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 Mas...

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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.

ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101-4311 United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NWl 7DX

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 .

. .

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

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

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

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 <2 t 2 t 2 <5

<3 <3 <3

c5 0-11.5 0-20.5 11.5-30 85-108' 85-108" <5

<2 <2 0-16 0-30 0 <3 15

0.77-1.64 1.77 2.30b 0.7-8.2 0.79-6.4b 1.0-2.1 0.22-1.34 10-17 12-20 0.72-2.05 0.65- 1.74 2.45-3.13 4.1 10.1 2.63 2.04-2.64 5.3-60 3.7-6.0 1.08-2.72 1.08-10.05 2.75-3.08 1.1-15.3 1.5-15.3 0.67-4.2 2.0-3.4 1.8-3.7 2.04-17.3 3.9-26.0 8.0-17.3 0.49-9.9 9.9-14.6 ~

0.8 4.76 4.76 0.76 0.28-0.75 6.2 0.6 2.2 2.2 2.40-3.81 2.40-3.81 2.40 0.56-0.77 0.71 0.71-1.60 2

-

1.067 1.07-1.81 1.o 10 10 1 1.1 1.1 2.0 2.0-2.5 2.0-2.5 1.1-4.13 2

0.5-2.2

-

90 13 8.0-17.3 15 0.6 0.3-0.6 0.3-0.6

-

3 3 30

-

-

Motion Motion

-

Yes YC.5 YeS -

-

-

2 2.14-3.62 1.0 15

Yes Yes Yes No

15

-

Wall jet

5 5

0.3-0.5 3 -

-

2-4 jets

L

Monde er a/. [64] Mudawar and Wadsworth [7] Nonn rt al. [46] Nonn et al. [37] Ruch and Holman [41] Sakhuja ef a/. [65] Sano et 01. [66] Sherman and Schwartz [SO] Shibayama et d.[30] Shibayama et of. [301 Struble and Witte [391 Taga ef al. [68] Toda and Uchida [28] Vader et al. [12, 361 Wadsworth [38] Wadsworth and Mudawar [72] Zumbrunnen e t a / . [69]

Circ-free Planar-conf Circ-free Circ-free Circ-free Circ-free Planar-free Circ-free Cic-free Circ-free Circsub Planar-free Planar-free Planar-free Planar-conf Planar-conf Planar-free

Water FC-72 FC-72 Mixturet R-113 Water Water Argon Water Mixture’ R-113 Water Water Water FC-72 FC-72 Water

<5 10-40f 20-37 20-30 27 18-77 0 -9.9 <2 <4 14-27 80

0-30 50-70 10-40 lo‘ 79

7.3-13.1 1-11 2.83- 12.7 1.59- 12.7 1.23-6.87

-

-

5.08 0.1-5.0 0.1-5.0 4.7-9.8 6.35-12.7

Yes

-

2 0.127-0.254 0.5-1.0 0.5-1.0 0.21-0.433 1.59-3.18

3.5

-

-

-

0.127 2.40-3.81 1.99-3.81 1.04 2.6 0.5-0.7 10.2 0.127-0.508 0.254 10.2

2.54

-

-

YeS YeS Yes

1.71-4.14 1.70-3.40 2.07-4.16 z 1.4 2-10 1.8-4Sb 1-11 5 2.60b

-

6.35 -

89.7 0.508-5.08 5.08 56

~

Yes -

~

~

No Yes Yes No

2-4 jets 1-9 jets 1-9 jets ?jets

Motion Wall jet

Motion

a 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. ‘Although Ishigai er a/. [53] obtained both transient and steady-state measurements, only the latter are considered here. Water mixed with a surfactant: Rapisool 8-80 (0.02% by weight) [u = 35.5 x N/m at 25”CI. Subcooling is based on the saturation temperature corresponding to the stagnation pressure (Pa p,V:). 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. ‘50% volumetric mixture of FC-72 and FC-87. Water mixed with various surfactants: puluronic types (0.02% by weight) F88 (a = 41.4 x lo-’ N/m), F98 (u = 39.4 x lo-’ N/m), and F208 (u = 41.6 x lo-’ N/m); Rapisool B-80 (0.02% by weight) (u = 35.5 x lo-’ N/m); Rapisool 8-80 (0.04% by weight) (u = 30.9 x lo-’ N/m); sodium oleate (15 ppm) (u = 63.8 x lo-’ N/m) (All values of u are given for 2s”C.)

+

TABLE 11 NUCLEATE BOILINGINVESTIGATIONS-EXPERIMENTAL APPARATUS’ Heater Information

Author Aihara ez al. [61] Chen and Kothari 1701 Chen et al. [71] Cho and Wu [13] Copeland [40] Ishigai ef al. [a]” Ishimaru et nl. [62] Kamata er al. [58, 591 Katsuta and Kurose [48] Katto and lshii [la Katto and Kunihiro [14] Katto and Monde [44]

Ma et a[. [49] Ma and Bergles [18] Ma and Bergles [35] McGillis and Carey [74] Miyasaka and Inada [34] Miyasaka er al. [19] Monde Monde and Furukawa [45] Monde and Katto [16,51,57]

[ln

%Area coverage

Angle (deg)

6-12

90

A

0.020 0.020

90 90 90

G G A

90

C

90 90 90

A A

90

A

0.14 0.021-0.042 12 16 1.2 0.63-6.5 0 0.50-2.6 4.9

>loo 3.6

< 30 1.9 > 100 > 100 0.15-0.23 0.034 0.91-3.2

Temp. loc.

B

15

E

90

A A A A A

90

90 90 90

90 90

90 90

90 90

A

B F A A A

Orientation

Material

Vertical Down Down Vertical Down

Copperb Low-carbon steel Low-carbon steel Copper Copper stainless steel

UP Vertical UP UP Down UP UP Vertical Vertical Vertical Vextical UP Down UP UP Up and down

-

Copper Copper Copper copper Stainless steel foil Nickel foil on SS Constantan foil Constantan foil Copper Platinum foil Pt foil on copper Copper Copper Copper

Heating scheme

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

Indirect Indirect Indirect Indirect

Sue (-)

D = 2.3 254 x 355; x6.35 254 x 355; x6.35 D = 20.5 D = 19.1 12 x 50 D = 1.5 D = 20 15 < D I 2 5 10 s L I 20 D = 10 8x8 D = 10; x 4 0 5x5 3 x 3 and 5 x 5 6.4 x 6.4 4x8

D

=

1.5

20.7 g D < 25.5 D = 59.8 11.2 5 D S 21.0

Surface finish -

25 - pm nickel plating

No. 100 emery; acetone No. 3000 emery; acetone No. lS00 emery; acetone No.800-0/6 emery; acetone -

No. 0 emery; acetone Acetone Polished; acetone Acetone Acetone -

No.5/0 emery; acetone No. 5/0 emery; acetone

-

-

#O emery; acetone

-

Monde and Okuma [47] Monde et al. [64] Mudawar and Wadsworth [7] Nonn et af. [46] Nonn et al. [373 Ruch and Holman [41] Sakhuja et al. [65] Sano et a/. [66] Sherman and Schwartz [So] Shibayama er al. [30] Struble and Witte [39] raga et 01. [68] Toda and Uchida [28] Vader et al. [12,36] Wadsworth [38] Wadswortb and Mudawar [72] Zwnbrunnen er al. [69]

0.034-0.47 0-2.6 1.0-2.0 0.49-4.4 0.12-4.4 0.026-0.1 1 -

0.084 0.82-3.0 4.2 0.37 0 8.6 1.O-4.0 2.0 3.2

9

0

A

0

E c C

90

9 90

9 o c 45-90 c 9

0

E

9 0 9 0 90

B A A

9 o c 9 0 G 20 c 9

0

B

9 o c 9 9

o o

c G

Down UP Verbcal Vertical Vertical Down Vertical UP Vertical UP UP UP UP UP Vertical Vertical UP

Copper Copper Copperb Brass Brass Copper Copper Copper Aluminum Copper MJ802 transistor BmSS

Copper Nickel alloy/SS Copperb Copperb Stainless steel

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

D = 60 D = 25 12.7 x 12.7 12.7 x 12.7 12.7 x 12.7 D = 12.9 51 x 152; x102 20 x 150; x 120 D = 4.38 D = 22 4.5 x 4.5 9 0 x 700; x 3 30 x 30 35.7 x 119 12.7 x 12.7 12.7 x 12.7 74 x 318; x 16

~

Vapor blasted -

25 pm nickel plating ~

t3 # 800- # 0/6 emery: acetone ~

#O/l-#O/8 emery Vapor blasted Vapor blasted Vapor blasted # 600 emery

” % Area coverage refers to the percenlage of the heater surface area covered by the nozzle area. Angle refers to the angle of impingement (90’ being normal). Orientation refers to the direction the heater surface is facing with respect t o gravity. Temp. loc. refers to the surface location of the temperature measurement: A, locally at the stagnation point; B, locally at the stagnation point and other surface locations; C, averaged over the surface; D, locally in the streamwise direction; E, locally at the center of the heater surface; F, locally at the perimeter of the heater surface; G. one or more surface locations which traverse the entire streamwise flow due to surface motion. Emery refers to polishing the surface with the given grade of emery paper. Acetone refers to cleaning the surface with acetone. Oxygen-free copper. ‘Sanded (emery No.500), machined, and mirror finishes. Although lshigai et al. [53] obtained both transient and steady-state measurements, only the latter are considered here. Milled 1.6 pm finish; cleaned with Freon, then alcohol, then water.

16

D. H. WOLF

ET AL.

Shibayama et al. [30] obtained heat flux and surface temperature measurements at the stagnation point for circular, free-surface jets of water and various water-surfactant combinations (Table I). Boiling incipience was associated with the first observable bubble while increasing the heat flux. The data, presented in terms of the heat flux and wall superheat at incipience, exhibited considerable scatter, making specific trends difficult to identify. The data were compared with the following modified incipience model of Hsu [31]:

Although the water data were reasonably well correlated by this expression, the incipient wall superheat for the water-surfactant data were generally underpredicted by Eq. (2). Moreover, the data for water and the water-surfactant mixtures exhibited similar inception characteristics (within the band of data scatter), despite nearly a factor of 2 difference in surface tensions. Observation of boiling cessation with decreasing heat flux revealed that nucleation persisted at wall superheats lower than those observed for increasing power. The overall accuracy of these findings may be limited, however, because of the difficulties associated with visually resolving the smallest bubbles (Jiji and Clark [32]). Moreover, bubble departure diameters are known to decrease with decreasing surface tension (Cole and Rohsenow [33]), making visual detection of the initial bubbles in the water-surfactant mixture more difficult. This may explain why the visually observed incipience results of the water-surfactant mixture were underpredicted by Eq. (2) and were similar to those of pure water. Miyasaka and Inada [34] obtained surface temperature and heat flux measurements at the stagnation point of a planar, free-surface water jet for both single-phase convection and fully developed nucleate boiling. The data were well correlated in both regions with independent expressions for the heat transfer coefficient. The authors defined boiling incipience to occur under conditions for which the correlated heat transfer coefficients for the respective regimes were equivalent. The resulting dimensional expression for subcoolings in the range of 85 5 ATubI 108°C (Tatbased on the stagnation pressure, P , ) was given by qGNB

= 1.40 x lo6

v:'56

(3)

where qLNBand V, have units of W/m2 and m/s, respectively. For the three velocities investigated (1.1, 3.2, 15.3 m/s), this expression correlated the data well. The authors acknowledged the presence of noncondensibles in the water supply from observed effects on the single-phase heat transfer coefficient at surface temperatures below saturation. However, no inferences were made concerning the effect of noncondensibles on boiling incipience.

JETIMPINGEMENT BOILING

17

Ma and Bergles [35] examined boiling incipience at the stagnation point and in the parallel-flow region for a circular, submerged jet of degassed R- 1 13. The onset of nucleate boiling was associated with the first significant increase in the heat transfer coefficient above that for single-phase convection. Figure 5 shows that this departure from the single-phase heat transfer coefficient is a function of the direction of heating (i.e., increasing or decreasing heat flux), consistent with Shibayama et al. [30]. For a jet velocity of 2 mjs and a subcooling of 12S°C, incipient boiling at the stagnation point was observed at a wall superheat of about 20°C for increasing power, and a decrease in power showed that boiling persisted for wall superheats as low as approximately 17°C. For the purpose of correlating the data, the conditions at which boiling ceased upon a decrease in power were used. The data were correlated by the following expression, where nucleation was postulated to begin at the largest active sites on the heater surface with a radius of imax:

All of the 23 data points for both the stagnation and parallel-flow regions in the range 0.4 to were well correlated by this expression for values of ima, 0.6 pm. Examination of the highly polished constantan heater with a scanning electron microscope revealed surface cavities with diameters in the range 0.2 to 1.2 pm, consistent with the maximum diameter (2imax)inferred from the correlation (0.8-1.2 pm). The range of operating parameters for the

-1

V, = 2.04m/s A

Test section No. 14

v

Increasing power Decreasing power

o4 9 8 7

'10

12

14

16

18

20

22

24

Tw - Tsat ("C) FIG. 5. Transition between single-phase convection and fully developed nucleate boiling as a function of the direction of heating for a submerged, circular jet of R-113.(Reprinted with permission from Ma and Bergles [35] 01986, Pergamon Press PLC.)

18

D. H. WOLF ET AL.

correlated data was not given; however, an approximate indication may be obtained from the overall parameter range listed in Table I. Vader et al. [12] obtained local temperature measurements as well as highspeed photographs of bubble formation along a uniformly heated impingement surface for a planar, free-surfacejet of water. The surface temperature at the onset of nucleate boiling was obtained by photographically identifying the location on the heater surface at which the first vapor bubble was visible and simultaneously measuring the surface temperature distribution. Details of the measurement procedure, as well as additional boiling photographs, are provided by Vader et al. [36]. All of the reported findings for boiling incipience were obtained in the parallel-flow region of the jet. The results were compared to temperatures calculated from the following expression proposed by Hsu [31] for a uniform heat flux surface:

The authors recognized that Eq. ( 5 ) would likely overestimate the wall superheat needed for incipience, since the presence of noncondensible gases, which often serve to stimulate incipience at lower wall temperatures, was neglected. Estimates of incipience from photographic observations were also expected to overestimate the actual value, due to the inability to resolve the smallest bubbles, particularly in a highly subcooled flow. For jet velocities of 1.8 to 2.5 m/s, subcoolings of 50 to 70°C, and surface heat fluxes of 0.24 to 1.72 W/m2, Eq. ( 5 ) predicts wall superheats between 7 and 17"C, while the data from visual observations yielded a range of superheats from 4 to 27°C. At low wall superheats, Eq. ( 5 ) overpredicted the temperature at which incipience was observed to commence, which was expected based on the effect of noncondensibles. At high wall superheats, the temperatures inferred from visual observations exceeded those calculated from Eq. (5). Vader et al. [12] speculated that the noncondensible gases, present when the surface was first flooded with liquid, had been driven from surface cavities with increased heating, thereby diminishing their effect on incipience at larger wall superheats. Arguably, Eq. ( 5 ) then becomes more representative of incipience, while the photographic observations are still plagued by the inability to resolve the initial bubbles. Nonn et al. [37] have shown the point of boiling incipience to be delayed to higher wall temperatures and heat fluxes with increasing velocity for a 50% mixture of FC-72 and FC-87. For circular, free-surface jets, the delay in boiling incipience with increasing velocity was independent of the number of jets (1, 4, or 9 jets) and nozzle diameter. Comparisons of boiling incipience between the aforementioned mixture and pure FC-72 were also addressed (Fig. 6). Incipience for the mixture was observed to occur at a lower heat flux

19

JET IMPINGEMENT BOILING lo6 -

I

I

--

-

I

I

I 1 1 1 1

I

I

Incipience Critical Heat Flux (CHF)

A

(u

-0

E

Z

fis

0--

-

-

V, = 3.18 m/s FC-72. Tsat = 57 OC, ATsub = 37 O C Mixture, Tsat = 41 "C. ATsub = 21 "C

0

0

~

;

e o e o eo e.-

8

-

1

1 1 1 1

I

e8e

105:

u

I

I

~

I

I 1 1 1 1 1 1

10

I

I

I

I 1 1 1 1

100

and wall temperature than that of the pure liquid for jet velocities of 3.2 and 6.4m/s (differences of approximately 20% for the heat flux and 10°C for the wall temperature). However, these findings may have been influenced by the differing saturation temperatures for the two liquids (41°C and 57°C for the mixture and the pure FC-72, respectively) and a common jet temperature of 20 "C. Mudawar and Wadsworth [7] examined both single- and two-phase heat transfer to a planar, confined jet of FC-72. Although incipience was not correlated, the authors noted that independent increases in the jet velocity and subcooling delayed the point of incipience to higher heat fluxes and surface temperatures. In addition, Wadsworth [38] noted that increases in the nozzle width (0.127-0.508 mm) appeared to increase the superheat required for incipience for a fixed nozzle velocity. However, despite an increase in the n o d e width by a factor of 4, increases in the incipient superheat were modest (4-5°C).

20

D. H. WOLFET AL.

Struble and Witte [39] examined the single-phase and boiling characteristics of a submerged jet of R-113 impinging on a heated transistor chip. Consistent with Mudawar and Wadsworth, their data reflect a shift of the incipience point to higher heat fluxes and surface temperatures with simultaneous increases in the jet velocity and subcooling. Comparison of Eq. (3) with Eqs. (l), (2), (4), and ( 5 ) reveals some striking differences with respect to their functional dependence, most notable being the presence or absence of the impingement velocity or the wall superheat. Incipience has been widely reported to depend only on qffand AT,,, for a given fluid and operating pressure (as, for example, Eq.( 5 ) proposed by Hsu [31]). The dependence on the velocity (convection coefficient) and subcooling is implicit and arises solely from the single-phase relationship involving h, q”, T,, and T . Hence, velocity and subcooling influence incipience only through their effect on the relationship between q” and AT,,,. These effects may be represented by applying Newton’s law of cooling at the point of incipience,

The wall superheat in Eqs. (l), (2), (4), and ( 5 ) could be replaced by the convective heat transfer coefficient. In the case of Eq. (3, for example, the wall superheat and heat flux may be expressed directly in terms of h o N B by

where

Hence, knowledge of the relationship between the convective heat transfer coefficient (hONB) and the impingement velocity would provide closure for the implicit effects of velocity and subcooling on incipience. A similar approach could be taken with Eq. (3) to obtain an expression which includes the effect of wall superheat. Both Eqs. (7) and (8) predict monotonic increases in the wall superheat and heat flux with increasing heat transfer coefficient (velocity) and degree of subcooling, as suggested by experimentation (Toda and Uchida [28]; Nonn et al., [37]; Mudawar and Wadsworth [7]; Struble and Witte [39]).

JET IMPINGEMENTBOILING

21

B. FULLYDEVELOPED NUCLEATE BOILING FOR SINGLE-JET IMPINGEMENT Results for fully developed nucleate boiling involving the impingement of a single jet may be delineated according to geometry (circular or planar) and ambient surroundings (free surface or submerged). For each special case, the effects of various independent parameters have been addressed. Several parameters, such as velocity, will be addressed for each case, while the consideration of others, such as impingement angle, is limited by the paucity of available data. This subsection is concluded by contrasting the various findings in an attempt to deduce common or distinct characteristics between the jet configurations. 1. Effects of System Parameters on Boiling Heat Transfer f o r Free-Surface, Circular Jets Free-surface, circular jets have clearly received the most attention in the open literature. Consequently, the number of independent parameters investigated has been the most comprehensive of all configurations. a. Jet Velocity. Copeland [40] performed boiling experiments for a jet of water impinging on a downward-facing, heated surface. It was found that, for fully developed nucleate boiling, the heat flux was independent of impingement velocity (0.79-6.4 m/s) and depended only on the wall superheat. Correlation of the nucleate boiling data was given by the expression

where 4 g N B and AT,,, have units of W/m2 and "C, respectively. This equation was said to be valid for wall superheats in the range of 8 to 31°C. Ruch and Holman [41], using an apparatus very similar to that used by Copeland, performed boiling measurements for a jet of R-113, also impinging on a downward-facing, heated surface. Consistent with Copeland, the heat flux in the fully developed nucleate boiling regime was found to be independent of the jet velocity (1.23-6.87 m/s) and a function only of the wall superheat. The results were correlated by 4kNB= 467 where 4& has units of W/m2,AT,,, has units of "C, and the wall superheat is bound between 17 and 44°C. This expression differs from that of Copeland, due, at least in part, to significant differences in the thermophysical properties of water and R-113. Ruch and Holman present the following more general

22

D.H.WOLFET AL.

correlation based on the guidelines outlined by Rohsenow [42] for nucleate pool boiling:

where C,, a constant dependent on the surface-fluid combination, and n were determined to be 3.07 x lo-’ and 1.95, respectively. Although this expression provides a good fit to the Ruch and Holman data, it overpredicts the Copeland heat flux data by as much as an order of magnitude. The authors suggest that the discrepancy may be attributed to the constant CSf. Although values of C,, for R-113 and nickel (the boiling surface employed in both investigations)could not be identified, the value for pool boiling with R113 and platinum has been reported to be 0.005 (Danielson et al. [43]), while that for water and platinum has been reported as 0.013 (Rohsenow [42]). In both cases, the exponent in the correlation corresponded to n = 3. The cubed ratio of these values of CBfis approximately 18, which could account for the aforementioned discrepancies, if the relative values of C,, are similar for nickel. Katto and Monde [44] found that, even for a jet velocity range as large as 5.3-60 m/s with saturated water, the fully developed nucleate boiling curve was independent of the velocity and simply an extension of the data for pool boiling to larger heat fluxes and wall superheats. Similar results were reported in subsequent publications by the same authors (Monde and Katto [16]; Monde [17]; Monde and Furukawa [45]), extending the range of velocities down to 0.67 m/s. Nonn et al. [46] have also shown the nucleate boiling heat transfer to be independent of the velocity (3.2 and 6.4 m/s) for a subcooled jet of FC-72. A later investigation (Nonn et al. [37]) found independence for the same velocities, but for a subcooled, 50% mixture of FC72 and FC-87. Monde and Okuma [47], examining the effects of low flow rates for a jet of saturated R-113, found that, under some circumstances, the heat flux was clearly affected by the impingement velocity (Fig. 7). The results show that the influence of velocity increases with increasing heater-to-jet diameter ratio (D/d). The authors did not specifically address the mechanisms surrounding this phenomenon. However, they consistently observed an effect of velocity on nucleate boiling under conditions for which the amount of heat extracted from the surface (q”nD2/4)is approximately equal to the latent heat required to evaporate completely the supply of saturated liquid (hfspfV,nd2/4). In Section 1V.A.l.a, this condition will be shown to correspond to the so-called L-regime of CHF. Under conditions for which nucleate boiling is influenced by the jet velocity, a significant portion of the impinging liquid is converted to vapor, thereby depleting the amount of liquid available to induce the high

JET IMPINGEMENT BOILING

10

30

6010

30

6010

23

30

60

ATsst ( “ C )

FIG. 7. Effect of velocity on nucleate boiling for a free-surface, circular jet of saturated R113 (d and D have units of mm). (Reprinted with permission from Monde and Okuma [47] 01985, Pergamon Press PLC.)

convective heat fluxes characteristic of nucleate boiling. This condition is precluded at large velocities, when the latent heat of the jet far exceeds heat transfer from the surface. This interpretation is consistent with the dependence on the heater-to-jet diameter ratio (Fig. 7), since the ratio of latent heat to heat removal from the surface varies with (D/d)-’. Hence, the effects of velocity may be expected to be most pronounced at large values of D/d. Shibayama et al. [30] investigated nucleate boiling for saturated jets of water and water-surfactant mixtures. Surprisingly, for a fixed heat flux, increases in the wall superheat were reported with increasing jet velocity for mixtures of both F88 and F98 (puluronic) surfactants (0.02% by weight). Figure 8 shows the boiling curve for the F98 surfactant. No explanation concerning this anomalous behavior was offered by the authors. Unfortunately, results for the other liquids were presented for cases in which more than one independent variable was altered. All of the nucleate boiling data for water and the water-surfactant mixtures (puluronic F88, F98, and F208; Rapisool B-80; and sodium oleate), as well as a limited amount of the saturated R- 1 13 data reported by Monde and Katto [161, were correlated to within f 15% (roughly 10 : 1 odds) by the following expression:

D. H. WOLF ET AL.

24

Working Fluid: F98 additive (0.02wt%)

lo6 ----

o

8

d (mm)

vn( 4 s )

ATsub ("c)

3.81

1.70

1.39

3.81

2.56

1.10

0

8 : :

8

:I

0

0

g

-

t

t

0, incipieni boiling

0

point

0

-

0

105

-c

0

L-J-410

5 1

0

FIG. 8. Eflect of velocity on nucleate boiling for a circular, free-surface jet of watersurfactant mixture. (Replotted from Shibayama et a/. [30], used with permission.)

where 6, is the mean liquid film thickness over the heated surface. No expression for 6, was provided. However, based on the assumption of a third-order polynomial velocity profile, the following equation for the local film thickness was given: 6

=d

[

(2+

0.25

-

0.710Rei 1'2

($)"']

where r is the radial coordinate. Presumably, the mean liquid film thickness is obtained by averaging Eq. (14) in the radial direction from 4 2 to D/2, introducing a dependence on the heater diameter, which was not varied in the investigation.Likewise, with the exception of the few data for R-113 obtained by Monde and Katto [16], the density, latent heat, and thermal conductivity did not vary. The surface tension was varied, however, by use of the different surfactants (0.0336 Io I0.0588 N/m),and Eq. (13) indicates an increase in the heat transfer coefficient with decreasing surface tension. A noteworthy feature of the foregoing correlation is the implied dependence of q;,, on

JET IMPINGEMENTBOILING

25

AT,,. Elimination of the heat transfer coefficient from Eq. (13) by use of Newton’s law of cooling yields a uncommonly large dependence of the heat flux on the wall superheat (&, AT,,,’.’). In a later investigation, Katsuta and Kurose [48] reported the nucleate boiling heat transfer to be independent of velocity for saturated jets of R-1 1 and R-113. However, for a fixed heat flux, results for a water-surfactant mixture (Rapisool B-80) showed a decrease in the wall superheat with decreasing velocity, which corresponds to the trend reported by Shibayama et a/. [30]. Again, no attempt was made to explain the related mechanism. Boiling curves for saturated R-113 and velocities in the limited range from 2.45 to 3.13 collapsed remarkably well but revealed the following strong dependence on the wall superheat:

-

where q:NB has units of W/m2 and AT,,, is in the range from 24 to 31°C. The only other authors to report increasing wall superheat with increasing velocity for a water jet were Ma et al. [49]. The results were obtained for a rapid quench from an initial temperature of 700°C,and the boiling curve was reported to shift slightly to the right with increasing jet velocity. However, because of the small temperature shift and the uncertainties often associated with quenching experiments, the findings are not considered to be conclusive. Cho and Wu [13] reported fully developed nucleate boiling data for a jet of R-113 at velocities ranging from 0.7 to 8.2 m/s (subcooling was not specified). For a fixed heat flux, increases in the jet velocity were shown to decrease the surface temperature monotonically by as much as 9°C over the range of velocities. However, the effect decreased with increasing velocity. All of the nucleate boiling curves were virtually parallel to each other but had an uncommonly low dependence on the wall superheat (&B Sherman and Schwartz [SO] reported nucleate boiling data for a circular, free-surface jet of liquid argon. However, the unusual circumstances surrounding their experiment suggest caution in interpreting the results. The jet was obstructed prior to impingement to enable measurement of the fluid temperature. A sheathed thermocouple of 0.305 mm diameter was placed within the path of the jet of 0.127 mm diameter. These disproportionate dimensions would seem to preclude a continuous jet at the heater surface. Moreover, based on an average jet temperature which exceeded T,,, (ATub z - 10°Cfor nucleate boiling and ATub FZ - 14°C for film boiling), the authors expressed uncertainty as to whether liquid or vapor was present on the surface but speculated that the negative subcoolings were caused by splashing of superheated liquid. In accordance with other investigators, the nucleate boiling data showed no dependence on the jet velocity.

-

26

D. H. WOLFET AL.

b. Subcooling. Copeland [40] investigated the effects of subcooling on fully developed nucleate boiling but could detect no differences in the boiling curves for water at subcoolings of 4 and 78°C. Monde and Katto [16] also examined the effects of subcooling for both water (ATubI30°C) and R-113 (Aqub I16°C) but did not present any of the data. Rather, the authors comment that the data deviate from the saturated results at lower wall superheats (the direction of deviation was not provided), with differences becoming larger as the degree of subcooling was increased. At higher wall temperatures, the data were independent of subcooling and coincided with results for saturated boiling. However, if the unfurnished data correspond to results presented in the Japanese version of this work (Monde and Katto [5 11, the aforementioned trends are difficult to identify. Moreover, if such trends do exist, they fall within the band of scatter associated with data for the saturated jets (Monde and Katto [16]). In accordance with Monde and Katto's [16] claim, however, Nonn et al. [37] have shown that moderate changes in the degree of subcooling (from 20 to 30°C) may affect the nucleate boiling heat transfer, depending on the heat flux (or wall superheat). For higher degrees of subcooling, data for a 50% mixture of FC-72 and FC-87 revealed lower surface temperatures for heat fluxes near incipience, while differences were reported to diminish with increasing heat flux. Similar trends have been reported for subcooled pool boiling of FC-72 (Carvalho and Bergles [52]). However, since these observations have been reported at heat fluxes near incipience,it is possible that the effects are due to a transition from a developed stage of boiling to a condition of partial boiling, where parameters such as subcooling can influence the rate of heat transfer. c. Fluid Properties. Monde and Katto [I61 presented results for saturated water and R-113 at atmospheric pressure. The authors did not correlate the data, but from their graphical representations the following relations were inferred: %

450 AT,,i2.7

(saturated water)

(16)

qgNB X

790 AKai2.'

(saturated R-113)

(17)

&NB

where q;NB has units of W/m2 and AT,,, has units of "C. The data represented by Eqs. (16) and (17) spanned wall superheats from 18 to 46°C and 15 to 30°C, respectively. These results indicate that, for a fixed wall superheat, the heat flux associated with R-113 is significantly less (by a factor of4 to 7) than that for water and that there is a weaker dependence of the heat flux on the surface temperature for the R-113 jet. Ruch and Holman [41] compared their nucleate boiling data for R-113 with those obtained by Copeland [40] for water, using nearly the same apparatus at atmospheric pressure. The correlations of the data [Eqs. (10)

JET IMPINGEMENT BOILING

27

and ( 1 l)] exhibit trends that are similar to those of Eqs. (16) and (17), namely that the water jet yields significantly higher heat fluxes for a given wall superheat, as well as a stronger functional dependence on the superheat. Shibayama et al. [30] investigated nucleate boiling for saturated jets of water and several water-surfactant mixtures, in order to examine the effects of surface tension. For surface tensions ranging from 0.0336 to 0.0588 N/m, the nucleate boiling heat transfer coefficient was reported to vary as a-0.4 [See Eq. (13)]. Invoking Newton's law of cooling and holding all other parameters fixed, this result would suggest that the heat flux varies as or the wall superheat as Nonn et al. [37] have compared their nucleate boiling data for a 50% mixture of FC-72 and FC-87 with that of an earlier investigation, where pure FC-72 was employed as the coolant (Nonn et al. [46]). Figure 6 provides a typical comparison for one set of operating conditions. It is noteworthy that the single-phase convection heat transfer coefficient is unaffected by the fluid. Also, for a fixed fluid temperature (20°C) and heat flux in the nucleate boiling regime, the mixture maintains a cooler surface than does the pure FC-72. The smaller temperatures are a direct consequence of the lower saturation temperature of the mixture (41°C compared to 57°C for FC-72). Marked differences in the critical heat flux are also evident. d. NozzlelHeater Dimensions. Several investigations have addressed the dependence of nucleate boiling heat transfer on the nozzle and/or heater diameter. Copeland [40] (0.28 I d I0.75 mm), Ruch and Holman [41] (0.21 I d I 0.43 mm), Monde and Katto [ 161 (2.0 Id I2.5 mm), and Katsuta and Kurose [483 (2.4 I d I 3.8 mm) all found that fully developed nucleate boiling was not affected by the jet diameter. Similarly, Monde and Katto [16] (1 1.2 ID I 21 mm), Monde [17] (20.7 I D I 25.5 mm), and Katsuta and Kurose [48] (15 ID I 25 mm) found no dependence on the heater diameter. In a subsequent study, however, Monde and Okuma [47] did find conditions in the nucleate boiling regime to be influenced by the heater-to-jet diameter ratio (14.6 5 D/d I54.5) when the supply of liquid to the surface was low. Figure 7 summarizes their findings, and a more detailed discussion of related mechanisms is given in Section 1II.B.l.a. e. Surface Orientationllmpingement Angle. Ruch and Holman [41] investigated the effects of impingement angle on nucleate boiling heat transfer for a circular jet of R-113 directed vertically upward. Heater orientations ranging from 0 to 45" with respect to the horizontal had no noticeable effects on the nucleate boiling curve. Monde and Katto [161 examined normal impingement of circular jets of R-113 and water on upward- and downward-facing heater surfaces, and regardless of the fluid, heater diameter, jet diameter, or velocity, no discernible effects of surface orientation were observed.

28

D. H. WOLF ET AL.

f. Nozzle-to-Surface Spacing. Nonn et al. [37] have investigated the effects of nozzle-to-surface spacing on nucleate boiling heat transfer for a jet composed of a 50% mixture of FC-72 and FC-87. For spacings of 0.5 to 5 nozzle diameters, no effects could be detected for jet velocities of 3.2 and 6.4 m/s. For spacings below 0.5 nozzle diameters, however, the radial flow became restricted and a declining surface temperature was observed (see Section III.B.3.d on confined jets for more details).

2. Effects of System Parameters on Boiling Heat Transfer for Free-Surface, Planar Jets a. Jet Velocity. Ishigai et al. [53] used steady and transient methods to investigate the effects of jet velocity on nucleate boiling heat transfer for a subcooled, planar jet of water. Definitive conclusions concerning the nucleate boiling regime were difficult to draw for the transient experiments; however, the steady-state results showed that for velocities of 1.0 and 2.1 m/s and a subcooling of 3 5 T , the heat flus was independent of velocity. An approximate relationship between the heat flux and the wall superheat was obtained from graphical results provided by Ishigai et al. [53] and is of the form

where q:NB has units of W/mZ and AT,,, has units of "C. The correlation is based on data for wall superheats in the range from 26 to 47°C. Miyasaka and Inada [34] and Miyasaka et al. [19] investigated the effects of jet velocity on nucleate boiling heat transfer for a highly subcooled (85 I ATub I 108"C),planar jet of water. The heat flux was unaffected by jet velocities ranging from 1.1 to 15.3 m/s and was correlated fairly well by the following expression obtained for pool boiling on the same experimental apparatus : qENB

= 79

(19)

where q k N B has units of W/m2, ATa, has units of "C,and the data correspond to wall superheats of 26 to 90°C. However, for a fixed wall superheat, Eq. (19) slightly underpredicted the surface heat flux, and differences became more pronounced with increasing velocity. Following a method proposed by Rohsenow [54], which involved superposition of the effects of single-phase convection and bubble motion, Miyasaka and Inada [34] achieved better agreement by replotting their data in terms of the difference between the applied heat flux and that predicted from an appropriate correlation for single-phase, forced convection. A unique feature of the Miyasaka et al. investigations was the fact that the heater was very small in comparison to the jet width, causing the entire heating surface to be engulfed within the

JET IMPINGEMENT BOILING

29

impinging flow. Under these circumstances,the saturation conditions on the surface are governed by the stagnation pressure P,. Hence, T,,, used in the subcooling and wall superheat [Eq. (1911 was based on the stagnation pressure (Inada [ S S ] ) . Vader et al. [12] have demonstrated the invariance of fully developed nucleate boiling at the stagnation point to impingement velocities ranging from 1.8 to 4.5 m/s for a subcooled water jet. The results are limited, however, to a small region of the nucleate boiling curve near the point of incipience, thereby precluding correlation of the data. b. Subcooling. Ishigai et al. [53] considered nucleate boiling heat transfer for a planar jet of water with a velocity of 2.1 m/s and subcoolings of 35 and 75°C. Despite doubling of the degree of subcooling, only minor enhancements in the heat transfer were observed for a fixed wall superheat, and the data for the different subcoolings were still well predicted by Eq. (18). Similarly, Vader et al. [lZ] have shown fully developed nucleate boiling at the stagnation point to be invariant with the degree of subcooling for a jet of water and 50 I ATub I 70°C. As with the effects ofjet velocity, however, the results are limited to a small region of the nucleate boiling curve near the point of incipience. c. Surface Orientation. Although the investigations by Miyasaka et al. considered upward-facing (Miyasaka and Inada [34]) and downward-facing (Miyasaka et al. [19]) surfaces, the results were found to be well correlated by the single expression given by Eq. (19).

3. ESfects of System Parameters on Boiling Heat Tranger for Submerged, Confined, and Plunging Jets a. Jet Velocity. Katto and Kunihiro [14] investigated the effects of jet velocity on fully developed nucleate boiling for a circular, submerged jet of saturated water. The relationship between the surface temperature and heat flux was shown to be unaffected by the velocity except to extend the results for pool boiling to higher values of heat removal and wall superheat. The range of nozzle velocities tested, however, was relatively limited (2.04-2.64 m/s). Based on the best-fit line drawn through their nucleate boiling data, the following approximate expression has been obtained: (IkNB Z

340

(20)

where q g N 8 has units of W/m2 and AT,,, has units of "C.The data represented by Eq. (20) spanned a range of wall superheats from 18 to 38°C. Ma and Bergles [18] also found the nucleate boiling heat transfer to be invariant with respect to velocity (1.08-2.72 m/s) for a circular, submergedjet

30

D.H. W O L F ET AL.

of saturated R-113, In accordance with other investigations, the forced convection boiling results were found to be well predicted by an extrapolation of the boiling curve for fully developed pool boiling. The authors do not provide an explicit correlation of their results but do provide what appears to be a best-fit line through the data. The following relation was extracted from Figures 6 and 7 of Ma and Bergles [181, qiNB z 0.15A T,a14.4

(21) where qkNe has units of W/m2 and AT,,, has units of "C. This expression represents their impingement data well for superheats in the range from 26 to 33°C.Although this expression is in reasonable agreement with that of Monde and Katto [16] and Ruch and Holman [41] within a small range of wall temperatures, the functional dependence of the heat flux on the wall superheat is large, with n = 4.4as compared to n = 2.0 for Monde and Katto and n = 1.95 for Ruch and Holman. Interestingly, in a separate figure without the impingement results (Fig. 5 of Ma and Bergles [ls]), additional data presented for pool boiling under the same conditions corresponded to an exponent of approximately n = 2.3. Ma [56] has suggested that the differences in slope are related to differing surface conditions, influenced by the impingement of R-113.Citing the work of Joudi and James [22], Ma argued that R-113,a strong cleaning agent, served to rejuvenate the surface with passing time, thereby generating different boiling curves. Examining the effects of surface contaminants on nucleate, pool boiling, Joudi and James [22] found that surface deposits left by water and methanol caused the boiling curve to shift to the right (increasing ATat) with increasing time; however, the slope remained essentially constant. It was further noted that subsequent boiling with R-113 removed the contaminants and returned the heater surface to a near-original state. In contrast, for repeated experiments with R-113,no surface contamination or effects on the boiling curve were observed. Hence, it seems unlikely that surface rejuvenation could have influenced the R-113 results of Ma and Bergles [18]. In a later investigation, Ma and Bergles [35] confirmed the independence of heat transfer on velocity but also identified a case in which the fully developed boiling region for forced convection did not coincide with the extrapolated results for pool boiling. The authors noted, however, that the extrapolation of the pool boiling data was extensive and that minor modifications to the slope would produce better agreement. As in their earlier publication (Ma and Bergles [18]), the pool and forced convection boiling results showed an uncommonly strong dependence of the heat flux on the wall superheat. Monde and Katto [57] investigated the effects of velocity on nucleate boiling heat transfer for a highly confined, circular jet of saturated water. A

31

JET IMPINGEMENT BOILING

glass plate was attached to the nozzle exit and parallel to the heater surface. Data were presented for confinement heights of 0.3 and 0.5 mm with a nozzle diameter of either 2.0 or 2.5 mm. Their results showed little effect of the jet velocity, in the range from 8.0 to 17.3 m/s. Monde and Furukawa [45] obtained data in the nucleate boiling regime for a circular, plunging jet of saturated R-113. The results are shown in Fig. 9, where H denotes the height of the liquid pool (the nozzle was consistently positioned 5 mm above the heater surface). For liquid pools less than 1 mm in depth, the effects of the jet velocity are seen to be negligible. However, as the depth of the pool increases, the effects ofjet velocity become discernible at higher heat fluxes near CHF. Through visual observations, the authors note that dryout on the heater surface occurred abruptly for low liquid levels, while dryout for the deeper pools with low jet velocities appeared and disappeared locally, without rapid spread of vapor across the surface. Monde and Furukawa suggest that the formation and collapse of this vapor layer may have caused the departures from the fully developed nucleate boiling curve shown in Fig. 9 for liquid depths of 2 and 4 mm. Like Monde and Katto [57], Kamata et al. [58,59] examined boiling for a circular, confined jet of saturated water which impinged into a narrow region ( < 1 mm) composed of the heater surface and a plate which was attached to the nozzle exit and was parallel to the heater. For velocities in the range 10 to 20 m/s, the stagnation point heat transfer in the nucleate boiling regime

10

30 5010

30 5010

30 t

ATsat (OC) FIG. 9. ElTect of velocity on nucleate boiling for a free-surface (H = 0) and plunging ( 1 IH 5 4 mm), circular jet of saturated R-113. (Monde and Furukawa [45], used with

permission.)

32

D. H. WOLFET AL.

exhibited no dependence on velocity. Similarly, Mudawar and Wadsworth [7] showed the nucleate boiling heat transfer regime to be insensitive to changes in the jet velocity (1-1 1 m/s) for a confined, planar jet of subcooled (10°C) FC-72. Struble and Witte [39] examined the single-phase and boiling characteristics of a submerged jet of R-113 impinging on a 4.5-mm-squareYsilicon power transistor (MJ802). The transistor was used as both a heater and temperature sensor. Although the nucleate boiling curve was presented for two different jet velocities (2.07 and 4.16 m/s), the doubling of the velocity was accompanied by a doubling of the degree of subcooling (14-27”C), making it impossible to isolate the effects of V, and ATub. Nonetheless, data for these separate operating conditions collapsed with data obtained for saturated pool boiling over much of the q”-AT,,, domain. However, small enhancements in the heat flux for a fixed wall superheat were evident for the higher jet velocity and subcooling. Overall agreement with the impingement boiling data of Ma and Bergles [18] was poor, but similar qualitative trends were demonstrated. The only other reference to impingement boiling on a silicon surface is that of Goodling et al. [a]. They cooled a 4 x 4 array of simulated chips mounted on a silicon wafer with an equal number of circular jets of R-12 (d = 0.79 mm) impinging on the opposite side. The simulated chips were 5-mm-square, p-type resistors (similar in size to a small VLSI chip) diffused into an n-type silicon wafer (75 mm in diameter). Despite several attempts to refine the measurement process, accurate estimates of the heat flux and surface temperature were unobtainable. Much of the heat generated by the simulated chips was conducted into the wafer and spread over a larger region, precluding determination of the surface area. The several different transducers employed to measure the surface temperature (both conductive and semiconductive elements) were generally fraught with poor sensitivity or electrical interference. Although boiling was said to occur, no boiling data were reported. Aihara and co-workers (Aihara et al. [61]; Ishimaru et al. [62]), have provided the only nucleate boiling data for a circular, submerged 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 to that of the incoming jet), As with the other fluids, little effect of jet velocity was evident in the range from 0.22 to 1.34m/s for impingement on a flat heater surface (Ishimaru et al. [62]). However, for impingement on a concave, hemispherical surface (Aihara et al. [61]), slightly lower wall superheats (- 15%) were reported for increases in the jet velocity from 0.77 to 1.64 m/s (4’’fixed). Information concerning the operating pressure or saturation temperature was not provided in either investigation.

JET IMPINGEMENTBOILING

33

b. Subcooling. In contrast to other investigations (Copeland, [40]; Ishigai et al. [53]; Vader et al. [12]), which have shown the degree of subcooling to have no effect on nucleate boiling heat transfer, Ma and Bergles [18,35] have shown that small differences do exist in the boiling curve for moderate amounts of subcooling (- 20°C). They found that increasing the subcooling of an R-113 jet from 0 to 11.5"C[l8] or to 203°C [35] caused the convective boiling curve to shift slightly to the left, as revealed in Fig. 10. Results for pool boiling were found to follow the same trend, which has also been reported for pool boiling from a horizontal plate (Duke and Schrock [63]). Mudawar and Wadsworth [7] varied the degree of subcooling in the range from 10 to 40°C for a confined, planar jet of FC-72. Although boiling incipience was affected by the subcooling (Section IILA), the nucleate boiling regime showed no dependence. c. NozzlelHeater Dimensions. Katto and Kunihiro [141 investigated the effects of nozzle diameter on fully developed nucleate boiling for a circular, submerged jet of saturated water. The relationship between the surface temperature and heat flux was shown to be unaffected by the diameter within 1o6

9 8

7 6

Test section No. 12

5 h

"E

s

:

0-

3

vn

2

ATsub

'

06/08/83 R2 1.08 0 $: 06/08/83 R4 0 20.5 :$ 06/08/83 R5 1.08 20.5

:c

A

1

1o5 10'

2

v

3

Increasing power Decreasing power

4

5

6

7 8 9102

ATsat ("C) FIG. 10. Effect of subcooling on nucleate boiling for a submerged, circular jet of R-113. (Reprinted with permission from Ma and Bergles [35] 01986, Pergamon Press PLC.)

34

D. H. WOLF ET AL.

the range 0.71 to 1.60 mm. Similarly, Wadsworth [38] investigated the effects of nozzle width for a planar, confined jet of subcooled (10°C) FC-72. For ajet velocity of 5 m/s and nozzle-to-surface spacing of 0.508 mm, the nucleate boiling heat transfer was shown to be independent of the nozzle width in the range 0.127 to 0.508 mm. Monde and Katto [57] investigated the effects of both nozzle and heater diameters on nucleate boiling heat transfer for a highly confined, circular jet of saturated water. The jet impinged into a narrow gap (0.3-0.5 mm) between a glass plate attached to the nozzle exit and parallel to the heater surface. For heater diameters of 11.2 and 20.2 mm and nozzle diameters of 2.0 and 2.5 mm, no discernible effect on the fully developed nucleate boiling curve was evident. d. Nozzle-to-Surface Spacing. Monde and Katto [57] investigated the effects of nozzle-to-surface spacing on nucleate boiling heat transfer for a highly confined, circular jet of saturated water. A glass plate was attached to the nozzle exit and parallel to the heater surface. Data were presented for confinement heights of 0.3 and 0.5 mm with a nozzle diameter of either 2.0 or 2.5 mm. Their results showed no effect of the separation distance over this limited range. The data were in very good agreement with those obtained for the same jet but without the glass plate, suggesting that the effects of confinement were minimal. Similarly, Kamata et aE. [58,59] examined boiling for a circular jet of saturated water that impinged into a narrow region where spacings were varied in the range 0.3 to 0.6 mm. The stagnation point heat transfer in the nucleate boiling regime exhibited no dependence on spacing. Nonn et al. [37,46] also examined the effects of separation distance for circular jets of FC-72 [46] and a 50% mixture of FC-72 and FC-87 [37]. A plate was attached perpendicular to the nozzle exit and parallel to the heater surface. At large nozzle-to-surface spacings, the jet had a free surface, and the nozzle plate had no influence on the hydrodynamics. At small nozzle-tosurface spacings, the nozzle plate could come in contact with the free surface of the radially flowing film and affect the hydrodynamics. For the jet of FC-72 (d = 1.0 mm and V, = 6 m/s), no effects on the nucleate boiling heat transfer were evident with nozzle-to-plate spacings in the range 0.1-5.0 mm, but spacings of less than 0.1 mm were reported to lower the surface temperature for a fixed heating rate. Results for a 50% mixture of FC-72 and FC-87 revealed moderate decreases in the surface temperature ( x 3-5°C) as the nozzle-to-surface spacing approached the free surface of the radially flowing film, which had a thickness of approximately 0.1 mm for jet and confinement diameters of 1 and 15 mm, respectively. The effect on the surface temperature of accelerating the radial film by reducing the separation was shown to be more significant for single-phase convection ( m 5-10°C). Wadsworth [38] also investigated the effects of nozzle-to-surface spacing for a planar, confined

JETIMPINGEMENT BOILING

35

jet of subcooled (10°C) FC-72. For a jet velocity of 5 m/s and nozzle width of 0.254 m, the nucleate boiling heat transfer was shown to be independent of the confinement channel height within the range 0.508 to 2.54 mm. Aihara et al. [61] investigated the effects of separation distance (0.5 I z 5 2.2mm) on nucleate boiling heat transfer for a circular, submerged jet (d = 0.8 mm) of saturated liquid nitrogen impinging on a concave, radially confined, hemispherical surface. For a fixed heat flux, decreasing nozzle-tosurface spacing caused an attendant decrease in wall superheat ( - 20% over the entire range of z), 4. Summary of Overall Trends and Conclusions The preceding discussion of fully developed nucleate boiling revealed that, typically, conditions are unaffected by parameters such as velocity, nozzle or heater dimensions, impingement angle, surface orientation, and possibly subcooling. However, conditions depend strongly on the type of fluid employed. A final consideration relates to the effects of nozzle geometry (circular or planar) and ambient surroundings (free surface or submerged). Such a comparison is made in Fig. 11, which contrasts free and submerged water jets with circular, planar, and wall (discussed in Section 1II.E) geometries. Despite significant differencesbetween the jet conditions, there is generally good agreement between their respective nucleate boiling characteristics. Excluding the correlation of Toda and Uchida [28], the remaining correlations fall within a heat flux band of f 41% for a given wall superheat. Figure 12 provides a similar comparison for R-113 jets, which includes free and submerged configurations for circular nozzles (the only type available for R113). Unlike the water jets, the agreement is poor. Further support for the insensitivity to the ambient surroundings is revealed in Fig. 13, which is based on the work of Katto and Kunihiro [14] and compares nucleate boiling data for three submerged jets, a free-surface jet, and a pool of saturated water.

5 Compendium of Correlations All of the correlations for fully developed nucleate boiling that have been presented in this section are summarized in Table 111. Although other literature has been cited in this chapter, the authors felt that suitable correlations of the corresponding data could not be developed with a reasonable degree of accuracf C. MULTIPLE-JET IMPINGEMENT

Monde et al. [64] examined the effects of multiple impinging jets on nucleate boiling heat transfer for both saturated water and R-113. Circular, free-

36

D. H. WOLF ET AL.

4 104

1

10 ATsat

100

["CI

FIG. 11. Comparison of nucleate boiling correlations for free-surface and submerged jets of water with circular, planar, and wall jet configurations.

surface jets, numbering between two and four, impinged at various locations on the heater surface, as depicted in Fig. 14a. Figures 14b and 14c show results for the fully developed nucleate boiling of the water and R-113jets, respectively. The solid lines are based on the results of Monde and Katto [16] for single, circular, free-surfacejets of saturated water and R- 113. The authors conclude that the degree of scatter in the data is typical for nucleate boiling and hence that the number or placement of jets has little or no effect on the heat flux. Careful scrutiny of the data, however, indicates that the scatter is not random. For comparisons with respect to position, where the number of jets is fixed (n, = nb = 3, for example), sizable and consistent differences in the heat flux exist for a fixed wall superheat. The maximum heat flux for the impingement of three or four jets of water (n = 3, n = 4) is achieved for flow configuration a, that is, for some kind of symmetrical impingement of all jets within the boundary of the surface. By contrast, a two-jet system of water (n = 2, n = 2*) yields higher heat fluxes if impingement occurs just outside

JET IMPINGEMENT BOILING

31

q” [w/m21

FIG. 12. Comparison of nucleate boiling correlations for free-surface and submerged, circular jets of R-113.

the boundary of the heated surface (configuration b). For R-113,the results are less conclusive, except for the three-jet configuration, which yields the highest heat flux for impingement within the boundary of the heater. Although related physical mechanisms are difficult to infer from this limited study, the results suggest that multiple jets and their relative placement do have an effect on nucleate boiling heat transfer but that additional investigation is clearly warranted. Sakhuja et al. [65] have examined cooling characteristics for an array of circular water jets used to quench a copper slab. The jets were equally spaced in a staggered geometry, with pitches ranging from 4 to 12 nozzle diameters. The impingement surface was positioned vertically, which caused the downward rows of jets to be washed from above, while the separation distance between the heater and the nozzle plate was kept small (four to eight nozzle diameters). The authors reported the nucleate boiling heat flux to be independent of the nozzle-to-plate spacing and nozzle diameter but depen-

D. H. WOLF ET AL.

38 7 ,

I

I

2.63 0

pool boiling

q"

[MWI m2

1

0.7

i

0.5

3

FIG. 13. Nucleate boiling data for a free-surface ( H = 0) and submerged ( H = 5 mm, z = 3 mm), circular jet of water. (Replotted from Katto and Kunihiro [14], used with permission.)

dent on the wall superheat, nozzle-to-nozzle spacing, and square root of the jet velocity. Maximum heat fluxes were observed for nozzle spacings of approximately 8 to 10 diameters. No information was provided concerning the number of jets or the velocities investigated, nor were any boiling curves provided to reveal the velocity dependence. Nonn et al. [46] examined boiling heat transfer for a configuration of one, four, or nine free-surface, circular jets 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. Their nucleate boiling data showed no dependence on the velocity (3.2 and 6.4 m/s) for a single jet. However, a subtle influence of the velocity became more apparent with an increase in the

JET IMPINGEMENT BOILING

39

TABLE I11 NUCLEATE BOILING CORRELATIONS (W/m*) = C ATJ'C)"] Range of AT,,, Author

n

('X)

740 42 2.93 x 130 340

2.3 3.2 7.4 3.0 2.7

8-31 26-47 24-31 21-33 18-38

Jet type

Fluid

C .~

Copeland [40] Ishigai et ul. [53]' Katsuta and Kurose [48]' Katto and lshii [15y Katto and Kunibiro [14]' Katto and Monde 1441. Monde and Katto [16], Monde [I 71" Ma and Bergles [18]' Miyasaka and lnada [34] Miyasaka et a/. [19] Monde and Katto 1611" Ruch and Holman [Sl] Toda and Uchida 1281 ~~~~

~~

a

~

Water Water R-I13 Water Water

Circular-free Planar-free Circular-free Planar-wall Circular-free and sub

Water R-I13

Circular-free Circular-sub

450 0.15

2.7 4.4

18-46 26-33

Water R-113 R-I13 Water

Planar-free Circular-free Circular-free Planar-wall

79 790 461 6100

3.0 2.0

26-90 15-30 17-44 16-68

~

~

~

~

~

~

~

~

~

1.95

1.42 ~

~~

~

Correlation has been obtained by graphical means and should be considered approximate.

number of jets. Results for the nine-jet configuration revealed somewhat higher heat fluxes with increasing velocity (2.8-11.3 m/s) for a fixed wall superheat. In a subsequent investigation by Nonn et al. [37] with a 50% mixture of FC-72 and FC-87, the same trends were evident. At a fixed jet velocity (3.1 m/s), however, the nucleate boiling data for the one-, four-, and nine-jet systems were indistinguishable (Nonn et al. [46]). Consistent with Monde and Katto [16], the effects of subcooling (AKa, = 25 and 37°C) were significant only at lower heat fluxes, while larger fluxes revealed a single boiling curve for the four-jet configuration. The authors also examined boiling heat transfer for three in-line heaters that were flush mounted in a channel and each cooled by a separate jet. The heater surfaces were oriented vertically and the direction of the in-line arrangement was either vertical or horizontal. For a fixed jet velocity and subcooling, little difference could be seen among the boiling curves for single or multiple jet-heater arrangements, regardless of the orientation of the heaters with respect to each other. Mudawar and Wadsworth [7] performed both single- and multiple-jet boiling experiments for confined impingement of FC-72. However, the design of their test apparatus was such that nine separate heaters were cooled by an equal number of independent jets. Each jet-heater module was separated from the others in a manner that rendered jet interactions negligible. The nucleate boiling curves for each of the nine modules were found to be equivalent to within 2.1% for jet velocities of 0.75 and 1.3 m/s.

D. H. WOLF ET AL.

40

a

domain

by an impinging jet

I

surface Poeition of impinging jets

b

I

I

Domain control led by an impinging jet

Heated surface position of impinging jets

C

n=4

n=3

n=2

n = 2*

Maximum flow length L (Lm)

a b c

n=4

n=3

n=2

n=2'

9.1 13.9 12.7

10.3 14.0

14.6 18.9 18.0

18.5 24.6 23.5

FIG. 14. Nucleate boiling for multiple impinging jets (free-surface, circular). Shown are (a) the various jet arrangements and maximum flow lengths, (b) nucleate boiling data for saturated water, and (c) nucleate boiling data for saturated R-113. (Monde et a!. [a], used with permission.)

JETIMPINGEMENT BOILING

41

Monde and Katto (1978) onb=3 Vn = 7.3 m/r .nb=4 =13.1 Anb=2 z11.2

10

30

83

50 K

ATsat

(b)

=11.7 =11.7

=10.2

Q

.f

. -

Monde and Katto (1978) 1

5

10

30 (4

ATsat

50 K

42

D. H. WOLF ET AL.

D. LOCALMEASUREMENTS ALONG SURFACE

THE

IMPINGEMENT

Miyasaka and Inada [34] measured local temperature distributions along the back surface of a 0.1-mm-thick platinum heater serving as the impingement surface for a planar, free-surface jet of water. As many as seven equally space thermocouples (spot welded to the heater) were used, spanning a distance of approximately 5 mm in the streamwise direction. The thermocouples were positioned symmetrically about the stagnation point, as well as 2 nozzle widths downstream to investigate the parallel-flow region of the jet. Data at the stagnation point showed little variation of the rear surface temperature (within 5OC) over the range of heat fluxes associated with nucleate boiling. Measurements in the parallel-flow region of the jet also showed little variation in the temperature (within 5°C) for large surface heat fluxes, but more significant variations developed (up to 40°C) at lower heat fluxes in proximity to boiling incipience. These findings should be viewed cautiously, however, since the spatial extent of the temperature measurements was relatively small (5 mm) compared to the width of the nozzle (10 mm). Hence, the reported local measurements comprised only a small fraction of the total area beneath the jet. With respect to the stagnation point and the location which was 2 $nozzle widths downstream, the fully developed boiling curves were nearly coincident for the velocities investigated (3.2 and 15.3 m/s). Although they did not report a local temperature distribution along the surface, Kamata et al. [ 5 8 , 5 9 ] did compare boiling curves at radial positions of 0 and 5 mm for a circular, confined jet of saturated water (d = 2.2 mm). 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 betwen 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 this flow geometry, the nucleate boiling curve was independent of the radial position (0 or 5 mm). In the transition boiling regime, however, spatial differences in the temperature were noted and attributed to the liquid exiting the confined area along the nozzle-plate, while the vapor traveled 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 nozzle-plate (between the nozzle-plate and heated surface) to prevent stratification of the liquid and vapor at large heat fluxes. With the same nozzle-to-surface spacing, the nucleate boiling heat transfer at the stagnation point was indistinguishable from that of the brimless nozzleplate [ 5 8 ] . However, differences existed between results obtained at radial distances of 0 and 5 mm for the brimmed nozzle-plate. For a fixed heat flux,

4

JET IMPINGEMENT BOILING

43

the surface temperature at a radial distance of 5 mm was approximately 5 to 10°C higher than at the stagnation point. Due to the transient nature of the experiment, however, it is also likely that the heat flux varied locally on the surface. Sano et al. [66], conducting transient experiments, reported nucleate boiling data at nine different streamwise locations (0 I x I 56 mm) for a free-surface, planar jet of saturated water (the nozzle dimension was not provided). No effect of streamwise distance on the nucleate boiling curve was evident. Vader et al. [121 have provided the most comprehensive investigation to date of local nucleate boiling for an impinging jet. They obtained local temperature measurements from the stagnation line to a downstream distance of up to 14 jet widths for a planar, free-surface jet of water. Surface temperatures were inferred from measurements on the rear of the heater and, for prescribed conditions, revealed coexistence of single-phase convection and nucleate boiling at different locations along the surface. The surface temperature distribution, in conjunction with high-speed photographs of the boiling process, enabled delineation of important features of boundary layer development, including the effects of nucleate boiling. Figure 15 shows the wall-to-fluid temperature difference as a function of the streamwise coordinate from the stagnation line for various heat fluxes and fixed values of the 105 90

75 60

F 45

+3

30

15

+ 1.00

A 2.45 0 2.19 0 2.01 0 1.72

0.73 0.49 A 0.25

10

12

0 0

2

4

6

8

14

XIWj

FIG. IS. Effect of heat flux on local surface temperature for a planar, free-surface jet of water when V, = 1.8 m/s and ATub= 70°C. (Vader et a/. 1121, used with permission of ASME.)

44

D. H.WOLF ET AL.

impingement velocity (1.8m/s) and subcooling (70°C).For the lower heat fluxes, the figure shows an increase in surface temperature with the streamwise coordinate as the thickness of the laminar boundary layer increases. In the vicinity of the temperature maximum, the boundary layer begins a transition to turbulent flow, and higher levels of mixing cause a decline in the surface temperature. The maximum in temperature (approximate boundary layer transition point) is seen to occur at smaller values of the streamwise coordinate as the surface heat flux is increased. The authors suggested that flow disturbances induced by vapor formation on the surface are sufficient to accelerate boundary layer transition, reducing the critical Reynolds number obtained under single-phase convection by as much as 75%. For moderate heat fluxes (1.00-1.23MW/m2), Fig. 15 shows nucleate boiling to be present but to be limited to downstream portions of the surface, where the temperature distribution is nearly isothermal. With increasing heat flux beyond 1.45 MW/m2, the entire surface experiences fully developed nucleate boiling. The data also reveal that increases in the heat flux from 2.01 to 2.45 MW/mZ do not appear to cause an attendant increase in the surface temperature, in apparent conflict with many other boiling investigations. However, according to the functional relationship qkNB AT,,,3, this modest increase in heat flux (- 22%) would have a small effect on the surface temperature. Namely, a 22% increase in heat flux would cause only a 7% elevation in wall superheat, which corresponds to an increase of only 2°C for the temperatures of Fig. 15. The data presented in Fig. 15 for the heat flux of 1.45MW/m2 demonstrate the unexpected trend of larger surface temperatures in the stagnation region than in downstream regions. Ma and Bergles [35] have reported the same type of behavior for a submerged, circular jet of R-113,where measurements in the parallel-flow region of the jet were achieved by displacing the impingement point from the center of the heater. The fully developed nucleate boiling curves were coincident for a jet velocity of 2.04 m/s, while an increase to 10.05 m/s shifted the parallel-flow curve to the left, causing this region to have a lower surface temperature than that of the stagnation point for a fixed heat flux. Vader et al. [12] suggested that the larger temperature at the stagnation point was a result of achieving fully developed nucleate boiling in that region, while partial boiling persisted downstream. By virtue of the sizable pressure gradient in the stagnation region, flow disturbances, such as those induced by bubble motion, are damped and the boundary layer remains laminar. Therefore, mixing near the surface is limited, and the transition from singlephase convection to fully developed nucleate boiling occurs over a small range of heat fluxes. Farther downstream, however, the flow is turbulent and the damping effects of the pressure gradient are absent. Hence there is considerable mixing between the highly subcooled fluid in the bulk flow and

-

JET IMPINGEMENT BOILING

45

the much warmer fluid at the heated surface, ultimately extending the partial boiling regime over a larger range of heat fluxes. The laminar stagnation region therefore reaches fully developed nucleate boiling prior to its occurrence at turbulent downstream locations. E. WALLJETS Toda and Uchida [28] examined single-phase and nucleate boiling heat transfer for a planar, free-surfacejet of water. The jet was oriented at an angle of 20" with respect to the surface, and impingement occurred at the leading edge of the heater. Independent of variations in the jet velocity (2-10 m/s), subcooling (O-3O0C), or nozzle width (0.5-0.7 mm), all of the nucleate boiling data were well correlated by the expression. q:NB

= 6100

(22)

where &NB has units of W/m2 and AT,, has units of "C. This correlation is based on superheats in the range from 16 to 68°C and is presented in Fig. 11 with the correlations for normal jet impingement. Toda and Uchida suggest that the weaker temperature dependence (n = 1.42) may be attributable to boiling occurring only on downstream portions of the heater (by observation). This premise is unlikely, however, since spatially limited boiling was reported to occur only at lower heat fluxes, while larger heat rates were said to induce boiling over the entire surface. The relationship q&e would then apply at low heat fluxes, while a stronger temperature dependence would apply at larger fluxes. To the contrary, however, all of the data were well represented by Eq. (22). Similarly, Katto and Ishii [l5] investigated boiling heat transfer to a planar, free-surfacejet of saturated water. The nozzle was oriented at an angle of 15" with respect to the surface, and impingement occurred at a location 2 mm upstream of the leading edge of the heater. For velocities in the range from 4.1 to 10.1 m/s and streamwise heater lengths spanning 10 to 20 mm, little effect on the nucleate boiling heat transfer could be detected. Consistent with other reports, results for pool boiling experiments were shown to approach asymptotically those for forced convective flow of the jet. Although no correlation of the nucleate boiling data was provided by the authors, the following approximate relationship was developed from Fig. 7 of their publication:

-

&NB

X

130

(23)

where q g N B has units of W/mZand AT,,, has units of "C. The correlation in Eq. (23) is based on the wall jet results for superheats in the range from 21 to 33°C. Equation (23) is plotted in Fig. 1 1 with results for impinging jets, and

D. H.WOLF ET AL.

46

differences are shown to be relatively small. This finding agrees well with those of Miyasaka and Inada [34] and Vader et al. [l2], who found little variation in the fully developed nucleate boiling curve with downstream distance from the stagnation point of an impinging, free-surface jet (see Section 1II.D earlier).

F. OTHERPARAMETERS INFLUENCING BOILINGHEAT TRANSFER 1. Temperature Excursions Several investigations have observed the phenomenon of a temperature excursion or temperature overshoot, which corresponds to a decrease in the wall superheat with increasing heat flux following incipience. An example of such behavior, taken from the work of Ma and Bergles [l8], is shown in Fig. 16 for a circular, submerged jet of saturated R- 113 and a velocity of 1.08 m/s, where the overshoot is approxiamtely 5°C. Additional observations of the overshoot ( s 3°C) were also reported at a velocity of 2.72 m/s, both at the stagnation point and at a radial distance of two nozzle diameters. In a subsequent publication encompassing larger ranges of velocity and subcooling, Ma and Bergles [35] again observed the temperature overshoot and found the most pronounced effects (AT. x 5°C) at lower jet velocities ( z1 m/s) and subcoolings ( NN OOC). An increase in the subcooling to 20°C for a fixed velocity was shown to decrease the overshoot to less than 3°C. Temperature overshoots were also observed by Miyasaka and Inada [34] for a planar, free-surface of jet of highly subcooled water. The overshoots were evident in both the stagnation and parallel-flow regions of the jet. The overshoot was greatest at lower velocities (1.1 and 3.2 m/s) and was virtually indiscernible at the maximum velocity of 15.3 m/s. Similarly, small temperature excursions were also evident in the data of Vader et al. [12] for a planar, free-surface jet of subcooled (ATTsub = 50-70°C) water. Consistent with the other reports, the excursions were most pronounced at lower impingement velocities (1.8 and 2.5 m/s) and nonexistent at higher velocities (4.5 m/s). Temperature excursions have commonly been reported in investigations involving the direct cooling of electronic components with nonwettng, dielectric fluids. Bar-Cohen and Simon [67] suggest that temperature overshoots are a consequence of an insufficient number of active nucleation sites on a surface. Most dielectric fluids typically have small wetting angles, making the entrapment of gas or vapor within surface cavities difficult, thus causing nucleation to occur either in the bulk liquid (homogeneous nucleation) or at a limited number of microcavities on the heated surface (heterogeneous nucleation). A deficiency in the number of nucleation sites

JET IMPINGEMENT BOILING lo6-

I

l

l

k"

01 0

:

1

O

-

j

ooo

lo5:

-

-

/

1'

0

0-

I

Burnout Overshoot

0

E

I

&-

-

01

I

I

-

47

i

i 0

0

Test section No. 6

ol4-

o

V, = 1.08 m/s r/d = 0 z/d = 2.0

0

Pool boiling

10' ATsat

("C)

FIG. 16. Nucleate boiling data for a submerged, circular jet of R-113 showing the temperature excursion at the point of incipience. (Ma and Bergles [18], used with permission of ASME.)

can delay incipience, allowing the wall temperature to surpass that which is typical of nucleate boiling. Once incipience has been initiated, the resulting vapor impregnates previously flooded cavities and the wall temperature declines in response to subsequent nucleation. Based on relationships for heterogeneous boiling incipience and nucleate boiling heat transfer, Bar-Cohen and Simon [67] have proposed the following approximate relationship for the magnitude of the temperature excursion

(AKA:

D. H. WOLF ET AL.

48

/

fully-developed

U

M

Wall Superheat

log AT,,,

FIG. 17. Schematic showing the temperature excursion at the point of incipience. (Adapted from Bar-Cohen and Simon 1671.)

where p e is the partial pressure of vapor within a surface cavity

P is the local pressure near a surface cavity, c is the surface tension, [ is the cavity radius, and pnc is the partial pressure of noncondensible gases. The equation may best be understood in terms of Fig. 17, which shows the temperature excursion to be the difference beween the incipient superheat (the term in square brackets) and the superheat that corresponds to the incipient heat flux (q&) but is based on a correlation for fully developed nucleate boiling (as in Table 111, for example, which provides values of C and n). The degree of the temperature excursion is then closely related to the surface structure (i), wetting angle of the fluid-surface combination (C, a), and presence of noncondensibles (pnc). In general, this relationship suggests that nonwetting fluids (R-113,FC-72) .are most susceptible to temperature overshoots. Moreover, Bar-Cohen and Simon [67] have shown through Eq. (24), as well as through limits for homogeneous nucleation and available data, that the degree of the excursion decreases with increasing velocity, subcooling and operating pressure.

JETIMPINGEMENT BOILING

49

2. Hysteresis Ma and Bergles [18, 351 obtained nucleate boiling curves by both increasing and decreasing heater power for a circular, submerged jet of R-113. In most cases, differences between results for the two directions of heating were negligible for both the nucleate boiling and single-phase regions of heat transfer. When differences did exist, however, the boiling curve for a decrease in power did not follow the overshoot (see Fig. 10, for example). Small levels of hysteresis were also shown to be evident in the vicinity of boiling incipience (Fig. 5). The temperatures at which boiling was initiated with increasing power and ceased with decreasing power differed by nearly 4°C for a jet velocity of 2.04 m/s and a subcooling of 12S"C. Nonn et al. [46] have also demonstrated hysteresis for a circular, freesurface jet of FC-72, with differences being most marked in the region of boiling incipience. Moreover, data are presented for two subsequent cycles of increasing and decreasing power. Although the results for decreasing power are coincident, two separate paths characterized the data for increasing power. This behavior is most likely caused by a change in the number of vapor embryos with cycling. Additional experiments involving four jets positioned at the corners of a square heater showed much smaller differences in the boiling curve as a function of the direction of heating. Minor amounts of hysteresis were evident for a single jet composed of a 50% mixture of FC-72 and FC-87 (Nonn et al. [37]), and the phenomenon was seen to diminish for multiple-jet impingement.

3. Surface Conditions Ma and Bergles [35] observed shifts in the nucleate boiling curve from dayto-day experiments for the same global operating conditions, despite regular surface cleaning and system degasification (Fig. 18). The authors noted that the general trend was toward improved boiling performance with time and cited similar findings for pool boiling by Joudi and James [22], who found R113 to reverse the contamination process experienced in some boiling experiments. Joudi and James showed that the cleansing action of R-113 dissolved surface contaminants left during the boiling of other liquids, thereby increasing the number of active nucleation sites. They did not, however, report day-to-day shifts in the boiling curve for experiments conducted exclusively with R-113. Aihara et al. [61] reported nucleate boiling data for a circular, submerged jet of saturated liquid nitrogen impinging on a concave, radially confined, hemispherical heater. Data were provided for sanded (emery paper no. SOO), machined, and mirror finishes of the copper surface. Whereas the nucleate

D. H. WOLFET AL.

50

I lo6987-

-E

N

2

2

R5

6-

54-

v

Decreasing power

0

U

Test

3-

2-

I 051 10'

Section Date 12 12 12 12 14 14 14 14

06/08/83 06/09/83 06/09/83 06/30/83 10/08/83 10/09/83 10/09/83 10/12/83

Run V, R3 R5 R8 R10' R2 R6

R7 R13

(mfS)

ATsub(oc)

1.08 1.08 2.72 2.72 5.83 10.05 5.83 2.04

I

I

1

I

2

3

4

5

10.5 20.5 20.5 12.0 12.5 12.5 1215 12.5

$: $:

1

1

:: $: $:

tc $: :$ 1

1

6 7 8 9102

ATsat ("C) FIG. 18. Effect of changing surface conditions on nucleate boiling for a submerged, circular jet of R-113. (Reprinted with permission from Ma and Bergles [35} 01986, Pergamon Press PLC.)

boiling curves for the machined and mirror finishes were indistinguishable, the sanded finish produced slightly lower wall superheats (- 10%)for a fixed heat flux and V, = 1 m/s.

4. Moving Impingement Surface Taga et al. [68] have examined boiling heat transfer for a planar, free-surface jet of water impinging on a moving surface. The test specimen consisted of a brass plate heated in an electric furnace and pulled beneath the jet at speeds in the range 0.03 to 0.83 m/s. Although the authors do not specify the nozzle velocity (held constant for all experiments), a value of approximately 1.4 m/s has been estimated from other given parameters. Three thermocouples were mounted to the rear surface of the test specimen and separated by equal intervals of 250 mm. Complete boiling curves (based on all of the thermocouples) are given as a function of the surface velocity (0.23-0.63 m/s) and the initial plate temperature (150-450°C). Although minor effects of the initial temperature were evident at low surface speeds, no discernible effects of

JET IMPINGEMENT BOILING

51

surface motion on nucleate boiling could be seen. These results should be interpreted cautiously, however, since each thermocouple on the plate is cooled by a different region of the jet with the passage of time. However, due to the considerable length of the plate (over 250 nozzle widths), each thermocouple was located within the parallel-flow region of the jet for the majority of the time. Although performing predominantly single-phase experiments, Zumbrunnen et al. [69] compared local convection coefficients for single- and twophase heat transfer from a moving surface. Results are reported for a planar, free-surface jet of subcooled water ( 5 = 2.60 m/s) impinging on a surface moving at a speed of 0.6 m/s. For surface temperatures near 150°C, the local heat transfer coefficient for nucleate boiling exceeded that for single-phase convection by more than a factor of 3. Moreover, the maximum heat transfer coefficient for nucleate boiling was located approximately one nozzle width upstream of the stagnation line, while the maximum single-phase coefficient coincided with the jet centerline. Data of Chen et al. [70,71] also reveal enhancement in the local heat transfer coefficient for boiling over that of single-phase convection on a moving surface, but the augmentation is considerably less than that reported by Zumbrunnen et al. [69]. They examined heat transfer from a moving, downward-facing plate to a circular, free-surface jet of subcooled water. For plate and jet velocities of 0.35 m/s and 1.77 m/s, respectively, Chen and Kothari [70] found the stagnation point convection coefficient to be 60% larger for an initial plate temperature of 122°C than for 85°C. For plate and jet velocities of 0.50 m/s and 2.30 m/s, respectively, Chen et al. [71] found less than a 20% increase in the convection coefficient in going from an initial plate temperature of 88 to 240°C. With regard to the effects of surface motion on boiling heat transfer, Chen et al. [71] found an increase in the stagnation point convection coefficient of approximately 15% for an increase in plate velocity from 0 to 0.5 m/s. Enhancement is based on the steady-state convection coefficient for the stationary plate, which was obtained several hundred milliseconds after impingement. At the time of impingement, the coefficient was observed to exceed the steady-state value by nearly 2074 which the authors attributed to the thermal shock of the jet striking the plate. Despite the foregoing studies, the effects of surface motion on boiling heat transfer are considered to be inconclusive at this date.

5. Nozzle Geometry Wadsworth and Mudawar [72] examined the effects of nozzle geometry on single-phase and nucleate boiling heat transfer for a planar, confined jet of FC-72. Figure 19 shows the different nozzle configurations and the corre-

D. H.WOLF ET AL.

52

+ 3.175mm

--I

n nIt

ri 'n-r

it-

-I -I

Wn

Flat Profile

+

3.175rnm

B

A

2

wn

Vee Profile

19

"E

3, *

I-3 . 1 7 5 m

10

0

-

0

d

U

P 0

B

O@O 0

B 0

1 .o

m

a

Curved Profile

Vee Profile

JET IMPINGEMENT BOILING

53

sponding data. It is evident that the nozzle geometry has no effect on singlephase convection and only a minor effect on nucleate boiling. However, the combination of a relatively low Reynolds number (Re, x 4300) and a straight development length of five nozzle widths for each configuration has no doubt damped the effects of the upstream geometries. Wolf et al. [73] showed the effect of nozzle type on single-phase impingement heat transfer to be significant and to be coupled to the turbulence level of the jet. The absence of any effect for the single-phase results of Fig. 19 suggests that the level of turbulence is approximately the same at the exit of each nozzle.

6. Heater Geometry Aihara et a f . [61] investigated the effects of heater geometry on nucleate boiling heat transfer with a circular, submergedjet (d = 0.8 mm) of saturated liquid nitrogen. The circular, radially confined heater configurations (Fig. 20a) consisted of a flat surface (A, = 4.15 mm'), a concave hemispherical surface (A, = 8.31 mm2), and a flat surface with a small, concentrically located, conical projection (needle) (A, = 6.32 mm2). The nucleate boiling data are presented in terms of a heat flux (Fig. 20b) &, which is based on a cross-sectional reference area that is located below the boiling surface and is common to all geometries (Aref= 7.55 mm2). For a fixed heat flux, the heater with the conical projection had wall superheats that were approximately 30% lower than those for the flat and hemispherical surfaces. However, the critical heat flux (based on Aref)for the hemispherical surface was approximately 70 and 90% above the value of CHF for the flat surface with and without the conical projection, respectively.

7. Jet Suction McGillis and Carey 1741 peformed nucleate boiling experiments with R-113 for both submerged jet impingement and jet suction. The jet geometries are shown in Fig. 21a, along with channel flow obtained on the same apparatus by Strom et al. [75]. The nucleate boiling data are shown in Fig. 21b, along with the pool boiling results obtained by Ma and Bergles [18]. Despite dramatic differences in the flow geometry, the effects on nucleate boiling are seen to be negligible, typifying the dominance of bubble motion in two-phase, convective heat transfer. IV. Critical Heat Flux

This chapter attempts to provide a comprehensive review and analysis of the current knowledge base on the critical (maximum) heat flux in impingingjet systems. This topic has been the most extensively examined in the impingement boiling literature; yet, for some jet configurations, comprehensive

54

D. H. WOLFET AL. A, = 0.55

A, = 1.1

Aref

Are1

Flat

A, = 0.84 Aref

Hemispherical

Flat with a Needle

All dimensions are in mm. (a)

A : FLAT WITH

-

2

EMERY 500 z = 0.5 mm

I 4

Go = 0.5 gls

0

6

4

8

10

20

ATsat (OC) (b)

FIG. 20. Effect of heater geometry on nucleate boiling. Shown are (a) the heater configurations and (b) nucleate boiling data for a submerged, circular jet of saturated liquid nitrogen. (Aihara et al. [61], used with permission.)

JETIMPINGEMENT BOILING

55

parametric investigations have not been reported and a fundamental understanding of the boiling process is lacking. Most publications dealing with the critical heat flux for impingingjets have included correlations of the data. In order to provide the broadest possible spectrum of information, virtually all of these expressions are contained in the following review, with efforts made to delineate their relative merits and/or shortcomings. The critical (maximum) heat flux literature cited in this chapter is summarized in Tables IV and V, along with particulars concerning the respective experiments. A. SINGLE-JET IMPINGEMENT 1. Efects of System Parameters on the Critical Heat Flux for Free-Surface, Circular Jets

The relationship between the critical heat flux and various system parameters depends on the specific flow conditions. Several authors (Katto and Shimizu [76], Monde et al. [77]; Monde and Okuma [47]; Monde [78]) have proposed and substantiated the existence of four different CHF regimes (referred to as the V-, I-, L-, and HP-regimes). In each regime, dependence of the critical heat flux on parameters such as the jet velocity, density ratio (pf/pg), and heater diameter has been shown to differ markedly. To date, however, speciJc demarcations between the respective regimes have not been proposed. Hence, in the following discussion, delineation of the different regimes in terms of generalities such as lowJlow rates or high pressures is unavoidable. At atmospheric pressure, only the L- and V-regimes have been observed. The L-regime (limiting condition of CHF) denotes the limiting condition for which the latent heat of a saturated, circular jet (hfgpfV,nd2/4)is approximately balanced by the heat extracted from the surface (circular) at the point of burnout (q&7cD2/4), thereby inducing nearly complete evaporation of the liquid. This condition occurs with low mass flow rates or large ratios of the heater to nozzle diameters (Did), and the dependence of CHF on jet velocity is of the form qCHF V,. By contrast, the V-regime (variance of CHF with respect to velocity) occurs at larger mass flow rates, where the fraction of liquid consumed by evaporation is small. Although this regime has been shown to exist at elevated pressures, the V-regime encompasses the majority of flow conditions at atmospheric pressure (the only other observed being the L-regime) and generally demonstrates a dependence on jet velocity of the form qgHF VIf/3. The I- and HP-regimes have been reported only at pressures above atmospheric. The I-regime (invariance with respect to velocity) has been reported to occur at moderate pressures and generally

-

N

56

D. H. WOLF ET AL.

FIG. 21. Effect ofjet suction on nucleate boiling. Shown are (a) the flow arrangements and (b) nucleate boiling data for a confined,circularjet of R-113. (McGillisand Carey [74], used with permission of ASME.)

57

JET IMPINGEMENT BOILING 106

h

N

E

s .U

1o5

h

I

Ma and Bergles (1983) saturated pool boiling flush elements

4 1

(b)

FIG. 21. Continued.

demonstrates little or no dependence of CHF on the jet velocity. At much larger pressures and large ratios of the heater to nozzle diameters, increases in the critical heat flux with increasing velocity were again evident (HP-regime). To avoid confusion in the following review, the V-regime, which is the most widely investigated, will be addressed first. To date, the specific mechanisms causing these different regimes are unknown. Moreoever, investigations into the effects of low mass flow rates and elevated pressures on CHF have been limited to the aforementioned studies, which have employed free-surface, circular jets of diameter smaller than that of the heater. Hence, these observations were limited to conditions

TABLE IV CRITICAL (MAXIMUM) HEATFLUXINVESTIGATIONS-OPERATING PARAMETERS.

Author Aihara et al. [61] Andrews and Rao [lo01 Baines et al. ClOS] Cho and Wu [13] Copeland [40] Ishigai and Mizuno [83] Ishigai et al. [531d lshimaru et al. [62] Kamata et al. [%Id Kamata et al. [59Id Katsuta and Kurose [48] Katsuta and Kurose [48] Katsuta and Kurose [48] Katto and Haramura [lOn Katto and Haramura [lo71 Katto and Isbii [l5] Katto and lsbii [lq Katto and Isbii [lg Katto and Kunihiro [I41 Katto and Kunihiro [14] Katto and Kurata [lo61 Katto and Kurata [lo61 Katto and Monde [44] Katto and Shimizu [76] Katto and Shimizu [76] Katto and Shimizu [76l Ma and Bergles [l8] Matsumura et al. [941d McGillis and Carey [74]

Jet type

Fluid

Circ-sub Planar-sub Planar-free Circ-free Circ-free circ-free Planar-free Circ-sub circ-conc Circ-conf Circ-free Cic-free Circ-free Planar-free Planar-free Planar-free Planar-free Planar-free Cic-free Circ-sub Planar-sub Planar-sub circ-free Cic-free Circ-free Circ-free Circ-sub CiC-free Circconf

Nitrogen Water Water R-113 Water Water Water Nitrogen Water Water Mixture‘ R-11 R-113 R-113 Water Water R-113 Trichloroethane Water Water Water R-113 Water R-12 R-113 Water R-113 Water R-113

A Tub (“C) 0 0-62 0

-

4-78 45-80 5-55 0 0 0 <2

<2 <2 0 0 <5 <5 <5 <3 13 <3 <3 <3 0 0 0 0-29.5 0-89 0-30

v,

d or w,

0.77-1.64 0.3-2.0 0.854.9 0.7-8.2 0.79-6.4’ 1.3-9.0 1.O-3.17 0.22-1.34 10-17 12-20 0.5-2.69 0.35-3.48 1.22-3.84 1.8-65 3.7-5.4 1.5-13 1.5-9 2.5-7.5 1-3 1-3

1.5-9.1

1.3-6.0 5.3-60 0.9-18 2-15 2-3 1.08-2.72 1.3-4.0 0.95-3.08

z

(mm)

(m/s) 0.8

-

1.33 0.76 0.28-0.39 5.7-17 6.2 0.6 2.2 22 2.38-4.04 2.&3.81 1.99-3.81 0.4- 1.5 0.4- 1.5 0.56-0.77 0.56-0.77 0.56-0.77 0.71- 1.60 0.7 1-1.60 5-10 5-10 2 2 2 2 1.067 2 1.O

Degasb

Comments

0.5-2.2 12-51

-

Wall jet

13 8.0-9.2

-

15 0.6 0.3-0.6 0.3-0.6 -

-

-

1-30 1-30

-

WaU jet Wall jet Wall jet Wall jet Wall jet

Some plunging Wall jet Wall jet

30

-

2 -

1.O

P, > 1 bar

Miyasaka and Inada [34] Miyasaka ef al. [19] Monde [171 Monde and Furukawa 1451 Monde and Furukawa 1451 Monde and Katto [16, 51, 571 Monde and Katto [16. 51, 571 Monde and Okuma [47] Monde and Okuma [47l Monde et al. [64] Monde er al. [64] Monde et a/. [77J Monde et al. [77] Monde el al. [77l Monde et al. [78] Mudawar and Wadsworth [7] Nonn et al. [46Ib Nonn et al. [37Ib Ochi et al. [93]' Skema and SlanEiauskas [99] Wadsworth [38] Wadsworth and Mudawar [lo91

Planar-free Planar-free Circ-free circ-free Circ-sub Circ-free Circ-free Circ-free Circ-free Circ-free Circ-free Cic-free Circ-free Circ-free Circ-free Planar-conf Circ-free Circ-free Circ-free Circ-sub Planar-conf Planar-cod

Water Water Water R-113 R-113 R-113 Water R-113 Water R-113 Water R-12 R-113 Water R-12 FC-72 FC-72 Mixtureh Water Water FC-72 FC-72

85-108' 85-108' 15 <2 t 2 0-16 0-30 1 3 1 3 t 5 t 5 0 0 0 0

4.5-406 20-37 20-30 5-80 85-150' 4.5-408 10-w

1.1-15.3 1.5-15.3 0.3-15 1-4 1-4 1.7-26 1.7-26 0.33-13.7 0.33-13.7 1-20 1-20 0.9-20 0.7-16 1.6-20 0.9-20 1-13 1.6- 12.7 1.6-12.7 3 1-35 1-13 2-13

10 10 0.7-4.15 1.1 1.1 2.0-2.5 2.0-2.5 0.7-4.13 0.7-4.13 2 2 2 2 2 2 0.127-0.508 0.5-1.0 0.5-1.0 5-20 3-18 0.127-0.508 0.254-0.508

15 15 -

5 5

Some plunging

-

3 3 ~

-

10 0.508-5.08 0.1-5.0 0.1 -4.0 25 6-72 0.508-5.08 2.54-5.08

2-4 jets 2-4 jets Pa > 1 bar Pa z 1 bar Pa > 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
No. loo0 emery -

No. 5/0emery; acetone No. 510 emery; acetone -

-

No.0 emery; acetone -

-

Vapor blasted -

No. 100 emery -

Vapor blasted Vapor blasted

" % Area coverage refers to the percentage of the heater surface area covered by the nozzle area. Angle refers to the angle of impingement (90" being normal). Orientation refers to the direction the heater surface is facing with respect to gravity. Emery refers to polishing the surface with the given grade of emery paper. Acetone refers to cleaning the surface with acetone. Oxygen-free copper. 'Sanded (emery no. 500), machined, and mirror finishes.

62

D. H.WOLFET AL.

where the heater surface was covered with a thin liquid film. It is unclear whether these regimes would also exist for submerged or confined arrangements. Similarly, classification of the regimes with respect to pressure may be superficial in that the actual distinction between regimes could, for example, be more fundamentally related to properties such as the density ratio (pf/pg). Hence, reports that a particular regime has been observed only at elevated pressures do not necessarily reveal a general result. a. Jet Velocity. Numerous publications by Katto and Monde have addressed the effects of jet velocity on the critical heat flux, with each subsequent report demonstrating a refinement in their understanding of the dependence. Initially, Katto and Kunihiro [14] showed CHF to increase linearly with increasing velocity at the nozzle exit, but the data spanned only a limited range of velocities (1-3 m/s) for a saturated jet of water. For a much larger range of velocities (5.3-60 m/s), Katto and Monde [44] showed CHF to vary with Monde and Katto [16] performed an extensive study of CHF for impingingjets of water and R-113 at atmospheric pressure, in which the jet velocity, subcooling, nozzle and heater dimensions, and heater orientation with respect to gravity were varied. For velocities ranging from 1.7 to 26 m/s, they were able to obtain a good correlation of all of their data, in addition to those of the previous study (Katto and Monde, [MI). Their expression is of the form

where the term (o/pfV;D) is an inverted Weber number and correction factor to account for the effect of subcooling:

&,,b

is a

The correlation reveals the dependence of the critical heat flux on jet velocity to be VAI3. Katto and Shimizu [76] sought to extend the range of the density ratio (pf/pg)to values below that of R-113 at atmospheric pressure (approximately 200). Employing saturated R-12 at pressures from 6 to 28 bar, density ratios in the range from 5 to 40 were achieved. Although these elevated pressures generated data in both the V- and I-regimes (to be discussed below), the Vregime data were best represented by the following expression for velocities in the range 0.9 to 18 m/s:

JET IMPINGEMENT BOILING

63

Monde [17] confirmed the cube-root power dependence of q&.,F on the velocity for a saturated jet of water with 0.3 I V , I15 m/s and also greatly extended the ratio of heater-to-nozzle diameters ( D / d ) to 36.4. Although the density ratio and Weber number terms of the Monde and Katto correlation [Eq. (26)] were preserved, a dependence on the diameter ratio was added: 1

+ 0.00113(D/d)’

This expression effectively correlated Monde’s results, as well as those of Katto and Kunihiro [14], Katto and Monde [44], Katsuta [79], and Monde and Katto [16]. Moreover, Eq. (29) also predicted CHF data when the jet impinged eccentrically on the heater surface to within f20%,provided D was replaced with twice the length from the stagnation point to the farthest edge of the heater. Monde et al. [77] also examined saturated R-12 at elevated pressures (6 to 28 bar), as well as saturated water (1 to 6 bar) and R-113 (1 to 3 bar). As found by Katto and Shimizu [76], the elevated pressures generated data in both the V- and I-regimes. The V-regime data were well correlated for velocities in the range 0.7 to 20 m/s by

However, the authors did not include the term from Eq. (29), which accounts for the ratio of heater to nozzle diameters. Monde [80] subsequently developed a new correlation based on a general CHF model proposed by Haramura and Katto [Sl]. The correlation was intended to account for the wide range of density and diameter ratios employed in the preceding investigations. The resulting expression, developed from the data of Katto and Kunihiro [14], Katto and Monde [44], Katsuta [79, 821, Monde and Katto [16], Katto and Shimizu [76], Monde [17], and Monde et ai. [77], is of the form =0.280(;) 0.645 tT (31)

[

where the characteristic length scale of the Weber number is now the difference between the heater and nozzle diameters (D - d). Equation (3 1) correlates about 94% of the aforementioned V-regime data to within & 20%. Ishigai and Mizuno [83] examined the effects ofjet velocity (1.3 I V, I 9.0 m/s) on CHF for a subcooled (45 I AXub I 80°C) jet of water. Increases in the jet velocity were shown to cause attendant increases in the critical flux according to the relation q&F V,0.34.Correlation of the data was achieved

-

D. H. WOLF ET AL.

64

with the following dimensional expression:

where q&, VJd, and A x u b have units of W/m2, l/s, and "C, respectively. [Note: Although the jet diameter ( d ) used in the foregoing correlation was specified by Ishigai and Mizuno to have a unit of millimeters, this specification caused the correlation to consistently underpredict the data shown in their figures by an order of magnitude. By using meters as the unit for nozzle diameter, agreement between the correlation and data is good. Hence, it is assumed that the unit of meters was intended.] Although the correlation provided a good fit to their data, its usage must be limited to water and to the specified range of subcoolings. Because of the direct dependence on ATub,the correlation clearly becomes unrealistic for near-saturated jets. Moreover, the inverse relationship between CHF and the nozzle diameter is inconsistent with much of the CHF impingement literature. Katsuta and Kurose [48] reported critical heat flux data for saturated jets of R-11, R-113, and a water-surfactant mixture (Rapisool B-80) in the velocity range 0.5 to 3.81 m/s. The data were well correlated by the following expression:

Nonn et al. [46] presented a limited amount of CHF data for a subcooled jet of FC-72. Although the ranges of velocity and subcooling were not provided for the CHF results, ranges for the investigation in general were reported to be 1.6 IVn I12.7 mfs and 20 I AT,,, I 37°C. Comparison was made between their data and Eqs. (26) and (27), proposed by Monde and Katto [16]. Although the cube-root dependence on the velocity was confirmed, good agreement could be achieved only by reducing the coefficient of Eq. (27) from 2.7 to 0.456. Hence, although Eq. (27) accurately predicted the effects of subcooling for jets of water and R-113, its applicability to other fluids may be suspect (this issue is addressed more fully in Section 1V.A.l.b). The authors also correlated their data in terms of an expression proposed by Lee et al. [84] for subcooled flow boiling over surfaces with short heating lengths. Although the qualitative trends were matched, the data were consistently underpredicted from 15 to 20% by this correlation, which is of the form

--

(34)

JET IMPINGEMENTBOILING

65

where &,,b is a correction factor for the effect of subcooling

0.g52(E)

0.118

&sub =

1.414

(7) CprA%b

(35)

(Nonn et al. [46], employing a square heater, replaced D with the length of the heater diagonal.) Equations (34) and (35) also provided a satisfactory correlation of the limited data (2 datum) obtained in a subsequent study by Nonn et al. [37] for a subcooled, 50% mixture of FC-72 and FC-87. [Although the saturation temperature of the mixture was measured (T,,, = 41 "C), no information was provided concerning the measurement of other properties. However, the difference in properties between the two fluids, such as pf, pe, hfg, cpr, and 6, is within 8% at the atmospheric boiling point (Mudawar and Anderson [SS]).] Again, the ranges of velocity and subcooling were not provided for the CHF data, although ranges for the investigation in general were reported to be 1.6 I V, I 12.7 m/s and 20 I ATub I 30°C. Cho and Wu [13] reported CHF results for both a free-surface, circular jet and a spray of R-113. Increases in the jet velocity from 0.7 to 8.2 m/s increased the critical heat flux from 0.217 to 0.541 MW/m2 (subcooling unknown). Correlation of data for the four largest velocities (4.2-8.2 m/s) yielded the following dimensional expression:

where q&F has the units of W/m2. The functional dependence of CHF on the velocity, heater dimension, and fluid properties implied by this correlation is in stark contrast to the majority of the literature. The velocity dependence given by qCHF V:.638is uncommonly high, and the suggestion that CHF increases with increasing heater diameter or decreasing surface tension is inconsistent with all other reports. The study used a single fluid (R-113) at atmospheric pressure and one heater dimension (D = 20.5 mm). Although most of the literature clearly demonstrates the relationship of q& V:l3 at atmospheric pressure, there is a lower limit of velocity below which this dependence is not longer valid. This lower limit is determined primarily by thermodynamic considerations, which for saturated conditions limit the maximum energy released by the heater (qZHFaD2/4)to the amount of latent energy stored in the jet (hfgp,V,,nd2/4).Hence, the requirement that

-

-

66

D. H. WOLF ET AL.

prescribes a linear variation of q& with velocity and is commonly referred to as the L-regime. Monde and Okuma [47] are the only authors to examine this regime thoroughly, and they did so with saturated jets of water and R113 impinging normal to a downward-facing surface. Rearranging Eq. (37), the authors proposed the following correlation:

where K is the ratio of liquid consumed by evaporation on the heated surface to the liquid supplied by the impinging jet. The authors suggested that K is related to the splashing of droplets away from the surface, thereby depleting the amount of liquid available for evaporation. Invoking dimensionless parameters that govern the Taylor instability of a liquid film, the following expression for K was proposed:

where [g(pf - p,)d2/a] is the Bond number. Equations (38) and (39) correlated all water and R-113 data to within approximately k40%.The authors note, however, that the expression for K is likely to change for upward-facing heaters, since its development depended on the splashing of liquid moving away from the surface under the influence of gravity. Figure 22 shows some of the R-113 data in dimensional form, along with Eq. (38) for the L-regime and Eq. (3 l), proposed by Monde [SO], for the V-regime. Demarcation between the two regimes was prescribed by the intersection of their respective CHF correlations. Hence, by setting Eq. (38) equal to Eq. (31) and approximating

P, (m/s) FIG. 22. Effect of velocity on CHF for a free-surface, circular jet of saturated R-113. (Reprinted with permission from Monde and Okuma [47] 0 1 9 8 5 Pergamon Press PLC.)

JET IMPINGEMENT BOILING

67

the last term on the right-hand side of (31) as (D/d)-0.364 (D/d > lo), it follows that

(Note: The density ratio exponent of +0.194 given by Monde and Okuma [47] should be -0.194, as shown in this equation.) Several other investigators have reported CHF behavior typical of the Lregime. Copeland [40], also employing a downward-facingheater, obtained a limited amount of CHF data (with considerable scatter) for subcooled and saturated jets of water. The author speculated, however, that some of the CHF data were characterized by heating that exceeded that required for complete evaporation of the jet (the L-regime). Applying Eq. (40) to Copeland’s parameters confirms that some of his low-velocity data were most likely obtained in the L-regime. Furthermore, despite significant scatter at larger jet velocities, the lower-velocity results revealed the linear dependence of CHF on V, that is suggested by Eq. (38). Katto and Kunihiro [14] reported monotonic increases in CHF with increasing velocity for all but one flow condition (data inside dashed circle in Fig. 23a). Namely, for the lowest rate of mass flow and the smallest nozzle diameter, a jet velocity of approximately 2.5 m/s yielded a critical heat flux that was lower than that for a jet velocity of 1 m/s. The authors attributed this behavior to an insufficient supply of liquid to the heater. However, the latent heat of the jet was approximately six to seven times larger than the heat transferred from the surface. Moreover, the same behavior is evident in Fig. 23b (data inside dashed circle) for the same mass flow rate and nozzle diameter, but with the jet plunging into a 5-mm-deep pool of water. Clearly, the abundant amount of latent energy stored in the pool greatly exceeds the heat transferred from the surface, which makes existence of the L-regime seem unlikely. For a saturated jet of R-11 impinging on an upward-facing surface, Katsuta and Kurose [48] also identified conditions at low mass flow rates where CHF occurred as a result of nearly complete evaporation of the liquid (identified by the authors as dryout). Correlation of data was achieved with an expression of the same functional form as Eq. (38) but with K = 0.04. However, this value of K is relatively small (Monde and Okuma [47] reported 0.08 I K Il), implying existence of the L-regime, even when only 4% of the impinging liquid is converted to vapor. A series of experiments by Katto, Monde, and co-workers revealed the existence of two additional CHF regimes at pressures above atmospheric (Iand HP-regimes). Katto and Shimizu [76] initiated the findings by operating at elevated pressures (6 to 28 bar) with saturated R-12 to expand the ranges

68

A

N

E

2 LL

:Io U

2A 1

"0

1

2

3

4

"0

1

2

3

4

A

N

E

2 U

I

:o U

-E

N

2

U

I

: 0

0-

-

Vn (m/sec)

0, (gmlmin) 60 d (mm) 0.71

150 180 270 1.60 1.165 1.60

FIG. 23. Effect of velocity on CHF for circularjets of saturated water with configurationsof (a) free surface (H = 0 mm, 1 5 z 5 30 mm), (b) submerged and plunging (H = 5 mm, 1 5 z I, 30 mm), and (c) submerged (H = 30 mm, 1 I, z < 30 mm). (Replotted from Katto and Kunihiro [14], used with permission.)

JETIMPINGEMENT BOILING

69

of both the Weber number and the density ratio employed in many of the CHF correlations. Their data revealed two distinct regimes of CHF dependence on the jet velocity, demarcated by the following critical value of the Weber number:

At Weber numbers less than this value, the relationship between the critical V,?,/3,corresponding to heat flux and the jet velocity was found to be (I&F the V-regime (variance with velocity) [Eq. (28)]. At Weber numbers beyond 6.10 x lo4,the critical heat flux was found to be insensitive to changes in the jet velocity, and the term I-regime was used to denote the invariance. Data for this region were correlated by

-

The notion of a critical Weber number implies the existence of a maximum velocity for a given fluid and heater diameter, above which no additional increase in CHF may be achieved. Monde et al. 1771 further considered the effects of elevated pressure by obtaining data for water (1-6bar) and R-113(1-3 bar), as well as for R-12 (6-28 bar). They showed that water, even at pressures as large as 6 bar, exhibits V-regime behavior, while the R-113 data were split between the Vand I-regimes, depending on pressure and velocity. Their I-regime data were well correlated by

where the coefficient and exponent of the density ratio differed significantly from those of Katto and Shimizu. Moreover, they reported the existence of an additional CHF regime (HP-regime), which occurred at very high pressures ( x 28 bar), and revealed increases in CHF with increasing velocity, much like the V-regime. Monde et a!. tentatively correlated their data as q& = 0.143 x lo6 V i / 2 ,where qGHF and V, have units of W/m2 and m/s, respectively. General demarcations between the V-, I-, and HP-regimes were not provided. Although Katto and Shimizu were the first to speculate about the existence of an HP-regime, they lacked supporting data, whereas Monde et al. clearly showed its existence at the same range of operating pressures. The only apparent difference between the two experiments was the ratio of heater to nozzle diameters (D/d), which were 5 and 10 for the Katto and Shimizu [76] and Monde et a/. [77] studies, respectively.

70

D. H. WOLF ET AL.

Monde [78] further quantified the characteristics of the HP-regime by noting that its existence depended not only on high pressures but also on large ratios of the heater to nozzle diameter (Did). Figure 24a through c shows the q&-I/n data of Katto and Shimizu [76] (D/d = 5), Monde et al. [77] (D/d = lo), and Monde [78] ( D / d = 20), respectively. Figure 24a reveals increases in CHF with increasing velocity at the lower pressures (6.00- 11.6 bar) and nearly complete invariance with respect to velocity at higher pressures (23.5-27.9 bar). At larger diameter ratios (Fig. 24b and c), the relationship between CHF and velocity is clearly one of increasing CHF with increasing V,, even at the largest pressure (27.8 bar). No suggestions concerning physical mechanisms were proposed. The general correlation given by Monde [78] for all three regimes was

where for the V-regime [as suggested by Monde [80], Eq. (31)] m = 0.645, n = 0.343, j = -0.364, C = 0.280; for the I-regime m = 0.466, n = 0.421, j = -0.303, C = 0.925; and for the HP-regime m = 1.27, n = 0.28, j = - 1.01, C = 0.209. The correlation reveals the velocity dependence on CHF to be V:.314 in the V-regime, V:.”* in the I-regime, and V:.44 in the HP-regime. Monde acknowledged that the limited data base for the HPregime (only 24 measurements) may not be sufficient to provide a high level of confidence in the recommended constants for this regime. Moreover, no general, quantitative demarcations between the regimes were proposed. It was noted, however, that all of the data with pr/pg > 67 fell within the V-regime with no apparent restrictions on the value of the Weber number as suggested by Katto and Shimizu [76] [Eq. (4111. Kandula [86] attempted to address the dichotomous nature of Katto and Shimizu’s [76] CHF data through an analytical approach (development presented in Section IV.A.6) based on physical mechanisms. The I-regime was suggested to be a function of the liquid viscosity and not the surface tension, as previously proposed. The following correlation was obtained:

The parameter r] results from a corresponding states analysis and is given by og,WT, =

( 7 )

where o is Pitzer’s acentric factor (Reid et al. [87]) and representative values are 0.348,0.253, and 0.175 for water, R-113, and R-12, respectively; W is the

JET IMPINGEMENTBOILING

71

0.5

1

2

3

5

10

20

0.5

1

2

3

5

10

20

I

I

1

2

3

5 L

0.5

I

'

I "

(a)

I

I

5

I

I

,

,

,

I

10

20

v. (nw

FIG. 24. Effect of velocity on CHF at various elevated pressures for a circular, free-surface jet of saturated R-12 with ratios of the heater-to-nozzle diameters of (a) D/d = 5 (Katto and Shimizu [76], (b) D/d = 10 (Monde [78]), (c) D/d = 20 (Monde [78]). (Used with permission of ASME.)

72

'^m D. H. WOLFET AL.

p, = 5.7 bar

Q5 0.3

P, = 9.6 bar

0.5

0.3 P, = 13.6 bar

05

03

1.o:

-

P. =27.8bar I

45-

V-reglme

- Eq. (31)

I

3

1.0

,

I

5

I

I

, . I

1 0 : 3 0, (m/s)

(b) FIG. 24. Continued.

'i-----_ JET IMPINGEMENT BOILING 0 P.=11.3bar

0.5

0.3 0.2 1.0 0 P, = 13.6bar

P. = 12.6bar

0.3

0.l

0.1

v.regtme. Eq. (31)

0.05

-

1

3

5

(4 FIG. 24. Continued.

10

Vm

30 (rn/s)

73

D. H. WOLF ET AL.

74

universal gas constant; T, is the absolute critical temperature; and A is the molecular weight. By equating Eq. (45) and the Katto and Shimizu correlation, Eq. (28), for the V-regime, the following expression was obtained:

:)1r+)1(z>

2.316

We, = 0.09,0(

(47)

For Weber numbers (We, = p,V:D/a) greater than this value, Eq. (45) was recommended, while Eq. (28) was suggested for smaller Weber numbers. Although predictions based on this correlation and the transitional Weber number were in good agreement with the results of Katto and Shimizu [76], a comparison with the broader data base of Monde [78] was not provided. Lienhard and co-workers attempted to correlate the CHF data for both the I- and V-regimes employing a single expression derived from a CHF model based on mechanical energy stability (addressed fully in Section IV.A.6). Lienhard and Eichhorn [88] originally proposed the model and correlation for CHF on cylinders in cross-flow and later extended it to impinging jets (Lienhard and Eichhorn [89]), utilizing the data of Katto and Monde [44] and Monde and Katto [16]. Subsequently, Lienhard and Hasan [go], benefiting from the published data of Katto and Shimizu [76], refined aspects of the correlation which depended on the density ratio. They reported the expression

where { N 0.4346

+ 0.1027(ln y) - 0.0474(1n 7)’ + 0.00426(1n y)3

(49)

and y is the ratio of densities (pf/pg). Correlation of 95% of the data reported by Katto and Monde [44], Monde and Katto [16], and Katto and Shimizu [76] was achieved to within &20%,independent of the respective regime. Later, Sharan and Lienhard [9l] reformulated the foregoing correlation in response to information from Monde that the results reported by Katto and Shimizu [76] contained an error. Monde [80] also noted the presence of an error in the results of Katto and Shimizu [76], which he attributed to their data reduction procedure. However, the magnitude of the error or its effect on their results was not provided. Monde et al. [77] performed experiments for some of the operating conditions of Katto and Shimizu and obtained Vregime data which were 10-30% lower. The difference was attributed to Katto and Shimizu’s inaccurate estimate of heat losses. Due to Sharan and Lienhard [91] the revised correlation is &HF = (0.21

PgkgVn

+ O.M171y)(-)

)(.

1000~ D Pf V l D

JET IMPINGEMENT BOILING

75

where

< = 0.486 + 0.06052(1n y ) - 0.0378(1n 7)’ + 0.00362(1n y)3

(51)

Again, correlation of the data utilized for Eq. (48), as well as that reported by Monde [17], was achieved to within +20%. Furthermore, in cases in which viscosity and gravity were suggested to have no influence [for (D/d)Re; < 0.40 and Fr, 2 81, the data are correlated to within k 8.66%. For fluids such as water and R-113 at atmospheric pressure, takes on values of 0.329 and 0.283, respectively (q&F,.,,, V:.342 and q&- ,,3 v : . ~ ~ ~ ) . Katto and Yokoya [92] formulated a CHF correlation to predict virtually all of the available data for circular, free-surface jets (Katto and Kunihiro [14]; Katto and Monde, [44]; Katsuta [79]; Monde and Katto [16]; Katto and Shimizu [76]; Monde [17]; Monde et al. [77]; Monde and Okuma [47] [not all data]; Monde [78] [not all data]). The resulting correlation was

<

-

-

where

< = 0.374(?)

-0.0155

< = 0.532(:)

-0.0794

with a root-mean-square (RMS) error of 15.7%. For water and R-113 at atmospheric pressure, 5 takes on values of 0.334 and 0.349, respectively 0.332 (qzHFwater and q&#HFR-I13 V,0’302). For comparison, Katto and Yokoya contrasted the correlations of Monde [78] [Eq. (44) for the V- (p,/p, > 71) and I-regimes (pr/pg < 71)] and Sharan and Lienhard [91] [Eq. (50)] with the same data. The Monde correlations had an RMS error of 20.2%, while that of Sharan and Lienhard had an RMS error of 16.3%. Although Katto and Yokoya concede that each of the three correlations provides a comparable fit to the data, they argue that their correlation follows the experimental trends better. Nevertheless, the present authors recommend the correlations of both Sharan and Lienhard [9l] and Katto and Yokoya [92] for use in predicting the critical heat flux for circular, free-surfacejets. Although the correlations for the I- and V-regimes proposed by Monde [78] provide comparable accuracy, there is some uncertainty associated with the demarcation between regimes. However, for atmospheric conditions, with jets of water or R-113, the V-regime correlation of Monde [80,78] performs well. The correlations of these three groups of investigators are summarized in Table V1, along with the applicable ranges.

-

TABLE VI RECOMMENDED CHF CORRELATIONS FOR CIRCULAR,

Author

Correlation

Katto and Yokoya [92]

Q = y(0.0166

+ 7.00y-1.'2)We&

(1

+

FREE-SURFACE, SATURATED JETS'

v,

d

Y

D (mm)

(mm)

D/d

(m/s)

5-1603

10-60.1

0.7-4.1

3.9-53.9

0.3-60

15-1603

10-60.1b

0.7-4.1b

5-57.1b

0.2-ab

1

5-1603

10-25.5

0.7-4.1

5-36.4

0.3-60

2 x 103-2 X lo6

We,orWe,-,

[ = 0 . 3 7 4 ~ - ~ .for " ~y ~2 248

6 = 0.532y-0.0794for y I 248

[ = 0.486 + 0.0.06052 (In y ) 4 Q -

q;rrr ,y=(:),weD4=(

PrVM

- 0.0378 (Iu 7)' + 0.00362 (h7)3

- d)

Pr v : D ),weD=(T)

P A K Estimates of widest range; specific ranges used in the correlation were not provided. 'No range provided nor a means by which to estimate. Rang of We,,

is most likely very similar to that listed for Monde [SO].

X

10'4

X

JETIMPINGEMENT BOILING

77

b. Subcooling. Ishigai and Mizuno [83] reported CHF data for a water jet with subcoolings in the range from 45 to 80°C. The data were well correlated AT:;;’. by Eq. (32), which revealed the unusual dependence of q& Clearly, Eq. (32) is restricted to water and the correlated range of subcoolings (45 I A z u b I 80°C); otherwise it would predict q& x 0 for a saturated jet. Monde and Katto [16] investigated the effects of subcooling on CHF for jets of water (3 I ATub I30°C) and R-113 (3 I AT,,, I 16°C). The results were well correlated by Eqs. (26) and (27), with the effect of subcooling represented by the parameter (1 &sub). The subcooling correction factor (&sub) depends on the square of the subcooling, as well as the selected fluid properties. At the highest investigated subcoolings of 16 and 30°C for R-113 and water, Monde and Katto reported enhancements in the critical heat flux of 41 and 34%, respectively. Nonn et al. [46] reported CHF data for a subcooled (20 I AT,, I 37°C) jet of FC-72. Monde and Katto’s [16] correlation [Eqs. (26) and (27)] was shown to predict the data well, provided the coefficient in Eq. (27) was changed from 2.7 to 0.456. Due to different thermophysical properties between FC-72 and the fluids employed by Monde and Katto (water and R113), the subcooling coefficients (aSu, = cPfAT,,,,/hfg) of Nonn et al. were significantlylarger (0.26 I@sub I 0.48) than those used to formulate Eq. (27) (0 I @sub S 0.11). Much better prediction of the data was achieved with a general CHF correlation (Lee et al. [84]) proposed for surfaces with short heating lengths [Eqs. (34) and (35)] and developed over a much larger range of subcooling coefficients (0.06 I @sub I 0.93). Equations (34) and (35) have also been shown (Lee et al. [84]) to provide a good fit to the R-113, subcooled data of Monde and Katto [16]. Subsequent results of Nonn et al. [37] for a subcooled (20 I AT,, I 30°C) jet composed of a 50% mixture of FC-72 and FC-87 were also well correlated by Eqs. (34) and (35); however, only two datum were provided. The effects of subcooling on the maximum heat flux, obtained by quenching a test specimen with a water jet, have been reported by Ochi et af. [93]. Increases in the subcooling ( 5 I ATubI 80°C) produced attendant increases in the maximum heat flux at the stagnation point. For a jet velocity of 3 m/s, a sixfold increase was reported for a change in subcooling from 5 to 80°C. However, no correlations were provided. Similar trends of higher maximum heat fluxes with increased subcooling were also reported by Matsumura et al. C941.

-

+

c. Fluid Properties. Several investigators have reported CHF data for more than one fluid and for a single fluid at various pressures (Monde and Katto [16]; Katto and Shimizu [76]; Katsuta and Kurose [48]; Monde et af. [77]; Monde and Okuma [47]; Monde [78]; Nonn et al. [37]), and the essence of

78

D. H. WOLFET AL.

the property dependence is provided by Eq. (31), which is representativeof Vregime data for a wide range of fluid properties. The critical heat flux depends on pf, pet 6,and hfg, as well as on cprfor a subcooled jet. For a saturated jet, the critical heat flux increases with increasing latent heat, density ratio (pJpe), and surface tension. d. NozzlelHeater Dimensions. Several investigators have examined the effects of nozzle diameter on CHF (Ishigai and Mizuno [83], 5.7 5 d I 17 mm; Monde and Katto [16], 2.0 5 d 5 2.5 mm; Monde [17], 0.7 S d 5 4.15 mm; Katsuta and Kurose [48], 1.99 5 d I 4.04 mm; Monde and Okuma [47], 0.7 5 d 5 4.13 mm; Ochi et al. [93], 5 I d I 20 mm). Nearly all of the previous correlations indicate an increase in CHF with increasing nozzle diameter (constant nozzle velocity), although Ishigai and Mizuno [83] [see Eq. (32)] and Ochi et al. [93] found CHF to increase with decreasing nozzle diameter. The effects of the heater diameter have been investigated by Monde and Katto [16] (11.2 5 D 521.0 mm), Monde [17] (11.9 5 D I 25.5 mm), Katsuta and Kurose [48] (15 5 D I 25 mm), Monde and Okuma [47] (405 D 5 60 mm), and Monde [78] (10 s D s 40 mm). The results of these investigations are well represented by the foregoing correlations, which indicate an increase in CHF with decreasing heater diameter. Such a dependence is related to the poor cooling characteristics of the parallel-flow region of the jet relative to those of the stagnation region, with the amount of convective transport declining monotonically with increasing radial distance. Indeed, due to an accumulation of vapor bubbles in the streamwise direction, initiation of the vapor blanket has been observed to occur at the perimeter of the heated section (Katto and Kunihiro [14]; Katto and Ishii [15]; 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 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, inducing additional blanketing. For a free-surface jet, Monde and Furukawa [45] have shown that, if the heater diameter becomes large enough, the critical heat flux can fall below the value predicted for pool boiling. With one exception obtained for limited operating conditions (Ishigai and Mizuno, [83]), all of the investigations cited thus far involved heater dimensions which were larger than the nozzle dimension, thereby ensuring the presence of a parallel-flow region for the jet. Presumably, a weaker dependence on D would be evident for cases in which the nozzle diameter exceeds that of the heater.

JET IMPINGEMENT BOILING

79

e. Surface Orientation. Monde and Katto [16] were the only investigators to obtain CHF data for both upward- and downward-facing surfaces. For saturated jets of water and R-113, no effects of the heater orientation could be detected. Furthermore, the correlations by Monde [80], Sharan and Lienhard [91], and Katto and Yokoya [92] have incorporated data from numerous investigations employing either upward- or downward-facing arrangements, and no dependence on orientation was demonstrated. Sharan and Lienhard [91] represented CHF data of Monde and Katto [16] for both configurations in terms of a Froude number VJ(gD)1/2.For Froude numbers as low as 10, there were no discernible differences in data for the two orientations. One possible exception to the foregoing behavior was suggested by Monde and Okuma [47], who showed a different type of CHF behavior to exist at very low liquid supply rates (the L-regime). For low jet velocities or large heater-to-nozzle diameter ratios, they showed that a significant fraction of the impinging fluid was converted to vapor, thereby depleting the amount of liquid available for effective cooling. Their results for a downward-facing heater were correlated by Eq. (38), which depends on K [Eq. (39)], the ratio of liquid consumed by evaporation on the heated surface to the liquid supplied by the impinging jet. They maintained that, since K depends largely on the amount of liquid which splashes from the surface and subsequently falls under the influence of gravity, a different expression for K would have resulted had the heater been upward-facing.

f. Nozzle-to-Surface Spacing. Katto and Kunihiro [14] have shown the nozzle-to-surface spacing to have little effect on the critical heat flux for a saturated jet of water (see Fig. 23a). Spacings in the range of 0.63-42 nozzle diameters were investigated for fixed velocities at the nozzle exit (1-3 m/s). Likewise, Nonn et al. [46] have shown spacings of 0.5-5 nozzle diameters to have no effect on CHF. However, when spacings decreased to the point where the fluid became confined between the face of the nozzle exit and impingement plane (0.1 and 0.3 nozzle diameters), enhancements in CHF were reported. An approximate 20% increase in CHF was induced by decreasing 'the spacing from 0.5 to 0.3 nozzle diameters for a jet of FC-72 (V, = 6 m/s). 2. Effects of System Parameters on the Critical Heat Flux for Free-Surface, Planar Jets a. Jet Velocity. Miyasaka et al. [19] reported nucleate and transition boiling data for a highly subcooled (ATub> 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

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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

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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

--

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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

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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.

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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

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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

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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].)

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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.)

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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).

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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

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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

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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

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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 (<0.5) was substantially less than the anticipated value. Monde and Katto [16] later reported G/Go% 0.9 at CHF for nearly the same operating parameters and attributed the differences to the type of heating method. The earlier investigation [44] employed a direct heating scheme with alternating current, while the latter [16] used an indirect method. Heating with alternating current induced periodic fluctuations in the surface heat flux and affected the amount of splashing from the free surface.

JET IMPINGEMENT BOILING

107

1.0 '-

0.5

0.3

,

D = 11.2 m m , d = 2.0 mm 0 Go = 2.39 kg/min Go = 1.70 D = 20.2 m m , d = 2.0 mm 0 Go = 2.22 kg/min W Go = 1.66

0.1

2

3

10'

5 q"

w/m2

FIG. 31. Effect of heat flux on the rate of splashing from the free surface of a circular jet of saturated water. (Reprinted with permission from Monde and Katto [16] 01978, Pergamon Press PLC.)

(Photographs of the splashing at different times during the heating cycle are provided by Katto and Monde [MI. A more detailed discussion of the relation between the heating cycle and rate of splashing is given, in Japanese, by Monde and Katto [57]). Results for the indirectly heated surface (Monde and Katto [16]) are shown in Fig. 31 for saturated water. The fraction of liquid lost to splashing increases sharply with increasing q" and approaches 0.9 at the critical heat flux. Interestingly, despite the significantly different splashing rates obtained for the two heating schemes, no discernible differences in the nucleate boiling or CHF data could be detected between the two investigations. Although not providing any quantitative splashing data, Katto and Ishii [151 reported qualitative differences in the splashing angles (relative to the impingement plane) between saturated jets of water, trichloroethane, and R113. The splash angle for the jets of trichloroethane and R-113 was less than that for water, and the departure of droplets from the free surface was generally less coherent.

108

D. H. WOLF ET AL. V. Transition Boiling

This chapter addresses jet impingement literature pertinent to the transition boiling regime. Specifically,information pertaining to surface wetting and the minimum heat flux and temperature have been included. The impingement transition and film boiling literature (experimental) cited in this and the following chapter is summarized in Tables VII and VIII, along with particulars concerning the respective experiments. The amount of time that elapses between the application of coolant to a surface and the subsequent wetting of the surface (wetting delay time) is important in the emergency cooling of nuclear reactor cores. Two publications that address this issue involve free-surface,circular jets of water used in one case (Piggott et al. [llo]) to quench a cylindrical surface and in the other (Owen and Pulling [l 111) to quench a planar surface. Both studies revealed a significant decrease in the delay time with increasing subcooling and jet velocity. Models based on analytical considerations of mass, energy, and .momentum transport were proposed to predict the delay times, and agreement between predictions and data was fair. Ishigai et al. [53] measured heat transfer to a planar, free-surface jet of water that quenched a surface with an initial temperature of approximately 1OOO"C. Simultaneous measurements of the liquid-solid contact were also reported for certain operating conditions. Figure 32a shows their q"-AT,., stagnation point data as a function of the jet subcooling. The data reveal that, with increased subcooling, the curve shifts toward higher heat fluxes and wall superheats. Moreover, characteristics of the transition region between the maximum and minimum heat fluxes are seen to differ markedly as a function of subcooling. At low subcoolings (AT,,, = 5 and 15 "C), the heat flux decreases after the start of the quench and the onset of film boiling, reaches a minimum at the onset of surface wetting, and increases monotonically to the maximum heat flux. At higher subcoolings (ATub = 25 and 35"C), the heat flux again declines at the start of the quench and reaches a minimum, but the subsequent increase to the maximum heat flux is not monotonic. Rather, a shoulder, or region of nearly constant heat flux, occurs. From visual observations, the authors suggest that, within this constant heat flux region, the surface is wetted intermittently, with complete wetting occurring as the heat flux again begins to increase toward its maximum value. At the largest subcooling (AT,,, = 55"C), the data reveal that no film boiling occurred at the stagnation point, despite surface temperatures as large as 1ooo"C. Figure 32b shows the boiling data as a function of the jet velocity at two subcoolings. At the lower subcooling (ATub= 15"C),the jet velocity has little effect on the transition boiling regime or Tmin,but the minimum and film boiling heat fluxes do increase with increasing jet speed. At the higher

TRANSITION AND

TABLE VII FILMBOILINGINVESTIGATIONS-OPERATING

PARAMETERS'

Author

L

Akimenko [ll2] Hanasaki et al. [114 Hatta et al. [ll5] Hatta et al. [116] Hatta et al. [1 1 71 Hatta et al. [I181 Hatta and Osakabe 1191 Ishigai et al. [113] Ishigai et al. [53Ib Ishimaru et al. [62] Kamata et al. [58,591 Kokado et al. [20] Lamvik and Iden [125] Nevins [126Ib Ochi et al. [93] Owen and Pulling [1 1 11 Piggot et al. [llO] Ruch and Holman [41] Sakhuja et al. [65] Sano et al. [66]

?-free Circ-free Circ-free Circ-free Circ-free Planar-free Planar-free Circ-free Planar-free Circ-sub Circ-conf Circ-free Circ-free Circ-free Circ-free Circ-free Circ-free Circ-free Circ-free Planar-free

Water Water Water Water Water Water Water Water Water Nitrogen Water Water Water Water Water Water Water

R-113 Water Water

80 -

80 50-80 85 82-92 10-80 5-55

0 0

8-29 88 70-80 5-80 < 10 < 80 27 18-77 0

3-20 0.042 0.21-1.49 0.025-1.43 0.21-1.49

-

-

-

0.83 1.5-12 0.65-3.5 0.22-1.34 10-20 0.32 11-35 0.8-1.0 2-7 2.9-6.5 1-15 1.23-6.87 -

1 3-12 6.2 0.6 2.2 10 0.7-2.0 12.7 5-20 1 .O-1.5 1.5-3.0 0.21-0.433 1.59-3.18

4.7-9.8 6.35-12.7

3.5

-

-

10 10 10 10

150 100 100-500 123-470 200 100 100 15

0.6 0.3-0.6 200 100 12.7-63.5 25 50

-

1050 900 900 900

900 900

900 400-800 loo0

Motion

~

300 900 500 640 1100 700 500-800

I-? jets

-

352 300

?jets

a Each range of operating parameters applies to transition and film boiling only; the range for the overall investigation, including other types of boiling data, may have been broader. Performed both transient and steady-state measurements.

TABLE VIII TRANSITION AND FILMBOILING INVESTIGATIONS-EXPERIMENTAL APPARATUS' Heater information

Author

L

L

Akimenko [1121 Hanasaki et al. [114] Hatta et al. [I151 Hatta et al. [116] Hatta er al. [1177 Hatta el al. [ l l S ] Hatta and Osakabe [119] Ishigai et al. [113] lshigai ez al. [53] Ishigai ef al. [53] Ishimaru et al. [62] Kamata et al. [58, 591 Kokado et al. [20] Lamvik and Iden [I251 Nevins [126] Nevins [126] Ochi et al. [93] Owen and Pulling [1113 F'iggott et al. [ l l O ] Ruch and Holman [41] Sakhuja et a/.[65] Sano et al. 1663

%Area coverage

Angle (deg)

-

90 90 90 90

0.20 0.20 0.20 0.20

0.42 0.014-0.23 7.8 1.8 16 1.2 0.20 >0.0022 25 25 0.22-3.5 o.oo~o-o.oo68

0.026-0.1 1

-

90 90 90 90 90 90 90 90

90 90

90

90 90 45-90 15-90 45-90

90 90

Orientation

Material

-

SS or copper stainless steel stainless steel stainless steel stainless steel stainless steel

UP UP UP UP UP UP UP

UP UP Vertical

stainless steel

Copper Stainless steel Stainless steel -

UP UP

Copper Stainless steel

Up, down, vertical Down Down UP Down Veriical Down Vertical UP

Aluminum Stainless steel stainless steel stainless steel ss/Nimonic Gold, inconel silica Copper Copper Copper

Heating scheme Transient Transient Transient Transient Transient Transient Transient Transient Transient Direct-ac Indirea Transient Transient Transient Transient Indirect Transient Transient Transient

Indirect Transient Transient

Si

Surface finish

(mm)

200 x 200 x 200 x 200 x 100 x 100 x

200; 200; 200; 200; 300; 240;

x x x x x

10 10 10 10 10 x 10 D=250; x40 12 x 80; x 2 12 x 80 D = 1.5 D = 20 200 x 200; x 10 D = 150; x 10 D = 25.4; x 16.2 D = 25.4 50 x 180; x 2 102 x 254; xO.5-0.91 6.3 5 D* I 25.4; x 200 D = 12.9 51 x 152; x102 20 x 150; x 120

-

Chrome plated No. 100 emery; acetone No. 100 emery; acetone No. 3000 emery; acetone No. 1500 emery; acetone

-

No. 100 emery -

Several types 25-jm nickel plating -

'% Area coverage refers to the percentage of the heater surface area covered by the n o d e area Angle refers to the angle of impingement (90" k i n g normal). Orientation refers to the direction the heater surface is facing with respect to gravity. Emery refers to polishing the surface with the given grade of emery paper. Acetone refers to cleaning the surface with acetone.

JET IMPINGEMENT BOILING

111

(a) 1o7

5

-E

N

g2 D

1 06

5

2 50

100

500

200

ATsat

1000

("C)

(b) 1o7

5 A

N

E

g2 0-

106

5

2

50

100

200

ATsat

500

1000

2000

("C)

FIG. 32. Boiling curve for a planar, free-surface jet of water showing the effects subcooling and (b) velocity. (Ishigai et al. [53], used with permission.)

D. H.WOLF ET AL.

112

subcooling (ATub = 55"C),the overall effects of the velocity are less evident; however, the velocity appears to have affected the shoulder in the transition region somewhat and precluded film boiling for the 1.55 m/s jet as compared to that of 1.0 m/s. Similar trends have been reported by Akimenko [112], who showed the minimum heat flux to increase monotonically with increasing jet velocity until no film boiling occurred for an initial plate temperature of 1050°C. For jet velocities and subcoolings in the ranges 0.65 I V, I 3.5 m/s and 5 IATub I55"C,Ishigai et al. correlated the minimum heat flux by the expression qkin = 0.054 x lo6 Vt.607(l

+ 0.527AT,,,)

(82) where qkin, V,, and A T u bhave units of W/m2, m/s, and "C, respectively. The consistently increased with increasingjet velociminimum temperature (Tmin) ty and subcooling. However, the effects of velocity were smallest at low subcoolings. The same trends were reported in an earlier publication (Ishigai et al. [113]), but for a free-surface, circular jet of water (1.5 IV,, I12 m/s and 20 IAT,,,,, I 9OOC). Ochi et al. [93] also examined transition boiling for a free-surface,circular jet. Likewise, the general trends reported by Ishigai et al. [113, 531 with respect to the effects of subcooling and jet velocity were replicated. Unlike Ishigai et al., however, the authors varied the nozzle diameter and found the largest sizes to induce the lowest minimum heat fluxes for a constant jet velocity (3 m/s). For velocities, subcoolings, and nozzle diameters in the ranges 2 IV, I7 m/S, 5 I AT,,, I 45"C, and 5 Id I 20 mm, their stagnation point data were correlated by &in

= 0.318 x

lo6

-

(1

+ 0.383AT,,b)

(83)

(:)""28

where qkin, V,, d, and ATub have units of W/m2, m/s, mm, and "C, respectively. The minimum temperature was also shown to be a function of the nozzle diameter and jet velocity, with the effects of velocity being least significant at smaller n o d e diameters. Approximate measurements of the velocity of the rewetting front showed the wetted region to expand most rapidly with increases in the nozzle diameter, jet velocity, and subcooling. Sakhuja et a1. [65] have examined cooling characteristics for an array of circular water jets used to quench a copper surface. Data for the transition boiling regime were correlated in terms of a heat transfer coefficient given by the following expression: h = 6.71 x 104V,,Pr,0.6(1.8AT,,,)"XS/d) where

n=

- 1.89

+ 0.328(S/d) - O.O25(S/d)*

(844 (84b)

JETIMPINGEMENT BOILING

113

and

f ( S / d ) = 27 - 0.35(S/d) - 1.51(S/d)'

+ 0.15(S/d)3

(844

The units of h, V,, and AT,,, are W/m'"C, m/s, and "C, respectively. Equation (84) correlated most the data to within 20% for AT,,, > 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

JET IMPINGEMENTBOILING

117

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

+

JET IMPINGEMENT BOILING

119

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

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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

RADIATIVE HEATTRANSFER IN POROUS MEDIA

135

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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M. KAVIANY AND B. P. SINGH

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<, - 1) = X,

X,

<,

<,

where and are random numbers between 0 and 1. This process is carried out for each layer using newly generated 5, and 5, for each layer. After tracing a small number of rays (say lOO), the process is repeated on the original center locations using freshly generated random numbers. Spheres of unit diameter result in a porosity of 0.476. To get a higher porosity, the sphere size can be reduced. In this case, the spheres will no longer touch each other. For a lower E, the distance between the layer centers for unit-diameter spheres must be y, - y,- = 0.524/(1 - E), where y, refers to the y coordinate of the nth layer. Alternate layers have a sphere at the square center. The layers are staggered by an amount limited by the physical constraint that no overlap is allowed. Thus the maximum distance by which a layer can be staggered varies from 0.5 for a porosity of 0.476 to 0 for a porosity of 0.26.Equations (91) and (92) must now be changed to x, = x, + y(25 - 1) (93)

+ y(25 - 1)

(94) where y is a function of porosity alone and represents the extent to which sphere centers can be displaced without overlap. Both of these models are used in conjunction with the periodic boundary condition in the x and z directions. z, = z,

A. OPAQUEPARTICLES

A ray is defined by the coordinates of its starting point Po(x,, yo, zo) and its direction cosines (I, m, n). The ray enters the bed at a random point in the x-z

RADIATIVEHEATTRANSFER IN POROUS MEDIA

163

plane [forming the lower surface (y = O)], i.e., (xo, Y o , z o ) = WY,, 0,WY,)

(95) where W is the lateral dimension of the box being used and 5, and tz are random numbers between 0 and 1. The angles and 0 are given by

4 = 2715

e = C O S - ~ C-~ ((1

- cOS

(96)

emax)]

(97) where Omax is the maximum angle that the incident radiation makes with the normal. For diffuse incident flux, cos Omax = 0. The direction cosines of the ray are I = sin 8 cos 4 (98) m = cos 0,

(99)

n = sin 0 sin 4

(100)

The coordinates of a ray after traveling a length S are given by x = xg

+ 1s

y=y,+mS z = zo

+ nS

(101) ( 102)

(103)

Substituting in the equation of the sphere, we have ( x - xc)2

+ ( y - yc)2(2 - zc)2 = R 2

(104)

i.e., a quadratic equation in S is obtained. A positive discriminant indicates that the ray intersects the sphere. Equation (104) is solved for all spheres for which it has a positive discriminant. The smaller of the two solutions obtained gives the actual point of intersection with a sphere. Also, the distance the ray travels before it intersects a bounding surface is determined. The minimum distance a ray travels before it is intercepted by a sphere or a bounding surface is then determined. The sphere or bounding surface corresponding to this solution is the surface that actually intercepts the ray. If the ray is intercepted by the side walls, the periodic boundary condition is applied. In case it passes through the upper or lower face, the energy associated with the ray is registered as transmission or reflection. If it is intercepted by a sphere, the point of intersection is determined and the direction cosines of the reflected ray for a specularly scattering sphere are found using the following laws of reflection: The incident ray, the reflected ray, and the normal to the surface all lie in the same plane. The angle of incidence is equal to the angle of reflection.

M. KAVIANY AND B. P. SINGH

164

If the sphere is assumed to be diffusely scattering, then the ray is scattered in a random direction from the point of interception under the restriction that the ray does not penetrate the sphere. After reflection, the energy of the reflected ray is given by E, = pE,. This process is repeated until the ray passes through either the upper or lower surface. The number of rays used for each simulation ranged from 100,OOO to 1,0oO,OOO. Packed beds with lower transmittance need more rays for the same accuracy. B. SEMITRANSPARENT PARTICLES Transmitting particles are dealt with by ray tracing inside the sphere, following the laws of reflection and refraction. The angle that the incident radiation makes with the tangent to the surface, i.e., Oi, is calculated. Then the angle of refraction is given by

n (105) nf Next, the Fresnel coefficients and the reflectivity are calculated in terms of the angles Oi and 8, (Siegel and Howell [l]), i.e., cos 6, = n cos 0,

tan(6, - e,) tan(6, 0,)

+

and p l d =

n =2

]

sin(8, - 6,) sin(Oi + 6,)

[

for K

+0

(106)

Thus, the reflected parts of energy are p’rldand p l A . The refracted parts are 1 - p\rdand 1 - p l d . Then the energy carried by the various rays is (van de Hulst [191) Ell,, = P

h

El,,, = (1

- pilJ2(pilL)p-1

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

( 107)

(108)

For the other polarization, replace with However if K , is not small, then (106) should not be used to calculate the reflectivity. Instead, an exact analysis should be followed (Siegel and Howell [l]), although ray tracing beyond (p = 0) will not be required because even moderate values of K , (for a large particle in the geometrical optics range) make the particle virtually opaque. For nonpolarized irradiation, the total energy carried by a ray is given by (109) E = &qP + El,,,) When a ray strikes a sphere, it is either reflected (p = 0) or transmitted (p = 1, 2,. . .) with a reduction in the energy due to absorption. The outcome is

RADIATIVEHEATTRANSFER IN POROUS MEDIA

165

decided by generating a random number. Let us define i

Pi

=

C Ej j=O

Then the ray is reflected (P = 0) if

5 < Po and is transmitted with P = i if

Pi < 5 5 P i + l

(1 12)

Generally, tracing up to P = 2 or 3 is sufficient. Figure 3 shows a sketch of a ray traced up to P = 2. The ray incident on the point Po can be either reflected or transmitted. If the ray is reflected, its direction cosines are calculated as in the case of opaque particles. However, the energy carried by the ray remains unchanged. are found If the ray is transmitted, the direction cosines of the ray using the laws of refraction. -The incident ray, the refracted ray, and the normal to the surface all lie in the same plane. -The angle of refraction is related to the angle of incidence by the Snell law.

.

The coordinates of point PI are found by using P O P , = 2R sin 8,. The direction cosines of the rays t l , Zand rl are found by applying the laws of reflection and refraction at point P I respectively. The coordinates of point P , and the direction cosines of ray rz are found by repeating the preceding steps. In the case of semitransparent particles (K # 0), the energy of the rays is reduced by an attenuating factor given by

F,

= exp(

- 4P m Rsin 0,)

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

(113)

Therefore, the energy carried by a transmitted ray is given by E,, = FpEi

for P = 1,2, . . .

(1 14)

C. EMITTINGPARTICLES The spheres are assumed to have a high enough thermal conductivity to be assumed isothermal. The case simulated here is of a bed of absorbing, emitting, and scattering spheres. If the sphere has a reflectivity p, the ray is reflected if p > ( (either diffuse or specularly reflecting particles may be considered), or is absorbed and emitted if p I (.

.

166

M. KAVIANY AND B. P. SINGH

The emission can take place from any randomly selected point on the surface. Also, the direction of the emitted ray is determined according to the Lambert cosine law as in the case of diffusely reflected rays.

VII. Radiant Conductivity

The radiative heat transfer for a one-dimensional plane geometry with emitting particles under the steady-state condition is given by (Vortmeyer C6l)

l-pw' d

where F is called the exchangefactor and the properties are assumed to be wavelength independent. If pw = 0 and the bed is several particles deep, then the first term of the denominator can be neglected. Then, for TI - T, < 200 K, a radiant conductivity is defined (Tien and Drolen {43]) k, = 4F dOTi

(1 16)

The approach has many limitations, but the single most important one is that the value of F cannot be easily calculated. Only the Monte Carlo method can be used for calculating F for semitransparent particles. The value of F also depends on the value of the conductivity of the solid phase. In the Kasparek experiment (Vortmeyer [ 6 ] ) and the Monte Carlo method infinite conductivity is assumed, which is justified for metals. Similarly, the case of zero conductivity can easily be treated by considering the rays to be emitted from the same point at which they were absorbed. However, the intermediate case (i.e., when the conductivity is comparable to the radiant conductivity) shows a strong dependence of radiant conductivity on the solid conductivity. The extent of this dependence may be seen by comparing the difference in the values of F in Table I corresponding to low and high emissivities. If the conductivity were small, all the F values would be close to those obtained for the E = 0 case. Thus, a simple tabulation of F as in Table I is of limited use. On the other hand, a large number of important applications lie in the large solid conductivity range for which Table I can be used.

A. CALCULATION OF F Many different models are available for prediction of F, and these are reviewed by Vortmeyer. Here, the main emphasis will be on examining the

RADIATIVE HEATTRANSFER IN POROUS MEDIA TABLE I RADIATION EXCHANGE FACTOR F (E

167

= 0.4)

Emissivity Model

0.20

0.35

0.60

0.85

1.00

Two-flux (diffuse) Two-flux (specular) Discrete ordinate (diffuse) Discrete ordinate (specular) Argo and Smith Vortmeyer Kasparek (experiment) Monte Carlo (diffuse) Monte Carlo (specular)

0.88 1.11 1.09 1.48

0.91 1.11 1.15 1.48

1.02 1.11 1.25 1.48

1.06 1.11 1.38 1.48

1.11 1.11 1.48 1.48

0.11 0.25

0.43 0.54

0.32 0.34

0.45 0.47

0.68 0.69

0.74 0.85 1.02 0.94 0.95

1.00 1.12

-

0.21 0.33 0.54

-

1.10 1.10

validity of the radiant conductivity approach by comparing the results of some of these models with the Monte Carlo simulations and with the available experimental results. A solution to this problem based on the two-flux model is given by Tien and Drolen [43]

which can be written as

For isotropic scattering, B = 0.5 and Eq. (1 18) becomes independent of the particle emissivity (for large particles). In the Kasparek experiment (described by Vortmeyer), measurements of radiation heat transfer through a number of planar series of welded steel spheres were made. Conduction and convection were eliminated by placing the layers a small distance apart and performing the experiment in a vacuum. The high thermal conductivity of the material ensured that its heat resistance was negligible and that the spheres were isothermal. Measurements were made using polished steel spheres (E, = 0.35) and chromium oxide-coated spheres ( E , = 0.85). The arrangements considered were a cubic ( E 21 0.5) and a porosity of 0.4. The Monte Carlo method was used to predict the heat transfer through the packed bed used in the Kasparek experiment (unlike the previous sections,

168

M. KAVIANY AND B. P.

SINCH

we encounter emitting particles). The results confirm the validity of the exchange factor approach as long as the emissivity is not close to zero. The change in the value of the exchange factor resulting from increasing the number of layers from 8 to 16 was less than 0.01 for E, 2 0.20. The simulation was performed for both specularly scattering and diffusely scattering spheres. The polished steel spheres can be considered specularly scattering, while the chromium oxide spheres would scatter diffusely. Table I shows the results of the Monte Carlo simulation as well as the results obtained from the two-flux model and the models of Argo and Smith (described by Vortmeyer) and Vortmeyer for a porosity of 0.4. The predictions by the Monte Carlo method match the experimental results fairly well, considering that some uncertainty is always present in the emissivity values. Diffuse spheres result in slightly smaller values of F although, as expected, the difference decreases with increasing emissivity and vanishes for E, = 1. The change in value of F with porosity also matched the experimental results. For cubic packing, the value of F increases from 0.47 to 0.51 for E, = 0.35,while the experimental value increases from 0.54 to 0.60. For E , = 0.85, the value of F increases from 0.94 to 0.97, while the experimental value increases from 1.02 to 1.06. The predictions based on the two-flux model for diffuse spheres show a very small sensitivity to the emissivity, while those for specularly scattering spheres show no change at all. Using the method of discrete ordinates, higher values of F are obtained than are predicted by the two-flux method. Also, for specularly scattering spheres, the heat transfer remains independent of the emissivity. This can be seen from the equation of transfer by applying the condition of radiative equilibrium. Physically, when the spheres are treated as point scatterers, there is clearly no difference between isotropic scattering from a point and emission from it. The mechanism that results in an increase in the radiative heat transfer with increase in the emissivity is that of transportation of the absorbed energy through each particle (by conduction); i.e., particles absorb radiation at one face and emit a part of it from the other face. For a dilute medium consisting of isotropically scattering small particles separated by large distances, the heat transfer is again expected to be independent of the particle emissivity. In a later section, we present a method that models the effect of solid conductivity on radiant conductivity, i.e., calculate F as a function of the solid conductivity.

VIII. Modeling Dependent Scattering In order to account for the interparticle interactions, the decrease in the efficiencies has been correlated to the porosity. The correlations are discussed by Tien and Drolen [43] and Tien [48]. For ctR + 0, the Percus-Yevick

RADIATIVE HEATTRANSFER IN POROUS MEDIA

169

model gives

which is not accurate for large aR. Another correlation is that of Hottel et al. [39], who recommend

C

0.905

> 0.069

(120)

These relationships, along with the experimental results of Ishimaru and Kuga [36] and the liquid model of Drolen and Tien [38], are plotted in Fig. 7. The results are for a, = 0.529 and n = 1.19. The Percus-Yevick model (which is for aR + 0) and the correlation of Hottel et al. [39] predict the behavior well. The model of Drolen and Tien also predicts the behavior relatively well for this small aRand E > 0.85. However, at higher values of aR, the predictions of this model deviate substantially from that of Hottel et al.

0 . 0 5 8 ~ ' pc.~

1.o

I

aR= 0.529, %= 1.19 nt

I

,

1

o Experiment (Ishimaru and Kuga)

Hottel et al. (empirical)

log log &b&dU?= U.3'

0.5

0.25 -3.83$

\

(V1s)dep

0.6

0.7

0.8

0.9

1.o

E

FIG.7. Effect of interparticle spacing (or porosity) on the normalized spectral scattering coefficient for spherical particles with optical properties shown.

170

M. KAVIANY AND B. P.SINGH

Kumar and Tien [18] model dependent scattering in a cloud of Rayleighsized particles. Singh and Kaviany [49] model dependent interactions in the geometric range. The model is presented in detail in the next section. DEPENDENT SCATTERING FOR LARGEPARTICLES A. MODELING One approach is to scale the independent properties so that dependent computations can be carried out using the equation of radiative transfer with these scaled properties. However, since the deviations from the independent theory are a function of the porosity and the complex index of refraction, we will show that a simple scaling of the extent of dependence is not feasible. This will be done by examining the probability density functions for independent and dependent scattering from both opaque and transparent particles. Singh and Kaviany [49] use an approach that separately accounts for multiple scattering in the representative elementary volume and the transportation of radiation through a particle (across a substantial optical thickness). Multiple scattering depends on the porosity alone and is accounted for by scaling the optical thickness using the porosity. The transmission through semitransparent particles is modeled by allowing for the transportation effect while describing the intensity field by the method of discrete ordinates. This is done by taking into consideration the spatial difference between the point where a ray first interacts with a sphere and the point from which it finally leaves the sphere. This spatial difference corresponds to an optical thickness (for a given porosity) across which the ray is transported while undergoing scattering by a particle. The results of the application of this dependence-included discrete-ordinates method are shown to be in good agreement with those obtained from the Monte Carlo method. The correct modeling of the physics results in the applicability over the full range of porosity and optical properties and obviates the need for calculating and presenting scaling factors in a threedimensional array. Kamiuto [SO] has proposed a heuristic correlated scattering theory that attempts to calculate the dependent properties of large particles from the independent properties. The extinction coefficient and the albedo are scaled as

RADIATIVEHEATTRANSFER IN POROUS MEDIA

171

The phase function is left unchanged. The results of this theory will be compared to those of the Monte Carlo simulation. 1. Scaling

In this section, we attempt to find scaling factors so that the independent radiative properties can be scaled to give the dependent properties of the particulate media. The scalingfactor S, is assumed to be scalar and scales the optical thickness leaving the phase function and albedo unchanged. a. S, for Opaque Spheres. Consider a plane-parallel particulate medium

subject to diffuse incident radiation at one boundary. The medium contains particles that are nonemitting in the wavelength range of interest. For opaque particles with nonzero emissivity, the slopes of the transmission curve on a logarithmic scale approach a constant value away from the boundary. The scaling factor S, is calculated by finding the ratio of the slopes calculated by the Monte Carlo method described in Section VI and by the independent theory. Figure 8 shows the scaling factor for opaque spheres as a function of porosity for different emissivities.The values of S, can be curve fitted as S, = 1

+ 1.84(1 - E ) - 3.15(1 - E)’ + 7.20(1 - E

for E > 0.3 (124)

) ~

3.5

&,=0.1

3.0

m

0.4 0

0.6 0 2.5

sr 2.0

1.5

1.o

0.3

I

I

I

I

I

0.4

0.5

0.6

0.7

0.8

I

0.9

1.0

& FIG. 8. Variation of the scaling factor with respect to porosity, for several emissivities.

M. KAVIANY AND B. P.SINGH

172

-

E =0.476

Monte Carlo

4

1.5

Pd f

1.o

0.5

0.0

0

1

2

Lfd

3

FIG. 9. Variation of the probability density function as a function of distance, for a bed of opaque particles (e = 0.476).

Since the effect of emissivity on S, is small, (124) can be used to obtain the value of S, for other emissivities. b. The Basis of Scaling. Figure 9 shows the probability density function (pdf) for a bed made of specularly reflecting opaque spheres with a porosity of 0.476. The pdf was obtained by a direct Monte Carlo simulation for packed beds of spheres as discussed by Singh and Kaviany [49]. Also plotted are the pdfs calculated from the theory of independent scattering and the pdf for the scaled properties. The effect of increasing the emissivity is to increase the relative importance of the right-hand side of the curve. This is because multiple reflections attenuate the energy of a ray undergoing a number of interactions, as a result of short path lengths. Thus, the net contribution to transmittance will come from the rays that include a greater number of longer paths and thus are transmitted with a lesser number of interactions. If the scaled pdf and the Monte Carlo pdf have different shapes, the scaling factor will change greatly with the particle emissivity. However, since the scaled pdf is found to conform closely to the pdf from the Monte Carlo simulation, the effect of the emissivity on S , is small, as seen in Fig. 8. Therefore, the scaling can be carried out, treating the scaling factor as a function of porosity alone.

RADIATIVEHEATTRANSFER IN POROUS MEDIA

173

FIG. 10. Schematic of ray transmission through a particle and its arrival at an adjacent particle.

Figure 10 shows a schematic of the interaction of a ray with a transparent sphere. The ray is intercepted by the first sphere at point P o . Part of the energy is transmitted through the sphere and interacts with a second sphere at point Qo. The distance PoQ, is the distance that this energy travels after interaction at point Po and before its interaction with the next sphere. Other parts of the incident energy at P o travel different paths, as explained in detail in the next section. Figure 11 shows the pdf for a bed of transparent particles (n = 1.5). The pdf for semitransparent particles is similar except that the fraction of rays passing through the sphere has to be modified to account for the energy attenuated on passing through the sphere. It is clear that the pdf for nonopaque particles and that obtained from the independent theory are basically dissimilar. Even though scaling factors can still be found for a prescribed set of n, ic, and E, a change in any one of the three parameters will change the pdf and thus affect S,. Therefore, a scaling approach necessitates calculation and presentation of scaling factors in a three-dimensional array and is not found to be suitable. 2. Dependence-Included Discrete-Ordinates Method (DIDOM)

DIDOM models radiation heat transfer in a packed bed of semitransparent spheres. The deviation from the independent theory takes place because of the following two distinct effects. Multiple scattering within a small elemental volume. Transportation across a substantial optical thickness.

M. KAVIAW AND B. P.SINGH

174

1.2 [

c

Monte Carlo --

E = 0.476

lndeeendent

0.8

Pdf

0.4

0.0 0

1

2

3

LId

FIG, 11. Same as Fig. 9, except for transparent particles (e = 0.476, n = 1.5).

Multiple scattering is a function of porosity alone and is accounted for by scaling, as shown in the previous section. The transportation effect is modeled by allowing for transmission through a sphere while solving the equation of radiative transfer. For this, the method of discrete ordinates has been Found to be most suitable. The key to understanding and modeling the transportation effect is that a ray may be scattered by a particle from a point that is different from the point at which the ray first interacts with the particle. This is because of transmission through a particle. In highly porous media (e 4 l), this effect is of no consequence because the particle size is small compared to the interparticle distance. However, in packed beds, the ray may be transported through a distance that corresponds to a substantial optical thickness. Thus, is it important not only to know the direction in which a particle scatters but also to know the displacement undergone by the ray as it passes through the particle. In this section, we first examine the properties of a single particle. Then the properties of beds are discussed. Finally, the DIDOM is presented. a. Properties of a Single Particle. The procedure for obtaining the properties of a single particle by geometric scattering has already been discussed. Here we extend this method to also calculate the points from where the rays are scattered. Figure 12 shows a sketch of an incident ray being scattered.

RADIATIVE HEATTRANSFER IN POROUS MEDIA

175

FIG. 12. Ray tracing through a single particle.

For independent scattering, 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. However, for cases in which the transportation effect is important, addition of gains for different values of P is not permissible. This is because rays scattered in the same direction from different points (Po for P = 0, PI for P = 1) will not have the same effect on transmission in a packed bed. Therefore, along with the gain, information regarding the point from which the ray leaves the sphere must be mentioned. Thus, the phase function will be reported as a three-column array, i.e., [@(8,, P), Ax', Ay']. Here @(O0, P) = G(8,, P)/qs, Ax' = xb - xb, and Ay' = yb - yb represent the displacement undergone by the ray in a direction perpendicular and parallel to the incident ray, respectively. Thus, A, = r p R 2 , A, = qsaRZ, and [@(8,, P), Ax', Ay'] are determined. Ax' and Ay' are given by Ax' = 0, Ay' = 0

for P = 0

(125)

and P

sinC(2P' - l)8, - O,]

AX' = d sin 8, p'=l

P

AY' = d sin 0, p'=l

cosC(2P' - l)O, - Oil

for P = 1,2, ...

(126)

176

M. KAVIANYAND B. P.SINGH

b. Properties of Beds. In this section, we will relate the radiative properties of a single particle determined in the previous section to the radiative properties of the particulate medium. We assume a one-dimensional planeparallel slab geometry. The required properties are (a,), (a,), and [(Q)(pj + pi), Ak]. The last one represents the phase function from a direction p j to a direction pi, and Ak represents the number of grids through which it is transported in the direction perpendicular to the slab boundaries. For monosized scatterers of porosity E, we have similarly, (oa) = N,A,,S,. The procedure for computing [(Q)(pj

+

pi), Ak] is outlined here.

(a) To find the phase function for scattering into a direction pi from a direction p j , we must integrate over the azimuthal angle Cp. For this purpose we employ a Gaussian quadrature and find discrete values of Cpi - Cpj between 0 and ~tat 24 points. (b) At every point, we find (c) Up to this point, the treatment is similar to that used when employing a standard DOM with the phase function available at discrete values of 8, except that each value of P has its own phase function for every B0. However, here the numerical integration over Cp to evaluate

is not performed. This is because we cannot add (@)[6,(pj, Cpj + pi, Cpi), P] terms unless they have the same Ak. (d) For every (@)[8,(pj, c$j + ply CpJ, PI, we find Ak = Integer(Ak,. + AkyJ),where Ak,, and Ak,. are the contributions of Ax' and Ay' to Ak. Ak is rounded off to the nearest integer. Figure 13 shows how Ax' and

FIG. 13. Demonstration of the transportation effect for semitransparent particles.

177

RADIATIVE HEATTRANSFER IN POROUS MEDIA Ay' contribute to Ak. Equations for Akx, and Ak,,, are written as

where radius of sphere

" = distance between two grids

- 0.75(1 - &)Sr AT

(130)

(e) We calculate (@)(p, + p i , Ak), i.e., the phase function from p j to pi that is transported by Ak number of grid points. Note that (QXPj

+

pi) =

C (Q)@j -,pi, Ak) Ak

(131)

c. DIDOM. The one-dimensional radiative transfer equation at x and in direction pi can be written as

i = - M , - M + l , . . . , M,i#O

(132)

where the T i term represents the in-scattering term and accounts for in-scattering into the direction pi at location x from all directions at x as well as from all directions at other x locations. After discretization, by evaluating (132) at the midpoint between two nodes ( k and k + 1) as in Fiveland [ S l ] , the in-scattering term can be written as n

ri,k+1/2

=

M

1

Apjlj,k'+1/2((D)(pj

pi, Ak)

for Ak = k - k',

k'= 1 j = - M , j # O

(133)

where A p j are the quadrature weights corresponding to the direction p j and M

C

Apj=2

(134)

j = -M,j#O

The boundary conditions are Ii = &,I,, + p s l - i + 2p,

Ii= &,Ib+ psi-,

-M

j=-1

ApjZjpj,

i= 1 , . .. ,M at x = 0

M

+ 2p, C A p j l j p j , j= 1

i=

(135)

- 1,. .. , - M a t x = L (136)

M. KAVIANY AND B. P. SINGH

178

In the case of incident radiation on a transparent boundary, this equation is used with E, = 1, ps = 0, and pd = 0. The intensity at the boundary, in a direction pi, is equal to the intensity of the incident radiation in that direction. As a first step, the values of (@)[8,(pj, $ j + pi, (bi), P ] are calculated at discrete values of ( $ j - &i) for all combinations of pi and p j , and the corresponding Aks are calculated to obtain (@)(pj -+ pi, Ak). For pi > 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.

RADIATIVE HEATTRANSFER IN POROUS MEDIA

185

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

Copyright 0 1993 by Academic Prcss, Inc. All rights of reproduction in any form reserved.

ISBN 0-12-020023-6

188

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

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

189

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

192

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

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

193

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

R. P. CHHABRA

194

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

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

195

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

R. P. CHHABRA

196

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-

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

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,
= (Rh AP/L)

(14)

(3,)

= ( K O V/Rh)

(1 5 )

and

R. P. CHHABRA

198

Equations (14) and (15) can now be used to describe the flow of timeindependent fluids in porous media and packed beds by substituting for R , from Equation (2), L, for L, and Yi for Y to obtain the following expressions for packed beds made up of monosize spheres.
= (d/6)(&/1- &)(AP/TL)

(?,>

= 6KOTCU

-W ) ( ~ / d )

(16) (17)

Further progress can be made only by choosing a specific model for the rheological behaviour of the fluid. Equations (16) and (17) together with suitable values of K O and Twill yield Eq. (9). By analogy with the flow of time-independent fluids in circular pipes, the shear stress at the wall of a capillary tube (in a packed bed) is related to the nominal shear rate at the wall by a power law-type relation as

(18) where m' and n' are the apparent fluid consistency and flow behaviour index, respectively, which in turn can be related to the true rheological parameters, as outlined by Metzner and Reed [181]. For instance, for a truly power law fluid, n' = n (1 9a)

= 111'

Similar relations for other fluid models are available in the literature but are much more involved than Eq. (19) [180, 2381. It is also worthwhile to add here that the Rabinowitsch-Mooney factor of [(3n 1)/4n]", appearing in Eq. (19), is strictly applicable for cylindrical geometry but, fortunately, the calculations of Miller [I851 suggest that it is nearly independent of the conduit geometry. Thus, for power law fluids, the Kozeny-Carman equation becomes

+

f

=

180/Re*

(204

where Re* = [pv'2-ndn/m'(l - &)"I(15 &e2)'-"

(20b)

Equation (20) is based on K O = 2.5 and T = @. Kemblowski et al. [134] reported the agreement between the predictions of Eq. (20) and their own limited experimental results to be satisfactory. Other significant contributions in this area are due to Christopher and Middleman [43], Sadowski and Bird [226], and Kozicki and coworkers [141-1491. By combining Eq. (1 1) with the Blake-Kozeny model,

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

199

Christopher and Middleman [43] obtained the following expression for friction factor:

f = 150/Re,-,

(21)

where Re,

= [ p V 2-"d/H (1 - E)]

9n

(22a)

+3

It is of interest to point out here that Eq. (21) not only employs a value of 150 instead of 180 but also uses T = (25/12), which is much greater than the commonly cited values in the literature [137]. Figure 2 shows a typical comparison between the predictions of Eq. (21) and experimental results culled from two different sources [43, 2261; the agreement is seen to be satisfactory. The literature abounds with numerous similar expressions all of which purport to correlate the experimental values of pressure drop for power law fluids satisfactorily. Most of these have been reviewed by Srinivas and Chhabra [243] and Chhabra [38]. More surprising is the fact that the predictions of different methods have rarely been compared with each other. Such a comparison should be instructive in the sense that it will illustrate the basic differences inherent in the three capillary models as well as those arising from the different values of the tortuosity factor used by different investigators. Irrespective of the model used, most expressions available in the literature for the flow of power law fluids are of (or can be reduced to the) following form: f = A(n, &)/Re*

(23)

The resulting values of A(n, E ) are summarised in Table I1 for a selection of methods. Figure 3 contrasts the predictions for packed bed conditions ( E = 0.4) for a range of values of the power law index. Included in this figure are the estimates based on two cell models which are discussed in detail in the next section. Inspection of this figure shows that the divergence of predictions increases with the increasing extent of non-Newtonian behaviour. For Newtonian fluids ( n = 1) the value of the loss coefficient ranges from 150 to 193 (about 29% variation), whereas it varies from 100 to 250 for n = 0.2. Besides, the different predictions are not even in qualitative agreement concerning the influence of n on the loss coefficient (A). As mentioned earlier, much less information is available on the flow of the other types of time-independent fluids. By way of illustration, we cite the landmark study of Sadowski and Bird [226]. These investigators have

200

R. P. CHHABRA

REYNOLDS NUMBER,Re,, FIG. 2. Typical comparison between experiments and predictions of Christopher and Middleman [43].

coupled the Blake-Kozeny model with the three-parameter Ellis fluid model to arrive at the following expression for friction factor:

f = 180/Re,, ResE=@[(l -I--(-) 4 PO

a

+3

(2,)

=-I

]

(24)

21/2

is given by Eq. (14). Sadowski and Bird [226] also used a tortuosity factor of (25/12). These investigators too reported the agreement between their predictions and experiments to be generally satisfactory except in the cases of some concentrated polymer solutions. The resulting discrepancies in the latter case were (2,)

TABLE I1 EXPRESSIONS FORA(&,fl) IN EQ. (23) Investigator

Value of A for n = 1

Expression"

Christopher and Middleman [43]

h,

150

Park et al. [202]

150M2

Brea et al. [23]

160

180

Kemblowski and Michniewicz [137]

180

0, Al-Fariss et al. (for T~

= 0)

[4]

Kawase and Ulbrecht [127]

150

[

270,,hF,~'-~.~'" 15$(1

4n

I'

- cX1 + 3n)

- 22n2 + 29n + 2 n(n + 2X2n + I )

1 , n" = 3n/(2

Dharmadhikari and Kale [56] Kozicki and Tiu [148] 'M=l+-

2d 30(1 - 8 ) '

180M"

+ n)

150

180M

R. P. CHHABRA

202 275

1

75

T.0

CURVE NO.

I

0.8

REF

I

I

I

0.6

0.4

0.2

POWER LAW

INDEX,n

FIG. 3. Predictions of different methods in creeping flow regime.

tentatively ascribed to wall-polymer interactions and viscoelastic effects. Similar expressions based on a variety of other time-independent fluid models, such as the Carreau model and the Meter model, are available in the literature [137, 2021. In contrast to the voluminous work on the flow of fluids without yield stress, the flow of viscoplastic fluids through packed beds has received only scant attention. Al-Fariss et al. [4] combined the Herschel-Bulkley fluid model with the Blake-Kozeny model to yield an expression of the form of Eq. (23) with the following definition of Reynolds number: Re, =

12pv2J/

2 r n d V " ~+~~ 3(1 - E )

~

$

2

~

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

203

and using their own experimental data gleaned with waxy crudes, the best value of A(n, E ) was found to be 150. Attention is drawn to the fact that Eq. (26) includes several known limiting cases, that is, the Newtonian result (n = 1, to= 0), Bingham plastic result (n = l), and power law result (to= 0). A more general and somewhat different approach for studying the flow of time-independent fluids in complex geometries including porous media has been developed by Kozicki et al. [1441. Recognising the qualitatively similar forms of the Rabinowitsch-Mooney equation for the laminar flow of timeindependent fluids in circular tubes and in between two parallel plates, they derived the so-called generalized form of the Rabinowitsch-Mooney equation for rectilinear flow in a conduit of arbitrary cross section. With appropriate choices of the two geometric parameters, one can obtain the pressure drop-flow rate relationship for the flow of any time-independent fluid through conduits of complex geometries such as open channels, packed beds, and porous media and the other noncircular but regular-shaped cross sections. Since detailed descriptions of this approach are available elsewhere [144, 2491, only its salient features are recapitulated here. Their analysis begins with the Rabinowitsch-Mooney equation for circular tubes written as

where a and b are two constants characteristic of a given geometry but are assumed to be independent of the fluid model; e.g., for a circular tube a = (1/4) and b = (3/4). Equation (27) can now be modified for flow in packed beds by noting D = Dh = 4Rh to rewrite it as

Since the wall shear stress (T,) may not be constant along the contour of the wetted perimeter, an average shear stress, defined as follows, may be used:

For constant values of a and b, Eq. (28) can be integrated with respect to (z,) to obtain

The integral appearing in Eq. (30) can be evaluated for a known form off(z), that is, for a specific fluid model, e.g., a power law fluid,f(T) = (~/m)'/", etc. As mentioned earlier, the values of a and b for a range of geometries, inferred from the corresponding Newtonian results, have been compiled by Tiu [249].

R. P. CHHABRA

204

More recently, Kozicki and Tiu [148] have contrasted the predictions of the Blake, the Blake-Kozeny, and the Kozeny-Carman models for the flow of power law and Ellis model fluids. At this juncture, it is also appropriate to mention that most investigators have used their own experimental data while establishing the validity of a specific model and the resulting correlations have rarely been tested rigorously using independent data available in the literature. Such a comparison will facilitate the assessment of the quality, reliability, and reproducibility of data in this field. Figure 4 shows friction factor-Reynolds number results drawn from seven different sources. The other relevant information pertaining to each set of data is presented in Table 111. Examination of Fig. 4 shows several orders of magnitude variation in the value off for a given value of

\

\

SYMBOL \

ld2 -

-

do

\

0 0

A A

\

\\ \ m

REF 163 53 a9 56 226 243 202

+

a- 8 10

V

i2 Z

P 106 -a V

b.

loL

lo2

I O - ~ 10+

10-2 100 102 REYNOLDS N U M B E R , R ~ *

lo4

FIG.4. Friction factor-Reynolds number relationship (based on literature data).

TABLE 111 EXPERIMENTAL DETAILS OF RESULTS SHOWN IN Investigator

h,

Sadowski and Bird [226]

28.88-57.4

Dauben and Menzie [53]

156-702

Srinivas and Chhabra [243]

6.43-39

0 v,

Dharamadhikari and Kale [56]

Park et al. [202]

7.83-15.8

Gregory and Griskey [89]

7.61-19.4

Levy [1631 'Aqueous solutions.

75-508

FIG.

4

Test media*

Range of n

Elvanol Natrosol 250-G Natrosol 250-H WSR-301, 35,205 coagulant (polyethylene oxide) CMC Polyacrylamide Oil-water emulsion 1.25%CMC Sodium alginate solution Polyacry lamide Polyvinylpyrrolidone Molten polymers Polyethylene CMC, Carbopol, Separan

0.498-0.943

0.01 14 to 5.95

0.459-1 .O

1.11 x 10-5 to 1.49 x

0.381-0.94

3.11 x

0.483-0.557

2.56 x 10-4 to 2.1

0.405-0.456

3.1 x

Range of Re*

to 11.5

to 1.16

1.42 x lo-' to 1.29 x 0.31-0.45

10-2

1.12 x 10-4 to 93

10-2

206

R. P. CHHABRA

Reynolds number. Essentially similar observations have been made by numerous other investigators [l, 137, 174, 179, 234, 2351. Several possible reasons have been suggested for such poor agreement; the most important ones are viscoelastic effects [12, 24, 56, 93, 102, 2021, wall effects [47, 2431; wall-polymer molecule interactions [45,46,95, 142-1491, gel formation [28, 2261, inadequacy of the power law model [60], experimental uncertainty [234], extensional effects [63-65,92,113,122], pseudodilatant effects [26-30, 531, and the dubious nature of the tortuosity factor [2,56]. Likewise, in spite of the fact that capillary models have enjoyed a great deal of success in describing the flow of Newtonian fluids and time-independentfluids (albeit to a lesser extent), they are not devoid of criticism [60,62,190]. For instance, the Hagen-Poiseuille equation and its equivalent forms for time-independent fluids are based on the assumption of fully developed flow (and hence constant shear stress at the wall). Clearly, this condition is not met in the case of flow in packed beds. Second, the notion of hydraulic radius has proved to be successful only under fully turbulent conditions, whereas here it is being used to describe streamline flow. Third, this approach does not take cognizance of the various kinds of nonuniformities (e.g., series and parallel) present in a porous medium. Last, the tortuosity factor is treated more or less as an adjustable parameter of dubious, physical value [2, 56, 134, 2351. Admittedly, some attempts have been made to rectify some of these deficiencies of this class of models [62, 88, 1231, but to date these have not been extended to the flow of time-independent non-Newtonian fluids. This section is concluded by reiterating that this class represents the simplest type of models for describing the complex flow phenomena in packed beds and porous media. It is perhaps safe to use this approach to estimate the frictional pressure loss for the flow of time-independent fluids when no anomalous effects are anticipated. There is no method, however, to predict a priori whether such effects will be encountered in a new application. Even in the absence of these unusual effects, the predictions of various methods differ increasingly as the non-Newtonian behaviour becomes more and more pronounced, as seen in Fig. 3. It is thus suggested that for a given application, the ‘upper’ and ‘lower’ bounds on pressure gradient be established while carrying out design calculations. b. Submerged Object Models. In this approach, the flow through a packed bed is viewed as being equivalent to the flow around an assemblage of submerged objects (spheres and cylinders) and the resulting fluid dynamic drag manifests as the frictional pressure drop across a packed bed. Thus, the central problem here is essentially that of calculating the drag force on a typical particle in the assemblage. Various ideas have been proposed to achieve this objective, all of which involve modification of the Stokes drag on

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

207

a single particle. Two distinct approaches can be discerned in the literature. In the first of these, purely dimensional considerations coupled with experimental data have been used to derive the modified drag on a particle, without presupposing any arrangement of particles. This approach has been used successfully for correlating the data for Newtonian fluids by Barnea and coworkers [13-151 and others [271]. However, as far as known to us, this approach has not yet been extended to encompass non-Newtonian fluid behaviour. In contrast, the second approach involving the modification of the Stokes drag, based on more rigorous considerations, has proved to be some what more successful, at least in the low-Reynolds-number flow region. Numerous models have been proposed which purport to provide the necessary correction to the Stokes drag expression for a single particle. In his landmark studies of Newtonian fluids, Brinkman [25] calculated the drag force on a typical particle of the bed by assuming it to be embedded in a homogeneous and isotropic porous medium. He expressed his result in the form of the permeability of the porous medium, which can be rewritten in terms of the usual dimensionless friction factor and Reynolds number. These results have been subsequently improved on by Tam [I2451 and Lundgren [164]. In another approach, the correction to the Stokes drag is obtained by employing so-called cell models. Here, the hydrodynamic influence of neighbouring particles is simulated by enclosing the particle in question in a hypothetical cell. Thus, the difficult many-body problem is converted into a conceptually much simpler one-body equivalent. This approach is also not completely devoid of empiricism, especially with regard to the shape of the cell and the associated boundary conditions. Consequently, several cell models of various shapes and employing different boundary conditions are available in the literature; most of these have been reviewed by Happel and Brenner [97] and more recently by Jean and Fan [116]. Finally, numerous investigators have analysed the flow of incompressible Newtonian fluids through a variety of periodic arrays of spheres and cylinders to simulate the flow through a packed bed, and the resulting body of knowledge has been reviewed by Zick, and Homsy [270] and by Sangani and Acrivos [228]. Among the various cell models available in the literature, there is no question that the free-surface [96] and the zero-vorticity cell models [156] have received the greatest amount of attention. Both are of sphere in sphere type and exactly identical in all respects except with regard to the boundary conditions imposed at the cell surface. In the free-surface cell model, the cell boundary is modeled as being frictionless (that is, shear stress vanishes), whereas Kuwabara [156] proposed the vorticity to be zero. The latter is thus tantamount to the exchange of energy between the cell and the surroundings, violating the basic premise of the noninteracting nature of the cells. In view of

208

R. P. CHHABRA

this, it has been argued [70,97] that the free-surfacecell model has a sounder physical basis than the zero-vorticity cell model. Figure 5 shows both these idealisations of flow in a multiparticle assemblage. In this approach, each particle is envisioned to be surrounded by a hypothetical spherical envelope of fluid and the particle moves with a velocity !b While no-slip boundary conditions are applied at the particle surface, both the radial velocity and the shear stress (or vorticity) vanish at the cell surface. The radius of the latter is so chosen that the voidage of each cell is equal to the mean voidage of the assemblage. Analytical results based on both these models for the creeping flow of incompressible Newtonian as well as nonNewtonian fluids relative to an assemblage of solid spheres [97, 171, 188, 1891 and fluid particles [81, 90, 91, 1721 are available in the literature. Limited results for assemblages of solid particles outside the creeping flow regime are also available [lo& 109, 128, 160, 1611. Extensive comparisons between predictions and experimental results for Newtonian fluids have been carried out, and the useful ranges of applicability of both these models have been identified [l09]. It will suffice to add that the zero-vorticity model always predicts values of pressure drop which are 10-20% higher than those yielded by the free-surface cell model. The free-surface cell model has also been extended to the creeping flow of a variety of time-independent fluids through assemblages of solid particles. For instance, Mohan and Raghuraman [188, 1891 employed the well-known variational principles to obtain rigorous upper and lower bounds on pressure drop for the flow of power law and Ellis model fluids through assemblages of spherical particles. Similar results for fluids adhering to the Carreau model are also available in the literature [39]. Subsequently, Kawase and Ulbrecht [1271provided an approximate closed-form solution for this flow configuration. More recently, numerical results based on the finite-element solution of the pertinent equations together with the free-surface cell model have also been reported [ll2]. The results are often expressed in the form of a drag correction factor (Y(n, E)) defined as follows: GRe, Y(n, E ) = ___ 24 Evidently, Y(n = 1, E = 1) = 1, whereas the values of Y(E= 1) for different values of n (0.1 I n I 1) have been reported by Gu and Tanner [89a]. Figure 6 shows a comparison of the various predictions of the drag correction factor based on the free-surface cell model for a range of values of power law index and bed voidage. There seems to be reasonable correspondence up to about n 0.6-0.8. Recently [171, 228a1, the predictions of the free-surface model have been contrasted with that of the zero-vorticity cell model as well as with appropriate experimental results available in the literature as shown in Fig. 7.

-

V

V

t 4 0

W

Z E R O VORTICITY MODEL

F R E E S U R F A C E MODEL

FIG. 5. Schematic representations of free-surface and zero-vorticity cell models.

R. P. CHHABRA

210

10'

> CK0

L lo1

i2 Z

Q t-

V W

C Y

0

0

c3

a

g 10( - -

112) 188) Ref.(127)

Ref. Ref.

0

+ 0-2 1.0

08

0-6

0.4

0.2

0

POWER LAW INDEX,n FIG. 6 . Various creeping power law model predictions for free-surface cell model.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS 120 100

\&

>

az

0

a LL

I

I

- ZERO VORTICITY --- F R E E SURFACE WUW

21 1

I

MODEL(^^^ MODEL(11;

BAND OF DATA

50

z

0 -

F

0 W

cr az

20

0 0

0

a az n

10 \ \

5 1

I

I

I

0.8

0.6

0.4

POWER LAW INDEX, n FIG. 7. Comparison between predictions (free-surface and zero-vorticity cell models) and experimental results for packed bed conditions ( E = 0.4). Replotted from ref. [171].

Only results which are believed to be free from anomalous effects were included in this figure. Notwithstanding the wide scatter present in the experimental results, the correspondence between predictions and experiments is seen to be satisfactory but it is not yet possible to put complete credence in one model. More recently, Zhu [269] has analyzed the creeping flow of Carreau model fluids through a regular array of spheres and the predictions are in good agreement with those based on the free-surface cell model [39]. c. Other Methoh. Besides the aforementioned two distinct approaches, several other ad hoc methods have been employed to develop the means of

212

R. P. CHHABRA

predicting pressure drop through packed beds. For instance, in a series of papers, Pascal [204-2 131 empirically modified Darcy’s law for different types of non-Newtonian fluid models. By way of example, Darcy’s law is rewritten for power law fluids as follows:

Based on Eq. (32) and the other similar modifications, Pascal has extensively investigated the steady as well as transient flow in porous media. It is appropriate to add here that expressions based on the capillary model can be rearranged in the same form as Eq. (32), but the latter does not involve any description of the porous medium. Similarly, Benis [17] has used the lubrication approximation to adapt Darcy’s law for power law fluids. McKinley et al. [171] and White [261] have employed what are essentially dimensional considerations to extend Darcy’s law for the flow of timeindependent fluids, whereas Hassell and Bondi [1001 have proposed completely empirical expressions for pressure drop through packed beds of beads, screens, and mats. Unfortunately, most of these methods and expressions have not been tested extensively and are too tentative to be included here. Some ideas on the averaging procedures of the field equations for power law media are also available in the literature [158, 162). Similarly, the transient flow of power law media in oil reservoirs has been simulated by a few investigators [lOS, 197, 2553. 2. Transitional and Turbulent Flow Regimes As with all other fluids, the nature of the flow of non-Newtonian fluids is

determined by the relative importance of viscous and inertial forces. Owing to generally high viscosities, the flow conditions for non-Newtonian systems in packed beds rarely extend beyond the so-called laminar or creeping flow region. There does not, however, appear to be a clear-cut definition of the critical Reynolds number denoting the cessation of laminar flow conditions in packed beds. It is reasonable to postulate that the laminar flow region is characterized by an inverse relation between the friction factor and Reynolds number. This yields a critical value of the Reynolds number somewhere in the range -1 to 30, depending on the specific definition of Reynolds number used in a particular analysis. This is also borne out by numerical simulations [log, 111,1121. Although, strictly speaking, the capillary model is not applicable outside the laminar flow regime, it provides a convenient method of correlating the experimental data in transitional and turbulent regimes. Brea et al. [23], Mishra et al. [186], and Kumar and Upadhyay [153] have introduced the

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

213

following expression for calculating the effective viscosity in a porous medium: 127/(1 - E ) peff = m'[

E2d

]

"'-l

(33)

to define the particle Reynolds number as

By analogy with the well-known Ergun equation [74] for Newtonian fluids, the friction factor for power law fluids was correlated with Re' through an expression of the form U

f=-+fl Re'

(35)

The best values of a and fl are evaluated using experimental data; e.g., Mishra et al. El861 and Kumar and Upadhyay [153] found u = 150 and = 1.75 (same as in the Newtonian case), whereas Brea et al. [23] reported a = 160 and fl = 1.75. Kemblowski and Mertl [135], on the other hand, extended their low-Reynolds-number work to embrace transitional and turbulent regions as 150

f =

where

ti

+ 1.75

tiH2

Jm

and H are further correlated as follows: H =[Re*

c aini 5

log ti =

i=O

a, = - 1.7838,

a2 = - 6.239,

a, = 5.219, a4

= 2.394,

a5

a3 = 1.559,

= - 1.12

5

log =

bini i=O

bo = - 4.9035,

b,

=

10.91,

b4 = 4.25,

b, = - 12.29, b, = - 1.896

b,

= 2.364,

(36e) Equation (36) is based on experimental data empracing the conditions 0.5 I n I 1.6 and 0.03 5 Re* I 115. Note that this cumbersome-looking expression does reduce to its proper Newtonian limit as the value of n

R. P. CHHABRA

214

approaches unity. Kemblowski et al. [134] asserted that this method predicts the value of the pressure drop with an uncertainty of about L 30%. Figure 8 shows the results in graphical form for E = 0.4. Limited results for the flow of power law fluids obtained using the freesurface cell model are also available up to about a Reynolds number of 20. Preliminary comparisons between simulations and measurements are affirmative [lOS]. By invoking the boundary layer flow approximation in the free-surface cell configuration, both Hua and Ishii [103a] and Kawase and Ulbrecht [l28] have presented the values of the drag coefficient for particle assemblages for Reynolds up to 1000, but these predictions are of doubtful validity owing to the unrealistic boundary conditions implicit in the freesurface cell model. Based on limited amounts of experimental data, Singh et al. [236] and Masuyama et al. [1751 have developed cumbersome empirical expressions for the turbulent flow of Bingham plastic fluids through packed beds of spherical particles. Little is known about their accuracy and reliability and hence they are not included here.

0 -

I

I

I

I

1oo

1o1

lo-7

10"2

'01

REYNOLDS NUMBER ,Re* FIG. 8. Friction factor-Reynolds number relationship for turbulent flow of power law media. Replotted from ref. [134].

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

215

3. Eflect of Particle Shape It has long been known that the pressure drop for flow through a packed bed is strongly influenced by a large number of parameters, such as particle size distribution and particle shape and roughness [62, 165). Particle shape is a much more important variable in determining the value of voidage (hence pressure drop) than the particle roughness. A considerable body of literature is available on the influence of particle shape and roughness [62, 1651 on pressure drop for Newtonian fluids. From the engineering applications standpoint, MacDonald et al. [165] have shown that the widely used Ergun equation, originally developed for beds of spherical particles, also yields satisfactory results for nonspherical particles provided the sphere diameter is replaced by the volume equivalent diameter (&) multiplied by a sphericity factor (+s). The limited work available on the flow of non-Newtonian fluids through beds of nonspherical particles (cubes, cylinders, gravel chips, etc.) also seems to conform to this proposal [40,153,168,233,251,265]. Figure 9

m

0

J

-3

-2

-1

0

1

Log (Re*) FIG. 9. Friction factor-Reynolds number relationship for fixed beds of nonspherical particles. Replotted from ref. [38].

216

R. P. CHHABRA

shows a typical comparison between experimental results and the predictions of Eq. (37) with d = deq&: 150

f = Re* + 1.75 ~

(37)

4. Effectof Containing Walls In most real applications, packed beds are of finite size in the radial direction and the confining walls are known to influence the flow phenomena in packed columns. Wall effects are manifest in two ways. First, the wall of the tube provides an extra surface which comes in contact with the moving liquid, and therefore the frictional losses occur over a larger area than that of the packing. The second effect, the more important one, is the variation of bed voidage in the radial direction, which reaches a value of almost unity near the wall. Three different approaches have evolved to account for the presence of walls. The simplest is the one in which the constants off-Re* relations [such as a and B in Eq. (35) or Eq. (37)] are determined as functions of the particle/ tube diameter ratio. This approach has been used for Newtonian [79,220] as well as power law fluids [243]. In the second approach, the contributions of the confining walls to the wetted perimeter and flow area are incorporated in the definition of the hydraulic radius. Mehta and Hawley [I781 thus obtained the following expression for R,: R -

Ed

- 6(1 - E)M

where

1 3Dl-E)

2d M = 1 +---

It is really tantamount to substituting ( d / M )ford in the expressions discussed in the preceding sections. This form of wall correction has been used successfully for the flow of power law [95, 2021 and Bingham plastic [175] fluids through packed beds. This method, however, does not take into account the variation of the bed voidage in the radial direction. Furthermore, Cohen and Metzner [47] point out that one would intuitively expect the wall effects to persist only in the wall region, whereas Eq. (38) is applied throughout the entire column. The third method takes into account the radial voidage profiles existing in a bed. The limited literature in this field has been critically and thoroughly reviewed by Cohen and Metzner [47]. The salient features of voidage

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

217

distribution are that it is almost unity at the wall and oscillates about the mean value as one moves away from the wall, eventually attaining a constant value equal to the mean voidage. Experimental data elucidating the effects of the ( d / D ) ratio on bulk voidage of packed beds as well as radial voidage profiles in beds of spherical particles have been presented by numerous investigators (e.g., see [18, 36, 57, 66, 101, 173, 218, 222, 2231). Cohen and Metzner [47] developed a three-zone model to account for the wall effects. They essentially employed different voidage functions for each region while calculating the volumetric flow rate. Based on extensive comparisons between predictions and experimental results, they concluded that if the wall effects are to be avoided, packed beds with ( D / d ) values greater than 30 should be used. This observation is consistent with the findings of Srinivas and Chhabra [243]. Subsequently,Nield [1961 developed a two-layer model whose predictions are nearly as accurate as those of the three-region model of Cohen and Metzner [47]. From the applications standpoint, however, the approach of Mehta and Hawley [1781 is more convenient.

B. VISCOELASTIC EFFECTS All materials exhibit varying levels of viscous and elastic characteristics. However, the extent to which the elastic properties are important varies from one application to another. It is now generally agreed that the flow of viscoelastic media in packed beds results in pressure drops larger than those expected from purely viscous considerations. At low velocities, the frictional pressure gradient is determined primarily by shear viscosity, with the viscoelastic effects being negligible. As the flow rate is gradually increased, viscoelastic effects begin to be seen. Consequently, when the loss coefficient (A) is plotted against a suitably defined Deborah (or Weissenberg) number, the pressure drop rises rather sharply beyond a critical value of the Deborah number. Figure 10 illustrates this type of behaviour. Although numerous investigators have obtained qualitatively similar results with a wide variety of chemically different polymers, there is little quantitative agreement regarding the critical value of the Deborah number marking the onset of viscoelastic effects as well as the magnitude of the increase in the value of pressure drop above that estimated from shear viscosity considerations. One possible reason for this lack of agreement in the reported values of the critical Deborah number is the arbitrariness inherent in the definition of Deborah number, especially with regard to the evaluation of a fluid characteristic time. The oft-used definition of Deborah number is

ev, De = 1,

(39)

R. P. CHHABRA

218

0

Carbobol

e ' PIB

I

o ET 597 WISSLER

I

0.001

0.01

(263)

I

0-1

I

1

DE BORAH NUMBER, De FIG. 10. Viscoelastic effects in packed bed flows. Replotted from ref. [134].

where 8 is a characteristic time of the fluid and (IJV,) is a characteristic time for the process. Considerable confusion exists in the literature regarding an appropriate choice of each of these variables. For instance, some researchers [39,138,226] have extracted the value of 8 from shear viscosity data, whereas others [24,167, 174,237,2501 have evaluated it using primary normal stress difference data in steady shear. Other methods, such as die swell measurements [136] and dilute solution theories [133], have also been used to calculate the value of 8. There is no question that all the aforementioned methods yield values of 0 which are a measure of the importance of viscoelastic effects but different methods are known to yield widely divergent values of 8 [69], precluding the possibility of rigorous comparisons. Likewise, the choice of a characteristic velocity (V,) is also far from obvious; some have employed the superficial velocity for this purpose, while others have preferred the average interstitial velocity. Finally, although a proper and unambiguous choice for 1, (e.g., maximum or minimum clear passage) is also not clear, the particle diameter has often been used as the characteristic linear dimension for spherical particles. Table IV provides a concise summary of the range of definitions of Deborah number used to interpret and/or correlate pressure drop data in packed bed flows.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

219

Examination of Table IV clearly shows two to three orders of magnitude variation in the reported critical values of De denoting the onset of viscoelastic effects. In view of this, it is not surprising that the empirical expressions (which are direct extensions of those developed for time-independent fluids) which purport to correlate the friction factor for viscoelastic media are equally diverse in form; a selection of these is shown in Table V. Evidently, most of these are of the following two forms:

f

A =E+F(Re,De)

f

= -(1

or A Re

-t BDe2)

Whereas the quadratic dependence on De, embodied in Eq. (40b), has some theoretical basis [263, 2691, correlations of the form of Eq. (40a) are of completely empirical character. Intuitively, it would be desirable for the function F(Re, De) to meet the following limiting requirements. For purely viscous (or negligible elastic) effects, F(Re, De) -P 0, and for elastic liquids F(Re, De) -, 0 with diminishing value of the Deborah number. Admittedly, although all expressions summarised in Table V as well as the others available in the literature ascribe the excess pressure drop to the viscoelastic behaviour (quantified in terms of De), there is some evidence that the inclusion of De alone is not adequate to account for viscoelastic effects [183, 2501. Besides, several recent studies seem to suggest that the flow in packed beds involves an appreciable amount of extensional flow, and hence it is an unsound practice to interpret or correlate results only in terms of steady shear rheological data [63-65, 68, 92). In order to elucidate the role of the successive contractions and expansions character of the flow geometry in packed beds, several investigators have carried out numerical simulations of viscoelastic flows in a variety of periodically constricted tubes (PCTs), the most commonly used configuration being sinusoidal variation of the cross section in the axial direction [SS, 104, 115, 170, 198,217,2681. Considerable controversy, however, surrounds the theoretical as well as experimental findings in this field; e.g. see [104,217]. Numerical simulations [115] seem to suggest that even in the presence of significant viscoelastic effects together with large amplitude ratio of PCTs, the resulting values of pressure drop are virtually indistinguishable from those predicted using shear viscosity considerations alone. To date, clearly, the debate is far from being over!

TABLE IV DEFINITIONS USEDAND CRITICALVALUESOF DEBORAH NUMBER REPORTED IN THE LKERATURE

Deborah number Investigator

h)

Sadowski and Bird [22q Gaitonde and Middleman [82] Marshall and Metzner [174] Siskovic et al. [237] Kemblowski el al. [133] Vossoughi and Seyer [256] Michele [1831 Park et al. [202] Kemblowski et a!. [136] Tiu et al. [2M]

(-1 VPOIds 1,2

8vd

ev&d 8ykd

8V&d

ev4m

Fluid characteristic time, 8, (s)

Range of 0 (ms)

Critical De

Estimated from viscosity data via the Ellis fluid model 8 = 12poM/n2CRT(Bueche theory) Based on N data Based on N, data 8 = 1201, - A)M/nZCRT Based on N, data

s 77 4.7-8.6 0.96-34 200-4000 33 10-20

Based on N data Based on dies well measurements Based on N , data

80-1200

0.1 1.2 0.05-0.06 0.3 0.2 0.02 3 > 0.13 > 0.07 No critical value

N,/2rjr 0vd 8ved and 8vul 0vf4

fi

-

N

-

TABLE V CORRELATIONS OF f FOR VISCOELASTIC FLUIDS’ Investigator

Equation forf

Observations

~~

f--= E3 (1 - E ) ’

Sadowski and Bird [226]

Wissler [263]

180

(

,c+( I - &

I+-.-a

+ 3 T1,2

= (1

?*

),4

+ lODe*)

Only weak viscoelastic effects

Used the experimental data of Metzner and co-workers to evaluate the constants.

(--El-)fRe,,

Vossoughl and Seyer [256]

1-&

N

2

Michele [183Ib

+M

= (5/Re,)

+ 90De2

+ (0.3/Reg1’) + (0.075De2/Re,)

Kemblowski and Dziubinski [133]

(&)Re*

Kemblowski and Michniewicz [137]

(->fRe’

a

=1

E3

= 15q1

+ 8De2)

= 180(1+

4De2)

1-&

See Table IV for corresponding definitions of De. pvdp

‘

perf evaluated at fCrr = 6~tV(2/3)~.~(1 - E)/G2d.

Datas for arrays of cylinders

222

R. P. CHHABRA

c. BEHAVIOUR OF DILUTE/~EMIDILUTE DRAG-REDUCING POLYMER SOLUTIONS Studies of the flow of dilute/semidilutepolymer solutions in packed beds and porous media have received impetus both from theoretical considerations, such as the fact that this flow configuration provides a good ‘testing ground’ for validating the predictions of dilute solution theories, and practical considerations, such as their applications in enhanced oil recovery processes [59, 117, 192, 2191. Beside improving the efficiency of the oil displacement process, the use of dilute polymer solutions is beneficial in at least two other ways, namely the reduction in the permeability of the rock and the retardation of flow at high flow rates which is brought about by viscoelastic effects [113]. Although most field applications use the commercially available partially hydrolyzed polyacrylamide solutions, laboratory tests have been carried out with dilute solutions of polyethylene oxide PEO) [68, 114, 125, 159, 1931 and of polysaccharide, a biological polymer [37], in addition to polyacrylamide (PAM) solutions. The main difficulty with PEO solutions is this susceptibility to mechanical degradation and the scission of PEO molecules at high deformation rates. Sometimes, this uncertainty has led to conflicting conclusions [114, 1251. Broadly speaking, the term ‘dilute’ solution refers to the condition of no or little entanglement of polymer molecules in a solution. Among the several quantitative criteria available to classify a solution as dilute, semidilute, or concentrated, the simplest one is that in which a solution is classified as dilute and/or semidilute as long as [q]C < 0.2-0.3. One can also make a distinction between a concentrated and a dilute or semidilute solution depending on whether their shear viscosity shows discernible variation with shear rate. Dilute and semidilute solutions (< 100-200 ppm, depending on molecular weight) are characterised by virtually constant shear viscosities, whereas concentrated solutions show marked shear thinning behaviour. On both these counts, it is safe to state that the preceding sections have dealt with the flow of concentrated polymer solutions. Most investigators have attempted to establish the friction factorReynolds number relationship for the flow of dilute or semidilute polymer solutions in packed beds, although the stability of flow has also been investigated in one instance [l20]. Dauben and Menzie [53] appear to be the first who worked with really dilute/semidilute solutions of PEO in porous media and documented up to 20 times larger values of pressure drop than could be attributed to the solution viscosity. Subsequently, qualitatively similar (even more dramatic) results have been reported by many other investigators [63-65, 68, 93-95, 114, 118-121, 125, 150, 151, 1931. The emerging overall picture can be summarised as follows: For a given polymer

-

-

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

223

solution-packed bed combination, the experimental values of friction factor are in line with the expected Newtonian behaviour up to a critical Reynolds number (i.e. velocity). Beyond this value, with increasing liquid velocity, the friction factor deviates increasingly from the Newtonian line, goes through a maximum value, and finally shows a weak downward trend. Typical experimental data exemplifying all these features are shown in Fig. 11 for a series of PEO solutions. Several plausible mechanisms, including gel formation and adsorption [30, 142, 191,219,2261, plugging of pores [98], and extensional and viscoelastic mechanisms [92, 93, 114, 122, 1251, have been advanced to explain the observed pressure drop behaviour, but the observed pattern is generally attributable to the viscoelasticity of solutions. Thus, the different regions present in Fig. 11 can be explained qualitatively as follows: at sufficiently low velocity (i.e., small values of De), the relaxation time of the fluid is shorter than that of the process ( - d/V), enabling a fluid element to

1 ," ::

10

\

* f2

1

Ncwtoniancurve-

1Cj1

loo

10'

REYNOLDS NUMBER,Re FIG. 1 1 . Friction factor-Reynolds number relationshipfor dilute PEO solutions in packed beds. Replotted from ref. [125].

R. P. CHHABRA

224

adjust almost instantaneously to its continually changing surroundings as it traverses the tortuous path whence no viscoelastic effects are observed. As the solution velocity increases, the value of ( d / V )decreases and the fluid element no longer has the time to adjust to its rapidly changing surroundings. This inability of fluid elements leads to the buildup of elastic stresses, which, in turn, show up as an increase in the resistance to flow. A thorough and systematic study of the flow of PEO solutions ( < 80 ppm) in glass bead packs is due to Kaser and Keller [125]. In another study with PEO solutions, Naudascher and Killen [1931 investigated the onset conditions and the saturation limit (i.e., maxima in friction factor). Based on simple heuristics, they developed the following criterion for the onset of viscoelastic effects: d

JE= constant

where 0 is estimated from the Rouse formula for dilute solution [224] and the constant on the right-hand side is a characteristic value of polymer-solvent combination. The results shown in Fig. 11 are replotted in Fig. 12 in accordance with Eq. (41).The agreement is seen to be about as good as can be expected in this type of work. Figure 12 also supports the contention of

= 20 o o

87fnm I

1tj5

I

I

40 60

1 1 1 1 1

I

-4

10

I

I

1 1 1 1 ,

lo3

FIG. 12. Data of Fig. 1 1 replotted in accordance with Eq. (41). Replotted from ref. [125].

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

225

Naudascher and Killen [193] that the maximum value of the loss coefficient A shows a linear dependence on the concentration of polymer solution. This approach, however, has not been tested for polyacrylamide and polysaccharide solutions. In recent years, extensive research efforts have been directed at understanding the behaviour of dilute PAM solution in porous media flows. Most contributions in this area have come from two research groups. Durst and co-workers [63-65, 93, 94, 150, 1511 have systematically elucidated the influence of molecular weight and concentration of PAM solutions, degree of hydrolysis, number and type of ions present in the solution, and particle size on the resistance to flow in random and orderly beds of spherical particles. The main focus of this research has been to delineate the importance of extensional effects in this flow configuration. It has been possible to reconcile most of the data using a reduced extensional viscosity and a modified Deborah number based on Bird's finitely extensible nonlinear elastic (FENE) model for dilute solutions of flexible molecules. Overall, their correlation is good but is not completely successful [113]. The work of the second group, led by Chauveteau [37,170], although less extensive, is also praiseworthy because they have employed not only the actual sandstone filters but also channels of other geometries to mimic a wide variety of porous media. Moreover, instead of using the conventional Ergun coordinates cf and Re), this group have presented their results in the form of shear stress-shear rate plots based on porous media and viscometric measurements. It is then argued that if the two sets of data collapse onto each other, the flow is assumed to be shear dominated and free from all other complications such as viscoelastic effects. Conversely, the lack of agreement between the two sets of data implies the presence of anomalous effects, e.g., slip. Furthermore, when the pore size is small, the macromolecules migrate away from the wall and the resistance to flow is found to be lower than that expected from the bulk solution viscosity considerations. Hence, this form of presentation facilitates the delineation of shear thinning, viscoelastic, and wall-surface interactions under appropriate circumstances. From a theoretical standpoint, Daoudi [52] has estimated the rate of energy dissipation by considering the deformation of randomly coiled macromolecules in a periodically constricted tube. Preliminary comparisons between the predicted and observed values [114] of the maximum pressure drop with PEO solutions appear to be encouraging. D. WALL-POLYMER MOLECULE INTERACTIONS Aside from the voluminous body of information referred to in the foregoing sections, the contemporary literature on the flow of polymer solutions

226

R. P. CHHABRA

through consolidated and unconsolidated porous media also abounds with numerous anomalous and hitherto unexplained phenomena. For instance, the results obtained under constant-pressure and constant-flow conditions do not agree with each other [226]; the apparent shear stress-shear rate data evaluated from porous media experiments are often at odds with the corresponding viscometric measurements [45, 53, 103, 2341. Similarly, numerous workers [12,26-30, 53, 118-1213 have documented pressure drop values well in excess of those anticipated from the rheological properties, even when the viscoelastic effects are believed to be unimportant. All these observations seem to suggest significant differences between the in situ and bulk solution rheological characteristics [61, 103, 2541. Many plausible mechanisms, based on a range of considerations such as gel effects [28,226], slip effects [46,48,95, 142-1491, adsorption [102, 118-1211, wall effects [Sly 199,200,219,227, 2301, pseudodilatant behaviour [26-301, and uncertainty in tortuosity factor [56], have been postulated; none, however, has proved to be entirely satisfactory. The presence of a solid boundary is known to alter the rheology of a macromolecular solution; consequently,polymeric systems show anomalous wall effects including steric hindrance (when the characteristic linear dimension of flow passage is comparable to the size of macromolecules) and polymer retention [199,200,230,262]. All such phenomena, although poorly understood, are believed to contribute to the observed differences between the in situ and bulk rheological characteristics in porous media flows. Since excellent comprehensive reviews are available on this subject [44,227,262], only the salient features will be recapitulated here. 1. Polymer Retention

Macromolecules are retained as a polymer solution flows through a porous medium, thereby reducing the solution viscosity and permeability, both of which adversely influence the efficiency of oil recovery processes [44,227, 2621. Broadly speaking, polymeric molecules are retained by two mechanisms, adsorption and mechanical entrapment. Often the total amount of polymer retained is estimated from the concentration of exiting solution, and hence the individual contributions of adsorption and mechanical entrapment are not known but can be inferred using additional information such as pore size distribution and size of polymer molecules in solution. For a given solid-solution combination, the amount adsorbed is influenced by a large number of variables such as the chemical nature of the polymer, its molecular weight distribution and concentration, the pH, the presence or absence of certain ions, porosity, permeability, nature of the surface, and the flow rate [1691. Moreover, the results obtained under dynamic conditions

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

-

0, Y

E"

16

v

z

-

I

I

0 2% N a C l

' r

I

227

I

0 10% NaCl

X 0 '10 NaCl

-

-

0 I-

-

0

160

320

400

EQUILIBRIUM CONCENTRATION ( ppm) FIG. 13. Typical adsorption data for polyacrylamidesolutions with different doses of NaCI. Replotted from ref. [244].

often do not match those expected under static conditions [244, 2621. Generally, the amount of polymer adsorbed per unit mass of solid increases with polymer concentration, with a propensity to level off beyond a critical polymer concentration, as illustrated in Fig. 13. In view of the large number of influencing variables, the results are strongly system dependent and therefore generalizations should be treated with reserve. Gogarty [U] and Smith [240] have examined the process of polymer retention by mechanical entrapment. This mechanism comes into play whenever the polymer molecules in solutions are of comparable size to the pores present in the porous medium, i.e., plugging of small pores by polymer molecules. Adsorption also promotes complete or partial blockage of pores. Thus, the two mechanisms go hand in hand. An excellent review article is available on this subject [262]. From a macroscopic standpoint, irrespective of the underlying mechanism, polymer retention has two effects. First, the retained molecules occupy a portion, however small, of the void volume present in the porous medium, thereby reducing its permeability. Second, the thin layer of solution in the vicinity of solid walls is depleted in polymer concentration compared to the bulk solution, thereby altering its rheology. Polymer adsorption has been modeled as a kinetic process obeying a Langmuir-type expression whose constants are found to be temperature and system dependent C262). This

R. P. CHHABRA

228

approach, however, has not yet been incorporated directly in the models for flow through porous media. Alternatively, based on the assumption of monomolecular adsorption, some estimates of the adsorbed layer thickness are available; typically, these are of the order of a few micrometers [46,142,143]. This, in turn, is employed to calculate the effective radius to be used in the usual capillary model [l02]. Detailed descriptions of this as well as other approaches are available in a recent book C241). 2. Slip Eflects Another phenomenon that has received considerable attention is the slip effects that stem from the fact that the macromolecular solutions do not appear to satisfy the classical no-slip condition at solid boundaries [44,199, 200, 2301. Although it is not clear whether true slip occurs, this notion has proved to be convenient in explaining and interpreting some of the anomalous results reported in the literature. Cohen [44] has presented a thorough and critical review of slip effects encountered in the flow of polymer solutions in a range of geometries, including small capillaries, down an inclined plate, and packed beds. In contrast to the phenomenon of polymer retention accompanied by the reduction in permeability, the slip effects, inferred from the observed abnormally high flow rates, result in an increase in the permeability. Moreover, the slip effects are known to be more pronounced in small pores, which are also more prone to polymer retention by adsorption and blockage. It is thus possible that, under appropriate circumstances, these two competing mechanisms may nullify each other and the permeability of the porous medium may even increase. Currently, two approaches are available for investigating the importance of slip effects in porous media flows. In the first method, the capillary model is modified by simply replacing the no-slip condition by a nonzero velocity at the wall (V,), and the resulting expression for the apparent shear rate, i.e., Eq. (30), can be rewritten as 2v

2V,

Rh

Rh

-= -

4 + -5

jF

r2f(z) dr

=w

Differentiation of Eq. (42) with respect to (I/&,) at constant zw yields

where the factor ( L J L )comes about through the use of Eq. (3). Equation (43) suggests that one can evaluate the effective slip velocity Y, provided data are

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

229

available for a range of values of R,, i.e., different particle diameters. This approach has been successfully employed by Kozicki and co-workers to rationalize the slip effects in packed bed flows [95, 142-147, 1493. In the second method, the significance of slip effects is ascertained by simply comparing the experimental and predicted Ap-Q relationship. The predictions are based on the choice of a fluid model as well as a description of the porous medium. For instance, Cohen and Cheng [45] have illustrated the utility of this approach and asserted that ( VexperimentaJ~p/predicted)< 1 indicates the presence of slip effects. As mentioned previously, the behaviour of macromolecular solutions on or near solid surfaces is greatly influenced by steric, repulsive, or attractive interactions between the surface and polymer molecules. It is reasonable to postulate that attractive forces will promote adsorption, whereas repulsive and steric effects are expected to facilitate the migration of molecules away from the solid surfaces, thereby giving rise to depleted layers in the wall region; the latter is thought to be the main mechanism for the slip effects. Such simple ideas coupled with the notions of stress-induced diffusion, thermodynamic equilibrium, etc. have been employed to develop theoretical frameworks to model slip effects. Although the preliminary results appear to be encouraging, none of these attempts has yet been refined to the extent to be used as a basis of calculations.

E. MIXING Limited results on mixing and dispersion of power law fluids in packed beds have been reported by a few investigators [loo, lola, 215,259,2601. In broad terms, the quality of mixing deteriorates with increasing degree of nonNewtonian behaviour. For weakly shear thinning fluids (n > OM), the Newtonian expressions are adequate to estimate the value of the Peclet number, at least within a first-order approximation. However, the particle shape seems to exert appreciable influence on axial dispersion [lola].

-

F. HEATAND MASSTRANSFER In contrast to the extensive literature on wall-to-bed and interphase heat and mass transfer in packed beds for Newtonian fluids (e.g., see [22]), very little is known about the analogous phenomena involving non-Newtonian fluid behaviour. The only study of forced convection in packed beds with power law fluids is that of Wang et a!. [258]. They combined the modified Darcy’s law with the energy equation and elucidated the significance of thermal dispersion. However, no results on Nusselt number, etc. are presented. Limited information is, however, available on interphase mass transfer in

230

R. P. CHHABRA

packed beds for power law fluids. For instance, using the fluid boundary layer approximation, Kawase and Ulbrecht [1271 have extended the analysis of Pfeffer [216] to predict mass transfer from beds of particles to power law fluids. Their results are, however, applicable only under creeping flow conditions and for weakly non-Newtonian effects (n 1). Indeed, their analysis predicts very little deviation from the corresponding Newtonian values of Sherwood number up to about n 0.7. In a subsequent study [1281, the same authors reported analogous results in the high-Reynoldsnumber region by employing the boundary layer flow approximation. In yet another paper, based on the penetration model [34,187] for interphase mass transfer, Kawase and Ulbrecht [130, 130a] proposed the following expression for Sherwood number for power law fluids in packed bed flows: ( n + 2 ) / 3 ( n + l ) Sci/3 Sh = Al(n)&-[l/(fl+l)lRe (44)

-

N

P

where the values of Al(n) are available in the original paper [130b]. An empirical equation for particle-fluid mass transfer based on the BlakeKozeny model of packed beds is also available in the literature [132]. More recent numerical simulations [228a], however, predict significant differences in mass transfer characteristics for power law and Newtonian media. Some experimental results on interphase mass transfer with power law fluids in fixed beds are available in the literature [49, lola, 153, 218a, 232, 2641. Kumar and Upadhyay [153] and Wronski and Szembek-Stoeger [264] have measured the rate of mass transfer from fixed beds consisting of spheres and cylindrical pellets of benzoic acid to aqueous solutions of carboxymethyl cellulose. Combined, these two investigations encompass wide ranges of I;Re, I;40 and 800 ISc, 5; 72,000). Based on experimenconditions tal data obtained with one polymer solution (n = 0.85), Kumar and Upadhyay [153] proposed the following empirical correlation for the j factor:

.

EJ

0.765

= Re0.82 + ~

1

0.365 Re0.386

(45)

1

Equation (45) was stated to reproduce their results with a mean deviation of 10%. To facilitate a direct comparison between the experimental results [153] and the predictions of Eq. (44), the latter can be rearranged in the following form: Ej = A , ( ~E)~e;(2n+i)/3(n+1) ,

where

(464

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

23 1

.-

loo

lo'

102

500

REYNOLDS NUMBER , Re1 FIG. 14. Interphase liquid-particle mass transfer results in fixed bed flows of power law fluids.

Figure 14 shows a comparison between the experimental [153] and predicted values [using Eq. (46)] as well as those of Kawase and Ulbrecht [128]. The agreement is seen to be satisfactory. In another experimental investigation [264], particle-liquid mass transfer data have been reported for packed beds of cylindrical pellets. An analysis of these results shows that one can use Eq. (45) or Eq. (46) for cylindrical particles also, provided a volume equivalent diameter is used in lieu of the sphere diameter, although the recent study of Hilal et al. [lola] seems to suggest the use of (6/a,) as the characteristic linear dimension rather than the volume equivalent diameter (a, is the specific area). The experimental results [264] shown in Fig. 14 substantiate this assertion. Included in the same figure are predictions based on the following empirical expression due to Wronski and Szembek-Stoeger [264] : Ej =

1 (0.097Rey.30+ 0.75Rey.61)

(47)

The agreement between the two independent sets of experimental results and the predici ons of Eqs. (45), (46), and (47) is seen to be satisfactory but further discrimina .ion must await additional work, especially for a range of values of the power law index. In an interesting study, Coppola and Bohm [49] investigated mass transfer with power law fluids flowing through a packed

232

R. P. CHHABRA

bed of screens. Based on two different approaches, namely flow around a cylinder or the capillary bundle model, they developed two separate correlations of the following generic form: Sh, = A ReBScC

(48)

For the flow past cylinders approach, based on the following definitions: Vd sc = D Re2/(n+1)' AB

pV2-"d

Re =

~

Perf

the best values of the constants in Eq. (48) were found to be A = 0.838, B = 0.33, and C = 0.37. For the capillary bundle model, the Schmidt number is defined as sc = Perf V"PDAB

and the resulting best values of the constants are A = 0.908, B = 0.33, and C = 0.34. The values of constants have been estimated using experimental data for two values of n only (n = 0.74 and 0.81). In Eq. (48a), M1 is the so-called Kozeny constant and is a function of bed voidage only. In spite of the two entirely different approaches, the resulting values of A, B, and C are seen to be nearly equal. The effect of drag-reducing additives on the rate of mass transfer in fixed bed reactors has been studied by Sedahmed et al. [232]. The rates of mass transfer were measured for the cementation of copper from dilute copper sulfate solutions containing trace amounts of PEO in a fixed bed of zinc pellets. Depending on the polymer concentration and the value of the Reynolds number, the rate of mass transfer was found to decrease by up to about 50% below the corresponding value for copper sulfate solutions without any polymer addition. The maximum reduction in mass transfer was found to occur at Re = 1400. Similar conclusions regarding gas-liquid mass transfer in packed absorption columns have been reported by others [218a]. Our understanding of these phenomena is far from satisfactory, however. IV. Non-Newtonian Effects in Fluidised Beds

A. TWO-PHASE SYSTEMS When a liquid flows upward through a bed of particles, one can discern three distinct flow regimes depending on the flow rate of the liquid. At low superficial velocities, it gives rise to a fixed bed; but if the velocity of the liquid

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

233

is sufficiently high, the solid particles will be freely supported in the liquid to give rise to what is known as a fluidised bed. At very high flow rates, the solid particles will be transported from the system. The bed in which the fixed bed conditions cease to exist is described as an incipientlyjuidised bed and the liquid velocity corresponding to this point is termed the minimumjuidisation velocity. When the flow rate of liquid is increased above this value, the bed continues to expand so that the average distance between the particles increases. Behaviour of this type is described as particulate juidisation. It is generally believed that this kind of fluidisation occurs with most liquid-solid systems (except when the solid particles are too heavy) and in gas-solid systems over a limited range of conditions, particularly with fine particles. This section, however, is concerned primarily with particulate fluidisation as encountered in liquid-solid systems involving non-Newtonian liquid media. It is widely acknowledged that the minimum fluidisation velocity and the extent of bed expansion as a function of liquid velocity (above the point of incipient fluidisation), together with the rates of interphase and wall-to-bed heat/mass transfer, are some of the important design variables in such systems. Consequently, these aspects have received considerable attention in the literature. Other important aspects of fluidization technology include the mixing and flow patterns, detailed particle trajectories, residence time distribution, particle attrition, and conversion and selectivity. But these have been studied less extensively even in the case of Newtonian fluidising media. Most of the information pertaining to Newtonian media is available in a number of excellent books on this subject [22, 54, 1551, and the corresponding body of information on non-Newtonian media is reviewed here. 1. Minimum Fluidisation Velocity a. Definition For a liquid flowing upward through a bed of particles, the pressure drop across the bed, Ap, initially increases as the superficial velocity of the liquid, V, increases as long as the bed behaves like a fixed bed. When the velocity of liquid reaches such a value that the pressure drop across the bed is equal to the buoyant weight per unit area of the particles, any further increase in velocity leads to rearrangement of the particles in such a manner that the resistance to flow remains the same; i.e., as the velocity increases, the bed undergoes an expansion but the pressure drop remains essentially constant. This is known as the point of incipient fluidisation, and the corresponding velocity and bed voidage are designated the minimum fluidisation velocity, Vmf, and the corresponding bed voidage, E , ~ . For V > Vmf,the pressure drop across the bed remains constant. If the- velocity of liquid is gradually decreased, the pressure drop remains constant up to the point of incipient fluidisation but the pressure drop values in the fixed bed region turn out to be

234

R. P.CHHABRA

Log

v

FIG. 15. Schematic pressure drop-superficial velocity relationship showing transition from fixed to fluidised bed region.

slightly smaller than that recorded when the velocity is being increased. This hysteresis is often attributed to the slight change in the value of bed voidage due to ‘repacking’ of the bed. Figure 15 shows schematically the aforementioned Ap-V behaviour. In practice, however, departures from such an ideal behaviour are observed, due mainly to interlocking of particles, channelling, etc. Moreover, the transition from fixed to fluidised conditions occurs gradually over a range of velocities rather than abruptly, as shown by a dotted line in Fig. 15 [221]. Since the minimum fluidisation velocity has no absolute significance, the generally accepted standard method for its determination from Ap- V plots involves drawing separate lines through fixed and fluidised bed regions, and the point of intersection of these two lines yields the value of V,. This procedure is also shown in Fig. 15. b. Prediction of V , Experimental determination of the minimum fluidisation velocity is neither always possible nor desirable. Hence the need for its estimation often arises while performing design calculations for fluidisation equipment. Most attempts at developing predictive expressions for the estimation of V, hinge on the fact that at the point of incipient fluidisation, the pressure drop per unit length of the bed is given by its buoyant weight, which, in turn, is equated to the value obtained by assuming the bed to behave like a fixed bed of mean voidage of emf, -AP =(IL

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

235

It is readily recognized that although an incipiently fluidised bed represents a slightly looser bed, it is justifiable to treat it as a fixed bed since the particles are still in contact with each other; this assumption has, however, been questioned by Barnea and co-workers [13-151. In principle, thus, one can use any of the methods described in the previous section for calculating the pressure drop across a fixed bed to substitute in Eq. (49). By way of example, this procedure is illustrated for Newtonian media and can then easily be extended to non-Newtonian systems. For instance, one can combine the Ergun equation [74] with Eq. (49) to yield

It is customary to rewrite Eq. (50) in dimensionless form:

where

sd3 3 Gamf= - (CDRe2)&= p(Ap) 4

P

and

Thus, for a given Liquid-solid system, one can evaluate Gamf,which, in turn, facilitates the estimation of Remf(hence Vmf) via Eq. (51) provided the value of cmf is known. The main drawback of this class of methods is that the predictions are very sensitive to the value of emf,which cannot be measured with very high levels of accuracy. A few investigators have attempted to circumvent this difficulty by correlating the minimum fluidisation velocity with the free settling velocity of particles. Consequently, there is no scarcity of predictive expressions employing both these approaches which purport to yield the values of the minimum fluidisation velocity for Newtonian media. Most of these have been reviewed by Couderc [SO]. In contrast, the corresponding literature on liquid-solid fluidized beds involving the use of non-Newtonian media (primarily power law fluids) is less extensive, as pointed out by Tonini [252]. Table VI provides a brief resumt of the studies on non-Newtonian liquid-solid fluidised bed systems available in the open literature. Evidently, very few investigators have dealt with the measurement and/or prediction of the minimum fluidisation velocity, and a listing of the available correlations for power law fluids is provided in Table VII. It will suffice to add here that,

TABLE VI SUMMARYOF STUDIESON TWO-PHASE FLUIDISATION WITH NON-NEWONIAN MEDIA Investigator

Type of work

Yu et al. [265]

Experimental

Details" D = 100

d = 2.5; 4.8; 9 ps = 1050-2450

Aqueous solutions of polyox Wen and Fan [259]

Experimental

D=50 d = 0.12-1.43 p8 = 1520-11,300

Aqueous solutions of CMC D = 70 d = 4.3, 6 pI = 1200,2500 PVA in water and greasekerosene mixtures D = 50 d = 1.1-3.1 ps = 2500-7790 Titanium dioxide slurries

Mishra and co-workers [186,236]

Experimental

Brea et al. [23]

Experimental

Tonini et al. [253]

Experimental

D=50

Experimental

d = 1.8 p s = 2940 Aqueous solutions of CMC D = 56.80, 126 d = 0.52-3.05 ps = 1300-2500 One CMC solution (n = 0.85)

!2 Q\

Kumar and Upadhyay [153, 1541

Main results Ad hoc modifications of the Richardson-Zaki correlation for predicting V, and V-E behaviour. Reasonable agreement between experiments and predictions in the range 0.81 5 n 5 1. Axial dispersion coe5cients almost identical to Newtonian values (0.86 I n I 1).

Significant non-Newtonian effects are neither expected nor observed.

Preliminary results on minimum fluidisation velocity and bed expansion characteristics. An ad hoc modification of the correlation of Richardson and Zaki is presented. Mainly concerned with the mass transfer aspects in electrochemical reactions. Only two moderately nonNewtonian test fluids were used. No results on V,: limited results on V-E behaviour and mass transfer data.

Kawase and Ulbrecht [I311

Theoretical

Briend et al. [24]

Experimental

Machac et al. [I661

Experimental

Lali et al. [I571

Experimental

Srinivas and Chhabra [243]

Experimental

4

Jaiswal er al. [1 1 11

Sharma and Chhabra [233]

D = 102 d = 0.23-1.86 ps = 2480-11,350 Aqueous solutions of Carbopol and Separan D = 20,40 d 1.46-3.91 ps = 2500-16,600 Aqueous solutions of Natrosol, CMC, and Separan D = 76 d = 1.65, 3.1 ps = 2500 Aqueous solutions of CMC D = 50, 100 d = 1.27, 2.6, 3.57. pE= 2500 Aqueous solutions of CMC

Theoretical

-

Experimental

D = 50, 100 Nonspherical gravel chips fluidized with aqueous CMC solutions

D and d are in mm, p. in kg/m3; CMC is carboxymethyl cellulose.

Based on the free-surface and zero-vorticity cell models, approximate closed-form expressions for V,, and Vmf-& are developed and agreement with the literature data was stated to be moderately good, especially for weakly shear thinning fluids. Preliminary results on V, and V-E behaviour which are in line with a non-Newtonian form of the BlakeKozeny equation.

Correlation for V,, and V-E behaviour for power law and Carreau model fluids.

Bed expansion behaviour is in accordance with Newtonian formulas.

Extensive results on minimum fluidisation velocity and bed expansion characteristics. Detailed comparisons with existing correlations. Numerical solution of the field equations in conjunction with the free-surface cell model. Good agreement with the literature data for V-E behaviour and V,, . Extensive data on V,, and V-E behaviour.

TABLE VII

EXPRESSIONS AVAILABLE FOR PREDICITNG Investigator

Yu et

v,

IN POWER

Expression"

af. [265]

my =

Remarks

= (aYGad)(*-")/"

Re,

LAWLIQUIDS

For creeping flow only.

G

12.5[(9n

+ 3)/n)(l - E,)]"

Mishra et al. [186Ib

No upper limit on the value of Re, was stated.

Brea et al. [23]

+ 1.75Re22-"'

&Gad

= (160/aB)Re~z-")

00

(

a, = 3n ~

Kumar and Upadhyay [I531 Kawase and Ulbrecht [131] Machac et al. [166] Jaiswal et al. [1 1 11

+

4n

(

>'

(1

~

8%

12(1 - E )

Same as Brea et al., exoept substitute 150 for 160. where Y =f(n,

E)

No upper limit on value of Re,.

is available in

the original publication (i) V,, = 0.019q (ii) V, = 0.015[1 0.73(d/D)]V; Same as Kawase and Ulbrecht [131]

+

Gad = (3/4)C,,

*

No upper l i t on the value of Re,,.

1-1

It has subsequently been corrected by Kumar and Upadhyay [153].

Creeping flow only. Error 14.6% Error 10.3% The drag correction factor Y(n, E ) is available up to Re, = 20.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

239

except for the works of Kawase and Ulbrecht [131], Machac et al. [166], and Jaiswal et al. [l 111, all other expressions have been obtained by combining Eq. (49) with Eq. (23). Whereas both Kawase and Ulbrecht [131] and Jaiswal et al. [I 1 11have employed the free-surfacecell model to evaluate the pressure drop through fixed beds, Machac et al. [1661 have used purely dimensional considerations to develop their correlation. Further inspection of Table VII shows that most of the experimental data pertain to the low-Reynoldsnumber regime and to beds of spherical particles, except the limited results for nonspherical particles [151a, 233, 2651. In a recent study [38a], the performance of the expressions listed in Table VII has been evaluated by carrying out extensive comparisons between the predicted and experimental values of the minimum fluidisation velocity. Table VIII summarises the ranges of variables encompassed by the data bank (with 47 data points), and Table IX gives a comparative summary of the results. Admittedly, none of these methods seems to work particularly well, but in assessing the results shown in Table IX it should be borne in mind that errors of the order of 50-100% are not uncommon in experimental determination of Vmr.Moreover, part of the discrepancy can also be attributed to the uncertainty associated with the evaluation of emt. In view of these factors, the methods due to Machac et al. [166] and Jaiswal et al. [ l l t ] entail overall minimum deviations; Fig. 16 contrasts the experimental and predicted values [1111 of V,, for individual data points. However, firm conclusions regarding the reliability of a particular method must await more experimental work. Preliminary comparisons for nonspherical particles [233] suggest that one can use the expressions developed for spherical particles provided the sphere diameter is replaced by a volume equivalent diameter multiplied by the sphericity factor. The resulting deviations, however, are somewhat larger than those for spherical particles. Finally, it is appropriate to mention here that analogous expressions for other fluid models, namely Bingham plastic,

TABLE VIII EXPERIMENTAL DATAON Investigator

Yu et ~ l [265] . Briend et al. [24] Kumar [151a]" Machac et ~ i .[166] Srinivas and Chhabra [242] Sharma and Chhabra [233]" LI

v,,

Range of conditions covered

No. of data points

n = 0.81 and 0.936, Re,, < 0.034 n = 0.64, 0.75, 0.76, and 0.765; Re,, 5 0.17 n = 0.85, Re,, I 15 0.34 5 n 5 0.96 Re,, I0.0395 0.34 s n < 1, RemfI7 0.78 s n S 0.91, Re,, I0.037

4 4 7 8 19 5

Pertains to or contains some results for nonspherical particles.

R. P.CHHABRA

240

TABLE IX RESULTS

SUMMARY OF

% Error" Method

Average

Maximum

44 87 32

146 96 174 146 105

28 37 24

76 111 99

Yu et al. [265] Mishra et al. [186] Brea et al. [23] Kumar and Upadhyay [153] Kawase and Ulbrecht El311 Machac et al. [166] Method I Method I1 Jaiswal et al. [ll 11

55

44

%-Error = lOO(experimenta1 - calculated)/calculated.

1

W

I?

r

I

l

l

I

16~

l

l

16~ V,,

I

l

l

I

16*

( EXPERIMENTAL)

I lo"

I

1

10O

mls

FIG. 16. Comparison between predictions [111] and experimental values of minimum fluidisation velocity. Replotted from ref. [38aJ.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

24 1

Carreau viscosity equation, and Ellis fluid, are available in the literature [131, 2363. 2. Bed Expansion Behaviour As mentioned earlier, once the superficial velocity of the liquid exceeds the minimum fluidisation velocity, the mean voidage of the bed gradually increases while the pressure drop across the bed remains constant at a value equal to its buoyant weight. It has been tacitly assumed that this kind of behaviour occurs and data are seldom reported to confirm it. Figure 17 shows typical pressure drop-superficial velocity data obtained for with beds of 3.57-mm glass beads fluidised by non-Newtonian polymer solutions. An anomalous effect has been reported by Machac et al. [166], who found that, depending on the value of the power law index (especially small values), the pressure drop across the bed may drop to a value lower than the buoyant weight of the bed, as shown in Fig. 18. This behaviour seems to suggest that there is a maximum achievable bed voidage, for a given liquid-solid system. The value of cmaxwas found to decrease with decreasing value of the flow behaviour index. This phenomenon has not been reported by any other investigator, nor are the possible reasons for this anomaly immediately obvious. No further reference will be made to this effect hereafter. By analogy with the behaviour observed with Newtonian liquid media, it is customary to depict the bed expansion behaviour of fluidised beds by

D = 101.6 m m

d=3.57mm

n = 0.845

:

1000 I

0

SUPERFICIAL

I

I

I

2 00

400

VELOCITY

,

v x 105(m/s>

FIG. 17. Experimental Ap-Vplots for fluidised beds. Replotted from ref. [243].

600

R. P. CHHABRA

242 1.2

a, 0

-

m

-

w

1.0

I

-

.

41 "I

I

0.6-

I

\

a

4 0.4

I I

-

%If 1

I

..U*U."YU

r

41

0.8-

I

v J

aa

"VUU,

"I.-

1"

UUUU"J

..I

I

& V I V U C Y U 111 C I I V V A L V I A ,

I

v.

"VU

VAyULIuIvII L U C l l V l

than the value of pressure drop, and often such data have been represented and correlated using dimensionless expressions of the following form :

For Newtonian fluidising media, the index Z has been found to be a function of Reta and (d/D) or only Re,. The literature abounds with empirical expressions for predicting the value of 2 in a new application. A thorough and critical review of these correlations has been presented by Khan and Richardson [139a]. In particular, two methods for the estimation of Z have gained wide acceptance. Based on a large body of experimental data, Richardson and Zaki [221a] presented a set of correlations, which in their

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS -

n 10,603

A D =101.6mm

d

=3.57mm

]

I

REE(242)

1

n =0.603

I

d

I---

MACHAC et.artl66) KAWASE AND ULBRECHT(~~~

1

I=

0 t-

8

I

-

I

a

I

>

I

&l

REE(242)

=357rnm

I

4

c

1

I A D =50.8mm

I -K-

>

243

I I I

0.1 -

-I

W

>

I

-

I I I I I

-

I

I I I

0.01 0.1

BED VOIDAGE,€

0.1 10

I

I

BED VOIDAGE,€

FIG. 19. Typical bed expansion data together with predictions. Replotted from ref. [243].

modified form are reproduced here: Z = 4.65 + 20(d/D) = 4.40 f = 4.40

= 2.40

18(d/D)Re;o.03

+ l8(d/D)ReL0.'

Re,m< 0.2

(544

0.2 _< Re,- I1

(54b)

1 I Re,_ _< 200

(544

Retrn2 200

(544

The second method is due to Garside and Al-Dibouni [83] and correlates (VK) instead of (VK-) as follows:

R. P. CHHABRA

244

where Z is now found to be dependent only on Re, via the following single expression: Z=

5.09 + 0.2839Rep.877 1 + 0.104Rep.877

Re, 5 lo4

One can, however, rewrite Eq. ( 5 5 ) in terms of (VVJ by introducing a function of (d/D), as proposed by Garside and Al-Dibouni [83]: V -

K

{+

= 1

2.35 %}-'E~

(57)

In essence, the new factor in front of ez on the right-hand side of Eq. (57) corrects the single-sphere velocity for wall effects. It is worth reiterating here that the performance of Eqs. (53) and ( 5 5 ) must be compared in terms of the resulting values of ( VKm)rather than the values of 2. Extensive comparisons have shown that the predictions of these two methods seldom deviate from each other by more than 10-15% [IIO]. Owing to the qualitatively similar nature of the bed expansion characteristics for Newtonian and non-Newtonian systems, it is worthwhile first to explore the possibility of extending the aforementioned two formulas to power law liquids. In this instance, intuitively, one would expect the index Z to show a possible additional dependence on non-Newtonian fluid model parameters. For power law liquids, the relevant definition of Re, becomes p V:-"d" Re; = ____ m

and one can now postulate: Z = Z(Re;, n, d/D)

(59)

To delineate any possible dependence of the fluidisation index Z on n, experimental values of 2 culled from different sources are compared with those calculated using Eqs. (53) and ( 5 5 ) in Table X and the resulting discrepancies, if any, can be unambiguously ascribed to the nowNewtonian behaviour of liquids, namely n. Examination of Table X clearly shows that for large values of ( d / D ) (the first three entries), the experimental values of Z are closer to the predictions of Garside and Al-Dibouni [83], whereas in the remaining cases, i.e., small values of (d/D), there is good correspondence with the predictions of Richardson and Zaki [221a]. Nor is there any discernible trend with respect to the power law index. Attention is drawn to the fact that exceptionally large values of Z have been reported in the literature when the fluidising medium exhibits viscoelastic effects [24, 2421 (e.g., see the entries denoted by +), for which neither of the aforementioned methods appears to

TRANSFER I N NON-NEWTONIAN MULTIPHASE FLUIDS

245

TABLE X VALUESOF 2 FOR FLUIDISATION WITH POWER LAWLIQUIDS Values of 2 -

Exp.

Eq. (53)

Eq. (55)

Eq. (60)”

Data source

~

0.782 0.74 0.786 0.845 0.835 0.900 0.941 0.382 0.382 0.603 0.603 0.603 0.603

0.156 0.156 0.156 0.0351 0.0703 0.0351 0.0703 0.0351 0.0703 0.0351 0.0703 0.0256 0.05 12

3.66 1.42 7.13 0.33 0.55 1.03 1.68 0.58 0.58 5.75 5.75 2.06 2.06

4.81 6.35 4.09 4.93 5.97 4.73 4.93 8.30+ 8.77 + 522 5.75 6.05 5.74

6.33 6.95 5.92 5.20 5.76 5.03 5.37 5.1 1 5.75 4.22 4.75 4.52 4.95

4.52 4.81 4.21 5.01 4.97 4.22 4.80 4.96 4.97 4.37 4.42 4.73 4.73

8.04 8.11 8.03 5.54 6.25 5.46 6.12 6.97 7.68 6.01 6.71 5.82 6.33

242

0.68 0.88

0.0523 0.0523

0.33 0.33

6.10 5.70

5.52 5.52

5.02 5.02

6.17 5.83

253

0.89 0.89 0.82 0.82 0.696 0.74

0.0408 0.02 17 0.0408 0.0217 0.0408 0.0217

22.8 5.0 6.14

3.76 4.08 4.28 4.58 4.94 4.86

3.68 4.41 4.34 4.79 4.81 4.95

5.59 5.21 5.69 5.31 5.90

157

1.48 0.61

3.58 3.97 3.95 5.02 4.93 4.80

0.75 0.76 0.77 0.64

0.0023 0.0023 0.0125 0.0125

0.0 134 0.029 16.08 19.60

4.90 6.60 5.10 5.70

4.70 4.70 3.49 3.43

5.09

0.82 0.78 0.83 0.91 0.89

0.123 0.061 0.06 1 0.123 0.061

0.489 0.844 1.571 3.03 1 4.26

6.30 5.78 5.57 4.91 4.92

5.63 5.53 5.57 4.93 4.76

1.58

-

5.44

24‘

3.83 3.73

5.03 5.03 5.20 5.46

4.97 4.90 5.26 4.58 4.45

7.33 6.15 6.07 7.15 5.99

233*

5.09

Strictly applicable for Re;_ 5 0.2. Nonspherical particles. has been estimated using the numerical results of Gu and Tanner [89a]. The value of

vm

be applicable. Based on the comparison shown in Table X, it is thus safe to conclude that either of the two methods mentioned above may be used to estimate the value of 2, at least for viscoinelastic power law fluids. The limited results for nonspherical particles also conform to this behaviour, as can be seen from the last five entries in Table X. In spite of such close

246

R. P. CHHABRA

agreement with the Newtonian predictions, both Brea et al. [23] and Machac et al. [166] developed new correlations based on their limited experimental data. Brea et al. [23] simply modified Eq. (54a) as follows: d 1-n 2 = 4.65 + 20- Re;, < 0.2

D

+

n

Note that this equation predicts increasing deviation from the Newtonian value of Z with decreasing value of the power law index. Although most of the data presented in Table X pertain to Re;, > 0.2, the predictions of Eq. (60) are included in this table for the purpose of qualitative comparison. It is interesting to see a fair match between the predictions and measurements in most cases, even for viscoelastic fluids. This agreement is believed to be fortuitous, however. Similarly, in the low-Reynolds-number regime (Re;, < 0.3) Machac et al. [1661 presented the following empirical expression for bed expansion: 0.218-0.404(d/D)

(-

V ~ ) 0 . 8 0 Z - 1 . 3 5 5(d/D)

- 0.862(1 - n)

(61)

The fact that Eq. (61) is based on experimental data showing anomalous pressure drop-velocity behaviour (referred to earlier), coupled with the observation that it does not reduce to the expected limiting behaviour for n = 1, questions its suitability. Nonetheless, the predictions of Eq. (61) as well as those of Kawase and Ulbrecht [131] are included in Fig. 19 for the sake of qualitative comparisons. Analogous developments for Carreau model fluids are also available in the literature [131]. Numerical predictions [ l l l ] of the flow of power law fluids through particle assemblages simulated using the free-surface cell model can also be used to obtain theoretical estimates of the bed expansion behaviour. From a knowledge of the drag coefficient of multiparticle systems as a function of voidage and the power law index, one can prepare so-called fluidisation charts by introducing the following normalized velocity (V') and diameter (d'):

Figures 20 to 22 show the results plotted in the form of V + versus d + for n = 1,0.8, and 0.6. For a given liquid-solid system (i.e., known m, n, p, Ap, d) the value of d + is known and the corresponding Re',, can be estimated using existing methods [38b, 89a], thereby fixing a point on the E = 1 curve. One can now generate (vY,)-&curves simply by drawing a line parallel to the

247

10

100

FIG. 20. V + - d + plot for n = 1. Replotted from ref. [lll].

y-axis and passing through (d', Re;-) on the E = 1 line. In the absence of wall effects, typical comparisons of predictions and experiments are shown in Figs. 23 and 24 for Newtonian and a power law liquid, respectively. In both cases, the agreement is seen to be about as good as can be expected in this kind of work. Extensive comparisons between predictions and experiments for Newtonian and power law media are available elsewhere [ l l O , 1111. Thus,

R. P.CHHABRA

I

I

I

I

10'

2-n C$"

2 Rep2+n

FIG. 21. Vt-df plot for n = 0.8. Replotted from ref. [lll].

I

249

i n =0.6

W

I

I

10-3 loo

10’

2-n 2+n CD

-

2 2+n

7

FIG. 22. Vt-dc plot for n = 0.6. Replotted from ref. [111].

lo2

R. P. CHHABRA

250

8

>* > \

0.3

0.5

0.7

0.9

1.0

BED VOIDAGE, € FIG.23. Comparison between theoretical predictions [l 1 11 and experimental results for n = 1. Replotted from ref. [llO].

this approach provides a theoretical vantage point for predicting the bed expansion characteristics for power law fluids. Aside from the aforementioned studies on fluidisation with power law media, Wen and Fan [259] have reported limited measurements on axial

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

25 1

'/

- THEORY (111)

, 0 DATA

0.001

0.2

0.4

0.6

(242)

0.8

1.0

BED VOIDAGE, & FIG. 24. Comparison between predictions [l 1 11 and experimental results for n = 0.6. Replotted from ref. [lll].

dispersion in beds of spherical particles fluidised by non-Newtonian media. Their results clearly show that the dispersion is virtually unaffected by nonNewtonian characteristics of the fluidising medium, at least for weakly shear thinning fluid behaviour. 3. Heat and Mass Transfer

As far as known to us, there has been no investigation of wall-to-bed or interphase heat transfer in fluidised systems involving non-Newtonian fluids.

252

R. P. CHHABRA

However, scant experimental results are available on the interphase [153, 1541 as well as the wall-to-bed mass transfer [253] in particulate beds fluidised by power law media. For instance, based on the assumption that the sole effect of the presence of neighbouring particles is to alter the flow field around a particle, as seen in Eq. (53), Kawase and Ulbrecht El311 developed the following semiempirical expression for interphase (particle-to-liquid) mass transfer in fluidised systems involving power law fluids:

Figure 25 shows a comparison between the predictions of Eq. (63) and the limited experimental results of Kumar and Upadhyay [154] for n = 0.85. Also included in the figure are the predictions of Eq. (45). The correspondence between predictions and experimental results is seen to be satisfactory in the limited range of conditions encompassed by the data. The only other study of mass transfer in non-Newtonian fluidised systems is due to Tonini et al. [253]. They measured the wall-to-bed mass transfer using an electrochemical technique. The particles were contained in the annular space in between the outer and inner walls. In this manner, mass transfer coefficients were measured for 10 different fluids (I 2 n 2 0.68) and in the bed voidage range 0.45 5 E I 0.90. Based on the capillary bundle

I

W

0.1

0.5

1

5

10

50 100

REYNOLDS NUMBER , Re1 FIG. 25. Predictions versus experimental results on liquid-particle mass transfer in fluidised beds.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

253

approach, the results were correlated via the following empirical correlation:

- .5)0.5* Sh = 1.45Rey.42Sc$0.33(1

(64) The average deviation between experiments and predictions was found to be about 10% in the following ranges of conditions: 0.0049 I Re, I190;

1800 ISc? 537,000,

5.8 I Sh, 5 72.

B. THREE-PHASE FLUIDISED BEDS A bed of solid particles fluidised by cocurrent upward liquid and gas flows gives rise to a three-phase fluidised bed (TPFB). In spite of the fact that the advantages of TPFBs over packed beds as well as slurry bubble columns have long been recognized [lo, 31, 32, 771, such systems have begun to receive systematic attention only during the past two decades or so. There is no question that the hydrodynamics and the nature of heat and mass transfer processes with and without a chemical reaction are much more intricate and complex in three-phase systems than in two-phase fluidised systems (discussed in the previous section). Excellent review articles thoroughly discussing the fundamental aspects, the existing body of information, and the aspects meriting further attention have been written by Ostergaard [201] and by Epstein [72]. However, a cursory inspection of both of these survey articles shows no mention of the instances wherein the liquid phase exhibited nonNewtonian behaviour such as that encountered in numerous biotechnological applications [lo]. This is primarily due to the fact that TPFBs involving non-Newtonian liquid media have been studied only during the last 10 years or so. Consequently, the available information is not only scant but also of a preliminary nature. We begin by providing a brief summary of the results on TPFBs available in the literature in Table XI. Inspection of this table shows that there have been no systematic developments in this field and, to date, attempts have been made to elucidate the role of non-Newtonian characteristics of the liquid phase in the following aspects of TPFBs: bed expansion, gas and liquid holdups, mixing and dispersion characteristics, gas-liquid and liquid-solid mass transfer processes, and wall-to-bed heat transfer. Each of these will be dealt with briefly in the ensuing sections.

1 . Bed Expansion The effect of system and process variables on bed expansion has been studied by Kato et al. [126], Patwari et al. 12141, and Zaidi et al. [266, 2673. To describe the bed expansion behaviour of a TPFB at least two independent holdup measurements are necessary. Patwari et al. [214] measured the extent

TABLE XI SUMMARY OF RESULTS AVAILABLE ON TPFBS WlTH NON-NEWTONIAN LIQUID PHASE Range of variables

Investigators

Systems

Kato a d. [12q

Air/CMC solutions/glass beads

D=52and120mm

Remarks Bed expansion and wal-to-bed heat transfer.

d = 0.420.66, 1.2 and 2.2 m m

Kim and Kim [I401

d

1.7, 3 and 6 mm

I

A d dispersion and mixing characteristics.

n = 0.946 and 0.836

-

Nguym-Tienet d.[I951 N

D=152mm d = 1.7, 4, 4 8 mm

Kaeg ef d.c12q

v,

P

Air/CMC solutions/*

bcads

D=140mm d = 3,5,

Schumgc et al. [UI]

AirfCMC and xanthan solutions/glapa txads

and 8 m m

D = 140-

Reanalysis of data from rd. [I401 Heal transfer from a conical surface.

Bcd expansion and gas-liquid mass transfer charactcristia Liquid-phase mas transfer cmtlicknt for 0,.

d = 8 mm; 12 n z 0.8 Air/xanthan solution/*

B u m and BriCos [31, 321

beads

Air/CMC and PAA s o l U t i O ~ / g l s S sbeads

D=100mm d = 3, 5 mm;0.7 2 n 2 0.21 D=17Omm d = S m n = 0.67, 0.87

' CMG. carboxymethylcellulose; PAA,plyacrylamide

Bed expansion and heat transfer characteristics. Particle-lquid interphase masz transfer character is ti^.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

255

of bed expansion for three different sizes of glass beads being fluidised by air and carboxymethyl cellulose solutions. Depending on the particle size and gas and liquid velocities, on introduction of gas into a liquid-solid fluidised bed, one may observe bed expansion or contraction or both over different ranges of operating conditions, as shown in Fig. 26. The phenomenon of bed contraction has often been explained by postulating that the introduction of gas causes part of the liquid flow to be diverted into the solids-deficient bubble wakes, thereby lowering the effective liquid velocity. This behaviour, however, is observed only with small particles, whereas relatively large particles always tend to result in bed expansion by reducing the wake liquid flux brought about by the rupturing bubbles. The limited available results appear to suggest that the critical particle size denoting the transition from ‘contraction’ to ‘expansion’ behaviour is strongly influenced by the liquid viscosity. For instance, in air-water systems, no bed contraction has been observed with particles larger than 3 mm, whereas even particles as large as 8 mm were found to result in bed contraction with a glycerol solution having viscosity of 26 cP. Similar results have been reported by Kim et al. [140] and Zaidi et al. [266,267]. In broad terms, the available bed expansion data seem to conform to the criterion proposed by Epstein [71], which, however, does not predict the initial contraction observed with large particles. 2. Gas Holdup Experimental measurements of gas holdup in TPFBs have been reported by Patwari et al. [214] and Zaidi et al. [266, 267). Figure 27 shows typical d(mm) -1

VL ( m l s ) 0.041

0.066

0

0-05 0.10 0.15 SUPERFICIAL AIR VELOCITY ,VG (mls)

FIG. 26. Typical bed expansion-contraction behaviour for a three-phase fluidised bed. Replotted from ref. [214].

R. P. CHHABRA

256

I

0

I 0.15

I

0.05

0.10

SUPERFICIAL GAS VE LOClTY , VG (rn/s) FIG. 27. Effect of rheological characteristics on gas holdup in three-phase fluidized bed systems. Replotted from ref. [214].

variations of gas holdup with system parameters. Qualitatively, the addition of solid particles lowers the gas holdup compared to that in a gas-liquid system under identical conditions. The larger the particles, the smaller the reduction in gas holdup. However, this effect diminishes with increasing liquid viscosity, as is seen in Fig. 27. Based on a limited amount of data obtained with air and xanthan solutions and 3- and 5-mm glass beads, Zaidi et al. [266, 2671 presented the following correlation for gas holdup: - 0 . 0 3 ~ ~ C & . 2 5 ~ ~ --0.07 0.24~ &G L (65) where the effectiveviscosity (perf)of xanthan gum solutions was estimated at a mean shear rate from the following expression:

(

?jeff = 2800

VG -

12V.5 3n + 1 +4n

2) d 2 i s ( )

V, -

~

Equation (65) was stated to correlate the experimental results with k 10% error, in the following ranges of conditions: 0.02 I V, I0.14 m/s, 0.013 s V' I 0.09 m/s, 3.7 IpeffI 300 mPa-s, D = 100 m, and d = 3, 5 mm.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

257

3. Liquid Hoidup Liquid holdup measurements in TPFBs involving carboxymethyl cellulose solutions have been reported only by Kato et al. [126]. With addition of gas to a liquid-solid fluidized bed, the liquid holdup was found to increase with increasing liquid velocity and viscosity and with decreasing gas velocity and particle size. Kato et al. [126] extended the correlation of Garside and AlDibouni [83], Eqs. (55) and (56), to represent their results as follows:

+ ($)

Z

=

EL* = 1-9.7 (350 + Re:.')-0.5

with

(67b)

In turn, the index 2 was correlated as follows: 5.1(1 + 16.9 Z - 2.7 where

-2

= 0.1(1

+ 4.43 K0.165)ReP;9

K = pLVi/gc.

(67c) (674

It is, however, not clear how the effective viscosities of the non-Newtonian carboxymethyl cellulose solutions were estimated in this study. 4. Axial Dispersion Axial dispersion and mixing characteristics in three-phase fluidised beds using air, carboxymethyl cellulose solutions, and glass beads have been investigated by Kim and Kim [140]. The values of Peclet number were found to increase with gas velocity in both gas-liquid and gas-liquid-solid systems, whereas the intensity of mixing was found to decrease with the addition of solid particles. The extent of axial dispersion increased with liquid velocity, at least with particles smaller than 3 mm, whereas the Peclet number showed a shallow maximum at E, x 0.35. Furthermore, the surface tension did not seem to exert any significant influence on either the intensity of mixing or the axial dispersion, but both these characteristics were strongly affected by the liquid viscosity. For instance, at low gas velocities, axial mixing improved with increasing liquid viscosity in TPFBs with 1.7-, 3-, and 6-mm glass beads, whereas at high gas velocities axial mixing deteriorated with increasing liquid viscosity in the presence of small particles (e.g., 1.7 and 3 mm). Kim and Kim [1401 developed the following comprehensive correlation for Peclet number of the liquid phase:

R. P. CHHABRA

258

Equation (68) encompasses the following ranges of variables: 0.0012 I Pe, s 0.62, 0.0022 5 ( d / D ) 5 0.08, and 0.148 I (VJV, + V,) I0.962. Subsequently, Nguyen-Tien et al. [195] suggested that Eq. (68) correlates the literature data much better if the numerical constant of 20.19 is replaced by 61. 5. Mass Transfer

It is widely acknowledged that the addition of solid particles to a gas-liquid system leads to higher values of the volumetric mass transfer and heat transfer coefficients (see Fig. 28 and 29). This enhancement is attributed solely to the increased level of turbulence brought about by rupturing of bubbles due to the presence of particles. However, the gas-liquid mass transfer not only is strongly dependent on the particle size but also goes through a maximum at a critical particle size of about 3 mm. Schumpe et al. [231] have proposed the following correlation for predict-

z w

30

-

2

LL

W

s

a

20-

w LL m

'"No

z

a

a I-

Part ides d

in

ln

ln a

I

0

0.05

0.10

0.15

SUPERFICIAL AIR VELOCITY, VG ( m k ) FIG. 28. Effect of particle size on mass transfer coefficient in three-phase fluidised bed systems. Replotted from ref. [214].

r-----l

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

3

259

- - - -NO PARTICLES n

m=0.097Pas n = 0 . 9

ln

ln

-

a

n ~0.8 I

I

I

I

I

I

I

ing the gas-liquid mass transfer coefficient in TPFBs: -2988~~44~~.42~,p.34~0.75

JDAB ‘La

1-

(69)

where peffis evaluated at the shear rate given by Eq. (66). Note that Eqs. (66) and (69) are not dimensionless and all quantities must be used in SI units. The correlation covers the following ranges of variables: 0.017 I V, I 0.1 18 m/s, IpeffI0.12 Pa-s,

0.03 I V, 50.16 m/s,

0.08 I Vm I0.6 m/s

In another study, Burru and Briens [31,32] measured particle-to-liquid mass transfer in TPFBs involving CMC and polyacrylamide solutions. These investigators found the mass transfer coefficient to decrease with the addition of polymer to water. This can perhaps be attributed to the higher viscosity of polymer solutions. However, no predictive expression was developed in this study.

R. P. CHHABRA

260

L

0.4

0.5

0.6

0.7

BED VOIDAGE, € FIG. 30. Effectof gas velocity on heat transfer coefficient in three-phase fluidised beds with d = 3 mm. Replotted from ref. [267].

d = 3 mm

i

2 VLx10 m/

I

,3.79 7 3.2

L

] 2-2 t-

a

Polymer Conc.

111 0

0.05

0.10

GAS VELOCITY

,

0.15 V~(mls)

FIG.31. Effect of particle size on heat transfer coefficient in three-phase fluidised bed systems. Replotted from ref. [267].

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

26 1

6. Heat Transfer

A small amount of information is available on the wall-to-bed heat transfer characteristics in three-phase fluidised beds [124, 126, 266, 2671. In general terms, the heat transfer coefficient increases with increasing fluid velocities and particle size, while it shows an inverse dependence on the liquid viscosity. Furthermore, the heat transfer coefficient goes through a maximum with respect to both liquid velocity and bed voidage at about -0.5 to 0.6, as shown in Figs. 30 and 31. Based on their experimental results, Zaidi et al. [266, 2671 developed the following expression: Nu, = 0.042 ReE.72PrE.86FrE067 Equation (70) was stated to be applicable in the following ranges of conditions: 0.0081 IV, I 0.144 m/s, 0.0127 IV, I 0.09 m/s, 3.7 I perfI 300 mPa-s, and d = 3, 5 mm. The effective viscosity is evaluated at shear rate given by Eq. (66). Similar expressions have been developed by Kang et al. [124] and Kato et al. [126] but are not included here simply because all of these are believed to be too tentative and limited in their applicability.

V. Sedimentation of Concentrated Suspensions Hindered settling of nonflocculated uniform suspensions of noninteracting monosize spherical particles represents an idealisation of several industrially important processes encountered in slurry handling, waste disposal, etc. The variable of central interest in all such applications is the rate of sedimentation in the initial constant-concentration zone. This information is needed when designing the equipment for handling suspensions and slurries, such as thickeners and settling ponds. It is readily acknowledged that the hindered settling velocity of a concentrated suspension of spheres is influenced by a large number of variables: liquid properties (p, p), particle characteristics (d, ps,shape), sphere/tube diameter ratio, and concentration (volume fraction) of solids. Further complications arise in the case of fine particles due to van der Waals and other surface forces; these systems are not considered here but have been discussed by other investigators; e.g., see Russell [225]. Based on purely dimensional considerations, Richardson and Zaki [221a] were probably the first to recognise that, in spite of the obvious differences in the detailed flow fields, a great deal of similarity existed at a macroscopic level between the settling characteristics of a suspension and the expansion behaviour of a particulately fluidised bed of uniform spheres. Moreover, they asserted that their formulas, initially developed for fluidisation, Eqs. (53) and (54), were also successful in correlating the hindered settling data. It is thus

262

R. P. CHHABRA

customary to use the same expressions for fluidisation and sedimentation [83, 139a, 221al. Very little experimental and analytical information is available on the sedimentation of clusters of particles in quiescent non-Newtonian media. CaswellC35a) investigated the settling behaviour of two spheres in viscoelastic media. Depending on the choice of rheological fluid model parameters and the initial separation, the two spheres may converge or diverge; subsequently, these predictions were confirmed experimentally [22 1b]. Kawase and Ulbrecht [1291 considered the sedimentation of concentrated suspensions in power law media using the free-surface cell model. Unfortunately, their final deviates from the literature values by almost 100% in expression for the limit of n = 1 and thus their predictions for power law media must be treated with reserve. Four independent experimental studies on hindered settling in power law media have been reported in the literature [5, 6, 11,411. Balakrishna et al. [l 11 presented preliminary results on the settling behaviour of fine sand particles and concluded that no non-Newtonian effects were present in their study. In spite of this assertion, a cumbersome expression for (V/V,_) was proposed.* Allen and Uhlherr [ 5 , 61 have studied the sedimentation behaviour of glass beads of different sizes in aqueous polyacrylamide solutions. They reported the existence of pockets of heterogeneities and aggregation, akin to that observed in gas-solid fluidisation. Besides, when their results are plotted in the form of (wV,_) versus E on double logarithmic coordinates, two distinct power law regions are present. Although the exact reasons for these anomalous phenomena are not immediately obvious, Allen and Uhlherr [5, 61 have tentatively attributed them to the viscoelastic behaviour of their test fluids. More recently [41], new experimental data on settling of glass spheres in viscoinelastic power law media have been reported. Figure 32 displays sample results on the sedimentation of glass beads in two different polymer solutions. Owing to the similar dependence of (VKJ on E in fluidisation and sedimentation, as seen in Figs. 19 and 32, Chhabra et al. [41] analyzed the hindered settling velocity data using Eqs. (54) and (55); a comparison between the predicted and observed values of Z is shown in Table XII. Once again, the correspondence is seen to be fair in the range of conditions shown in the table. Based on these limited comparisons, one is tempted to conclude that the reconciliation of fluidisation and sedimentation data in power law and Newtonian media seems possible. However, more work is needed to substantiate or refute this generalization.

(v/vJ

*Buscall et a1 . [32a] studied the sedimentation of dilute suspensions (- 5% by volume) of monodisperselatex particles (1.55 Im)in weakly shear thinning solutions of ethyl hydroxyethyl cellulose. Due to the small size of particles coupled with the relatively low concentration of dispersions, the sedimentation velocity was not influenced by the non-Newtonian properties of the suspending medium.

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS 0.1

d = 3.3 mm

7

d l D = 0.041

I

0.2

d

0.5

263 1.0

= 3.3mm

I

I

d l D = 0.0703

I

n =0.8

0.02' 0.1

I

I

0.2

0.5

I

I

n = 0.93

I

I

1.0

FIG. 32. Typical hindered settling velocity data in power law media. Replotted from ref. [41].

VI. Concluding Summary In this chapter, the voluminous literature available on momentum, heat, and mass transfer processes in multiphase particulate systems involving nonNewtonian liquid media has been critically and thoroughly reviewed. The flow of time-independent fluids in unconsolidated packed beds and porous media has been studied most extensively. Consequently, it is now possible to estimate the frictional pressure drop incurred in the flow of purely viscous fluids through packed beds of spherical and nonspherical particles under most conditions of practical interest provided no anomalous effects are present. Unfortunately, there is no way of knowing a priori whether or not

R. P.CHHABRA

264 VALUESOF Z

FOR

TABLE XI1 SEDIMENTATION IN POWERLAWMEDIA[41] Values of Z )

1

0.8

0.925

0.797

0.95

a

0.0167 0.0083 0.0323 0.0294 0.0413 0.0164 0.0703 0.0500 0.0272 0.0413 0.0294 0.0164 0.0703 0.0500 0.0272 0.0294 0.0413 0.0164 0.0703 0.0500 0.0272 0.0294 0.0413 0.0164 0.0703 0.0500 0.0272

4.11 x 10-3 7.38 x 10-4 0.0384 0.606 2.000 0.072 2.000 0.606 0.072 0.484 0.165 0.0243 0.484 0.165 0.0243 0.0388 0.127 4.64 x 10-3 0.127 0.0388 4.64 x 10-3 0.233 0.671 0.0354 0.671 0.233 0.0354

Exp.

Eq. (53)

Eq. ( 5 5 )

4.95 f 0.11" 4.80 f 0.08 5.18 f 0.12 5.20f 0.09 4.75 f 0.16 5.13 f 0.10 5.37 f 0.18 5.26 f 0.09 5.34 f 0.11 5.37 f 0.10 5.28 f 0.07 5.10 f 0.06 6.0f 0.21 5.68 f 0.11 5.30f 0.08 5.11 f 0.12 5.28 f 0.15 5.00 f 0.10 5.51 f 0.21 5.35 f 0.18 5.20 f 0.11 5.08 f 0.15 5.16 f0.13 5.00 0.10 5.40f 0.25 5.40& 0.19 5.20f 0.17

5.00 4.81 5.31 5.00 4.80 4.97 5.29 5.38 5.20 5.26 5.24 4.97 5.80 5.65 5.20 5.24 5.48 4.98 606 5.65 5.19 5.15 5.21 4.98 5.73 5.54 5.19

5.09 5.09 5.08 4.95 4.73 5.07 4.71 4.94 5.07 4.97

*

5.04

5.12 4.98 5.04 5.12 5.08 5.05 5.09 5.06 5.08 5.09 5.03 4.94 5.08 4.95 5.03 5.08

95% confidence bands.

such effects would be encountered in a new application. In contrast, the literature on the flow of viscoelastic fluids is less extensive and is inconclusive at present. There is some evidence that it is not sufficient to correlate or interpret the pressure drop data for viscoelastic media purely in terms of steady shear rheological measurements. The flow of dilute polymer solutions in packed beds shows several unusual features that await the development of suitable theoretical frameworks for rationalization and/or interpretation. Notwithstanding the overwhelming pragmatic significance of wall-polymer interactions in polymer flooding, no unique and universally applicable methodology has yet been developed to integrate them into the design strategies. One impediment to achieving this objective seems to be that the results are strongly system dependent, precluding the possibility of general-

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

265

isations. However, the slip effects and polymer retention phenomena warrant further investigations to develop a better understanding of the underlying processes. Liquid-solid fluidised beds involving non-Newtonian, mostly power law media have received very little attention. Most studies have focussed on hydrodynamic aspects, namely the minimum fluidisation velocity and bed expansion characteristics. Preliminary results seem to indicate that satisfactory methods are available for the prediction of both these parameters, at least in the low-Reynolds-number region. The important and rapidly growing area of three-phase fluidised beds has received even less attention and only preliminary results are available on their macroscopic behaviour, including bed expansion, liquid and gas holdup, mixing, and heat and mass transfer aspects. All available trends and expressions are of a tentative nature and no definitive conclusions can yet be drawn. Finally, the phenomenon of hindered settling in non-Newtonian media remains an unexplored subject. Aside from yielding complementary information on the behaviour of fluidised beds, hindered settling merits further attention in its own right owing to its wide-ranging implications in understanding the rheological behaviour of filled systems and the disposal of waste slurries. From the foregoing description, it is abundantly clear that the subject of non-Newtonian effects in multiphase particulate systems is in its embryonic stage and merits much more attention than it has received in the past. In particular, the following is a (partial) list of the related topics which, in the opinion of author, are worthy of systematic investigations: 1. Mixing, heat, and mass transfer aspects in single-phase non-Newtonian flow through packed beds. 2. Cocurrent flow of a gas and non-Newtonian media in packed beds, as encountered in polymer flooding processes. 3. Detailed kinematics as well as macroscopic behaviour of TPFBs. 4. Hindered settling behaviour of spherical and nonspherical particles. 5. Viscoelastic effects in all these configurations.

Thus, considerable scope exists for both experimentalistsand theoreticians in this rugged and challenging branch of non-Newtonian technology.

Nomenclature a, 6 A A(n, E ) A,@, E ) A,@, E )

geometric parameters, Eq. (27) constant, Eq. (40) constant, Eq. (23) constant, Eq. (44) constant, Eq. (46)

C

CD

c, d d+

polymer concentration drag coefficient specific heat particle diameter normalised diameter

R. P. CHHABRA volume equivalent diameter column or tube diameter Deborah number, Eq. (39) hydraulic diameter binary diffusion coefficient axial dispersion coefficient friction factor (= d ~ d / p L V ~ ) ( ~-? E/ )l) Froude number (= V2/gd) acceleration due to gravity Galileo number, Eq. (51) mass transfer factor =

5sc*2'3

( v

Schmidt number [= ~ ( D A B / ~ ~ ~ - ' / P D A L I ~ Schmidt number [= m'/p(DAB)(12V(1- &)/e2d)"-'], Eq. (64) Schmidt number [= ( m / P D ~ d ( d / v ) ' - " IEq. , (4) modified Schmidt number [= Sc,(3n + 1/4n)" (12( 1 - & ) / E Z ) ] " - 1 Sherwood number (= k,d/D,,) tortuosity factor (= L,/L) superficial liquid velocity or as subscripted slip velocity particle settling velocity normalised velocity, Eq. (62) drag correction factor, Eq. (31) fluidisation index, Eq. (53), etc.

)

permeability thermal conductivity convective mass transfer coefficient volumetric mass transfer coefficient bed height effective length of capillaries power law consistency index apparent power law consistency index Morton number ( = gp&/pLu3) flow behaviour index apparent flow behaviour index primary normal stress difference Nusselt number (= hds,/&s,) pressure Peclet number ( = Vd/D,) Prandtl number (= C , p / k ) polar coordinate particle radius Reynolds number [ p Vd/p(l - E ) ] Reynolds number, Eq. (25) Reynolds number, Eq. (22) Reynolds number, Eq. (24) particle Reynolds number (= pV2-"d"/m) Reynolds number, Eq. (20b) modified particle Reynolds number [= ReP(4n/3n + 1)"(12(1 - &)/&2)'-"] modified Reynolds number C P L ~ ~ / P-Ld( ~l modified Reynolds number, Eq. (34) particle Reynolds number for free fall conditions hydraulic radius

GREEKLETTERS u, B constant, Eq. (35) U Ellis model parameter it, shear rate at wall E porosity in two-phase systems/ holdup in three-phase fluidised beds e fluid characteristic time A loss coefficient (=JRe*) I Carreau model parameter P viscosity PO zero shear viscosity Pm infinite shear viscosity P density U surface tension 7 stress 70 yield stress Ellis model parameter 5, shear stress at wall sphericity factor

*

SUBSCRIPTS eff effective mf corresponding to incipient fluidisation G gas L liquid S solid t terminal fall conditions terminal fall conditions in the tm absence of wall effects

TRANSFER IN NON-NEWTONIAN MULTIPHASE FLUIDS

267

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ADVANCES IN HEAT TRANSFER VOLUME 23

Advances in Heat Flux Measurements

T. E. DILLER Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia

I. Introduction

A. OVERVIEW The decade of the 1950s saw great advances in heat transfer measurement techniques. Optical methods became popular, along with several new heat flux gages that are still in wide use today, as evidenced by the commercial heat flux gage manufacturers. In the last 10 to 20 years a number of new techniques have been developed and applied that have greatly increased the resolution and operating range of heat flux instrumentation. Although the older methods will be discussed as background, the focus of this chapter will be on the newer techniques and novel uses of the older techniques. In addition, an effort will be made to bring together information from a variety of fields that deal with heat transfer but which often haven’t communicated. Three areas of new capability with important applications are timeresolved heat flux measurements, simultaneous measurement of spatially distributed heat flux, and heat flux measurement at high-temperature conditions. Examples of recent advances in these areas will be discussed in detail. Such new capabilities, when applied to real-world problems, make the field of heat transfer exciting. Numerous articles and sections in instrumentation books have been written which review available heat flux measurement techniques. Some cover general operating principles [1-51, while others focus on specific applications or specific methods of measurement. Because of the wealth of knowledge already assembled, these reviews will be used as a starting point and basis for much of the ensuing discussion.

279

Copyright 01993 by Academic Press, Inc. All rights of reproduction in any form reserved.

ISBN 0-12-020023-6

T. E. DILLER Englund and Seasholtz [6] describe the need for sensors that can measure high heat fluxes ( 1 MW/m2) at high surface temperature and under large transverse gradients in the material of gas turbine blades. The standard gages generally do not perform well under these conditions because of the large gradients of temperature. Limitations of current sensors for gas turbine applications were also discussed by Bennethum and Sherwood [7]. Alwang [8], Paulon et al. [9], and Godefroy [lo] also discuss techniques for turbomachinery, particularly for high-temperature operation. An extensive review of methods commonly used for aerothermodynamic testing has been given by Neumann [ll]. Details of the problems encountered during hypersonic testing are elaborated by Neumann et al. [12]. Kidd [13] describes some successful techniques at these high-temperature, high-heatflux conditions. Trimmer discusses methods for use in high-speed flow facilities [141. Methods for application to ceramic components of propulsion systems are reviewed by Atkinson et al. [lS]. A review of standard methods for application to the severe conditions of the National Aerospace Plane found none that were sufficient [16]. On the opposite end of the heat flux range, van der Graaf [17] has reviewed heat flux sensors for application with buildings and insulation measurements. Time-resolved heat flux measurement capability and examples of applications were the focus of Diller and Telionis [18]. Application of heat flux sensors to fluidized beds was reviewed by Saxena et al. [19], with particular emphasis on separating the radiation transfer from the total heat transfer (radiation plus convection). A number of specific gage designs are discussed. Moffat has published a general review of temperature and heat flux measurement [20], a specific review for application to electronic cooling [21], and a review emphasizing spatially distributed heat flux measurements [22]. Hay [23] discusses methods useful for wind tunnel testing of turbomachinery components, emphasizing surface temperature measurements. Much recent work has used transient temperature measurements to calculate surface heat flux in short duration flows. Schultz and Jones [24] wrote an extensive article on how to use surface temperature for heat flux measurements. Scott [25] reviewed the general field of transient temperature measurements for steady heat flux. Application of the transient temperature method to turbomachines was reviewed by Arts and Camci [26]. Extension of the same technique to unsteady flows to also obtain the unsteady heat flux was reviewed by Oldfield [27] and specifically for turbomachinery by Dunn C281. From these reviews it is clear that no one gage or method is good for every application. The limitations of previous heat flux gage performance are highlighted, and the need for better gage characteristics is often expressed. Recent advances have been able to overcome some of these limitations and N

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provide measurements with improved accuracy under conditions previously not possible. It is the purpose of the present review to showcase these advances for the heat transfer community. Liberal use of references has been made to give the original sources for readers to obtain additional information.

B. IMPORTANT ISSUES Heat transfer is a field which is generally recognized to embody a good portion of both art and science. This is particularly true of the measurement of heat transfer. To make good heat transfer measurements it is mandatory to have a good understanding of the physics involved. It is also most important, however, to have good engineering design skills to produce real, working devices. If the measurements are not well designed, they have little chance of accuracy and reliability. There are many possible ways of measuring heat transfer and many potential pitfalls. The accumulated knowledge and established wisdom of the well-seasoned masters, therefore, can be important. At the same time, however, applications of new techniques made possible by advances in other scientific fields can be equally important. Two examples are the advent of microsensors [29] and high-speed digital data acquisition. The impact of both of these areas is evident in the methods and results reviewed here. Because there are many good textbooks covering the basic thermodynamics and physics involved in heat transfer, such background will not be covered here. It is necessary, however, to clarify the terminology used and the corresponding nomenclature. In thermodynamics heat is defined as the movement of thermal energy across a system boundary. The rate of this energy transfer is commonly termed “heat transfer” or “heat transfer rate”. It is given the symbol q, and the standard SI units are watts. Correspondingly, the heat transfer per unit surface area or “heat flux” is given the symbol q” and has the SI units of watts/m2. To convert to the usual English units, 1 Btu/ft2.s = 11.36 kW/m2. Because heat flux is simply equal to the heat transfer divided by the surface area, however, these terms can be used somewhat interchangeably. All three modes of heat transfer are involved. Convection and radiation provide the heat to or from the surface which is to be measured. Conduction provides the means for the heat to go to or from the surface within the material. It is important to remember that the first law of thermodynamics requires that this balance be maintained at the surface, as illustrated in Fig. 1 for heat transfer to the surface. For the control volume indicated

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FIG. 1. Surface energy balance.

The method used for most heat transfer measurements is to measure qcond and use Eq. (1) to infer qconyand/or qrad. One of the goals of a good heat transfer measurement is that it be nonintrusive to the system that is being measured. First, the surface should remain smooth so that the flow over the surface is not altered or disturbed. Second, it should provide minimal disruption of the conduction pattern in the material so that the temperature distribution along the surface is not altered. If a convective boundary layer is formed over the surface, small disruptions of the surface temperature can affect the heat transfer. Radiation measurements tend to be less sensitive to the condition of the surface because this is a thermal exchange between a local point on the surface and the environment. In convection the thermal history of the boundary layer is carried with the fluid and affects all of the downstream heat transfer. The two most common characterizations of the surface thermal boundary condition are constant temperature and constant heat flux. An example of the effect on the heat transfer is illustrated in Fig. 2 for cross-flow over a circular cylinder. The first curve was taken from the constant-heat-flux measurements of Giedt [30] and appears in virtually every undergraduate heat transfer text as the curve for a cylinder in cross-flow. The second curve is typical of constant-surface-temperature conditions [3 11. Although they were both taken at the same Reynolds number (Re = lo5), the difference becomes quite large as the bundary layer develops around the cylinder from the front stagnation point. The same trends were also found by Baughn and Saniei [32] and Papell [33] using the same wind tunnel for both boundary condition cases. Baughn and Saniei [32] also demonstrated a good match with the corresponding boundary layer analysis. Chyu [34] discusses the difficulties in establishing a truly constant condition of either temperature or heat flux at the surface. Most surfaces are actually not constant in temperature or heat flux, but are somewhere in between.

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I.50

Re= 16

1.25

-__ Constant Heat Flux [30] - Constant Temperature [311

1.00

0.75

Nu -

JRe

LXO

0.25

0.00

20

40

60

80

I00

120

140

160

180

Angle from Stagnation Point (deg)

FIG. 2. Effect of surface thermal boundary condition for cross-flow over a cylinder, from [30,311 with permission of ASME.

Consequently, the surface thermal boundary condition is an important part of any convective heat transfer measurement and needs to be specified as part of the measurement. The presence of a heat flux gage necessarily alters the temperature and heat flux distributions of the surface. This tends to cause a number of errors in any resulting measurements, even though the gage may accurately measure the heat flux that it observes. It can best be visualized if a hypothetical gage is placed in or on an otherwise isothermal surface, and the resulting surface temperature distribution is illustrated in Fig. 3. For convective heat flux to the surface, the surface is cold relative to the fluid flow and the gage creates a local hot spot. There are two effects. First, the local heat flux at the gage location is smaller because of the smaller temperature difference between the surface and fluid. This assumes that the local heat transfer coefficient is

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284

XI

x2

X FIG. 3. Typical effect of heat flux gage on the surface temperature.

unaffected by the presence of the temperature disturbance,

The situation is actually considerably more complicated because the presence of the gage causes a three-dimensionaldisturbance of the temperature field in the wall. The case of a gage with a thickness of 6 and thermal conductivity of k , placed on the surface has been analyzed by Wesley [35] with the conclusion that the one-dimensional assumption is valid when

Baba et al. [36] report an equivalent analysis for the case of combined radiation and convection with similar results. The second part of the problem, however, is that the convective heat transfer coefficient is different from the isothermal coefficient, h,. This is a well-documented effect that is usually analyzed as a superposition of unheated starting length solutions. The unheated starting length problem is illustrated in Fig. 4. Reynolds et al. [37] and later Taylor et al. [38] documented this effect for a turbulent boundary layer on a flat plate with no pressure gradient.

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x

285

”

FIG. 4. Convection boundary layer with unheated starting length.

The corresponding laminar case is derived analytically in some undergraduate heat transfer texts [39,40] (5)

To illustrate this effect, the local heat flux was calculated for the temperature distribution shown in Fig. 3 using superposition of the results in Eq. (5) r411

The results are plotted in Fig. 5 for a 10% drop in temperature difference between the fluid and surface where the gage is located. For comparison, only a 10%drop in heat flux over the gage surface is predicted if h = h, as in Eq. (2). Figure 5 shows a considerably larger effect localized over the l-cm length of the model gage when the boundary layer effects are included. Similar results have been calculated for turbulent boundary layers [42,43] and measured by Bachmann et ~ l [44]. . These results give graphic illustration of the problems that can be encountered due to the temperature perturbations caused by heat flux gages. In addition, when large temperature differences are present between the wall and the flow, the boundary layer can be affected by changes in the fluid properties. One important way to characterize heat transfer from surfaces is with the convection heat transfer coefficient. The most common definition for heat flux to the surface is

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1.4

0.0

0

2

4

6

8

10

12

14

16

18

20

x (cm) FIG. 5. Typical heat distribution for surface temperatures shown in Fig. 3.

For high-speed flow the fluid friction effects are included by using the recovery temperature in place of the free-stream temperature.

h = - q'kv (8) T, - Tw For internal flow the fluid temperature can no longer be referenced to outside the boundary layer, so the bulk mean temperature is generally used.

As discussed earlier, the value of the heat flux at any location is a function of the thermal boundary condition imposed. Because constant wall temperature and constant wall heat flux are easiest to specify, the heat transfer coefficient has its greatest utility under these conditions. In fact, when more than one of the quantities on the right-hand side of Eqs. (7)-(9) is not constant, the value of the heat transfer coefficient becomes ambiguous. In such conditions the fluid temperature may be replaced with the adiabatic wall temperature. This has been used with good success for film cooling and jet

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impingement cooling when the reference fluid temperature changes. Moffat [22] suggests also using it when the wall conditions of heat flux or temperature are not constant, such as in electronic cooling components. To clarify the problem he suggestss labeling this heat transfer coefficient accordingly : had

=

421,"" - Tw

(10)

Tad

C. ORGANIZATION To organize the discussion of heat transfer measurement, the many different methods can be placed into three or four general categories [l, 181. One unifying factor is that all of the methods require accurate temperature measurements to be made somewhere as the key to good heat transfer measurements. Heat transfer (flux) measurement categories: 1. A temperature difference is measured over a spatial distance with a known thermal resistance. 2. A temperature difference is measured over time with a known thermal capacitance. 3. A direct measure of the energy input or output is made at steady or quasi-steady conditions. Temperature measurement is required to control or monitor conditions of the system. 4. A temperature gradient is measured in the fluid adjacent to the surface. Properties of the fluid are required.

The discussion of measurement methods in Section I1 is organized by the first three categories (types of gages). The fourth category is not considered further because it is not widely used and has limited application. Recent measurements using the methods discussed are showcased in Sections IV and V to demonstrate the advances made in measurement capability. Calibration is briefly addressed in Section 111.

TI. Measurement Methods A. TYPE 1 METHODS-SPATIAL TEMPERATURE DIFFERENCE All of these gages have the advantage of giving output signals proportional to heat flux either into or out of the surface. Because the gages measure continuously, the heat flux through the gage can be measured as long as the signal is monitored. If thermocouples are used for the required temperature

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measurements, the output voltage is self generated by the gage. When resistance temperature sensors are used, an excitation voltage is required. 1. Layered Gage Conceptually, the simplest of the type 1 methods is the layered or sandwich gage as illustrated in Fig. 6. The temperature is measured on either side of a thermal resistance layer and is proportional to the heat flux in the direction normal to the surface. Gages vary according to the method of temperature measurement, physical size, and materials of construction. A number of books and articles have been written which cover various aspects of temperature measurement [S, 20,45-511. The two most common methods of measurement are resistance temperature devices (RTDs) and thermocouples. Any material that is an electrical conductor and changes resistance with temperature can in theory be used for an RTD. Some of the more common materials are listed in Table I [SZ] with the corresponding fractional change in resistance per centigrade degree. Typically, the value for metals is nearly constant over a wide temperature range, but is small (a fraction of a percent change in resistance per "C). Semiconductors have a much higher change in resistance over a narrow temperature range and become highly nonlinear outside that range. The advantage of an RTD is that it can generally be used to read temperature at 10 times the accuracy of a thermocouple. One of its disadvantages is that it is also sensitive to strain. Metal strain gages typically have a gage factor of 2, which means that the percent change in resistance is twice the strain experienced. Consequently, it requires a strain of 0.1% to give a resistance change approximately equal to a 1.o"C temperature change for many metals. Although this is a relatively large strain, it may still be important to control the strain in any gages that use RTDs for the temperature measurements. The temperature sensitivity of different combinations of materials for thermocouples is indicated by the Seebeck coefficient, S,. Values for common

Temperature Sensors

T-

I

Thermal

- Resistance Adhesive Surface

FIG.6 . Example of a type 1 layered gage, from [76] with permission.

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TABLE I RESISTANCE-TEMPERATURE COEFFICIENTS AT 25°C ("C- ') [52] Resistance-temperature coefficient

Material Nickel Iron (alloy) Tungsten Aluminum Copper Lead Silver Gold Platinum Mercury Manganin Carbon Electrolytes Semiconductor (thermistors)

0.0067 0.002 to 0.006 0.0048 0.0045 0.0043 0.0042 0.004 1 0.004

0.00392 0.00099 rf. o.ooo02 -0.0007 -0.02 to -0.09 -0.068 to f 0 . 1 4

materials relative to platinum at 0°C are listed in Table I1 [52]. Excluding the semiconductor materials, the maximum sensitivity for any of the possible pairs is 50 to 100 pV/T. Kinzie [SO] lists details of many nonstandard thermocouple pairs. Moffat [20] and Benedict [49) give good summaries of the important points on thermocouple theory and practical issues in their use, including common problems that are encountered. A sketch of a typical thermocouple circuit is shown in Fig. 7 using copper and nickel for the thermocouple materials. These were purposely chosen as a nonstandard pair to illustrate that most any materials can be successfully used. It is important to note that the junction of any nonsimilar materials at different temperatures in the circuit can affect the measured voltage. Therefore, the same material should be used as leads to connect to the voltagemeasuring device. In Fig. 7 copper is used. The two copper-nickel junctions are labelled TI and T, and the voltage output is proportional to the temperature difference between these junctions. E = S,(T, - Tz)

(11)

If TI is known by some means, such as an ice point reference, then the absolute value of T2 can be determined from the thermocouple voltage output. To summarize, there are a number of advantages of using either thermocouples or RTDs to make the required temperature measurements. RTDs give absolute temperature without a reference temperature. Thermocouples

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TABLE I1 THERMOELECTRIC SENSITIVITY OF MATERIALS RELATIVE TO PLATINUM AT 0°C (pv/”c) [52] Material

Thermoelectric sensitivity

Bismuth Constantan Nickel Potassium Sodium Platinum Mercury Carbon Aluminum Lead Tantalum Rhodium Silver Copper Gold Tungsten Cadmium Iron Nichrome Antimony Germanium Silicon Tellurium Selenium

cu I

-72 - 35 - 15 -9 -2 0 0.6 3

3.5 4 4.5

6 6.5 6.5 6.5 7.5 7.5 18.5 25 47 300 440 500 900

I

cu

FIG. 7. A copper-nickel thermocouple circuit.

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generate an electrical output without the excitation current required of RTDs. Thermocouples are insensitive to strain and most other factors besides temperature, unlike RTDs. The output from an RTD can be read much more precisely than a thermocouple. For the heat flux gage in Fig. 6 the reference temperature is not needed, however, because the heat flux is proportional to the temperature difference, TI - T2.Consequently, if thermocouples are used the signal can be amplified on the gage by using a thermopile circuit design, as illustrated in Fig. 8. Here three pairs of thermocouples are placed in series electrically across the thermal resistance layer. Each produces an output ' - T2.The total voltage that is proportional to the temperature difference, 7 output voltage is then proportional to the number of thermocouple pairs, N. E = NS,(T, - 7'2)

(12)

Using a thermopile, therefore, can easily overcome the greater sensitivity of individual RTDs. The sensitivity of the heat flux gage illustrated in Fig. 6 is a function not only of the sensitivity of the temperature measurement but also of the thickness and thermal conductivity of the thermal resistance material. At steady state the one-dimensional conduction equation reduces to

k

q" = - (TI - 7'2) b

as indicated in Fig. 6. The corresponding sensitivity of the ideal layered gage is therefore

s

'

E q"

=-=-

NS,S k

I

I I I -I I

I

E

1

. I

- Ni --- c u FIG. 8. A thermopile circuit for differential temperature measurement.

292

T. E. DILLER

The transient response of the gage is a function of the thermal resistance layer thickness and the thermal diffusivity of the material. Hager [53] has analyzed the one-dimensional transient response and gives the time required for 98% response as t=--

1.56' a

(15)

It should be noted that the sensitivity increases linearly with the thermal resistance layer thickness, but the time response increases as the square of the thickness. Consequently, sensitivity versus time response is one of the major trade-offs in design of these gages. The other factor is the temperature disruption of the surface discussed in Section I.B. To minimize errors due to this disruption the temperature change across the gage should be small

For the case shown in Fig. 6, T, - T, = TI - T3.If the heat transfer is pure convection, the requirement of Eq. (16) can be expressed as h6 k

- 6 1

To minimize the temperature disruption the thickness, 6, should be kept small, particularly if the convection coefficient to the surface, h, is large. Because of these design trade-offs, the values of 6 used in different gages vary widely depending on the range of heat fluxes to be measured. Thermal resistance layers with thicknesses of 1 mm or more have generally been used for heat fluxes less than 1 kWfm'. The time response is on the order of a second. Wire resistance elements [54, 551 and thermopiles [56, 57) have been used for the temperature difference measurement. Applications are typically conduction heat flux in structures or insulation and natural convection. Bales et al. [58] have published a book of articles discussing the design, calibration, and use of heat flux gages for building applications. Haupin and Luffy [59] used a sealed air gap between two plates of the same metal. Thermocouple wires through the air insulator provided the heat transfer path and the measurement of the temperature difference between the plates. For heat fluxes of up to 100 kWfm', thermal resistance thicknesses of 25-100 pm have been used. The corresponding time response is as low as 50 ms.Although the resulting frequency response is low, some improvement is apparently available with appropriate signal conditioning [60]. To measure the temperature difference, TI - T,, Hager [53, 611 used one thermocouple

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',

\

'.\ \

'-

293

Upper Thermocouple Junctions at T,

'-,Lower - Thermocouple Junctions at T2

FIG. 9. Microfoil heat flux gage, from [62]. Reprinted by permission. Copyright 0 1983, Instrument Society of America. From ISA/83 Advances in Instrumentation,Volume 38, Part 11.

pair while Ortolano and Hines [62] used a thermopile for greater sensitivity, as illustrated in Fig. 9. For low-temperature applications a Mylar or Kapton sheet is used for the thermal resistance layer. It is very versatile because it easily conforms to most surface shapes. Applications have included many types of conduction, convection, and radiation. Examples of uses are to measure heat flux in fluidized beds [63], fouling in combustion chambers [64], and forced convection [65-671. One effective method of eliminating the problem of the thermal and physical disruption from gluing the gage onto the surface during forced convection measurements is to mask the entire surface with an equivalent thickness of the gage material (in this case Kapton) [67]. Farouk et al. [68] measured heat fluxes of 1 MW/mZ in the continuous casting of metals by using chrome1and alumel for the thermocouple pair with alumel for the thermal resistance layer. A number of new gages using the layered design have been produced in the past few years based on advances in processing techniques for thin materials. Thick-film technology was used by Van Dorth et al. 169) to put over 500 thermocouple pairs on a heat flux sensor that was 15 by 30 mm in size. This gave good sensitivity for demonstrated heat fluxes up to 200 kW/m2 and temperatures up to 500°C. Hayashi et al. [70, 711 produced thin-film heat flux gages using vacuum evaporation. The structure of the sensor is the same as shown in Fig. 6 except that a silicon monoxide layer is used in place of the adhesive layer to provide electrical insulation from the metal substrate. Two layers of nickel 0.2 mm wide and 3 mm long are deposited on either side of a second silicon monoxide

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layer, which provides the thermal resistance. Nickel layers are used as RTDs to measure the temperature difference across the silicon monoxide. A bridge circuit is used with a 1-V excitation across the two resistances to provide two output voltages which can be linearly related to the heat flux. The resulting sensitivity is S, = 2.1 pVjlcW amZ. The frequency response of the gage, obtained from tests with a shuttered light source, was estimated to be 600 Hz. Tests were performed in supersonic flow to observe shock passage and the heat flux from the associated turbulent boundary layer. An ongoing French program to develop high-temperature heat flux gages for turbomachinery application has been reported by Godefroy et al. 172-741. Their gage, as illustrated in Fig. 10, consists of a pair of platinel thermocouples on either side of a zirconia thermal resistance layer. Additional layers are added for electrical and physical isolation and adhesion. All layers are deposited by RF diode cathodic sputtering. They have studied in detail the fabrication process and the resulting structure of the materials and their survivability at high temperatures. Reports of actual heat flux measurements have apparently not yet appeared, however. The design for a layered gage using optical temperature measurement methods has appeared [75]. Two different layers of thermographic phos-

"i

0.3 pm

/

,

\

. 4

2pm

' Platinel type thermocouple

Zirconia / /

'0"

"

~ G m Z p m

T

Alumina layer

NiCoCrAlY coating

+

Nickel base superalloy

*

25 pm

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phors with different wavelength characteristics are deposited above and below a transparent thermal resistance layer. Illumination with ultraviolet light causes the phosphors to emit light as a function of the temperature, as discussed later in Section II.B.6. By using the intensities of the spectra produced by the two different phosphor layers, a differential temperature measurement can be produced proportional to the heat flux. The main advantage is that no electrical connections are required to the measurement surface, which is ideal for rotating assemblies. Unfortunately, the method has not yet been put in practice. Epstein et al. [76, 77) have produced a gage that is quite useful for turbomachinery research. They use a 25-pm-thick sheet of polyimide (Kapton) with nickel RTDs deposited on either side, as shown in Fig. 6. The pattern of the resistance elements used for temperature measurement is illustrated in Fig. 11. As shown, the sensing area is 1.0 mm by 1.3 mm. The nickel resistance element is in contact with gold leads because of the much lower resistance of gold. This isolates the voltage drop of the measurement at the sensor location. The leads from the bottom element are brought through the polyimide sheet so that all four leads can be taken to the edge of the sheet together. Originally, the nickel elements were vacuum deposited with dc sputtering. More recently a process of electroless plating has been used. To avoid the physical and thermal disruption caused by placement of the gage on the measurement surface, the entire surface is completely covered with a polyimide sheet to match the gage thickness, as was done by Hollworth and Cole [67]. Up to frequenciesof about 20 Hz the gage responds directly to the heat flux, as indicated in Eq. (13). For frequencies above 1 kHz the polyimide resistance layer appears infinitely thick, and the top resistance element (TI)

FIG. 11. Layered resistance gage pattern, from [76] with permission.

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responds like a type 2 transient heat flux gage. To cover the entire range from dc to 100 kHz, a numerical data reduction technique is used to reconstruct the heat flux signal. Calibration is done using a pulsed laser with the gage immersed in a heated bath of dibutylphthalate. The thermal properties of the fluid are used with an appropriate heat transfer model to determine the heat flux generated by the laser at the surface of the gage. Very successful measurements have been obtained with these gages, as discussed in Section IV.A.2. One of the advantages of these gages is that they can be wrapped onto curved surfaces, although the temperature calibrations change during this process, which may necessitate in situ calibration. A much thinner gage yet has been produced by Hager et al. [78-821. Their Heat Flux Microsensors are fabricated using thin-film sputtering techniques with a thermal resistance layer of silicon monoxide that is only 1 pm thick. Because the gage is deposited directly onto the surface to be measured, the thermal disruption and temperature drop across the gage, Ti - T2, is extremely small even at very high heat fluxes. The gage, which does not need an adhesive layer, is illustrated in Fig. 12. The signal from this small temperature difference is effectively amplified on the gage by a differential thermopile with up to hundreds of thermocouple pairs. The thermopile is formed by narrow metal strips (fingers) which overlap above and below the thermal resistance layer, forming the many junctions. As illustrated in Fig. 13, the fingers are also connected outside the thermal resistance layer to complete the thermopile circuit. Precise registration of the five layers hvolved is performed with microcircuit techniques. The overlay pattern of the heat flux gage and an associated RTD for surface temperature measurement is shown in Fig. 14 [80). To keep the surface free of physical disturbance, the connections are made through the substrate material using pins located in holes through the large pads seen in Fig. 14. Because the resulting gage is so thin ( < 2 pm), the thermal response time is as low as 20 ps [79]. Use of high-temperature materials in the fabrication has allowed operation at wall temperatures exceeding 1000°C [82]. Successful measurements have been made using both ceramic and stainless steel substrates for

Ti T2

Thermal Resistance

FIG. 12. Thin-film thermopile layered gage, from [79] with permission of ASME.

ADVANCES IN HEATFLUXMEASUREMENTS Lower Themocouple

297

Upper Thermocouple

FIG. 13. Heat Flux Microsensor detailed section, from [79] with permission of ASME.

temperature --c sensor

-

-

-

FIG. 14. Heat Flux Microsensor overlay pattern, from [SO]. Reprinted by permission. Copyright 0 1991, Instrument Society of America.

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the wall materials [Sl]. Sample results are discussed in Section IV.A.3 and

1v.c.

2. In-Depth Temperature Instead of making the temperature difference measurement for Eq. (11) across a thermal resistance layer placed on top of the heat transfer surface, the temperature measurement can be made in the actual material of the wall. The difficulty then becomes one of attaching the temperature-measuring devices at precisely known locations and interpreting the results. The heat transfer and temperature gradients are usually no longer one-dimensional. The operating principle, however, is very similar to that of the layered gages. Liebert et al. [83] sputtered-coated alumel on both sides of a stainless steel plate to create a differential thermocouple with the entire wall acting as the thermal resistance. Heat flux measurements were successfully made at wall temperatures up to 91 1 K. Atkinson et al. [84-861 embedded alumel wires by resistance welding on the two sides of a high-temperature alloy, as illustrated in Fig. 15. The resulting gage was calibrated and tested to 1200 K. Heat flux was successfully measured on a cylinder placed in a combustor rig [87], but further tests on a rotating gas turbine blade were unsuccessful because of lead problems [88]. Multiple thermocouple temperature measurements were made in the direction of the heat transfer by Hanneman and Mikic [89] for condensation and Gador et al. [90] in magnetohydrodynamic flows. The experiments were designed to limit the temperature gradients to one dimension. Lopata [91], Wittig et al. [92], Dullenkopf et al. [93], and Neumann et al. [12] report two- and three-dimensional modeling techniques to infer the local heat flux from multiple temperature measurements in the wall material. Neumann et al. [12] discuss the advantages and disadvantages of this approach in detail. The data reduction is complex and costly. The required modeling is highly material dependent and it is hard to properly define the Alumel

I Cold Side

Hot Side

\

Alumel

I

FIG. 15. Schematic of an embedded thermocouple gage, from [86] with permission.

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boundary conditions. The model conditions may be different for every thermocouple installation. George and Smalley [94] have taken the multiple thermocouple technique one additional step to include time-dependent heat transfer. They used an eroding thermocouple at the surface and an embedded thermocouple deep in the material. The long-time average temperature difference was used to determine the average heat flux through the material, while the short-time changes of temperature recorded by the surface thermocouple were used to infer the fluctuating component of the heat transfer as a type 2 gage, based on the transient of the temperature. Eppich and Kreatsoulas [95] incorporated an optical method of surface temperature measurement to measure the heat flux distribution over an entire surface at steady-state conditions. They fabricated a composite wall of a copper base with a thin sheet of stainless steel bonded on top. Thermocouples at the interface gave the temperature in the wall, which was uniform because of the thick copper base. The surface temperatures were measured with an infrared camera with a high-emissivitycoating applied to the surface, as discussed in Section II.B.6. Equation (13) was used with corrections for the background radiation to calculate the heat flux distribution for an impinging jet flow.

3. Wire- Wound Gage (Schmidt-Boelter) The review article by Neumann [11] gives some of the historical background on what is commonly known as the Schmidt-Boelter gage. A sketch of the gage as currently used is shown in Fig. 16. This is similar to the

TOP VIEW

SIDE VIEW FIG. 16. Schematic of a wire-wound gage, from [97] with permission.

T. E. DILLER thermopile layered gages described in Section II.A.l except for the method of producing the thermocouplejunctions around the thermal resistance layer. A fine wire of one of the thermocouple materials, for example, constantan, is wrapped around the thermal resistance layer N number of turns. One-half of the wire is then electroplated with the other thermocouple material, for example, copper. The result is a set of thermocouple junctions where the electroplating stops on the top and bottom of the thermal resistance layer. This forms a type of thermopile having N pairs of thermocouple junctions. Relative to the layered gages, the thermal resistance layer (wafer) is very thick (-0.5 mm). Therefore, to minimize the thermal disruption, it is made of a high-thermal-conductivity material, such as anodized aluminum. The nonconductive coating is necessary to provide electrical insulation with the bare wire of the thermopile. The entire wafer is then placed on a heat sink and surrounded by potting material to give a smooth surface to the top of the gage. The sensitivity of the gage is relatively high because of the thermopile and the operating temperature can also be high, depending on the materials used. One of the drawbacks is that one-dimensionalheat transfer is not really maintained. As demonstrated by Hayes and Rougeux [96] and Kidd [97], two-dimensional effects are significant. Kidd [97] has reported extensive numerical analysis, calibrations, and test results for the Schmidt-Boelter gages. He reports a transient response of 1 s.

-

4. Circular Foil Gage (Gardon) The circular foil gage was originated by Robert Gardon [98] to measure radiation heat transfer, as demonstrated in furnace tests [99]. Since Gardon [loo] introduced it for use in convectiveflows in 1960, it has been used to measure heat flux in a wide variety of situations. Some details are needed to specify when these gages can be used with good results and when they can be expected to perform poorly. They continue to be used extensively with mixed results because of their potential for inappropriate use. A sketch of the gage geometry is shown in Fig. 17, with an example of the temperature distribution in the disk. The temperature difference is measured between the center and edge of the disk using a differential thermocouple. Unlike the previous gages discussed, however, the heat transfer in the gage is not in the direction that it enters the surface. One of the key elements of the gage is that thermal energy is collected by the disk and transported to the heat sink connected to the edge of the disk. The recorded temperature difference is, therefore, a function of the total heat transfer to the disk, but also a function of the distribution of heat flux over the disk surface. A number of analytical models of the circular foil gages have been presented in the literature. Gardon [98] did the original solution for uniform

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30 1

CONSTANTAN FOIL

COPPER

BODY

FIG. 17. Schematic of a circular foil gage, from [108). Reprinted with permission of ASME.

radiation heat flux to the gage. He presented gage sensitivities and time constants for a range of sizes using a constantan disk and a copper body and center wire. The resulting chart is presented in Fig. 18. The exponential time constant, t,is a function only of the foil radius and thermal diffusivity, a

R2

t=-

4a

Experimental tests by Gardon [98] substantiated these relationships. Ninetyfive percent time response requires three exponential time constants. Consequently, for small-diameter foils (0.3 mm) the response time can be as low as 3 ms. The problem is that even for thin foils (6 = 0.025 mm) the corresponding sensitivity is very low. Consequently, this response time can be achieved only in a high-heat-flux environment. Later analytical models have included the additional effects of convection and heat losses down the center wire [101-1031, the two-dimensional temperature distribution in the disk [104, 1051, and variable property effects as a function of temperature [106]. Ash and Wright [lo61 demonstrated a good match of their analysis with experimental results for the response time. The usefulness of these analyses is limited by the knowledge of model parameters for actual gages. Each gage must still be calibrated, and the factors such as the center wire heat loss and variable properties will generally be the same in use as in the calibration. The more important question is what effects appear differently between the calibration and the application. Because the calibrations are almost exclusively done with a radiation source to

T. E. DILLER

302

6.0 4 .O E

E

2 .o

a

: I

W

1.0 0.8 4 6 0.6

(u

0 0

0

*0

0 0

wa-

0 0 0 0 0 0 0 0

01

8

FOIL THICKNESS

* 8

m a -

000 0 0

n!

0

mm

FIG. 18. Design chart for circular foil gage, from [98] with permission.

the gage, applications involving convection heat transfer can give different results due to the different temperature distribution in the disk. This problem is often not addressed, even by the ASTM standard [107]. The convection and radiation analytical solutions for steady-state conditions are presented and compared by Malone [102], Bore11 and Diller [lOS], and Kuo and Kulkarni [l09]. Reference should be made to Fig. 17 for the geometric parameters. The temperature distribution for a uniform heat flux condition is parabolic:

When the gage is calibrated with radiation at a cold wall condition, which is the usual case, there is negligible radiation from the gage. Consequently, the heat flux is uniform. The corresponding equation for a constant heat transfer coefficient, h, over the surface of the gage is

ADVANCES IN HEATFLUX MEASUREMENTS

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where L2 = R2h/k6. Because the heat flux is now proportional to the temperature of the gage, which is not uniform, the heat flux is no longer uniform either. q” = h(T, - T ) (21) The corresponding temperature difference center to edge, which is proportional to the gage signal, is To - T,

=

4rad

(22)

~

4nk6

for the constant heat flux (radiation) case and

for the constant heat transfer coefficient case. The additional term in Eq. (23) relative to Eq. (22) should be noted. For the same total heat transfer the gage will give less output for the convection case than for radiation because of the different temperature distribution. In addition, the heat transfer to the gage is less than that to the surrounding wall because the temperature of the disk is closer to the temperature of the fluid, T, . Consequently, the following correction is required if the convective heat transfer is determined from the gage output using a radiation calibration.

A plot of this equation by Kuo and Kulkarni [lo91 for typical gage parameters is shown in Fig. 19 for a range of values of the heat transfer 1.4 1.3 -

1.2

-

4whg

0

10

20

30

40

50

60

h (W/m2.K) FIG. 19. Circular foil gage correctionfor convection,from [lo91 with permission of ASME.

T. E. DILLER

304

coefficient. Kuo and Kulkarni [lo91 also demonstrated that this correction applies for a mixture of convection and radiation. A similar correction is also needed if the wall is hot and the gage is radiating to the environment [lOS]. These corrections are all based on the assumption that the gage makes perfect contact with the wall. Because the gage is normally slip fit into place, there may be an additional temperature drop between the heat sink of the gage and the wall. This would cause an additional drop in gage output and would require another correction of the results. As can be seen, using a radiation calibration for convection measurements can easily lead to large errors. The surface temperature disruption caused by the presence of a circular foil gage also affects the local thermal boundary layer [llo], as illustrated in Fig. 5 and discussed in Section I.B. The effect causes the gage to give an artificially low output. The best solution to these problems is to keep the sensitivity of the gage as low as possible to maintain the gage temperature close to isothermal. Therefore, circular foil gages cannot be used to measure convection at the manufacturer’s recommended heat flux [2]. One way to increase the sensitivity without aggravating the nonisothermal problem is to use thermopiles for the temperature difference measurement, as suggested by Trimmer et al. [14]. Such a gage is shown in Fig. 20. Overlapping of Antinomy and

- Sensing Foil /

Copper Wire Copper Tubing Nylon Insulator

0.38 in.

Copper Heat Sink Solder EPOXY Potting

FIG. 20. Thermopile circular foil gage, from [14] with permission. 0 1973 IEEE.

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305

High-temperature, high-heat-flux operation causes a different set of problems for the circular foil gages. If high-temperature materials are used, they can withstand temperatures up to at least 800°C [2, 111, 1121. It is more common practice, however, to use water cooling. An example of such a gage is shown in Fig. 21. The problem is that this causes a local “cold spot” and results in an erroneously high heat flux measurement, the opposite of the foil temperature disruption effect just discussed. Placing gages in insulating materials or as free-standing gages has also been shown to cause severe measurement problems [12, 1131. It is also known that large temperature gradients side to side [6] or behind the gage [114] cause erroneous results that are hard to correct. In spite of these many problems, successful measurements can be performed with the circular foil gages at high heat fluxes, as in jet flames [1151 and in boiling [1161. The tendency simply to buy a gage and use it must be avoided, however. Care must be taken to properly design and execute the measurements with these gages. Another approach to high-temperature gages is to fabricate the gage directly into the part by machining a cavity and attaching wires to form a differential thermocouple from center to edge of the remaining thin disk, as illustrated in Fig. 22. Radiation calibrations and oven tests to 1200 K were successfully performed using Hastelloy-X for the base material and alumel for

Foil Iconstantan) ~\

Gage Base (Heat Sink) \

FIG. 21. Water-cooled circular foil gage, from [lo91 with permission of ASME.

T. E. DILLER

306

I I

Hot Side

Alumel Alumel

I

FIG. 22. Cavity circular foil gage, from [86] with permission.

the wires [84-861. Tests on a cylinder placed in a combustor rig, however, gave very unrealistic results over most of the cylinder, supposedly because of a poor gage design when large temperature gradients were present near the gage [87]. Some success with this type of circular foil gage has been recorded in combustion tests at low wall temperatures (500 K) [117, 1181. 5. Radiometer

With the proper surface coatings any of the heat flux gages can in principle be used to measure radiation. When convection is present along with the radiation, they generally measure the total heat flux. To separate the two modes of heat transfer, a gage is designed to measure only radiation or only convection. It is relatively easy to measure only radiation by placing a transparent window or lens between the radiation source and the heat flux gage. A number of common devices are based on this approach [48], such as pyrometers, bolometers, and infrared cameras, representing a well-developed and specialized field. Consequently, it is not the intent of this review to detail these devices. The incident heat flux is measured with gages using resistance devices or thermocouples [191, or the radiation is measured directly with photoconductive detectors [Sl]. In a different approach, the total heat flux and the convection component can be measured by using two heat flux gages, one with a high-emissivity coating and one with a low-emissivity coating. The radiation is then the difference between the measurements of the two gages [119]. Problems arise with the radiation-only devices when the environment of the measurement is dirty enough to leave deposits on the window that degrade its transmission. One solution that has been developed to measure radiation heat flux in combusting environments is the transpiration radiometer. Atkinson et al. [lZO] developed one for use in the hot sections of gas turbine engines. Matthews et al. [121-1231 developed a version for use at Sandia in large sooty pool fires. The basis of operation is to blow air through

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a porous plug which composes the exposed surface of the gage. There are several purposes of the air flow. First, if the flow is sufficient it will literally blow off any fluid boundary layer from the surface of the gage. This eliminates convection with the combustion air. Second, the flow provides the temperature difference measured by the gage. The signal is proportional to the temperature of the porous plug minus the temperature of the inlet air. The porous plug temperature is a function of the flow rate of the air, the specific heat of the air, and the net radiation heat flux to the porous plug. The balance between gas cooling and radiation heating determines the porous plug temperature, although calibration is necessary to establish the efficiency of the heat transfer process between the gas and the plug. Brajuskovic et al. [124- 1261 have developed a slightly different version to operature in high-ash boilers. Instead of going through the plug, the air is directed through slots around a solid plug. The air flow keeps the ash particles from reaching the heat flux gage and it takes away the thermal energy, as in the transpiration gages. The path of the heat transfer is radial through the plug, however, as in a circular foil gage. The heat flux is proportional to the temperature difference measured between the center and edge of the plug. This “clean” heat flux gage can be used with a second gage that has no air flow to form a system to measure particle deposition in combustors. As deposition occurs on the “dirty” gage surface, the heat transfer resistance increases and the measured heat flux drops [64,127]. The thickness of the deposit can then be related to the difference in heat transfer between the two gages. B. TYPE2 METHODS-TEMPERATURE CHANGEWITH TIME

These are the simplest gages to make-only a single temperature measurement is needed to infer the heat flux at a given location. The effort instead must be expended on recording and interpreting the temperature versus time history that is generated. This involves solving the conduction heat transfer problem in the gage and the wall material. Because the solution starts with the measured temperatures as a function of time and calculates backward to find the needed heat flux boundary condition, it is in the form of an inverse heat transfer problem. The type of solution needed differs according to the gage used and the corresponding assumptions. 1. Slug Calorimeter

A calorimeter is a device used for measuring the quantity of absorbed thermal energy. As shown by the schematic in Fig. 23, a slug calorimeter uses one temperature measurement on the back surface to represent the entire mass of

308

T.E. DILLER Insulation

/A

FIG. 23. Slug calorimeter schematic.

the slug. The corresponding assumption is that the internal thermal resistance is negligible, which implies a large thermal conductivity for the material. If a control volume is taken around the slug, application of energy conservation yields

where q is the heat transfer to be measured entering the gage surface and mC is the thermal mass of the slug. The losses are minimized by insulating all of the surfaces other that the front surface. If the losses can be neglected, several solutions can easily be obtained, depending on the nature of the incoming heat flux. If q” is constant with time and constant spatially over the gage, the gage temperature increases linearly with time. q”A

T=‘I;+-t

mC

If the incoming heat flux is due to convection with a uniform heat transfer coefficient, however, the solution is exponential.

The time constant is z=-

mC hA

The negligible internal resistance assumption is valid to within a few percent when the Biot number in terms of the gage length, L, is smaller than 0.1. hL k

- < 0.1

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309

Heat Source Off

Heat Source

Y

t FIG. 24. Calorimeter response to constant heat flux, from [128]. Copyright ASTM. Reprinted with permission.

A series expansion of Eq. (27) shows that for small times ( t / t < 1) the temperature response can be approximated as linear, in which case Eq. (27) reduces to Eq. (26). The temperature rise is within 5% for values of t/t < 0.1. When a slug calorimeter is used, a temperature history is generated corresponding to the input heat flux. A constant heat flux case generates a straight line on a linear plot as given by Eq. (26) and illustrated in Fig. 24 from the ASTM standard [128]. The convection case of Eq. (27) gives a straight line on a semilog plot [21], as illustrated in Fig. 25. In each case the slope of the line can be used to determine the desired heat flux or heat transfer coefficient. The ASTM standard [l28] recommends allowing a time of t = L2/(2a) (although the equation is misprinted) for the initial transients before data are used to calculate the slope. It also recommends calculating the slope for the cooling process after the heat source is removed as an indication of the losses during heating. To neglect losses, the rate of cooling should be less than 5% of the rate of heating [128]. Although a number of different styles of slug calorimeters have been used, as reviewed by Schultz and Jones [24], there are several inherent problems [4]. Because the thermocouple or resistance element (RTD) is mounted on the back of a slug, the temperature measurement is not the average temperature of the slug. The material of the gage should be the same as the wall to minimize nonuniform temperature effects. The heat losses are usually hard to control in models with high-heat-flux conditions. Although slug

T. E. DILLER

310

T-T

1

I

0

2

,l ‘l

,

I

4

6

8

1‘21

10

I

i?,

-L

>-

14

t FIG. 25. Calorimeter response to constant heat transfer coefficient,from [21]. Reproduced by permission. All rights reserved.

calorimeters are rarely used currently, the concept is simple and the analytical models are easy to solve. Therefore, they form a good starting point to consider the other transient temperature gages. One recent advance of the slug calorimeter concept is what has been termed the plug-type heat flux gage, developed by Liebert at NASA Lewis [129-1311. This started out as a copper slug that had three thermocouples placed at known locations from the surface replacing the one thermocouple on the back of the usual slug calorimeter. The advantage for high-heat-flux environments was to give a measure of the actual temperature distribution in the direction of transfer. This allows a distributed measure of the temperature increase to give a better estimate of the heat flux.

To estimate the surface temperature, a curve fit of the temperatures was used plus the gradient at the surface determined using the measured heat flux from Eq. (30).

ADVANCESIN HEATFLUX MEASUREMENTS 0

311

Th.mocoupkkuuon

In c o v d Jot

Acthre suriace Surfaw heat flux

FIG.26. Plug-type heat flux gage from [131] with permission.

The real value of the technique was realized when the plug could be fabricated directly in the model material by electrical discharge machining. Four thermocouples were attached from the back side as illustrated in Fig. 26 to leave the front surface completely smooth and untouched. Application for turbine blade testing is described [130, 1311.

2. Null-Point Calorimeter One improvement that has been made to the slug calorimeter is to drill a hole in the back so that the thermocouple can be located closer to the surface. Not only does this give a temperature measurement that is more representative of the surface, but also the initial transient before data can be used is much shorter. The time response of the gage is, therefore, much faster. Because of these improved characteristics, it has been named a null-point calorimeter. Much analytical modeling has been performed to determine the optimum geometry for the size and depth of the hole and subsequent placement of the thermocouple, illustrated conceptually in Fig. 27. The goal of the design is to produce the same temperature history at the null point (thermocouple location) as would occur on the surface of a semi-infinite slab of the model if the gage were not present. One-dimensional transient solutions for the temperature and heat flux in a semi-infinite material are well documented. A number of data reduction schemes have been developed based on these solutions to determine the heat flux from a set of surface temperature measurements. One common method based on linear spline fits of the temperature history [13, 1321 is given by

Kidd [131 used a finite-element computer code to generate temperature solutions for different null-point calorimeter geometries for a step-change surface heat flux. He used the temperature results as input to Eq. (32) and

T. E. DILLER

312

1 L

1

r------

FIG. 27. Null-point calorimeter schematic, from [133]. Copyright ASTM.Reprinted with permission.

compared the resulting heat flux of the gage, denoted qi, to the input heat flux used in the finite-elementmodel, 4.: The ratio of these heat fluxes is plotted as a function of time in Fig. 28 for several values of the ratio of the radius to the thickness, u/b. The effect on the initial transient response is clear. There is some discrepancy over the best value to use for the ratio u/b. The ASTM standard [133] recommends either 1.0or 1.1. Kidd [13] recommends a larger value of 1.375 because the initial transient is eliminated faster. The time for this initial response (95%) is estimated as 3b2/a [l]. For the parameters in Fig. 28 with high-conductivity copper this initial transient is less than 1 ms.

YInw n

111 w m 111 mu

0.25 0

0

0.0025

0.0050

0.0075

0.0100 0.0125

0.0150

Time (sec) FIG.28. Null-point calorimeter analytical time response, from [13]. Reprinted by permission. Copyright 0 1990, Instrument Society of America.

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313

The longest time of useful measurement is determined by the time for the temperature to penetrate the total thickness, L. This is estimated as 0.3 L2/u, which corresponds to 300ms for the parameters of Fig. 28. The time for useful data collection can be summarized as

The details of a typical null-point calorimeter El341 are illustrated in Fig. 29. The flange is provided to give a small air space around the gage to minimize radial heat losses. The usual thermocouple pair is chromel-alumel. Its attachment to the gage is one of the major fabrication difficulties [13]. To accomplish a test within the recommended test time, the model is often swept through the flow. This also minimizes the time that the model and gages are exposed in high-heat-flux environments and extends their useful life. Because the null-point temperature on the back of the gage is designed to match the undisturbed wall temperature, the front side of the gage is necessarily hotter. Although this creates a “hot spot” effect, these gages are often used at the stagnation point of models where the effect would be small. An example of use in arc-heated facilities is given by Carver and Kidd [135].

3. Coaxial Thermocouple The goal of the coaxial thermocouple is to measure the surface temperature of the model wall as a function of time, the same as that of the null-point calorimeter. The results can then be used in an equation like Eq. (32) to determine the corresponding heat flux. Coaxial thermocouples are used more widely because they are easier to fabricate than the null-point calorimeters. As shown in Fig. 30, the concept is simple. One thermocouple material forms the center wire, which is surrounded by an electrical insulator. The second

f. AWSl IYPt

I(,

0.02 IN. O.D.

FIG. 29. Section view of a null-point calorimeter, from [lS]. Reprinted Copyright 0 1990, Instrument Society of America.

permission.

314

T. E. DILLER

111 DlYINSlOWS IN INtHFS

I F R O I i ~( O W 0 LFM WN IS AllMHtD 10 lHEIMOtLFYtW1S IIV SlOl WtlOlNG 01 Sllytl SOLDfllNG

FIG. 30. Coaxial gage schematic, from [136]. Reprinted by permission. Copyright 0 1990, Instrument Society of America.

thermocouple forms a sheath around these two layers. The final assembly is often drawn down to a smaller diameter [136). Wires are attached at the end and the completed unit is press fit into the model. The actual thermocouple junction is formed at the top end by plating a thin layer of one of the materials [137], vacuum deposition [138], or simply lightly sanding to mix the two materials together and bridge the thin insulating layer. Because the thickness of the top junction layer can be on the order of 10 pm, the initial response time can be much less than 1 ms. Equation (33) provides a good estimate of the useful measurement time if the onedimensional semi-infinite solution [Eq. (32)] is the basis of the data reduction. More sophisticated methods can be used [132, 139, 1401, including a finite-thickness material solution to extend the useful measurement time. Neumann [l 11 also describes the use of a second thermocouple on the back of the gage to further extend the measurement time with appropriate heat transfer modeling. The most important parameter in specifying the coaxial thermocouple gages is to match the material thermal properties with those of the wall. As seen from Eq. (32), the property of importance is the product kpC, which for the coaxial thermocouple is taken as the average of the three different material layers. In most cases this can be matched quite well by judicious choice of the thermocouple pair and the ratio of their thicknesses. A few examples of the use of coaxial thermocouples are given by Neumann [13], Carver and Kidd [135], and others [132, 137, 1401. 4. Thin-Skin Method

The thin-skin method is like a continuous slug calorimeter that forms the entire body of a model. It is referred to as thin because the temperature is

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315

assumed constant through the material but not transversely along the skin. Thermocouples are attached to the back side of the skin to measure the local temperature. The heat flux is calculated at each measurement location using the transient temperature in Eq. (25), just as for a slug calorimeter. Neumann [l 11 and Schultz and Jones [24] include extensive sections reviewing thin-skin measurements. Jones [3] gives a summary of the latter review. The review of Trimmer et al. [14] includes a description of the method with examples for the case of constant heat transfer coefficient. Neumann [l 1) argues that the thin-skin method is outdated for today’s aerothermodynamic facilities. Standard procedure is to use 0.125-mm-diameter thermocouple wire of low thermal conductivity, such as chromel-alumel[141]. Copper should not be used because its high thermal conductivity will cause excessive thermal disruption. The wires are attached by laser, spot, or electron beam welding. A common material for the skin is 0.75-mm-thick stainless steel. The time required for the initial transient is L2/a[l], which corresponds to 50 ms for this material. The main errors in using the thin-skin method are caused by (1) heat loss from the back side, (2) transverse conduction through the skin, and (3) conduction down the thermocouple wires. These errors have been analyzed [142- 1461, with suggestions to optimize the measurement accuracy [l, 141). The analysis of Kidd [143] for the thermocouple conduction error is summarized in Fig. 31 as a function of thermocouple material and wire size. Experimental data were used to confirm the analytical results. As expected, the error for copper is much higher than for the other materials. Miller [147] used an effective thickness for the skin in highly curved regions to get the proper thermal mass and a second-order least-squares curve fit of the temperature-time data. Measurements performed in flows with Mach numbers up to 10 matched within 5% of predicted values. Barry et al. [148] have reported a variation of the thin-skin method that uses internal cooling in place of the usual adiabatic condition on the inside surface. Instead of injecting the model into the flow, the change in heat flux is provided by a change in the temperature of the coolant fluid. Matching the response harmonics with the imposed temperature perturbation allows calculation of the heat transfer coefficient on the external surface. The method was used on hollow models of gas turbine blades at gas temperatures up to 1300 K [149]. Hay [23] reviews details of the method. 5. Thin-Film Method

The thin-film method has largely replaced the thin-skin method for aerodynamic testing for a number of reasons. The advancement of thin-film deposition techniques has made the fabrication of large numbers of resistance elements for temperature measurement on the surface of models much

316

T. E. DILLER Skin Material: Stainless Steel

Wire Diameter x

Id. in

FIG. 31. Thin-skin thermocouple errors, from [143]. Reprinted by permission. Copyright

0 1985, Instrument Society of America. simpler. Because the thin-film devices can be made a fraction of a micrometer (pm) thick, the initial response is very fast (< 1 p).With the gages placed on models with thicks walls of ceramic insulating material, the temperature response is good and the thermal disruption caused by the gage is usually negligible. The biggest drawback for high-temperature flows is that the gages are exposed on the surface of the model and may experience degradation during testing. The operating principle of the thin-film gages is to measure the surface temperature of a semi-infinite material (substrate) in response to the applied surface heat flux. Therefore, the data analysis is similar to that used for nullpoint calorimeters and coaxial thermocouple gages, as described in Sections II.B.2 and II.B.3. The difference is that the thin-film gages are mounted on the surface of a homogeneous substrate. The coaxial thermocouple measures temperature at the surface of a nonhomogeneous plug which is inserted into the wall. The null-point calorimeter measures temperature on the back of a specially designed slug of material which is inserted into the wall.

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Thin Film Temperature Sensor,

FIG. 32. Semi-infinite geometry for thin film. Reprinted by permission of the publisher from [152]. Copyright 1991 by Elsevier Publishing Co, Inc.

The requisite transient conduction solutions for the one-dimensional semiinfinite model are given in books [41, 1501 and explained in detail for this application by Schultz and Jones [24]. The starting point is the onedimensional heat diffusion equation, with the geometry illustrated in Fig. 32.

The initial condition is T = T at t = 0 and the two boundary conditions are initial temperature T = T at x -+ 00 and a prescribed temperature or heat flux at the surface, x = 0. For a prescribed surface heat flux the surface temperature is given by

The function ql(r) gives the time variation of the heat flux at the boundary, with z representing time at which the heat flux occurs for the integration. If the surface heat flux is given a step change from zero to a constant value, 4; at time t = 0, the solution reduces to

The surface temperature increases proportional to the square root of time. The material properties of the substrate enter as the square root of the product of kpC. The inverse of Eq. (35) is often desired, the heat flux corresponding to a known (or measured) surface temperature.

318

T. E. DILLER

This solution assumes that the prescribed (or measured) surface temperature is a continuous function of time, T. Often this is not the case, particularly when the input is digitized data. Therefore, the following form is generally more convenient:

For the standard case of a step-function change in surface temperature from to T,,, Eq. (37) does not give a convenient answer, while Eq. (38) gives

Data analysis for the thin-film gages can be performed by several methods, depending on the type of heat flux input and the desired form of the output. If the heat flux is constant, the temperature difference, T, - T , will be linear with the square root of time, and the slope of the line will be proportional to the heat flux. Most cases are not so simple, however, and the desired output may be the complete time-resolved heat flux record. For this more general case, two approaches have been developed. One is to do the numerical evaluation of the data using an equation like Eq. (38). Unfortunately, this equation has a singularity when t = T, which requires additional numerical techniques. A number of approaches have been used, the most popular of which is that of Cook and Felderman [lSl] as already expressed by Eq. (32). This assumes constant thermal properties for the substrate, which generally are a function of temperature. To include the variable properties, numerical techniques are required for the time-dependent data reduction [152,153], or a correlation factor as used by Miller [147]. The second method for data reduction is to use an analog circuit of resistors and capacitors to mimic electrically the thermal response of the substrate. The equations for the circuit can be made to match exactly the model expressed by the heat transfer equations. Schultz and Jones [24] have described this technique in detail and have extensively demonstrated its utility. It should be noted that the thermal boundary condition at the surface of the substrate (usually low thermal conductivity) varies with time. At the initiation of flow the surface is isothermal at the initial temperature, Ti. As time progresses the surface temperature changes in response to the local heat flux conditions. Unless the heat flux is uniform over the model, the surface temperature will have a different response at different positions. The boundary will generally be neither constant heat flux nor constant temperature. Because the measurements occur over such a short time, however, the temperature changes are generally very small compared to the temperature

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difference from the free stream (or recovery) to the surface. Therefore, the surface will usually appear as nearly constant in temperature to the convective boundary layer. Thin-film gages are relatively easy to fabricate. All that is required is the application of a thin temperature-measuring layer on the model with appropriate lead connections.The usual choice for temperature measurement is a resistance temperature device (RTD). The thin-film method has been used, however, with thermocouples that were fabricated either by vapor deposition of gold on germanium [154] or by sputtering platinum and platinum-rhodium [lSS]. The most popular method has been to apply a thin layer of platinum paint to an insulating substrate. It must then be fired at over 600°C to remove the solvent and create a hard, thin layer of platinum. Several layers may be necessary to ensure a uniform thickness and width and the desired resistance value. Sputtering processes that have been used more recently produce a more uniform layer with the possibility of intricate patterns. Because it is deposited in a controlled high vacuum, the purity of material is very high and the resistance is repeatable. With a typical thickness of 0.1 pm the transient response of the RTDs is of the order of 0.01 ps [28]. This causes negligible effect on the heat flux measurements for times greater than a few microseconds [24]. Therefore, results that are properly processed are essentially instantaneous even for high-speed flows. There is a limit on the length of the useful test data, however, due to the semi-infinite substrate assumption used in the modeling. The time before the thermal boundary layer reaches the back surface of the substrate is [24] t=-

L2 16u

For a typical insulating substrate (MACOR) of L= 1 mm thickness the corresponding maximum time is 80 ms. This makes the technique useful for many types of short-duration flows, as in shock tubes or tunnels or highspeed blow-down facilities. To determine the transient temperature history, a small current is used to drive a bridge circuit to measure the electrical resistance of the RTD. The constant-current source must be large enough to provide a measurable output voltage but small enough to keep the self-heating negligible. This optimization is made easier if the sensor resistance is a high value. For a resistance of at least 100 ohms, a current of 1 to 2 mA provides good results. One of the advantages of vacuum-deposited resistance films is that the resistivity is considerably higher (by a factor of 2 or more) than that of the bulk material [24].

T. E. DILLER Calibration of the sensors for heat flux measurements requires two steps. First, the resistance versus temperature relationship must be determined for each sensor. This can simply be done by putting the model in an oven and monitoring the resistance of each gage over a set temperature range. The relationship is generally not linear, which means that the temperature coefficient of resistance is not constant as a function of temperature. In addition, the resistance curve may shift over time [1471, particularly if high temperatures are used. The second part of the calibration is to determine the thermal properties of the substrate. The most popular method is to measure the response of the sensor to a pulse of energy provided by an electrical discharge through the resistance film. The value of kpC can be calculated from the response in two fluids with different values of kpC, such as air and glycerin. Other heating regimes can be used [156], or even a pulsed radiation source to the surface. It is important to measure the value of kpC for each model because properties can vary significantly for many of the substrate materials between batches. The number of papers reporting experimental heat transfer results with the thin-film method attests to its usefulness. A partial list includes ref. [26, 157-1711. Although most used a ceramic substrate, Polat et al. [172] deposited a thin gold film on a porous glass substrate. The resulting sensor was used for studies of air throughflow on moving surfaces to model paper drying. , Several groups have refined the techniques to make detailed measurements of the time-resolved heat flux. A group working with J. E. OBrien at NASA Lewis developed a method of injecting a preheated cylinder that was instrumented with thin-film resistance gages into a steady low-speed flow [173]. Time-resolved heat flux was obtained using an analog circuit to observe the detailed effect of rotor wake passageC174, 1751 and grid turbulence at a stagnation point [176]. A group at Calspan working with M. G. Dunn has been measuring heat flux to gas turbine blades [e.g., 28,177-1841. A shock tube is used to provide the flow conditions for 20 to 25 ms to entire gas turbine stages. The heat flux gages are painted onto Pyrex inserts that are placed into the blades. For data reduction both analog circuits and digital data processing are used, including Fourier transforms to optimize the frequency response and accuracy of the time-resolved results [185]. This allows detailed measurement of the heat flux on a rotating turbine [186], as illustrated in Section IV.A.2. A rather large group at Oxford University, associated with the late D. L. Schultz and with T. V. Jones, M. L. G. Oldfield, and R. W. Ainsworth, has been instrumental in developing the thin-film techniques for many years. Their application has been turbine blades, which were tested in an isentropic light piston tunnel [187-1891. They used analog circuits to convert the measured temperature signal into time-resolved heat flux. A sample of the

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AT F

0

50

I00

I50

200

250

300 mS

FIG. 33. Transient thin-film temperature and heat flux, from [24] with permission. The original version of this material was first published by the Advisory Group for Aerospace Research and Development, North Atlantic Treaty Organisation (AGARD/NATO) in AGARDograph 165 “Heat-transfer Measurements in Short-duration Hypersonic Facilities” in February 1973.

temperature signal and the resulting heat flux is illustrated in Fig. 33. Their results include extensive testing of the effects of rotor wakes and shock wave passing on the time-resolved turbine blade heat transfer [190- 1961. The gages were platinum resistance painted onto ceramic blades. The timeresolved heat flux measurements allowed important extensions of the state of understanding of unsteady effects, as discussed in Section IV.A.2. The Oxford University group has also developed a technique to apply the thin-film gages to metal substrates using a vitreous enamel coating to provide electrical insulation [1971. Although this gives much greater flexibility for experimental application of the method, it also complicates the data analysis [198]. The technique that has been shown to provide good results [199] is to use the analog circuit plus digital processing of the signals, including filters and a frequency-boosting circuit. The calibration is also more complicated because of the multiple layers making up the substrate. The metal blades give better capability to rotate the turbine blades while measurements are being made [200,201].

6. Optical Method Optical techniques for measuring the transient surface temperature have the advantage of being nonintrusive to the surface. In addition, some of the methods can simultaneously provide the entire surface temperature distribution. This obviously has benefits relative to installing hundreds of point gages on a model to measure the heat flux distribution. The common optical methods can be classified as either temperature-sensitive coatings (phasechange paint, liquid crystal, and thermographic phosphor) or infrared emission. The measurement theory is the same as in the thin-film technique, with the same equations applicable. Complexity is added, however because of the optical determination of temperature, particularly to obtain quantitative

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results. The qualitative capability to “see” the heat flux distribution is very attractive, however. The use of phase-change paint is discussed by Trimmer et al. [14], Schultz and Jones [24], and Newmann [ll]. The latter argues that it is not cost efficient because of the limited amount of information that is obtained in each run and because it is not reusable. It must be reapplied for each use. The advantage of the phase-change paint method is that it is an easy visualization technique. Quantitatively, however, it gives only one temperature measurement (the phase-change temperature) at one time at each location on the surface. In addition, it is difficult to automate the measurement. Phase-change paint is available commercially as a liquid that is simply spread over the surface of interest. At a specified temperature the material undergoes a phase change during which it changes from an opaque solid to a transparent liquid. If the underlying surface is made a different color than the initial opaque color, the time of the phase change is easily observed. The temperature of the phase change can be specified to occur at a variety of temperatures. Metzger and Larson [202] describe the use of phase-change paint in a lowspeed channel flow. Because only one temperature-time point is obtained, the heat transfer coefficient was assumed to be constant throughout the test. The corresponding equation for one-dimensional transfer into a semi-infinite substrate is

Although Eq. (41) assumes a step change in the air temperature from T to T,, the actual time variation of the air temperature was used in the corresponding series solution. The melting patterns during the runs (- 30 s) were recorded photographically at discrete times after the flow was suddenly initiated. The data from visual comparisons of the photographs were reduced at several hundred grid points. Horvath et al. [203] successfully used the phase-change paint technique to measure heat flux on models at hypersonic flow velocities. The phase-change paint method, however, is not as widely used as liquid crystal or thermographic phosphor methods. Although liquid crystals have been used for measuring temperatures for many years, their use for measuring heat flux is only recent. Moffat [22] has given a good review of the liquid crystal methods for heat transfer measurement, including the transient technique discussed here. Several types of liquid crystals reversibly change the color of reflected light as a function of their temperature. The color change is a result of changes in the molecular structure. Unfortunately, the color change can also be caused

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by changes in other properties, such as pressure and mechanical shear [24]. In addition, the crystals are easily contaminated and lose their sensitivity with time. Fortunately, if the crystals are microencapsulated their properties are stabilized and their sensitivity to factors other than temperature is diminished. For transient temperature measurement the chiral-nematic form is preferred because it has a relatively fast time response on the order of milliseconds for the required molecular rearrangement of its structure [204]. The material can be obtained commercially as a paint which can be sprayed or screen printed onto a surface in layers the thickness of the microcapsules, 10pm. Although the liquid crystals change their color reflection through several distinct colors, the most accurate measurements are done by following the change of a single color. Liquid crystals can be obtained with the temperature of the color sensitivity specified over a variety of temperatures typically in the range of 25 to 40°C. When they are microencapsulated, crystals with different temperature sensitivities can be mixed together to give several temperature points for the same color change. This allows measurement of the same color shift sequentially to indicate different surface temperatures. The ready availability of video cameras makes the data acquisition easy, particularly if the test section is constructed of transparent walls. To provide the temperature difference to drive the heat transfer, either the flow is heated relative to the surface or the surface is initially heated to a uniform condition and then allowed to cool in the flow. As with the thin-film method, the surface boundary condition changes with the time of the experiment. For the thin-film method of Section II.A.5 this is usually a negligible effect and the boundary appears to be nearly constant temperature. Because of the longer run times for the liquid crystal method (seconds instead of milliseconds), this may not always be true, however. It depends on how much the surface temperature changes relative to the total free-stream-tosurface temperature difference. It is desired that

-

Whenever Eq. (42) is valid, the surface can be considered to have a constant-temperature boundary condition. If the heat transfer coefficient is assumed constant as a function of time, the semi-infinite solution in Eq. (41) can again be used. Ireland and Jones [205,206] appear to have been the first to use the liquid crystal transient method. They measured the heat transfer in complicated flow geometries around a pedestal and ribs. Jones and Hippensteele [207] used what they called the double crystal method, with two liquid crystals of

324

T. E. DILLER

FIG. 34. Transient liquid crystal temperature distribution (time in seconds after start of flow), from [207] with permission of ASME.

different temperature sensitivities, to include the effect of nonuniform initial temperature in their heat transfer coefficient calculation. An example of one of their temperature versus time distributions is shown in Fig. 34. This was taken on the floor of a duct expansion in a low-speed wind tunnel. The numbers indicate the time in seconds when each curve reached the specified temperature. Larger numbers, therefore, indicate lower heat flux. Baughn et al. [ZOS] estimated a total uncertainty for their heat transfer coefficient measurement of 7.2% using three different liquid crystals. The monitored color changes occurred at 31,35, and 41.2"C. Measurements were made on the surface of a pin fin. The diversity of possible applications of the method is illustrated by two additional reports. Metzger et al. [209] used it to measure the jet impingement heat transfer to a rotating disk. Wang et al. [210] used it to measure the convective heat transfer to a grain-roughened flat surface. To date, the only results reported using the transient liquid crystal method have been for time-averaged or steady heat transfer coefficients. An interesting application for the future would be to develop techniques for measurement of the unsteady heat flux. Some of the applicable color processing techniques are discussed in Section ILC.1. The result would have the potential for simultaneous presentation of the two-dimensional spatial distribution of heat flux as a function of time. Use of thermographic phosphors for temperature measurement has been recognized for many years, although their use for heat transfer has been

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recent and is still limited. Schultz and Jones [24] and Neumann [11] give brief reviews of their use for transient surface temperature measurement. Trimmer et aE. [14] give a detailed description of the method and some sample results of heat transfer measurements in high-speed flows. When illuminated with ultraviolet light, electrons in the phosphors are excited and subsequently radiation is emitted in the visible spectrum. Impurities in the crystal structure are instrumental in the energy transitions that give the characteristic wavelengths emitted. The log of the intensity of emitted radiation is nearly linear with temperature over a fairly wide useful temperature range. Mixing different phosphors allows adjustment of the operational temperature range. The phosphors can easily be applied to a surface using a suitable carrier. Temperature calibration is usually achieved by mounting a thermal gage on the surface for simultaneous monitoring of the absolute surface temperature. The most recent report of phosphor use for heat flux measurement is in hypersonic flows at NASA Langley [203,211]. Two-color bandwidth-filtered images were recorded and processed using a color video system. Digital image processing was used to convert the measured values to thermal images using intensity look-up tables. The camera framing rate was 30 frames per second for run times of 0.5 to 3 s. Surface heating distributions were determined from curve fits of the temperature data and application of Eq. (41) to determine the best average heat transfer coefficient. The corresponding experimental uncertainty was estimated to be f 15%. One problem noted was that the surface roughness due to the phosphor coating might cause premature transition in the Mach 10 flows. The angle of the incident light relative to the surface was also important. The potential for transient analysis of the results to create a time-resolved map of the surface heat flux was discussed. Determination of temperatures of heated bodies from measurements of the infrared radiation emission requires no special coatings or materials as in the previous optical methods. A coating to establish a known, uniform emissivity is usually applied, however. Infrared techniques for surface temperature measurement are very old and have become well established. Their use for heat transfer measurement is, however, more recent and coincides with the advent of high-speed ( 20 ps) infrared scanning radiometers. These cameras can provide real-time measurement of an entire surface in detailed color images. The emissivity of the surface and the background radiation must be known to convert the absorbed radiation of the camera into absolute temperatures. With the resulting temperature versus time images, the surface heat transfer coefficient or heat flux distributions can be calculated using Eqs. (32), (38), (41), or similar solutions. Short summaries of the method are provided by Neumann [111, Schultz and Jones [24], and Hay [23].

-

326

T. E. DILLER

Simeonides et al. [212] have demonstrated good results with the infrared method in hypersonic flows. Fort et al. [213] used an interesting variation of the method by measuring the infrared emission from the back side of the wall due to transient combustion on the front side of the wall. They used an appropriate conduction solution to correlate the measured temperature with the heat flux, resolving a 30-ms burst of combustion. Spot heating of the surface to provide the heat source during convection has been reported with accompanying transient infrared measurements [214, 2151. Results were erratic, and the experiments did not account for the disruption of the thermal boundary condition discussed in Section I.B. Many more applications of the transient infrared method should be seen in future years. C. TYPE3 METHODS-ACTIVE HEATING A direct measure of the energy transfer under steady conditions jnto or out of a surface can provide a very accurate measure of the surface heat transfer. The two most important issues that must be considered with this method are the effect of the thermal capacitance of the surface material if the conditions are not truly steady and the heat transfer at the edges of the measurement region [18]. Very small rates of temperature change may cause large errors in the measured heat transfer, and it may take a long time to achieve steadystate conditions under which the temperature is no longer changing. Examples of these effects will be discussed with the presentation of the specific techniques. The usual method of energy exchange is through electric heating. The power can be easily measured and easily controlled. The one drawback is that the transfer is alwaysfrorn the surface. Consequently, this method is most useful in well-controlled laboratory experiments. Because of the time constraints and power limitations, it is not used in high-heat-flux or hightemperature situations. This eliminates most high-speed flows. The methods can be grouped according to the two boundary conditions that normally results, constant heat flux and constant wall temperature. In constant heat flux cases the experiment usually runs until a steady-state temperature distribution is achieved. For constant wall temperature, manual or active control of the power is required to stabilize the system at the prescribed constant temperature. In both cases the heat transfer is measured as the required power minus losses as a function of position and/or time. 1. Constant Heat Flux

Measuring convective heat transfer coefficients with an electrically heated constant heat flux method has been done for many years. The most common technique is to use a heater with uniform resistance, insulate the back surface,

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and minimize lateral conduction by the design. A thin metal sheet is often used as the heater to provide uniformity of resistance (and heat flux) and to minimize lateral conduction. Several centimeters of good insulation as the backing material usually ensures acceptable heat loss from the back surface. Unfortunately, the response time for this arrangement is typically very long, depending on the surface convection and substrate properties. With a surface heat transfer coefficient of 100 W/m2 K and a good substrate insulator, the time from a cold start to reach steady-state conditions (within 2%) is typically over 1 h. The time limitation is the result of transient heat transfer into the insulation backing layer. If the heated metal film covers the entire surface, the model may closely resemble the physical arrangement of a thin-skin model (Section II.B.4). In addition to the different method for determining the surface heat flux, however, the usual range of heat flux is much lower (< 10 kW/mz) than for the thin-skin method. The surface temperature measurement is not needed to determine the heat flux, but it is required to quantify the conditions under which the heat flux occurs. It is particularly necessary for the determination of the heat transfer coefficient, as specified in Eq. (7). Although any of the previously discussed methods for measuring temperature are possible candidates, only thermocouples, resistance temperature devices, infrared cameras, and liquid crystals are commonly used. Their use is discussed in that order. When thermocouples are used to measure the surface temperature, their attachment and layout are important because of the potential problems caused by heat transfer through the wires. Because the wires have a much larger thermal conductivity than the insulation, the measured temperature can be significantly lowered. Running the wires parallel to the surface for a distance and using small-diameter wire can minimize the effect. The lead wire conduction effect is more important for the heated foil technique than for the earlier thin-skin method (Section II.B.4) because of the lower heat flux range measured with the heated foils. Wang and Simon [216] developed a bendable surface to study the effect of wall curvature on the heat flux in a turbulent boundary layer. They used a composite wall with a 0.1-mmstainless steel sheet and 0.076-mm thermocouples on a flexible heater, as illustrated in Fig. 35. Baughn et al. [217] used a gold-coated plastic sheet to generate the uniform heat flux. Copper-constantan thermocouples (0.076-mm-diameter wire) were used to measure the temperature distributions in several geometries for both internal and external flows [217-2191. Estimated uncertainties ranged from k 2 to f 4%. Dielsi and Mayle [220] used stainless steel heater strips to model the leading edge of a turbine blade. Their estimated uncertainty was & 2.5%after an 8-h time to reach steady state. A similar method was used by Ligrani et al. [221], using 126 thermocouples to measure the effect of film cooling holes.

328

T. E. DILLER

FIG. 35. Composite heated wall for constant heat flux measurements, from [216] with permission. 0 AIAA.

Oker and Merte [222] used a thin gold film deposited on Pyrex both to generate the heating of the surface and to measure the average temperature of the surface. The temperature was determined from the resistance, which was found from the measured current and voltage. The response to a step power input was used to study the transient boiling of refrigerant 113 and liquid nitrogen. In the first few seconds most of the heat flux went into conduction in the substrate. Analysis of the convection and conduction was used to identify the different boiling regimes. A similar arrangement of a thin-film resistance element sputtered onto quartz was used by Samant et al. [223] for boiling studies of refrigerant 113. Because the surface of the heated film was only 0.25 mm by 2.0 mm, significant heat was transferred laterally through the quartz substrate, giving the heater a larger effective surface area by an estimated 12 to 15%. Normally the edge effect is much larger, but the high heat fluxes in boiling kept it to a minimum. Because this was forcedconvection boiling, however, the temperature nonuniformity of the wall could also have a large effect (Section 1.B). Jet impingement heat transfer was studied optically by Carlomagno [224] with a heated stainless steel foil and an infrared camera to measure the resulting temperature distributions. The advantages include no lead wire errors and a wider temperature range than with liquid crystals. Temperature measurements using liquid crystals were mentioned briefly in Section II.B.6 for heat flux measurement from the transient temperature. The

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majority of liquid crystal experiments, however, have used steady-steady heating of the surface, giving a constant heat flux condition. Both the cholesteric and chiral-nematic forms of liquid crystals have been used with good results. They have either been spray painted in microencapsulated form or applied in self-contained plastic sheets. Quantitative information is most often obtained by visually observing color transitions corresponding to specific temperatures. The heat flux and surface heat transfer coefficient must be carefully matched with the fluid temperature and the desired temperature limits of the observable colors of the liquid crystal. This can be expressed by [225]

where TL and Tu are the lower and upper temperature limits of the liquid crystal. When using liquid crystals, the surface must be calibrated for the temperature relation to the color. This should be done in the actual test apparatus with the lighting level and viewing angle used in the actual experiment. Hollingsworth and Moffat [225] temporarily attached five thermocouples to the surface for calibration. In later work, Hollingsworth and Watwe [226, 2271 used a thin-film nickel resistance element permanently mounted on the thin-film nickel heater. This allowed calibration and continual monitoring of the temperature during the tests. Boiling of refrigerant 11 was observed from the back side of the surface with cholesteric liquid crystals by Hollingsworth and Watwe [226,227]. The temperature overshoot in incipient boiling was clearly observed by the RTD and on a video of the liquid crystals. One of the main attractions of liquid crystals is the ability not only to produce quantitative results but also to “see” heat transfer phenomena as they occur. This can be rather spectacular. One of the drawbacks of the constant heat flux surface was also apparent. The time response of the system was limited by the thermal capacitance of the heater and liquid crystal layers to about 1 s. This is too slow to observe many transient phenomena. Kenning [228] used a similar physical arrangement for studying boiling of water. He used unencapsulated liquid crystals covered with a thin film and with a color play over the range 100 to 134°C. Calibration was done with a thermocouple at 2°C intervals. The upper limit of the frequency sensitivity of their system was estimated at 20 Hz. Some of the first measurements of heat transfer coefficients with liquid crystals were made in simple flows for comparison with established results. Impinging jets were a popular subject [229-2311, along with a cylinder in cross-flow [232] and recirculating pipe flow [233]. Simonich and Moffat

330

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[234] made measurements in water on a concave-surface turbulent boundary layer. A streaky structure was observed to move back and forth across the plate, but the time response of the surface was too slow to measure details. Taslim et al. [235] mounted a camera with the liquid crystals on a rotating channel to study the internal cooling in gas turbine blades. This eliminated the problem of taking signals off a rotating rig. Baughn et al. [236, 2371 measured heat transfer from an impinging jet. For better quantitative measurements the issue of color interpretation must be resolved. Moffat [20, 221 gives detailed discussions of the possible methods for image processing. Two main techniques are used for processing a digitized color image. The first employs narrow-band pass filters and measures the intensity of light in each wavelength segment. This spectral intensity-based interpreter is used to define the isothermal lines on the model. The method preferred by Moffat [20] uses the relative fractions of the three primary light colors. This trichromic decomposition is a standard method used by industries dealing with color and is referred to as a true-color or chromatic interpreter. Either method is limited to about 0.5"C resolution for a standard temperature range of 6°C for the liquid crystals. Better resolution is possible if a narrower temperature band is used. The true-color method was used by Hollingsworth et al. [225] for a flat plate with protrusions. Microencapsulated chiral-nematic liquid crystals were applied to the back of a 0.0254-mm-thick stainless steel foil. A layer of air was used as insulation and also the viewing port to record the temperature. The estimated uncertainty of the resulting heat transfer coefficients was - 10% on average. Baughn et al. [238] performed an analytical study of the frequency response limit of the heated liquid crystal technique. They used a thin layer of microencapsulated chiral-nematics directly applied to a gold film heater on plastic and insulation. The response for a minimum detectable temperature change of the liquid crystal layer was 13 Hz. It is important to note the difference between this response time of detectable temperature change and the time required to reach a new steady-state heat flux condition, which would be orders of magnitude longer. Analytical conduction solutions can be used to correct the amplitude and phase of the time response [239], but this has apparently not been tried for liquid crystal experiments. In a separate study Baughn et al. [208] compared the heat transfer coefficients measured with the transient liquid crystal method to the heated liquid crystal methods for a pin fin in cross-flow. The results were close even though the boundary conditions were different, constant wall temperature for the first and constant heat flux for the second. Jones and Hippensteele [207] performed a similar comparison of steady-state and transient liquid crystal methods. They measured differences of 30-50% in the heat transfer coeffi-

+

ADVANCESIN HEATFLUX MEASUREMENTS

33 1

r TRANSPARENT RYLAR I

--

BLACK SEAL COAT ADHESIVE SHEET

-

COPPER BUS BAR CONDUCTIVE ADHESIVE

'-GOLD ON POLYESTER

'- ADHESIVE

SHEET

END WALL TEST SURFACE -'

FIG. 36. Liquid crystal heater composite, from [243] with permission.

cients between methods, which they attributed to the different boundary conditions. At NASA Lewis, Hippensteele has developed a liquid crystal method using a composite heater [240-2431. Cholesteric liquid crystals were encapsulated in between a black plastic layer and a transparent Mylar layer. This composite was attached to a thin-film gold heater on polyester, which was adhered to the test surface. As illustrated in Fig. 36, a copper bus bar was used to attach the electric power to the end of the gold film heater. The temperature of the yellow line was calibrated and used to reduce the liquid crystal pictures to isothermal lines. The corresponding Stanton number (dimensionless heat transfer coefficient) lines were determined from the heat flux minus losses and the temperature difference [Eq. (7)]. An example on the end wall of a turbine cascade using multiple pictures with different heating levels is shown in Fig. 37. Results were also generated from the same data by computer interpolation of digitized liquid crystal photographs. As shown in Fig. 38, the resulting contours show much more detail. Computer-generated color contour plots were also reported [243].

2. Constant Surface Temperature The design and operation of constant surface temperature experiments are different than those for constant heat flux. Instead of measuring the temperature that results from an imposed heat flux, the heat flux that is required for a set temperature is measured. This typically requires a control system to maintain the temperature constant spatially and/or temporally, depending on whether steady or unsteady measurements are desired. Electric resistance heating is almost exclusively used, providing only heat transfer from the surface. A device capable of providing measured heat transfer in either

T. E. DILLER

332

FLw

9

FIG. 37. Liquid crystal results (Stanton numbers), from [243] with permission.

CONTOUR

STANTON NUMBER

F H

0.00355 0,00398 0.00436 0.00508 0.00553 0,00609 0.00664 0.01070

FIG. 38. Liquid crystal results, computer data reduction, from [243] with permission.

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direction, however, was constructed by Shewen et al. [244]. Their design, which used a number of Peltier devices under the measurement surface, was limited to heat fluxes less than 500 W/mZ. Many time-averaged measurements of spatial distributions of heat transfer have been made using segmented plates and heaters. If the plates are made of a thick, high-conductivitymaterial, the surface will be nearly isothermal, even if the heat transfer coefficient has large spatial variations. Some thermal isolation is required between segments and is often provided by insulation strips. Usually the individual heaters are adjusted manually to achieve a uniform temperature [245-2483, although automatic temperature controllers [249] can speed the stabilization process by an order of magnitude. The controllers operate phase-angle-fired silicon-controlled rectifiers (SCRs), which control the power of each of the 60-Hz electrical waveforms. On-off controllers are generally not adequate to maintain a steady-state condition. One of the most detailed descriptions of the use of the segmented constant temperature plate method is the review of the turbulent boundary layer experiments at Stanford by Moffat and Kays [250]. The method was used and continually refined for over 25 years. To achieve the quoted accuracy of f.2% required meticulous detail to experimental procedure and refinement of the data reduction programs. The latter is a lengthy process done to eliminate any small errors and to build up an experience base to include all of the small corrections needed. Figure 39 illustrates the typical geometry for the segmented plate geometry. The measured heat flux occurs from the front surface, while the back surface is placed on the heater, followed by insulation. The heat loss from the edges usually causes the major errors. The two modes of heat loss [25 13 are shown as q’, , the loss per unit width through the insulation by convection from the insulation surface, and q;, the loss per unit width by conduction through the insulation to the adjoining plates because of the temperature mismatch T,. HEATED PLATES

FIG. 39. Geometry of constant wall temperature systems. Reprinted by permission of the publisher from [251]. Copyright 1989 by Elsevier Science Publishing Co., Inc.

T. E. DILLER

334 10.0000

-

@,/(I-+l

'

I

I

I

I

0.10

0.01

1.00

I

_

10.00

Bi,

FIG. 40. Analytical corrections for insulation strips. Reprinted by permission of the publisher from [251]. Copyright 1989 by Elsevier Science Publishing Co., Inc.

VandenBerghe and Diller [251] analyzed this problem assuming a constant heat transfer coefficient on the surface and uniform temperature within the high-conductivity plates. The results for convection loss, q;, are shown in Fig. 40 in dimensionless form. -=E l 1-

2q;

hL(T. - T,) + 2q; This represents the ratio of convection from the insulation to the convection from the plate plus the insulation. The two Biot numbers are Bi, = hL/k and Bi, = hw/k, using the thermal conductivity of the insulation and lengths l E ~ represents ) the error due to given in Fig. 39. The parameter ~ ; / ( assuming the heat transfer from the insulation surface is the same as from the plate. The insulation is used to isolate the two plates thermally to allow separate heat transfer measurements from each. The second potential error arises from conduction between the plates when the temperatures are not perfectly matched. It was demonstrated that the one-dimensionalconduction solution for q; kd 4; = A T (45) W

is almost always sufficient to represent this measurement uncertainty. The corresponding dimensionless loss, E ~ is, equal to q; divided by the total power supplied to the plate.

ADVANCES IN HEATFLUXMEASUREMENTS

LI 0

0.0

0.3

335

"C

I

0.4

w (em) FIG. 41. Sample design calculation for constant wall temperature systems. Reprinted by permission of the publisher from [251]. Copyright 1989 by Elsevier Science Publishing Co., Inc.

To illustrate these effects, an example is presented in Fig. 41 for a plate of given length, L, and heat transfer coefficient, h. As expected, the dimensionless conduction error, e2, decreases as the insulation width, w, increases. Conversely, the correction for the convection loss, e l , increases with increasing w, as the centerline temperature of the insulation

The best design is one in which is depressed below the wall temperature, 'Iw. these competing effects are minimized. Most experiments are designed with an insulation thickness much larger than needed for negligible conduction effects. The convection corrections and temperature disruption of the surface, therefore, are much larger than necessary. In addition, there is a limit to how small the errors can be made for heater plates of a given size. The smaller the plates, the larger the relative errors, as illustrated in Fig. 40 as BiL becomes small. This limits the theoretical resolution of the method. Figure 40 also gives an indication of what happens if this method is used without guard plates. The edges then appear to have an infinite insulation

336

T. E. DILLER

width, w. As Bi, becomes very large, the dimensionless loss reaches an asymptote which is a strong function of Bi,. Only if the measurement plate is large and the heat transfer coefficient high is the relative convection loss through the insulation small. Consequently, in many cases the measurement accuracy is limited if guard plates are not used. Small gages for local heat flux measurements have been produced to actively match the surrounding wall temperatures. Achenbach [252] and Groehn [253] used copper plugs with a 0.5-mm to 1.0-mm layer of thermal insulation around them and separate heaters and thermocouples inside. They were used for steady-state heat flux distributions around a circular cylinder. Kraabel et al. [254] also used a copper plug, but with a tighter gap and a differential thermocouple to control the power input to a thermistor. The design is illustrated in Fig. 42. Steady heat flux measurements on a cylinder were reported with k 2% estimated uncertainty [255]. It should be noted that this low uncertainty is due to a well-designed sensor and detailed experimental procedure. Such experiments, however, are very time consuming to perform. Baughn et al. [256-2581 designed and used a similar sensor for internal flow by replacing the copper plug with a nichrome ribbon. Four sets of differential thermocouples were used to match the temperature with the surrounding aluminum wall. The sensor is illustrated in Fig. 43 and has an estimated uncertainty of i-2%. It was used downstream of a backward-

FIG.42. Differential thermocouple gage, from [254] with permission of ASME.

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differential thermocouple /

aluminum

337

,

electric heater

ribbon

FIG.43. Heated ribbon gage, from [258] with pemssion of ASME

facing step in a tube. Heat flux distributions were obtained by sliding the step in the tube relative to the sensor. To measure time-resolved heat flux, thin-film gages are required along with high-frequency temperature controllers. The thin films can be fabricated by the techniques discussed in Section II.B.5. A thin resistance element serves as both the heater and the temperature measurement device. A constanttemperature hot-wire anemometer bridge can be used to supply regulated dc power and to control the temperature at the set point. This is very similar to the operation of a thin-film heated skin friction gage [259-2611. The difference is the thermal boundary conditions along the wall [lS, 2621. For skin friction gages the thin film is given a large overheat with a large adiabatic region around the heated element. This keeps the thermal boundary layer over the heated element thin and in the fluid sublayer as illustrated in Fig. 44. The skin friction can then be related by calibration to the heat lost from the thin-film heater. One of the problems noted with this type of skin friction gage is that a large fraction of the heat is lost by convection from the surrounding substrate, similar to the results presented in Fig. 40. The effective sensor area, therefore, appears to be larger than the actual thin-film sensor. As the surface heat flux changes, the effective area changes, which affects the amplitude and phase of the gage frequency response. The skin friction gage does measure the total heat transfer from the heated element, but it has little correlation with the heat flux from a surface with a uniform thermal boundary [18,2621. The difference in the thermal boundary layers is illustrated in Fig. 44.The goal of a skin friction gage is to create a heat source as small as possible, whereas the goal of a heat flux gage is to have a heat source controlled to match exactly the surrounding wall temperature.

T.E. DILLER

338

-

7, ;

-

u

n

d

a

r

y

Layer

.wF

a) Heat Flux Gage

>

U m, Tm , ,Thermal Boundary Layer

b) Skin Friction Gage FIG. 44. Boundary layer comparison between skin friction and heat flux.

Consequently, the control problem for heat flux gages is more difficult, which is usually seen in problems when a hot wire bridge is used for heat flux gages. Several investigators have used thin-film skin friction gages for unsteady heat flux [263-2651. Corrections of the steady and unsteady portions of the signals were applied by Creff and Andre [264] and Rosiczkowski and Hollworth [265]. The latter group performed transient calibrations that demonstrated the strong attenuation effect of the substrate on the transient heat flux. Beasley and Figliola [266] performed a numerical solution of the transient response of the actively heated thin films. They analyzed two basic geometries, shown in Fig. 45. As indicated, there are three components to each

A

Section Ad.

B Design I

m Section 6-6

Design I1

FIG.45. Heated thin-film gage geometries, from [266] with permission.

ADVANCES IN HEATFLUXMEASUREMENTS

339

TABLE111 HEATEDTHIN-FILM MODELPARAMETERS [266] Dimension Substrate half-width Sensor element half-width Sensor element thickness Substrate thickness Coating thickness over sensor element

Design I Olm)

Design I1 Olm)

3000 350 2 847

3000 2500 2 847 9

9

probe: the heated thin film, the substrate on which it is deposited, and a protective coating layer overtop. The sensor was assumed to be mounted in an isothermal plate, which defined all of the boundary conditions except at the surface. At zero time the heat transfer cofficient at the surface was given a step change in its value. The geometric parameters are listed in Table 111. The probe transient response is summarized by the results in Fig. 46 for a probe m2/s) and deposited onto a Pyrex substrate (k = 1.2 W/m * K, a = 61 x covered by an aluminum oxide coating (k = 119.2 W/m K, CL = 8.0 x m2/s). The controller was assumed to give ideal response to maintain the thin film at constant temperature during changes in the surface heat transfer coefficient. Figure 46 gives the resulting fractional error of the power as a function of time. Two plots are shown because the response appears to have two time constants, one for the coating and one for the substrate. The first design is typical of the skin friction gages used and the second is an improvement that eliminates much of the surrounding adiabatic substrate. a

100 ‘Design

I

I\

10-‘

10-2

-

10-31

0

,

,

20

,

,

40

,

,

60 Time Ips)

,

Time , (nsl ,

80

1 100

FIG. 46. Transient thin-film model response, from 12661 with permission.

T.E. DILLER

340

The coating acts as a fin which keeps the surface closer to isothermal but also increases the effective area of the sensor. Clearly, design I1 is better than design I. A number of investigators have used gages similar to design I1 of Fig. 45 with success [267-2751. There are nagging problems, however. Matching the temperature of the sensor with the surrounding material is difficult. If the temperature of the surrounding material goes slightly above the heatedelement temperature, the conduction into the heated element may be large enough for the controller to turn off the power. To counter this problem a bias voltage is often used which effectively operates the sensor temperature slightly higher (up to 3°C) than the surrounding surface. This offset temperature causes an additional heat loss from the sensor by conduction and causes a nonisothermal surface boundary. The controllers (usually hot wire anemometer bridges) are not ideal and often have some droop in the terhperature response. To observe the temperature of the heater film independently, Campbell et al. [276] embedded a thermocouple made with 76-pm wire at the surface of the substrate next to the sensor film. They also designed the active film as shown in Fig. 47, to cover the entire substrate in an attempt to minimize heat loss through the substrate. To match the sensor and surrounding surface temperatures, a second matched thermocouple was embedded in the surrounding surface and a slow-response differential controller was used to match the temperatures. COPPER CONTACTS

ALL DIMENSIONS IN m m

FIG. 47. Heated thin-film heat flux gage, from [276]. Reprinted with permission of ASME.

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341

Consequently, the hot wire bridge controlled the power to the sensor during fast surface heat flux transients and the second controller matched temperatures over long times by adjusting the power to the surrounding material. This system was capable of maintaining the temperature match within - 0.1"C. The resulting heat flux uncertainty was estimated to be 5%. Measurements of the effect of vortex shedding on the time-resolved heat flux around a cylinder in cross-flow were reported. Because of the uncertainties in the various losses and transient response of these heated gages, detailed calibrations are particularly important. Steady calibrations must be done by convection because radiation to the gage is in the wrong direction. The method used for transient calibration is to chop a radiation source, either mechanically [269, 2761 or optically [79, 2773. The system used by Campbell et al. [276] is shown in Fig. 48. Convection from the surface is used to maintain the positive heat flux alwaysfrorn the surface. The sample results shown in Fig. 49 clearly demonstrate a cycle of lower and higher heat transfer corresponding to the on and off radiation to the surface. A small overshoot provided by the controller is also obvious. The amplitude of the step function response, however, is much more important than the details of the shape. This amplitude was compared with the steadystate radiation response for different models and makes of bridge circuits for the entire range of compensation settings. Under most of the conditions tested the amplitude of the response for all frequencies was several times smaller than for steady state (up to a factor of 10). The thermocouple output on the gage [276] was also recorded to help interpret these results. It was found that in all of the cases with smaller amplitude the temperature was not maintained at the set point. The gage

+

(MOUNTED IN PLATE)

LENS

Zf

'-Y

LENS LAMP 300r FLOOD

FIG. 48. Unsteady calibrationapparatus,from [276]. Reprinted with permission of ASME.

0

TIME (msec)

80

FIG.49. Unsteady calibration results, from [276]. Reprinted with permission of ASME.

temperature would be up to 1°C higher with the radiation on than off. For the few cases in which the amplitude of the gage response matched the steady values, there was no measurable temperature change of the surface. The compensation tuning of the bridge had to be adjusted close to the instability point and only certain bridge circuits were satisfactory. Because the normal hot wire probe operates with 250°C overheat, this small amount of control droop is negligible. The overheat for the heat flux gages, however, is usually only 30 to 40°C. In conclusion, the actively heated gages can be used for unsteady heat transfer measurement, but it is difficult to produce proper operation of the gage and it must be thoroughly calibrated using an unsteady calibration method.

In. Calibration Calibration has been addressed in previous sections in connection with specific gages. A number of overriding issues, however, should be discussed. First, many calibration methods have been used for heat flux. Individuals have been very innovative in designing methods suitable for their gages and applications. All of the heat transfer modes have been utilized-radiation, convection, and conduction. Radiation has been used most, usually because it is easier to establish a known heat flux level and the available heat flux range is so large. The review by Whitmore [278] and the number of papers devoted to the method of radiation calibration attest to its popularity [279-2841.

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343

The second point is that the different calibration methods may give different results, particularly if radiation methods are compared with convection [lo81 or conduction methods. Third, the calibration method is no better than the heat flux standard that is used. Currently there is no NET-certified standard for heat flux, although NIST [285] does perform radiation calibrations up to heat fluxes of 35 kW/m2. Fourth, all of the current calibration facilities are designed to calibrate gages at room temperature. Because many of the applications involve hot walls, calibrations with the gage at high temperature are important and becoming more so as temperature limits are increased. Fifth, a method for in situ calibration of heat flux would be very beneficial [286]. Although calibration of the transient response of heat flux gages is rarely done, lasers provide a good on-off energy source for this purpose. Optical switching with a Bragg cell is a good method for achieving rise times significantly less than 1 p s [79, 2771.

IV. Applications A. TIME-RESOLVED MEASUREMENTS The development of the fabrication techniques for thin-film heat flux gages has opened the door to a vast new experimental world of time-resolved heat flux measurement. The gages have been described in Section 11, with their operating characteristics. In this section samples of the results are presented to document some of the capability of these methods. 1. Reciprocating Engines

Because of the unsteady temperatures naturally created in a reciprocating engine during each cycle, the conditions necessary for the type 2 gages (temperature change with time) are easily created. Heat flux from transient surface temperatures has been measured with a variety of the techniques described in Section 1I.B. Gatowski et al. [277] evaluated these methods for application to engines. The thermal match of the sensor with the substrate was found to be important during the tests in a rapid-compression machine and under transient heat flux from a switched laser. Investigators have made numerous measurements in different types of engines of the time-resolved heat flux as a function of the position in the cycle [287-2921. Fig. 50 shows an example of the results for different engine speeds taken from the work of Alkidas and Cole [287]. The surface temperature was measured on the cylinder head between the valves and was used to calculate the unsteady portion of the heat flux. The steady heat flux was determined with a second

T. E. DILLER

344

1000r/min 1500 r/min

2500 r/min

---------

hl

E 2.0

8-4

5

1.6

0 I 80

‘ffn 1.0 0.5

0 300 320 340 360 380 400 420 440 460 Crank Angle-Degrees

FIG. 50. Example of time-resolved heat flux in a reciprocating engine, from [287] with permission of ASME.

thermocouple buried in the wall along with the time-averaged signal of the surface thermocouple, as discussed in Sections II.A.2 and II.B.3. In addition, the paper by Woschni and Spindler [293] contains comments from many of these investigators relevant to heat transfer and measurement in diesel engines. 2. Gas Turbine Engines

One of the limitations of gas turbine engines is the maximum temperature that can be used in the combustor without damaging the blades in the turbine hot section. Numerous schemes have been developed to cool the blades and increase their temperature limits. Therefore, knowledge of the blade heat transfer is critical to the design of new high-performance engines [294,295]. Experimental results are needed for prediction capability and computer code validation. It is also well known that the heat transfer coefficient between the

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345

combustion gases and the blades is increased by the large amount of fluid unsteadiness present in a gas turbine engine. One of the sources of this unsteadiness is the wakes of the rotor blades [296]. Detailed measurements of the heat flux have given indications of the nature and source of the increased heat transfer [297]. Dunn et al. [298] have reported what is referred to as phase-resolved heat flux on turbine blades. These are time-resolved data that are referenced to the rotor blade position rather than time. Since the rotor speeds were 27,000 rpm, one blade event lasts less than lops. A high-pressure turbine stage was instrumented with the transient thin-film resistance gages described in Section II.B.5 and a shock tube was used as a short-duration source of heated air. A sample of their experimental heat flux results is shown in Fig. 5 1 at the stagnation point of the blade. The plot shows data as one blade passes over four of the 23 vane passages. Although there is variation between passages, the maximum heat transfer consistently occurs at the point identified as when the vane is intercepting the rotor wake. This type of time resolution at these rotational speeds is impressive. The group at Oxford University has reported basic work with a stationary turbine cascade and a rotating-bar mechanism to simulate the flow unsteadiness of rotor wakes. An example of the heat transfer results [298] using

5f3

1

N” x‘ 73 Y

0

7.83

15 :w

PHASE, OEGREES

FIG. 51. Example of phase-resolved heat flux in a turbine stage, from [298] with permission of ASME.

T. E. DILLER

346

7'c-

-8.

.

0

.

.

.

.

i

Cycle Time msec

1.k'

1:s

FIG. 52. Example of wake-passingeffects in a turbine cascade, from [190] with permission of ASME.

transient thin-film gages (Section II.B.5) is shown in Fig. 52. The regions marked A, B, and C refer to the effect of the incident shock wave, a turbulent spot induced by the shock, and the transition to turbulence from the arrival of the bar wake. It should be noted that the large increase in heat flux of region C corresponds to the lower velocity but higher turbulence of the wake region. Averaged over the cycle, these increases explain the observed increase in time-average heat flux. The group at MIT performed time-resolved heat flux measurements on rotating blades in a blow-down facility [299-3011. The gages were described in Section II.A.l. They also compared results with those from the Oxford University group [191] at comparable conditions [300], as shown in Fig. 53. Although there is an additional spike in the Oxford cascade results, the overall quantitative match is good. 3. Basic Research Many flows traditionally treated as steady actually have significant components of unsteadiness. From a fundamental standpoint, how this fluid unsteadiness affects surface heat transfer is an important question. To develop the basic understanding necessary to answer this question, detailed, time-resolved measurements of surface heat flux will be required to match with the characteristics of the fluid motion. The correlation between unsteady fluid motions and the corresponding heat flux over a range of flow length and time scales is a necessary step toward the more difficult task of understanding the mechanisms of turbulent heat exchange. With this type of detailed understanding comes the possibility of a level of prediction and performance characterization and control not previously achievable. Examples of progress

ADVANCESIN HEATFLUXMEASUREMENTS 5000 -

'

I M I T Rotor, x / s = O . I O -Oxford Coscode, X I S0.08 = 1

347

I

-

0

0

I

2

3

Time (Blode Passings I

FIG. 53. Comparison of turbine heat flux results, from [ 3 0 ] with permission of ASME.

in correlating unsteady fluid measurements with unsteady flux measurements are presented. Ching and OBrien [176] used the transient thin-film method (Section II.B.5) in a low-speed wind tunnel to study the effect of free-stream turbulence on heat transfer at the stagnation point of a cylinder. A constant-temperature hot-wire anemometer was located near the heat flux gage (2.5 mm in front) to obtain simultaneous velocity measurements. For grid-generated turbulence of S%, the unsteady excursions of the heat flux often reached 40%. Crosscorrelation of the heat flux and velocity signals indicated correspondence of the signals with little phase lag. Simmons et al. [302] performed a similar study using a Heat Flux Microsensor (Section II.A.l) at the stagnation point of a large free jet. Again, grid-generated turbulence created unsteadiness measured simultaneously with the heat flux gage and a constant-temperature hot-wire anemometer. A sample of the detailed results is shown in Fig. 54 for a free-stream turbulence level of 11.2%. The RMS of the heat flux was 10.3%of the mean, nearly the same magnitude as the free-stream velocity. The phase of the heat flux signal did lag the velocity by up to 90". Although the heat flux and velocity signals visually appear to correlate, the coherence function was used as a means of quantifying the relationship between the two signals over a range of frequencies. The coherence function relates the degree of linear correlation between two signals as a function of frequency using many simultaneous samples. The magnitude of the coherence function is defined between zero and one. As shown in Fig. 55, the coherence of the heat flux and velocity

T. E. DILLER

348 46.0

9

(16s w/m2)

30.0

0.0

Time (mire)

8.0

Time (mrrc)

8.0

a) HEAT FLUX 7.0

Velocity (4.8 mls)

3.0 0.0

b) VELOCITY

FIG.54. Effect of turbulence at a stagnation point, from [302]. Reproduced by permission. A11 rights reserved.

0.5

Coherence

0.o 0.0

Frequency

(kHz1

1.0

FIG. 55. Coherence function of heat flux and velocity (50 averages), from [302]. Reproduced by permission. All rights reserved.

ADVANCESIN HEATFLUXMEASUREMENTS

349

signals of Simmons et a.1. [302] peaks around 200 Hz at a value of about 0.2. Although this is not a strong correlation, there is a definite linear relationship between the two signals. The method of using the correlation of unsteady velocity and heat flux was further tested by Mancuso and Diller [303] using the same instrumentation placed in flow behind a cylinder. The regular pattern of vortex shedding produced was clearly demonstrated by both the Heat Flux Microsensor and the velocity probe. As seen in Fig. 56, the coherence between the signals is quite high over a narrow bandwidth around the natural shedding frequency of 650 Hz. Swisher et al. [304] reported similar measurements in the unsteady flow created by a turbulent junction vortex and in a two-dimensional turbulent boundary layer. A coherence of up to 0.35 was found at frequencies below 50 Hz in the low-speed boundary layer with the RMS of the heat flux 20% of the mean. Mori et al. [66,305] measured the unsteady heat flux in separated flow regions near forward- and backward-facing steps. They used a lowfrequency layered gage (Section 1I.A.1) for frequencies below 50 Hz and found large heat flux unsteadiness. B. SPATIALLY DISTRIBUTED MEASUREMENTS Because of the limitation to near room temperatures for liquid crystals, most of the spatially distributed heat flux measurements have been for basic 1.o

COHERENCE

0.0

FIG. 56. Coherence function of heat flux and velocity for vortex shedding from a cylinder at 650 Hz (50 averages), from [303]. Reprinted with permission of ASME.

350

T. E. DILLER

research. VanFossen and Simoneau [306] published a most interesting study with direct application to gas turbine engine heat transfer. They used the liquid crystal technique developed at NASA Lewis (Section ILC. 1) to study the heat transfer augmentation of free-stream turbulence in the stagnation region of cylinders and blade models. A set of 0.05-cm-diameter wires were placed perpendicular to the cylinder and the air flow direction. The wires created pairs of counterrotating vortices which impinged on the cylinder and in the process were stretched. Smoke visualization was used along with the liquid crystal-covered surface to obtain pictures of the vortices and their relationship to the surface heat flux, as shown in Fig. 57. In addition, a hot-wire probe was used to measure the distribution of the mean velocity and turbulence intensity. A schematic illustrating the interpretation of the results is shown in Fig. 58. Increased surface heat flux corresponds to regions where the vortices are feeding fluid down to the surface. Surprisingly, this is a region of low turbulence. Lower mean velocity and higher turbulence intensity are directly behind the wires, where the heat flux is lower. Understanding of the phenomenon was greatly

FIG. 57. Smoke visualization and liquid crystal heat transfer on a cylinder,from [3M]with permission of ASME.

ADVANCESIN HEATFLUX MEASUREMENTS

.

I

351

.

I

I

VORTU: PAIR

‘-CYLINDER

I Z-DIRECTION

FIG. 58. Schematic of the relationship of vortices and heat flux at the stagnation point of a cylinder, from [306] with permission of ASME.

enhanced by the simultaneous flow visualizations and measurements taken along with the heat flux measurements. C . HIGH-TEMPERATURE MEASUREMENTS

Flight vehicles such as the National Aerospace Plane are pushing the temperature and heat flux limits of instrumentation. Heat flux levels of 100 MW/mZ are expected at many locations within the propulsion system [297] and on the structure of the vehicle. The cooling problems will be severe and the expected surface temperatures will exceed 1000°C. Operation within actual gas turbine engines is also desired over the same range of temperature. An example of one response to this challenge is the Heat Flux Microsensor

T. E. DILLER

352

900 800 700

600 500 400 300

g

T* ("C)

0

-100

-200 -300 -400 0

2

4

6

8 10 12 Time (sec)

14

16

18

26=

FIG. 59. Supersonic combustion tests, from [82] with permission.

discussed in Section II.A.l. Heat flux tests with one of these gages deposited on ceramic and exposed to an oxyacetylene torch were successful to surface temperatures of 750°C [81]. In more recent tests a Heat Flux Microsensor deposited on stainless steel was placed on the wall of the supersonic combustion tunnel at NASA Langley [82]. A sample of the measured temperature and heat flux is shown in Fig. 59. The steps in heat flux are indicative of the increase in conditions as the tunnel is stepped up to the full hydrogen combustion conditions. The last step is when the hydrogen injectors are lit. Not only did the gage survive high wall temperatures (exceeding 1ooo"C in some of the tests), but the heat flux measurements demonstrated large amounts of unsteadiness during combustion. Only because of the high frequency response of these gages were they able to measure the unsteadiness.

V. Conclusions As seen in Section IV, some very exciting research in heat transfer has been made possible by the recent advances in measurement capability. The application of microsensor fabrication techniques and high-speed digital data acquisition have been keys to these advances in heat flux instrumentation. Future improvement of optical methods for spatial distribution of heat flux appears very promising. A few researchers have demonstrated the capability

ADVANCESIN HEATFLUXMEASUREMENTS

353

for quantitative measurement of heat flux with the infrared method for high temperatures and liquid crystals for room temperatures. Future developments should continue to be exciting.

Acknowledgments Appreciation is expressed for the support from DOE, NSF, and NASA over a number of years which has allowed the author to pursue the study of experimental heat transfer leading to this review. Appreciation is also offered to all of the researchers who allowed the reproduction of their work.

Nomenclature null-point calorimeter dimensions (m) surface area (mZ) specific heat (J/kg * K) gage output voltage (V) convection heat transfer coefficient (W/mz K) thermal conductivity (W/ m.K) wall thickness, plate length, or gage length (m) mass (kg) number of thermocouple junctions heat transfer (W) heat transfer per unit width (W/d heat flux (W/m2) constant surface heat flux (W/m2) radial coordinate (m) radius (m) heat flux sensitivity (pV/kW * m2) thermocouple temperature sensitivity (pV/T) time (s) center temperature (”C) temperature (“C) temperature mismatch (“C) initial temperature or temperature at a specified time (“C) temperature limits (“C) mean fluid temperature (“C)

T, Tw

Tm,

W X

-fluid recovery temperature (“C) wall temperature ( “ C ) free-stream fluid temperature (“C) free-stream velocity (m/s) insulation thickness (m) coordinate in flow direction

(m) x,,, x,. xz 2

distances (m) coordinate normal to wall (m)

GREEKLETTERS

thermal diffusivity (m2/s) thickness (m) difference dimensionless heat losses dimensionless temperature dimensionless parameter, Eq. (20) density (kg/m3) time variable or time constant

6) SUBSCRIPTS

ad CL cond conv g rad S

T W

adiabatic centerline conduction convection gage radiation surface isothermal conditions wall

354

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1. Thompson, W. P. (1981). Heat transfer gages. In Fluid Dynamics (Mefihodsof Experimental

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151. Cook, W. J., and Feldman, E. M. (1966). Reduction of data from thin film heat-transfer gages: a concise numerical technique. AIAA J. 4, 561-562. 152. George, W. K., Rae, W. J., Seymour, P. J., and Sonnenmeier, J. R. (1991). An evaluation of analog and numerical techniques for unsteady heat transfer measurement with thin film gauges in transient facilities. Exp. Thermal Fluid Sci. 4, 333-342. 153. Cook, W. J. (1970). Determination of heat-transfer surface temperature measurements, AIAA J. 8, 1366-1368. 154. Elrod, J. C., Gochernaur, J. E., Hitchcock, J. E., and Rivir, R. B. (1985). Investigation of transient technique for turbine vane heat transfer using a shock tube. ASME Paper 85-IGT- 17. 155. Gladden, H. J., and Proctor, M. P. (1985). Transient technique for measuring heat transfer coefficients on stator airfoils in a jet engine environment. AIAA Paper No. 85-1471. 156. Keltner, N. R., Bainbridge, B. L., and Beck, J. V. (1988). Rectangular heat source on a semiinfinite solid-an analysis for a thin film heat flux gage calibration, ASME J. Heal Transfer 110,42-48. 157. Jessen, C., and Groenig, H. (1991). A new method for manufacture of thin film heat flux gauges. Shock Waves 1, 161-164. 158. Miller, C. G. (1984). Experimental and predicted heating distributions for biconics at incidence in air at Mach 10. NASA TP-2334. 159. Miller, C. G., Micol, J. R., and Gnoffo, P. A. (1985). Laminar heat-transfer distributions on biconics at incidence in hypersonic-hypervelocity flows. NASA TP-2213. 160. Miller, C. G. (1985). Refinement of an ‘alternate’ method for measuring heating rates in hypersonic wind tunnels. AIAA J . 23, 810-812. 161. Micol, J. R. (1991). Experimental and predicted pressure and heating distributions for aeroassist flight experiment vehicle. J. Thermophys. Heat Transfer 5, 301-307. 162. Tsou, F. T., Chen, S. J., and KO,S. Y. (1983). Measurement of heat transfer rates using a transient technique. ASME Paper No. 83-HT-87. 163. Consigny, H., and Richards, B. E. (1982). Short duration measurements of heat-transfer rate to a gas turbine rotor blade. ASME J. Eng. Power 104, 542-551. 164. Nagamatsu, H. T., and Choi, K. Y. (1987). Endwall heat transfer in the junction region of a circular cylinder normal to a flat plate at 30 and 60 degrees from stagnation point of the cylinder. AIAA Paper No. 87-0077. 165. Geller, A. S. (1989). Heat flux from a fuel-air detonation to a structure. In Heat Transfer Measurements, Analysis and Flow Visualization (R. K . Shah, ed.), pp. 9-14. ASME, New York. 166. Ratzel, A. C., and Shepherd, J. E. (1985). Heat transfer resulting from premixed combustion. In Heat Transfer in Fire and Combustion Systems (C. K. Law el al., eds.), pp. 191-201, ASME, New York. 167. Camci, C., and Arts, T. (1985). Short-duration measurements and numerical simulation of heat transfer along the suction side of a film-cooled gas turbine blade. ASME J. Eng. Gas Turbine Power 101, 991-997. 168. Camci, C., and Arts, T. (1991). Effect of incidence on wall heating rates and aerodynamics on a film-cooled transonic turbine blade, ASME J. Turbomachinery 113,493-501. 169. Kercher, D. M., Sheer, R. E., and So, R. M. E. (1983). Short duration heat transfer studies at high free-stream temperatures. ASME J . Eng. Power 105, 156-166. 170. Van Heiningen, A. R. P., Douglas, W. J. M., and Mujumdar, A. S. (1985). A high sensitivity fast response heat flux sensor. Int. J. Hear Mass Transfer 28, 165771667, 171. Harasgama, S. P., and Wedlake, E. T. (1991). Heat transfer and aerodynamics of a high rim speed turbine nozzle guide vane tested in the RAE isentropic light piston cascade (ILPC). ASME J. Turbomachinery 113, 384-391.

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172. Polat, S.,Van Heiningen, A. R. P., and Douglas, W. J. M. (1991).Sensor for transient heat flux at a surface with throughflow. Int. J. Hear Mass Transfer 34, 1515-1523. 173. OBrien, J. E. (1990).A technique for measurement of instantaneous heat transfer in steadyflow ambient-temperature facilities. Exp. Thermal Fluid Sci. 3, 416-430. 174. OBrien, J. E., Simoneau, R. J., La Graff, J. E., and Morehouse, K. A. (1986).Unsteady heat transfer and direct comparison to steady-state measurements in a rotor-wake experiment. In Heat Transfer 1986, Vol. 3 (C. L. Tien et al., eds.), pp. 1243-1248. Hemisphere, Washington, DC. 175. OBrien, J. E.(1990).Effects of wake passing on stagnation region heat transfer. ASME J. Turbomachinery 112,522-530. 176. Ching, C. Y., and OBrien, J. E. (1991). Unsteady heat flux in a cylinder stagnation region with high freestream turbulence. In Fundamental Experimental Measurements in Heat Transfer (D. E. Beasley and J. L. S. Chen eds.) pp. 57-66.ASME, New York. 177. Dunn, M. G. (1986). Time-resolved heat-flux measurements for the rotor blade of a TFE731-2H P turbine. In Convective Heat Transfer and Film Cooling in Turbomachinery, von Karman Institute for Fluid Dynamics, Rhode St. Genese. 178. Dunn, M. G., and Stoddard, F. J. (1979).Measurement of heat transfer rate to a gas turbine stator. ASME J. Eng. Power 101,275-280. 179. Dunn, M. G., Rae, W. J. and Holt, J. L. (1984).Measurement and analyses of heat flux data in a turbine stage: Part 11-Discussion of results and comparison with predictions. ASME J. Eng. Gas Turbines Power 106,234-240. 180. Dunn, M. G., and Hause, A. (1982).Measurement of heat flux and pressure in a turbine stage. ASME J. Eng. Power 104,215-223. 181. Dunn, M. G. (1985). Heat flux measurements and analysis for a rotating turbine stage. AGARD CP-390. 182. Dunn, M. G. (1986). Measurement for the rotor of a full-stage turbine: part l-timeaveraged results. ASME J. Turbomachinery 108,90-97. 183. Dunn, M. G., Martin, H. L., and Stanek, M. J. (1986). Heat-flux measurements and comparison with prediction for a low aspect ratio turbine stage, ASME J. Turbomachinery 108,108-115. 184. Dunn, M. G., Rae, W. J., and Rigby, D. L. (1987). Experimental and theoretical studies of time-averaged and time-resolved rotor heat transfer. In Structural Integrity and Durability of Reusable Space Propulsion Systems (NASA N87-22770). pp, 29-32. 185. Dunn, M.G., George, W. K., Rae, W. J., Woodward, S. H., Moller, J. C., and Seymour, P. J. (1986).Heat-flux measurements for a full-stage turbine: part 11-Description of analysis technique and typical time-resolved measurements. ASME 3. Turbomachinery 108,98-107. 186. Dunn, M. G. (1990). Phase and time-resolved measurements of unsteady heat transfer and pressure in a full-stage rotating turbine. ASME J. Turbomachinery 112,531-538. 187. Jones, T. V., Schultz, D. L., Oldfield, M. L. G., and Daniels, L. C. (1978).Measurement of heat transfer rate to turbine blades and nozzle guide vanes in a transient cascade. In Sixrh Iniernational Heat Transfer Conference, Vol. 2,pp. 73-78.Hemisphere, New York. 188. Nicholson, J. H., Forest, A. E., Oldfield, M. L. G.and Schultz, D. L. (1987)Heat transfer optimized turbine rotor blades-an experimental study using transient techniques, ASME J. Eng. Gas Turbines Power 106,173-182. 189. Horton, F.G., Schultz, D. L., and Forest, A. (1985).Heat transfer measurements with film cooling on a turbine blade profile in cascade. ASME Paper No. 85-GT-117. 190. Doorly, D. J., and Oldfield, M. L. G. (1985).Simulation of the effects of shock wave passir.; on a turbine rotor blade. ASME J. Eng. Gas Turbine Power 107,998-1006. 191. Ashworth, D. A., LaGraff, J. E., Schultz, D. L., and Grindrod, K. J. (1985). Unsteady aerodynamic and heat transfer processes in a transonic turbine stage. ASME J. Eng. Gas Turbines Power 107,1022-1030.

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192. Doorly, D. J., Oldfield, M. L. G., and Scrivener, C. T. J. (1985). Wake-passing in a turbine rotor cascade. AGARD CP-390. 193. Doorly, D. J. (1988). Modeling the unsteady flow in turbine rotor passage, A S M E J. Turbomachinery 110,27-37. 194. Johnson, A. B., Rigby, M. J., Oldfield, M. L. G, Ainsworth, R. W., and Oliver, M. J. (1989). Surface heat transfer fluctuations on a turbine rotor blade due to upstream shock wave passing. ASME J. Turbomachinery I l l , 105- 115. 195. Ashworth, D. A., LaGraff, J. E., and Schultz, D. L. (1989). Unsteady interaction effects on a transitional turbine blade boundary layer. ASME J. Turbomachinery 111, 162-168. 196. LaGraff, J. E., Ashworth, D. A., and Schultz, D. L. (1989). Measurement and modeling of the gas turbine blade transition processes disturbed by wakes. ASME J. Turbomachinery 111, 315-322. 197. Doorly, J. E., and Oldfield, M. L. G. (1986). New heat transfer gages for use on multilayered substrates. ASME J. Turbomachinery 108, 153-160. 198. Doorly, J. E., and Oldfield, M. L. G. (1987). The theory of advanced multilayer thin film heat transfer gauges. Int. J. Heat Mass Transfer 30, 1159-1168. 199. Doorly, J. E. (1988). Procedures for determining surface heat flux using thin film gages on a coated metal model in a transient test facility. ASME J. Turbomachinery 110, 242-250. 200. Ainsworth, R. W., Allen, J. L., Davies, M. R. D., Doorly, J. E., Forth, C. J. P., Hilditch, M. A., Oldfield, M. L. G., and Sheard, A. G. (1989). Developments in instrumentation and processing for transient heat transfer measurement in a full-stage model turbine. ASME J. Turbomachinery 111,20-27. 201. Hilditch, M. A., and Ainsworth, R. W. (1990). Unsteady heat transfer measurements on a rotating gas turbine blade. ASME Paper No. 90-GT-175. 202. Metzger, D. E., and Larson, D. E. (1986). Use of melting point surface coatings for local convection heat transfer measurements in rectangular channel flows with 90 degree turns. ASME J. Heat Transfer 108,48-54. 203. Horvath, T. J. (1990). Aerothermodynamic measurement on a proposed assured crew return vehicle (ACRV) lifting body configuration at Mach 6 and 10 in air. AIAA Paper No. 90- 1744. 204. Ireland, P. T., and Jones, T. V. (1987). The response time of a surface thermometer employing encapsulated thermochromic liquid crystals. J. Phys. E. Sci. Instrum. 20, 1195-1199. 205. Ireland, P. T., and Jones, T. V. (1986). Detailed measurements of heat transfer on and around a pedestal in fully developed passage flow. In Hear Transfer 1986, Vol. 3 (C. L. Tien et aL, eds.), pp. 975-980. Hemisphere, Washington, DC. 206. Ireland, P. T., and Jones, T. V. (1985).The measurement of local heat transfer coefficients in blade cooling geometries. AGARD CP-390. 207. Jones, T. V., and Hippensteele, S. A. (1987). High-resolution heat-transfer coefficient maps applicable to compound-curve surfaces using liquid crystals in a transient wind tunnel. In Dew in Exp. Tech. in Heat Transfer and Combustion (R. 0.Warrington et al., eds.), pp. 1-9. ASME, New York. 208. Baughn, J. W., Ireland, P. T., Jones, T. V., and Saniei, N. (1989). A comparison of the transient and heated-coating methods for the measurement of local heat transfer coefficients. ASME J. Heat Transfer 111, 877-881. 209. Metzger, D. E., Bunker, R. S., and Bosch, G. (1991). Transient liquid crystal measurement of local heat transfer on a rotating disk with jet impingement. ASME J. Turbomachinery 113, 52-59. 210. Wang, Z., Ireland, P. T., and Jones, T. V. (1990).A technique for measuring convective heat transfer at rough surfaces. ASME Paper No. 90-GT-300.

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211. Buck, G. M. (1991). Surface temperature/heat transfer measurement using a quantitative phosphor thermography system. AIAA Paper No. 91-0064. 212. Simeonides, G., Wendt, J. F., Van Lierde, P., Vander Stichele, S., and Capriotti, D. (1989). Infrared thermography in blowdown and intermittent hypersonic facilities. A I M Paper NO. 89-0042. 213. Fort, C., Kageyama, T., and Saulnier, J. B. (1990). Identification of heat flux in the wall of a combustion chamber by solving an inverse conduction problem. In Heat Transfer 1990, Vol. 5. (G. Hetsroni, ed.), pp. 473-478. Hemisphere, Washington, DC. 214. Porro, A. R., Keith, T. G., Jr., and Hingst, W. R. (1991). A laser-induced heat flux technique for convective heat transfer measurements in high speed flows. ICASE 91 Record, IEEE, pp. 146-155. 215. Crowther, D. J., and Padet, J. (1991). Measurement of the local convection coefficient by pulsed photothermal radiometry. Int. J. Heat Mass Transfer 34, 3075-3081. 216. Wang, T., and Simon, T. W. (1989). Development of a special-purpose test surface guided by uncertainty analysis. J. Thermophysics Heat Transfer 3, 19-26. 217. Baughn, J. W., Takahashi, R. K., Hoffman, M. A., and McKillop, A. A. (1985). Local heat transfer measurements using an electrically heated thin gold-coated plastic sheet. ASME J. Heat Transfer 107, 953-959. 218. Baughn, J. W., Hoffman, M. A., Takahashi, R. K., and Lee, D. (1987). Heat transfer downstream of an abrupt expansion in the transition Reynolds number regime. ASME J. Heat Transfer 109, 37-42. 219. Baughn, J. W., Dingus, C. A., Hoffman, M. A., and Launder, B. E. (1989). Turbulent heat transport in a circular duct with a narrow strip heat flux boundary condition. ASME J. Heat Transfer 111, 864-869. 220. Dielsi, G. J., and Mayle, R. E. (1984). A composite constant heat flux test surface. In Heat and Mass Transfer in Rotaiing Machinery (D. Metzger and N. H. Afgan, eds.), pp. 259-268. Hemisphere, Washington, DC. 221. Ligrani, P. M., Subramanian, C. S., Craig, D. W., and Kaisawan, P. (1991). Effects of vortices with different circulations on heat transfer and injectant downstream of a single film-cooling hole in a turbulent boundary layer. ASME J. Turbomachinery 113, 433-441. 222. Oker, E., and Merte, H. Jr. (1978). A study of transient effects leading up to inception of nucleate boiling. Sixth Int. Heat Trans. Con$. Part 1, pp. 139-144. 223. Samant, K. R., Simon, T. W., and Stuart, R. V. (1984). Using thin-film technology to fabricate a small-patch boiling heat transfer test section. In New Experimental Techniques in Heat Transfer (0.C. Jones and N. M. Farukhi, eds.), pp. 33-38. ASME, New York. 224. Carlomango, G. M., and DeLuca, L. (1987). Heat transfer measurements by means of infrared thermography. In Flow Visualization IV (C. Veret, ed.), pp. 611-616. Hemisphere, Washington, DC. 225. Hollingsworth, D. K., Boehman, A. L., Smith, E. G., and Moffat, R.J. (1989). Measurement of temperature and heat transfer coefficient distribution in a complex flow using liquid crystal thermography and true-color image processing. In Coltected Papers in Heat Transfer. (W. J. Marner et a!., eds.), pp. 35-42. ASME, New York. 226. Hollingsworth, D. K., and Watwe, A. A. (1992). Thermal images of the transition from natural convection to nucleate boiling. AIChE Symposium Series 88(288), 10-17. 227. Watwe, A. A., and Hollingsworth, D. K. (1992). Thermal images of the transition from bubble-forced convection to nucleate boiling. AIChE Symposium Series 88(288), 1-9. 228. Kenning, D. B. R. (1992). Wall temperature patterns in nucleate boiling. Int. J. Heal Mass Transfer 35, 73-86. 229. Hoogendoorn, C. J. (1977). The effect of turbulence on heat transfer at a stagnation point. Int. J. Heat Mass Transfer 20, 1333-1338.

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230. den Ouden, C., and Hoogendoorn, C. J. (1974). Local convective heat transfer coefficients for jets impinging on a plate; experiments using a liquid-crystal technique. Proc. 5th Int. Heat Transfer ConJ, Vol. V , pp. 293-297. 231. Goldstein, R. J., and Timmers, J. F. (1982). Visualization of heat transfer from arrays of impinging jets. Int. J. Heat Mass Transfer 25, 1857-1868. 232. Cooper, T. E., Field, R. J., and Meyer, J. F. (1975). Liquid crystal thermography and its application t o the study of convective heat transfer. ASME J. Heat Transfer 97,442-450. 233. Kang, Y., Nishino, J., Suzuki, K., and Sato, T. (1982). Application of flow and surface temperature visualization techniques t o a study of heat transfer in recirculating flow regions. In Flow Visualization 11 (W. Merzkvich, ed.), pp. 71-81. Hemisphere, New York. 234. Simonich, J. C., and Moffat, R. J. (1984). Liquid crystal visualization of surface heat transfer on a concavely curved turbulent boundary layer. ASME J. Eng. Gas Turbines Power 106, 619-627. 235. Taslim, M. E., Bond, L. A., and Kercher, D. M. (1991). An experimental investigation of heat transfer in an orthogonally rotating channel roughened with 45 degree criss-cross ribs on two oppsosite walls. ASME J. Turbomachinery 113, 346-353. 236. Baughn, J. S., Hechanova, A. E., and Yan, X. (1991). An experimental study of entrainment effects on the heat transfer from a flat surface to a heated circular impinging jet. ASME J. Heat Transfer 113, 1023-1025. 237. Baughn, J. W., and Shimizu, S. (1989). Heat transfer measurements from a surface with uniform heat flux and an impinging jet. ASME J. Heal Transfer, 111, 1096-1098. 238. Baughn, J. W., Hoffman, M. A., and Makel, D. B. (1986). Improvements in a new technique for measuring and mapping heat transfer coefficients. Rev. Sci. Instrum. 57, 650-654. 239. Kalinin, E. K., and Drietser, G. A. (1970). Unsteady convective heat transfer and hydrodynamics in channels. Adv. Heat Transfer 6, 367-502. 240. Hippensteele, S. A., Russel, L. M., and Stepka, F. S. (1983). Evlauation ofa method for heat transfer measurements and thermal visualization using a composite of a heater element and liquid crystals. ASME J. Heat Transfer 105, 184-189. 241. Hippensteele, S. A., Russell, L. M., and Torres, F. J. (1985). Local heat transfer measurements on a large, scale-model turbine blade airfoil using a composite of a heater element and liquid crystals. ASME J. Eng. Gas Turbines Power 107, 953-960. 242. Hippensteele, S. A., Russel, L. M., and Torres, F. J. (1987). Use of a liquid-crystal, heaterelement composite for quantitative, high-resolution heat transfer coefficients on a turbine airfoil, including turbulence and surface roughness effects. NASA TM-87355. 243. Hippensteele, S. A., and Russell, L. M. (1988). High-resolution liquid-crystal heat-transfer measurements on the end wall of a turbine passage with variations in Reynolds number. NASA TM 100827. 244. Shewen, E. C., Hollands, K. G . T., and Raithby, G. D. (1989). The measurement of surface heat flux using the Peltier effect. ASME J. Heat Transfer 111, 798-803. 245. Watts, J., and Williams, F. (1984). A technique for the measurement of local heat transfer coefficients using copper foil. Firsf UK National Conference on Heat Transfer, Institution of Chemical Engineers (Symposium Series), Vol. 86, pp. 919-932. 246. McCormick, D. C., Test, F. L., and Lessman, R. C. (1984). The effect of heat transfer from a rectangular prism. ASME J. Heat Transfer 106, 268-275. 247. Florschuetz, L. W., Berry, R. A,, and Metzger, D. E. (1980). Periodic streamwise variations of heat transfer coefficients for inline and staggered arrays of circular jets with crossflow of spent air. ASME J. Heat Transfer 102, 132-137. 248. Florscheutz, L. W., and Su, C. C. (1987). Effects of crossflow temperature on heat transfer within an array of impinging jets. ASME J. Heat Transfer 109, 74-82. 249. Vanden Berghe, T. M., and Diller, T. E. (1986). Heat transfer in a perpendicular

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arrangement of cylinders in steady and pulsating crossflow. In Heat transfer 1986, Vol. 3 (C. L. Tien et al., eds.), pp. 1029-1034. Hemisphere, Washington, DC. 250. Moffat, R. J., and Kays, W. M. (1984). A review of turbulent-boundary-layer heat transfer research at Stanford 1948-1983. A h . Hear Transfer 16,241-365. 251. VandenBerghe, T., and Diller, T. E. (1989). Analysis and Design ofexperimental systems for heat transfer measurement from constant temperature surfaces. Exp. Thermal Fluid Sci. 2, 236-246. 252. Achenbach, E. (1975). Total and local heat transfer from a smooth circular cylinder in cross-flow at high Reynolds number. Inf. J. Heat Mass Transfer 18, 1387-1396. 253. Groehn, H. G. (1986). Integral and local heat transfer of a Yawed Single Circular Cylinder up to Supercritical Reynolds Numbers. In Hear Transfer 1986, Vol. 3. (C. L. Tien et al., eds.), pp. 1013-1018. Hemisphere, Washington, DC. 254. Kraabel, J. S., Baughn, J. W., and McKillop, A. A. (1980). An instrument for the measurement of heat flux from a surface with uniform temperature, ASME J. Heat Transfer 102,576-578. 255. Kraabel, J. S., McKillop, A. A., and Baughn, J. W. (1982). Heat transfer to air from a yawed cylinder. Int. J. Heat Mass Transfer 25, 409-418. 256. Baughn, J. W., Hoffman, M. A., and Lee, D. (1986). Heat transfer measurements downstream of an abrupt expansion in a circular duct with a constant wall temperature. ASME Paper No. 86-WA/HT-100. 257. Baughn, J. W., Hoffman, M. A., Launder, B. E., Lee, D., and Yap, C. (1989). Heat transfer, temperature, and velocity measurements downstream of an abrupt expansion in a circular tube at a uniform wall temperature. ASME J. Heat Transfer 111, 870-876. 258. Baughn, J. W., Hoffman, M. A., and Lee, D. (1987). An instrument for the measurement of the heat flux distribution along a contour of a surface at uniform temperatures. In Deu. Exp. Tech. in Heat Transfer and Combustion (R. 0.Wamngton et al., eds.), pp. 11-17. ASME, New York. 259. Bellhouse, B. J., and Schultz, D. L. (1966). Determination of mean and dynamic skin friction, separation and transition in low-speed flow with a thin-film heated element. J. Fluid Mech. 24, 379-400. 260. Winter, K. G. (1977). An outline of the techniques available for the measurement of skin friction. Prog. Aerosp. Sci. 18, 1-57. 261. Hanratty, T. J., and Campbell, J. A. (1983). Measurement of wall shear stress. In Fluid Mechanics Measurements (R. J. Goldstein, ed.), pp.. 559-615. Hemisphere, New York. 262. Van Heiningen, A. R. P., Mujumdar, A. S., and Douglas, W. J. M. (1976). On the use of hot film and cold film sensors for skin friction and heat transfer measurements in impingement flows. Lett. Heat Mass Transfer 3,523-528. 263. Bernis, A., Vergnes, F., Le Goff, P., and Mihe, J. P. (1977). Une sonde a film chaud pour mtsure locale et instantanee du coefficient de transfert de chaleur dan un lit fuidise gaz-solide. Power Technol. 17,229-234. 264. CretT, R., and Andre, P. (1986). Influence of a periodically recirculating flow on convective unsteady heat transfer. In Heat Transfer 1986, Vol. 3 (C. L. Tien et al., eds.), pp. 1005-1010. Hemisphere, Washington, DC. 265. Rosiczkowski, J., and Hollworth, B. (1991). Local and instantaneous heat transfer from an isothermal cylinder oscillating in a cross-flow. In Fundamental Experimental Measurements in Heat Transfer. (D. E. Beasley and J. L. S.Chen, eds.), pp. 49-56. ASME, New York. 266. Beasley, D. E., and Figliola, R. S. (1988). A generalized analysis of a local heat flux probe. J. Phys. B Sci. Instrum. 21, 316-322. 267. Suarez, E., Figliola, R. S.,and Pitts, D. R. (1983). Instantaneous azimuthal heat transfer coefficients from a horizontal cylinder to a mixed particle size air-fluidized bed. ASME Paper No. 83-HT-93.

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268. Figliola, R. S., Beasley, D. E., and Subramanian, C. (1984). Instantaneous heat transfer between an immersed horizontal tube and a gas fluidized bed. ASME Paper No. 84-HT111.

269. Kawamura, T., Tanaka, S., Macbuchi, I., and Kumada, M. (1987-88). Temporal and spatial characteristics of heat transfer at the reattachment region of a backward-facing step. Exp. Heat Transfer 1, 299-313. 270. Wu, R. L., Lim, C. J., and Grace, J. R. (1989). The measurement of instantaneous local heat transfer coefficients in a circulating fluidized bed. Can. J. Chem. Eng. 67, 301-307. 271. Wu, R. L., Lim, C. J., Grace, J. R., and Brereton, C. M. H. (1991). Instantaneous local heat transfer and hydrodynamics in a circulating fluidized bed. Int. J. Heat Mass Transfer 34, 201 9-2027. 272. Fitzgerald, T. J., Catipovic, N. M., and Javanovic, G. N. (1981). Instrumented cylinder for studying heat transfer to immersed tubes in fluidized beds. Ind. Eng. Chem. Fundam. 20, 82-88. 273. Boulos, M. I., and Pei, D. C. T. (1974). Dynamics of heat transfer from cylinders in a turbulent air stream. Int. J . Heat Mass Transfer 17, 767-783. 274. Hayward, G. L., and Pei, D. C. T. (1978). Local heat transfer from a single sphere to a turbulent air stream. Int. J. Heat Mass Transfer 21, 35-41. 275. Gundappa, M., and Diller, T. E. (1987). Unsteady heat transfer measurements around a cylinder in pulsating crossflow. In I987 ASMEIJSME Thermal Engineering Joint Conference (P.J. Marto and I. Tanasaua, eds.). pp. 629-634. ASME, New York. 276. Campbell, D. S., Gundappa, M., and Diller, T. E. (1989). Design and calibration of a local heat-flux measurement system for unsteady flows. ASME J. Hear Transfer 111,552-557. 277. Gatowski, J. A,, Smith, M. K., and Alkidas, A. C. (1989). An experimental investigation of surface thermometry and heat flux. Exp. Thermal Fluid Sci. 2,280-292. 278. Whitmore, R. A. (1967). History of heat measurement calibration. 22nd Annual ISA Conference Proceedings, Part 11, Paper No. P6-3-PHYMMD-67. 279. ASTM Standard E638-78. (1988). Standard test method for calibration of heat transfer rate calorimeters using a narrow-angle blackbody radiation facility. Annual Book of ASTM Standards, Vol. 15.03, pp. 396-400. 280. Liebert, C. H., and Weikle, D. H. (1989). Heat flux measurements. ASME Paper 89-GT107. 281. Liebert, C. H. (1985). Heat flux sensor calibration. In Structural Integrity and Durability of Reusable Space Propulsion Systems. NASA Conf. Pub. 2381. 282. Stempel, F. C. (1969). Basic heat flow calibration. Instrum. Control Syst. 42(5), 105-107. 283. Bernstein, F. V., and Kinchen, B. (1969). The development and evaluation of a radiative heat source for the calibration of transient and steady-state heat flux sensors. ISA Trans. 8(2), 110-116. 284. Kidd, C . T. (1983). Determination of experimental heal-flux calibrations. AEDC-TR-83-13. 285. Steckler, K. (1992). Personal communication, Building and Fire Research Laboratory, NIST. 286. Kipp, H. W., and Eisworth, E. A. (1981). In-situ calibration of flight heat transfer instrumentation. ASME Paper No. 81-ENAS-131. 287. Alkidas, A. C., and Cole, R. M. (1985). Transient heat flux measurements in a dividedchamber diesel engine. ASME J. Hear Transfer 107,439-444. 288. Alkidas, A. C. (1980). Heat transfer characteristics of a spark-ignition engine. ASME J. Heat Transfer 102, 189-193. 289. Alkidas, A. C. (1987). Heat release studies in a divided-chamber diesel engine. ASME J. Eng. Gas Turbines Power 109, 193-199. 290. Morel, T., Wahiduzzaman, S., Fort, E. F., Tree, D. R., DeWitt, D. P., and Kreider, K. G. (1989). Heat transfer in a cooled and an insulated diesel engine. SAE Paper No. 890572.

T.E. DILLER 291. Assanis, D. N., and Badillo, E. (1989). On heat transfer measurements in diesel engines using fast-response coaxial thermocouples. ASME J. Eng. Gas Turbines Power 111, 458-465. 292. Klell, M. (1989). Measurement of instantaneous surface temperatures and heat flux in internal combustion engines and comparison with process calculations. In Heat and Mass Transfer in Gasoline and Diesel Engines (D. B. Spalding and N. H. Afghan, eds.), pp. 699-710. Hemisphere, Washington, DC. 293. Woschni, G., and Spindler, W. (1988). Heat transfer with insulated combustion chamber walls and its influence on the performance of diesel engines. ASME J. Eng. Power Gas Turbines 110,482-502. 294. Graham, R. W. (1980). Impact of new instrumentation on advanced turbine research. In Measurement Methods in Rotating Components of Turbomachinery (B. Lakshminarayana and P. Runstadler, eds.), pp. 289-302. ASME, New York. 295. Metzger, D. E., and Mayle, R. E. (1983). Gas turbine engines. Mech. Eng. 105(6), 44-52. 296. Simoneau, R. J., Morehouse, K. A., VanFossen, G. J., and Behning, F. P. (1984). Effect of rotor wake on heat transfer from a circular cylinder. ASME Paper No. 84-HT-25. 297. Simoneau, R. J., Hendricks, R. C., and Gladden, H. J. (1988). Heat transfer in aerospace propulsion. NASA-TM-100874. 298. Dunn, M. G., Seymour, P. J., Woodward, S. H., George, W. K., and Chupp, R. E. (1989). Phase-resolved heat-flux measurements on the blade of a full-scale rotating turbine. ASME J. Turbomachinery 111, 8-19. 299. Guenette, G. R., Epstein, A. H., Norton, R. J. G., and Yuzhang, C. (1985). Time resolved measurements of a turbine rotor stationary tip casting pressure and heat transfer field. AIAA Paper No. 85-1220. 300. Guenette, G. R., Epstein, A. H., Giles, M. B., Haimes, R., and Norton, R. J. G. (1989). Fully scaled transonic turbine rotor heat transfer measurements. ASME J . Turbomachinery 111, 1-7. 301. Epstein, A. H. (1989). Short duration testing for turbomachinery research and development. In Transport Phenomena in Rotating Machinery (J. H. Kim and W. J. Yang, eds.), pp. 243-271. Hemisphere, Washington, DC. 302. Simmons, S. G., Hager, J. M., and Diller, T. E. (1990). Simultaneous measurement of timeresolved surface heat flux and freestream turbulence at a stagnation point. In Heat Transfer 1990, Vol. 2 (G. Hetsroni, ed.), pp. 375-380. Hemisphere, New York. 303. Mancuso, T., and Diller, T. E. (1991). Time-resolved heat flux measurements in unsteady flow. In Fundamental Experimental Measurements in Heat Transfer (D. E. Beasley and J. L. S. Chen, eds.), pp. 67-74, ASME, New York. 304. Swisher, S. E., Diller, T. E., and Pierce, F. J. (1992). Time-resolved heat flux measurements in a turbulent junction vortex. In Topics in Heat Transfer (M. Keyhani et al., eds.), pp. 55-63. ASME, New York. 305. Mori, Y.,Uchida, Y., and Sakai, K. (1986). A study of the time and spatial structure of heat transfer performances near the reattaching point of separated flows. In Heat Transfer 1986, Vol. 3. (C. L. Tien et al., eds.), pp. 1083-1088. Hemisphere, Washington, DC. 306. VanFossen, G. J., and Simoneau, R. J. (1987). A study of the relationship between freestream turbulence and stagnation region heat transfer. ASME J. Heat Transfer 109,lO-15.

ADVANCES IN HEAT TRANSFER, VOLUME 23

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

MICHEL QUINTARD L,E.P.T.-ENSAM ( U A CNRS), Esplanade des Arts et Mktiers, Talence, France

STEPHEN WHITAKER Department of Chemical Engineering, University of California at Davis, Davis, California

I. Introduction In the study of heat transfer in two-phase systems, one is immediately confronted with the choice between a one-equation model and a twoequation model. The choice is almost always made on the basis of either intuition or convenience, neither of which provides a completely reliable foundation for the analysis which follows. Intuition certainly motivates the use of a one-equation (single-temperature) model for the study of heat conduction in porous media, provided that the particles or pores are small enough and that the physical properties are close enough. Truong and Zinsmeister [l] found that the one-equation model gave good results for transient processes only when the thermal conductivities of the two constituents did not differ widely. On the other hand, Batchelor and O’Brien [2] were content to employ the one-equation model when the thermal conductivities differed so widely that an asymptotic analysis could be employed to study the point-contact porous medium. In a discussion of the two-equation model (or the two-medium treatment), Kaviany [3] has noted that when there is a significant heat generation in any of the phases the system will be far from local thermal equilibrium. Along the same lines, Nield and Bejan [4] have pointed out that at sufficiently large Rayleigh numbers local thermal equilibrium will break down. 369

Copyright 01993 by Academic Pms, Inc. All rights of reproduction in MY form n ~ e ~ c d . ISBN 0-12-020023-6

370

MICHELQUINTARD AND STEPHEN WHITAKER

Our objective in this chapter is to begin a systematic investigation of what is meant by small enough, significant, sufficiently large, widely different, etc. Alternatively, our objective is to move beyond the order-of-magnitude estimates that currently define the conditions of local thermal equilibrium in order to develop precise and useful constraints. The question associated with the principle of local thermal equilibrium as applied to both multiphase and multiregion transport processes can be stated as follows: Under what conditions can intrinsic phase average temperatures, concentrations, etc. or intrinsic region average concentrations, pressures, etc. be assumed to be equal, thus allowing the use of one-equation models to describe transport processes at the Darcy scale (Whitaker [S]) or at the region average scale (Quintard and Whitaker [S])? The need for two-equation models has long been recognized in chemical engineering (Vortmeyer and Schaefer [7]; Vortmeyer [8]; Schliinder [9]; Froment and BischofT, Ch. 1 1 [lo]; Pereira Duarte et al. [ll]; Glatzmaier and Ramirez [12]) and other areas of engineering science (Ingram [13]; Brusseau et al. [14]; Ramesh and Torrence [lS]; Sozen and Vafai [16]; Powers et al. [17]; Kaviany, Ch. 7 [3]). However, the conditions referred to that control the choice between one- and two-equation models have not yet been clearly identified. Some help is available in terms of a recent study of the principle of local thermal equilibrium (Whitaker [18]) in which a series of constraints are given that must be satisfied in order that a one-equation model be acceptable. Although these constraints provide a consistent framework for the study of one and two-equation models, they are based on orderof-magnitudeanalysis and thus can provide only an indication of the validity of a one-equation model. In the final analysis, one can make a reliable choice between one and two-equation models only by solving exact versions of both models and comparing the results. This is not a trivial task since exact versions of two-equation models are limited to the development of Zanotti and Carbonell [19-211 for passive heat transport and the extension of their work (Whitaker [22]) to include homogeneous and heterogeneous thermal sources. Neither Zanotti and Carbonell [19-211 nor Whitaker [22] provided solutions to the set of closure problems appropriate for general porous media. However, Zanotti and Carbonell [19-2 11 have developed solutions for the general case of laminar flow in a capillary tube, and Kaviany (Ch. 4 [3]), following a development by Koch et al. [23], presents a solution to the closure problem for the special case in which the fluid and the solid properties are identical. The situation at this time is that general solutions of the closure problems are available for neither the one-equation model nor the twoequation model when convective transport is important. The choice between one- and two-equation models is made for a wide variety of processes and an enormous range of length scales. For the process

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

371

of diffusion and reaction in micropore-macropore systems, it has been traditional to assume local mass equilibrium in order to arrive at a oneequation model (Wakao and Smith [24]); however, Carberry [25] has proposed a two-region model for this process and the choice between the two models could be made on the basis of estimates (Whitaker [26]) associated with local mass equilibrium. In the analysis of mass transport and adsorption in chemically heterogeneous soils (Plumb and Whitaker [27]), one is faced with the choice between one- and two-equation models and all too often the choice is made on the basis of convenience rather than physics. For the case of single-phase flow of a slightly compressible fluid in a mechanically heterogeneous porous medium (Chen [28]; Douglas and Arbogast [29]), one is also confronted with the problem of one- and two-equation models and the results from this work will provide some insight into the problem of the flow of gases in fractured reservoirs. The mechanical and/or chemical heterogeneities found in geological systems can have an important influence on the effective transport coefficients associated with the process of two-phase flow (Quintard and Whitaker [30]) and on the form of the macroscopic equations (Quintard and Whitaker [3 13). This transport process is traditionally treated with a one-equation model (Saez et al. [32]); however, recent studies by A1 Hanai et al. [33] suggest that region average equations may be necessary for each distinct region under consideration. This means that, at a minimum, two-equation models will be required for a precise description of some oil recovery processes. In this initial effort to compare exact formulations of one- and twoequation models, we are restricted to pure diffusion processes. Our analysis of the two-phase, transient heat conduction problem provides a two-equation model given by /?-phase:

a-phase:

in which the coupled transport coefficients are equal

K,,

= Ku,

Closure problems are formulated and solved in order to predict all of the transport coefficients in these equations, and our studies indicate that K,, is

372

MICHELQU~NTARD AND STEPHEN WHITAKER

generally on the order of the smaller of either K,,.or K,,. Under certain circumstances convective transport terms appear in the fi- and a-phase transport equations, but the analysis suggests that these terms are of minor importance. When the principle of local thermal equilibrium is valid, the twoequation model reduces to the classic result given by

q

<e>,c,

= V.CK,ff .V
The results of this study have direct application to diffusion-like processes such as diffusion and reaction in micropore-inacropore systems or two-phase flow in heterogeneous porous media. In addition, this work provides a basis for the obvious extension to convective-diffusion processes.

A. THE THERMAL CONDUCTION PROBLEM As a prototype diffusion process, transient heat conduction in a two-phase system offers some attractive advantages. It is the simplest process to study in laboratory experiments, and numerous experimental results are available for comparison with one-equation models (see Nozad et al. [34] for a summary). In addition, at least one experimental study associated with a two-equation model is available (Glatzmaier and Ramirez [l2]). The process under consideration is described by the following boundary value problem (ec ) a? = V.(k,Vq) p B

B.C.1

at

q=T,

in the b-phase at A,,

(2)

at A,,

(3)

in the 0-phase

(4)

T, = F,(rp),

T, = F,(r,), t = 0

(5)

B.C.3

Tp = I&rp, t )

at A,,

(6)

B.C.4

T, = I&,,

at A,,

(7)

B.C.2

n,,.k,Vq

= n,b.k,VT,

aT,

(ec,), at = V.(k,VT,)

I.C.

t)

Here we have used A,, to represent the area of the p-a interface contained in the macroscopic region illustrated in Fig. 1, while A,, and A,, represent the entrances and exits of the b- and a-phases, respectively, for that region. In general, the conditions at A,, and A,, are known only in terms of average quantities (Prat [35,36]); however, experiments can be done in which 5 and T, are specified at the boundaries of the macroscopic system.

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 373

FIG. 1. Macroscopic region and averaging volume for a two-phase system.

A little thought will indicate that the structure of this mathematical problem is essentially identical to the problem of conduction in a two-region model of a heterogeneous porous medium provided the principle of local thermal equilibrium is valid for the conduction process in the w - and qregions illustrated in Fig. 2. While it is not so obvious, the structure of Eqs. (1) through (7) contains the problem of single-phase flow in the heterogeneous porous medium illustrated in Fig. 2 (Quintard and Whitaker [S]), and the general structure of the boundary value problem is similar in many ways to the problem of compressible flow in a fractured porous medium (Chen [28];

FIG. 2. Two-region model of a heterogeneous porous medium.

374

AND STEPHEN WHITAKER MICHELQUINTARD

Douglas and Arbogast [29]). The analysis of transport processes in heterogeneous porous media can be carried out by means of large-scale averaging (Quintard and Whitaker [30]; Plumb and Whitaker [37]), while our analysis of the two-phase heat conduction problem will be based on local volume averaging (Anderson and Jackson [38]; Marle [39]; Slattery [40]; Whitaker ~411). B. AVERAGETEMPERATURE In previous studies of the heat conduction process, the superficial phase average temperature for the /?-phase was defined as I

P

Here V is the averaging volume having a radius r,, shown in Fig. 1, and V, is the volume of the /?-phase contained in V. It has been assumed that this definition provided a macroscopic quantity devoid of small-scale fluctuations provided that the following length scale constraint (Whitaker [42]) was satisfied: la < Yo < L (9) Here L should be thought of as the characteristic length scale associated with average quantities, and this idea is illustrated in Fig. 1. In a recent study of transport processes in ordered and disordered porous media (Quintard and Whitaker [43-44a]), certain difficulties associated with the definition given by Eq. (8) were identified.These difficulties can be illustrated for the case of steady thermal conduction in a periodic system for which the temperature field takes the form

T, = h.rp + fa@@) + T,,

(10) In this representation, h and 5,are constants and fais a periodic function of zero mean over the /?-phase.We can construct the average indicated by Eq. (8) according to 1 (T,> = 7 h.ra dV + &#qo (1 1)

J,,

in which

is the P-phase volume fraction defined by

=vv, Since h is a constant vector, Eq. (11) takes the form

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 375 and we can use the definition of the intrinsic phase average temperature

to extract the following version of Eq. (13): (T,), = h.(r,),

+ Tg0

(15)

We now represent r, in terms of the position vector, x, that locates the centroid of the averaging volume, and the relative position vector ya as indicated in Fig. 3, so that Eq. (15) can be expressed as (TP>@= h.x

+ h.(ya)@+ TBo

(16)

The gradient of the intrinsic phase average temperature is given by V(T,), = h

+ h.V(y,)@

and here we see that V( T,)@will be a constant only if V(yS>@is zero. The properties of (yo), and V(y,)fl have been studied extensively by Quintard and Whitaker [43-44a] for ordered (spatially periodic) and disordered systems. The results provided by that study can be summarized as

FIG. 3. Position vectors associated with the averaging volume.

3 76

MICHELQUINTARD

AND

STEPHEN WHITAKER

follows: (i) In order that (y,), and V(y,)@ make negligible contributions to Eqs. (16) and (17), the constraint expressed by Eq. (9) is not sufficient. For example, in a spatially periodic system the averaging volume consists of a unit cell, Vcell,and the tensor V(yS)@ has the following characteristics: 1

__ Kcell

J

V(ys)@ d V = 0

VCd

V(Y,>@ = O(1 - 6,)

( W (18b)

Under these circumstances the gradient of (q), given by Eq. (17) contains fluctuations at the small length scale that are on the order of V(y,)B itself. (ii) In spatially periodic systems, the derivative of (q), may not exist everywhere. This possibility was discussed earlier by Mls [45] and examples are given by Prat [35]. These difficulties can be avoided by following the work by Made [39], who used a smooth weighting function, m(r), in order to define volume averages as convolution products. The weighting function method is particularly useful when one wants to compare theoretical predictions obtained from macroscopic equations with average values determined on the basis of numerical simulations at the microscopic scale. Examples concerning the thermal conduction problem under consideration in this chapter are provided in Section V. A generalized volume average associated with the temperature in the /?phase can be represented as

in which T@(r)is zero when r locates a point in the a-phase. With this definition, T, must be treated as a generalized function or a distribution (Schwartz [46]; Richards and Youn [47]). We refer the reader to the earlier references as well as to Marle [39,48] and Quintard and Whitaker [43-43b] for more details of the mathematical backgrodnd necessary to implement these ideas, and we will provide a brief summary of the theory of distributions in the next section. The average represented by Eq. (19) should be thought of as a generalized superficial average. This becomes more clear if we consider the special

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

377

weighting function defined by

for under these circumstances Eq. (19) reduces to

<

5)mv

Ix

= (mv

* 5) = Ix

In order to construct a generalized intrinsic average, we define a generalized volume fraction by

in which yo(') is the /&phase indicator function defined by Yp(') =

1, 0,

r locating a point in the /%phase r locating a point in the o-phase

We now define the generalized intrinsic phase average temperature according to B,= m * Tpim * Y p =
(24)

in which it is understood that all averaged quantities are associated with the centroid of the averaging volume. At this time we will require that the weighting function, m(r), have the following properties: H1

m(r)ECm

(254

H2

m(r) has a compact support

(25b)

H3

m(r) is normalized according to

m(r) d K L

=

1

(25c)

3

In subsequent sections we will impose further constraints on the weighting function that are associated with the type of porous media under consideration. In expressing the weighted average by Eq. (19), we have followed the development of Marle [39]; however, other approaches are possible. Anderson and Jackson [38] found it to be convenient to form the weighted average by integrating m(x - r)Tp over the /%phase contained in R3,and a similar approach was employed by Whitaker [49] in a study of dispersion. In their studies of multiphase transport phenomena, Gray and Lee [SO] and Gray

378

MICHELQUINTARD AND STEPHEN WHITAKER

and Hassanizadeh [Sl] used the indicator function defined by Eq. (23). This led them to the integral of y,(r)T,; however, this still demands that one say something about T, in the a-phase in order that yaT, have any significance in the a-phase. At the very least one must say that TsIS bounded in the a-phase in order to use y,T, in the evaluation of the weighted average. Our situation concerning the development of weighted averages for the /?-phase is this: either we form the integral of m(x - r)T, over only the /3-phase contained in R3,or we are forced to make some statement about T, in the a-phase (such as T,is bounded). Our motivation for defining the weighted average by Eq. (19), with T, being zero in the a-phase, is based on Marle’s [39] observation that weighted averages in R3can be viewed as convolution products and this leads to considerable mathematical simplification. It is of some interest to note that the use of weighting functions plays an important role in the development of theories that provide exact correspondence with experimental measurements. Baveye and Sposito [52] and Cushman [53] have pointed out that the instrument weighting function (Maneval et al., [54]) can be used in place of m(x - r) in Eq. (19) to produce an average temperature that is identical to that measured during an experiment. This point of view has been explored by Whitaker [49] for the case of multiple weighting functions; however, there is very little quantitative information currently available concerning the use of instrument weighting functions. 11. Volume Averaging

In order to use the definition given by Eq. (19) with the boundary value problem expressed by Eqs. (1) through (7), we need to provide a few results from the theory of distributions (Schwartz [46]; Richards and Youn [47]) to clarify our use of weighting functions with the method of volume averaging. We begin with a distribution 1(1 associated with any physical quantity in the two-phase system illustrated in Fig. 1, i.e.,

’ {t:: =

in the /?-phase in the a-phase

In the analysis of multiphase transport phenomena, the interfacial region is often sufficiently thin that it can be replaced by a singular surface at which there may be jump discontinuities (Whitaker [SS]). These jump discontinuities are constructed in a manner that is entirely consistent with the physical phenomena under consideration, and we identify the jump in the distribution $ as C$lp = $I7 - $, at A,, (27)

ONE- AND TWO-EQUATION MODELS IN TWO-PHASE SYSTEMS

379

Here A,, locates the singular surface, which may be moving with a speed of displacement to be determined by the governing differential equations, the jump conditions, and the boundary conditions (Crank [56]). From the theory of distributions we have

v* = w)"+ ",,C$l,,~,,

(28)

in which the distribution (V$)" corresponds to derivatives in the usual sense in each phase, i.e., V+,,

(")"

{

= VI)~,

in the 8-phase in the cr-phase

The theorem for the divergence operator takes the form

v.

*

= (V. $Y

+ "pa. [*l,ud,u

(30)

which is analogous to Eq. (28). In both Eqs. (28) and (30) we have used d,, to represent the Dirac distribution associated with the singular surface A,,. This provides

for any function cp. The convolution product between two distributions is given by r

( S * T)(x)= J

S(x - r)T(r) d K

R3

and in our definition of a generalized average temperature represented by Eq. (19), we have used S to be the weighting function, m. Given the restrictions imposed on the distributions with which we are dealing, we have the following relations: S*T=T*S

S*T* U=S*(T* U)=(S* T)*U V(S * T) = (VS * )T = S * (VT)

(334 (33b)

(33c) These theorems are central to the theory of distributions, and once they have been proved, a variety of other results can be obtained with ease. AVERAGINGTHEOREM Given Eqs. (33), one can quickly derive the spatial averaging theorem. We consider the distribution defined by Eqs. (26) through (29) along with the

380

MICHELQUINTARD AND STEPHEN WHITAKER

average expressed by ($>m=m*

$

(34)

From Eq. (33c) we have

V($>m = m * (V$) and the use of Eq. (28) immediately leads us to

(35)

m * (Vll/)U = V($>m - m * (npuC$lpuapu)

(36) At this point we are ready to develop the weighted average forms of Eqs. (1) and (4) and thus produce the Darcy-scale version of the boundary value problem given by Eqs. (1) through (7). We begin by expressing Eq. (1) as

and we form the weighted average, or the convolution product, to obtain ((4cp)p

z)m

= ((V.(kpvq)y)m

(38)

We ignore variations of physical properties within the support of m(r) so that the left-hand side of Eq. (38) takes the form

and we make use of the intrinsic phase average temperature defined by Eq. (24) to obtain

Here we have been able to interchange the convolution product with the time derivative since the /?-a interface is assumed to be fixed in space, and for the same reason we have taken csrnto be independent of time. Substitution of Eq. (40) into Eq. (38) provides

7 = ((V.(ksVT,))U)m

Epm(PCp)p a(Tfl)L

(41)

and we are ready to use the averaging theorem represented by Eq. (36) to obtain Epm(QCp)fi

7 = V.((ksVTB)U)m - m * (np,.kp[(VTp)”]SuSs,)

(42)

ONE- AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

381

Here we note that (VTJ conforms to

,

in the 8-phase in the a-phase

(43)

Thus the definition of the jump given by Eq. (27) allows us to express Eq. (42) as

It is consistent with our treatment of (ecp)pto also neglect variations of k, within the support of m(r) and use the averaging theorem a second time to express Eq. (44) as

In order to obtain a governing equation for (T,)$, we must eliminate the point temperature from this result. We begin this process by introducing the decomposition defined by Tp = Y,(T,>fl, + T ,

(46)

in which is a generalized spatial deviation temperature (Gray [573. When we introduce this decomposition into the second term on the right-hand side of Eq. (45) we obtain

m * (npaTpJpu) =m

* (npu~p(~p>B,hpu) + m * (npuTp6pu)

(47)

and here we are confronted with three problems: 1. To obtain a governing differential equation for (T,)$, we must be able to represent T , in terms of (T,);. This will be achieved in the closure problem that is developed in Section 111. 2. To obtain a solution to the closure problem we require the approximation that m * Tp= 0. 3,. To obtain a local theory for (6):we must be able to remove (5): from the area integral on the right-hand side of Eq. (47). The second and third problems were first studied in detail by Carbonell and Whitaker [58, Sec. 21, and on the basis of a recent study (Quintard and Whitaker [43-44a]) we are now able to deal with this problem in a more satisfactory manner. We begin with a Taylor series expansion of (T,)$ to

MICHELQUINTARD AND STEPHEN WHITAKER

382

obtain from Eqs. (46) and (47) the following two results:

All quantities that are evaluated at the centroid can be removed from the convolution product, thus allowing us to write Eqs. (48) as m

* T, = C(n: * Y ~ ) I < ~ ) ! ++ICm , * (y,y)l.V(~>B,l, + +[m * (ysyy)] :VV(T,>!+I. + + m * T,

(49a)

m * (nsuYs(Ts>B,8,u) = L-m * (~,un,uY,)l(~>!+Ix + Cm * (~,un,ursY)l.V(~,)!+l, (49b) + tCm * (d,un,oY,YY)l :vv
. n-times ...y)]

=

-V[m * (y&y ...n-times.. .y))] (50)

Thus the influence of the terms on the right-hand side of Eq. (49b) depends on the characteristics of m * (ys(y ... n-times . . . y)). Before continuing with our study of Eqs. (49) we need to present some ideas concerning the convolution product and periodic functions. In order that the spatially smoothed equation given by Eq. (45) be devoid of lengthscale fluctuations on the order of I,, we require that the terms on the right-hand side of Eq. (49) contain no fluctuations at the small length scale. For disordered systems Quintard and Whitaker [44, 44al have shown that this can be accomplished by the length scale constraint 1,

< r,,,

(51)

where r,,, is the support of the weighting function m(r). For ordered (spatially periodic) systems the small length scale fluctuations can be removed from m * (y,(y ... n-times . . . y)) by use of a weighting function given by

m = m, * m y* m y

(52)

Here m pis any weighting function having the characteristics identified by Eqs. (25), and m y corresponds to an average over a unit cell in a spatially periodic

ONE- AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

383

system. We define my explicitly by

in which Kel, is the volume of the unit cell. We are now in a position to return to Eq. (10) and to examine the weighted average of that particular temperature field. In order to do so we need a representation in R3,and this is given by

T, = YphJ

+ f y r ) + r&9qo

(54)

in which fp(r)is taken to be zero when r locates a point in the a-phase. Using the weighting function given by Eq. (52), we obtain

(T,), About

* my * my * (yph.r) + m, * my * m y * f p + m, * mv * mv * (rpT,o)

= m,

(55)

f,&) we know

j

ff,(r) dl/,= o

(56)

v8. c s l l

where V&9,cell represents the volume of the j?-phase contained in a unit cell. Given that fa(')is zero in the a-phase, we have

m v * pp=-f1 Kelt

Fp(r)dl/, = o

(57)

v+)

and we can use this result along with Eq. (22) to simplify Eq. (55) to the form (q), = m,

* my * mv * (yph.r) + m, * m y * ( E ~ T ~ o )

(58)

Since ~~~q~ is a constant, we can use

mg * mv * (constant) = constant to represent

(q),

(59)

as

The position vector can be expressed as

r=x+y

(61)

to obtain

mv * (yph.r) = cB,h.x

+ h.mv * ( y p y )

From Quintard and Whitaker [43-44a] we know

(62)

384

MICHELQUINTARD AND STEPHEN WHITAKER

However, my * (yay) is spatially periodic with a zero mean, which leads us to

mv * mv * (Yay) = 0 Use of this result along with Eq. (62) in Eq. (60) yields
(65) in which eamh.x is a constant with respect to the convolution product with m, * m y . This means that Eq. (59) can be employed to show that the generalized average of Eq. (54), using the weighting function given by Eq. (52), provides the relation (Tp)t= b.x -I- 5,

(66) The gradient of the intrinsic weighted average temperature takes the form

V(q)t =h (67) and when we compare these two results with Eqs. (16) and (17) we see that the weighting function given by Eq. (52) can be used to remove all the smallscale fluctuations by the generalized averaging of the temperature field given by Eq. (10) or Eq. (54). For a disordered system this is accomplished simply by imposing the constraint indicated by Eq. (51). We now return to the problem presented by Eqs. (49) and make use of Eqs. (24) and (50) to obtain m*

Tp= -Cm * ( Y ~ Y I I . V ( T-~ tCm ) ~ I *~ ( y a y y ) ~ : ~ ~ ( ~ a ) ~ ~(68a) x-...

m * (npu?'p!Ppu)

=

-(VCm * YpIKqX - (VCm * (YsYII)-V(~)$, (68b) -t(VCm * (YpYY)I): Wq)t - . - *

At this point we consider only Eq. (68b), leaving Eq. (68a) to be discussed in the context of the closure problem which is presented in the next section. For spatially periodic systems, with na specified by Eq. (52), we have m * (yay.. . n-times . . .y) =

{instant,

n=l n>1

and the right-hand side of Eq. (68b) is zero. For disordered porous media, we retain the terms on the right-hand side of Eq. (68b) and make use of Eqs. (22) and (47) in Eq. (45) to obtain

at

&pm(Qcp)p ___

at

= v-Ckfi(~pmV(q)fa- VCm

- $V[m

* (ysyy)]

5)i - . ..) + m * (nau.k,VT,S,,)

:VV(

+ k,m * ( n J y @ u ) l

* (~p~ll-v(T')& (70)

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS To obtain a simplified form of the transport equation for

(q):,

VCm * (Yay)] 4 Epm

385

we require (71)

Quintard and Whitaker [43-44a] have shown that the constraint given by Eq. (71) is satisfied whenever Eq. (51) is valid, and they have shown that Eq. (72) leads to the following length scale constraint:

Here L, is the characteristic conductive length associated with the estimate

while L, is derived from

VEp, = o r e )

(75)

We can now simplify Eq. (70) to obtain

+m * (np,. kpV q6p.J

(76)

provided either of the following conditions is met: (i) The porous medium is spatially periodic and the weighting function is given by Eq. (52). (ii) The porous medium is disordered (see Quintard and Whitaker [43-43b] for a definition) and Eqs. (51) and (73) are satisfied. In addition, one must assume that the Taylor series expansion converges over the support of the weighting function. The last term in Eq. (76) is the interfacial flux, which would typically be expressed as

m * (na,,.kpVTpap,)

= -a,h((T’,>’ - (T,)“)

(77)

where a, is the interfacial area per unit volume and h is the film heat transfer coefficient. In this study we wish to develop a theoretical constitutive equation for the interfacial flux and we wish to determine theoretically the coefficients that appear in that equation. To do so, we use the decomposition given by Eq. (46) and repeat the development given by Eqs. (68) through (76)

MICHELQUINTARD AND STEPHEN WHITAKER

386

’

to obtain

* (ns,.k,V7&,)

= k,C-VEBm.V(q)Bm

+ m * (n,,.V~,8,,)]

(78) In general, the part of the interfacial flux that is proportional to Vepmwill be quite small compared to the total flux; however, we will keep this term for the sake of completeness. Use of Eq. (78) in Eq. (76) leads to the unclosed form of our spatially smoothed transport equation for heat conduction in the 8-phase. m

The analogous result for the o-phase is given by

+ kuC -VEum *V( Tu); + * (nu, *vTuJ,u>I

(80) and we are now in a position to develop the closure problem which will and provide representations of Tfi and T,,in terms of V ( q ) t , V(T,):, (L). 111. Closure

In the closure problem we will assume that either the system is ordered and the weighting function is given by Eq. (52), or the system is disordered and the support for m(r) is constrained by Eqs. (51) and (73). The formulation of the closure problem will provide the correct form of the macroscopic equations given by Eqs. (79) and (80) when the boundary conditions imposed at A,, and A,, [see Eqs. (6) and (7)] have no influence on the local T, and p, fields. For spatially periodic (ordered) media, spatial periodicity can be imposed in the closure problem to produce theoretical values of the effective transport coefficients. For disordered media this procedure represents an approximation that provides excellent agreement between theory and experiment for the one-equation model (Nozad et al. [34]). To obtain a governing differential equation for p,, we use the decomposition given by Eq. (46) in Eq. (l), leading to (@CP)B

a( q>B,+ (ec,), 7

2

= V.(k,V(q)i)

+ V.(k,VT,),

in the 8-phase

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

ONE- AND

387

In order to eliminate the terms involving (T,)k, we note that Eq. (79) can be expressed as

+

k,m

E;,

* (n,, .VT:,S,,)

(82)

and when this result is subtracted from Eq. (81) we obtain the governing differential equation for 7, given by

a?

(ec ) 9 = V.(k, p B

at

V q ) - E;,

V.(kpm * (no,& dBu)) *

- Eg-,l

k, m * (n,,.vq

J,,)

(83)

This procedure can be repeated for other equations in the original statement of the boundary value problem given by Eqs. (1) through (7), thus providing the closure boundary value problem

q = z +
B.C.1. B.C.2.

n,,,.k,VG

at A,,

(84)

at A,,

(85)

-1 - &urn k, m * (n,,.VZ 60,) =fa(rs), T , =f,(r,>, t =0

(86) (87)

at A,e

(88)

at Aue

(89)

= n,,.k,Vz

+ n,,.k,V(T,);

- n,.k,V(T,)b9

I.C. B.C.3. B.C.4.

5

5 q = 9,(’,, t ) , t = ga(ro,0,

Since and ?, are not generally known at the boundaries of the macroscopic system, Eqs. (88) and (89) represent a reminder of what we do not know about the closure problem, and often the same can be said about the initial conditions given by Eqs. (87). It should be clear that we have no intention of solving the closure problem as posed by Eqs. (83) through (89), for if that were the case we would solve the simpler boundary value problem given by Eqs. (1) through (7). With the T, and ;lb fields, one can associate two length scales. The first is the small length scale that influences ?;s and T. through the variation of n,,. The second is the large length scale which influences $ and 7, through the variation of (q); and (T,);. For disordered media we have already

388

MICHELQUINTARD AND STEPHEN WHITAKER

required that these two length scales be constrained by [see Eqs. (151) and ( 173)l

I,, 1, 4 L,

(90)

and we now impose this constraint on ordered media. Given this length scale constraint, we can think of (T,)", (T,)$, V(T,)$, and V(T,)C as slowly ., in some local varying functions and we need only determine Tp and ? representative region such as we have illustrated in Fig. 4. This is possible because the boundary conditions given by Eqs. (88) and (89) will affect the and T, fields only in a region of thickness I, or 1, near the macroscopic boundary of the system. In addition to discarding the influence of B.C.3 and B.C.4, we impose the constraints

5

so that the closure problem can be treated as quasi-steady. Even though the macroscopic heat conduction process may be unsteady, the closure problem will generally satisfy the constraint given by Eq. (91) since 1 ; or 1," will be much, much less than L,2 on the basis of Eq. (90). In the quasi-steady local closure problem we can neglect variations in k, and k, as we did earlier in going from Eq. (44) to Eq. (45). In addition, we can discard the terms given by V . [ m * (ns,k,Tp Spa)] and V . [ m * (nupkupuS,)] so that our local closure problem takes the form

k, VzT, = E& m * (np,.kg V$ S,) B.C.1. B.C.2.

7,

=

T, + ( T U X

-

(T&,

in the P-phase

(92)

at A,,

(93)

at A,, in the n-phase

(94) (95)

n,,.k,Vq = n,,.k,Vz

+ n,,.k,V(TXl - n,,.k,V
k, V2T, = E;: m * (n,,.k, V p ,,)a, Here we have used A,, to represent the interfacial area associated with the local closure problem, and it should be clear from Fig. 4 that the boundary conditions given by Eqs. (88) and (89) are going to be replaced by spatially periodic conditions. In order to justify the neglect of the terms V . [ m * (n,,k, and V . [ m * (n,,k,~S,,)] we note the following orderof-magnitude estimates

5Spu)]

ONE- AND TWO-EQUATION MODELS IN TWO-PHASE SYSTEMS

I+---4

389

__*I

UNIT CELL FIG.4. Local representative region of a two-phase system.

In Eq. (96) the lengths I, I,, and 1, are all order I,. In Eq. (97) the length L results from the fact that we have the divergence of an average quantity, and the factor A,,JV comes from the fact that tn * (n,,k,$tp,,) is a surface area integral divided by the volume of the averaging domain. Since the surface area per unit volume can be estimated as

A,, = 0 V

(i)

We can use Eq. (90) to conclude that

v.Cm * (n,uka$8Ba)l B ka$/l2

(99)

and Eqs. (83) and (86) can be simplified to Eqs. (92) and (95). In the closure problem given by Eqs. (92) through (99, there are three nonhomogeneous terms or sources, i.e., (( T,): - (T,)!), V( and V( q)fa.This suggests the following representations for $ and F,:

z):,

$ = b,,.V(T,)!

+ b,,.VB,- (TI):) + 5 p T,= b,,.V(T~>fa + b,,.Vb, + sX(T,)$ - (T@>8,>+ 5,

in which 5, and 5, are arbitrary functions.

(100) (101)

390

AND STEPHEN WHITAKER MICHELQUINTARD

A. CLOSURE VARIABLES We refer to bPs,b,, ... , s,, and s, as the closure variables and we think of V ( q ) ; , V(T,);, and ((To); - (T,);) as sources in the boundary value problems for the six closure variables. If the sources were constant, we could prove that g, and 5, are zero and we would think of Eqs. (100) and (101) as representing solutions to be obtained by the method of superposition. In the two-equation model of Zanotti and Carbonell [21] the sources were treated as constants in the closure problems; however, an analysis of Eq. (93) will indicate that this simplification should be viewed with some caution. To explore this idea, we expand (T,); and (T,); in a Taylor series about the centroid x indicated in Fig. 3. This allows us to express Eq. (93) as

- Y@,.v
(102)

.*.

Here we have used y,, to represent the position vector that locates points on the p-o interface relative to the centroid. If it were acceptable to treat the source (( T,); - (T,);) as a constant in Eq. (93), it would be acceptable to neglect the terms (103a) Y,a*VL= 0 (lv(~,)L)

(103b)

in which 1 is the characteristic length of the unit cell shown in Fig. 4. However, examinations of Eqs. (100) and (101) will indicate that we are retaining contributions to 5 and Tathat are on the order of 1 V( T,); and 1 V( T,)L . Under these circumstances it seems imprudent to treat the sources as constants. The variation of V(T,);, V(T,)G, and ((T,); - (T,);) was considered by Nozad et al. [34] and they eventually concluded that the variation of the sources was of no importance for a one-equation model. In this treatment we will avoid this approximation and develop the closure problem in greater detail, and we will find that it is difficult to make a priori judgments concerning the importance of the higher-order terms that arise from the representations given by Eqs. (100) and (101). We begin our analysis of Eqs. (92) through (95) and the representations given by Eqs. (100) and (101) by specifying the closure variables in terms of a plausible set of boundary value problems. This will then provide a boundary value problem for t, and 5, which will lead to estimates for these functions. The first problem for b,, and b,, is given by Problem I in the j3-phase

(104a)

at * p a

(104b)

ONE-AND TWO-EQUATION MODELSI N TWO-PHASE SYSTEMS 391

b,, = b,,,

B.C.2.

k , V’b,, = 8;;

baa(’

at A,,

in the a-phase (104d)

c,,

+ li) = baa(rX

(104c)

ba,(r

+ li)

= b,,(r)9

i = 1,2,3

m * b,, = 0, m * b,, = 0

( 104e)

(1040

c,, = m * (n,,.k, Vb,,S,,)

(104g)

(104h) c,, = m * (no,. k, Vb,,J,,) Here we have replaced the boundary conditions given by Eqs. (88) and (89) with the spatially periodic conditions given by Eqs. (104e). These conditions are consistent with ordered systems when Eq. (90) is in effect and they represent an approximation for disordered systems. Nozad et al. [34] found that periodic boundary conditions in the closure problem provided excellent agreement between theory and experiment for disordered systems, thus we view Eqs. (104e) as an acceptable approximation. It is consistent with the use of a spatially periodic model to write m * (n,,kaS,,) = - k, VE,,,, = 0 ( 105) and this permits us to express c,, as =m

* C(n@,*k,Vb,, + n , , k ~ ) ~ , , ]

(106) This represents an important simplification since it allows one to use the boundary condition given by Eq. (104b) to obtain C/IB

c,, = - cap (107) This same type of simplification can be applied to the second boundary value problem which takes the following form.

Problem I1

k, V’b,, = ca;,’ c,, B.C.1.

n,,.k, Vb,, = n,.

k, Vb,,

+ n,,k,,

b,a = boa,

B.C.2.

k, V‘b,, = c;,

bp,(r

+ li)

c,,,

= b&),

in the /3-phase

(108a)

at A,,

(108b)

at A,,

(108c)

in the a-phase

(108d)

i=l,2,3

(108e)

baa@ + l i )

= bao(r),

m * baa = 0, m * b,, = 0

( 1080

392

MICHELQUINTARD AND

STEPHEN WHITAKER

Here we can follow Eqs. (105) and (106) to show that c,

=

- c,o

(109)

The final boundary value problem for the two scalar fields is given by Problem I11

k, V2s, = EiLh,, B.C.l.

n,,.k, Vs, = n,,.ko Vs,,

B.C.2.

s, = 1

+ s,,

k, V2s, = - &;:ha, s,(r

+ li) = s,(r),

m*s,=O,

s,(r

=

(110a)

at A,,

(1 lob)

at A,,

(110c)

in the a-phase

(110d)

i = 1,2,3

(1loel (1 100

m*s,=O

h, = m * (n,,.k,

h,

+ I,) = s,(r),

in the 8-phase

Vs, S,)

- m * (n,,. k, V s, J,,)

Here one can use the boundary condition given by Eq. (110b) to show that h, = h, = a,h (111) in which a, is the interfacial area per unit volume given by m * (J,,). Before moving on to the estimation of (,and t,, we need to comment on the general mathematical structure of Problems I, 11, and 111.If, in Problem I, we assume that c,, = - c,, is known, we see that Eqs. (104a) through (104e) specify b, and b,, to within an arbitary vector. Thus if baa and b, represent solutions to Eqs. (104a) through (104e), we know that b,, + c and b,, + c are also solutions. Since c, = - c,, is an unknown vector, the two constraints on the averages are necessary to specify completely the b,, and b,, fields. For example, one can use m * ,b = 0 to remove the arbitrary constant from the solution forb,, and bas, and one can use m * b,, = 0 to determine the vector cg, =

- c,,.

The next step in our analysis of the closure problem requires that one use Eqs. (100) and (101) in Eqs. (92) through (95) along with the constraints on the closure variables imposed by Problems I, 11, and I11 in order to develop the boundary value problem for t, and 5, which is given by Problem IV B.C.1.

B.C.2.

k, V25, = &,& * (n,,k,.Pt,J,,) ,h n,,k,. Vt, = n,,k,.Vt, + R,,,

5,

=

+ @,

t,,

k, V 2 t , = c;2m

* (n,,k,.Vt,G,,)

+ @,

(112a) at A,,

(112b)

at A,,

(1 12c) (1 12d)

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS If

(P,

393

R,,, and 0,were zero, one could easily prove (Nozad [91]) that

5, = 5 ,

= 0 and the representations given by Eqs. (100) and (101) would be simplified. However, these three terms are not zero and they are given by @p

= 4 k m * (nfi&,-Cb,,.VV(T,)i+ - k,[2Vb,p:VV(

+bp,W2(

+ 2Vb,,:VV(

V(T,);)]~~,)

T,);

T,>B,>+ b,,.V(V2( TUX)

- 2Vs,.(vq)B,

a,,

T,):

bs,.VV(T,);-s,(V(Tg)B,-

- v(T,);)

- s,F2(T,)i - V2(T,)91

(113)

+ b,,.vv(T,);

= n,,k,.Cb,,VV
+ s,(V(T,X - V(T,>L)I - n,,k,.Cb,,.VV(T,)L + b,,.VV(T,>; - s,(V(T,X - V(TU)bm)l + b,,.vv(T,); + s,(V(TuX

0,= GLm * (n,,k,.Cb,,.VV(q?>L

- k,[2Vb,,:VV(

T,)i

+ 2Vb,,:VV(

(1 14)

-V(~,>914,>

5);

+ bUp:V(V2(TpX) + bu,.V(V2('G>b,) + 2vs,.(vi) + s,(v2(T,>: - V2S,)l

(115)

These representations for 0,,R,,, and @, contain terms proportional to V( q ): and V( T,); which one might be tempted to include in Problems I and 11; however, if that is done we are confronted with constraints on the s, and s, fields in Problems I and I1 that cannot be satisfied. This means that terms such as 2Vs,.(V(T,)i - V(T,);) must be included in the boundary value problem for 4, and 5,. This gives rise to a situation in which Problem IV can produce values of 5, and 5, which are estimated to be on the same order as terms in the representations for and T,,. When local thermal equilibrium occurs the one-equation model is valid and the source terms in the boundary value problem for 5, and 5, depend only on second-order terms such as VV( T,):. Under these circumstancesone can show that t, and 5, can be neglected in Eqs. (100) and (101). In order to develop order-of-magnitude estimates of 5, and t,, we begin with Eq. (112a) and estimate the contribution of 0,to 5, as

5

5,

= 0(1,2@,/k,)

(116)

and the same type of estimate based on Eq. (1 12d) leads to

5,

= O(C@U/kU)

(1 17)

These estimates are consistent with the boundary condition given by Eq. (1 12b) and from Eq. (112c) we conclude that t8and 5 , should be the same

394

MICHELQUINTARD AND STEPHEN WHITAKER

order of magnitude. A little thought will indicate that we can estimate 5, and 5, according to 5p5, = O : V V ( T ) , )

+ O(W
- V(T,)3)

(118)

Here I should be interpreted as either I, or I,, while s represents either s, or s, indicates either VV( T,); or VV( Tux. The vector b in Eq. (118) and VV( can be thought of as any one of the four vectors in Eqs. (100) and (101). If we use the above estimate for [ a in Eq. (100) and impose constraints on the basis of analogous terms, we consider 5, to be negligible when

lpbp,:VV(

5);4 bp,:V( 5);

- V(T,)3 The first of these would lead to l,S,(V
@ Sg((5);

- (Z>3

(1 19a) (1 19b)

h 4 1 (120) Lc on the basis of Eq. (74), and this constraint was already imposed by Eq. (90). In order to extract something useful from Eq. (119b), we make use of the estimate

Under these circumstances the constraint given by Eq. (119b) immediately leads to Eq. (120) and one might conclude that t pand t, can be discarded from the representation for T, and T,, on the basis of the constraint given by Eq. (90). On the other hand, one might also wish to impose the constraint.

l,S,(V(T,>fK - V(T,)E) 4 b,p.V(q?); (122) before ignoring the effect of 5, on T,. Since b,, is on the order of I, and sp is on the order of one [see Eq. (lloc)], this constraint leads to

wq>;- V(T,)L e v;

(123) This result is quite similar to one of the constraints associated with the condition of local thermal equilibrium (Whitaker [18], Eq. (3.13)), and this might suggest that t,., can be neglected only when local thermal equilibrium is achieved. In the final analysis it is not the contribution of t pand [, to Tpand F, in Eqs. (100) and (101) that is important, but rather the effect of these functions on the closed form of the volume-averaged transport equations. Thus we retain the general form of the representations forrs and T, and move on to the development of the macroscopic equations for (5);and

a>:.

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 395

B. CLOSEDFORM OF

THE

VOLUME-AVERAGED EQUATIONS

In order to obtain the closed form of Eq. (79) we make use of Eq. (100) for T, and express the result as

in which averaged quantities have been removed from the convolution products on the basis of Eq. (73). Our situation concerning the importance of would appear to be unchanged; however, we are now in a position to indicate how difficult it is to make judgments concerning the importance of higher-order terms. For example, about the term containing ,b we know that

<,

and we also know that the terms containing b,, and Vs, are nontrivial. On the other hand, in the next section we will show that symmetric unit cells give rise to the following conditions: (126a) (1 26b) ( 126c)

The result given by Eq. (125) is consistent with the order-of-magnitude estimate

b,

= W,)

(127)

and the fact that a, = m

* (a),

= O(I - l )

(128) as we indicated earlier in Eq. (98). We say consistent with, because if b,, were a constant vector we would have

m * (n,,b,,6,u) = Cm * (n,,~,,)lb,, = 0 (129) whenever eBmis treated as a constant. This type of result was used earlier in Eqs. (105) and (106).

396

MICHELQUINTARD AND STEPHEN WHITAKER

The ideas presented in Eqs. (125) through (129) indicate that it is extremely difficult to make judgments concerning the influence of t pin Eq. (124). In the introduction we pointed out that intuition is often used to make the choice between a one-equation model and a two-equation model, and at this point we draw on our intuition and adopt the plausible intuitive hypothesis (Birkhoff [59]) that t pand 5, make negligible contributions to the volumeaveraged equations whenever the constraint given by Eq. (90) is satisfied. Under these circumstances we simplify Eq. (124) in the obvious manner and express the result in compact form according to & p r n ( e ca(Ta)’ ,)pat

- V.[Kaa.V(q)f,

+ upu.v(Tu>;

+ Kpu.V(T,);] + uSs.V(Tp)B, - a,hKTp>B, - < T U X )

(130)

Here the transport coefficients are given by Kpp = RpCEpmI + m * (npubap8pu)l

(131)

Kpu = kpm * (npubpudpu)

(132)

up8

= cpp - kpm * (npuspdpu)

(133)

+ kpm * (npuspJpa)

( 134)

up0 = c p u

while the heat transfer coefficient, h, was defined earlier by Eqs. (1log), (110h), and (111). The a-phase transport equation for (T,); is analogous to that for (T,)’ and we list the result as

+ uu,.V(Tp)’ + u u u . v < ~ u ~ ; a,h((T,X

-

B,> (135)

in which the coefficients are defined by Kup = kum * (nugbupdpu) Kuu = kuCEumI uuu

= cuu

+ m * (nupbuu6pu)I

+ kum * (nupsudpu) - kum * (n,*s,Jp,)

(136) (137) (138)

(139) In Eqs. (130) and (135) we see both conductive and convective coupled transport, and these coupled transport terms are usually neglected in the traditional approach to two-equation models. One motivation for discarding uug = cup

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 397 these terms is that they are difficult to measure experimentally; however, we are in a position to calculate these coefficients by means of the closure problems presented in the previous section and this will allow us to make some rational decision concerning the simplification of Eqs. (130) and (135). In Section IILC we demonstrate some important results associated with the closure problems that are summarized here. We define two continuous [see Eqs. (104c) and (108c)l and periodic vectors b, and bll according to

b,=p {t: b,

bll

=

in the /I-phase in the a-phase in the /I-phase in the a-phase

and we use these definitions to prove: Theorem 1

Theorem 2

Theorem 3 If b is the mapping vector for-the closure problem associated with the one-equation model, then

b = b1 + b,,

(144)

We can use these results to demonstrate two important characteristics of the two-equation model. From the definition of K,, given by Eq. (132) we obtain

K,, = k," * (n,b,,d,,) = k,m * (n,,b11~,,) and use of the theorem given by Eq. (142) leads to

(145)

K,, = - k,m * (n,b,d,,) = k,m * (n,,b,6,,) (146) and this provides the following important results concerning the conductive cross-coeflcients in Eqs. (130) and (135) K,, = Ku, (147) This has an appearance of an Onsager reciprocal relation (De Groot and Mazur [60]); however, it is best to think of Eq. (147) as a natural consequence of the heat conduction process described by Eqs. (1) through (7) along with the various constraints that led to the closure problems given in Section 1II.A.

398

AND STEPHEN WHITAKER MICHELQUINTARD

The second important result that we can obtain from the foregoing three theorems is associated with the convective transport terms in Eqs. (130) and (135). We can sum the coefficients in those terms to obtain upp

+ up. + Uup + u,

= cflp - k,m

* (npuSflSp,) + Cpa

+ kpm * (npuSp8pu) + cup - kum * (nupsu8,u) + c, + kum * (n,fls,~p,) (148) This immediately leads to (149) upfl + up0 + Uufl + u, = Cfl8 + Cfla + cup + c,, and on the basis of Eqs. (107) and (109) we obtain the following constraint on the convective transport:

+ U p a + U,p + u,

uflfl

(150)

=0

In terms of the two-equation model, this is not as useful as Eq. (147); however, it does prove that there is no convective transport term in the oneequation model for transient heat conduction. C. SYMMETRIC UNIT CELLS Closure calculations are often carried out using symmetric unit cells (Ryan et al. [61]; Eidsath et al. [62]; Nozad et al. [34]; Ochoa et al. [63]; Kim et al. [64]; Quintard and Whitaker [6]; Quintard [65]); thus it is important to identify the special characteristics of the closure problems given in Section 1II.A for symmetric unit cells. In Section II1.E we demonstrate some important properties of symmetric unit cells that provide the following results. Cp,

= Cap = 0,

upp = up, = ucp =,u

=0

(151)

the first two of which also require that cflpand c,, be zero. Under these circumstances our two-equation model takes the form

- avh('

- (G>fz,>

(1 53)

and this represents an important simplification of Eqs. (130) and (135). Symmetric unit cells can be extremely complex and are not limited to the

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

399

classic case of a uniform array of cylinders (McPhedran and McKenzie [66]; Perrins et al. [67]) or that of a uniform array of spheres (Zick and Homsy [68]); thus it seems likely that many systems could be adequately represented by symmetric unit cells. Highly anisotropic systems may be an exception; however, even in those cases the results given by Eqs. (151) would suggest that convective-like transport terms would be negigible for the diffusive processes studied in this work. Equations (152) and (153), along with the closure problems represented in Section III.A, will form the basis for our studies presented in Sections IV and V. When making use of Eqs. (152) and (153) one should remember that the coupling coefficients, K,, and K,,, are equal as indicated by Eq. (147). Our limited study of the coefficients K,,, K,, = K,,, and K,, suggests that the coupling coefficients are on the order of the smaller of K,,, and K,,, and this means that the coupled conductive terms should not be omitted in any detailed two-equation model of conductive or diffusive processes. The mathematical problems associated with solving transport equations that are coupled by conductive transport are discussed in Section V. 1. The One-Equation Model The one-equation model is valid whenever the two temperatures, ( K ) ; , are sufficiently close to each other so that we can write

(q); and

Under these conditions one can add Eqs.(l30) and (135) and make use of Eq. (150) to obtain (Q>mcp

7 - V.[Keff.V(T),]

(155)

Here ( Q ) , is the spatial average density defined by ( ~ > m= EBmQp

+ EumQu

( 156)

and C , is the mass fraction weighted heat capacity given by

The effective thermal conductivity tensor can be expressed as

and one can use previous representations for K,,, K,,, and K,, to obtain

400

MICHEL QUINTARD AND STEPHEN WHITAKER

On the basis of Theorem 3 this takes the form

K,,

=

(&,As + ~,,k,Y+

(k, - k,)m

* [naub~~,l

( 160)

which is identical to the result given by Nozad et al. [34] for the one-equation model effective thermal conductivity tensor. In Section V we will compare solutions of Eq. (155) with solutions of Eqs. (152) and (153) in order to develop a better understanding of what is meant by suficiently close. Eventually this will allow us to develop precise constraints with the condition of local thermal equilibrium. 2. Special Characteristics of the Closure Problems

In Section 1II.B we presented the important result K,u

= Ku/J

(161)

which results from the three theorems given by Eqs., (142), (143), and (144). In this section we wish to prove those three theorems, and we begin by recalling the definition of the continuous, periodic vector field b, given by

{:

in the P-phase in the a-phase

( 162)

b1 = Here b, is a distribution in the same sense as J/ in Eq. (26), and with this distribution we can express the closure Problem I [see Eqs. (104)] as Problem I V.(kVb,) = - V.(y,kpI)

m * (Ysbl) = 0,

+ m * (rub,) = 0

(163) ( 164)

Here ya and y, are the p- and 0-phase indicator functions respectively, and k is the distribution defined by k = { z

in the /3-phase in the a-phase

In order to express the second closure problem [see Eqs. (108)] in compact form, we let b,, be the continuous, periodic vector field given by

1I'

I:{ =

in the P-phase in the a-phase

This leads to the following version of Eqs. (108).

O N E - AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

401

Problem I1

V.(kVbIl) = - V.(y,k,I) -:&(

* ( ~ p 4 J= 0,

m

m

):

--

-c

(167)

@ ,

* ( ~ , b d= 0

(168)

We continue this procedure with the third closure problem [see Eqs. (1 lo)] and define s to be a continuous, periodic scalar field according to

.=r"l + s,

1

in the B-phase in the cr-phase

(169)

This leads to the compact form Problem I11

V.(kVs) = - a $

m

* ( Y ~ s=) 0,

(170) m * (yus) = turn

(171)

At this point we return to Problem I1 and define a vector field f as

f = -(2)bI1 Since k, and k, are considered to be constants, we can used Eq. (172) in Eq. (167) to obtain

k V.(kVf) = 3 V.(yuk,I) k,

+

(173)

in which the vector c is given by [see Eq. (log)]

The first term on the right-hand side of Eq. (173) can be expressed as

ks V.(y,k,I) kU

= V.(y,k,I)

= - V.(y,k,I)

thus the boundary value problem for f is given by

(175)

MICHELQUINTARD AND STEPHEN WHITAKER

402

In order to prove that f is equal to b,, as indicated by Eq. (142), we need to prove that c is equal to c,, so that Eqs. (163) and (164) can be compared with Eqs. (176) and (177) to prove that

f=b,

(178)

From Eqs. (174) and (108g) we have

c =k, c,, = k, - [m * n~a.kgVb~,c&,)l ( 179) ka k, and here we must be very careful to note that b,,, like T,, is defined in the usual sense in the 8-phase and is zero in the o-phase. Equation (179) can be arranged in the form c = m * n,,.k,V

(

[“.I> f b,, ,a,

and on the basis of Eqs. (166) and (172) we have

-

k),, kjnu

in the 8-phase

in the a-phase

Under these circumstances, Eq. (180) can be expressed as c = - m * (n,,.k,Vf,S,,)

( 182)

and our boundary value problem for f takes the form

V.(kVf) = - V.(y,k,I) -

(”- $)m ,E

* (n,,.k,Vf,G,,)

(183)

* (raf)= 0,

m * (y,f) = 0 ( 184) Here we have used Vf, to represent the gradient off evaluated in the 8-phase We now return to Problem I and use Eqs. (104g), (104h) and (107) to express c,, as

m

c,,

=

- m * (n,,.k,Vb,,&)

(185)

On the basis of Eq. (162), this permits us to write Eqs. (163) and (164) as Problem I

V.(kVb,) =

- V.(y,k,I) -

(e e>-

* (n,,.k,Vb,a6pu)

(186)

O N E - AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

403

Here we have used Vb,, to represent the gradient of b, evaluated in the 8-phase. If the solution to Eqs. (183) and (184) is unique, we see that f is equal to b, and Eq. (178) has been proved. This means that Theorem 1, given by Eq. (142), has been proved. An alternate route to this result is available. It consists of multiplying Eqs. (108) by k,/k, and using the result along with Eqs. (104) to prove that

These results can be used with Eqs. (104g), (104h), (107), (lOSg), (IOSh), and (109) to easily prove Theorem 2, which was given as Eq. (143). We now move on to the proof of Theorem 3, which begins by adding Eqs. (163) and (167) to obtain

+

V.(kVb) = -V.(kl)

(&rm i.> 2-

C,,

Here b is defined by

b = bl

+ b,!

( 190)

and our objective is to prove that the boundary value problem for b gives us the previously derived result for the one-equation model of heat conduction. In Eq. (189) the constant vector C,, is given by in which Vb,, and Vb,,, have been used to represent the gradients of b, and bl in the a-phase. It may be of some help to remember that the terms in Eq. (191) originate from the last term in Eq. (44) with the gradient of T, given by Eq. (43). We can also express Eq. (191) as CUD= m * (n,,.k,Vb,G,,)

=

-m

* (n,,.k,Vb,G,,)

=

-C,,

(192)

The boundary value problem for the periodic vector b consists of Eq. (189) and the following conditions derived from Eqs. (164) and (168):

m * (7,b) = 0,

m * (7,b) = 0

(193)

In the classical sense, Eq. (189) can be written in an expanded form given by

k,V2b, = &i:m B.C.l. B.C.2.

np,.k,Vb,

+ n,,k, b,

* (n,,.k,Vb,G,,)

= n,.k,Vb,

+ n,,k,

= b,

k,V2b, = &:,m

* (nUp.k,Vb,G,,)

in V,

(194a)

at A,,

(194b)

at A,,

(194c)

in V,

(194d)

404

MICHELQUINTARD AND STEPHEN WHITAKER

This provides the boundary value problem for the mapping vectors defined by

T, = b,.V(

T),

T,,= b,.V(

T)

(195)

which are associated with the principle of local thermal equilibrium and the one-equation model of heat conduction. Nozad et al. [34] developed arguments indicating that the terms on the right-hand side of Eqs. (194a) and (194d) could be neglected, and for the case of symmetric unit cells their arguments are confirmed by the results given in Eqs. (151). The results given by Eqs. (151) are of considerable importance since they allow us to simplify Eqs. (130) and (135) to Eqs. (152) and (153), and they provide an important simplification of closure Problems I and 11.At this point we want to prove the results for symmetric unit cells given by Eqs. (151). As indicated in the previous paragraphs, closure problems for diffusive or conductive transport have the general form

in which k, cp, and $ are distributions and @(g) for the heat conduction problem takes the form

This last expression can also be written as

or as

In addition to the constraints placed on g by Eqs. (196) and (197), we note that g is spatially periodic and has specified phase averages associated with the p- and a-phases. Equations (196) and (199) apply to general spatially periodic systems, and in order to develop results for symmetric unit cells we consider a twodimensional system for simplicity. Our results can then be extended to three dimensions. To be specific, we consider a unit cell which is symmetric with respect to the x,-axis. This leads to k(x,, x2) = 4x1, -%), $(XI,

cp(x,,

x2)

x2) = $(XI¶ -x2)

= d x , , -x,),

(2oo)

In two dimensions, Eq. (196) for the x,-component is written explicitly as

ONE- AND TWO-EQUATION

MODELSIN TWO-PHASE SYSTEMS

405

At this point we make the change of variables given by Yl

= x1,

Y2 = -x2

(203)

and introduce a new notation defined by Bl(Yl9

Bz(Y13

Y2)

= Sl(X1,

YZ) = -92(x1,

-x2)

-x2)

(204) (205)

Using the symmetry condition identified by Eq. (200) along with the definition provided by Eqs. (203) through (205), we can express Eqs. (201) and (202) as

406

MICHELQUINTARD AND STEPHEN WHITAKER

Here we should note that symmetry with respect to the xz-axis leads to Y&I,

(208)

XZ) = YAY,, Y2)

and that the conditions concerning periodicity and average values for g are also applicable to g on the basis of Eqs. (204) and (205). This means that the boundary value problem for jj is identical to the boundary value problem for g, and in terms of the components this leads to Bl(X1, x2) = c7l(Y,, Yz) = Bl(X1, -xz) g2(x1, XZ) = BAY,, Yz) =

-&1,

-xz>

(209) (2 10)

A key idea here is that the Laplacian operators in Eq. (196) are symmetric while the gradient operators are skew-symmetric,and it is this characteristic that leads to Eqs. (209) and (210) for systems that are symmetric in x2. The closure problem for s given by Eqs. (170) and (1 71) (see also Eqs. 110) differs only by the fact that q(r) = 0, hence the skew-symmetric part in the equation is suppressed. Thus we can use the previous development to demonstrate that s(x1, x2) = d X 1 , -xz)

(211) for systems that are symmetricin xz,and the obvious symmetry condition for s(xl, xz) can be developed for unit cells that are symmetric with respect to xl. Without providing further details we list the results for a two-dimensional unit cell with symmetry with respect to both x1 and x2. Unit cell symmetric with respect to xz:

Unit cell symmetric with respect to x 1: brxl(X1,

x2) = - blX1(- x19 xz)

blx2(X1,x2) = b1x2(-X19 XZ)

(213a) (213b)

(213c) 4x1, x2) = s(-x1, XZ) We can now consider the consequences of symmetric unit cells on some of the terms that appear in the averaged equations. For skew-symmetric functions the averages will be zero and this is also the case when symmetric operators are applied to skew-symmetric functions. In the averaged equations we find

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

407

that the following terms are zero for symmetric unit cells: c,,, c,,, UB,, u,, Up,, u, and this means that the convective transport terms in the /?- and a-phase equations are the consequence of nonsymmetric unit cells. In addition to certain terms being zero in the averaged equations, all skewsymmetric functions associated with symmetric unit cells have zero values on the boundaries of the unit cells. This provides an important simplification in the numerical solution of the closure problem (Quintard and Whitaker [6)). C,,?

Cpal

IV. Prediction of the Effective Transport Coefficients In this section we describe the numerical method used to solve the closure problems presented in Section III.A, and we present selected results for stratified systems and uniform arrays of cylinders. Analytical results are given for stratified systems and a finite-volume numerical method is used for the nodular systems. In Section V we will make extensive use of the methods described in this section; however, the discussion of numerical methods in that section will be restricted to problems associated with the coupled nature of the macroscopic equations. The effective transport coefficients in the macroscopic or volume-averaged equations are determined by the solution of pore-scale closure problems for the two-phase system illustrated in Fig. 1 (Ryan et al., [61]), or by the solution of Darcy-scale closure problems for the two-region system illustrated in Fig. 2 (Quintard and Whitaker [6]). It is of some importance to recognize that the essential features of a unit cell are process dependent and parameter dependent. This means that the features that are necessary for the satisfactory prediction of an effective thermal conductivity tensor are not necessarily the same as those required to predict the Darcy’s law permeability tensor. This process and parameter dependence is nicely illustrated in terms of the effective thermal conductivity for a two-phase system. If the ratio of conductivities, k,/k,, is less than 10, any reasonable model will satisfactorily predict the effective thermal conductivity (Kaviany [3, Ch. 31). On the other hand, for values of k d k , greater than 100 the process becomes increasingly sensitive to the nature of the particle-particle contact (Batchelor and OBrien [2]; Shonnard and Whitaker [69]). In our studies we have not included the effects of particle-particle contact, thus our results have validity only when k,/k, is less than 10 or the system under consideration is a suspension of nontouching particles. We recall that the three closure problems under consideration can be expressed in compact form when the closure variables are viewed as distributions. The first of these problems is given by

408

MICHEL QUINTARD AND

STEPHEN WHITAKER

Problem I

+ (” -*),,

V.(kVb,) = -V.(y,kl)

E,,

bI(r + It) = bl(r),

i = 1,2, 3

m * (Ypb,) = 0,

m

(214a)

E,m

* (Yub,) = 0

(214b) (214c)

Here b, is the continuous vector field defined by in the j3-phase in the a-phase

(214d)

in the 8-phase in the a-phase

(214e)

and k is the distribution defined by

k=(: The constant vector cupis given by

in which Vb,, represents the gradient of b, evaluated in the a-phase and Vb,, is used to represent the gradient of b, evaluated in the j3-phase. In Eq. (214b) we have used li to represent the three lattice vectors needed to describe a unit cell. For the mapping vectors b,, and b,, the closure problem can be expressed as Problem I1

V.(kVbII) = -V.(y,kl) b,,(r m

+ li) = bIl(r),

* (Y,b,l)

-

(215a)

i = 1, 2, 3

= 0,

m

* (YUhI) = 0

(215b) (215c)

Here b,, is the continuous vector field defined by

b,,

in the j3-phase in the a-phase

(215d)

and the constant vector c,, is given by

The boundary value problem for the scalar functions, s, and s,, determine the heat transfer coefficient is given in compact form as

that

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

409

Problem 111 (216a)

s(r

+ li) = s(r),

m * (ras>= 0,

i = 1, 2, 3

(216b)

m * (yes) = E,,

(216c)

In this problem s is the continuous scalar field defined by

.=(SI s,

+1

in the 8-phase in the o-phase

(216d)

and the constant a,h is given by a,h = m * (n,,

.k,Vs,G,,)

= -m

* (nu,. k,Vs,G,,)

(216e)

In this derivation of the closure problems it is assumed that the essential features of the closure variables can be determined using a spatially periodic model of a porous medium. This means that Eqs. (213a), (215a), and (216a) must be invariant to a transformation of the type r = P + li, and this requires that k be spatially periodic when the geometry is spatially periodic. Normally this condition is satisfied by assuming that k, and k, are constant. For disordered systems, one must think of the spatially periodic closure problems as approximations that have provided reasonably good agreement between theory and experiments for diffusion (Ryan et al. [61]; Quintard [65]), heat conduction (Nozad et al. [34]), and single-phase flow (Barrkre et al. [70]), while the agreement between theory and experiment for dispersion (Eidsath et al. [62]; Edwards et al. [71]) leaves something to be desired. Once one has accepted the use of a spatially periodic model for the purposes of completing the closure, some specific simplifications can be made concerning the weighting function. In an analysis of spatially periodic systems, Quintard and Whitaker [43-44a] demonstrated that the weighting function must be of the form

m

= m g * m,

*my

(217)

in which m, is a general weighting function having a compact support with m, E C", and satisfying the normalized condition

The weighting function, m,, is defined by

410

MICHELQUINTARD AND STEPHEN WHITAKER

and m y * m y produces what is known as a cellular average. Since the average of any periodic function, i,hp, over a unit cell is a constant, it follows that

Thus, in the closure problems and in the calculation of the effective transport coefficients,but not in the determination of (q); and (T,)", one can replace m with my.Under these circumstances the effective transport coefficients can be expressed as

In obtaining these representations we have used the following results from Section III.C.2:

and when symmetric unit cells are used to determine the effective transport coefficients the above relations are considerably simplified.

A. SYMMETRIC UNIT CELLS When the unit cell used to solve the three closure problems is completely symmetric, the analysis presented in Section III.C.2 provides the following simplifications Cpu

= cup = 0,

upp = U,p = upu = , u

=0

(231)

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS and the macroscopic equations take the form

The presence of the coupled flux terms involving K,, and K,, makes these equations more difficult to solve than the model equations proposed by Glatzmaier and Ramirez [121, and the mathematical properties of Eqs. (232) and (233) are discussed in detail in Section V. In this article we focus our attention on the determination of the coefficients that appear in Eqs. (232) and (233) and we present no solutions for nonsymmetric unit cells. In Section 1V.A analytical solutions are provided for stratified systems, and a general algorithm for the solution of the closure problems is given in Section 1V.B. In Section 1V.C numerical results are provided for a two-dimensional array of cylinders embedded in a continuous phase.

B. STRATIFIED SYSTEMS Although stratified systems are not commonly associated with the unconsolidated porous medium suggested in Fig. 1, they are important in the study of transport processes in heterogeneous porous media, especially those encountered in geological formations (Quintard and Whitaker [31]). Since the mathematical problem represented by Eqs. (1) through (4) is identical to the problem of heat conduction in a heterogeneous porous medium when local thermal equilibrium is valid at the Darcy scale, we can obtain some useful results for geological formations by examining the stratified system illustrated in Fig. 5. This system belongs to the set of symmetrical unit cells since it is invariant along the y-axis and it is symmetric with respect to the planes at x = +la f n(I, + I,) or x = I, + $1, k n(l, l,). To determine the effective transport coefficients for a stratified system, we need to solve Problems I, 11, and 111 as given by Eqs. (214), (215), and (216).

+

Problems I and I1 There are no source terms in Problems I and I1 for the y-component of b, and b,,; thus we have

j.b, = 0,

j.b,, = 0

(234)

412

MICHELQUINTARD AND STEPHEN WHITAKER

FIG. 5. A two-phase system.

and this quickly leads to j.Kpp.j = Eaks

(235a)

j.Kpa.j = j.Kap.j = 0

(235b)

j.Kaa.j = eaka

(235c)

In addition, the off-diagonal terms of these tensors are all equal to zero. The solution of Problems I and I1 for the x-component of b, and b,, is straightforward since i.b, and i.b,, are linear in each phase. The constants in these linear relations are given by the boundary conditions and the constraints on the averages given by Eqs. (214c) and (215c). The effective transport coefficients determined from the x-component of bI and bl, are given by i*Kpp*i= kp

Eirn(ka/kp) Earn

+ Eprn(ka1kp)

(236a) (236b) (236c)

The dependence of these components on cam and kalkp is illustrated in Figs. 6 through 8. In order to determine the heat transfer coefficient given by Eq. (228) we need to solve Problem 111. Problem I11 Since this problem is invariant along the y-axis, s is a function only of x. Integration of Eq. (216a) gives s in terms of two second-order functions of x. The constants of integration, as well as a&, are determined by the boundary

ONE-AND TWO-EQUATION MODELS IN TWO-PHASE SYSTEMS 413 “Kpp

.a$

08

0.7

-

0.6

-

0.5

-

04

-

0.3 -

0.2

-

0.1

-

0

OM)(

001

01

10

1

IW

1wo

FIG.6. Effective transport coefficient i.K66.i for a stratified system.

0s

04

03

02

01

0

0.WI

0.01

0.1

10

1

ka’

lw

Iwo

$3

FIG.7. Effective transport coefficient i.K6A for a stratified system.

414

MICHELQUINTARD

AND STEPHEN WHITAKER

__--

, /

/

/

/'

0001

001

1

01

10

FIG. 8. Effective transport coefficient i.KJ

1w

low

for a stratified system.

conditions and the constraints on the averages indicated by Eqs. (216c). The solution for a,h is given by

a,h(l, + 4J2 12(kdk,) (237) &urn + &pln(ku/k,> k, in which the area per unit volume for the stratified system shown in Fig. 5 is given by

w, +

(238) The dependence of the heat transfer coefficient on the thermal conductivity ratio, k,/k,, and E,,,, (which is referred to as the porosity) is illustrated in Fig. 9. From this study of stratified systems we can draw two conclusions about the heat transfer coefficient, or more importantly the product a,h: a, =

(1) a,h is a linear function of (la

+

(,)-2.

(2) For given values of k, and k,, a,h increases with decreases with tBmif k, > k,.

+,,, if k, < k,

and

For stratified systems the p- and a-phases play the same role in the transport process; however, this will not be the case for the two-dimensional array of cylinders studied in the next section. In that case the fl-phase will be continuous and the 0-phase will be discontinuous; thus the two phases will influence the transport process in a different manner.

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

60

-

40

-

30

-

20

-

I0

-

0 001

0

or

D l

3

(0

100

415

too0

FIG. 9. Heat transfer coefficient as a function of the thermal conductivity ratio for a stratified system.

C. NUMERICAL METHODS For a two-dimensional array of cylinders (the a-phase) imbedded in a continuous /&phase, the closure problems given Eqs. (214), (215), and (216) must be solved numerically. The choice of a particular numerical algorithm is somewhat arbitrary, and Ochoa (1988) has presented an extensive comparison between finite-difference, finite-element,and boundary integral methods. Here we present a simple algorithm based on a description of two-phase systems in terms of finite volumes. We consider the two-dimensional unit cell shown in Fig. 10 and in our numerical algorithm this system is replaced by the grid-block description illustrated in Fig. 11. This method has been used by Quintard [65] to solve the closure problem associated with diffusionin porous media and the results are in excellent agreement with other theoretical calculations (Kim el al. [64]) and with experimental data. In addition to being relatively simple, this approach is especially well suited for use with data obtained from image analysis (Ehrlich et at. [72]). Although the numerical calculations presented in this section are for the system represented in Fig. 10, it is important to note

416

MICHELQUINTARD AND STEPHEN WHITAKER

FIG. 10. Unit cell for a two-phase system.

that the algorithm introduced here is valid for any kind of geometry including nonsymmetrical unit cells. Referring to Fig. 11, let ( x i , y i ) be the coordinates of the centroid of the grid-block (i, j) which has the uniform size Sx Sy. The numerical procedure will be illustrated for the x-component of the vector b, which we denote by

i.bI = b

(239)

The equation for b can be extracted from Eq. (214a) as

a(k g) + $ ( k ; ) -axaf + =

gc

ax

FIG. 11. Grid block representative for a two-phase system.

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

417

in which c is the constant i.co8to be determined. For symmetric unit cells, c is zero. Integrating Eq. (240) over the block (i, j), we use the classical approximations to obtain

kg)i.

= sx s y QijC j- 112 + e

Here E has been used to indicate on which side of the interface the function is to be evaluated. The key step at this point is the discretization of the four terms on the left-hand side of Eq. (241), and we begin with the first term on the left-hand side. In the neighbourhood of an interface between two grid-blocks, we can use either Eq. (214a) or Eq. (240) to obtain

in which

[fli+

is the jump in f defined by

we can approximate Eq. (242) by

and this provides an estimate of the value of b at the interface between the grid-blocks (i,j) and (i 1, j ) . This can be used to obtain the following result for the first term in Eq. (241):

+

(245) This procedure can be repeated for the other three terms on the left-hand side of Eq. (241) in order to obtain an algebraic representation of Eq. (240) for each grid-block in the unit cell. These results, along with the periodicity conditions, form a linear set of equations. If (b} = {bl, b2, ..,bi,j , . . . } is the vector corresponding to the natural ordering of the b-field discrete values, the structure of the matrix, A = {ui,j}, appearing in the linear system is shown in Fig. 12. Because of the sparseness and the relative complexity of this matrix due to periodic boundary conditions, it is particularly appealing to solve the

418

MICHEL QUINTARD AND

STEPHEN WHITAKER

FIG. 12. Matrix structure (8 x 6 nodes; crosses indicate nonzero values).

associated linear system by using an iterative technique. We chose a conjugate gradient method taking into account the particular structure of the sparse matrix (Hesteness and Stiefel [73); Golub and Van Loan [74]; Gambolati and Perdon [75]; Van der Vorst and Dekker [76]). The choice of a conditioning matrix is particularly sensitive in the implementation of the conjugate gradient method. We tested conditioning with the diagonal matrix D = {q,, S,}, and with the incomplete Choleski decomposition. Numerical tests showed that both choices are efficient, although conditioning with the incomplete Choleski decomposition gives good convergence in less iterations than conditioning with the diagonal matrix. However, in terms of computational time required to achieve a given accuracy, conditioning with the diagonal matrix proved to be slightly more efficient than conditioning with the incomplete Choleski decomposition. Whatever the linear solver used, the solution procedure can be described as follows: 1. The linear system of equations was solved for a given constant, c. The solution involves an unknown constant of integration which is determined in step 2. 2. Let b’ be the b-field obtained at step 1; then the b-field is modified according to

b=b’-m*b’

(246)

in order to satisfy the condition m*b=O which represents the sum of Eqs. (214~).

(247)

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 419

3. The function 6(c) defined by &c) = m

* (y,b)

(248)

is calculated. 4. Iteration with respect to c is carried out by repeating steps 1 through 3 in order to satisfy the condition 6(c>= rn * (y,b) = o

(249) The iteration on c was performed using Brent’s method [77], which requires a small number of iterations to achieve reasonable accuracy. In the case of symmetrical unit cells, no iterations were performed since in that case c is equal to zero. The computation of s on the basis of Eqs. (216) is performed using the same algorithm described in the preceding paragraphs. The function f(r) is set equal to zero and Eq. (248) is replaced by ri(c = a,h) = E,,

Examples of the fields obtained for the unit cell shown in Fig. 10 are presented in Figs. 13 and 14.

FIG. 13. Three-dimensionalplot of (b,)&

as a function of x and y (egn = .727).

MICHELQUINTARDAND STEPHEN WHITAKER

420

__

FIG. 14. Three-dimensionalplot of s as a function of x and y ( E ~ , , ,= ,727).

The iterative method described above provides a convenient and powerful technique for the solution of the closure problems described by Eqs. (214), (215), and (216), and it can be extended to achieve a direct solution for the coefficients cop,cpoyand a,h. Using Problem I as an example, we decompose the distribution bl according to

*

bl = bp

+

(251)

$1

and we require that bp satisfy the problem given by Problem Ia V.(kVbp) = -V.(yakl)

bp(r + li) = bf(r), VI

* (7,bp)

=0

(252a)

i = 1,2, 3

(252b) (252c)

When we substitute Eq. (251) into Eqs. (214a) and (214b) and the first of Eqs. (214c), we find the following problem for @,.

ONE- AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

421

Problem Ib (253a) (253b) (253c) This suggests a solution for

+,of the form *I

(254)

= BI.C,@

and substitution of this result into Eqs. (253) gives the following boundary value problem for the tensor B,. Problem Ic (255a) B,(r

+ li) = B,(r),

i = 1,2, 3

(255b)

(255c) m * (YUB,) = 0 One can follow the type of analysis presented by Ryan et al. [61] to prove that the off-diagonal components of B, are zero. Under these circumstances Eq. (254) takes the form *I

(256)

= BICu/l

in which B, is a scalar determined by Problem Id

V.(kVBI) =

-!5

B,(r

s)

--

(E:m

+ li) = B,(r),

i = 1, 2, 3

(257a) (257b)

The representation given by Eq. (251) now takes the form in which b, satisfies Eqs. (214a), (214b), and the first of Eqs. (214c). In order to determine the constant vector cub, we impose the first of Eq. (214c) and arrange the results to obtain

422

MICHELQUINTARD AND STEPHEN WHITAKER

The advantage of this scheme is that c,,, or c = Lc,, in Eq. (240) is determined directly without the iterative step indicated by Eqs (246) through (248). The disadvantage of this approach is that one must solve both Eqs. (252) and (257), and since the solution to Eqs. (214) converges in three to four iterations the two methods are essentially equivalent. The analysis of Problem I11 given by Eqs. (216) can also be altered to obtain an explicit expression for a,h. One begins with the representation s = (a,h)S

(260)

in order to obtain Problem I11 (261a) S(r + li) = S(r)

m * (y$) = 0,

m * (y,S)a,h = E, In this case the exchange coefficient is given directly by

(261b) (261c)

This approach provides a definite advantage over the iterative scheme represented by Eq. (250); however, Problems I, 11, and 111 were all solved using the same numerical method, thus the results presented in the next section were all obtained by means of the iterative solution.

ARRAYSOF CYLINDERS D. RESULTSFOR TWO-DIMENSIONAL In this section results are presented for the two-dimensional system illustrated in Fig. 10. The system is representative of a fluid (8-phase)-solid (cphase) system when particle-particle contact is unimportant. This can occur if the solid phase is suspended in the fluid phase or if particle-particle heat transfer is inhibited by the presence of a low-conductivity oxide film on the solid particles (Nozad et al. [34]). The general problem of particle-particle contact has been discussed by Shonnard and Whitaker [69]. Since the system illustrated in Fig. 10 is isotropic with respect to the heat conduction process, there is only a single distinct component of K,, and this component is presented in Fig. 15 as a function of k J k , and E,, which is referred to as the porosity. The results for K,,/k, are somewhat reminiscent of the values for effective thermal conductivity for systems of nontouching particles (Perrins et al. [67]; McPhedran and McKenzie [66]; Nozad et al. [34]); however, one

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 423

FIG. 15. Effective transport coefficient K,, as a function of kJk, and E~,,,.

must remember that K,, represents only a part of the one-equation model effective thermal conductivity, which was given in Section II1.B as For most fluid-solid systems k,/k, is greater than one; thus in a typical heat transfer process one would not be concerned with the values of k,/k, that are less than one. However, the mathematical problem given by Eqs. (1) through (4)also describes the process of diffusion in a micropore-macropore model of a porous catalyst pellet (Whitaker [26, 78]), and in that case the values of kJk, less than one are useful for the determination of effective diffusivities. In Fig. 15 we see that K,, approaches a constant value for large values of k,/k,. This occurs because the cr-phase particles become isothermal for large values of k , relative to k,, and when this occurs the heat transfer process becomes independent of k d k , . In Fig. 16 we have shown the single distinct value of KBU= K,, and it is clear from those results that this cross-coefficient associated with the coupled heat flux cannot be ignored in any two-equation model of the heat conduc-

424

MICHELQUINTARD AND

STEPHEN WHITAKER

FIG. 16. Effective transport coefficientK,, = K , as a function of k,/k, and E , ~ .

tion process. At high values of k,/k, we see that K,, approaches an ~ asymptotic value that depends on E , ~ .Although the range of values of E , is limited, the results suggest that K,, tends toward zero when either E,,,, or E,, approaches zero. This is consistent with the behavior of i.K,,.i for the stratified system, as indicated by Eq. (236b); however, one must be careful to note that in general the behavior of the nodular system shown in Fig. 10 is different from the behavior of the stratified system shown in Fig. 5. The calculated values of K,, are shown in Fig. 17, and when the a-phase particles are not touching (eBm> 0.214) the behavior of K,, is similar to that of K,, and K,, in that an asymptotic value is approached at high values of kulk, . The theoretical results for the heat transfer coefficient, in terms of a,hli/k,, are shown in Fig. 18. We again see asymptotic behavior at large values of k,/k,, and we also see a maximum value with respect to E,,,, for a fixed value of k,/k,. The behavior illustrated in Fig. 18 is somewhat similar to the behavior illustrated in Fig. 9 for stratified systems; however, the results for stratified systems do not illustrate a maximum with respect to E,,,,.

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

FIG. 17. Etfective transport coefficient K,, as a function of kJk, and

E

~

425

~

.

V. Comparison of One- and Two-Equation Models In this section we compare results from the two-equation model for transient heat conduction in a two-phase system with results from the one-equation model. Both models are exact in the sense that the effective transport coefficients have been determined theoretically and this allows us to draw some definite conclusions concerning the assumption of local thermal equilibrium. In order to test the theoretical developments leading to the volume-averaged equations and the respective closure problems, we have performed numerical experiments on stratified and nodular systems. From these experiments we have extracted exact values of volume-averaged temperatures which are compared with those determined by the one- and twoequation models. Two numerical methods are proposed to solve the twoequation model which contains both an exchange term proportional to (7''); - (T,); and flux terms proportional to V( T,)fR and V( T,):. The first method makes use of the classical Fourier series; the second method is a finite-difference scheme combined with operator splitting. Comparison of the

426

MICHELQUINTARD AND STEPHEN WHITAKER

FIG. 18. Heat transfer coefficient as a function of k,/&, and ePn.

two-equation model with the numerical experiments clearly establishes the validity of the volume-averaged equations and the closure problems. Comparison of the one-equation model with the two-equation model indicates that the assumption of local thermal equilibrium is valid for transient, onedimensional conduction processes except at very short times. The underlying principle associated with the one-equation model is that of local thermal equilibrium. For the two-phase system shown in Fig. 18, one assumes that local thermal equilibrium is valid when the two average temperatures, (T,J[ and (Tu>i,are suficiently close to each other so that they can be replaced by a single temperature, (T),,,. Under these circumstances the transport equations for (7''); and (T,); can be added to obtain the greatly simplified one-equation model. The concept of local thermal equilibrium was first discussed in terms of the drying process (Whitaker [79, Sec. 1111) and constraints for three-phase systems undergoing simultaneous heat and mass transfer were subsequently derived (Whitaker [80, Sec. 3.33). The original method used to develop the

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

427

constraints associated with local thermal equilibrium was also applied to packed bed catalytic reactors containing both homogeneous and heterogeneous thermal sources (Whitaker [8 11). Subsequently it became clear that the constraints were overly severe and a new approach was developed for estimating the difference between ( T , ) ; and (T,): (Whitaker [183). This consisted of deriving an approximate governing differential equation for the temperature difference, (3); - (T,):, and from this equation one can extract a more reliable set of constraints. These constraints are based on order of magnitude analysis and are thus qualitative in nature. The length scales associated with conductive transport in multiphase systems usually lead to conditions for which local thermal equilibrium is valid; however, this is often not the case when convective transport is important. In the analysis of packed bed catalytic reactors, the need for two-equation models has long been recognized (Vortmeyer and Schaefer [7]; Vortmeyer [8]; Schliinder [S]), and recently the importance of two-equation models for the simulation of heat and mass transfer processes in ventilated food grains has been examined by Thorpe and Whitaker [82]. While the tradition has been to accept local thermal equilibrium for conduction processes and to speculate about its validity for processes with convection (Froment and Bischoff [lo, Ch. ll]), Glatzmaier and Ramirez [I21 have suggested that local thermal equilibrium may fail for transient conduction processes. They proposed equations of the form

Eu(@Cp)u

7 a(T,)" - V.(Ku,.V( q)')

- a&( T,)" - (q),)

(265)

and used transient experimental results to determine K , K,,, and a,h. On the basis of the analysis presented here, we now know that the coupled heat flux terms, Kp,.V(Tu)u and KuB'V(T,)B, cannot be discarded in the exact representation of the two-equation model given by Eqs. (152) and (153). However, if one is confronted with a heat transfer process in which V( T,)" x V(T,)B, one could argue that Eqs. (264) and (265) represent a reasonable approximation of Eqs. (152) and (153). Under these circumstances K,, in Eq. (264) would be given by K,, K,, while K,, in Eq. (265) should be interpreted as K,, + KUu.

+

AND STEPHEN WHITAKER MICHELQUINTARD

428 A. ONE-

AND

TWO-EQUATION MODELS

The problem under consideration is described at the pore level by the following equations and boundary conditions: aT (ec ) 2 = V.(k,VT,)

in the 8-phase

T, = T, n,,.k,VT, = n,,.k,VT,

at A,, at A,,

p B

B.C. 1. B.C.2.

(ec,),

at

a T, = V.(k,VT,) at

in the a-phase

in which A,, represents the area of the /3-a interface contained in the macroscopic region under consideration. To be complete, we could specify the initial conditions and the boundary conditions at the entrances and exits of the macroscopic system; however, we need not do so at this point. According to the theoretical developments presented in Section I of this chapter, phase average temperatures are defined in a general manner according to B,= m * T,/m

*Yp

< T X = m * T,lm * Ye where the convolution product m * T, is defined explicitly as

(270a) (270b)

r

m * T, = J

m(x - r)T,(r) d K R3

The indicator function, y,, in Eq. (270a) is defined by

1, 0,

r E the P-phase r 4 the P-phase

and y, is determined in an analogous manner. The need for the generalized average given by Eq. (271) is discussed in Section I. In particular, it is shown that for functions having a periodic component the weighting function should take the form m = mg * my * my

(273) in which my yields the average over a unit cell and my * my provides the cellular average. This latter average is discussed in detail by Quintard and Whitaker [43-43b] along with the general weighting function, mg, which is necessary to ensure that m * T, is differentiable. In the analysis presented by Quintard and Whitaker [43-43b] it is shown that averages such as my * T,

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

429

do not provide regularized temperatures for spatially periodic systems; i.e., these averages contain fluctuations on the length scale of a unit cell. This will be evident in Section V.C, where m y * 5 is used as the first estimate of the macroscopic temperature for the 8-phase. In Section II1.B we have shown that the generalized average forms of Eqs. (266) and (269) are given by

provided that the unit cell used in the closure calculations is completely symmetric. In Eqs. (274) and (275) we have used a more traditional nomenclature which is given explicitly by

(Tp)k = m * TB/m * YS,

(C>;= m * T,/m * Ya

(276)

and we have used generalized volume fractions defined by Egm = m

* Ye,

Earn

=m

* Ya

(277)

In Eqs. (274) and (275), the effective transport coefficients and the exchange coefficient, a&, are determined directly from the pore-scale properties by solving three closure problems for unit cells that are representative of the two-phase system under consideration. Examples for stratified systems and periodic arrays of cylinders are provided in Section IV. The one-equation model can be extracted from Eqs. (274) and (275) whenever the difference between (T,); and (T,)fa is negligible. Here the word negligible has process connotations since what is negligible in a pure heat conduction process may not be negligible in a process involving heat transfer, mass transfer, and chemical reaction. When the difference between (To); and (Ts)fit is indeed negligible, we make use of (Tp>fl,= (To>",= ( T ) m

and add Eqs. (274) and (275) to obtain

(278)

430

MICHELQUINTARD AND STEPHEN WHITAKER

Here we have made use of the definitions (Q>mcp Keff

= EpAecp), = K,,

+ Eum(eCp)u

(280)

+ 2K,, + Kuu

(28 1)

the second of which draws upon the proof given in Section 1II.B that K,u

= K,

(282)

In the next section transient, one-dimensional solutions of Eqs. (274) and (275) are investigated. B. ONE-DIMENSIONAL MACROSCOPIC PROCESSES In this section we consider two systems, the first of which is illustrated in Fig. 19. At the pore scale, the conduction process in the system shown in Fig. 19 will be transient and two-dimensional, while in terms of the two-equation model it will be transient and one-dimensional. The process is described by Eqs. (266) through (269) in addition to the boundary conditions B.C.3 B.C.4 & 5

B.C.6 I.C.

T, = Ti,

x=L

(283)

y=O,H

(284)

Tp = TZ,

x=o

(285)

TB=T,=Tl,

t=O

(286)

n.VT, = 0,

Equations (266) through (269) along with Eqs. (283) through (286) can be solved numerically to produce the TBand Tufields, which can, in turn, be used to produce average temperatures. That calculation is described in Section V.C, and at this point we are concerned with the associated macroscopic problem. The macroscopic equations are given by Eqs. (274) and (275); however, the boundary conditions present a problem since the length-scale constraints inherent in the derivation of Eqs. (274) and (275) fail in the neighborhood of a macroscopic boundary. On the basis of a study by Quintard and Whitaker [31] it seems clear that in the neighborhood of a boundary nonlocal theories will be required in order

FIG. 19. Two-dimensionalnodular system.

ONE- AND TWO-EQUATION MODELS IN

TWO-PHASE SYSTEMS

431

to obtain a precise description of the transport process. This means that volume-averaged equations such as Eqs. (274) and (275) will be replaced with integrodifferentialequations. One of the first attempts to generate acceptable boundary conditions at a porous medium-fluid interface is due to Beavers and Joseph [83]; however, their development was largely intuitive and completely outside the framework of volume averaging. In a series of papers, Prat [35, 36, 84, 851 has attacked the problem of boundary conditions in terms of the method of volume averaging coupled with numerical experiments using spatially periodic systems. On the basis of Prat's results we are reasonably comfortable with the following volume-averaged boundary and initial conditions: (T,)i = (To): = T I , x = L (287) B.C.3'

(T,)$

B.C.6'

=

(T,):

=

T2,

x =0

(288)

(T,): = (To): = Tl, t =0 (289) LC However, we expect to find differences between theory and experiment near x = 0. For the one-equation model given by Eq. (279), it should be obvious that the boundary and initial conditions take the form ( T ) , = T,, B.C.3' x =L (290) B.C.6

( T ) , = T,,

x =0

(291)

( T ) , = T,, t =0 I.C.' (292) In order to compare the one-equation model, the two-equation model, and the numerical experiments,it is convenient to use the following dimensionless variables and parameters: x* = x/L (293a) t*

= Kefft/L2(e>mCp

(293b)

Ts* = <B,- Tl)/(T2 - TI)

(293c)

T,* = ( ( T , ) : - Tl)l(T, - Tl)

(293d)

T* = (
(293e)

9, = &,,(ecp),/(e>,C,

(2930

(pa =

&am(Vp)o/(Q)mCp

(293g)

A,,

= K,,/Keff

Aaa

= Kaa/Kefr

(293i)

= Aop = KpaIKe,, = KaBlKeff

(293J)

h* = avhL2/Ktff

(293h)

(293k)

MICHELQUINTARD AND STEPHEN WHITAKER

432

With these definitions we have the following constraints 'pp

+ (p,

A,,

= 1,

+ Ap, +

J,)q

+ 4,= 1

(294)

and the two macroscopic boundary value problems take the form 1. Two-Equation Model

B.C.l

TZ = T,* = 1,

X*

B.C.2

T$ = T,* = 0,

X* =

I.C.

TZ = T,* = 0,

t* = 0

(29%)

=O

(295d)

1

(295e)

2. One-Equation Model aT* -=at*

a2T* ax*~

(296a)

B.C.l

T* = 1,

X* = 0

(296b)

B.C.2

T* = 0,

X* = 1

(296c)

T* =O,

t* = O

(296d)

I.C.

It should be obvious that the system under consideration shown in Fig. 19 is isotropic with respect to the heat conduction process; thus we have used

Kerf = Me,,,

K,,

Kpp = IKpp,

= IK,,,

Kp, = K,)q = lKp, = IK,p

(297) On the basis of the constraints given by Eq. (294), we know that T$ and T,* will depend on the four parameters h*, A,,, Ap, whereas T* has no parametric dependence. (Pp,

(298)

3. Solution Using Fourier Series The solution of Eq. (296) is well known (Carslaw and Jaeger [86)) and the dimensionless temperature for the one-equation model is given by n= m

T*(x*, t*) = (1 - x*) -

1 ( ~ > e - n z ~ z ~sin(nm*) *

n=l

(299)

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

433

The solution of the two-equation model given by Eqs. (295) is somewhat more complex; however, a solution in terms of Fourier series can be obtained. We begin by representing the two dimensionless temperatures according to n=m

C

T$(x*, t*) = (1 - x*) -

an(t*)sin(nnx*)

(300a)

bn(t*) sin(nnx*)

(300b)

n=l n=

T,*(x*,t*) = (1

03

- x*) n=l

Introducing these series into Eqs. (295), multiplying by sin(rnnx*), and integrating over [O, 13 with respect to x*, we obtain the following pair of ordinary differential equations and initial conditions: dam pa = -(n2n21as ~

dt*

I.C. cp

+ h*)a, + ( --n2n2APa+ h*)b,

a,,, = -2fmn,

d b m = (-n2n2AaP + h*)a, dt*

~

a

I.C.

t* = 0

(301b)

+ -(nZnZRaa+ h*)b,

b, = -2/mn,

(301a)

(302a) (302b)

t* = 0

We can express these results in matrix form according to

Here [a], is a column vector and A, is a two-by-two matrix, both of which can be represented explicitly as

+

+ h*)

nZx2Aaa h*)]

(- % n2nzAz + (

+ h*)

- n2n21aP

h*)

(305)

-

The solution of Eq. (303), subject to the initial condition given by Eqs. (301b) and (302b), is given by --2 mn [a],,, = e A m f *

I

--2

mn

434

AND STEPHEN WHITAKER MICHELQUINTARD

For computational purposes, the explicit anaIytica1 form of Eq. (306) was obtained using the symbolic calculus program MathematicaTM[87]. 4. Solution Using Finite Differences

Since the analytic solution for T ; and TZ involving Eqs. (300) and (306) is rather complex, we verified our results using a finite-difference solution based on the following splitting procedure: 1. First, we solve the following parabolic equations for a single time step (307a) (307b) These equations were solved using a &scheme. The value of 8 was slightly greater than 3, which is the value for the Crank-Nicholson method. 2. Second, we solve the following equations: (308a) (308b) for the same time step with the temperature fields obtained from Eqs. (307) as initial conditions. This leads to the following analytical solutions for Eqs. (308): T,*(x*,( j + l)At*) = qpT,*(x*,jAt*)

+ q,T,*(x*,j At*)

+ 'P6e-h*A1*/Q8(pu[Tg*(x*, j At*) - T,*(x*,jAt*)] (309a) T,*(x*,( j + l)At*) = qsT,*(x*,j At*)

+ q,T,*(x*,jAt*)

+ rpae-h*AIb'Q8Q.rT,*(~*, j At*) - T* &*,j At*)] (309b) in which At* is the dimensionless time step and the dimensionless time is given by t; = j At*

(3 10)

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

435

From Eqs. (309) one can see that if the time step is too large, i.e.,

At* s qaqc/h* (31 1) the solution of Eqs. (308) forces the equality of the two temperatures. This analysis provides the characteristic time associated with the operator in the two-equation model which corresponds to the exchange of heat between the two phases. This characteristic time decreases with decreasing values of 'psor cp,, and it decreases with increasing values of h*. This provides an important qualitative explanation of the quantitative results that are presented in Section V.D. In addition, this analysis provides a means of controlling the choice of the time step in the numerical analysis. Several numerical tests showed that both the Fourier series solution and the finite-difference solution converge to the same results. For our particular applications, we used the Fourier solution with the number of terms ranging from 16 to 64 in order to achieve reasonable accuracy. We believe that the proposed numerical method provides a general framework for more complex boundary value problems of the type illustrated by Eqs. (295). Solutions are provided in the next section along with results from our numerical experiments. C. NUMERICAL EXPERIMENTS

Two-dimensional numerical simulations were performed for the nodular system shown in Fig. 19 and the stratified system shown in Fig. 20. The system illustrated in Fig. 19 is representative of a porous medium when particle-particle contact is unimportant (Shonnard and Whitaker [69]). In addition, that system is representative of a two-region model of a heterogeneous porous medium (Quintard and Whitaker [6]; Plumb and Whitaker [27]) as is suggested by Fig. 2. The stratified system shown in Fig. 20 is typical of heterogeneous porous media encountered in geological systems (Bertin et al. [88]). The boundary value problem that forms the basis for our numerical experiments is given by Eqs. (266) through (269) and Eqs. (283) through

FIG.20. Stratified system.

436

MICHELQUINTARD AND STEPHEN WHITAKER

(286). Since the numerical solution of transient heat conduction problems in heterogeneous systems is well known (Prat [35]), we will not discuss the numerical procedure. However, for clarity and completeness we will outline the principal steps of the numerical algorithm. In the sense of distributions (Richards and Youn [47]), the heat conduction equation is simply aT (ec ) - = V.(kVT) at

The heterogeneous system is discretized in grid blocks with each grid block having an assigned value of the thermal conductivity, k. Integration of Eq. (312) over the grid block (i, j) gives

- A y ( k Eax )

i-l/2+E,j

+ Ax(k$)

- Ax( k i, j+ 112 - e

2)

i,j-l/z+e'

(313)

Here (i, j) refers to the centroid of the grid block (i, j). On the right-hand side of Eq. (313), the derivatives are evaluated at the edges of the grid block with the small parameter E indicating that the derivatives are associated with the interior of the grid block domain. Equation (313) is integrated over a time step At with the integrals of the terms on the right-hand side being approximated by a &scheme with 0 E [i,11. The spatial derivatives in Eq. (313) are approximated (see Section 1V.B) by formulas of the type

The discretized equations for all the grid blocks form a set of linear equations that was solved at each time step using a conjugate gradient method (Van der Vorst and Dekker [76]). The resulting fields were used to compute local volume averages over a unit cell. These are defined as

(Tg>iv = (mv * Tdlmv * Yg

(315a)

* Ta)/mv * Y a

(315b)

= (mv

in which the weighting function my is given explicitly by 1 mv = -g(x, h d y , 412

Here g(x, l1) is defined by

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 437 and g(y, I,) is defined in an analogous manner. In Eq. (316) we have used I , and I, to represent the magnitudes of the lattice vectors, li, that are used to characterize the spatially periodic system. For the stratified system m y was taken as

where 6(x) is the Dirac distribution. This latter weighting function corresponds to a cross-sectional area average. From the discussion in Section 1.B and from the studies of Prat [35], we know that the averages defined by Eqs. (315) will not be free of small-scale fluctuations. For periodic systems one needs to use the cellular average (Quintard and Whitaker [43-44a]) in order to produce average values that are devoid of small length scale fluctuations, and the cellular average of the P-phase temperature is given by temperature

(3 19)

and this average can be represented as a double convolution product as described in Section ILA. The one-dimensional, transient profiles for the systems shown in Fig. 19 and 20 were obtained by direct numerical simulation in order to compute the averages defined by Eqs. (315) and by solution of the one- and two-equation models given by Eqs. (295) and (296). The predictions based on Eqs. (295) and (296) were obtained using the following procedure: The effective transport coefficients are calculated by solving the closure problems for the unit cells associated with the two-dimensional system under consideration. The unit cell associated with the nodular system is represented in Fig. 21 and that for the stratified system is given in Fig. 22. Some care must be taken in the interpretation of the results for the stratified system. Since the boundaries at y = 0, H are adiabatic (and not periodic), the problem corresponds to heat flow in a periodic, stratified system for which the strata are twice as thick as those illustrated in Fig. 22. The solution of the closure problems is described in Section IV, and the results will be used without further comment. The one-dimensional macroscopic equations are solved with the effective transport coefficients determined in step 1 using the algorithm described in Section V.B. Since the theoretical predictions using the macroscopic equations, and the numerical experiments using the point equations, were carried out in a completely independent manner, the comparison represents a strong test of

438

MICHELQUINTARD AND STEPHEN WHITAKER

FIG. 21. Unit cell for the nodular system.

FIG. 22. Unit cell for the stratified system.

the theory. In the following paragraphs we present this comparison for three experiments. 1. Experiment I

Our first numerical experiment is based on the laboratory experimental study of Glatzmaier and Ramirez [12], who used the traditional two-equation model to interpret laboratory measurements obtained by the hot wire method. We were unable to obtain a precise comparison with the laboratory results, since they were interpreted in terms of the traditional two-equation model. This differs from the model given by Eqs. (274) and (275) in that the coupled fluxes, involving the coefficients K,, and K,,, are neglected. From the results presented in Section IVC, we know that the coupled fluxes are not

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 439 negligible for the systems studied by Glatzmaier and Ramirez. Because of this, the energy transported by the neglected coupled fluxes is distributed between the traditional flux and the exchange term in an arbitrary manner. This introduces an uncertain error in the two effective thermal conductivities and the exchange coefficient, a,h, all of which were determined experimentally by Glatzmaier and Ramirez [12]. The physical properties and the effective properties for our first numerical experiment are listed in Table I, and there one can see that the effect of the coupled fluxes is nontrivial. In order to determine the one-equation model effective thermal conductivity, we have made use of the proof presented in Section III.C.l that led to Eq. (281), and it is of some interest to compare this result with the theory of Rayleigh [89], which provides

This is in good agreement with the value listed in Table I, indicating that our solution of the two-equation model closure problem is consistent with wellknown one-equation model results. The dimensionless form of the heat transfer coefficient for the stratified system is given by Eq. (237), and it leads to the following result for the conditions listed in Table I:

TABLE I PROPERTIES OF THE TWO-PHASE SYSTEM:NUMERICAL EXPERIMENT 1

One-equation model properties

2.14 10-3

0.62

10

0.026

0.5

1202

0.603

0.340

0.823

KefdkR

(e>c, (J/m’-K)

2.106

6.45

1.7

25.8

= 2.5 105t* a,r/l: = 7.1 103t* a,tp;

440

MICHELQUINTARD AND STEPHEN WHITAKER

This is comparable to the value of 25.8 for the nodular system. Zanotti and Carbonell [21] have analyzed the problem of laminar flow in a tube of radius ro when heat is being exchanged with the tube wall of thickness ri. In terms of their nomenclature one obtains

and if we choose I, to be equal to 2r0 we obtain (323) Z&C

after a bit of algebra the result of Zanotti and Carbonell can be expressed as

(F?)

=

z&c

32~p(ka/k& [(I - ~g2)/~jIAlu + (ka/k,)

(324)

in which A , , is a function of E~ given by

For the conditions given in Table I, the result obtained by Zanotti and Carbonell yields

(F?)

=

19.81

Z&C

This is also comparable to the value given in Table I, so we are confident that our numerical solutions of the three closure problems have provided us with reliable values of the effective properties for the nodular system illustrated in Fig. 21. In Fig. 23 we present the one-equation model results obtained from the solution of Eqs. (296), along with the experimental results for the /3- and 6phase dimensionless temperatures defined by (327a) (327b) Here it is important to note that the dimensionless temperatures are

O N E - AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS I

441

.

0.9

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

0.8 0.7 0.8

T 0.5 0.4

0.3

t*=0.0345

0.2 0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f

x*

FIG. 23. Comparison of the one-equation model with experimental values of the ternperatures in the 8- and u-phases, experiment 1.

constructed in terms of the local volume averages, which are given explicitly by (328a) (328b) Here V, and Vu represent the volume of the p- and o-phase, respectively, contained in any unit cell. The use of these averages, which are the convolution products mv * T, and m v * To, gives rise to the small length scale fluctuations that were originally observed by Prat [35]. We have chosen to use these averages in order to illustrate the small differences between the temperatures in the /Iand o-phases and the temperature predicted by the one-equation model. If one forms T,* and T,* in terms of the double or cellular average (Quintard and Whitaker [43, MI), the difference between T z , T,*,and T* in Fig. 23 is insignificant. In thinking about the results shown in Fig. 23, one should keep in mind that the time has been scaled in such a way that t* will be on the order of one when steady state is reached. We express this idea as t* = 0(1)

at steady state

(329)

to make it clear that the comparisons shown in Fig. 23 are for relatively short times. For a system comparable to that described in Table I, Glatzmaier and

MICHELQUINTARD AND STEPHEN WHITAKER

442

Ramirez [121 have published a single set of calculated temperature profiles for a process having a characteristic process time of 1 s. At a time of 0.02 s, which would correspond to t* x 0.02, they found a significant difference between the gas-phase temperature and the solid-phase temperature, whereas the results presented in Fig. 23 suggest that no such difference should exist. The parameters given in Table I closely resemble those for the air-glass bead system studied by Glatzmaier and Ramirez with the exception that their value of E~ was 0.40. Under these circumstances there is no significant parameter mismatch between our results presented in Fig. 23 and the calculations of Glatzmaier and Ramirez [12, Fig. 71. Our model, illustrated in Fig. 19, does not take into account the effect of particle-particle contact; however, the studies of Nozad et al. [34] and Shonnard and Whitaker [69] clearly indicate that particle-particle contact is unimportant in the air-glass bead system. Under these circumstances there is no significant mismatch in the pore-scale modeling and our calculations should be in good agreement with those of Glatzmaier and Ramirez; however, they are not. In order to understand why, we return to our two-equation model given by Eqs. (274) and (275) and note that our model differs from that of Glatzmaier and Ramirez [12, Eqs. 34 and 351 only because of the coupled transport terms. This means that a comparison of our calculated effective properties with the measured values of Glatzmaier and Ramirez can only be approximate. Within the spirit of this approximation we express the correspondence as &,k,(l - z), s Kgg

(330a)

&,k,(l - 7,)2 KO,

(330b)

c,k,a z a,h

(330c)

Here k, and k, are the gas- and solid-phase thermal conductivities, E, and E, are the gas- and solid-phase volume fractions, and z, and z, are parameters that were determined experimentally by Glatzmaier and Ramirez [121. In Eq. (330c) the parameter c,, was determined experimentally and a is identical to a,. If we express these results in the same form as we have used in Table I, we find Gas-phase effective thermal conductivity: = 0.603,

this work

(331a)

Glatzmaier and Ramirez

(331b)

k, cg( 1

- zg) = 0.012,

ONE- AND TWO-EQUATION MODELS IN TWO-PHASE SYSTEMS

443

Solid-phase effective thermal conductivity: Kaa __ = 0.82,

this work

k, EM1 - zs)

= 0.0017,

Glatzmaier and Ramirez

(332a) (332b)

k, Heat transfer coefficient:

a,h12

2= 25.8,

this work

(333a)

k, Chad: = 0.27,

Glatzmaier and Ramirez

(333b)

It seems clear that the measured values of the gas- and solid-phase thermal conductivities given by Eqs. (331b) and (332b) are much too small and entirely inconsistent with what we know about heat conduction in porous media (Kaviany [3, Ch. 31). The extremely low value for the dimensionless heat transfer coefficient given by Eq. (333b) is inconsistent with the value for stratified porous media given by Eq. (321), the value obtained by Zanotti and Carbonell [2l] given by Eq. (326), and the value for the nodular system given by Eq. (333a). It is undoubtedly the low value of chwhich led Glatzmaier and Ramirez [12, Fig. 71 to calculate values of the gas- and solid-phase temperature that were significantly different for t* z 0.02. Once again we note that this is in direct conflict with our results shown in Fig. 23, and we believe that further laboratory experiments are necessary. These should be carried out with the objective of testing the theory in the absence of adjustable parameters. 2. Experiment 2 This experiment represents a numerical simulation of the nodular system shown in Fig. 19 with the physical and effective properties given in Table 11. There one sees that (ec,) is the same for both phases, while the ratio of thermal conductivities is given by kJk, = 0.01. This represents a highly artificial system; however, it does provide an interesting comparison of the one- and two-equation models with a numerical experiment. In Fig. 24a we have illustrated the one- and two-equation model results for t* = 0.00437, and in this case we see a distinct difference between ( T a x and (Tp)fa with (T), being bounded by the two phase average temperatures. In Fig. 24b we have presented similar results for t* = 0.0131, and there we see that the difference between (T,); and ( Tp)fahas diminished but it is still significant. Once again, we remind the reader that steady state is reached when t* is on

MICHELQUINTARD AND STEPHEN WHITAKER

444

TABLE I1 PROPERTIES OF THE TWO-PHASE SYSTEM: NUMERICAL EXPERIMENT 2

Physical properties

Effective properties

One-equation model properties

1

0.62

10

k, (W/m-K)

k, (W/m-K)

(ecp)p (J/m3-K)

1

0.01

1. lot6

1.10+6

K,Jk#

K,.llk,

K,dk#

auhli/k,Q

0.429

0.0019

0.0038

KefC/kR

(eW, (J/m3-K)

0.4366

(ec,). (J/m’-K)

0.25

= 2.2 W t + a,t/lJ = 4.7t*

a&$

1

the order of one; thus the profiles illustrated in Fig. 24a and b are for very short process times. In Fig. 25a we have compared the two-equation model results with numerical experiments, and it is easy to imagine that if (q)t’,,were averaged a second time to produce the cellular average temperature for the /3-phase we would see excellent agreement between theory and experiment for the 8phase. The o-phase is another matter, however, and if we used the numerical results to produce m, * m y * T,, rather than m y * T, which is shown in Fig. 25a, we would find that the experimental result would be higher than the value of (T,): predicted by Eqs. (274) and (275). This suggests that the constraints inherent in the development of Eqs. (274) and (275), and the associated closure problems, are violated for this particular set of conditions. Given the repeated use of the length-scale constraints

I , a L,,

I, a L,

(334)

it is not surprising that the volume-averaged equations would begin to fail when I, = O(4 L,) and k , and k, differ by a factor of 100. In addition to the length scale constraints given by Eq. (334), one must also remember that the closure problems were taken to be quasi-steady on the basis of Eqs. 111-11. Those time-scale constraints were expressed as

(335a) (335b)

ONE- AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

445

FIG. 24. (a) Comparison of the one-equation model with the two-equation model for the nodular system, experiment 2. (b) Comparison of the one-equation model with the two-equation model for the nodular system, experiment 2.

446

MICHELQUINTARD AND STEPHEN WHITAKER

FIG. 25. (a) Comparison of the two-equation model with numerical experiments for the nodular system, experiment 2. (b) Comparison of the two-equation model with numerical experimentsfor the nodular system, experiment 2.

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 447 and from the information given in Table I1 we have

+', M t

= 2.2 lo%*

MUt

13

= 4.7t*

(336a) (336b)

These results suggest that the quasi-steady constraint is not satisfied for the results shown in Fig. 25a or for the results shown in Fig. 25b. However, the agreement between theory, in terms of m y * m, * T, and m y * m, * T,,and experiment for the conditions shown in Fig. 25b is excellent. This indicates that the quasi-steady closure problems are satisfactory for dimensionless times significantly smaller than suggested by Eqs. (335). About the difference between theory and experiment for the a-phase temperatures shown in Fig. 25a, we say only this: either length-scale effects have caused the theory to fail, or time-scale effects have caused the theory to fail, or both. Length-scale constraints first appear when the averaged equations are simplified from the nonlocal form to the local form. This procedure was discussed in Section II.A, and it is possible that a more robust theory can be developed by retaining the nonlocal form of the averaged transport equations. Time-scale constraints appear in the closure problem described in Section 111, and it may be possible to develop an improved theory by retaining the transient form of the closure problem. This will lead to timedependent effective transport and exchange coefficients. This approach has been explored by Paine et al. [90] for mass transfer and reaction processes during flow in capillary tubes. Both length- and time-scale constraints are associated with the neglect of laand 5, in Eqs. (100) and (IOl), and if these terms cannot be neglected the macroscopic equations will contain higherorder derivatives. The resolution of this difficulty is not clear. 3. Experiment 3 This numerical experiment represents a simulation for the stratified system illustrated in Fig. 20; and the system parameters are presented in Table 111. The comparison between the one-equation model and the two-equation model is shown in Fig. 26a and b for the dimensionless times t* = 0.0152 and t* = 0.0606. At both times there are significant differences between (T,); and (T,)fl, indicating that the one-equation model is not acceptable if an accurate description of the phase average temperatures is required. By comparing Fig, 26a with Fig. 24b, one can see that the difference between (T,); and (5); is greater for the stratified system than it is for the nodular system. This is in agreement with our intuition.

448

MICHELQUINTARD AND STEPHEN WHITAKER TABLE 111 PROPERTIES OF THE TWO-PHASE SYSTEM: NUMERICAL EXPERIMENT 3 ~

e8

LIH

1

0.5

10

0.5

0.0

Unit cell

50.0 ~

One-equation model properties

K,,lk,

(e>c, (J/m3-K)

50.5

23.76

~~~

a8t/$ = 7.9t* a,t/l: = 7.9 W t *

1.7

In Fig. 27a and b we have shown the comparison between theory and numerical experiment, and in Fig. 27a one sees a detectable departure between theory and experiment for the 8-phase temperatures. If we think about the results shown in Fig. 27a and those shown in Fig. 25a, we develop the impression that it is the length-scale constraint that controls the failure of the volume-averaged equations since the difference between theory and experiment is greatest where the temperature gradients are the steepest. However, for the processes under consideration the length scales, such as x* = O(0.2) for the &phase temperature profile shown in Fig. 27a, are closely tied to the time scales for the individual phases and the time scale for the macroscopic process. Under these circumstances it is difficult to do much more than identify the conditions for which the theory fails. The general conclusion that can be drawn from the results presented in Figs. 24 through 27 is that the two-equation model given by Eqs. (274) and (275), along with the three closure problems given by Eqs. (104) through (1 10) or in compact form by Eqs. (214) through (216), can be confidently used to predict the phase average temperatures. The two constraints that we would suggest at this time are the length- and time-scale constraints given by L, 2 4Up I,) (7,?)20.3

(337a) (337b)

Given that the two-equation model is generally reliable, we can use it to produce a quantitative study of the assumption of local thermal equilibrium.

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 449

FIG. 26. (a) Comparison of the one-equation model with the two-equation model for the stratified system, experiment 3. (b) Comparison of the one-equation model with the twoequation model for the stratified system, experiment 3.

450

MICHELQUINTARD AND STEPHEN WHITAKER 1 0.9

0.8 0.7

0.8

T* 0.5 0.4

0.3 0.2

0.1 0

0

0.1

0.2

0.4

0.3

0.5

0.8

0.7

0.8

1

0.9

X*

(4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X* (b) FIG. 27. (a) Comparison of the two-equation model with numerical experiments for the stratified system, experiment 3. (b) Comparison of the two-equation model with numerical experiments for the stratified system, experiment 3.

ONE- AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

451

This is done in the following paragraphs; however, before moving on to Section V.D we need to mention Problem IV, which has been largely ignored in terms of our comparison between theory and experiment. Our conclusion concerning t8and 5, in Eq. (124), and in the representations for T, and p,, given by Eqs. (100) and (101), was in the nature of a plausible intuitive hypothesis. The test of that hypothesis is the comparison between the theory for the two-equation model and the numerical experiments illustrated in Fig. 25% 25b, 27a, and 27b. When the constraints indicated by Eq. (337) are satisfied it would appear that one could describe the agreement between theory and experiment as excellent. This suggests that our plausible hypothesis concerning t, and 5, is correct or we have been victimized by an unidentified cancellation of errors somewhere in the theory. This seems unlikely; however, our study of this problem is far from exhaustive, and questions concerning Problem IV remain to be answered.

D. THE PRINCIPLE OF LOCALTHERMAL EQUILIBRIUM

Up to this point the concept of local thermal equilibrium has been based on the idea that certain terms in the sum of the two averaged energy equations could be neglected (Whitaker [18, Eq. 2.131). These terms were directly related to the temperature difference, (q)i - ( and previous studies (Whitaker [go, Section 3.3; 181) were directed toward order-of-magnitude estimates of this difference. The condition of local thermal equilibrium was assumed to exist when the unwanted terms involving (q)$ - (T,>; were judged to be negligible. At this time we are in a position to take a more direct approach to the study of local thermal equilibrium since we can calculate (T& - (T,); directly and compare it to the change in the temperature for the one-equation model, A( T),,,. When the following constraint is satisfied:

c);,

(Tp)i - (Ta);

A(T)rn

(338)

the one-equation model should be an acceptable representation of the heat conduction process. To study the convergence of the two-equation model to the one-equation model, we construct the following norm: @(t*) =

(lo1

[T$x*,

112

t*)

- T,Yx*, t*)] d x * )

(339)

in which TZ and T,* are given by Eqs. (293c) and (293d). This norm is computed easily from the Fourier series described in Section V.B, and from the analysis presented in that section we know that @(t*)depends on only the

452

MICHELQUINTARD AND STEPHEN WHITAKER

following four dimensionless parameters: qps, h*, Afipy

a,a

(340)

The norm is also constrained by the initial condition and the boundary conditions, leading to @(t*) = 0,

t* = 0

lim @(t*) = 0 F-

(341) (342)

Q,

For the particular problem under consideration, the influence of the boundary at x* = 1 is negligible for t* < 0.1; thus many of the results presented in this section are applicable to the problem of transient heat transfer in a semiinfinite system. While the four dimensionless parameters given by Eq. (340) are convenient for the compact presentation of results, it is also convenient to remember that they can be expressed as ‘P, = Efim(ec,)~/(e>,CP

(343)

To remind the reader of the range of values of K e f f / k ,for nonconsolidated porous media we have shown in Fig. 28 the results compiled by Nozad [91] for a wide range of values of ka/k,. The line for the continuous a-phase refers to the case of no particle-particle contact, while the line for the continuous ophase refers to the case of particle-particle contact. The parameter c/a represents the fraction of surface area that is involved in the particle-particle contact. Subsequent calculations (Whitaker [22]) have revised the value downward to 0.01, and Kaviany [3, Ch. 31 suggests that the value should be 0.002. From our numerical experiments presented in Section V.C,it would appear that h* is a particularly important parameter. Our studies in Section V.C would suggest that

r?)

= O(1

- 20)

(347)

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 453 I

I

kdka FIG. 28. One-equation model effective thermal conductivity for nonconsolidated porous media.

whenever k,/k, 2 1, and from Fig. 28 we have a reasonable idea how K,,,/k, varies with k d k , . These results suggest that h* has a lower bound of one and that it is dominated by (L/l,)’. In order to explore the effect of h*, we have fixed the following parameters according to cp, = 0.50,

I,,

= 0.90,

A,,

= 0.01

(348)

and calculated the norm defined by Eq. (339). The results are presented in Fig. 29 for values of h* ranging from 1 to 1000, and it is best to follow the curves from left to right to see how the condition of nonequilibrium develops and then diminishes as t* approaches one. For very small values of t* the initial condition given by Eq. (341) forces 0 to zero, while for values oft* greater than one the condition given by Eq. (342) controls the norm and 0 returns to zero. In thinking about the results illustrated in Fig. 29, one must keep Eqs. (295) in mind along with the definitions given by Eqs. (293) and (294). For the case under consideration, we have cpe = cp, and if A,, were equal to I,, one could use Eqs. (295) to prove that 0 = 0 for all times and all values of h*. Clearly, the dependence of 0 upon h* and t* depends on the choice of pa,I,,, and A,, and the results that we have shown in Fig. 29 are purely illustrative. However, it is clear that large values of L/l, produce large values of h* and this leads to the condition of local thermal equilibrium in the sense of the norm defined by Eq. (339). The role of the specific heat capacities is controlled by the parameter cp, which ranges from zero to one. The influence of cp, on the norm defined by

454 0.3

0.25

0.2

8 0.15

0.1

0.05

0 0 . m 1

o.oooo1

o.Ooo1

0.001

0.01

0.1

1

t+

FIG. 29. Influence of the dimensionless heat transfer coefficient on the norm, 0.

Eq. (339) is illustrated in Fig. 30 for the following values of h*, A,,

and A,,:

h* = 10, A,, = 0.9, A,, = 0.01 (349) To begin with, one should trace the curves in Fig. 30 from left to right to see how the departure from local equilibrium evolves with time. Next it is best to consider the curves in a sequential manner beginning with the result for rp, = 0.01. This condition can also be expressed as E,(ecp)s + c,(ecp),, and it leads to large values of 0 for t* on the order of 0.01. When 'p, is increased to 0.5 we have the condition Ea(ecp),= E,(ecp), or q, = rp, and Eqs. (295) tell us that any departure from local thermal equilibnum must be caused by differences between A,, and A,,. On the basis of Eqs. (293) and (294) we can represent the various effective thermal conductivities as

-K @ - 0.90, Kerf

= 0.01, Keff

__ Kuu

- 0.08

(350)

Kerf

and it is these differences which generate the nonzero values of 0 for the condition rp, = 0.5. As rp, is increased to 0.8 we have the condition E,(ecp), >

ONE- AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

455

0.4

0.35 0.3 0.25

8

0.2 0.15

0.1

0.05 0

0.0000001 0.000001

o.Oooo1

0.0001

0.001

0.01

0.1

1

t* FIG. 30. Influence of the dimensionless heat capacity on the nom, 0.

E,(ecp)a and this increased difference in the heat capacities gives rise to decreased values of 0.What happens in this case is that the difference in the heat capacities actually cancels the effect of the differences in the effective thermal conductivities and the values of 0 for 'ps= 0.8 are smaller than those for 'pa = 0.5. When qa is increased to 0.999 we have the condition ca(ec,)s >> E,(ecp)u and this causes an increase in the value of the norm defined by Eq. (339). The time at which the maximum value of 0 occurs increases with increasing values of 'ps up to 0.5; however, for large values of 'ps we see that the time at which the maximum occurs decreases with increasing values of 'ps. From Fig. 30 it is clear that the behavior of the norm, 0, is complex, and it should be obvious that the local departure from local thermal equilibrium, as determined by T; - T,*,is very complex indeed. The results presented in Fig. 29 and 30 are based on Eqs. (295) and are valid for any topology, including the nodular system illustrated in Fig. 19 and the stratified system illustrated in Fig. 22. We could continue with this degree

456

MICHELQUINTARD AND STEPHEN WHITAKER

of generality in a study of the effect of I,, and I,,, which depend on the ratio of conductivities, k,/k,, and the geometrical characteristics of the unit cell. However, in this case we have chosen to limit our presentation to a study of the nodular system shown in Fig. 19, subject to the following choice of parameters: pe= 0.40

(351) which requires that (ec,,), be equal to (ec,,),. Having chosen E, and the geometry illustrated in Fig. 21, we have a situation in which I,, and I,, depend only on k,/k,. All the other parameters in the two-equation model given by Eqs. (274) and (275) or Eqs. (295) were determined by the methods presented in Section TV. The time evolution of 0 is shown in Fig. 31. The greatest departure from local equilibrium is observed for k,/k, = 0.01; i.e., the thermal conductivity of the discontinuous phase is much, much less than that of the continuous phase. When k,/k, is increased to 0.1 the maximum value of 0 is decreased from about 0.14 to about 0.04, and when k,/k, is further increased to 1.0 the E, = 0.40,

0.14

0.12

-

0.1

-

0.08

-

0.06

-

0.04

-

0.02

-

--.__ ...... ---.

- - - -100.

-

Ma.

e

0' 0.oO00001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

t*

FIG. 31. Time evolution of the temperature difference represented by 0 as a function of

k,fk,.

ONE- AND

TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS

457

value of 0 is independent of time and equal to zero. For values of k,/k, larger than one we see nonzero values of 0; however, for values of k,/k, greater than 100 there is no noticeable change in the behavior of 0.The asymptotic behavior for 0 at large values of k,/k, is reminiscent of the behavior of K,,Jk, shown in Fig. 28 and it occurs because the temperature in each 0-phase particle takes on a local constant value when k,/k, >> 1. The maxima of the curves shown in Fig. 31 are presented in Fig. 32 as a function of the conductivity ratio, k,/k,. The solid line represents the value of the maxima and the dashed line indicates the dimensionless time at which the maxima occur. A point of particular interest is the one corresponding to k, = k,, i.e., identical phases. The maximum of 0 is identically zero for this case, and the curve for t* features a minimum for this particular value. For k, < k, the maximum of 0 increases significantly for decreasing values of k,/k,. The value of t* corresponding to the maximum of 0 increases significantly with decreasing values of k,/k,. For k d k , > 1, both the maximum of @ and the value oft* corresponding to the maximum increases with increasing values of k,/k,; however, the variations are much smaller for

0.01

0.0014 r

0.1

1

10

100

1

I

I

I

0.0012

loo0 1 0.14 0.12

0.001

0.1

O.ooo8

0.08

O.ooo6

0.06

0.0004

0.04

0.0002

0.02

e

t*

0

0.01

0.1

1

10

100

0 loo0

Vq3 FIG. 32. Coordinates of the maxima of 0 as a function of the conductivity ratio.

458

MICHEL QUINTARD AND STEPHEN WHITAKER

k,/k, > 1. This is expected for the nodular system shown in Fig. 21, since the role of the two phases is not interchangeable. For the stratified system shown in Fig. 22, the roles of the two phases are interchangeable and the curves in Fig. 32 would be symmetric about the point k,/k, = 1. The results shown in Fig, 29 through 32 provide some quantitative information about the condition of local thermal equilibrium, and they are in agreement with intuitive considerations. In addition, the results shown in Fig. 24 through 32 are in qualitative agreement with previously published constraints associated with local thermal equilibrium (Whitaker [181). A detailed comparison of our numerical results with constraints based on order of magnitude analysis will be the subject of a subsequent publication.

VI. Conclusions In this article we have examined the process of transient heat conduction for a two-phase system in terms of the method of volume averaging. Both one- and two-equation models have been developed along with the relevant closure problems which allow one to determine theoretically the effective transport coefficients. For the one-equation model only the effective thermal conductivity tensor, Kerf, needs to be determined, while the two-equation model requires the four coefficients, K,,, K,, = K,,, K,,, and a,h. Both length- and time-scale constraints were imposed in the course of the theoretical development. These were based on order-of-magnitude estimates as indicated by Eqs. (73), (90), and (91). In addition, the two-equation model was simplified by the neglect of higher-order terms, which we were unable to identify as negligible on the basis of order-of-magnitude estimates. In order to test the theoretical analysis, we have performed numerical experiments from which values of (T,); and (T',)B, can be extracted for comparison with the one- and two-equation models. The results clearly indicate that the length- and time-scale constraints proposed during the development of the theory are overly severe and can be replaced with those given by Eqs. (337). The comparison between theory and experiment indicates that the higher-order terms contained in Problem IV can be neglected when the constraints given by Eqs. (337) are satisfied. Given reliable one- and two-equation models, we have explored the principle of local thermal equilibrium in Section V.D. The results are not exhaustive but a relatively wide range of parameters have been studied and the results can be used to determine when the one-equation model can be used to analyze transient diffusion processes.

ONE-AND TWO-EQUATION MODELSIN TWO-PHASE SYSTEMS 459

Acknowledgment This work was initiated while M. Quintard was on sabbatical leave at the University of California at Davis and completed while S. Whitaker was on sabbatical leave at L.E.P.T.ENSAM. Partial financial support for S. Whitaker was provided by NSF grant 8812870. The idea of using operator splitting to solve numerically the two-equation model was suggested by P. Fabrie (University of Bordeaux I-CEREMAB and L.E.P.T.-ENSAM).

Nomenclature interfacial area per unit volume (m-') area of entrances and exits for the /%phase (m2) area of the P-u interface (m') area of entrances and exits for the a-phase (m2) vector field that maps V( T), onto ?: in the one-equation model (m) b,, in the 8-phase b,, in the u-phase b,, in the 1-phase b,, in the a-phase vector field that maps V( T,); onto T, (m) vector field that maps V( T,); onto T, (m) vector field that maps V( T,)! onto Tq(m) vector field that maps V( T,); onto T, (m) = m*(n,,.kpVb,,b8.) (W/m2-K) =m*(n,,.k,Vb,,S,,) = -c,, (W/m2-K) = m*(n,,.k,Vb,,b,,) (W/mz-K) =m*(n,,.kpVb,,bpa) = -csa (W/m2-K) mass fraction weighted heat capacity (J/kg-K) constant-pressure heat capacity in the w-phase (J/kg-K) effective particle diameter (m) film heat transfer coefficient (W/m'-K) =a,hL2/K,,,, dimensionless film heat transfer coefficient = m*(n,.k,Vs,6,,) (W/m3-K)

=i ={

= -m*(n,,.kuVs,Sp,) = h, (W/m3-K) thermal conductivity of the ophase (W/m-K) one-equation model effective thermal conductivity tensor (W/m-K) two-equation model effective thermal conductivity tensor associated with V(T,); in the &phase equation (W/m-K) two-equation model effective thermal conductivity tensor associated with V(T,); in the 1-phase equation (W/m-K) two-equation model effective thermal conductivity tensor associated with V(T,); in the a-phase equation (W/m-K) two-equation model effective thermal conductivity tensor associated with V( q)c in the u-phase equation (W/m-K) characteristic length for the ophase (m) characteristic length associated with local volume-averaged quantities (m) characteristic length associated with Vz(T,); (m) characteristic length associated with VeBm weighting function associated with averages over a unit cell in a spatially periodic system = -no,, outwardly directed unit normal vector pointing from the p-phase toward the u-phase position vector (m)

460

SP

t

t*

To T*

MICHELQUINTARD AND STEPHEN WHITAKER characteristic length associated with the averaging volume (m) characteristic length associated with the support of the weighting function m (m) scalar field that maps ((To): (qx)onto 'li, scalar field that maps ((T,): (q>C)onto To time (s) = K e f f r / L 2 ( ~ ) , , , Cdimensionless p, time point temperature in the w-phase (K) = ( ( T ) , - T,)/(T, - TI),dimensionless average temperature =((T,); - T,)/(T, - T,),dimensionless phase average temperature for the w-phase =ea( T,)@+ E,( T,)", spatial average temperature (K) = eP,( q); generalized spatial average temperature (K) =E,( T,)", phase average temperature for the w-phase (K) intrinsic phase average temperature for the w-phase (K) = m * T, = &,,,,(To):, generalized phase average temperature for the w-phase (K) generalized intrinsic phase average temperature for the w-phase (K) = qo- y,(T,):, spatial temperature deviation in the w-phase (K) two-equation model transport coefficient associated with V(T,)B, in the @-phaseequation (W/mZ-K)

+

T,,x,

UP.

U.?,

uu8

V

Kc,,

v, X

X'

Y

two-equation model transport coefficient associated with V( T,,); in the 8-phase equation (W/m2-K) two-equation model transport coefficient associated with V( T,); in the a-phase equation (W/m2-K) twoLequation model transport coefficient associated with V(3); in the a-phase equation (W/m2-K) averaging volume (m3) volume of a unit cell (m') volume of the w-phase contained within V (m') position vector of the centroid of the averaging volume (m) = x/L, dimensionless spatial variable position vector relative to the centroid of the averaging volume (m)

GREEK LETTERS ~o

Yo 6,s

6 ,

&om

9"

49, 4% b 2,.

e, (e),,, Ax, Ay At*

= k,/(ec,),, thermal diffusivity for the o-phase (m2/s) o-phase indicator function Dirac distribution associated with the /?-a interface volume fraction of the w-phase =rn * yo, generalized volume fraction of the w-phase =&om(ecp),/(e)mCp = ~@PlK,ff = KPJK,,, = K,i3/Kerr

= A,,

= K,,IK,ff

density in the w-phase (kg/m3) spatial average density (kg/m') mesh size dimensionless time step

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O N E - A N D TWO-EQUATION

MODELSIN TWO-PHASE SYSTEMS

461

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49. Whitaker, S. (1986). Multiphase transport phenomena: matching theory and experiment. In Advances in Multiphase Flow and Related Problems (G. Papanicolaou, ed.), pp. 273-295. S A M , Philadelphia. 50. Gray, W. G., and Lee, P. C. Y. (1977). On the theorems for local volume averaging of multiphase systems. Int. f. Multiphase Flow 3, 333-339. 51. Gray, W. G., and Hassanizadeh, S. M. (1989). Averaging theorems and averaged equations for transport of interface properties in multiphase systems. Int. J. Multiphase Flow 15,81-95. 52. Baveye, P., and Sposito, G. (1984). The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers. Water Resour. Rex 20,521-530. 53. Cushman, J. H. (1984). On unifying the concepts of scale, instrumentation and stochastics in the development of multiphase transport theory. Water Resour. Res. 20, 1668-1676. 54. Maneval, J. E., McCarthy, M. J., and Whitaker, S. (1990). Use of NMR as an experimental probe in multiphase systems: determination of the instrument weighting function for measurements of liquid-phase volume fraction. Water Resour. Res. 26,2807-2816. 55. Whitaker, S. (1992). The species mass jump condition at a singular surface. Chem. Eng. Sci. 47, 1677-1685. 56. Crank, J. (1984). Free and Moving Boundary Problems. Oxford University Press, New York. 57. Gray, W. G. (1975). A derivation of the equations for multiphase transport. Chem. Eng. Sci. 30,229-233. 58. Carbonell, R. G., and Whitaker, S. (1984). Heat and mass transfer in porous media. In Fundamentals of Transport Phenomena in Porous Media (J. Bear and M. Y. Corapcioglu, eds.), pp. 121-198. Martinus Nijhof, Boston. 59. Birkhoff, G. (1960). Hydrodynamics, a Study in Logic, Fact, and Similitude. Princeton University Press, Princeton, NJ. 60. De Groot, S. R., and Mazur, P. (1962). Non-Equilibrium Thermodynamics. North-Holland, Amsterdam. 61. Ryan, D., Carbonell, R. G., and Whitaker, S. (1981). A theory of diffusion and reaction in porous media. AIChE Symp. Ser. 77,46-62. 62. Eidsath, A., Carbonell, R. G., Whitaker, S., and Herrmann, L. R. (1983). Dispersion in pulsed systems. 111. Comparison between theory and experiment for packed beds. Chem. Eng. Sci. 38, 1803-1816. 63. Ochoa, J. A., Stroeve, P., and Whitaker, S. (1986). Diffusion and reaction in cellular media. Chem. Eng. Sci. 41,2999-3013. 64. Kim, J.-H., Ochoa, .IA. ., and Whitaker, S. (1987). Diffusion in anisotropic porous media. Transport Porous Media 2, 327-356. 65. Quintard, M. (1992). Diffusion in isotropic and anisotropic porous systems: three-dimensional calculations. Transport Porous Media, in press. 66. McPhedran, R. C., and McKenzie, D. R. (1978). The conductivity of lattices of spheres. I. The simple cubic lattice. Proc. R. Soc. Lond. Ser. A 359,45-63. 67. Perrins, W. T., McKenzie, D. R., and McPhedran, R. C. (1979). Transport properties of regular arrays of cylinders. Proc. R. Soc. Lond. Ser. A 369,207-225. 68. Zick, A. A., and Homsy, G . M. (1982). Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115, 13-26, 69. Shonnard, D. R., and Whitaker, S. (1989). The effective thermal conductivity for a pointcontact porous medium: an experimental study. Inr. f. Heat Moss Transfer 323, 503-512. 70. Barrbre, J., Gipouloux, O., and Whitaker, S. (1991). On the closure problem for Darcy’s law. Transport Porous Media 7, 209-222. 71. Edwards, D. A., Shapiro, M., Brenner, H., and Shapira, M. (1991). Dispersion of inert solutes in spatially periodic, two-dimensional model porous media. Transport Porous Media 6, 337-358. 72. Erlich, R., Crabtree, S. J., Horkowitz, K. O., and Horkowitz, J. P. (1991). Petrography and

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reservoir physics. I. Objective classification of reservoir porosity. Am, Assoc. Pet. Geol. Bull. 75( lo), 1547- 1562. 73. Hesteness, M. R., and Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409-435. 74. Golub, G. H., and Van Loan, C. F. (1983). Matrix Computations. North Oxford Academic, Oxford. 75. Gambolati, G., and Perdon, A. (1984). The conjugate gradients in subsurface flow and land subsidence modelling. In Fundamentals of Transport Phenomena in Porous Media (J. Bear and M. Y. Corapcioglu, eds.), pp. 953-984. Martinus Nijhoff, Boston. 76. Van der Vorst, H. A., and Dekker, K. (1988). Conjugate gradient type methods and preconditioning. J. Comput. Appl. Math. 24, 73-87. 77. Brent, R. P. (1973). Algorithmsfor Minimization withoui Derivatives. Prentice-Hall, Englewood Cliffs, NJ. 78. Whitaker, S. (1988). Diffusion in packed beds of porous particles. AfChE J. 34(4), 679-683. 79. Whitaker, S. (1977). Simultaneous heat, mass, and momentum transfer in porous media: a theory of drying. Adv. Heat Transfer 13, 119-203. 80. Whitaker, S. (1980). Heat and mass transfer in granular porous media. In Aduances in Drying, Vol. 1 (A. S. Mujumdar, ed.), pp. 23-61. Hemisphere, New York. 81. Whitaker, S. (1986). Local thermal equilibrium: an application to packed bed catalytic reactor design. Chem. Eng. Sci. 41,2029-2039. 82. Thorpe, G. R., and Whitaker, S. (1992). Local mass and thermal equilibria in ventilated grain bulks. I. The development of heat and mass conservation equations. J. Stored Prod. Res., in press. 83. Beavers, G. S., and Joseph, D. D. (1967). Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30,197-207. 84. Prat, M. (1991). 2D modelling of drying of porous media: influence of edge effects at the interface. Drying Tech. 9, 1181-1208. 85. Prat, M. (1992). Some refinements concerning the boundary conditions at the macroscopic level. Transport Porous Media I, 147-161. 86. Carslaw, H. S., and Jaeger, J. C. (1946). Conduction of Heat in Solids. Oxford University Press, New York. 87. Wolfram, S. (1991). Mathematica: a System for Doing Mathematics by Computer. AddisonWesley, Reading, MA. 88. Bertin, H., Quintard, M., Corpel, Ph. V., and Whitaker, S. (1990). Two-phase flow in heterogeneous porous media. 111. Laboratory experiments for flow parallel to a stratified system. Transport Porous Media 5, 543-590. 89. Rayleigh, R. S. (1892). On the influence of obstacles arranged in rectangular order upon the properties of the medium. Philos. Mag. 34,481-489. 90. Paine, M. A., Carbonell, R. G., and Whitaker, S. (1983). Dispersion in pulsed systems. I. Heterogeneous reaction and reversible adsorption in capillary tubes. Chem. Eng. Sci. 38, 1781-1793. 91. Nozad, I. (1983). An experimental and theoretical study of heat conduction in two and three phase systems. PhD Thesis, Department of Chemical Engineering, University of California, Davis, California.

SUBJECT INDEX

A

Average temperature, 374 generalized volume average, 376 intrinsic phase average, 375 superficial phase average, 374 weighting function, 377 Axial dispersion, see also Mixing and Peclet number liquid-solid fluidized beds, 250 packed beds, 229 three-phase fluidized beds, 257

B Bed expansion liquid fluidized beds, 241 three-phase systems, 253 Bingham plastic fluids, 190 fluidized beds, 239, 241 packed beds, 202,214 Boiling curve saturated liquid, 6 transition boiling, I 1 1 C

Calibration, heat flux measurements, 342-343 Calorimeter null-point, 311-313 response to constant heat transfer coeficient, 309-310 slug, 307-311 Cavity circular foil gage, 305-306 Cell models free surface, 207, 209-210 packed beds, 207,209 zero vorticity, 207, 209, 211 Circular foil gage, 300-306 analytical models, 301-302 cavity, 305-306 convection correction, 303-304 design chart, 302

exponential time constant, '301 schematic, 301 temperature distribution, 302-303 thermopile, 304 water-cooled, 305 Coaxial thermocouple, 313-3 14 Coherence function, heat flux and velocity, 348-349 Combustions, heat flux measurements, 352-353 Conduction equation, one-dimensional, 291 Conjugate gradient method, 418 Constant heat flux method, 326-331 color interpretation, 330 composite heated wall, 327-328 liquid crystals, 328-331 Constant surface temperature method, 331, 333-342 convection loss, 334-335 differential thermocouple gage, 336 heated ribbon gage, 336-337 heated thin-film heat flux gage, 338-340 hot wire anemometer bridges, 340-341 segmented plate heaters, 333 skin friction gage, 337-338 time-averaged measurements, 333 unsteady calibration, 341-342 Critical heat flux, 53, 55-107 droplet splashing from free-surfacejet, 106-107 jet suction effects, 106 multiple-jet impingement, 99-101 research needs, 122 single-jet impingement, 55-99 boiling curves, 80-82 CHF correlations, 65-67, 74-76,95-98 CHF model, 62-64 critical Weber number, 69 data correlation, 93-99 differences between circular and planar jets, 93 differences between free-surface and submerged jets, 93 FC-72,64-65

465

SUBJECT INDEX

466

fluid properties, 77-78 free-surface, circular jets, 55, 57, 62-79 free-surface, planar jets, 79-85 Helmholtz critical wavelength, 97 I- and HP-regimes, 67, 69-70 investigations, 58-61 jet velocity, 62-76, 79-90 L- and V-regimes, 55, 57 liquid film thickness, 97-99 liquid-vapor structure, 98 liquid viscosity-controlledregime, 95 mechanical energy balance, 93-94 nozzle/heater dimensions, 78,91 nozzle-to-surface spacing, 79,91-93 pool boiling, 80-83 pool height and jet velocity, 85-86 stagnation point, 95 subcooling, 77, 84-85,90-91 submerged, confined, and plunging jets, 85-92

surface orientation, 79 surface tension-controlled regime, 95 transition regions, 83-84 surface conditions effects, 103- 106 wall jets, 101-103

D Darcy’s law, 212 Deborah number critical value, 217-218 definition, 217, 220 Dependence-included discrete-ordinates method, 173-179 bed properties, 176-177 single particle properties, 174-175 Differential thermocouple gage, 336 Directional spectral specular reflectivity, 148 Discrete-ordinate equations, solution,

F Film boiling, 9, 117-120 convective heat transfer coefficient, 118-119

investigations, 109-110 quenching, 118-119 research needs, 123 Fixed beds, see Packed beds Fluid flow fluidized beds, 232 hindered settling, 261 packed beds, 192 periodically constricted tubes, 219 three-phase systems, 253 Fluidized beds, 187 axial dispersion, 257 bed expansion, 241,253 gas holdup, 255 heat transfer, 251, 258, 261 liquid holdup, 257 mass transfer, 251, 258, 261 minimum fluidization velocity, 233, 239, 241

nowNewtonian effects, 232 three-phase systems, 253 Forced convection, single-phase, jet impingement boiling, 7 Friction factor Bingham plastics, 202 Carreau fluids, 211 Ellis model, 199 packed beds, 196 power law fluids, 196

G Gas holdup, three-phase systems, 255 Gas turbine engines, heat flux measurements, 344-346

140-141

Discrete-ordinates approximation, 139- 141 Drag coefficient, packed beds, 207 Drag reducing polymers mass transfer, 222 packed beds, 232

E Emitting medium, two-flux approximation, 138-139

Exchange factor, 166-168

Generalized Newtonian fluids fluidized beds, 232 hindered settling, 261 packed beds, 193-194 Geometric-optics scattering, 147-151

H Hagen-Rubens law, 143 Heat conduction, two-phase system, 372 Heated ribbon gage, 336-337

SUBJECT INDEX Heat flux, see also Critical heat flux to surface, 285-286 Heat flux gage, 283-284, see also Layered gage plug-type, 311 thin-film, 293-294, 315-321, 337 calibration, 320 data analysis, 318 heated, 338-340 semi-infinite geometry, 317 Heat flux measurements, 279-353 active heating, 326-342 constant heat flux, 326-331 constant surface temperature, 331-342 calibration, 342-343 categories, 287 combustions, 352-353 convection boundary layer, 284-285 heat flux gage, see Heat flux gage high-temperature, 351-352 issues, 281-287 overview, 279-281 spatially distributed measurements, application, 349-35 1 spatial temperature difference, 287-307 circular foil gage, 300-306 in-depth temperature, 298-299 layered gage, 288-298 radiometer, 306-307 wire-wound gage, 299-300 surface energy balance, 281-282 surface thermal boundary condition, 282-283 temperature change with time, 307-326 coaxial thermocouple, 313-3 14 null-point calorimeter, 31 1-3 13 optical method, 321-326 slug calorimeter, 307-31 1 thin-film method, 315-321 thin-skin method, 314-315 terminology, 281 time-resolved measurements, 343-349 basic research, 346-349 gas turbine engines, 344-346 reciprocating engines, 343-344 turbulence at stagnation point, 347-348 Heat flux microsensor, 296-297 Heat transfer fluidized beds, 251 packed beds, 229 radiative, see Radiative heat transfer

467

three-phase systems, 261 Heat transfer coefficient convective, film boiling, 18-119 impinging liquid jets, 1 Helmholtz critical wavelength, 97 Hemispherical isotropy, 136 Heterogeneous porous medium, 373 Hindered settling, see also Sedimentation; Settling nowNewtonian effects, 261 Hot wire anemometer bridges, 340-341 Hydrodynamics, jet impingement, 2-6 Hysteresis, nucleate boiling, 49

I Incipience model, modified, 16 Interphase heat transfer liquid fluidized beds, 251 packed beds, 229 three-phase systems, 261 Interphase mass transfer liquid-solid beds, 258 packed beds, 229 three-phase beds, 259

J Jet impingement heat transfer, 328-331 hydrodynamics, 2-6 inviscid pressure and velocity distributions, 4-5 jet configurations, 2-3 local saturation conditions, 5 Jet impingement boiling, 1-123, see also Heat flux, critical; Nucleate boiling; Transition boiling film boiling, 9, 117-120 nucleate boiling, 7-8 research needs, 120-123 single-phase forced convection, 7 system-specificeffects, 9 transition boiling, 8-9

L Laminar flow in packed beds, 194 Layered gage, 288-298 advantages, 291, 298 Heat flux microsensor, 296-297

468

SUBJECT INDEX

pattern, 295 resistance temperature devices, 288 sensitivity, 291-292 thermal resistance thickness, 292-293 thermocouples, 288-290 thermopile, 291 transient response, 292 Liquid crystals constant heat flux method, 328-331 heat flux measurements, 323 Liquid-solid fluidized beds, 23 1 axial dispersion, 250 bed expansion, 241 heat transfer, 251 mass transfer, 251 minimum fluidization velocity, 233 Liquid viscosity-controlled regime, 95 Local thermal equilibrium, 370,426,451-455

M Mass transfer liquid-solid beds, 251 packed beds, 229 three-phase beds, 258 Microfoil heat flux gage, 293 Mie theory, 141-142, 145. Minimum fluidization velocity definition, 233 prediction Bingham plastics, 239, 241 Carreau fluids, 239, 241 power law, 234 Mixing, see also Axial dispersion and Peclet number liquid-solid fluidized beds, 250 packed beds, 229 three-phase systems, 257 Monte Carlo simulation, radiative heat transfer, 161- 166 N Nonemitting medium, two-flux approximation, 137 Non-Newtonian effects liquid-solid fluidized beds, 232 packed beds, 192 sedimentation, 261 three-phase beds, 253

Non-Newtonian fluids, 189 Nucleate boiling, 7-8, 10-53 circular, free-surface jets, 18-t9 FC-72 and FC-87 mixture, 18-19 heater geometry, 53-54 hysteresis, 49 incipience, 17-18 temperature excursions, 6 - 4 8 investigations, 12- 15 jet suction, 53, 56-57 local measurements along impingement surface, 42-45 modified incipience model, 16 moving impingement surface, 50-5 1 multiple-jet impingement, 35-41 circular water jets, 36-38 nine-jet configuration, 39 saturated water and R-113,40-41 nozzle geometry, 51-53 onset, 11, 16-20, 121 research needs, 121-122 single-jet impingement, 21-35 correlations, 35, 39 fluid properties, 26-27 F98 surfactants, 23-24 free-surface, circular jets, 21-28 free-surface, planar jets, 28-29 jet velocity effects, 21-25, 28-32 local film thickness, 24 nodelheater dimensions, 27, 33-34 nozzle-to-surface spacing, 28, 34-35 saturated liquid nitrogen, 32 saturated water and R-113,26-27, 30-31 subcooling effects, 26,29,33 submerged, confined, and plunging jets, 29-35 surface orientation/impingement angle, 27,29 trends, 35-38 wall superheat, 23, 25, 28 stagnation point temperature, 44 subcoolings, 16 surface conditions, 49-50 surface temperature distributions, 10- 11 temperature excursions, 46-48 thermal shock, 51 transition from single-phase convection, 17 wall jets, 45-46 Null-point calorimeter, 311-313

SUBJECT INDEX 0

One-equation model, 369, 399 comparison with experimental values, 441 comparison with the two-equation model, 449,455 effective thermal conductivity tensor, 399, 453 mapping vectors, 404 Optical method, 321-326 liquid crystals, 322-324 phase-change paint, 322 thermographic phosphors, 324-325

P Packed beds axial dispersion, 229 capillary models, 195 Blake-Kozeny, 195, 197 Blake model, 195, 197 Kozeny-Carman, 195, 197 cell models, 206 free surface, 207, 209-210 zero vorticity, 207, 209-210 dilute polymer solutions, 222 drag coefficient, 207 flow, laminar, 194 friction factor, 196 heat transfer, 229 mass transfer, 229 mixing, 229 nowNewtonian fluid flow, 192 nonspherical particles, 215 polymer adsorption, 226-227 polymer retention, 226 pressure drop, 196 slip effects, 228 submerged object model, 206 tortuosity, 195-196 viscoelastic effects, 206-207 voidage profiles, 216 wall effects, 216 wall-polymer molecule interactions, 194, 206,225 Peclet number fluidized beds, 241-242 packed beds, 229 Penndorf extension, 146-147 Phase-change paint, 322

469

Phase indicator function, 377 Polymer retention in packed beds, 226-227 Porosity, see ako Voidage fluidized beds, 241-242 packed beds, 216 Porous media, see Radiative heat transfer Power law fluid flow liquid-solid fluidized beds, 236 packed beds, 197 sedimentation, 262 three-phase systems, 253 Pressure drop liquid-solid fluidized beds, 241 packed beds, 194 particle shape effect, 215 shearthinning fluids, 194 viscoelastic effects, 217 wall effects, 216

Q Quadrature, 139

R Radiant conductivity, 166-168 Radiative heat transfer, porous media, 133-183 continuum treatment, 134- 136 discrete-ordinates approximation, 139-141 modeling dependent scattering, 168-179 bed properties, 176-177 dependence-included discrete-ordinates method, 173-179 opaque spheres, 171-172 Percus-Yevick model, 169 scaling, 171-173 single particle properties, 174-175 noncontinuum treatment, 161-166 emitting particles, 165-166 opaque particles, 162-164 semitransparent particles, 164-165 phase function, 135 radiant conductivity, 166-168 radiative properties, 152-161 independent scattering limits, 154- 156 opaque spheres, 160-161 particle interactions, 153-156 spectral scattering coefficient, 152-153 transmittance, 156-159

470

SUBJECT INDEX

volumetric size distribution function, 153 single particle properties, 141-151 geometric- or ray-optics scattering, 147-151 Hagen-Rubens law, 143 Mie theory, 141-142, 145 Penndorf extension, 146-147 prediction comparisons, 141-147 Raleigh theory, 142-143 Snell law, 147, 149 solid conductivity effect, 179-181 two-flux approximations, 136- 139 Radiative transfer equation, one-dimensional, 177 Radiometer, 306-307 Rayleigh theory, 142-143 Ray-optics scattering, 147-151 Reciprocating engines, heat flux measurements,343-344 Research, basic, heat flux measurements, 346-349 Resistance-temperaturecoefficients, 289 Resistance temperature devices, 288, 319 Rheological properties, 189 bulk solutions, 226 in situ packed beds, 226 S

Saturated liquid, boiling curve, 6 Scattering, quasi-isotropic, 136-139 Schmidt-Boelter gage, 299-300 Schuster-Schwarzchid approximation, 136-139 Sedimentation, see also Hindered settling; Settling concentrated suspensions, 261 hindered settling velocity, 261 power law fluids, 261 viscoelastic fluids, 261 Segmented plate heaters, 333 Settling, see also Hindered settling; Sedimentation concentrated suspensions, 261 in shearthinning media, 261 velocity, 261 Shearthinningeffects liquid-solid fluidized beds, 232 bed expansion, 241,253

minimum fluidization velocity, 234, 239, 24 1 packed beds, 194 settling, 261 three-phase systems, 253 Shearthinning fluids, 189-190 Sherwood number liquid-solid fluidized beds, 253 packed beds, 230 Skin friction gage, 337-338 Slip effects, packed beds, 228 Slug calorimeter, 307-31 1 Snell law, 147, 149 Solid conductivity, radiative heat transfer, 179-1 8 1 Spectral absorption efficiency, 141 Spectral scattering coefficient, 152-153 Spectral scattering efficiency, 141 Surface tension-controlled regime, 95

T Theory of distributions, 378 convolution product, 379 Thermocouples, 288-290 coaxial, 313-314 differential, 336 thin-skin, 314-316 Thermoelectric sensitivity, 290 Thermographic phosphors, heat flux measurements, 324-325 Thermopile circular foil gage, 304 layered gage, 291 thin-film, 296 Three-phase fluidized beds, 253 axial dispersion, 257 bed expansion, 253 gas holdup, 255 heat transfer, 261 liquid holdup, 257 mass transfer, 259 Tortuosity factor, packed beds, 195-196 Transition boiling, 8-9,108-117 array of circular water jets, 112-113 boiling curve, 111 circular, confined jet, 117 free-surface, circular jet, 112-113 free-surface, planar jets, 116 investigations, 109-1 10

47 1

SUBJECT INDEX planar, free-surfacejet, 108 research needs, 123 subcooling, 108, 112 wetted region size, 113 wetting temperature, 115 Transpiration radiometer, 306-307 Turbulent flow, packed beds, 212 Two-equation model, 370-371 arrays of cylinders, 422 boundary conditions, 430 closure, 386 closure problems, 390,400,404,407 numerical methods, 415 closure variables, 390 comparison with one-equation model, 445, 449 conductive cross-coefficient, 397,423 deviations, representation, 389 dimensionless variables, 431 finite-difference solution, 434 Fourier series solution, 433 heat transfer coefficient, 396,424 interfacial flux, 385 nodular system, 443,456 numerical experiments, 435 stratified systems, 411, 447 symmetric unit cells, 395, 398, 404,410 transport coefficients, 371, 396, 410 volume-averaged equations, 395

Two-flux approximation, radiative heat transfer. 136- 139

V Viscoelastic effects packed beds, 217 sedimentation, 262 Viscoelastic fluids, 189, 191 Voidage profiles, packed beds, 216 Volume averaging, 378 averaging theorem, 379 cellular average, 410 disordered porous media, 384 geometrical theorem, 382 length scale constraint, 382, 385 spatial deviation, 381 spatially periodic system, 384 Volumetric size distribution function. 153

W Wall effects, packed beds, 216 Wall jets, 45-46 critical heat flux, 101-103 Wall-polymer molecule interaction, packed beds, 194,206, 225 Water-cooled circular foil gage, 305 Weber number, critical, 69 Wire-wound gage, 299-300

ISBN 0-12-020023-6