presented earlier as Eq. (45). Although it provided a good correlation of the results of Katto and Shimizu [76], its ability to predict the more extensive high-pressure data of Monde [78] was not tested. Monde [ S O ] proposed a CHF correlation for circular, free-surface jets of saturated liquid which is based on a hydrodynamic model recommended by

-

96

D. H.WOLF ET AL.

Haramura and Katto [Sl]. The schematic of the assumed flow condition is shown in Fig. 28. At large heat fluxes, it is suggested that vapor leaves a liquid sublayer on the surface as discrete jets or columns, while liquid is supplied from the jet to the sublayer primarily through a passage that is approximately

FIG. 28. Schematic depicting the liquid-vapor structure near the heated surface at high heat fluxes. (Adapted from Haramura and Katto [Sl] and Monde [SO].)

JET IMPINGEMENT BOILING

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xu' in circumference and of thickness 6, (the thickness of the sublayer decreases in the streamwise direction due to evaporation). The model assumes CHF to occur when the latent heat of the liquid is equal to the heat generated by the surface at the point of burnout, that is,

The velocity of vapor flowing through the liquid film may eventually become large enough to induce an instability at the interface causing breakdown of the discrete columns. However, Haramura and Katto contend that the vapor columns within the liquid film are stable, whereas outside the film the columns are likely to collapse and form common pockets of vapor (see Fig. 28). The stable region is assumed to result from the suppression of wave disturbances at the solid boundary. Furthermore, the thickness of the film is postulated to depend on the Helmholtz critical wavelength (A,), which is obtained by prescribing stability requirements at the vaporbiquid interface. That is,

where us and uf are the vapor and liquid velocities, respectively, at the interface of the vapor column. Although the jet induces liquid flow parallel to the wall, this flow is ignored in estimating A,. The thickness of the liquid film was taken as the average of likely values between 6, = 0 and 6, = ad2 (i.e., 6, = &/4). Since the liquid velocity (uf) was shown to be substantially less than that of the vapor (ud, uf was deleted from Eq. (71). The velocity ug was estimated from a heat balance on the vapor, q"A, = peugqghfg,where the ratio of vapor to heater areas was obtained from semiempincal results for pool boiling, AJA, = 0.0584(pJpf)o.2.The liquid film thickness 6, could then be expressed in terms of known variables

where 6, decreases with increasing heat flux. Moreover, the combination of Eqs. (70) and (72) yields the following expression for the critical heat flux of a circular, free-surfacejet :

This equation has been successfully used as the foundation for the correlations proposed by Monde [80] [Eq. (31)] and Katto and Yokoya [92] [Eqs. (52) and (53)].

D. H.WOLF ET AL.

98

Mudawar and Wadsworth [7] modeled CHF for a confined, planar jet by adapting an earlier analysis of falling films (Mudawar et al. [102]), which was similar to that of Haramura and Katto [81] but incorporated the effects of moderate subcooling. They assumed the surface to be wetted with a thin liquid film, penetrated by columns of vapor as shown in Fig. 29a. The supply of liquid to the surface was said to occur at the upstream region of the film through the thickness 6,. An energy balance that equates the latent and sensible heat of the liquid film to the surface heat transfer at CHF yielded

--

Vapor Release Jet

Liquid Sublayer

I

1

FIG. 29. Schematic depicting the liquid-vapor structure near the heated surface at high heat fluxes for (a) low-velocity jets and (b) high-velocity jets. (Reprinted with permission from Mudawar and Wadsworth [7] 01991, Pergamon Press PLC.)

JETIMPINGEMENT BOILING

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The value for the film thickness (6,) was obtained from the analysis of Mudawar et al. [102], who, like Katto and Haramura, assumed applicability of the Helmholtz instability, but also included the sensible energy in the formulation of the vapor velocity. Their expression for 6, was

where JI is an empirical constant which accounts for the ratio of the vapor to heater areas (A$&) (estimated by Haramura and Katto from pool boiling results) and Csub is an empirical subcooling constant. The resulting CHF expression was

where

Like most of the correlations, Eq. (76) indicates a dependence of CHF on the cube root of the jet velocity. However, experimentally, Mudawar and Wadsworth found a dependence of the form q&F V:,’. They suggest that their flow conditions corresponded to those shown in Fig. 29b, where the vapor blankets are isolated. The appropriate characteristic dimension for this regime is the size of a discrete blanket, as opposed to the entire heater (L - wn). Being unable to characterize the size of a discrete blanket, they multiplied Eq. (76) by the geometrical term [w& - w.)]” and determined the exponent on the inverted Weber number from the correlation of experimental data. The resulting correlation was given by Eqs. (63) and (64).

-

B. MULTIPLE-JET IMPINGEMENT Monde et al. [64] examined the effects of multiplejets (two to four), and their relative positions, on the critical heat flux. Circular, free-surface jets of saturated water and R-113 were employed at velocities ranging from 1 to 20 m/s for both upward- and downward-facing surfaces. Figure 14a shows the various arrangements. Each of the circular heaters is schematically divided into regions over which a particular jet is most dominant. The maximum distances from the impingement point to the edge or center of the heater are also shown. The authors content from visual and photographic observations that film boiling is initiated at these locations. The data for all 11 configurations and both upward- and downward-facing heaters were well correlated

100

D. H. WOLF ET AL.

by the expression

where L is the distance shown in Fig. 14a. The coefficient C was found to be 0.150 if burnout began at the edge of the heated surface (n, = 4, 3, 2, 2*; nb = 2,2*; and n, = 2,2*) and 0.1 14 when burnout was initiated at the center (nb = 4,3 and n, = 4). This correlation is similar to that proposed by Monde [17] [Eq. (29)] for a single, circular jet, with the characteristic dimension (heater diameter, D) replaced by 2L. As was evident from the single-jet literature, the critical heat flux varies inversely with the characteristic dimension of the heater surface. The implication is that multiple jets yield higher values of CHF than single-jet configurations (for a fixed fluid, nozzle diameter, and jet velocity) due to shorter flow lengths on the impingement surface. Monde and Inoue [lo41 have reevaluated the da. 1. of Monde et al. [MI, as well as additional unpublished data, and have proposed a correlation based on the hydrodynamic CHF model of Haramura and Katto [Sl] (see Section IV.A.6 for discussion of the model). As opposed to Eq. (78), they recommended Eq. (31), developed by Monde [80] for single-jet impingement, as the appropriate correlation of the foregoing multiple-jet results. Equation (3 1) correlated the data with an RMS error that was typically within 20% by replacing the characteristic dimension D in Eq. (31) with 2L, where L is the maximum flow length on the surface, as shown in Fig. 14a. Nonn et al. [46] reported CHF data for configurations of one, four, and nine free-surface, circularjets of FC-72 impinging on a single heat source. The single jet impinged at the center of the square heater (12.7 x 12.7 mm), while the four- and nine-jet configurations were equally distributed over an equal number of smaller squares. The critical heat flux was found to be well correlated for all combinations of jets by the expressions of Lee et al. [84], Eqs. (34) and (35), where the characteristic dimension (heater diameter, D ) was replaced by twice the maximum flow length on the heater surface (the diagonal of the square region allotted to each jet), in accordance with Monde et al. [MI. In a subsequent investigation by Nonn et al. [37] with a 50% mixture of FC-72 and FC-87, the CHF data were again correlated by Eqs. (34) and (35). [Although the saturation temperature of the mixture was measured (AT,,, = 41"C), no information was provided concerning the other properties. However, the difference in properties between the two fluids, such as pf, pg, h,,, cpf, and 6,is within 8% at the boiling point (Mudawar and Anderson, [85]).] The ranges of velocity and subcooling were not provided for the CHF data in either study, but ranges for the investigations in general were reported to be 1.6 < V, I12.7 m/s and 20 IAT,,, I 37°C.

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Skema and SlanEiauskas [99] also reported CHF data for an array of four circular, submerged jets of subcooled water. As with Monde et al. [64] and Nonn et al. [46,37], the critical heat flux was shown to decline with increased spacing between the jets. In contrast, however, the critical heat flux for the most densely packed array was slightly less than that for a singlejet (V, fixed).

C. WALLJETS This section addresses wall jets of two forms: those that are ejected from a nozzle in a direction approximately parallel to the surface and those that develop downstream of a stagnation point due to impingement normal to the surface. Although some may classify gravity-driven films (falling films) as wall jets, they are considered to be beyond the scope of this review. Katto and Ishii [l5] reported CHF data for a planar (0.56 I w, 50.77 mm), free-surfacejet of saturated water, R-113, and trichloroethane impinging at angles of 15" and 60" to a heated, downward-facing surface. The jet impinged 2 mm upstream of the heater, whose length in the streamwise direction was varied to study various degrees of boiling development (10 I L I 20 mm). The critical heat flux was well correlated over a range of velocities from 1.5 to 13 m/s by the expression

which is similar to the many other correlations presented thus far for normally impinging jets [see Eq. (26) of Monde and Katto [16], for example]. Furthermore, this correlation is independent of the angle of impingement and the thickness of the jet within the investigated ranges. The result concerning angular independence may be related to the fact that the jet did not actually impinge on the heater. Rather, impingement occurred at least 2.6 nozzle widths upstream of the heater's leading edge, which is sufficient for hydrodynamic differences due to different impingement angles to be substantially reduced (McMurray et al. [105]). It is not know whether this finding could be extended to cases in which impingement occurs on the heated surface. Katto and Kurata [lo61 similarly examined CHF for a planar ( 5 Iw, 5 10 mm), submerged jet of saturated water and R-113 flowing parallel and upward along a vertically oriented heated surface. For nearly the same range of heater lengths and jet velocities considered by Katto and Ishii [l5] (10 I L I 20 mm and 1.3 I V, I 9.1 m/s), the CHF data were well correlated by

102

D. H. WOLF ET AL.

which, like Eq. (79), is independent of the nozzle width. Katto and Kurata have suggested that the significant differences in the exponents of the density ratio and, to a lesser extent, the inverted Weber number between Eqs. (79) and (80) may be related to the different liquid-vapor flow patterns observed at CHF for the two cases. Katto and Ishii [lS], employing a free-surface jet, have reported the vapor to induce a near-complete separation of the jet from the surface, while only a thin film of liquid remains in contact with the heater. Within the film, nucleate boiling is sustained until complete evaporation causes vapor blanketing. By contrast, Kurata and Katto, employing a submerged jet, have observed the vapor to leave the surface in the form of discrete bubbles, swept downstream by the jet flow and growing in size with increasing heat flux. Bubbles on the order of the heater dimension were reported to occur near CHF; however, separation of the bulk flow and formation of a liquid film were not observed. Employing saturated, free-surface jets of water and R-113, Katto and Haramura [1071 greatly extended the range of velocities investigated by Katto and Ishii [l5] to 1.8 5 V , I 65 m/s. The data were correlated by the expression

-(IgHF - 0.0106(2)

P*h&"

0.867

0.281

(2) PfV3

which, although very similar to that of Katto and Ishii, has a smaller dependence on the Weber number, much like that of Katto and Kurata [106]. However, correlation of the data was achieved only to within &40%; the CHF results for the shorter heating length (L = 10 mm) were consistently underpredicted by Eq. (81), whereas those for the longer heating length ( L = 40 mm) were consistently overpredicted. Clearly, the term L-o.281did not adequately represent the effects of heater length in Eq. (81). Flow behavior identical to that described by Katto and Ishii [l5) at the point of burnout was also reported by Baines et al. [lo81 for a planar, freesurface jet of saturated water, flowing parallel to the heater surface. They not only observed separation of the bulk liquid from the surface but also noted the thin liquid film to be replenished by droplets from the bulk flow. Moreoever, when this replenishment was artifically disrupted by a plate, partially inserted between the bulk and film flows, the critical heat flux was seen to decline by about 30%.In the undisturbed state, CHF was shown to be independent of the heater orientation (vertical or upward-facing) and to be well predicted by Katto and Ishii's Eq. (79) for jet velocities ranging from 0.8 to 2.5 m/s. However, the critical heat flux exhibited a weaker dependence on velocity beyond this range (2.5-5 m/s). The authors suggested that a transition between the V- and I-regimes (described earlier in Section 1V.A.l.a) may have occurred and that the transitional velocity of 2.5 m/s is

JETIMPINGEMENT BOILING

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approximately equal to that calculated from Katto and Shimizu's critical Weber number [Eq. (41)]. However, based on information published later by Monde [78], this transition is not likely to occur for water at atmospheric pressure. For impingement occurring normal to the heated surface, several authors have reported CHF data in the parallel-flow region of the jet. Miyasaka and Inada [34] positioned a small heater (4 x 8 mm) directly beneath a planar, free-surface jet of water (w, = 10 mm) and 25 mm downstream. At a single subcooling of 85"C, CHF at the downstream region was 50% and 10% below that at the stagnation point for jet velocities of 3.2 and 15.3 m/s, respectively. In contrast, Kamata et al. [58, 591 reported transient temperature measurements at the stagnation point and at a radial location of 5 mm for a circular jet of water (d = 2.2 mm) that impinged into a confined region. Although the maximum heat flux at the downstream location was generally below that at the stagnation point, the differences were minimal.

D. OTHERPARAMETERS INFLUENCING THE CRITICAL HEATFLUX 1. Surface Conditions

Wadsworth and Mudawar [1091 examined the effects of enhanced surfaces on CHF for a planar, confined jet of FC-72. Figure 30a shows the microgroove and microstud surfaces, along with the direction of flow and various dimensions. Figure 30b shows CHF data for the enhanced and smooth surfaces as a function of jet velocity, where the heat flux is based on the planform area common to all of the arrangements. The dependence, qEHF V:.7, is maintained for all surfaces; however, the critical heat flux of the smooth surface was increased 122% and 230% at 3 m/s and 78% and 133% at 1 1 m/s by the microstud and microgroove configurations, respectively. Figure 30c'shows the same CHF data with the heat flux based on the total wetted area. The smooth surface achieved the highest fluxes, which implies that the attractive performance of the enhanced surfaces resulted primarily from the additional area. Interestingly, the monotonic increases in CHF with increased subcooling demonstrated for the smooth and microstud surfaces was not evident for the microgroove surface. Rather, a local minimum in CHF with respect to AXub was observed, below which a decrease in subcooling yielded an increase in CHF. Based on visual observations, Wadsworth and Mudawar speculated that at low subcoolings large volumes of vapor may be entrained within the grooves, accelerating the flow across the surface and enhancing CHF.

-

0.305 mm

Flow

Flow Direction

Direction

Centerline

Impingement Centerline

G I

MICRO-GROOVE SURFACE

MICRO-STUDSURFACE

1000

I

,

1000

.

I

surface structure

'", c I o

0 Smooth 0 Micro-Stud

-

A

A Micro Groove

N

E

4.444

Y

2?

100

LL

I

-

.v

U

w,= 0.254 mm

10

10 10

1

vn

(b)

I

10

1

(ds)

Vn

(ds)

(c)

FIG. 30. Effects of velocity on CHF for a planar, free-surfacejet of FC-72 impinging on enhanced surfaces. Shown are (a) the enhanced surface geometries,(b) results with heat flux based on reference area, and (c) results for heat flux based on wetted area. (Wadsworth and Mudawar [loS], used with permission of ASME.)

106

D. H. WOLF ET AL.

Aihara et al. [61] reported the effects of surface finish on CHF for a submerged, circular jet of saturated liquid nitrogen impinging onto a radially confined, concave, hemispherical heater. Differences in CHF between a sanded (emery paper no. 500) and a machined finish were minimal over the entire range of velocities (0.77 IV, < 1.64 m/s). The critical heat flux for a mirror finish was 12% below the CHF values for the sanded and machined surfaces; however, only one datum point was provided (V, = 1 m/s). 2. Jet Suction

McGillis and Carey [74] investigated the effects of submerged jet impingement and jet suction on the critical heat flux for subcooled R-113 (shown schematically in Fig. 21a). For comparable jet velocities and subcoolings, CHF values for the two configurations were in very good agreement over the investigated range of parameters (0.95 IV, I3.08 m/s; 0 IATub I30°C; protruding and flush-mounted heaters). 3. Droplet Splashing from a Free-Surface Jet While numerous observations of the hydrodynamics associated with CHF have been reported (Katto and Kunihiro [14]; Andrews and Rao, [loo]; Ishigai et al. [53]; Matsumura et al. [94]; Miyasaka et al. [19]; Monde, [17]; Monde et al. [64]; Taga et al. [68]; Ochi et al. [93]; Monde and Okuma [47]; Cho and Wu [13]; Monde and Furukawa [45]), a particularly lucid description has been provided by Katto and Monde (Katto and Monde [44]; Monde and Katto [16]), who related CHF to droplet splashing. They examined nucleate boiling and CHF for free-surface, circular jets of saturated water and R-113 and noted splashing at the jet’s liquid-ambient interface due to penetration of the interface by vapor flowing normal to the heater. Splashing increased with increasing q”, thereby depleting the supply of coolant to the heater surface. Katto and Monde [44] developed an apparatus to measure the mass rate of fluid lost to splashing (G) and compared the result with the mass rate of the impinging jet (Go). The measurements supported their observations of increased splashing rates with increased heating, although the maximum value of G/Goat CHF ( 42°C and is based on jet spacings in the range 4 I S/d I 12; the range of jet velocity was not specified. The specified dependence on the Prandtl number was not discussed by the authors; however, since the correlation is based on experiments with water only, there is no evidence to justify its presence in Eq. (84a). Hatta, Kokado, and co-workers have addressed the phenomenon of surface wetting. in numerous publications for a free-surface, circular jet of water (Hanasaki et al. [114], Hatta et al. [115-1171; Kokado et al. [20]). Experiments were conducted with the jet (d = 10 mm) impinging vertically downward onto a stainless steel plate (200mm square and 10mm thick), initially heated to 900°C. Temperature measurements were made at five locations on the back surface of the plate, equally spaced in the radial direction at 20-mm intervals, beginning at the stagnation point. The temperature distributions generally demonstrated a gradual decline with time during the early stages of the quench due to blanketing of the surface by vapor. At later times, the temperature became low enough to allow the liquid to penetrate the vapor film and wet the surface, inducing a more substantial cooling rate. Wetting of the surface occurred initially at the stagnation point and traveled radially outward with increasing time. The evolution of the wetting front is demonstrated in Fig. 33, which shows a plate, initially at 900"C, being cooled by a water jet (AT,ub = 80°C).The sequence begins with the plate appearing red in color over the entire surface. With increasing time, a circular, dark region, whose perimeter is closely related to the wetting front, develops and grows throughout the quench. Beyond the dark region, the liquid accumulates into discrete pools and is suspended above the surface by the underlying vapor. Hatta et al. [115, 1161 attempted to quantify the size of the wetted region based on their measured temperatures and recorded growth of the dark zone from visual observations for a highly subcooled jet (AT,",, = 80°C)of water. Numerically, they specified two different heat flux boundary conditions on the impingement surface, demarcated at the presumed location of the wetting front. The heat flux within the wetted area significantly exceeded that of the nonwetted area. They attempted to reconstruct the measured temperatures on the opposite side of the plate through systematic adjustments of the imposed heat fluxes and location of the wetting front. They concluded that the radius of the wetting front increased with time according to the relation C1 + C2t'/2,where C , is a constant and C, is a function of the mass flow rate of the jet and convection coefficient of the wetted region.

FIG.33. Sequence of photographs showing the quench of a stainless steel specimen with a free-surface, circular jet of subcooled water (4 = 2.0 m/s, AT,,, = 80°C). (Hatta et al. [ l l q 01983, The Iron and Steel Institute of Japan.)

JET IMPINGEMENT

BOILING

115

Kokado et al. [203 correlated wetting results at the stagnation point only, as a function of the jet temperature (71 I T, I 92"C), for impingement velocities ranging from 2.0 to 2.5 m/s (1.0 IQ I7.0 L/min), and proposed the following expression

T,,, = 1150 - 8T, where T f is the jet temperature and, along with the wetting temperature T,,,, has units of "C.At wall temperatures below Twel,the surface is assumed to be in contact with the liquid (wetted), while for wall temperatures in excess of Twcl,the surface is assumed to be insulated from the liquid due to vapor blanketing (nonwetted). Equation ( 8 5 ) correlated their data well but was limited to conditions where T, > 68°C. At temperatures below 68"C,wetting at the stagnation point was reported to occur immediately upon impingement, despite an initial wall temperature of nearly 900"C, which is consistent with results reported by Ishigai et al. [53] for a highly subcooled jet (T, = 45°C). The simplistic relationship between and T,,, depicted in Eq. ( 8 5 ) does not imply that T,,, is independent of other parameters such as jet velocity, nozzle dimension, and fluid properties. Rather, it represents a relationship that is most likely unique to the conditions of the Kokado et al. investigation. Hatta et al. [117] extended the use of Eq. ( 8 5 ) to predict the wetting temperature at radial locations other than the stagnation point. They applied the methodology of earlier investigations (Hatta et al. [115, 116]), iteratively imposing wetted and nonwetted heat flux boundary conditions on the impingement surface and attempting to reconstruct numerically the temperatures measured on the opposite side of the plate. They accounted for heating of the liquid along the impingement surface and evaluated Eq. ( 8 5 ) at each incremental location to determine whether the local wall temperature exceeded T,,, . The mean temperature across the liquid film was used in Eq. (85) for T,. Heat fluxes in the wetted and nonwetted regions were based on the product of a convection coefficient (different for each region) and local temperature difference between the wall and free stream (Newton's law of cooling), with the nonwetted region also accounting for radiative cooling. The imposed convection coefficient for the wetted region was based on a correlation of the form hr/k, = CRerPr;, which is typical of single-phase convective transport. However, their data clearly indicate surface temperatures in the wetted region to be signijcantly larger than the saturation temperature, thereby precluding single-phase convection. Despite the fact that such a boundary condition enabled favorable reconstruction of the temperature on the opposite face, its general applicability for quenches with a wetted surface is unlikely.

116

D. H. WOLF ET AL.

More recently, Hatta et al. [118] have extended the investigation to freesurface, planar jets of water. They employed the same experimental approach as that used for the circular jet and effectively reconstructed the measured temperature distribution from assumed heat flux boundary conditions on the impingement surface. The heat flux for the wetted region was again based on a convection coefficient typical of single-phase heat transfer, and the location of the wetting front was determined from Eq. ( 8 5 ) with T, corresponding to the local liquid film temperature. Hatta and Osakabe [1191 performed similar experiments with a planar jet, but for a moving surface. Plate speeds (V,) ranged from 0.48 to 2.4 m/min for a fixed impingement velocity ( 5 )of 1.63 m/s. They utilized the same experimental approach as that of Hatta et al. [I 181, but introduced an expression for the wetting temperature which was somewhat different than that given by Eq. (85). They state that the plate surface is nonwetted if either of the following two conditions is satisfied:

T, > 1100 - 8.5 T, T, > 710°C

(86b) where both the wall and fluid temperatures have units of "C. No justification was given for the use of Eq. (86) instead of Eq. (85). However, they were able to reconstruct the measured temperature distribution reasonably well from this equation and assumed heat flux boundary conditions on the impingement surface. As for the effects of surface motion, Hatta and Osakabe present numerically obtained temperature distributions across the thickness of the plate as a function of time that are clearly a function of the plate speed. At low plate velocities, temperatures from the impingement surface to the opposite face all declined monotonically with time. By contrast, although higher plate velocities induced monotonic cooling of the impingement surface, the opposite face cooled only as the jet impinged and, due to conduction from highertemperature regions within the plate, subsequently increased in temperature after the jet had passed. These differences in the temperature distributions as a function of plate speed, however, are due almost entirely to the amount of time available for impingement. Since the process is transient, the time of cooling will clearly alter the temperature field within the plate. Although the authors suggest that the plate motion causes hydrodynamic differences in the flow field (such as thinning of the liquid film on the surface with increasing speed), this effect is highly improbable in the context of their experiments, which were conducted at extremely small relative plate velocities (0.005 5 V,/vj 50.025). Equations (82), (83), (85), and (86), which correlate the minimum heat flux and wetting temperature, were obtained for water impinging on a stainless steel surface. Klimenko and Snytin [25] have shown qLin and Tmln(T,,, would be expected to follow the trends of Tmin)to be strong functions of

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combinations of the wall material and coolant properties. They correlated numerous pool boiling data from a wide spectrum of experimental conditions and accounted for the effects of the fluid-wall combination with a ratio of thermal effusivities (pc,k),/(pc,k),, where increases in this ratio induced monotonic increases in both q l i nand Tmin.Although employing only water, similar trends of decreasing minimum temperature with increasing wall effusivity have also been reported for spray cooling (Jeschar et al. [1201). Hence, applicability of the aforementioned equations is most likely limited to fluid-wall combinations for which the ratio of effusivities is similar to that of water-stainless steel. Kamata et al. [58, 591 reported @-AT,,, data in the transition boiling regime for a circular, confined jet of saturated water. Part one of this investigation [58] used an arrangement in which a circular plate was attached to the nozzle exit and was parallel to the heater surface. Clearances between the nozzle-plate and heater were small (0.3-0.6 mm), and the diameter of the confinement area (20 mm) was the same as that of the heater. Temperature measurements at the stagnation point and at a radius of 5 mm revealed differences in the transition boiling curves between the respective locations. The most notable difference pertained to the minimum heat flux, which was nearly 60%larger at the stagnation point than at the downstream location. The difference was attributed to liquid exiting the confined area along the nozzle-plate, while the vapor flowed along the heater surface. Part two of the investigation [59] employed the same nozzle arrangement, but with the addition of a 0.2-mm brim around the circumference of the nozzleplate to prevent stratification of the liquid and vapor at large heat fluxes. Consequently, only minimal differences were evident between the transition boiling curves at the stagnation point and the downstream location. Moreover, with the brimmed nozzle-plate, the heat flux increased by more than 50% within the transition region, all other parameters being fixed. Sano et al. [66], conducting transient experiments, reported transition boiling data at nine different streamwise locations (0 I x I 56 mm) for a free-surface, planar jet of saturated water (V, = 3.5 m/s; the nozzle dimension was not provided). Marked differences in the transition boiling curves were evident with respect to surface location. For a fixed wall superheat, the heat flux decreased monotonically with increasing streamwise distance from the stagnation point. VI. Film Boiling

Ruch and Holman [41] performed film boiling experiments for a circular, free-surface jet of R-113 at a single subcooling of 27°C. The jet impinged vertically upward onto a heater surface, with inclination angles (relative to

118

D. H. WOLF ET AL.

the heater surface) ranging from 45 to 90". The jet velocity and nozzle diameter were also varied. Correlation of the film boiling data at the stagnation point was achieved to within k 35% for the wall superheat range 60 IAT,,, 2 340°C by the dimensional expression

where q:ilmI denotes the total (convective and radiative) heat flux from the surface. The units of q&mI, V,, and AT,,, are W/m2, m/s and "C,respectively. No dependence on jet diameter or impingement angle was detected. Using dimensional analysis based on the Rayleigh method, the authors recast the correlation into the more general form

However, the applicability of this expression to other fluids was not investigated. Moreover, the effect of subcooling, which has been shown to influence strongly the rate of film boiling heat transfer (Zumbrunnen et al. [121]), was neglected. Ishigai et al. [53] obtained film boiling data for the quench of a surface, initially heated to approximately loOo"C, by a planar jet of water (some steady-state data are also reported). They also proposed an analytical model for film boiling at the stagnation point of the jet (a more detailed presentation of the model is given, in Japanese, by the same authors in Nakanishi et al. [122]). They solved the conservation equations (mass, momentum, and energy) for both the liquid and vapor phases using the similarity transformation typically employed for stagnation (Hiemenz) flows (Burmeister [1231). Solution of the equations yielded the vapor film thickness and the convective heat flux. As suggested by Bromley [124], the total heat flux (qiilmI)from the surface was computed by combining convective (q:ilm,) and radiative (q:ilrnr) contributions in the form q;lilrnt= q'ilrn, + 0.75 q;lilmr.Although the experimental heat flux data consistently exceeded the analytical predictions, trends with respect to subcooling, surface temperature, and jet velocity were well modeled. Nakanishi et d.[122] showed that the model accurately predicted experimental data (1.0 IV, I;3.17 m/s and 5 s AT,,, s 35"C), provided the total heat flux was computed as q:ilmt = 1.74 q;ilmo 0.75q:i,m,, where q:ilrn, was still obtained from the similarity solution. The vapor film thickness was estimated analytically to be of the order of 10 to 100pm, for the range of operating parameters. Lamvik and Iden [l25] measured the average convective heat transfer coefficient from a heated aluminum surface (T = 500°C) to a single jet and multiple circular jets of water (free surface). Measurements of local surface

+

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temperatures during quenching were not reported, but based on the experimental conditions it is presumed that both film and transition boiling occurred during the course of the experiment. Average coefficients are reported for a single horizontal jet impinging on a vertical surface, for a vertical jet impinging on a horizontal surface from below, and for a vertical jet impinging on a horizontal surface from above. It was suggested that the functional relationship between the average coefficient and the jet velocity depended on the orientation of the jet with respect to the gravitational field. The strongest dependence was for the horizontal jet impinging on a vertical surface, while the weakest dependence was for downward impingement of a vertical jet on a horizontal surface. Although these trends may provide some indication of the effects of gravity on the boiling process, the scatter in the data precludes a definitive judgement. Nevins [1261also measured the average convective heat transfer coefficient from a heated stainless steel surface to a circular, free-surface jet of water, employing both transient (100 IT, < 640°C) and steady-state (40 I T, I 115OC) techniques. The convection coefficients from the two different measurement schemes were reported for nearly the same operating condition (V, = 0.81 m/s and AT,,, x 81°C for the transient technique; V, = 0.93 m/s and ATubx 70°C for the steady-state technique); however, the convection coefficient from the transient measurement exceeded the steady-state value by a factor of 5. Nevins speculated that the disparity was related to the differing hydrodynamics between a jet whose flow is well established on the surface (the steady-state experiment) and one whose flow develops during the course of the measurement (the transient experiment). Although this reasoning is plausible, it is likely to be a second-order effect. The disagreement in convection coefficients has most likely resulted from operating at different locations on the boiling curve. For the steady-state experiment, heat transfer occurs by single-phase convection, whereas for the transient experiment, heat transfer occurs in the presence of film, transition, and nucleate boiling, the last of which is capable of generating very large convection coefficients. Kokado et al. [20] measured rear-surface temperatures for a stainless steel plate (T = 900°C) quenched by a circular jet of water. Surface wetting began at the stagnation point and spread radially with time. Outside the wetted region, the liquid was suspended above the surface by the underlying vapor and cooling occurred through convective film boiling and radiation to the surroundings. Heat flux boundary conditions on the impingement side of the plate were iteratively applied to a two-dimensional conduction model, in an attempt to reconstruct the measured temperature distribution on the opposite surface. Based on heat fluxes imposed in the nonwetted region of the plate and accounting for radiation losses, the following empirical fit was proposed for the conuectiue heat transfer coefficient associated with film boiling in the

D. H. WOLFET AL.

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parallel-flow region: h=200

2420 - 21.7'& AT,,p8

(89)

where h, T,, and ATa1have units of W/m2-OC, "C,and "C,respectively. The effect of surface motion on forced-convection film boiling heat transfer in the parallel-flow region of planar jet has been analyzed by Zumbrunnen et al. [121]. In an integral analysis of the laminar vapor and liquid boundary layer flows, they determined the extent to which plate motion can affect heat transfer upstream and downstream of the impinging jet. For cocurrent motion of the fluid and surface with a dimensionless plate velocity of 20 (8, = V,/u,), the convective heat transfer coefficient was predicted to increase by a factor of 5 relative to that for a stationary surface. For countercurrent motion with modest plate velocities (Pp = - 0.6), convective heat transfer was approximately one-half of that for a stationary surface. In terms of film boiling on a moving surface with jet impingement (typical for the cooling of primary metals), increasing plate velocities will suppress and enhance heat transfer, respectively, upstream and downstream of the impinging jet. Heat transfer is inhibited by thickening of the upstream vapor layer due to vapor being dragged with the plate and against the bulk flow; heat transfer is augmented by thinning of the downstream vapor layer due to vapor being dragged in the direction of the bulk flow. Significant enhancements in heat transfer with increased subcooling were also reported for both stationary and moving surfaces For a fixed surface temperature, the relative contribution of radiation to the total heat transfer was shown to be largest on the upstream side of the jet due to the poorer convective transport. On the downstream side, radiation became less significant as P, increased. Ishimaru et al. [62] have reported film boiling data for a circular, submergedjet of saturated liquid nitrogen. Results were given for velocities of 0.22 and 1.34m/s, with the heat flux of the latter being twice that of the former (ATaIfixed). The approximate relationship between qff and AT,, was of the form q:i,m, ATa1'.'.

-

VII. Research Needs

From the foregoing review, it is apparent that considerable progress has been made toward establishing a data base on the subject of impingement boiling. However, while a good deal of information has been obtained for particular jet geometries and modes of boiling, other important conditions have received only marginal consideration. Moreover, despite focused attention in certain areas, conflicting results have been obtained concerning the effects of

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various parameters. Consequently, additional research is needed, and possible topics are summarized according to the mode of boiling.

A. ONSETOF NUCLEATE BOILING The onset of nucleate boiling (ONB) and the oft-accompanying temperature excursion (AZJ have received only minimal attention. Moreover, results that have been reported were the by-products of broader nucleate boiling studies. Hence, a thorough, focused investigation of the effects of jet hydrodynamics, turbulence, fluid-surface combinations, and noncondensible gases on boiling incipience and temperature excursions should be conducted for impinging jets. Such results would, for example, be useful in designing liquid immersion cooling schemes for high-power electronic components. The large values of ATx reported for pool boiling in dielectric liquids have thus far precluded commercial applications in the cooling of electronic components (Bar-Cohen [1271). However, temperature excursions observed for impingement boiling are significantly smaller than those for pool boiling (Bar-Cohen and Simon [67]), and an improved understanding of ONB and ATx for impinging flows could advance prospects for eventual applications. BOILING B. NUCLEATE

For many cooling applications, nucleate boiling is the desired mode of heat transfer due to the large range and magnitude of heat fluxes accompanying only small changes in the surface temperature. However, despite the current volume of jet impingement literature, the knowledge base pertaining to nucleate boiling is comparatively sparse. Moreover, nucleate boiling data are often presented as merely a precursor to CHF results, which are of greater interest, and detailed discussions of trends, as well as the resolution of anomalies in the data, are often omitted. For example, while most of the impingement literature supports the invariance of nucleate boiling heat transfer to parameters such as velocity and subcooling, contradictory data have been reported. The wall superheat has been reported to decrease, as well as increase, with increasing velocity, and increased subcooling has been shown to reduce surface temperatures at heat fluxes near incipience. There is also evidence to support a strong effect of velocity at low mass flow rates, where the sensible and latent energy stored in the jet is comparable to that transferred from the surface. It has yet to be resolved how such effects can be observed in one study and not in others. Nucleate boiling data have been reported predominantly for R-I13 and water. Although both fluids are abundant and inexpensive, their potential applications as coolants are limited. In light of growing concerns over the

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environmental consequences of chlorofluorocarbons, continued widespread, commercial use of R-113 is unlikely (Bar-Cohen [127]). Moreover, the electrically conductive nature of water and its large saturation temperature preclude its used in applications such as direct electronic cooling. Due to their highly inert, dielectric characteristics,perfluorinated liquids such as FC72 and FC-87 are currently the coolants of choice for electronic components. However, there is a paucity of nucleate boiling data for such liquids. C. CRITICAL HEATFLUX Although the critical heat flux has received the most attention, many issues remain unresolved and some jet configurations have been virtually ignored. Several shortcomings are: 1. Few investigators have examined CHF over a wide range of subcoolings. Consequently, a CHF correlation has yet to be proposed that is valid over an extended range of subcooling coefficients (Osub = cpf AKub/hfg)*

2. Although a low mass flow rate CHF regime has been observed (socalled L-regime), general correlating equations for CHF and the limits of this regime have not been proposed, 3. Likewise, although high-pressure regimes (so-called I- and HP-regimes) have been observed and correlated, there is a poor understanding of the physical mechanisms that render these regimes different from conditions at atmospheric pressure. Moreover, correlations of the data are of limited use due to the inability to predict general demarcations between the regimes. Also, the current data base of CHF results at elevated pressures has been limited to free-surface jets. It is unclear whether similar trends could be expected for submerged or confined jets. 4. The majority of the CHF data base was obtained for free-surface, circular jets. Some configurations, such as free-surface, planar jets and confined jets, have received only limited attention, precluding the development of a reliable CHF correlation. Free-surface, planar jets are used widely in the cooling of processed metals, where the liquid is water and the subcooling is typically very large (AT,,, 2 75°C). However, only two publications dealing with the maximum heat flux for jets of this type are currently available. Confined jets (either between the nozzle-plate and the heater, as shown in Fig. Id, or in the streamwise direction) are likely candidates for future electronic cooling applications due to restrictions on available space. Available CHF data for this geometry are also limited to a few publications. 5. No information is available concerning the effects of nozzle geometry (planar or circular) on CHF, and the few comparisons between freesurface and submerged arrangements were not comprehensive.

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AND FILMBOILING D. TRANSITION

Information concerning transition and film boiling for impinging jets is sparse and is limited to the most fundamental quantities (qkin, Tmin).An important application is related to the cooling of metals during processing. The production of steel, aluminum, and other metals having desirable mechanical and metallurgical properties requires accurate temperature control. In a typical hot steel rolling mill, for example, steel plate leaves the last finishing stand at temperatures ranging from 750 to 1000°C and is rapidly transported along a runout table, where it is quenched prior to coiling. Cooling is often achieved with a series of highly subcooled water jets (ATub 2 75°C). Due to the large plate temperatures, nucleate boiling is typically confined to a small region beneath the jet, while film boiling exists over the majority of the surface at locations upstream and downstream of the stagnation point. There is considerable interest in controlling the local temperature of the plate as a function of time in order to achieve desired metallurgical and mechanical properties. However, the numerical prediction of the plate's time-temperature history relies heavily on knowledge of the heat transfer rates in each boiling regime and the spatial demarcations between regimes, all in the presence of surface motion. Although modeling efforts have been undertaken, uncertainty in the surface boundary conditions is the dominant limitation (Filipovic et al. [128]). Additional levels of uncertainty can develop, however, due to complications in modeling more complex geometries, such as those obtained through extrusion. Research needs in this area include (Viskanta and Incropera [3]): 1. Information concerning the spatial demarcation between boiling regimes on a stationary and, more important, moving surface. 2. Nucleate and film boiling data in the stagnation and parallel-flow regions, respectively, of a highly subcooled jet with cocurrent and countercurrent plate motion. 3. The development and validation of numerical models to enable the results of small laboratory experiments to be scaled to prototypic mill conditions.

Acknowledgments The authors are grateful for the many constructive suggestions made by Dr. David Vader of IBM and Dr. h a m Mudawar of Purdue University, which served to enhance the overall clarity of this review. The authors are also thankful to Dr. Akira Yamada of Nagasaki Research and Development Center, Mitsubishi Heavy Industries, Ltd. for his valuable assistance in translating the Japanese literature, and to Dr. Masao Takuma of Mitsubishi Heavy Industries, Ltd. for providing those Japanese publications that otherwisewould have been inaccessible.The support of the National Science Foundation under grant CTS-8912831 is also gratefully acknowledged.

124

D. H. WOLF ET AL. Nomenclature cross-sectional area of vapor stems cross-sectional reference area of heated surface total area of heated surface available for cooling radial velocity gradient (du,/dr) a constant constant in Rohsenow's [42] nucleate pool boiling correlation to account for different surface-fluid combinations empirical subcooling constant used by Mudawar and Wadsworth [7] specific heat of the liquid heater diameter heated cylinder diameter nozzle diameter droplet diameter (Sauter mean) Froude number ( V,/(gD)1/2) mass flow rate of splashed droplets mass flow rate of jet gravitational acceleration gravitational constant pool height for a plunging jet convection heat transfer coefficient [q"/(T, - T,)] convection heat transfer coefficient at the onset of nucleate boiling ([4bNdTWows latent heat of vaporization thermal conductivity of the liquid thermal conductivity of the vapor length of the heated surface molecular weight local pressure along the impingement surface ambient pressure stagnation pressure (Pa + kV,")

Prandtl number of the liquid partial pressure of the vapor partial pressure of noncondensible gases volume flow rate of the jet heat flux based on heater surface area A, critical heat flux heat flux at the departure from nucleate boiling heat flux for fully developed nucleate boiling heat flux at the onset of nucleate boiling critical heat flux for pool boiling convective heat flux from the surface during film boiling radiative heat flux from the surface during film boiling total heat flux (convective and radiative) from the surface during film boiling minimum heat flux heat flux based on reference area AIcf heat flux within the first transition region as defined by Miyasaka et al. [19] for jet impingement boiling heat flux within the first transition region as defined by Miyasaka et al. [19] for pool boiling heat flux within the second transition region as defined by Miyasaka et al. [19] for jet impingement boiling heat flux within the second transition region as defined by Miyasaka et al. [19] for pool boiling universal gas constant critical value of the Reynolds number associated with the onset of boundary layer turbulence [u,x,/vf] Reynolds number [V , d/vf]

JET IMPINGEMENT BOILING Reynolds number [V, wJvr] Reynolds number [V, r/vr] radial coordinate on the impingement surface nozzle-to-nozzle spacing in the case of multiple jets absolute critical temperature liquid temperature initial surface temperature minimum surface temperature corresponding to &in

saturation temperature wall temperature wall temperature at the onset of nucleate boiling wetting temperature of the surface time velocity of the liquid velocity of the vapor inviscid, local, streamwise velocity of the liquid along the impingement surface specific volume of the liquid difference of specific volumes (us - or) specific volume of the vapor impingement velocityvelocity of the jet at the point of impingement nozzle velocity-velocity of the jet at the nozzle exit plate velocity dimensionless plate velocity (V,/u,) width of the jet width of the nozzle Weber number (pr V.' D/a] Weber number (pr V.' ( D -

WJI

Weber number (pr V.' L/a] Weber number (Pt v .' ( L - W")/OI streamwise coordinate along the impingement surface with origin at the stagnation point critical value of the streamwise coordinate x associated with the onset of

2

125

boundary layer turbulence nozzle-to-surface spacing

GREEK LEITERS mass fraction of the liquid from the impinging jet which is converted into droplets Y density ratio (pr/p,) AT,, temperature excursion AT,,, wall superheat (T, - T,,,) (AT,al)ONB wall superheat at the onset of nucleate boiling (Two,, - T,,,) subcooling (T,,, - T,) local liquid film thickness on the impingement surface critical liquid film thickness on the impingement surface mean liquid film thickness on the impingement surface correction factor for the effect of subcooling on CHF radius of active nucleation site maximum radius of active nucleation sites parameter used in corresponding states analysis by Kandula [86] (mgc9T,/"41'* subcooling coefficient rc, AT,"b/h,l empirical constant used by Monde and Okuma [47] correction factor for the effect of subcooling on CHF Helmholtz critical wavelength dynamic viscosity of the liquid dynamic viscosity of the vapor kinematic viscosity of the liquid density of the liquid density of the vapor surface tension thickness of the platinum foil in Miyasaka et al. [I91 a

D. H. WOLFET AL. Y Q

x

(hrg kf/8 D T a t u* h o d dimensionless critical heat flux C4&/P* h,, KI pressure and subcooling correction used by Miyasaka

$

0

et al. [19] empirical coefficient used by Mudawar and Wadsworth c71 Pitzer’s acentric factor

References 1. Incropera, F. P., ed. (1986). Research needs in electronic cooling. Proceedings of a Workshop Sponsored by the National Science Foundation and Purdue University, Andover, MA, June 4-6. 2. Incropera, F. P. (1988). Convection heat transfer in electronic equipment cooling J. Heat Transfer 110, 1097-1111. 3. Viskanta, R., and Incropera, F. P. (1992). Quenching with liquid jet impingement. In Hear and Mass Transfer in Materials Processing (I. Tanasawa and N. Lior, eds.), pp. 455-476. Hemisphere, New York. 4. Milne-Thomson, L. M. (1955). Theoretical Hydrodynamics, 3rd ed., pp. 279-289. MacmilIan, New York. 5. Gardon, R.,and Akfirat, J. C. (1965). The role of turbulence in determining the heattransfer characteristics of impinging jets. Int. J. Heat Mass Transfer 8, 1261-1272. 6. Miyazaki, H., and Silberman, E. (1972). Flow and heat transfer on a flat plate normal to a two-dimensional laminar jet issuing from a nozzle of finite height. Int. J. Heat Mass Transfer 15, 2097-2107. 7. Mudawar, I., and Wadsworth, D. C. (1991). Critical heat flux from a simulated chip to a confined rectangular impinging jet of dielectric liquid. Int. J. Heat Mass Transfer 34, 1465-1479. 8. Martin, H. (1977). Heat and mass transfer between impinging gas jets and solid surfaces. Adu. Heat Transfer (T. F. Irvine, Jr. and J. P. Hartnett, eds.), Vol. 13, 1-60, Academic Press, New York. 9. Downs, S. J., and James, E. H. (1987). Jet impingement heat transfer-a literature survey. ASME Paper No. 87-HT-35. 10. Polat, S., Huang, B., Mujumdar, A. S.,and Douglas, W. J. M. (1989). Numerical flow and heat transfer under impinging jets: a review. In Annual Reuiew of Numerical Fluid Mechanics and Heat Transfer (C. L. Tien and T. C. Chawla, eds.), Vol. 2, pp. 157-197. Hemisphere, New York. 11. Collier, J. G. (1981). Convective Boiling and Condensation, 2nd ed., p. 156. McGraw-Hill, New York. 12. Vader, D. T., Incropera, F. P., and Viskanta, R. (1992). Convective nucleate boiling on a heated surface cooled by an impinging, planar jet of water. J. Heat Transfer, 114, 152-160. 13. Cho, C. S. K., and Wu, K. (1988). Comparison of burnout characteristics in jet impingement cooling and spray cooling. In Proceedings of the 1988 National Heat Transfer Conference (H. R. Jacobs, ed.), HTD-96, Vol. 1, pp. 561-567. ASME, New York. 14. Katto, Y., and Kunihiro, M. (1973). Study of the mechanism of burn-out in boiling system of high bum-out heat flux. Bull. JSME 16, 1357-1366. 15. Katto, Y.,and Ishii, K. (1978). Burnout in a high heat flux boiling system with a forced supply of liquid through a plane jet. Proceedings of the 6th International Heat Transfer Conference, Vol. 1, FB-28, pp. 435-440. (Also published in Trans. JSME 44, 2817-2823, 1978.)

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16. Monde, M., and Katto, Y. (1978). Burnout in a high heat-flux boiling system with an impinging jet. Int. J. Heat Mass Transfer 21,295-305. (Similar results published in Trans. JSME 43, 3399-3407 and 3408-3416, 1977.) 17. Monde, M. (1980). Burnout heat flux in saturated forced convection boiling with an impinging jet. Heat Transfer-Japanese Res. 9 (l), 31-41. (Originally published in Trans. JSME 46B,1146-1 155, 1980.) 18. Ma, C. F., and Bergles, A. E. (1983). Boiling jet impingement cooling of simulated microelectronic chips. In Heat Transfer in Electronic Equipment-1983 ( S . Oktay and A. Bar-Cohen, eds.), HTD-Vol. 28, pp. 5-12. ASME, New York. 19. Miyasaka, Y., Inada, S., and Owase, Y. (1980). Critical heat flux and subcooled nucleate boiling in transient region between a two-dimensional water jet and a heated surface. J. Chem. Eng. Jpn. 13,29-35. 20. Kokado, J., Hatta, N., Takuda, H., Harada, J., and Yasuhira, N. (1984). An analysis of film boiling phenomena of subcooled water spreading radially on a hot steel plate. Arch. Eisenhiittenwes. 55, 113-118. 21. Rohsenow, W. M. (1985). Boiling. In Handbook of Heat Transfer Fundamentals, 2nd ed. (W. M. Rohsenow er a/.,eds.), Chapter 12. McGraw-Hill, New York. 22. Joudi, K. A., and James, D. D. (1981). Surface contamination, rejuvenation, and the reproducibility of results in nucleate pool boiling. J. Hear Transfer. 103, 453-458. 23. Fisenko, V. V., Baranenko, V. I., Belov, L. A., and Korenevskiy, V. A. (1988). Effect of dissolved gas on nucleate boiling and critical heat flux. Heat Transfer-Sov. Res. 20, 294-299. (Originally published in Kipeniye Kondensafsiya, Riga, pp. 23-29, 1985.) 24. Guglielmini, G., and Nannei, E. (1976). On the effect of heating wall thickness on pool boiling burnout. Inr. J. Heat Mass Transfer 19, 1073-1075. 25. Klimenko, V. V., and Snytin, S. Yu. (1990). Film boiling crisis on a submerged heating surface. Exp. Thermal Fluid Sci. 3,467-479. 26. Houchin, W. R., and Lienhard, J. H. (1966). Boiling burnout in low thermal capacity heaters. ASME Paper No. 66-WA/HT-40. 27. Bergles, A. E., and Thompson, W. G., Jr, (1970). The relationship of quench data to steadystate pool boiling data. Int. J . Heat Muss Transfer 13, 55-68. 28. Toda, S., and Uchida, H. (1973). Study of liquid film cooling with evaporation and boiling. Heat Transfer-Japanese Res. 2( I), 44-62. (Originally published in Trans. JSME 38, 1830-1837, 1972.) 29. Toda, S. (1971). Preprints of JSME, No. 713-5, 77 (in Japanese). 30. Shibayama, S., Katsuta, M., Suzuki, K., Kurose, T., and Hatano, Y. (1979). A study on boiling heat transfer in a thin liquid film (Part 1: In the case of pure water and an aqueous solution of a surface active-agent as the working liquid). Heat Transfer-Japanese Res. 8(2), 12-40. (Originally published in Trans. JSME 44,2429-2438, 1978.) 31. Hsu, Y. Y. (1962). On the size range of active nucleation cavities on a heating surface. J. Heat Transfer, 84, 207-216. 32. Jiji, L. M., and Clark, J. A. (1962). Incipient boiling in forced-convection channel flow. ASME Paper No. 62-WA-202. 33. Cole, R., and Rohsenow, W. M. (1969). Correlation of bubble departure diameters for boiling of saturated liquids. In Chemical Engineering Progress Symposium Series (W. R. Martini, ed.), Vol. 65, No. 92, pp. 211-213. AIChE, New York. 34. Miyasaka, Y., and Inada, S. (1980). The effect of pure forced convection on the boiling heat transfer between a two-dimensional subcooled waterjet and a heated surface, J. Chem. Eng. Jpn. 13,22-28. 35. Ma, C. F., and Bergles, A. E. (1986). Jet impingement nucleate boiling. Int. J. Heat Mass Transfer 29, 1095-1101.

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36. Vader, D. T., Incropera, F. P., and Viskanta, R. (1991). A method for measuring steady local heat transfer to an impinging liquid jet. Exp. Thermal Fluid Sci. 4, 1-11. 37. Nonn, T., Dagan, Z., and Jiji, L. M. (1989). Jet impingement flow boiling of a mixture of FC-72 and FC-87 liquids on a simulated electronic chip. In Proceedings of the 1989 National Heat Transfer Conference-Heat Transfer in Electronics, HTD-Vol. 111, pp. 121-128. ASME, New York. 38. Wadsworth, D. C. (1990). Single and two-phase cooling of a multichip electronic module by means of confined two-dimensional jets of dielectric liquid. MSME Thesis, Purdue University, West Lafayette, IN. 39. Struble, C. L., and Witte, L. C. (1991). Emitter-base voltage measurement technique for jet nucleate boiling on power transistors in dielectric liquids In Proceedings of the ASME-JSME Thermal Engineering Joint Conference (J. R. Lloyd and Y. Kurosaki, eds.), Vol. 2, pp. 405-412. 40. Copeland, R. J. (1970). Boiling heat transfer to a water jet impinging on a flat surface (- lg). Ph.D. Thesis, Southern Methodist University, Dallas, TX. 41. Ruch, M. A,, and Holman, J. P. (1975). Boiling heat transfer to a Freon-113 jet impinging upward onto a flat, heated surface. Int. J. Heat Mass Transfer 18, 51-60. 42. Rohsenow, W. M. (1952). A method of correlating heat-transfer data for surface boiling of liquids. Trans. ASME 74, 969-976. 43. Danielson, R. D., Tousignant, L., and Bar-Cohen, A. (1987). Saturated pool boiling characteristics of commercially available perfluorinated inert liquids. In Proceedings of the 1987 ASME-JSME Thermal Engineering Joint Conference (P. J. Marto and I. Tanasawa, eds.), Vol. 3, pp. 419-430. 44. Katto, Y., and Monde, M. (1974). Study of mechanism of burn-out in a high heat-flux boiling system with an impinging jet. Proc. Sth Int. Heat Transfer Conf IV, B6.2,245-249. (Also published in Trans. JSME 41, 306-314, 1975.) 45. Monde, M., and Furukawa, Y. (1988). Critical heat flux in saturated forced convective boiling with an impinging jet coexistence of pool and forced convective boilings. Heat Transfer-Japanese Res. 17(5), 81-91. (Originally published in Trans. JSME 53B,199-203, 1987.) 46. Nonn, T., Dagan, Z., and Jiji, L. M. (1988). Boilingjet impingement cooling of simulated microelectronic heat sources. ASME Paper No. 88-WA/EEP-3. 47. Monde, M., and Okuma, Y. (1985). Critical heat flux in saturated forced convective boiling on a heated disk with an impinging jet-CHF in L-regime. Int. J. Heat Mass Transfer 28, 547-552. 48. Katsuta, M., and Kurose, T. (1981). A study on boiling heat transfer in thin liquid film (2nd report, the critical heat flux of nucleate boiling). Trans. JSME 47B, 1849-1860 (in Japanese). 49. Ma, C. F., Yu, J., Lei, D. H., Gan, Y. P., Auracher, H., and Tsou, F. K. (1989). Jet impingement transient boiling heat transfer on hot surfaces. In Second International Symposium on Multiphase Flow and Heat Transfer (X.-J. Chen et al., eds.), Vol. 1, pp. 349-357. Hemisphere, New York. 50. Sherman, B. A., and Schwartz, S. H. (1991). Jet Impingement boiling using a JT cryostat. In Proceedings of the 1991 National Heat Transfer Conference-Cryogenic Heat Transfer (A. Adorjan and A. Bejan, eds.), HTD-Vol. 167, pp. 11-17. ASME, New York. 51. Monde, M., and Katto, Y. (1977). Study of burn-out in a high heat-flux boiling system with an impingingjet (Part 2, generalized nondimensional correlation for the burn-out heat flux. Trans. JSME 43,3408-3416 (in Japanese). 52. Carvalho, R. D. M., and Bergles, A. E. (1990). The influence of subcooling on the pool nucleate boiling and critical heat flux of simulated electronic chips. Proceedings of the 9th International Heat Transfer Conference (G. Hetsroni, ed.), Vol. 2, pp. 289-294.

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53. Ishigai, S., Nakanishi, S., and Ochi, T. (1978). Boiling heat transfer for a plane water jet impinging on a hot surface. Proceedings of the 6th International Heat Transfer Conference, Vol. 1 , FB-30, pp. 445-450. [Similar findings published by Nakanishi, S., Ishigai, S., Ochi, T., and Morita, I. (1980). Cooling of a hot surface by a plane water jet. Trans. JSME 46B, 714-724 (in Japanese).] 54. Rohsenow, W. M. (1952). Heat transfer with evaporation. In Heat Transfer, a Symposium Held at the University of Michigan During the Summer of 1952 (B. A. Uhlendorf and W. W. Hagerty, eds.), pp. 101 -149. Engineering Research Institute, University of Michigan. 55. Inada, S. (1991). Private communication, August 22. 56. Ma, C. F. (1991). Private communication, June 14. 57. Monde, M., and Katto, Y. (1977). Study of burn-out in a high heat-flux boiling system with an impinging jet (Part 1, behavior of the vapor-liquid flow). Trans. JSME 43, 3399-3407 (in Japanese). 58. Kamata, T., Kumagai, S., and Takeyama, T. (1988). Boiling heat transfer to an impinging jet spurted into a narrow space (Part I, space with an open end). Heat transfer-Japanese Res. 17(5), 71-80. (Originally published in Trans. JSME 53B, 183-187, 1987.) 59. Kamata, T., Kumagai, S., and Takeyama, T. (1988). Boiling heat transfer to an impinging jet spurted into a narrow space (Part 11, space with a limited end). Heat Transfer-Japanese Res. 17(4), 1 - 1 1 . (Originally published in Trans. JSME 53B,188-192, 1987.) 60. Goodling, J. S., Jaeger, R. C., Williamson, N. V., Ellis, C. D., and Slagh, T. D. (1987). Wafer scale cooling using jet impingement boiling heat transfer. ASME Paper No. 87-WAIEEP-3. 61. Aihara, T., Kim, J.-K., Suzuki, K., and Kasahara, K. (1993). Boiling heat transfer of a micro-impinging jet of liquid nitrogen in a very slender cryoprobe. Znf.J. Heat Mass Trans. 36, 169-175. [Originally published in Trans. JSME 57B,2112-21 17, 1991 (in Japanese).] 62. Ishimaru, M., Kim, J.-K., Aihara, T., and Shimoyama, T. (1991). Boiling heat transfer characteristics due to a micro-impingingjet of LN,. Proceedings of the 28th National Heat Transfer Symposium of Japan, pp, 730-732 (in Japanese). 63. Duke, E. E., and Schrock, V. E. (1961). Void volume, site density and bubble size for subcooled nucleate pool boiling. In Proceedings of the I961 Heat Transfer and Fluid Mechanics Institute (R. C. Binder et al., eds.), pp. 130-145. Stanford University Press, Stanford. 64. Monde, M., Kusuda, H., and Uehara, H. (1980). Burnout heat flux in saturated forced convection boiling with two or more impinging jets. Heat Transfer-Japanese Res. 9(3), 18-31. (Originally published in Trans. JSME 46B, 1834-1843, 1980.) 65. Sakhuja, R. K., Lazgin, F. S., and Oven, M. J. (1980). Boiling heat transfer with arrays of impinging jets. ASME Paper No. 80-HT-47. 66. Sano, Y., Kubo, R., Kamata, T., and Kumagai, S. (1991). Boiling heat transfer to an impinging jet in cooling a hot metal slab. Proceedings of the 28th National Heat Transfer Symposium of Japan, pp. 733-735 (in Japanese). 67. Bar-Cohen, A., and Simon, T. W. (1988). Wall superheat excursions in the boiling incipience of dielectric fluids. Heat Transfer Eng. 9(3), 19-31. 68. Taga, M., Ochi, T., and Akagawa, K. (1983). Cooling of a hot moving plate by an impinging water jet. Proceedings of the ASME-JSME Thermal Engineering Joint Conference (Y. Mori and W.-J. Yang, eds.), Vol. 1, pp. 183-189. 69. 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. Thermal Fluid Sci. 3,202-213. 70. Chen, S.-J., and Kothari, J. (1988). Temperature distribution and heat transfer of a moving metal strip cooled by a water jet. ASME Paper No. 88-WA/NE-4. 71. Chen, S.-J. Kothari, J., and Tseng, A. A. (1991). Cooling of a moving plate with an impinging circular water jet. Exp. Thermal Fluid Sci. 4, 343-353.

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72. 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. 73. Wolf, D. H., Viskanta, R., and Incropera, F. P. (1990). Local convective heat transfer from a heated surface to a planar jet of water with a nonuniform velocity profile. J. Heat Transfer 112,899-905. 74. McGillis, W. R., and Carey, V. P. (1990). Immersion cooling of an array of heat dissipating elements-an assessment of different flow boiling methodologies. In Cryogenic and Immersion Cooling of Optics and Elecironic Equipment (T. W. Simon and S. Oktay, eds.), HTD-Vol. 131, pp. 37-44. ASME, New York. 75. Strom, B. D., Carey, V. P., and McGillis, W. R. (1989). An experimental investigation of the

critical heat flux conditions for subcooled convective boiling from an array of simulated microelectronic devices. In Proceedings of ihe 1989 National Heat Transfer ConferenceHeat Transfer in Electronics, HTD-Vol. 111, pp. 135-142. ASME, New York. 76. Katto, Y., and Shimizu, M. (1979). Upper limit of CHF in the saturated forced convection boiling on a heated disk with a small impinging jet. J. Heat Transfer 101, 265-269. 77. Monde, M., Kusuda, H., and Nagae, 0. (1982). Critical heat flux of saturated forced convective boiling with an impinging jet (in the high pressure region). Proceedings of ihe 19th National Heat Transfer Symposium of Japan, pp. 496-498 (in Japanese). 78. Monde, M. (1987). Critical heat flux in saturated forced convection boiling on a heated disk with an impinging jet. J. Heat Transfer, 109, 991-996, and Monde, M., Nagae, 0..and Ishibashi, Y. (1987). Critical heat flux in saturated forced convective boiling on a heated disk with an impinging jet. Heat Transfer-Japanese Res. 16(5), 70-82. (Originally published in Trans. JSME SZB, 1799-1804, 1986.) 79. Katsuta, M. (1977). Boiling heat transfer of liquid fdm (6th report, test fluid is Freon R113). Proceedings of the 14th National Heat Transfer Symposium of Japan, pp. 154-156 (in Japanese). (Majority of this work was included in a later publication by Katsuta and Kurose [48].) 80. Monde, M. (1985). Critical heat flux in saturated forced convective boiling on a heated disk with an impinging jet, a new generalized correlation. Warme Stoffierirag. 19, 205-209. (Similar findings published in Trans. JSME SOB, 1392-1396, 1984.) 81. Haramura, Y.,and Katto, Y. (1983). A new hydrodynamic model of critical heat flux, applicable widely to both pool and forced convection boiling on submerged bodies in saturated liquids. Int. J. Hear Mass Transfer 26, 389-399. 82. Katsuta, M. (1978). Boiling heat transfer of liquid film (7th report, critical heat flux of impinging jet of liquid with surfactant). Preprint of JSME, No. 780-18, pp. 68-70 (in Japanese). (Majority of this work was included in a later publication by Katsuta and Kurose [48].) 83. Ishigai, S., and Mizuno, M. (1974). Boiling heat transfer with an impinging water jet (about the critical heat flux). Preprint of JSME, No. 740-16, pp. 139-142. 84. Lee, T. Y., Simon, T. W., and Bar-Cohen, A. (1988). An investigation of short-heating-length effects on flow boiling critical heat flux in a subcooled turbulent flow. In Cooling Technologyfor Electronic Equipmeni (W. Aung, ed.), pp. 435-450. Hemisphere, New York. 85. Mudawar, I., and Anderson, T. M. (1989). High flux electronic cooling by means of pool boiling-Part I: Parametric investigation of the effects of coolant variation, pressurization, subcooling, and surface augmentation. In Proceedings of the 1989 National Heat Transfer Conference-Heat Transfer in Electronics, HTD-Vol. 111, pp. 25-34. ASME, New York. 86. Kandula, M. (1990). Mechanisms and predictions of burnout in flow boiling over heated surfaces with an impinging jet. rnt. J. Hear Mass Transfer 33, 1795-1803. 87. Reid, R. C., Prausnitz, J. M., and Poling, B. E. (1987). The Properties of Gases and Liquids 4th ed., pp. 23-24, 34. McCraw-Hill, New York. 88. Lienhard, J. H., and Eichhorn, R. (1976). Peak boiling heat flux on cylinders in a cross flow.

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Int. J. Heat Mass Transfer 19, 1135-1142. 89. Lienhard, J. H., and Eichhorn, R. (1979). On predicting boiling burnout for heaters cooled by liquid jets. Int. J. Hear Mass Transfer 22, 774-716. 90. Lienhard, J. H., and Hasan, M. Z. (1979). Correlation of burnout data for disk heaters cooled by liquid jets. J. Hear Transfer 101, 383-384. 91. Sharan, A., and Lienhard, J. H. (1985). On predicting burnout in the jet-disk configuration. J. Heat Transfer 107, 398-401. 92. Katto, Y., and Yokoya, S. (1988). Critical heat flux on a disk heater cooled by a circular jet of saturated liquid impinging at the center. Inr. J. Heat Mass Transfer 31, 219-227. 93. Ochi, T., Nakanishi, S., Kaji, M., and Ishigai, S. (1984). Cooling of a hot plate with an impinging circular water jet. In Multi-Phase Flow and Heat Transfer III. Part A: Fundamenrals (T. N. Veziroglu and A. E. Bergles, eds.), pp. 671-681. Elsevier, Amsterdam. 94. Matsumura, S., Kumagaya, T., and Takeyama, T. (1979). Coolings of a high temperature object by an impinging jet of water. Proceedings of the 16rh National Hear Transfer Symposium of Japan, pp. 322-324 (in Japanese). 95. Miyasaka, Y. (1991). Private communications, June 28 and July 30. 96. Kutateladze, S. S. (1952). Heat Transfer in Condensation and Boiling. U.S. AEC Report AEC-tr-3770. 97. Zuber, N., Tribus, M., and Westwater, J. W. (1961). The hydrodynamic crisis in pool boiling of saturated and subcooled liquids. International Developments in Heat Transfer, Part 11, pp. 230-236. 98. Kutateladze, S. S., and Schneiderman, L. L. (1953). Experimental study of the influence of the temperature of a liquid on the change of the rate of boiling. Problems of Heat Transfer During a Change of State. U.S. AEC Report AEC-tr-3405. 99. Skima, R. K., and Slantiauskas, A. A. (1990). Critical heat fluxes at jet-cooled flat surfaces. In Hear Transfer in Electronic and Microelectronic Equipment (A. E. Bergles, ed.), pp. 621-626. Hemisphere, New York. 100. Andrews, D. G., and Rao, P. K. M. (1974). Peak heat fluxes on thin horizontal ribbons in submerged water jets. Can. J. Chem. Eng. 52, 323-330. 101. Grimley, T. A., Mudawar, I., and Incropera, F. P. (1988). CHF enhancement in flowing fluorocarbon liquid films using structured surfaces and flow deflectors. Int. J. Heat Mass Transfer, 31, 55-65. 102. Mudawar, I., Incropera, T. A., and Incropera, F. P. (1987). Boiling heat transfer and critical heat flux in liquid films falling on vertically-mounted heat sources. Int. J. Heat Mass Transfer 30,2083-2095. 103. Nukiyama, S., and Tanasawa, Y. (1939). An experiment on the atomization of liquid, (4th report, the effect of the properties of liquid on the size of drops). Trans. Soc. Mech. Eng. (Jpn.) 5, 68-75 (in Japanese). 104. Monde, M., and Inoue, T. (1991). Critical heat flux in saturated forced convective boiling on a heated disk with multiple impinging jets. J. Heat Transfer 113, 722-727. 105. McMurray, D. C., Myers, P. S., and Uyehara, 0. A. (1966). Influence of impinging jet variables on local heat transfer coefficients along a flat surface with constant heat flux. Proceedings of the 3rd International Heat Transfer Conference, Vol. 11, pp. 292-299. 106. Katto, Y., and Kurata, C. (1980). Critical heat flux of saturated convective boiling on uniformly heated plates in a parallel flow. Inr. J. Multiphase Flow 6, 575-582. 107. Katto, Y., and Haramura, Y. (1981). Effect of velocity (Weber number) on CHF for boiling on heated plates cooled by a plane jet. Proceedings of the 18th National Heat Transfer Symposium of Japan, pp. 382-384 (in Japanese). 108. Baines, R. P., El Masri, M. A., and Rohsenow, W. M. (1984). Critical heat flux in flowing liquid films. Int. J. Hear Mass Transfer 27, 1623-1629. 109. Wadsworth, D. C., and Mudawar, I. (1992). Enhancement of single-phase heat transfer and

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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. 110. Piggott, B. D. G., White, E. P., and Duffey, R. B. (1976). Wetting delay due to film and transition boiling on hot surfaces. Nucl. Eng. Des. 36, 169-181. 111. Owen, R. G., and Pulling, D. J. (1979). Wetting delay: film boiling of water jets impinging hot flat metal surfaces. In Multiphase Transport: Fundamentals, Reactor Safety, Applications (T. N. Veziroklu, ed.), Vol. 2, pp. 639-669. Hemisphere, Washington, DC. 112. Akimenko, A. D. (1966). Features of film boiling in surface water cooling. NASA Report TT F-10184, N66-33689. (Translated from Znz. Fiz. Zh. 7(6), 32-34, 1964.) 113. Ishigai, S., Nakanishi, S.,Mizuno, M., and Sone, M. (1971). Heat transfer ofwater on a high temperature surface (2nd report). Proceedings of the 8th National Heat Transfer Symposium of Japan, pp. 145-148 (in Japanese). 114. Hanasaki, K., Kokado, J., and Hatta, N. (1981). Numerical method for violent change of temperature and application to a simple model. Tetsu-to-HaganC 67, 1972-1980 (in Japanese). 115. Hatta, N., Kokado, J., Hanasaki, K., Takuda, H., and Nakazawa, M. (1982). Effect of water flow rate on cooling capacity of laminar flow for hot steel plate. Tersu-to-Hagank 68, 974-981 (in Japanese). 116. Hatta, N., Kokado J., and Hanasaki, K. (1983). Numerical analysis of cooling characteristics for water bar. Trans. Iron Steel Insr. Jpn. 23,555-564. (Originally published in Tetsu-toHagant! 67,959-968, 1981.) 117. Hatta, N., Kokado, J., Takuda, H., Harada, J., and Hiraku, K. (1984). Predictable modelling for cooling process of a hot steel plate by a laminar water bar. Arch. Eisenhiittenwes. 55, 143-148. 118. Hatta, N., Tanaka, Y., Takuda, H., and Kokado, J. (1989). A numerical study on cooling process of hot steel plates by a water curtain. ISIJ Int. 29, 673-679. 119. Hatta, N., and Osakabe, H. (1989). Numerical modeling for cooling process of a moving hot plate by a laminar water curtain. ISlJ Int. 29, 919-925. 120. Jeschar, R., Reiners, U., and Scholz, R. (1986). Heat transfer during water and water-air spray cooling in the secondary cooling zone of continuous casting plants. 5th International Iron and Steel Congress-Proceedings of the 69th Steelmaking Conference, Vol. 69, pp. 511-521. Iron and Steel Society, Warrendale, PA. 121. Zumbrunnen, D. A., Viskanta, R., and Incropera, F. P. (1989). The effect of surface motion on forced convection film boiling heat transfer. J. Heat Transfer 111, 760-766. 122. Nakanishi, S., Ishigai, S., Ochi, T., and Morita, I. (1980). Film boiling heat transfer of impinging plane water jet. Trans. JSME 468,955-961 (in Japanese). 123. Burmeister, L. C. (1983). Convective Heat Transfer, pp. 297-312. Wiley, New York. 124, Bromley, L. A. (1950), Heat transfer in stable film boiling. Chem. Eng. Prog. 46, 221-227. 125. Lamvik, M., and Iden, B.-A. (1982). Heat transfer coefficient by water jets impinging on a hot surface. Proceedings of the 7th International Heat Transfer Conference (U. Grigull et al., eds.), Vol. 3, FC64, pp. 369-375. 126. Nevins, R. G., Jr. (1953). The cooling power of an impinging jet. Ph.D. Thesis, University of Illinois, Urbana. 127. Bar-Cohen, A. (199 1). Thermal management of electronic components with dielectric liquids. Proceedings of the ASME-JSME Thermal Engineering Joint Conference (J. R. Lloyd and Y. Kurosaki, eds.), Vol. 2, pp. xv-xxxix. 128. Filipovic, J., Viskanta, R., Incropera, F. P., and Veslocki, T. A. (1991). Thermal behavior of a moving steel strip cooled by an array of planar water jets. In Proceedings of the 1991 National Heat Transfer Conference-Heat Transfer in Metals and Coniainerless Processing and Manufacturing (T. L. Bergman et al., eds.), HTD-Vol. 162, pp. 13-23. ASME, New York.

ADVANCES IN HEAT TRANSFER. VOLUME 23

Radiative Heat Transfer in Porous Media*

M. KAVIANY A N D B. P. SINGH Deparrment of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, Michigan 48109

I. Introduction

Radiative heat transfer in porous media can be a significant mode of heat transfer in many applications, ranging from low-temperature insulation to combustion. A porous medium is defined as a heterogeneous system made up of a solid matrix with its voids filled with fluids. For radiative heat transfer, it serves as an absorbing, scattering, and emitting medium in which radiative heat transfer is often coupled with other modes of heat transfer. The porosity of the medium can be very low, as in consolidated particle systems, to intermediate, as in closely packed spheres ( E = 0.26) to greater than 0.98 for some foam-type structures. The solution to the radiative heat transfer problem can be obtained either from a discrete model which considers the system as being made up of individual particles or by following a single continuum treatment and solving the equation of radiative transfer. This continuum versus noncontinuum treatment is discussed here. Here the focus is on the physical aspects of radiative heat transfer in porous media, and therefore the solution for multidimensional systems is not addressed. A porous medium may be considered as a collection of elements or particles. The shape and size of these particles may be obvious (as in a packed bed of spheres), or some approximations may have to be made to break the structure down into a collection of particles. Here different sizes of particles and the application of the Raleigh, Mie, and geometric scattering theories are

* This review is in parts based on a treatment given by M. Kaviany in Principles o f f f e a t Transfer in Porous Media, Springer-Verlag, New York 1991. 133

Copyright 01993 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-020023-6

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examined. Since a large number of porous media applications lie in the largeparticle range, these systems are discussed in a greater detail. The question of dependent versus independent scattering (or absorption) is also taken up here in detail, Independent scattering (absorption) is said to exist when the interaction of a particle with the incident radiation is not influenced by the presence of its neighboring particles. Limits on the validity of the independent scattering are shown to be a minimum value of porosity and a minimum value of the ratio C/A, where C is the average interparticle spacing based on rhombohedra1 packing [ C / d = 0.9047/(1 - E)~’’ - 11. If both of these conditions are satisfied, then the bulk (away from the bounding surfaces) behavior of the bed can be predicted, from the equation of radiative transfer, by the theory of independent scattering. Direct simulation methods are discussed here as a technique for solving the class of problems that cannot be solved by continuum methods and to establish the range of validity of independent scattering and develop correlations to model dependent scattering. The ray-tracing Monte Carlo method is used to examine the thermal radiative transfer through packed beds of large (geometric range) particles. Opaque, semitransparent, and emitting particles are considered. A technique for continuum modeling of dependent radiative heat transfer in beds of large (geometric range) spherical particles is developed. It is shown that the dependent properties for a bed of opaque spheres can be obtained from their independent properties by scaling the optical thickness while leaving the albedo and the phase function unchanged. The scaling factor is found to depend mainly on the porosity and is almost independent of the emissivity. We show that such a simple scaling is not readily applicable to nonopaque particles. The transparent and semitransparent particles are treated by allowing for the ray displacement across a finite optical thickness (because of the transmission through the particle) while solving the equation of radiative transfer. When combined with the scaling approach, this results in a powerful method of solution that we call the dependence included discrete ordinates method (DIDOM). A novel method for modeling the effect of solid conductivity on radiative heat transfer is also presented. This method combines the Monte Carlo method of solving the radiation problem with a finite-difference solution for the temperature distribution in a spherical particle, to model the effect of solid conductivity on radiative heat transfer. 11. Continuum Treatment

For simplicity,the treatment is restricted to a one-dimensionalplane-parallel geometry. Azimuthal symmetry is assumed so that ZA(8,4) = ZA(0). The

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one-dimensional, steady-state, equation of radiative transfer for radiation in a direction p, in an absorbing, emitting and scattering continuum is

where I, is the spectral intensity, S is the distance traveled, (aAa)and (a,,) are the spectral absorbing and scattering coefficients, I,, is the blackbody emission, and (@,(pi, p ) ) is the phase function for scattering from a direction p i to a direction p ( p = cos 0) where 8 is the azimuthal angle in the plane-parallel geometry assumed (Siege1 and Howell [1)). The phase function, as written, is the integral of the phase function from a direction (Oi, Cpi) to (0, Cp) and accounts for scattering contributions for different values of Cp and Cpi. The phase function is given by

where the phase function ( Q A ) (0,) is defined as the ratio of the actual scattered radiation intensity in the direction 8,, to the intensity that would be scattered if the scattering were isotropic. Equation (1) is derived by making an optical energy balance on a representative elementary volume and then letting the volume go to zero in the limit. Implicit are the assumptions that an elementary volume can be found that contains enough particles to be representative of the porous medium and at the same time is much smaller than the overall dimensions of the system and a volume across which the intensity does not vary greatly. Therefore, the continuum treatment fails when the system contains only a few particles so that the particle size is comparable to the linear dimension of system and we have to resort to a direct simulation. Also, the foregoing equation is not strictly true if the variation in intensity across a representative elementary volume is large (as is often the case with closely packed particles). However, this difficulty can sometimes be overcome by making suitable adjustment to the optical properties of the medium. Equation (1) can be written for an absorbing, emitting, and scattering medium. In a nonparticipating medium, (a,,) and (a,,) are both zero. Other simplifications include a pure scattering medium ((a,,) = 0), a nonscattering medium ((aIs) = 0), a cold medium (ZAb= 0), and isotropic scattering (@,(pi, p)) = 1. In a typical porous medium, no such simplifications can be made and Eq. (1) has to be solved in its most general form.

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The radiative properties of the medium, i.e., (cia), (a,,}, and (@,(pi, p ) ) , are in general very difficult to obtain. They can be either measured experimentally or calculated from the theory of independent scattering or from some dependent scattering model. This is discussed in detail in Sections IV and V. 111. Solution Methods for Equation of Radiative Transfer

The approximate solution methods are reviewed by Davison [2], Sparrow and Cess [3], Siege1 and Howell [l], and Ozisik [4]. The integration of the equation of radiative transfer is made difficult when this integrodifferential equation includes the following effects. Emission is significant and coupling of this equation with the energy equation is required. Both absorption and scattering are significant. Scattering is highly anisotropic. The coefficients are highly wavelength dependent. Radiation in more than one dimension must be considered. Boundary conditions include emission, reflection (diffuse and specular), and transmission. The approximate solution to the equation of radiative transfer has been (and continues to be) attempted using various mathematical techniques. Here we will briefly review the two-flux method and the method of discrete ordinates. For the purpose of this section, it is assumed that the radiative properties of the medium are known and only the mathematical solution to the equation of radiative transfer is discussed. A. TWO-FLUX APPROXIMATIONS, QUASI-ISOTROPIC SCATTERING

The two-flux approximation (or Schuster-Schwarzchild approximation) for plane-parallel geometry has been discussed by Chandrasekhar [S], Ozisik [4], Vortmeyer [S], and Brewster and Tien [7], among others. The principle is the division of the radiation field into forward I: and backward In components. The two-flux method is based on the assumption of hemispherical isotropy and fails to give good results whenever this assumption is violated. This results in the case of a highly anisotropic phase function as noted by Brewster and Tien [7] and by Menguc and Viskanta [S]. Singh and Kaviany [9] note that hemispherical anisotropy is destroyed in a nonemitting but absorbing bed and show that under these conditions, the two-flux

RADIATIVEHEATTRANSFER IN

POROUS

MEDIA

137

model fails even for an isotropic phase function. Under the two-flux approximation, the radiative transfer equation can be integrated over the forward direction to give dl: dx ~

=

-(2(aAs)B

+ 2(aAex))IT + 2(aAa)aAb + 2(a,s)BI;

(3)

where B is the backscattering fraction given by 1 f l YO

B = i JJ-lo (Oa)(Oi

--*

0) d cos Oi d cos 0

(4)

1. Nonemitting Medium

The transmittance for a nonemitting bed is found by applying the two-flux approximation to the equation of radiative transfer. Chen and Churchill [lo] use dl: __ = -(@& + CFAa)IT+ @& (7) dx dl; dx -

--(@As

+ @Aa)IY+ @SIT

(8)

with the boundary conditions 1:

= Iai

(9)

at x = 0

I , = 0 (infinite radiation absorption) at x = L. The solution for the transmittance T, is I ' = cosh(@,2,+ 2@Aa@As)1/2L Ti' = 2 + Id x sinh(@:a

@as

+ @La

+ 2@Aa@As)1/2

+ 2@Aa@As)1/2L

and the reflectance R, is given by -

R , = T,

(53,

+

6.4s

sinh(@:,

2@As@Aa)1'2

+ 2@,a@,s)1/2L

(10)

M. KAVIANY AND B. P. SINGH

138

2. Emitting Medium When the medium is emitting, Tong and Tien [l 13 formulate the problem as given here. Under radiation and local thermal equilibrium (no other mode of heat transfer, i.e., evacuated with k, = 0) assumptions, we have Vq,, = 0 or

(

1

471(0Aa)z,b = 2°C

(0,Ia)IA d p = 2n(gAa)(z:

-1

or

Ilb = ;(IT

+ I,)

+ I,)

(13) (14)

The two-flux approximation with this replacement leads to

Defining and we have

dZ: = dz

~

--P(Zt

- I,)

The boundary conditions are (1 is for the lower surface and 2 is for the upper surface) I:(o) = &AlIAbl+ ( l I,(L) = EAZIAbZ

- E1l)I;(0)

+ ( l - EAZ)l:(L)

The solution to these is given as

The net radiative spectral heat Jlux is given by

+ 27c j-Z i p dp 0

1

= 271

I : p dp

1

(21) (22)

RADIATIVE HEATTRANSFER IN POROUS MEDIA

139

and for a quasi-isotropic scattering, using the definition of I: and I;, we have 4Ar = 274iz:

- $1;) = n(1: - I ; )

which gives

B. DISCRETE-ORDINATES (S-N) APPROXIMATION Following the approximation of Schuster-Schwarzchild in dividing the radiation field into an inward and,an outward stream (two-flux approximation), Chandrasekhar [5] increased the number of these discrete streams or discrete ordinates. The integral in the equation of radiative transfer is approximated by division into increments; e.g., the integrals - 1 and + 1 for p are divided into 2N increments. Since the process of determining the area (integration) is called quadrature, the various numerical approximations of the integrals have been referred to as quadratures, e.g., Gaussian quadrature. The result of the approximation is to reduce the integrodifferentialequation to a set of coupled ordinary linear differential equations, which is then either reduced to algebraic equations by direct integration (e.g., Hottel et al. [12], Rish and Roux [13]) or solved numerically (e.g., Carlson and Lathrop [14], Truelove [lS], Fiveland [16], Jamaluddin and Smith [17], Kumar et al. [18]). The in-scattering term (the integral) is approximated by a quadrature, where p i represents the quadrature points between - 1 and + 1 corresponding to a 2N-order quadrature and Api (solid angle increment) is the corresponding quadrature weight. Then the one-dimensional radiative transfer equation for intensity at x and in the direction p i becomes

fori

= -M,-M+l,...,

M,i#O,

M

1 j=-M,j#O

where

Apj = 2

(27)

M. KAVIANY AND B. P. SINGH

140

The boundary conditions are

at x = L,

I l j = &AIlb

+ pAslA-i M

+2p,d

C1 ApjI~jpj,

i = - 1,

* . a

,-M

(31)

j=

where I,(x) = I(x, p i ) and i = 0 (corresponding to the lateral boundaries) has been avoided because of the one-dimensional geometry assumed. Equation (27) is called the discrete-ordinates equation. 3. Solution of Discrete-Ordinate Equations The finite-difference solution of the discrete-ordinate equations has been discussed by Carlson and Lathrop [14] and Fiveland [16]. The general scheme consists of evaluating the intensity at the cell center by relating it to the intensities at the cell faces. The source term comprising in-scattering from other directions and the emission is calculated using this intensity. For one-dimensional (with no emission) radiation, the finite-difference approximation of the equation of radiative transfer leads to the following equation: l;j+l=

PiIAX + ( o A e x > P In cases in which emission is important, the condition of radiative equilibrium gives (Kumar et al. [18])

The difference equation becomes 1;. j + 1

=

M

1

RADIATIVEHEATTRANSFER IN POROUS MEDIA

141

Iteration begins from x = 0 ( j = 1) for pi > 0 and proceeds in the direction of actual irradiation. For pi < 0, iteration starts from x = L ( j = n) and proceeds toward x = 0.

IV. Properties of a Single Particle

For incidence of a planar radiation on a spherical particle, the spectral power (W/pm) arriving at the sphere is nR21Ai.The fraction that is scattered can be found by integrating the local scattered spectral intensity I,, over a sphere with a radius larger than the particle radius (since the intensity decays as r P 2 , the location r is irrelevant), that is, [4n I , j 2 dn. Then a spectral scattering efficiency qASis defined as I d 2 dQ nR21,,

[4n

‘Is=

(35)

The spectral scattering cross section is defined as A,, = VAsnR2

(36)

Similarly, the spectral absorption efficiency and cross section are defined as

(38) Measurement of I,, is difficult, but it can be predicted through analysis such as the Mie theory. Finally, the spectral extinction eficiency and spectral extinction cross section are defined as Aia = VianR2

The scattering-absorption of incident beams by a long circular cylinder has also been studied by van de Hulst [19]. He also considers other particle shapes. For small particles, a simplified approach to modeling the spectral scattering and absorption coefficient is given by Mengiic and Viskanta [20].

A. COMPARISON OF

PREDICTIONS

We expect the Mie theory to be applicable for all values of n, K , and aR. The Rayleigh theory is applicable for small aR and small values of )maR\,m is the complex refractive index, and the geometric treatment is expected to be valid

142

M. KAVIANY AND B. P. SINGH

for ctB %= 1. Here, we consider spherical particles only. Because of faster computers and improved subroutines, carrying out a full Mie solution is no longer limited by computer time. The problem lies more in making practical use of it, because no method of solution can handle the sharp forward peak produced for large particles. Thus this peak has to be truncated for geometric-sue particles and the phase function renormalized to ensure energy conservation. The computation involved increases with increasing aR. Therefore, for very large values of aR, the theory of geometric scattering provides a convenient alternative. For small particles, the Rayleigh theory can be used. Its main advantage is q&), and that it provides a closed-form solution. Here, we compute q&), @,(A) for a 0.2-mm sphere using the available experimental results for n,(A) and .,(A) for glass and iron (carbon steel).The computations are based on the Rayleigh, Mie, and geometric treatments. Then comparisons are made among the results of these three theories, and the limits of applicability of the Rayleigh and geometric treatments for these examples are discussed. The results of single particle scattering for 0.2-mm glass and steel spheres are shown in Fig. la-c and Fig. 2a-c, respectively. The optical properties of glass and iron are taken from Hsieh and Su [21] and Weast [22]. The data

628

100 50

'

(4

I

'

Glass Sphere,

2.0 1.0

20 10 5.0 1

'

n, , K, are constant p

I

0.5 10.314 I '

d = 0.2 rnrn

2.0-*-"..\ 1.6

-

\

%e

1.20.8 0.4. I

8

1

2

5 1 0

102

.

103

z

:lo3

h, wn FIG. 1. (a) Variation of the spectral scattering efficiency with respect to wavelength for a glass spherical particle of diameter 0.2 mm. When appropriate, the Rayleigh, Mie, and geometrical treatments are shown. Also shown is the Penndorf extension. (b) Same as (a), except for the variation of the spectral absorption efficiency. (c) Same as (a), except for the distribution of the phase function.

RADIATIVEHEATTRANSFER IN POROUS MEDIA 0.5

aR 100 50

628

I

2.01

20 10 5.0

'

I

2.0 1.0

'

)[

I

'

Glass Sphere, d = 0.2mm

t

1.6-

I 1

are

I/

I

r4

n,, constant T a t values for I h=206.6um 1 /-e\ '

Mil Mie

y

1.2-

143

/

I

I I

0.8*

r 0.4. 0.4

Geometric

/ / /

0. 1

I

2

I

I

I

5 1 0

,

I

1O3 2 xl03

102

h, p m

(a

r -

Glass Sphere, d = 0.2m m

weo) ,aR = 3.041

't

20.28

FIG. 1. Continued.

for iron are not available for wavelengths greater than 12.4 pm. Therefore, beyond this wavelength, the values at 12.4 pm are used along with the Hagen-Rubens law to extrapolate to higher wavelengths. ns = ns.o&

where 1,

=

12.4 pm.

(41)

M. KAVIANY AND B. P. SINGH

144

aR

10 5.0

100 50

628 2.8

I

(

I

1.0

,

0.5 10.314

I

t

I

Penndorf

0.81 Iron Spheres d = 0.2mm

o.4

I

1

2

5 1 0

I

,

102

8

1032x103

k,w

100 50

Iron Spheres d = 0.2mm

0.30

1

4

10 5.0

2

5 1 0

102

1032x103

k (Pm) FIG. 2. (a) Same as Fig. l(a), except for iron spheres of 0.2 mm diameter. (b) Same as Fig. lb, except for iron. (c) Same as Fig. lc, except for iron.

RADIATIVE HEATTRANSFER IN POROUS MEDIA

145

305.0 I

I

102

/

'y3

I

I

lo4

lo5

50.67

10

J

\

3.142

n

L

FIG.2. Continued.

For glass, the optical properties are available for wavelengths less than 206.6 pm. Since the optical properties do not show much change near this limit, the values at I = 206.6 pm are used for higher wavelengths. The experimentally obtained optical constants for metals may be greatly in error (Siege1 and Howell [l]); e.g., Weast gives the indices for iron and for I = 0.587 pm as n, = 1.51 and rc, = 1.63. The corresponding values obtained from the 68th edition are n, = 2.80 and K, = 3.34. For glass, the vlaues of n, are fairly well documented (Palik [23]). However, the value of K for the wavelength range where glass is almost transparent (I < 2.5 pm) is difficult to measure and may contain large experimental errors (Palik [23]). Small glass spheres in this range may be treated as transparent. However, as the sphere size increases, absorption becomes significant. Also, because of differences in composition, the properties vary with the type of glass used. For Figs. 1 and 2, the Mie scattering calculations are done using the Mie theory subroutine of Bohren and Huffman [24] with minor modifications. In particular, if this subroutine is to be used for very large values of the size parameters, double precision must be used. For particles of arbitrary shapes with linear dimensions small compared to A (i.e., I , and A,), the scattering of the waves is done by the oscillating induced dipole moment. The problem was formulated by Rayleigh and is reviewed in Chandrasekhar [5] and van de Hulst [19]. The phase function is

qe,)

= :(I

+ cos2 e,)

(43)

146

M. KAVIANY AND B. P. SINGH

which is symmetric around the Bo = 4 2 plane. The range of validity of the Rayleigh scattering has been investigated by Kerker et al. [25], Ku and Felske [26], Selamet [27], and Selamet and Arpaci [28]. The Rayleigh and Rayleigh-Penndorf scattering and absorption efficiencies are computed as follows (Selamet and Arpaci [28]). In the Rayleigh limit,

and

Inserting m = n - ilc = nJnf

- ilc,/n,

into these equations,

where M , = N:

+ (2 + N2)2

M 2 = 1 + 2N2 N, = 2nlc N, = n2 - 'K

(48) (49) (50)

(51)

The Rayleigh limit can be extended to higher particle sizes by using the Penndorf extension (Penndorf [29]). The Penndorf extension can be expressed as

and where M3=N3-4

M4 = N:

+ (3 + 2N,)'

RADIATIVE HEATTRANSFER IN POROUS MEDIA M5 = 4(N2

+ 7N3

(57)

+ N3 - 2)’ - 9N: N3 = (n2 + K ~ =) N:~ + N i

M6

and

- 5)

147

= (N2

(58) (59)

For glass, Figs. la and b show the results computed from these equations as well as the exact Mie calculation. For iron (Fig. 2), even at small size parameters, the particle does not lie in the Rayleigh limit because of the very high refraction index. The condition for Rayleigh scattering is not only aR Q 1 but also /ma,/4 1 (van de Hulst [19]). Even though this is clearly violated for iron, some agreement with Mie calculation is seen for the scattering efficiency. The absorption efficiencies for iron given by Rayleigh scattering and the Penndorf extension are highly inaccurate. B. GEOMETRICOR RAY-OPTICS SCATTERING The method involves ray tracing and reviews are given in van de Hulst [19] and Born and Wolf [30]. The restrictions in applying geometric optics are the following. The size parameter must be large, aR p 1. The phase shift given as (27r/A)d(n - 1) = 2c(,(n - I), i.e., the change of the phase of a light ray passing through the sphere along the diameter, must be large. Geometric optics fails at extreme incidence angles. Only half of the total scattering, i.e., that due to reflection and refraction, is considered. The other half arising from diffraction around the object must be included separately, leading to the Fraunhofer diffraction pattern. Note that the Mie theory includes reflection, refraction, and diffraction and that the Rayleigh theory does not distinguish between these three. Generally, only up to three internal reflections are included (Liou and Hansen [31]) in order to account for 99% of the scattered energy. Figure 3 gives a schematic of the ray tracing in a sphere where P stands for the number of internal reflections before the ray leaves the sphere. The angle of refraction 8, is related to the angle of incidence and the index of refraction through the Snell law. This states that

n, cos Oi = n, cos 8,

for IC,-+ 0

(60)

ns =-

(61)

or cos ei = n cos e,,

nf

148

M.KAVIANY AND B. P. SINGH

FIG. 3. Schematic of ray tracing for multiple internal reflection in a sphere.

In general, the polarization is decomposed into components parallel and perpendicular to the plane of incidence. For each direction the directional spectral specular reflectivity is found, i.e., piln and p l l . For nonpolarized irradiation, the directional spectral specular rejectioity is

Pi =W

L Z

+ P;Il)

(62)

where in terms of the angles Bi and 8, we have (Siege1 and Howell [l])

or in terms of 0, and n,/n, = n as

ei (02 - cos2 e p + sin ei n2 sin ei - (n2 - cos’ n2 sin ei + (n2 - cos2 e,)ll2 (n2 - cos2 B y 2

- sin

1

1

forrc-0

(64)

Thus the reflected parts of energy are p\Il and p l l . The refracted parts are

RADIATIVEHEATTRANSFER IN POROUS MEDIA

149

1 - p i I Aand 1 - p l A .Then the energy carried b y various rays is (van de Hulst ~191)

PI1 = Pi1 A

for P = 0

(65)

PI1

for P = 1, 2, 3 , . . . if IC = 0

(66)

= (1 - P\lA)2(PilJp-1

and

bll = (1 - pi,A)2(pjlA)p-1 e x p ( - 4 1 ~ ~sin t l ~8,)

for P = 1,2,3, ... ifK

+o (67)

For the other polarization replace with I.However when K, is not small, then Eq. (63) should not be used to calculate the reflectivity. Instead, an exact analysis should be followed (Siege1 and Howell [l]). The total deviation from the original direction is (see Fig. 3)

8' = 28, - 2P8,

(68)

The scattering angle in the interval (0,n) is given by

8, = k2n

+ 48'

(69)

+

where k is an integer and 4 = 1 or - 1. Differentiation and use of the Snell law leads to d8' _ - 2 - 2 p - tan Oi dei tan 8,

The emergent pencil spreads into an area r2 sin 8, d8, d4, where r is a large distance from the sphere. Dividing the emergent flux by this area, we obtain the intensity

where

D=

sin Bi cos Oi sin 8, Id8'/dOi I

(73)

and similarly for I l ( p , ei). Defining the gain G relative to the isotropic scattering as the ratio of scattered intensity to the intensity that would be found in any direction if the sphere scattered the entire incident energy isotropically, we have

150

M. KAVIANY AND B. P.SINGH

The gain for nonpolarized incident radiation is G = i(Gi,

+ GJ

(75)

The sum of the gains in a particular direction resulting from various values of P gives the phase function @ for a nonabsorbing sphere. For an absorbing sphere, the resulting values of the phase function must be divided by the scattering efficiency. The fraction of energy scattered, or the scattering efficiency, can be calculated as

-1 p=o

As mentioned earlier, a value of n = 3 is generally sufficient. The integral over do is replaced by a summation for carrying out the calculation. The absorption efficiency is given by VAa

(77)

= 1 - ?As

The phase function for large opaque specularly reflecting spheres is obtained by Siege1 and Howell [l] and is

where p;[(n - 8,)/2] is the directional and p A is the hemispherical specular reflectivity. For diffuse reflection, they show

@kr(Oo)

8 3n

= - (sin do - 8,

cos 8,)

(79)

The diffraction component of the phase function given by van de Hulst [191) is F(8) = a,J,(a, sin Bo)/sin 8,, which leads to

where J , is the Bessel function of the first order and the first kind. This has a very strong forward component (lobe around 8, = 0) with lobes for do 0 decreasing exponentially in strength.

=-

RADIATIVEHEATTRANSFER IN POROUS MEDIA

151

The scattering and absorption efficiencies for specularly or diffusely scattering large spheres are also given in Siege1 and Howell [l] and, along with that of the diffraction, are given here: specular or diffuse reflection: diffraction:

qAs= pA,qla = 1 - pA

qlsd = 1

(81)

(82)

where p 1 is the hemispherical spectral reflectivity. Since the diffraction scattering is dominantly forward for large particles, it is customary to exclude the diffraction contribution from the phase function and the scattering coefficient simultaneously. The geometric scattering calculations are done as explained above. The value of Bi is varied in discrete steps, and the ray is traced through the sphere. The values of 8, and G(P, 6,) are calculated for P = 0, 1,. .. . The tracing is stopped at some value of P depending on the accuracy required. Although P = 2 or P = 3 is accurate enough for most purposes, higher accuracies can be obtained by continuing to trace to about P = 6. Two different types of points of singularity are encountered in these calculations. Glory occurs when sin 6, = 0 but sin Oi cos 6, # 0. Rainbow occurs when )d6'/d6iI = 0. Both of these make the denominator on the right-hand side of Eq. (73) zero. However, the solid angle affected is extremely small, and by making the step size for Oi small enough, a fairly accurate computation can be carried out. Figures la and b, and 2a and b show the scattering and absorption efficiencies for 0.2-mm glass and iron spheres, respectively. Also plotted are the Rayleigh, Rayleigh-Penndorf, and geometric approximations. In general, changes in efficiencies at smaller size parameters (a, < 10)are due to changes in size parameter, whereas changes in efficiencies at larger size parameters (aR> 10) are mainly due to variation in optical properties (n, and IC,)with the wavelength. Figures lc and 2c show some phase functions for glass and iron spheres at different size parameters. The extremely forward character of diffraction at large size parameters justifies the neglect of diffraction for larger particles. Most porous media applications involve particles even larger than the 0.2 mm diameter considered here. Also, in some applications the wavelengths are generally in the combustion (1-6 pm) range. This results in very large size parameters. The diffraction peak that is included in the Mie phase function has to be removed because the methods for solution of the equation of radiative transfer cannot handle the extremely sharp peaks produced by large particles. Geometric optics provides a convenient alternative where the computation required is independent of the particle size.

152

M. KAVIANY AND B. P. SINGH V. Radiative Properties: Dependent and Independent

The properties of an isolated single particle were discussed in the previous section. However, the equation of radiative transfer requires knowledge of the radiative properties of the medium, i.e., (a,), (a,), and (a). The scattering and absorption are called dependent if the scattering and absorbing characteristics of a particle in a medium are influenced by neighboring particles and are called independent if the presence of neighboring particles has no effect on absorption and scattering by a single particle. The assumption of independent scattering greatly simplifies the task of obtaining the radiative properties of the medium. Also, many important applications lie in the independent regime; therefore, the independent theory and its limits will be examined in detail in this section. In obtaining the properties of a porous medium, the independent theory assumes the following. No interference occurs between the scattered waves (far-field effects). This leads to a limit on the minimum value of CIA, where C is the average interparticle clearance. Porous media applications involving large particles can be assumed to be above any such limit. Point scattering occurs; i.e., the distance between the particles is large compared to their size. Thus a representative elementary volume containing many particles can be found in which there is no multiple scattering and each particle scatters as if it were alone. Then this small volume can be treated as a single-scatteringvolume. This leads to a limit on the porosity. The variation of intensity across this elemental volume is not large. Then the radiative properties of the particles can be averaged across this small volume by adding their scattering (absorbing) cross sections. The total scattering (absorbing) cross section divided by this volume gives the scattering (absorbing) coefficient.The phase function of the single-scatteringvolume is the same as that for a single particle (for similarly oriented identical particles) or is a weighted sum (with scattered energy) of the individual phase functions. Using the number of the scatterers per unit volume Ns (particles/m3) and assuming independent scattering from each scatterer, the spectral scattering coe8cient for uniformly distributed monosize scatterers is defined as Similarly, (ala) = N,A,, and ( n l e x ) = (aAs)+ (a,,). For spherical particles the volume of each particle is 4nR3/3 and, in terms of porosity E,

RADIATIVE HEATTRANSFER IN POROUS MEDIA

153

we have 4 .nN,R3 = 1 - E 3

or

~

N, =

kE

4.n R 3

(84)

Then we have

or

When the particle diameter is not uniform, we can describe the distribution N,(R) dR, i.e., the number of particles with radius between R + d R per unit volume (number density). Note that N,(R) d R has the dimension of particles/m3. Then assuming independent scattering, we can define the average spectral scattering coeficient as (a,,) = JOrn rlls(R).nRZN,(R)dR.

(87)

A similar treatment is given to the absorption coefficients and the particle phase functions. The volumetric size distribution function satisfies

N,=

J

N,(R)dR 0

where N , is the average number of scatterers per unit volume. Wang and Tien [32], Tong and Tien [ll], and Tong et al. [33] consider fibers used in insulations. They use the efficiencies derived by van de Hulst and examine the effects of K, and d on the overall performance of the insulations. The effect of fiber orientation on the scattering phase function of the medium is discussed by Lee [34]. Whenever the particles are placed close to each other, it is expected that they interact. One of these interactions is the radiation interaction. In particular, the scattering and absorption of radiation by a particle are influenced by the presence of the neighboring particles. This influence is classified by two mechanisms: coherent addition, which accounts for the phase difference of the superimposed far-field scattered radiations, and disturbance of the internaljeld of the individual particle due to the presence of other particles (Kumar and Tien [35)). These interactions among particles can in principle be determined from the Maxwell equations along with the particle arrangement and interfacial conditions. However, the complete solution is

154

M. KAVIANY AND B. P.SINCH

very difficult, and therefore approximate treatments (i.e., modeling of the interactions) have been performed. This analysis leads to prediction of the extent of interactions, i.e., dependence of the scattering and absorption of individual particles on the presence of the other particles. One possible approach is to solve the problem of scattering by a collection of particles and attempt to obtain the radiative properties of the medium from it. However, the collection cannot in general be assumed to be a single-scattering volume. For closely packed particles, even a small collection of particles is not a single-scattering volume. Thus, some sort of regression method might be required to obtain the dependent properties of the medium. For Rayleigh scattering absorption of dense concentration of small particles, the interaction has been analyzed by Ishimaru and Kuga [36], Cartigny et al. [37], and DroIen and Tien [38]. Hottel et al. [39] were among the first to examine the interparticle radiation interaction by measuring the bidirectional reflectance and transmittance of suspensions and comparing them with the predictions based on Mie theory, i.e., by examining (qLcx)exp/(qlex)Mlc. They used visible radiation and a small concentration of small particles. An arbitrary criterion of 0.95 has been assigned. Therefore, if this ratio is less than 0.95 the scattering is considered dependent (because the interference of the surrounding particles is expected to redirect the scattered energy back to the forward direction). Hottel et al. [39] identified the limits of independent scattering as CIA > 0.4 and C/d > 0.4 (i.e., E > 0.73).Brewster and Tien [40] and Brewster [41] also considered larger particles (maximum value of aR = 74). Their results indicated that no dependent effects occur as long as C/A > 0.3, even for a close pack arrangement ( E = 0.3). It was suggested by Brewster that the point scattering assumption is only an artifice necessary in the deviation of the theory and is not crucial to its application or validity. Thereafter, the C/A criteria for the applicability of the theory of independent scattering was verified by Yamada et al. [42] (C/A > 0.5) and Drolen and Tien [38]. However, Ishimaru and Kuga note dependent effects at much higher values of C/A. In sum, these experiments seem to have developed confidence in the application of the theory of independent scattering in packed beds consisting of large particles, where C/A almost always has a value much larger than the mentioned limit of the theory of independent scattering. Thus, the approach of obtaining the radiative properties of packed beds from the independent properties of an individual particle has been applied to packed beds without any regard to their porosity (Brewster; [41]; Drolen and Tien [38]). However, all these experiments were similar in design and most of these experiments used suspensions of small transparent latex particles. Only in the Brewster experiment was close packing of large semitransparent spheres considered.

RADIATIVE HEATTRANSFER IN POROUS MEDIA

0.0

0.2

0.4

0.6

0.8

155

1.0

E

FIG. 4. Experimental results for dependent versus independentscatteringshown in the aR-E plane. Also shown are two empirical boundaries separating the two regimes.

Figure 4 shows a map of independent/dependent scattering for packed beds and suspensions of spherical particles (Tien and Drolen [43]). The map is developed based on available experimental results. The rhombohedra1 lattice arrangement gives the maximum concentration for a given interparticle spacing. This is assumed in arriving at the relation between the average interparticle clearance C and the porosity. This relation is

C - 0.905 d - (1 -

-

-1

or

-

(89)

156

M. KAVIANY AND B. P.SINGH

where C/L > 0.5 (some suggest 0.3) has been recommended for independent scattering (based on the experimental results). The total interparticle clearance should include the average distance from a point on the surface of one particle to the nearest point on the surface of the adjacent particle in a close pack. This average close pack separation should be added to the interparticle clearance C obtained when the actual packing is referred to rhombohedra1 packing (E = 0.26). This separation can be represented by a,d, where a, is a constant (al N 0.1). Therefore we suggest that the C/1 condition for independent scattering be modified to C + O.ld > 0.51

(90) where C is given earlier. This is also plotted in Fig. 4. As expected, for E 3 1 this correction is small, while for E 3 0.26 it becomes significant. In Fig. 4 the size parameters associated with a randomly packed bed of 0.2mm-diameter spheres at very high (combustion), intermediate (room temperature), and very low (cryogenic) temperatures are also given. Note that, based on Eq. (90), only the first temperature range falls in the dependent scattering regime (for d = 0.2 mm and E = 0.4). Singh and Kaviany [9] examine dependent scattering in beds consisting of large particles (geometric range) by carrying out Monte Carlo simulations (these simulations involve ray tracing and application of local laws of optics and avoid volume averaging: thus they model dependent scattering in large particle beds). Details of the Monte Carlo method are discussed in Section VI. They argue that the C/1 criterion accounts for only the far-field effects and that the porosity of the system is of critical importance if near-field effects are to be considered. According to the regime map shown in Fig. 4, a packed bed of large particles should lie in the independent range. This is because a very large diameter ensures a large value of C/A even for small porosities. However, Singh and Kaviany show dependent scattering for very large particles in systems with low porosity. Figure 5a shows the transmittance through a medium consisting of large (geometric range) totally reflecting spheres. The scattering is assumed to be specular. The transmittance through packed beds of different porosities and at different values of qnd was calculated by the method of discrete ordinates using a 24-point Gaussian quadrature. It is clear from Fig. 5a that the independent theory fails for low porosities. As the porosity is increased, the Monte Carlo solution begins to approach the independent theory solution. For E = 0.992, the agreement obtained is good. The bulk behavior (away from the bounding surface) predicted by the Monte Carlo simulations for E = 0.992 and the results of the independent theory are in very close agreement. A small difference occurs at the boundaries, where the bulk properties are no longer valid. However, although this difference occurs at the boundary, the commonly made

RADIATIVE HEATTRANSFER IN

-

POROUS

MEDIA

Specularly -Reflecting Particles @ =I)

8

-

Monte Carlo

H &=0.476 0 &=0.732 0 E=0.935 0 &=0.992

(b)

F-- ,

10’

-

10-I

Tr

--

... 8

-

3

Specularly -Reflecting Particles (p0.7)

---

Independent Monte Carlo H E=0.476

0 E4.732 &=0.935

l -

--

-

:

Z

-

-

10-2 T

---

10”

--

1o

-~

-

157

158

M. KAVIANY AND B. P. SINGH

assumption that the prediction by the continuum treatment will improve with increase in the optical thickness is not justifiable because this offset is carried over to larger optical thicknesses. Figure 5b shows the effect of the porosity on the bed transmittance for absorbing particles ( p = 0.7). Again, the independent theory fails for low porosities although the agreement for dilute systems is good. Thus, the transmittance for a packed bed of opaque particles can be significantly less than that predicted by the independent theory. This is due to multiple scattering in a representative elementary volume, so that the effective cross section presented by a particle is more than its independent cross section. Figures 6a-c show the effect of change in the porosity on the transmittance through a medium of semitransparent particles. The particles considered are large spheres with n = 1.5. For these particles, the only parameter that determines the radiative properties of a particle is the product m R(as long as IC is not too large). Figure 6a is plotted for the case of IC = 0 (transparent spheres). Differences from opaque particles (Fig. 5 ) are obvious. The violation of the independent theory results in a decrease in the transmittance for opaque spheres, but for transparent spheres it results in an increase in the transmittance. This is because the change in the optical thickness across one particle in a packed bed is large. Therefore, a transparent particle, while

1oo

h.

Transparent Parlicles (n=l.5)

---

Independent Monte Carlo

W E-0.476

lo-’ 0.0

2.0 4.0 6.0 8.0 ‘in,

10.0

12.0

14.0

RADIATIVE HEATTRANSFER IN POROUS MEDIA

159

10”

10-2

t 1o

1

-~ 0.O

2.0

4.0

6.0

8.0

10.0

12.0 14.0

‘ind

l oo e -

---

10-l

lo-*

.

..

Semitransparent Particles (qa=0.763) Independent

---

Monte Carlo

---

&=0.476

0 &=0.732

3 Z

--

! -

0 &=0.935

--

-

Tr

--1 0 - ~-

--

o-~

1

4

‘ind

FIG. 6. (a) Same as Fig. 5a, except for transparent spheres (n = 1.5, via = 0). (b) Same as Fig. Sa, except for semitransparent spheres (n = 1.5, qAa= 0.287). (c) Same as Fig. Sa, except for semitransparent spheres (n = 1.5, qls = 0.763).

160

M. KAVIANY AND B. P.

SINGH

transmitting the ray through it, also transports it across a substantial optical thickness. In a dilute suspension, a particle, while allowing transmission through it, does not result in transport across a substantial optical thickness. Figure 6b is plotted for semitransparent particles with K a R = 0.1, which gives qla = 0.287. The absorption decreases this effect (transportation across a layer of substantial optical thickness) to the extent that it is exactly balanced by the decrease due to multiple scattering in the elementary volume for E = 0.476. As a result, the Monte Carlo prediction for E = 0.476 shows very good agreement with the prediction from the independent theory. The results for dilute systems are exactly as expected: giving slightly less transmittance than the independent theory solution but showing the same bulk behavior. Therefore, due to these two opposing effects, the magnitude of deviation from independent theory for packed beds of transparent and semitransparent particles is smaller than that for opaque spheres. Figure 6c shows the effect of variation in porosity on transmittance through a medium of highly absorbing semitransparent particles ( K Q ~= 0.5, via = 0.763). Here, the multiple scattering effect clearly dominates over the transportation effect. The transmittance for low porosities predicted by the Monte Carlo method is far less than that predicted by the independent theory, while the most dilute system ( E = 0.992) again shows good agreement with the independent theory. It is encouraging to note that the E = 0.992 system matched the independent theory results for all cases considered. However, the effect of the porosity on transmittance is noticeable even for relatively high porosities ( E = 0.935). As seen earlier, the failure is more drastic for transmission through a bed of opaque spheres than for transparent and semitransparent spheres with low absorption. Also, the deviation from the independent theory is shown to increase with a decrease in the porosity. This deviation can be significant for porosities as high as 0.935.The independent theory gives good predictions for the bulk behavior of highly porous systems ( E 2 0.992) for all cases considered. Two distinct dependent scattering effects were identified. Multiple scattering of the reflected rays increases the effective scattering and absorption cross sections of the particles. This results in a decrease in transmission through the bed. Transmission through a particle in a packed bed results in a decrease in the effective cross sections, resulting in an increase in the transmission through a bed. For opaque particles, only the multiple scattering effect is found, whereas for transparent and semitransparent particles both of these effects are found and they tend to oppose each other. Fundamentally, when the porosity of the medium is small, it is impossible to find a representative elementary volume which is single scattering and across which the intensity does not vary by a large amount. Therefore, the independent scattering assumption cannot be made for low-porosity medium

RADIATIVEHEATTRANSFER IN

POROUS

MEDIA

161

applications. Continuum treatment may still be used, but some adjustment has to be made to the properties of the medium. Finally, we note that both the C/A criterion and the porosity criterion must be satisfied before the independent theory can be used with confidence.

VI. Noncontinuum Treatment: Monte Carlo Simulation

Chan and Tien [44] use ray optics in a simple cubic cell and assume specular reflection. Then they apply the results of the simple cubic cell to obtain the multicell transmission using the layer theory used in the analysis of multilayer coated surfaces. Their model underpredicts the transmittance significantly. Yang et al. [45] used a random arrangement of spheres ( E = 0.42) and ray optics along with a Monte Carlo technique. They show that the entering ray is most likely to have its first interaction at a distance of half the radius and a mean penetration of 4/3 of the radius. They compute the probability distribution function and use this information for computing the transmission through a packed bed. They also find that almost all the rays hit a sphere surface after traveling a distance of three diameters. Ray optics combined with a modified Monte Carlo method is used by Kudo et al. [46] on a unit cell basis for prediction of the transmittance through a packed bed of spheres with gray and diffuse reflection. All of these simulations use diffuse irradiation (not collimated) as the boundary conditions. Using an adustable parameter l/d (with 1 being the cell size and l/d of lo), Tien and Drolen [43] show that the prediction of Kudo et al. gives satisfactory agreement with the experimental results of Chen and Churchill [lo] assuming that the incident radiation in this experiment is diffuse. However, for this, E = 0.906 where the experiments are for E = 0.4. The results of Yang et al. are in good agreement with the experiment of Chen and Churchill (assuming diffuse irradiation) for packed beds that are only a few particles deep (i.e., L/d < 3). Examination of the experiment of Chen and Churchill shows that their beds are irradiated with a nearly collimated beam. This nondiffuse boundary condition when used in the Monte Carlo simulations is expected to change the transmittance significantly. All of these raytracing techniques neglect diffraction, which is justifiable for large particles. Singh and Kaviany [9] extended the Monte Carlo technique to accommodate semitransparent particles as well as emitting particles. They examine randomly packed beds as well as arrangements of variable porosity based on simple cubic packing. Their method is reviewed in the following. The first packing is a bed of randomly packed spheres. The bed was generated by the computer program PACKS (Jodrey and Tory [47]) and was

M. KAVIANY AND B. P. SINGH

162

previously used by Yang et al. The bed of randomly packed spheres generated by this method has a porosity of 0.42. The second bed is based on simple cubic packing. The layers are, however, staggered with respect to each other. This can be significant when considering a packed bed of particles with large absorption. The regular simple cubic structure would result in some rays being transmitted directly through the voids in this regular structure. Also, from a practical standpoint, irregular arrangements are more relevant. The domain of interest consists of a box with a square cross section bounded by x = 0, x = 1, z = 0, and z = 1 with a depth equal to the depth of the bed. The irregular arrangement is achieved by generating sphere centers at four corners of the square [(O,O), (OJ), (l,O), (1,1)] in the x-z planes at y = 0.5, 1.5,. . . . The centers are then staggered by applying the following transformation to all four spheres in the layer:

+ 0.5(25, - 1) Z, = Z, + 0.5(2 0, the intensities at x = 0 (k = 1) are known from the boundary conditions. 1, (pi > 0) is evaluated at k = 1, ... , n. Similarly, Zi (pi < 0) is evaluated at k = n , ..., 1. The in-scattering term ri,&+ 1,2 is stored in a two-dimensional array, which is updated at every point, e.g., while calculating the scattering phase function from direction j to direction i at point k + 1/2 for pj > 0 ri.k+ l j 2 + A k

= ri,&+

112 +A&

+ Apjzj,k+1,2(@%j

-+

pi, Ak)

(137)

This calculation is carried out for scattering into other directions, i.e., for different values of i. It is then repeated for all positive values of p j . Then the Z j , k + l @ j > 0) are calculated and updated and q k + l , 2 ( p > j 0) [used to calculate Zj,k+l(pj> O)] is set to zero. This procedure is carried out for k = 1,. . . , n. After the sweep for pj > 0 is complete, the calculation for Zi ( p j < 0) is carried out at k = n, . .. , 1 in a similar manner. Figure 14 shows the transmittance through a bed of specularly reflecting opaque spheres ( E = 0.476) as a function of the bed thickness. The particles are assumed to have a constant reflectivity. As expected, the scaled results show the same bulk behavior as the Monte Carlo results. However, the results are offset by a difference that occurs at the boundaries where the bulk properties are no longer valid. The difference is more pronounced for E, = 0.1 than for E, = 0.4. This is because for E, = 0.1 a large amount of energy is reflected at the surface of the bed (before continuum treatment becomes applicable). The results obtained from the Kamiuto correlated theory are found to overpredict the transmission. Figure 15 illustrates the change in the transmittance as a function of bed thickness for transparent and semitransparent particles ( E = 0.476). The spheres have a refractive index n = 1.5. Three different absorptivities are considered,i.e., rcaR = 0, iCaR = 0.05, and lcaR = 0.2,giving qp = 0, qa = 0.158, and qa = 0.479, respectively. The results of the DIDOM are in good agreement with those of the Monte Carlo method for all these cases. The results from the Kamiuto correlated theory underpredict the transmission for transparent particles. As the absorption of the particle is increased, the results become closer to the Monte Carlo results. In the limiting case of opaque particles, the correlated theory overpredicts the transmittance.

RADIATIVE HEATTRANSFER IN POROUS MEDIA

179

loo

lo-’

1o

-~

FIG.14. Transmittance through a bed of specularly reflecting opaque spherical particles (8

= 0.476).

Thus, the dependent properties for opaque particles are obtained by scaling the optical thickness obtained from the independent theory. Radiative transfer through semitransparent particles is modeled by allowing for the transmission through the particle while solving the equation of radiative transfer, resulting in the dependence-included discrete-ordinates method.

IX. Effect of Solid Conductivity The effect of thermal conductivity of particles, in a packed bed, on the radiative heat transfer is examined by Singh and Kaviany [52]. They use a method that combines the Monte Carlo method for the radiation with a finite-difference solution for the conduction to solve for the temperature distribution and the radiative heat transfer in a packed bed of large (geometric) spherical particles. Both diffuse and specularly scattering particles were investigated. It is assumed that the fluid phase is nonconducting and that there is no heat transfer at the contact points between particles. The method of solution is similar to that discussed in Section VI.However, unlike Section VI.C, where the particles were treated as being isothermal

M. KAVIANY AND B. P. SINGH

180

lo-’

0

Monte Carlo

1o4

0

4

8

12

16

L id FIG. 15. Same as Fig. 14, except for transparent and semitransparent particles (e = 0.476, n = 1.5).

(infinite conductivity assumption), here we actually solve for the temperature distribution in the particles. Because of the periodic boundary conditions, the temperature distribution has to be obtained for only one sphere per layer of particles. The amount of radiation absorbed by the particle at its surface serves as the boundary condition for the conduction solution. The particles are allowed to emit radiation from their surface. The number of these rays depends on the surface temperature distribution. Thus, the conduction and radiative solutions are obtained iteratively. Figures 16a and b show the effect of the normalized solid conductivity on the normalized radiant conductivity for a porosity of 0.476 for diffuse and specular particles, respectively. The diffusely reflecting particles show a slightly lower radiant conductivity, because the phase function is more backscattering. For an emissivity of 1, the radiant conductivity is the same for both cases, because no energy is scattered. The radiant conductivity increases with the solid conductivity, because of the transportation of the energy through the sphere by conduction. Since more energy is absorbed at higher emissivities, this effect is enhanced at higher emissivities.

RADIATIVEHEATTRANSFER IN POROUS MEDIA

181

1

0.O 0.01

0.10

1

10

100

kJ4doTm3

FIG. 16. (a) Effect of the normalized solid conductivity on the radiant conductivity for a packed bed of diffusively reflecting opaque spherical particles (e = 0.476). (b) Effect of the normalized solid conductivity on the radiant conductivity for a packed bed of specularly reflecting opaque spherical particles (e = 0.476).

182

M. KAVIANY AND B. P.SINGH

X. Conclusions In conclusion, we offer some suggestions on how to model the problem of radiative heat transfer in porous media. First, we must choose between a direct simulation and a continuum treatment. Wherever possible, the continuum treatment should be used because of the lower extent of computations. However, the volume-averaged radiative properties may not be available, in which case the continuum treatment cannot be used. If the continuum treatment is to be employed, we must first identify the elements that make up the system. The choice of elements might be obvious (as in the case of a packed bed of spheres) or some simplifying assumptions might have to be made. The common simplifying assumptions are that the system is made up of cylinders of infinite length (for fibrous media) and that arbitrary convex-surfaced particles are represented by spheres of equivalent cross section or volume. Then the properties of an individual particle can be determined. If the system cannot be broken down into elements, we have no choice but to determine its radiative properties experimentally. On the other hand, if we can treat the system as being made up of elements, then we must identify the system as independent or dependent. In theory, all systems are dependent, but it the deviation from the independent theory is not large, the assumption of independent scattering should be made. The range of validity of this assumption can be approximately set at C/A> 0.5 and e > 0.95. If the problem lies in the independent range, the properties of the bed can be readily calculated. If the system is in the dependent range, some modeling of the extent of dependence is necessary for obtaining the properties of the packed bed. Models for particles in the Rayleigh range and the geometric range are available. However, no approach is yet available for particles of arbitrary size, and experimental determination of properties is again necessary. Direct simulation techniques can be used to provide a wealth of information about the radiative transfer in porous media. Results obtained from these models can be used in simpler approaches like the radiant conductivity approach and for scaling of the optical thickness in low-porosity systems. Also, direct simulation techniques should be used in case the number of particles is too small to justify the use of a continuum treatment, and also as a tool to verify dependent scattering models. Except for the Monte Carlo technique for large particles, direct simulation techniques have not been developed to solve any but the simplest of problems. Finally, we note that the thermal conductivity of the solid phase influences the radiation properties. The Monte Carlo technique along with a solution for temperature distribution in a sphere has to be used to model this effect. Further techniques need to be developed for the simultaneoussolution of the

RADIATIVE HEATTRANSFER IN POROUS MEDIA

183

energy equation and the equation of transfer in porous media. Mathematical solutions of these equations are well developed for homogeneous media. However, a physically realistic treatment of low-porosity porous media is still lacking. Direct simulation is invaluable as a tool for studying and understanding physical phenomenon, and we expect it to provide further insight into the complex phenomena of combined heat transfer in porous media.

Acknowledgments We would like to thank Dr. Melik Sahraoui for his assistance in the preparation of this manuscript.

Nomenclature A B C

d E

F I L 1, m, n m n

N P

Pdf 4

r R S

sr

T x, Y , z, x', Y'

of sphere centers in-scattering term spectral efficiency porosity emissivity index of extinction polar angle angle between incident and scattered beam wavelength (m) random number between 0 and 1 reflectivity Stephan-Boltunann constant, 5.6696 x lo-* (W/m2-K4) absorption coefficient (l/m) extinction coefficient, o, a, (1/ m) scattering coefficient (l/m) optical thickness cos e azimuthal angle (rad) particle scattering phase function scattering albedo, o,/(a, + a,) solid angle (sr)

area, cross section (m2) backscatter fraction for a slab average interparticle clearance (m) diameter (m) fraction of energy carried radiation exchange factor, attenuating factor radiation intensity (W/m2) depth of the slab (m) direction cosines complex refraction index n - iK index of refraction number of layers in the bed integer that defines the reflected or the refracted rays for a transparent sphere probability density function heat flux (W/mz) ray radius (m) distance traveled (m) scaling factor temperature (K) coordinate axes (m) coordinate axes with y'-axis along the incident radiation

+

SUPERSCRIPT -

GREEK

a, Y

size parameter, 2nRI.A maximum allowable displacement

+ -

average value directional quantity forward backward

M. KAVIANY AND B. P. SlNGH SUBSCRIPTS a b C

d e ex f I

ind

absorption blackbody radiation center diffuse effective, emission extinction fluid phase incident independent

normal reflected, or radiation solid, or scattering, or specular wall wavelength dependent axial (or longitudinal) component lateral (or transverse) component

OTHER

0

volume average

References 1 . Siege], R., and Howell, J. R. (1981), Thermal Radiation Heal Transfer, 2nd ed. McGraw-Hill,

New York. 2. Davison, B. (1957), Neutron Transport Theory, Oxford University Press, New York. 3. Sparrow, E. M., and Cess, R. D. (1978), Radiative Heat Transfer, McGraw-Hill, New York. 4. Ozisik, M. N. (1985), Radiative Transfer and Interaction with Conduction and Convection, Werbel and Peck, New York. 5. Chandrasekhar, S. (1960), Radiation Transfer, Dover, New York. 6. Vortmeyer, D. (1978), ‘Radiation in packed solids,’ In Proceedings of 6th International Heat Transfer Conference, Toronto, vol. 6, pp. 525-539. 7. Brewster, M. Q., and Tien, C.-L. (1982), Examination of the two-flux model for radiative transfer in particular systems, Int. J. Heat Mass Transfer 25, 1905-1907. 8. Mengiic, M. P., and Viskanta, R. (1982), ‘Comparison of radiative heat transfer approximations for highly forward scattering planar medium’, ASME Paper No. 82-HT-20. 9. Singh, B. P., and Kaviany, M. (1991), ‘Independent theory versus direct simulation of radiative heat transfer in packed beds,’ Int. J. Heal Mass Transfer, 34, 2869-2881. 10. Chen, J. C., and Churchill, S. W. (1963), ‘Radiant heat transfer in packed beds,’ AIChE J. 9, 35-41. 11. Tong, T. W., and Tien, C.-L. (1983), ‘Radiative heat transfer in fibrous insulations-Part 1: analytical study,’ ASME J. Heat Transfer, 105, 70-75. 12. Hottel, H. C., Sarofim, A. F., Evans, L. B., and Vasalos, I. A. (1968), ‘Radiative transfer in anisotropically scattering media: allowance for Fresnel reflection at the boundaries,’ ASME J. Heat Transfer, 90, 56-62. 13. Rish, J. W., and Roux, J. A. (1987), ‘Heat transfer analysis of fibreglass insulations with and without foil radiant barriers’, J. Thermophys. Heat Transfer, 1, 43-49. 14. Carlson, B. G., and Lathrop, K. D., (1968), ‘Transport theory-The method of discrete ordinates’, In Computing Methods in Reactor Physics, Gordon & Breach, New York. pp. 171 -266. 15. Truelove, J. S. (1987), ‘Discrete ordinates solutions of the radiative transport equation,’ ASME J. Heat Transfer, 109, 1048-1051. 16. Fiveland, W. A. ( 1 988). ’Three-dimensional radiative heat transfer solutions by the discrcteordinates method,’ J. Thermophys. Heat Transfer 2, 309-316. 17. Jamaluddin, A. S., and Smith, P. J. (1988), ‘Predicting radiative transfer in axisymmetric cylindrical enclosure using the discrete ordinates method,’ Combust. Sci. Technol. 62, 173- 186.

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18. Kumar, S., Majumdar, A., and Tien, C.-L. (1990), ‘The differential-discreteordinate method for solution of the equation of radiative transfer’, ASME J. Heat Transfer 112,424-429. 19. van de Hulst, H. C. (1981), Light Scattering by Small Purticles, Dover, New York. 20. Menguc, M. P., and Viskanta, R. (1985), ‘On the radiative properties of polydispersions: a simplified approach’, Combust, Sci. Technol.,44, 143-149. 21. Hsieh, C. K., and Su, K. C. (1979), ‘Thermal radiative properties of glass from 0.32 to 206 pm,’ Solar Energy 22,37-43. 22. Weast, R. C., ed. (1987), Handbook of Chemistry and Physics, 68th ed. CRC Press, Boca Raton, FL. 23. Palik, E. D., ed. (1985), Handbook of Optical Constants of Solids,Academic Press, Boston. 24. Bohren, G. F., and Huffman, D. R. (1983), Absorption and Scattering Light by Small Particles, J. Wiley, New York. 25. Kerker, M., Scheiner,P., and Cooke, D. D. (1978) ‘The range of validity of Rayleigh and Mie limits for Lorentz-Mie scattering’, J. Opt. SOC.Am. 68, 135-137. 26. Ku, J. C., and Felske, J. D. (1984), The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies,’J. Quant. Spectrosc. Radial. Transfer,31,569-574. 27. Selamet,A. (1989), Radiation affected laminar flame propagation, PbD. thesis, University of Michigan. 28. Selamet,A., and Arpaci, V. S. (1989), ‘Rayleigh limit Penndorfextension,’Znt. J. Heat Mass Transfer, 32, 1809-1820. 29. Penndorf, R. B., (1962), ‘Scattering and extinction for small absorbing and nonabsorbing aerosols,’ J. Opt. SOC.Am. 8,896-904. 30. Born, M., and Wolf, E. (1988), Principles of Optics, Pergamon, Oxford. 31. Liou, K.-N., and Hansen, J. E. (1971), Intensity and polarization for single scattering polydisperse spheres: a comparison of ray-optics and Mie scattering, J. Atmos. Sci. 28, 995-1004. 32. Wan& K. Y., and Tien, C.-L. (1983), ‘Thermal insulation in flow systems: combined radiation and convection through a porous segment,’ ASME Paper No. 83-WA/HT-81. 33. Tong, T. W., Yang, Q. S., and Tien, C.-L. (1983), ‘Radiative heat transfer in fibrous insulations-Part 2: experimental study,’ ASME J. Heat Transfer, 105, 76-81. 34. Lee, S.C. (1990). ‘Scattering phase function for fibrous media,’ Int. J. Heat Mass Transfer, 33,2183-2190. 35, Kumar, S., and Tien, C.-L. (1990). ‘Dependent scattering and absorption of radiation by small particles,’ ASME J. Heat Transfer 112, 178-185. 36. Ishimaru, A., and Kuga, Y. (1982), ‘Attenuation constant of a coherent field in a dense distribution of particles,’ J. Opt. SOC.Am. 72, 1317-1320. 37. Cartigny, J. D., Yamada, Y., and Tien, C.-L. (1986) ‘Radiativeheat transfer with dependent scattering by particles. Part 1 -Theoretical investigation,’ ASME J. Heat Tramfer, 108, 608-613. 38. Drolen, B. L., and Tien, C.-L. (1987), ‘Independent and dependent scattering in packed spheres systems,’J. Thermophys. Heat Transfer, 1,63-68. 39. Hottel, H. C., Sarofim, A. F., Dalzell, W. H., and Vasalos, I. A. (1971), ‘Optical properties of coatings, effect of pigment concentration,’ AIAA J., 9, 1895-1898. 40. Brewster, M. Q., and Tien, C.-L. (1982), ‘Radiative transfer in packed and fluidized beds: dependent versus independent scattering,’ ASME J. Heat Transfer 104, 573-579. 41. Brewster, M. Q. (1983), ‘Radiativeheat transfer in fluidized bed combustors,’ASME Paper NO. 83-WAIHT-82. 42. Yamada, Y., Cartigny, J. D., and Tien, C.-L. (1986), ‘Radiative transfer with dependent scattering by particles. Part 2-Experimental investigation,’ ASME J. Heat Transfer, 108, 614-618.

M. KAVIANY AND B. P.SINGH 43. Tien, C. L., and Drolen, B. L. (1989, ‘Thermal radiation in particulate media with dependent and independent scattering,’ Annu. Rev. Nwner. Fluid Mech. Heat Transfer, 1, 1-32. 44. Chan, C. K., and Tien, C.-L. (1974), ‘Radiative transfer in packed spheres,’ ASME J. Heat Transfer, 96, 52-58. 45. Yang, Y. S., Howell, J. R., and Klein, D. E. (1983), ‘Radiative heat transfer through a randomly packed bed of spheres by the Monte Carlo method,’ ASME J. Heat Transfer, 105, 325-332. 46. Kudo, K., Yang, W., Tanaguchi, H., and Hayasaka, H. (1987), ‘Radiative heat transfer in packed spheres by Monte Carlo method,’ In Hear Transfer in High Technology and Power Engineering Proceedings, pp. 529-540, Hemisphere, New York. 47. Jodrey, W. S., and Tory. E. M. (1979), ‘Simulation of random packing of spheres,’ Simulation, January 1- 12. 48. Tien, C.-L. (1988), ‘Thermal radiation in packed and fluidized beds,’ ASME J. Heat Transfer, 110, 1230-1242. 49. Singh, B. P., and Kaviany, M.(1992), ‘Modeling radiative heat transfer in packed beds,’ fnt. J. Hear Mass Transfer, 35, 1397-1405. 50. Kamiuto, K. (1990), ‘Correlated radiative transfer in packed bed-sphere systems,’ J. Quant. Spectrosc. Radiat. Transfer 43, 39-43. 51. Fiveland, W. A. (1987), ‘Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,’ ASME J. Heat Transfer, 109, 809-812. 52. Singh, B. P., and Kaviany, M. (1992), ‘Effect of particle conductivity on radiative heat transfer in packed beds,’ h6.J. Hear Mass Transfer, submitted.

ADVANCES IN HEAT TRANSFER. VOLUME 23

Fluid Flow, Heat, and Mass Transfer in Non-Newtonian Fluids: Multiphase Systems R. P. CHHABRA Department

of'

Chemical Engineering, Indian Institure of Technology Kanpur, Kanpur, India

I. Introduction

Interest in studying the phenomena of momentum, mass, and heat transfer in particulate multiphase systems stems from both fundamental considerations, such as to develop better understanding of the underlying physical processes, and practical considerations, such as to develop suitable methods for the design of fixed and fluidized bed reactors for an envisaged application. It is indeed difficult to think of a chemical plant which does not employ packed or fixed beds to carry out a range of unit operations, including catalytic and noncatalytic chemical reactions, absorption, adsorption, and distillation; sometime packed beds are employed simply to achieve well-mixed conditions. Likewise, fluidised beds are used extensively to attain enhanced rates of heat and mass transfer between the different phases present. Other examples involving momentum transfer between fluids and particles include the filtration of water and of industrial suspensions using sand filters [43a, 1411, underground hydraulics, the flow of oil through rocks, dewatering of slurries by gravity settling, and the flow in coffee filters and cigarette filters. Thus, there is no dearth of examples involving relative motion between a viscous medium and a particulate phase. This chapter, however, is concerned primarily with the transfer processes occurring in the fixed (or packed) and fluidised bed configurations and in the gravity settling of concentrated suspensions. Over the years, considerable research effort has been expended in exploring and understanding the physics of momentum, heat, and mass transfer processes in such particulate systems when the fluid exhibits simple 187

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R. P. CHHABRA

Newtonian behaviour. Indeed, scores of books, research monographs, and review papers providing comprehensive accounts of developments in these areas are now available (e.g., see [22,35,54,77,88, 1551). Unfortunately, not all liquids of industrial significance display simple Newtonian flow behaviour. Indeed, it is now widely acknowledged that most materials encountered in chemical, biochemical, and mineral processing applications do not adhere to classical Newtonian behaviour and are accordingly classed as nonNewtonianjuids. One particular class of fluids of considerable interest is that in which the ‘effective’ (or apparent) viscosity depends on shear rate or, crudely speaking, on the rate of flow. Most particulate slurries (coal in water, china clays in water, sewage sludges, etc.), emulsions (water-oil), and gas-liquid dispersions (foams and froths etc.) are non-Newtonian, as are melts of high-molecular-weight polymers and solutions of polymers or other large molecules such as soap or protein. Further examples of substances exhibiting nowNewtonian behaviour include foodstuffs (soup, jam, marmalade, meat extracts, jellies, etc.), paints, cosmetics and toiletries, synthetic lubricants, pharmaceutical formulations, and biological fluids (blood, synovial fluid, saliva, etc.). Clearly, non-Newtonian fluid behaviour is so widespread that Newtonian fluid behaviour might be regarded as an exception rather than the rule. Although the earliest reference to non-Newtonian behaviour dates back to 700 B.C. [239], the importance of non-Newtonian characteristics and their influence on process design and operations have been recognized only during the past 30 years or so. Thus considerable research effort has been devoted to what might be called engineering analysis of non-Newtonian fluids, as is evidenced by the number of books on this subject (e.g., see [20, 38, 86, 180, 184, 238, 241, 2471). Particulate multiphase systems involving non-Newtonian flow properties are encountered in numerous chemical and biochemical processing applications. Typical examples include the filtration of polymer melts and waste slurries using sand packs [43a, 1411, the flow of highly non-Newtonian polymer solutions through rocks in polymer flooding processes, the use of fixed and fluidised bed reactors to carry out catalytic polymerisation reactions, and some ion exchange operations carried out in fixed bed reactors to recover metals like uranium from slurries and sludges before their disposal. Highly non-Newtonian microbial masses are also produced using fixed bed reactors. Considerable interest has also been shown in three-phase fluidised beds involving a non-Newtonian liquid phase, as encountered in biotechnological applications [lo]. In view of this overwhelming number of existing and potential applications, a vast body of knowledge is now available on the hydrodynamics of such multiphase systems; heat and mass transport processes have been studied less extensively, however. It is somewhat surprising

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that, in spite of its considerable industrial significance, this subject has not been dealt with in any of the recent review articles and books in this field. This is presumably so partly due to the fact that the major research effort in this area has emerged over the last decade or so. It was therefore considered desirable to undertake a critical and exhaustive review of this area. In general, the scope of this chapter is not only to summarize, in a comprehensive manner, the existing literature on non-Newtonian effects in multiphase particulate systems but also to identify the gaps in our knowledge that merit additional work. In paiticular, this chapter sets out to elucidate the influence of non-Newtonian fluid behaviour on momentum, heat, and mass transport processes as encountered in packed beds, liquid-solid and threephase fluidised beds, and hindered settling in concentrated suspensions. 11. Rheological Considerations

As mentioned earlier, many fluids that are encountered in industrial practice exhibit flow behaviour which is not normally experienced when handling simple Newtonian fluids. Clearly, it is beyond the scope of this chapter to undertake a detailed discussion of the non-Newtonian flow behaviour per se and of the large number of rheological equations of state which have been devised to portray the behaviour of real fluids. Indeed, excellent and comprehensive accounts of developments in this field are available in a number of outstanding books [20,21,238,247] and review articles [19]. It is, however, instructive and desirable to present a brief description of rheological complexities or peculiarities of the class of fluids considered here as they would be pertinent to the ensuing treatment of non-Newtonian effects in multiphase particulate systems. Depending on complexities and the form of constitutive relations required to describe the behaviour of real fluids, non-Newtonian fluids may be conveniently classified into three broad categories:

1. Substances for which the rate of shear is dependent only on the current value of shear stress; this class of materials is variously known as purely viscous, time independent, or generalized Newtonian j7uids (GNF). 2. More complex materials for which the relation between shear stress and shear rate also depends on the duration of shearing; such materials are classed as time-dependent fluids. 3. Materials exhibiting combined characteristics of both a solid and a fluid and showing partial elastic recovery after deformation; these materials are termed viscoelastic fluids. This classification (shown schematically in Fig. 1 ) is quite arbitrary in the sense that most real materials often display a combination of two or even all

R. P.CHHABRA

I90

NON-NEWTONIAN FLUIDS

I TIME INDEPENDENT BEHAVIOR

I TIME DEPENDENT BEHAVIOR

---+--I

Shearthinoinp (Pssudoplaslicily) (Dileiancy)

Thixolropic

I

VISCOELASTIC BEHAVIOR

Mamell Model Oldroyd Model

Phan Thien-Tanner Model Power Law (nc1)LPower Law Ellis Modal Carfew Model Meter Model

Cas6an Model Herschel-Bulkley Model

FIG.1. Classification of non-Newtonian fluid behaviour.

the three types of non-Newtonian characteristics. In most cases, it is, however, generally possible to identify the dominating non-Newtonian feature and to use this as a basis of subsequent process calculations. The extent to which non-Newtonian properties as such influence design also varies from one application to another. For instance, where the variation of apparent viscosity with shear rate is small compared with the effects of concentration and temperature gradients within the fluid, it is not really worthwhile to introduce the additional complications associated with nonNewtonian behaviour. Purely viscous fluids deviate from the classical Newtonian postulate in the sense that their apparent viscosity (shear stress divided by shear rate) may decrease with increasing shear rate (shear thinning or pseudoplastic fluids); it may increase with increasing shear rate (shear thickening or dilatant fluids); or the fluid may possess a yield stress (Bingham plastic or viscoplastic fluids). Until recently, it was believed that the shear thinning and viscoplastic behaviours are encountered much more frequently than shear thickening, but with the renewed interest in the processing of highly concentrated suspensions and pastes this is no longer so [16]. Numerous mathematical expressions of various forms and complexity are available in the literature which purport to model time-independent fluid behaviour. Some of the commonly used fluid models are listed in Table I, together with some representative examples. More complete and detailed descriptions of such models are available elsewhere [19, 38, 2381. Time-dependent fluids, on the other hand, are characterised by shear rate-dependent viscosity which further depends on the duration of shearing. Depending on whether the value of viscosity decreases or increases with the time of shearing, these fluids are called thixotropic or rheopectic, respectively. Aqueous suspensions of red mud C194) and bentonite, crude oils, and certain foodstuffs are known to exhibit thixotropic behaviour, whereas suspensions of ammonium oleate and vanadium pentaoxide [246] are believed to show rheopectic behaviour in a certain concentration and shear rate range. The

TABLE I SELECTION OF COMMONLY USED RHEOLOGICALMODELS ~

Fluid behaviour

Model

Pseudoplastic

Power law fluid Ellis fluid model

Expressions for apparent viscosity p = m(9Y-l PO

P=

1 +(Th,*Y-l

~~

Remarks and examples Polymer melts and solutions (n < 1) Includes zero shear viscosity Polymer melts and solutions ( a > 1)

Carreau fluid model

P-Pm

Po - P m = [I

+ (&I)*]("1)/2

Includes both zero and infinite shear viscosities. Polymer solutions (n < 1)

Viscoplastic

Bingham plastic fluid model (Izl > zo) Hershel-Bulkley fluid model (151 T ~ ) Casson equation (IT1 > 50)

5 = To

r

= ro

+ Po9

+ m(3)"

&=&+A

Shear thickening

Power law fluid

P =m ( r l

Viscoelastic

Maxwell model

r+I.-=/.iD

ET

E5

Suspensions and slurries Foodstuffs, suspensions (n < 1) Blood Concentrated suspensions (n > 1) Polymeric solutions and melts

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R. P. CHHABRA

rheological equations of state developed to describe time-dependent behaviour not only are much more complex than those for time-independent fluids but also are custom-built for a specific material. Excellent review articles are available on the rheology and fluid mechanical aspects of timedependent fluids [84, 86, 1821. As mentioned earlier, viscoelastic fluids show a combination of properties of elastic solids and viscous fluids. Other phenomena displayed by such materials include stress buildup and relaxation, strain recovery, and nonzero normal stress differences in simple shear motion and die swell. The rheological equations of state for viscoelastic behaviour are also much more involved than those for time-independent fluids, but the simplest of all, the so-called Maxwell model, is included in Table I. It is appropriate to mention here (and will be seen later also) that within the general framework of time-independent fluids, the behaviour of shear thinning and viscoplastic fluids in multiphase particulate systems has been studied most thoroughly, followed by that of viscoelastic substances. Very little information is available on the hydrodynamics of shear thickening fluids in general and in multiparticle assemblages in particular, and no corresponding results are available for time-dependent materials. It is our objective now to review the vast literature that has built up over the past two to three decades on the momentum, heat, and mass transfer aspects of non-Newtonian multiphase systems. An attempt is made to elucidate the role of rheological complexities in a given application, and often reference will be made to the corresponding Newtonian behaviour for comparison as well as to draw qualitative conclusions. The available literature on multiphase particulate systems can conveniently be divided into three categories depending on the configuration, namely packed or fixed beds, fluidised beds, and hindered settling of concentrated suspensions. 111. Non-Newtonian Effects in Packed Beds

A wealth of information on the different aspects of non-Newtonian fluid flow in packed beds or porous media is now available. Unfortunately, the growth of the contemporary literature in this fast-growing field has been somewhat disjointed and the emerging scenario is of interdisciplinary character. The importance of the flow of non-Newtonian fluids through packed beds can easily be gaged from the number of review articles available on this subject. Savins [229] provided the most thorough and thought-provoking account of the developments in this field prior to 1969, and this was supplemented subsequently by Kumar et al. [152] and Kemblowski et al. C134-J. Most

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recently, Chhabra [38] has provided a critical summary of the pertinent extensive and rich literature available on this subject. Indeed, a wide range of liquid media including polymer melts [89, 203, 237, 2571, polymer solutions [9,43,81,82], crude oils [3,4], foams [76,107], sewage sludges [139], emulsions [7, 106, 176,2483, and particulate slurries [23,43a, 175) has been employed to simulate a wide variety of non-Newtonian characteristics, especially shear thinning, viscoplastic, and viscoelastic fluid behaviours. An equally diverse variety of packed beds consisting of monosize spherical [12, 259, 2601 and nonspherical particles [33,40, 2331, sintered metal filters [67], bundle of cylinders [42, 2561, fritted disks [98], cylindrical cartridges [1391, screens and mats [loo], and chromatographic columns [S] has been used as model porous media. Some investigators have even employed actual rock and core samples[26-30, 58, 75, 871. Each example of a model porous medium is unique in its geometric morphology, contributing in some measure to formidable problems of assigning precise geometric descriptions and of intercomparisons between different media. Additional complications arise from the wide variation in macroscopic descriptions (e.g., the value of permeability) of nominally similar porous media. For instance, Christopher and Middleman [43], in their pioneering study, employed a 25-mm-diameter tube packed with uniform-size glass spheres (710 and 840 pm) and the resulting value of permeability was found to be of the order of 450 darcies, whereas the glass bead packs (53 to 300pm) used by Dauben and Menzie [53] had an order of magnitude lower permeabilities (2-18 darcies). Such large variations in the characteristics of packed beds make comparisons exceedingly difficult. There is no question that the major research efforts have been expended in elucidating the following facets of non-Newtonian flow phenomena in packed beds: 1. The development of generalised scale-up relations for predicting pressure drop for a given bed and time-independent fluids by coupling a specific fluid model with that for a packed bed. It is tacitly assumed that the rheological measurements carried out in the well-defined viscometric flows describe the flow in packed beds, although there is some evidence that this is not so [61, 2541. Preliminary results on the effects of containing walls and particle shape on frictional pressure drop are also available. 2. The response of viscoelastic fluids in packed beds is known to differ, both qualitatively and quantitatively, from that of purely viscous fluids in a variety of ways. The literature on this aspect, as will be seen shortly, is less extensive and also inconclusive. 3. Significant efforts have been devoted to understanding the behaviour of

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the so-called dilute drag-reducing polymer solutions in packed bed flows. Such studies have been done primarily to gain better insight into the fundamental aspects of the flow at the molecular level. 4. The literature on nowNewtonian flow in packed beds and porous media abounds with anomalous phenomena, including slip and pore blockage by adsorption and entrapment, none of which are encountered in the flow of simple Newtonian fluids. Owing to the wide-ranging implications of these processes for oil recovery by polymer flooding, concentrated efforts have been directed at developing a basic understanding of them. Hereafter, these phenomena will be referred to as wall-polymer molecule interactions. 5. Limited results are also available on heat and mass transfer processes in packed bed flows involving non-Newtonian fluids.

A.

PRESSURE LOSS FOR

TIME-INDEPENDENT FLUIDS

Undoubtedly, the practical problem of predicting the pressure drop necessary to maintain a specified flow rate of a liquid of given rheological characteristics through a packed bed of known porosity and/or permeability has received the greatest amount of attention in the literature. By analogy with the behaviour observed for Newtonian fluid flow in packed beds, it is convenient to divide the available body of knowledge into two categories-the laminar or viscous regime and the transitional turbulent regimedepending on the value of the Reynolds number. A recent literature survey [38] revealed that at least three distinct approaches have been employed to predict the frictional pressure drop in the low Reynolds number region: the capillary model; the submerged object model; and other methods based on the use of field equations, empirical/dimensional considerations, etc The developments in the transitional and turbulent regime are of a completely empirical nature. Broadly speaking, the rheological complexity of the fluid, the complexity of the packed bed or porous medium, and the extent of wall-polymer molecule interactions determine the general applicability of any given method. In the ensuing sections, the aforementioned approaches are described by presenting a selection of the more successful and widely used methods from each category. 1. Laminar Flow

a. The Capillary Model. In this approach, the interstitial void space present in the packed bed is envisioned to form tortuous conduits of complicated cross section but with a constant area on the average. Thus, the flow in a packed bed is equivalent to that in a conduit whose length and diameter are chosen so that it offers the same resistance to flow as that encountered in the

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actual packed bed. Undoubtedly, the conduits or capillaries so formed are interconnected in a random manner, but the simplest models of this class do not take this complexity into account. Within the general framework of this category, there are three different models, the Blake, the Blake-Kozeny, and the Kozney-Carman models, which differ from each other in minor details. In the Blake model, the bed is simply replaced by a bundle of straight tubes of complicated cross section (characterised by an average hydraulic radius Rh) and the interstitial pore velocity (V,) is related to the superficial (empty tube) velocity (V) through the well-known Dupuit equation,

4 = lq&

(1)

For a bed consisting of uniform-size spherical particles, the hydraulic radius (in the absence of wall effects) is given by the following expression [21]: Rh

= ~ d / 61( - &)

(2)

In this model, the length of the capillary tubes is assumed to be equal to that of the bed in the direction of flow. The Blake-Kozeny model, on the other hand, postulates that the effective length (L,) of the tangled capillaries is somewhat greater than that of the packed bed (L), thereby introducing the concept of the so-called tortuosity factor T, defined as (L,/L).Finally, the Kozeny-Carman model is exactly identical to the Blake-Kozeny model except that it also corrects the average velocity for the tortuous nature of the flow path as K = ( V 4 (L,/L) (3) In Eq. (3), the factor of (L,/L) stems from the fact that a fluid particle (in macroscopic field equations) moving with the superficial velocity V traverses the distance L in the same time as taken by an actual fluid particle moving with velocity to cover the average effective length L,. A relation between superficial velocity and pressure drop for the flow through a packed bed can now be derived simply by introducing the aforementioned modifications into the expressions for flow in circular tubes. For instance, the fully developed laminar flow of an incompressible Newtonian fluid in a circular pipe can be described by the well-known HagenPoiseuille equation as

v

(4)

Ap = 32 p V L / D 2

Equation (4) is adapted for the laminar Newtonian flow through a packed bed by writing D h ( = 4Rh) for D,L, for L, and the average interstitial velocity for V , which, in turn, leads to the following relation: 2

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This is the well-known Kozeny-Carman model. It can easily be seen that Eq. ( 5 ) includes both the Blake model (L, = L) and the Blake-Kozeny model [only (L,/L) will appear instead of (L,/L)’ on the right-hand side] as special cases. Despite these inherent differences, the predictions of the Blake and the Kozeny-Carman models are virtually indistinguishable from each other, whereas those based on the Blake-Kozeny model differ by about 20% [148]. Based on heuristic considerations, Carman [35] suggested the value of (L,/L) = and further argued that the cross section of capillaries lies somewhere in between that of a circular tube and a parallel slit and, consequently, used the mean value of 40 instead of 32 in Eq. (4). With these modifications, Eq. ( 5 ) can be rewritten as

fi

Ap = SSOpVL(1 - E ) ’ / ~ ’ E ~

(6)

It is customary to introduce the usual dimensionless variables, namely a friction factor (f)and Reynolds number (Re), defined as

f = (AP/PV2)(d/L)(E3/(1 - 4)

(7)

Re = pVd/p( 1 - E )

(8)

and which allow Eq. (6) to be rewritten in its more familiar form as

f

-

= 180/Re

(9)

Experience has shown that Eq. (9) compares favourably with experimental results up to about Re 10. It is, however, appropriate to add here that considerable confusion exists regarding the value and meaning of the tortuosity factor [73]. For instance, Sheffield and Metzner [235] argued that if a fluid element faithfully follows the surface of a spherical particle, the minimum value of T is (7c/2), which when combined with the numerical constant of 32 in Eq. (4) yields a value of 178, which is near enough to 180. On the other hand, based on intuitive arguments, Foscolo et al. [SO] approximated the value of the tortuosity factor by ( l / ~ )This . assertion has received further support from the work of Agarwal and O”eill[2]. Likewise, much uncertainty surrounds the value of the numerical constant in Eq. (9), as evidenced by the wide-ranging values (118 to 220) reported in the literature [4, 78). Even the two so-called best values of 150 and 180 differ from each other by 20% [62]. One can now parallel this treatment in an exactly identical manner for the laminar flow of time-independent fluids. In this case, depending on the choice of a fluid model, all that is needed is a relation between the average velocity and pressure gradient, akin to Eq. (4). Examination of the literature shows that the usual two-parameter power law model has been used most extensi-

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197

vely. For simple unidirectional shearing motion (as encountered in the fully developed laminar flow in circular tubes), the power law fluid model can be written as z,

= m( - d VJdr)"

(10)

It can readily be shown that the average velocity for a power law fluid is given by c211

V

+ 1)(D Ap/4mL)''"

= (D/2)(n/3n

(1 1)

Equation (1 1) can readily be modified to model the laminar flow of power law fluids in packed beds by substituting Dh for D, L, for L, Vi for V, etc. Evidently, different expressions will result depending on whether one uses the Blake or the Blake-Kozeny or the Kozeny-Carman model. Some of the early studies were based on the Blake-Kozeny model, whereas in recent years it has been argued that the Kozeny-Carman approach provides a more accurate description of flow in packed beds [137,148]. Expressions similar to Eq. (11) for a variety of fluid models are available in the literature [21, 86, 2383. Thus, in principle, it is straightforward to derive the non-Newtonian version of Eq. (9) for a specific fluid model. However, some progress can be made in developing a general framework for the flow of time-independent fluids and the choice of a specific model can be deferred to a later stage, as shown in the following. Equation (4) can be rearranged as:

( D AP/4U

= P(8V/D)

(12)

Clearly, the term on the left-hand side is the average shear stress at the wall and the quantity ( 8 V / D ) can be identified as the corresponding shear rate at the wall. Kemblowski et al. [134] asserted that Eq. (12) is also applicable to the laminar flow in packed beds and hence modified it as follows: Rh(Ap/L)= / d K O

V/Rh)

(13)

where K Ois a constant and depends on the geometry, e.g., K O = 2 for circular tubes. For time-independent non-Newtonian fluids, it can easily be shown that the average shear stress at the wall is still given by (R,, Ap/L), whereas, by analogy with the case of flow in cylindrical pipes, ( K OV/Rh) can be identified as the nominal shear rate at the wall. Thus,