Advances in Heat Transfer, Volume 9

ADVANCES IN

HEAT TRANSFER

Volume 9

Contributors to Volume 9 G. R. CUNNINGTON CREIGHTON A. DEPEW B. GEBHART D. JAPIKSE TED J. KRAMER HERMAN MERTE, JR. C. L. TIEN

Advances in

HEAT TRANSFER Edited by Thomas F. Irvine, Jr.

James I?. Hartnett

State University of New York at Stony Brook Stony Brook, Long Island New York

Department of Energy Engineering University of Illinois at Chicago Circle Chicago, Illinois

Volume 9

@ 1973 ACADEMIC PRESS

NewYork

London

COPYRIGHT 0 1973, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMIITED IN ANY FORM OR BY A N Y MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, N e w York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Ova1 Road. London N W l

LIBRARY OF

CONGRESS CATALOG CARD

NUMBER:63-22329

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS

. . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . Contents of Previous Volumes . . . . . . . . . . . . . . . List of Contributors

viii

ix xi

Advances in Thermosyphon Technology

D.

JAPIKSE

I . Introduction . . . . . . . . . . . I1. Open Thermosyphons . . . . . . 111. Closed Thermosyphons . . . . . IV . Closed-Loop Thermosyphons . . . V . Two-Phase Thermosyphons . . . . VI . Turbine Applications . . . . . . . VII . Future Research . . . . . . . . Appendix: Current Contributions . Nomenclature . . . . . . . . . References . . . . . . . . . . . .

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

. . . .

. . . .

. . . .

2

8 40 72 77 91 98 98

. . . . . . . . . . . . . . . . . . . . . 105 . . . . . . 106

Heat Transfer to Flowing Gas-Solid Mixtures

CREIGHTON A . DEPEWAND TEDJ . KRAMER I. I1. I11. IV . V.

Introduction . . . . . . . . . . . . . . . . . . . . Experimental Observations and Heat Transfer Correlations Fluid Mechanics of Suspensions . . . . . . . . . . . . Analysis . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . v

113 116 134 167 175 176 177

vi

CONTENTS Condensation Heat Transfer

HERMANMERTE.JR.

I . Introduction . . . . . . . . . . . . . . . I1. Nucleation . . . . . . . . . . . . . . . . 111. Liquid-Vapor Interface Phenomena . . . . IV . Bulk Condensation Rates . . . . . . . . . V . Surface Condensation Rates . . . . . . . VI . Mixtures . . . . . . . . . . . . . . . . . VII . Similarities between Boiling and Condensation Nomenclature . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

. . . . .

181 183 . . . . . . 215 . . . . . . 222 . . . . . . 227 . . . . . 264 . . . . . . 266 . . . . . . 267 . . . . . 268

. . . . .

Natural Convection Flows and Stability

B. GEBHART

I . Introduction . . . . . . . . . . . . . . . I1. T h e Relevant Equations . . . . . . . . . I11. Boundary-Layer Simplifications . . . . . . IV . Steady Laminar Boundary-Layer Flows . . . V . Combined Buoyancy Mechanisms . . . . . VI . Flow Transients . . . . . . . . . . . . . VII . Instability and Transition of Laminar Flows VIII . Instability in Plumes . . . . . . . . . . . IX . General Aspects of Instability . . . . . . . X . Separating Flows . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

. . . . . 273 . . . . . . 275

. . . . . .

280

. . . . . .

303 310 321 335 339 342 346

. . . . . . 282

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

Cryogenic Insulation Heat Transfer

C . L . TIENAND G . R . CUNNINGTON 1. Introduction . . . . . . . . . . . . . . . . . . . . I1. Cryogenic Insulation . . . . . . . . . . . . . . . . . I11. Fundamental Heat Transfer Processes . . . . . . . . . IV . Evacuated Powder and Fiber Insulation . . . . . . . .

350 352 356 365

vii

CONTENTS

V . Evacuated Multilayer Insulation . . . . . . . . . . . VI . Test Methods . . . . . . . . . . . . . . . . . . . . VII . Applications . . . . . . . . . . . . . . . . . . . . Symbols . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

Author Index . Subject Index

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

.

381 399 405 413 414

419 429

LIST OF CONTRIBUTORS G. R. CUNNINGTON, Lockheed Palo Alto Research Laboratory, Palo Alto, California CREIGHTON A. Washington

DEPEW,

University of

Washington, Seattle,

B. GEBHART, Sibley School of Mechanical & Aerospace Engineering, Upson Hall, Cornell University, Ithaca, New York D. JAPIKSE, Pratt and Whitney Aircraft, East Hartford, Connecticut TED J. KRAMER, Boeing Company, Seattle, Washington HERMAN MERTE, JR., Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan C. L. TIEN, Department of Mechanical Engineering, University of California, Berkeley, California

viii

PREFACE T h e serial publication “Advances in Heat Transfer” is designed to fill the information gap between the regularly scheduled journals and university level textbooks. T h e general purpose of this series is to present review articles or monographs on special topics of current interest. Each article starts from widely understood principles and in a logical fashion brings the reader up to the forefront of the topic. T h e favorable response to the volumes published to date by the international scientific and engineering community is an indicatior, of how successful our authors have been in fulfilling this purpose. T h e editors are pleased to announce the publication of Volume 9 and wish to express their appreciation to the current authors who have so effectively maintained the spirit of the series.

ix

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CONTENTS OF PREVIOUS VOLUMES Volume 1

T h e Interaction of Thermal Radiation with Conduction and Convection Heat Transfer R. D. CESS Application of Integral Methods to Transient Nonlinear Heat Transfer THEODORE R. GOODMAN Heat and Mass Transfer in Capillary-Porous Bodies A. V. LUIKOV Boiling G. LEPPERTand C. C. PITTS The Influence of Electric and Magnetic Fields on Heat Transfer to Electrically Conducting Fluids MARYF. ROMIG Fluid Mechanics and Heat Transfer of Two-Phase Annular-Dispersed Flow MARIOSILVESTRI AUTHOR INDEX-SUBJECT

INDEX

Volume 2

Turbulent Boundary-Layer Heat Transfer from Rapidly Accelerating Flow of Rocket Combustion Gases and of Heated Air D. R. BARTZ Chemically Reacting Nonequilibrium Roundary Layers PAULM. CHUNG Low Density Heat Transfer F. M. DEVIENNE Heat Transfer in Non-Newtonian Fluids A. B. METZNER Radiation Heat Transfer between Surfaces E. M. SPARROW AUTHOR INDEX-SUBJECT

INDEX

xi

xii

CONTENTS OF PREVIOUS VOLUMES Volume 3

The Effect of Free-Stream Turbulence on Heat Transfer Rates J. KESTIN Heat and Mass Transfer in Turbulent Boundary Layers A. I. LEONT’EV Liquid Metal Heat Transfer RALPHP. STEIN Radiation Transfer and Interaction of Convection with Radiation Heat Transfer R. VISKANTA A Critical Survey of the Major Methods for Measuring and Calculating Dilute Gas Transport Properties A. A. WESTENBERG AUTHOR INDEX-SUBJECT

INDEX

Volume 4

Advances in Free Convection A. J. EDE Heat Transfer in Biotechnology ALICEM. STOLL Effects of Reduced Gravity on Heat Transfer ROBERTSIECEL Advances in Plasma Heat Transfer E. R. G. ECKERT and E. PFENDER Exact Similar Solution of the Laminar Boundary-Layer Equations C. FORBES DEWEY,JR., and JOSEPHF. GROSS AUTHOR INDEX-SUBJECT

INDEX

Volume 5

Application of Monte Carlo to Heat Transfer Problems JOHNR. HOWELL Film and Transition Boiling DUANEP. JORDAN Convection Heat Transfer in Rotating Systems FRANK KREITH Thermal Radiation Properties of Gases C. L. TIEN Cryogenic Heat Transfer JOHNA. CLARK AUTHOR INDEX-SUBJECT

INDEX

CONTENTS OF PREVIOUS VOLUMES

...

Xlll

Volume 6

Supersonic Flows with Imbedded Separated Regions A. F. CHARWAT Optical Methods in Heat Transfer W. HAUFand U. GRIGULL Unsteady Convective Heat Transfer and Hydrodynamics in Channels E. K. KALININand G. A. DREITSER Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties B. S. PETUKHOV AUTHOR INDEX-SUBJECT

INDEX

Volume 7

Heat Transfer near the Critical Point W. B. HALL T h e Electrochemical Method in Transport Phenomena T. MIZUSHINA Heat Transfer in Rarefied Gases GEORGE S. SPRINGER T h e Heat Pipe E. R. F. WINTERand W. 0. BARSCH Film Cooling RICHARD J. GOLDSTEIN AUTHOR INDEX-SUBJECT

INDEX

Volume 8

Recent Mathematical Methods in Heat Transfer I. J. KUMAR Heat Transfer from Tubes in Crossflow A. ZUKAUSKAS Natural Convection in Enclosures SIMON OSTRACH Infrared Radiative Energy Transfer in Gases R. D. CESSand S. N. TIWARI Wall Turbulence Studies 2. Z A R I ~ AUTHOR INDEX-SUBJECT

INDEX

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Advances in Thermosyphon Technology+ .

D JAPIKSE Pratt & Whitney Aircraft. East Hartford. Connecticut

I . Introduction

I1

.

I11.

IV . V.

VI .

VII .

. . . . . . . . . . . . . . . . . . . . . . . 2 A . Classification and Application of Thermosyphon Systems . . 2 B. Property Modeling for Thermosyphon Systems . . . . . . 5 Open Thermosyphons . . . . . . . . . . . . . . . . . . . 8 8 A . General Behavior . . . . . . . . . . . . . . . . . . . 11 B . T h e Circular Open Thermosyphon, Pr > 0.7 . . . . . . . 33 C . T h e Circular Open Thermosyphon. Liquid Metals . . . . . D . Noncircular Open Thermosyphons . . . . . . . . . . . 36 E . Coriolis and Inclination Effects . . . . . . . . . . . . . 37 Closed Thermosyphons . . . . . . . . . . . . . . . . . . 40 A . General Behavior . . . . . . . . . . . . . . . . . . . 40 B . T h e Vertical Closed Thermosyphon . . . . . . . . . . . 41 C . Coriolis and Inclination Effects . . . . . . . . . . . . . 65 Closed-Loop Thermosyphons . . . . . . . . . . . . . . . 72 Two-Phase Thermosyphons . . . . . . . . . . . . . . . . 77 A . General Behavior . . . . . . . . . . . . . . . . . . . 77 B . Two-Phase Phenomena with Small Fillings . . . . . . . . 78 C . Two-Phase Phenomena with Moderate Fillings . . . . . . 84 D . Critical State Operation . . . . . . . . . . . . . . . . 90 Turbine Applications . . . . . . . . . . . . . . . . . . . 91 91 A . Thermosyphons for Turbine Cooling . . . . . . . . . . . B. Review of Thermosyphon Cooled Turbines . . . . . . . . 93 Future Research . . . . . . . . . . . . . . . . . . . . . 98 Appendix: Current Contributions . . . . . . . . . . . . . . 98 Nomenclature . . . . . . . . . . . . . . . . . . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . 106

+ This work was initiated while the author was an N D E A Title IV Graduate Fellow at Purdue University. continued while conducting postgraduate research at the Technische Hochschule Aachen. W . Germany as an N S F Postgraduate Fellow and concluded while working as an Assistant Project Engineer. Pratt & Whitney Aircraft. East Hartford. Connecticut .

.

1

2

D.

JAPIKSE

I. Introduction

A. CLASSIFICATION AND APPLICATION OF THERMOSYPHON SYSTEMS A thermosyphonl is a circulating fluid system whose motion is caused by density differences in a body force field which result from heat transfer. Mechanical inputs have so far been excluded from all thermosyphon studies. Davies and Morris (24) have suggested that thermosyphons can be categorized according to (a) the nature of boundaries (is the system open or closed to mass flow?), (b) the regime of heat transfer (is the process purely natural convection or is it mixed natural and forced2 convection ?),(c) the number or type of phases present (is the system in a single- or two-phase state ?) and (d) the nature of the body force (is it gravitational or rotational ?) Unfortunately a definition as broad as the one given above would require the preparation of a book, not a review article, to do it justice. I n fact, the above definition, suggested by Davies and Morris in 1965, is so broad as to include all natural convection processes, plus others, and thus it is well to note that all systems to which the name thermosyphon has been applied in formal studies (except the discussion by Davies and Morris (24)) are in fact systems which have the intrinsic function of removing heat from a prescribed source and transporting heat and mass over a specific path (frequently a recirculating flow) and rejecting the heat and or mass to a prescribed sink. That is, the path of the circulating flow which transports the thermal energy is or can be totally prescribed. Thus, for example, while ordinary free convection from plates and cylinders may tacitly meet these criteria, they generally are of interest only from the standpoint of rejecting heat and the subsequent transporting is of secondary or of little interest. Indeed, in industrial applications the path of heat flow in such a free convection process is rarely prescribed and will vary considerably. Furthermore, thermosyphon flows are intrinsically driven by thermal buoyancy forces, either locally or in an overall sense. A simple loop flow may well be the result of local buoyancy forces alone, but a multibranched flow circuit can easily incorporate sections in which the flow direction is contrary to the local buoyancy force resulting from pressures created by the overall system buoyancy forces. Based on these factors, the following definition will be used in T h e origin of the name “thermosyphon” is uncertain; however, the name appeared as early as 1928 in the sales literature of Deere and Co. to aptly describe their cooling system. Mixed convection requires a dividing partition across which pressure differences can be established.

ADVANCESI N

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3

this review (roughly following the definition used also by Lock (82)): A thermosyphon is a prescribed circulating fluid system driven by thermal buoyancy forces. This definition includes all basic studies to which the name thermosyphon has been applied in the literature (with the exception of parts of Davies and Morris (24),which is not a study of any particular system but rather a general discussion) and clearly defines a class of thermal systems which have become industrially important. T h e preceeding distinction notwithstanding, Davies’ subcategories are still very convenient and will be used. T h e most common industrial thermosyphon applications include gas turbine blade cooling (3, 9, 14, 20-22, 27, 33, 36, 37, 39-42, 44, 54, 65, 67, 93, 97, 98, 101, 107, 112, 113), electrical machine rotor cooling (25, 38, 95, 96), transformer cooling (68, 71), nuclear reactor cooling (23, 48, 92, 114, heat exchanger fins (73, 74, 85), cryogenic cool-down apparatus (10, I1,43, 69),steam tubes for bakers’ ovens (94),and cooling for internal combustion engines (70, 111, 115). Other intriguing thermosyphon (or very closely related) problems include the convection in the earth’s mantle (102), the temperature distribution in earth drillings in steam power fields (28), plus the use of thermosyphons for the preservation of permafrost under buildings in the Canadian northland (66, 76,84), and the maintenance of icefree navigation buoys (74). A variety of thermosyphon characteristics are responsible for the applications found to date and can lead to numerous future applications. For example, a thermosyphon can behave as a thermal conductor with either a small or a large thermal impedence depending on system choice; it can be used as a thermal diode or rectifier (43, 74); or even as a thermal triode (43), permitting a variation in heat flow based on small changes in temperature. Table I shows a large variety of thermosyphons which have been studied and/or are in use today. T h e application of thermosyphons to gas turbine blade cooling has clearly played a key role in thermosyphon research and will receive special attention later. T h e first section of this review considers a common single-phase, natural-convection open system in the form of a tube open at the top and closed at the bottom; the second section considers a simple singlephase, natural-convection closed system in the form of a tube closed at both ends; the third section considers various single-phase, mixedconvection thermosyphons, so-called closed-loop thermosyphons; the fourth section reviews two-phase3 and critical state thcrmosyphons and A note about semantics is in order. These systems have occasionally been called “wickless heat pipes” which is unfortunate since a wick is an integral and important part of a heat pipe. Any such system without a wick should certainly he considered a two-phase thermosyphon.

TABLE I: CLASSIFICATION OF THERMOSYPHONS AND EXAMPLES OF THEIR APPLICATIONS” h

. r

Open systems Heat-transfer regime

Body force

Single phase

Two phase

Closed systems Single phase

Two phase

Static

Hot springs Warming kettles

Washing machine boilers Kettles

Electric immersion domestic hot-water heaters Ovens Oil-filled electric convector heaters (internal)

Fire-tube boilers Hydrometeorology Baker’s ovens Ice prevention system for navigation buoys Heat exchanger fins Cryogenic cool-down equipment

Rotating

Axial-flow gas-turbine blade cooling

Axial-flow gas-turbine blade cooling

Axial-flow gas-turbine blade cooling

Axial-flow gas-turbine blade cooling Rotary condensers

Free convection

Static

Mixed convection

Cooling of encased Steam fields electrical equipment Fireplace and chimney

Rotating

Adapted from Davies and Morris (24), according to the revised definition.

Gas-fired domestic hot-water heaters Gravity-flow central heating (internal) Transformer cooling (internal) Car-engine cooling (internal) Nuclear reactor cooling Heat exchanger fins

Water-tube boilers Hydrometeorology with water power Transformer cooling (internal) Coffee percolators Annular jet mercury vapor pumps

Axial and radial-flow gas-turbine blade cooling

Electrical-machine rotor cooling

U cl

>

2 E m

ADVANCES IN THERMOSYPHON TECHNOLOGY

5

finally a review of the turbine blade cooling problem is given in the fifth section. I t is hoped that the review of these systems, which includes all basic thermosyphon studies, will provide a background of information for related and new thermosyphon problems. However, before examining these systems it is profitable to consider the matter of suitable property modeling in all thermosyphon problems.

B. PROPERTY MODELING FOR THERMOSYPHON SYSTEMS With the exception of density, all thermosyphon analyses to date have assumed constant properties; hence it is quite important to make a wise choice of reference temperature; indeed, poor choices have led to very sizeable errors in calculating heat transfer. Table I1 shows a few property variation ratios which illustrate the nature of variations possible. Table I1 shows clearly that thc most important property variation for ordinary liquids is that of viscosity. Hence Lock (82) neglected all property variations except p (and of course included p( T ) )and found that the integral momentum and energy equations can be reduced directly [see Eq. (9)] to show that the wall temperature is the appropriate property reference temperature. This somewhat unusual reference temperature has fortunately been used in nearly every open thermosyphon study. It might also be mentioned that this choice is also the most practical since the use of, say, the core temperature, is often difficult to predict. I n one case, Foster (34), the core temperatures were measured and a film temperature employed; regrettably this choice led to the conclusion or result that Nu decreased with increasing Pr, contrary to all other thermosyphon findings and general free-convection knowledge. In short, the use of the core temperature is undesirable; the wall temperature has proven most reliable. For treating liquids in the closed thermosyphon, it has been shown by Japikse and Winter (59, 60) that the wall temperature in each tube half should be used to model the flow process in that tube half. This is of considerable importance because not only can heat transfer rates be in error by as much as 50% if only one reference temperature is employed, but it is occasionally impossible to recognize the mode of flow which exists if this rule is not employed (see Japikse (62) for a discussion of two such cases). For gases, Table I1 shows that property variations do not appear to be too large; but they are sufficiently subtle to make u p the difference. Consider for the moment the Gr number, now based on the film temperature for purposes of discussion: Gr

= gp

OTa3/v2

6

D.

9 -

9

m

3

m

9

m -

-

-

o * 9 9 9 3

JAPIKSE

2

8

p!

M

2

00

3 3

W

*

8

00

2

ADVANCES IN TIIERMOSYPHON TECHNOLOGY

7

using the perfect gas equation and the viscosity power law relation,

rearranging and using the definition of the film temperature

If we consider a variety of different T , cases, assuming a constant pressure and a constant T , , we find that G r has a maximum with respect to T , at T,/T, = 5.4/3.4; i.e., increasing T , (or A T ) in order to increase G r has a point of no return: P/v2 begins to decrease faster than AT increases, and thus heat transfer is no longer increasing for increasing A T past this point. This concept has been recognized for some time in free-convection problems. On the other hand, for a given T , case we can consider what happens when T , varies, which we will see later can occur at times. I n this event, G r has a maximum at T J T , = 5.413.4, which obviously is never obtained for heating since then T , > T , but for cooling, such as occurs in one half of a closed thermosyphon, a maximum is obtained beyond which an increase in T , or AT will cause a decrease in G r and hence heat transfer. T h is latter point was recently pointed out by Biggs and Stachiewicz (12) and of course the former case was formally generalized from it. This use of the film temperature for gases was chosen for illustration. So far the wall temperature has been used for a reference temperature for convenience alone when considering open and closed thermosyphons operating with a gas. T h e previous argument based on strong p variations common to liquids for using the wall temperature is no longer pertinent for gases but the film temperature is very difficult to use due to the need to know T , . For closed-loop thermosyphons, the film temperature should be a very suitable choice since, as will be seen, the flow is now a mixed-convection problem where forced convection is significant. Furthermore, a mean fluid temperature can now be calculated. It would be wise, however, to model each loop segment using its own fluid properties. Two-phase system studies have so far used the film temperature definition for liquid film properties and suitable bulk temperatures for vapor phases and boiling pools. With the preceding categorization scheme and property reference conventions established, attention can now be given to the four basic thermosyphon systems and applications. A vast amount of information has been obtained for these systems, which permits one to achieve a rather

8

D. JAPIKSE

broad understanding of them. However, the reader will soon note that several fundamental questions still need to be resolved and divergent views exist in some areas. Thus the following pages present both a general picture, as complete as possible, of the phenomena involved plus a case by case review of the fundamental studies so as to compare various findings. Undoubtedly, the reader will wish to formulate additional conclusions himself and perhaps conduct further research in the areas of greatest need.

II. Open Thermosyphons A. GENERALBEHAVIOR T h e open thermosyphon shown in Fig. 1 provides a basic starting point for considering thermosyphon systems. Although other open Acceleration Field Reservoir

w

v -0-1

FIG. 1 .

T h e open thermosyphon.

thermosyphons exist, as shown in Table I, no other version has been so thoroughly studied or is so basic to the general subject. Hence this section is devoted entirely to the simple open thermosyphon as depicted in Fig. 1. As can be readily appreciated by studying Fig. 1 , the primary effect of heating the wall of an open thermosyphon should be to cause some type of flow upward along the wall due to buoyancy effects and an associated return flow downward in the core via continuity. Specifically, it has been found that for large heat fluxes, the buoyancy forces are sufficiently intense near the wall so that a boundary layer regime is obtained. For

ADVANCES IN THERMOSYPHON TECHNOLOGY

9

weaker heat fluxes, the buoyancy forces are less and the effect of shear is relatively enhanced causing the boundary layer to try to fill the entire tube or, in other words, for the effects of wall shear to be significant throughout the tube. Hence, compared with boundary layer flow, the flow is impeded. For sufficiently weak fluxes of thermal energy through the wall, this effect has been found to become rather uniform and a similarity flow is realized with, for still weaker fluxes, a stagnant bottom region. T h e effect of geometry, as expressed by L/a, is to accentuate the trend so that for increasing L / a larger values of Ra, are necessary in order to attain any given heat flux level. T h e effect of property variations, as expressed by Pr, is to increase heat transfer for increasing Pr under boundary layer conditions and to decrease it for the impeded or similarity flow conditions. Figure 2, in which mainly experimental results from

loglo t,l

+

Heat transfer in the open therniosyphon, Pr > 1 . Analysis, Lighthill (80); ethylene glycol; glycerine; - - - -, water. Martin (87): a, L / a = 7.5; b, L / a = 32.5; c, L / a = 47.5. Hasegawa (52): d, L / a = 31.6; e, L / a = 45.8; f,L/a = 96.8 and 99.8. Freche and Diaguila (36): slanting lines, rotating test, L / a = 40; see text, Section 1II.C.

FIG. 2.

-,

--

various workers for Pr > 1 are exhibited, shows these various regimes for laminar flow and various other results, presumably turbulent. This brief description of the flow processes applies directly to laminar flow, but also has some bearing on turbulent flow.

D.

10

JAPIKSE

Boundary layer transition to turbulent flow has been reported (63) to follow the trend Grcrit cc P r 4 I 3which is quite different from normal free-convection boundary layer transition which varies directly as Pr. It is argued, as will be discussed later, that the velocity profile inflection point and the adverse pressure gradient imply, to a first approximation, a variance such as Grcrit a Pr-l. Transition can also occur from laminar impeded flow, but the conditions for such transition have not been systematically investigated. T h e problem of describing turbulent flow processes in the open thermosyphon is at once far more complex than the laminar one. Again, one can expect to find similarity, impeded, and boundary layer flows if the appropriate conditions for transition are achieved. A general picture can be obtained by noting (as did Lighthill (80)) that for laminar flow Q cc v1l2 in the boundary layer regime and Q a v-l in the impeded regime. Since turbulence is, in a gross sense, a large increase in v (the other properties remaining fixed), one would anticipate a reduction of heat transfer in the impeded regime under turbulent flow conditions and an increase in heat transfer in the boundary layer regime. However, if transition occurs in the impeded regime, this boundary layer regime might not be attainable at all (63,87)and any subsequent increase might be only a general trend or tendency. Figure 2 shows experimental results which have been reported for turbulent flow for fluids with Pr > 1 and data for liquid metals are given later in Fig. 13. Three types of turbulent flows have been reported (52,87): turbulent boundary layer with a laminar core, laminar boundary layer with a turbulent core and fully turbulent (impeded) flow. Many of the impeded results for a vertical system were successfully correlated in the form Nu,

=

C1Ra,m(a/L)C2

(4

where C , and C, were found to be functions of radius. This dependence on radius is evident in Fig. 2 where lines b and c are in poor agreement with lines d and e, respectively. T h e difference is successfully correlated by including the radius. T h e above three paragraphs provide a very brief description of the major laminar, transition, and turbulent open thermosyphon characteristics, respectively. T h e remainder of this section gives a chronological description of the work done on the open thermosyphon, subdivided into recognized basic areas. Although the following remarks have been greatly reduced to include only those fibers which weave the general fabric of knowledge, divergent views will be found and the reader will want to keep the preceding concepts clearly in mind.

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B. THECIRCULAR OPEN THERMOSYPHON, 1% >, 0.7 T h e first experiments4 on a static open thermosyphon were reported in 1951 by Foyle (35),but they were of very limited extent. Foyle noted that eddy motion could be observed by fluid passing between the tube and reservoir. H e also observed that the heat transfer was independent of tube length beyond a certain critical value, although it increased quickly with increasing diameter. Several intensive studies followed these preliminary findings and they appeared almost simultaneously. Foster (34) presented an experimental study, and Lighthill (80) presented a detailed analytical study. A more detailed experimental study followed soon after by Martin (86,87) in 1954 and 1955. Foster (34) carried out a number of experiments using water, a light oil, and a medium weight oil as test fluids. His thermosyphons were constructed with fixed length and various diameters. Heat was supplied to the thermosyphon by convection from a circulated heated oil which generally gave a nearly constant wall temperature, although several cases showed moderate wall temperature variations. T h e heat transfer rate was measured by an energy balance first on the heating oil and later on the cooling water. T h e latter measurement was best but still yielded considerable scatter. A traversing probe was used for radial temperature profiles, and thermocouples free to move in longitudinal wall holes were used to measure the wall temperature distribution (apparently no correction was made to correct these wall values to the true inside surface temperature). T h e laminar radial temperature profiles which he measured showed a fairly constant core temperature. Those profiles were plotted in a general manner by using the independent variable c Y ; X 1 J 4where , c is an arbitrary constant, as is commonly done for free convection on a flat plate. Although no evidence was found for mixing between the central core flow and the boundary layer flow in the largest (3-in.) diameter tube, such mixing did occur in all the other tubes. T h e turbulent temperature profiles showed that the heat transfer rate was greater near the open end than near the closed (the opposite being true for laminar flow). T h e profiles further demonstrated that this core temperature increased from the open end t o the closed end in roughly an exponential manner and For accuracy it should be noted that Holzwarth (54) in 1938 and Schmidt et al. (106, 107) during W.W. I1 considered the open thermosyphon for turbine blade cooling and Eckert and Jackson (30) in 1950, plus Jackson and Livingood (57), prepared further such considerations; but with the exception of the specialized critical-state investigations of Schmidt, the basic thermosyphon fluid How and heat transfer studies began after these initial application oriented investigations-which largely prompted the subsequent work.

12

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faster for the smaller-diameter tubes. Hence the effect of mixing of the downward flowing core with the rising wall flow was evident and significant. For water, transition to turbulence was observed at GrLPr e 10’0 for the 3-in. tube and equal to about lo9 for the thinner tubes. For light oil in the smaller tubes, transition occurred at a value of about 1O1O. Foster’s primary results were heat transfer correlations for turbulent flow conditions of the form, using his nomenclature, Nu,

=

C’(GrLPrm)n,

where the temperature difference is based on the average wall temperature minus the average centerline or core temperature. His choice of temperature difference prevents direct comparison with any other studies and is not well founded. Property values were based on an average film temperature. T h e problems of using such an average film temperature have been noted in the introduction. Different values of C’, m, and n were needed for each tube and hence a dependence on diameter was evident. T h e results for the 3-in. tube showed good agreement with an expression given by Eckert and Jackson (30) for turbulent free convection on a vertical flat plate. Foster further observed that since the values of m were less than unity, the viscous forces dominate the inertial forces. It is unfortunate that Foster did not choose other coordinates for presenting his heat transfer results, because further trends could be found and because later investigators have come to accept alternative variables. Much of his data has been recomputed and replotted (59); using the conventional AT = Tl - To and Tref= Tl , it was found that N u increases with Pr, contrary to the result obtained by Foster using the unconventional data reduction method. Furthermore, his heat transfer data (when replotted) agrees quite well with subsequent experimental and analytical investigators. It is also interesting that his equation for N u L , which is similar in form to Eq. (2), had constants which likewise depended on diameter (virtually all subsequent studies varied length, not diameter). However, it is difficult to accept this study as conclusive evidence of diameter dependence since, as Foster himself was aware, considerable scatter existed in his data; it is not clear to what extent his values of C‘, m, and n reflected a statistical average of experimental inaccuracies or to what extent they reflected true dependencies. For example, the values he obtained for n (0.4,0.372,0.492,0.322 for d = 3, 1, $, =$-in.) seem, upon examination of his work, to reflect experimental uncertainties rather than real trends (in most experiments, n tends to be a nearly uniform constant). T h e analytical study presented by Lighthill (80) has served as the

ADVANCES IN THERMOSYPI-ION TECHNOLOGY

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foundation for most thermosyphon analyses (laminar) u p to the present time; in it the basic flow regimes were introduced. His treatment of flow in the thermosyphon yielded three laminar regimes of flow (which have been very well confirmed experimentally) and three corresponding turbulent regimes (for which significant differences have been found). These flow regimes and the corresponding solutions will now be examined in detail. I t is interesting to note that Lighthill formulated all these solutions without experimental results to guide his way. T h e basic laminar flow equations are of the boundary layer type since 1. These the flow is generally either boundary layer flow or L/a equations are5

>

apuR -+ax

U-

ac,T ax

apvR = aR

+ I/” aaRc

o

(3)

~1 1 a p R aR

- - --

p-l = P F ( 1

(Kk --)aT aR

+ P(T - TI));

(7)

and the corresponding boundary conditions are R

=

T T

a,

x = 0,

=

T,(x) or

q

=

q,(x) and

=

T,

q

=

qb

or

and

U U

= =

V V

=0 =

0 (8)

X = L and R = O , T = To T h e term aP/ax can readily be eliminated by using the value of aP/ax at R = a obtained from Eq. (4) and the equations can be integrated without any approximation across the tube to give

sl

pRU dR

a sapRU2 dR ax

=

=

0 pgP(T

-

T,)R dR

0

a a

ax I,,pUc,TR dR

=

aka

These equations are presented (only once) in this review in a more general form than given by Lighthill in order to include other results.

14

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Now the important point to observe is that all p terms appear only at T , and since for liquids p is by far the most important property to model (see Table 11), T , is the obvious choice for a property reference temperature. T h e relative effect of p, cp , k variations can be neglected (for liquids) and neglecting ap/aR l a as well, these equations integrate to give

/: R U d R

=

0

where the wall temperature property reference is employed; this reduction is quite similar to that formulated by Lock (82). T h e equations are also used for gases by simply assuming constant properties at the outset. Using nondimensional variables one obtains

--I

l a 1

Pr ax

,

ruZdr=

-1

1 0

T h e differential energy equation at Y = 1 and the momentum integral equation at Y = 0 are very usable. These are, respectively,

.,[

a2t

,_I,

at

+

=

O

and the boundary conditions become (in a more restrictive form than Eqs. (8), since only the following have been treated): at

Y =

1,

x = 0, x =

1 and r

=

0,

t = tl(x)

and

u = ZJ = 0

atlax =

o

and

u = ZJ =

t

to

=

0 (15)

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Lighthill began with laminar boundary layer flow and treated only the case of Pr = CO, thus he neglected the inertia terms [by setting Pr = 00 in Eqs. (11) and (14)] and estimated that this would give only about 10% error for Pr = 2. Using a Pohlhausen integral technique with equal thermal and momentum boundary layer thicknesses, he was able to obtain a direct solution to Eqs. (10)-(15) (see Fig. 3) without

Loglo t o t

FIG. 3. Heat transfer in the open thermosyphon with laminar flow (80). Notes: ( 1 ) similarity flow with stagnant bottom portion; (2) similarity solutions; (3) nonsimilarity flow with boundary layer filling the tube; (4)to the left of the cross involves a physical impossibility; ( 5 ) boundary layer flow; ( 6 ) limiting case of free convection on a vertical flat plate.

having to perform a detailed integration. He used a parabolic temperature and a cubic velocity profile in this analysis. T h e author stated that the boundary layer flow ceases before it fills the entire tube. In fact it ceases when the volume flow rate is no longer a maximum at the orifice. This was shown to occur for t , < 3400. This entire solution should be asymptotic to the solution for flow on a flat plate for large u/L values. Indeed, he noted that the entire solution is but little different from this asymptotic solution. He attributes this to the balancing of two effects. First the flow of cold fluid down the core increases the heat transfer, as in forced convection, but, secondly, the entire scale of motion is diminished by the larger viscous stresses. For smaller values of the to, parameter, Lighthill again used integral techniques to solve the problem where the viscous effects are significant throughout the entire tube. Velocity and temperature profiles were chosen as even functions of r and general functions of x. He found that the flow would still be similar to that shown in Fig. 1, but that a standing ring vortex should exist near the bottom of the tube. This flow could exist up to values of to, = 5600. The centerline temperature distribution was found to increase from the orifice to the closed end of the thermosyphon,

16

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and that the local heat transfer rate decreased from the orifice on down. He noted that the tot ranges for boundary layer flow and flow filling the tube have a definite overlap and that there exists a gap between the two N u vs. to, curves. Lighthill pointed out that a hysterisis effect thus might be possible or at least unsteady flow conditions should prevail. Finally, for very small tot values, the flow shows no change in profile type, only in scale. Hence a similarity solution became possible and he demonstrated that the velocity and temperature profiles must be linear functions of x. This solution occurred at a value of tot = 311. For smaller values of to, , a stagnant region (or a region with negligible motion) was predicted. Before presenting the corresponding solutions for turbulent flow conditions, it is necessary to review several of Lighthill’s general hypotheses concerning these cases. Lighthill noted first that two conditions promoting turbulence are the cross-sectional shear distribution which has a maximum in the thermosyphon flow and the upward temperature gradient which is negative. Opposing the development of turbulence were the viscous damping effect and the effect of the walls’ proximity. Although he believed that wall roughness was insignificant due to the fairly low velocities, he strongly stressed the importance of large inlet variations on the early development of turbulence. This later factor lies strongly behind all his predictions of transition to turbulence. I n searching for a turbulent exchange coefficient he ruled out mixing length theories due to the three stationary values in the velocity profile and he also discounted the possibility of small-scale turbulence that would reach an equilibrium intensity dependent only on local flow parameters. Rather he hypothesized that the exchange coefficient in confined turbulent flows may depend solely on position and some factor representing the scale of turbulence, perhaps the wall shear stress. Such a hypothesis allowed him to use the exchange coefficient for a straight pipe; he also asserted that only the average value over a cross section of factors causing turbulence is now necessary. Again, he ignored the inertia terms relative to the buoyancy force and shearing stress. It should be recalled that this is in agreement with Foster’s findings. For turbulent boundary layer flow, Lighthill carried out no new analysis but rather cited Saunders’ (103) experiments which gave the Nusselt number as Nu, = 0.1 lGr:’3 for Gr, < 1011. Lighthill predicted that for sufficiently large to, this boundary layer regime would appear.

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For turbulent flow filling the entire tube, Lighthill used the abovementioned exchange hypothesis plus Reynolds’ analogy to obtain a solution for Pr = 1, as depicted in Fig. 4. He used as a boundary 3

-

-

1

0

~~

Boundaryloyer not filling tube

-/ I

-I

-

~

4

5

6

7

8

9

1

1

I

1

2

log,oGr.

FIG. 4. Heat transfer in the open thermosyphon with laminar and turbulent flow (80). Notes: The x denotes a similarity solution; transition has been judged by assuming large entrance effects.

condition the fact that the turbulent flow should subside as the tube bottom is reached where laminar flow would prevail. This is borne out, at least qualitatively, by Foster’s centerline temperature profiles where the centerline temperature becomes nearly constant near the bottom, presumably for laminar flow. For the turbulent similarity solution, he used a general argument to show that the temperature scale must vary with eAx and the velocity scale with eAx/2.Due to the approximate nature of his argument, he suggested that h varies and then solved for h which was a weak function of x. In completing this solution, he again applied the boundary condition of laminar flow at the tube end. This quasi-exponential variation agrees, qualitatively, with Foster’s temperature profile results. All of Lighthill’s solutions are presented in Fig. 4 which also reveals the results of his expectations concerning transition from laminar to turbulent flow. I t should be noted that he placed great emphasis on the importance of strong entrance effects in causing laminar to turbulent transition. Martin, who made a large number of studies on the open thermosyphon, completed his first studies (86, 87) soon after Lighthill’s analytical models were presented, and many areas of agreement were found. Martin’s studies were conducted in an opaque test cell of variable length heated by five electrical heaters each of which could be controlled

D.

18

JAPIKSE

separately so as to assure a reasonably isothermal wall temperature distribution. H e was able to measure wall temperature distributions and centerline temperatures as well as local and total heat fluxes, accounting for losses to the environment. His indicators for turbulence consisted of observing reductions in heat transfer, changes in local heat transfer along the tube, and variations in wall temperature readings. Martin's laminar flow heat transfer results provided an excellent verification of Lighthill's models as shown in Fig. 5. T h e local heat Pr I

5000 2500 1000 500 200 100

1000

Pr 500 2 5 0

100

I (b) Ot'; -0.5' 4

5

6

'

' 5

'

'

'

' 1 7

6 loglo Roo

loglo Ra,

FIG. 5. Heat transfer measurements with laminar flow (87). Notes: Lighthill (80); o glycerin, L / a = 47.5; 0 rapeseed oil, L / a = 47.5.

~

analysis,

transfer results for low values of tot also verified the existence of a laminar impeded regime (analogous to Lighthill's second regime) and indicated an effectively stagnant bottom portion for sufficiently low values of t o t . As the parameter increased, distinct fluctuations arose in the region between the impeded and boundary layer flows and were quite periodic in nature. Martin observed them from looking into the reservoir and reported up-flow and down-flow surges. He suggested that these were caused by temporarily higher velocities near the wall as the flow tried to form a boundary layer, with the core flow similarly increasing in magnitude. As this process would only affect the part of the tube near the orifice, the core flow would soon be decelerated and, correspondingly, the boundary layer flow as well. Preheating of the core was thus possible and the overall Nusselt number was consequently increased tending toward the laminar boundary layer flow again. Finally, his boundary layer results were in good agreement with Lighthill's predictions, though Martin noted that Nu varied with to, to the 0.28 power as opposed to the 0.25 power predicted by the Pohlhausen solution for free convection on a vertical flat plate.

ADVANCES IN THERMOSYPI-ION TECHNOLOGY

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Martin found two distinct modes of turbulent flow. For fairly viscous fluids, the boundary layer would become turbulent with the core remaining laminar. It was established that two different relations correlated the data quite well, the first for large L / a values and the second for smaller ones. These relations are

> 75 (L/a = 32.5, 40, 47.5)

Nua

=

0.0325Ra:/5

200 > Pr

N~~

=

o.IoR~;/~

200 > Pr > 60 (L/a = 15, 22.5).

(16) (17)

Equation 17 was also found to be valid for water in short tubes. Turbulent boundary layer flow is, however, ultimately unstable, and for larger to, a fully mixed turbulent flow develops with lower heat transfer due to mixing of hot and cold fluid. These effects are evident in Fig. 6a. Pr Pr

O

W

5

.

6'

7

"

"

4

LogloRa,

8

7 6 4 3

5

6

7

loglo Ra,

FIG. 6. Heat transfer measurements with turbulent flow (87). Notes: -laminar turbulent analysis, Pr = 1 , Lighthill (80); analysis, P r = CO, Lighthill (80); o water, L / a = 47.5; ethylene glycol; x air; single arrow, transition to turbulence; double arrow, transition to fully mixed or impeded flow.

For less viscous fluids (air or water), transition occurs directly from the similarity or nonsimilarity impeded flow regimes as shown in Fig. 6b. A sharp crevice was observed in the heat transfer rate approaching the results of Lighthill's turbulent nonsimilarity flow solution giving fair agreement for air. Then the heat transfer rate rose sharply, contrary to Lighthill's prediction. Local heat transfer results indicated that the minimum might correspond to the turbulent similarity solution published by Lighthill. Martin suggested that the inherent tendency toward boundary layer flow causes the increase. Once past the crevice the fully mixed regime gave a rather constant value of the Nu number. T h e Nu number clearly is not as high as either in laminar or turbulent boundary layer flow, but it is at least twice as large as Lighthill's nonsimilarity turbulent (impeded flow) solution. Finally, Martin found no evidence of Lighthill's purely turbulent boundary layer flow ever occurring beyond the fully mixed turbulent flow regime.

20

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Martin investigated quite closely the problem of predicting the onset of turbulence and concluded that only rough guides could be given since transition is a complex function of viscosity and geometry. He noted that for ethylene glycol, transition to a turbulent boundary layer usually occurred for Pr rn 200 and to fully mixed turbulence for Pr m 60. A necessary but not sufficient condition for turbulence appeared to be that one of the curves given by the previous Q- or +power heat transfer laws must first intersect the laminar heat transfer curve. Based on RaL , he found that transition occurred for values of RaL between 109.45and 1010.55for ethylene glycol, 10s.75-109.75for water, and 107.0-109.4for air. Each range corresponds to increasing Llu. These values agree basically with those reported previously by Foster, but the effect of increasing Lja is the opposite. T h e effect of the L/a parameter is extremely important for design purposes and Martin investigated it extensively. For turbulent ethylene glycol tests, the $-power heat transfer law held for L/a = 40.0, whereas for L/a = 22.5 and 15.0 the +-power law held. For L/a = 32.5, the Q-power law held initially followed very soon by a :-power law as the parameter to, was increased. Fully mixed flow eventually developed but only for L/a greater than about 35 did the resulting heat transfer drop below the laminar boundary layer curve for the range of to, considered. For his water tests, fully mixed flow was obtained immediately for Lja 7 20. With L/u = 15 a turbulent boundary layer was set up giving rise to a Q-power heat transfer law prior to the fully mixed regime, whereas for L/a = 7.5, transition was delayed until fully mixed flow was established. T h e Nusselt number was always less for water than in the corresponding ethylene glycol L/a experiment and never greater than in the laminar boundary layer flow experiments. Hence for given Ra and L/a numbers, Nu increases as Pr increases, whereas if Ka and Pr are constant Nu, varies inversely as Lja. Hartnett and Welsh (49) conducted the first thermosyphon experiments using a constant wall heat flux condition and water as a test fluid. T he authors plotted their heat transfer results on a Nu vs. G r Pr plot, using an average wall temperature minus the inlet temperature ( T o ) for the basic temperature differences. By comparing these results to other studies, they were able to assert that the average performance for the constant flux case is equivalent to that for the isothermal wall case. Ostrach and Thornton (99) have extended the laminar similarity solution of Lighthill (80) to include the case of linearly increasing or decreasing wall temperature. T h e first detailed analytical study concerned with the effects of variable Prandtl number was presented by Leslie and Martin (78) in 1959. They

ADVANCES IN TIIERIMOSYFIION TECHNOLOGY

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considered in detail the similarity and boundary layer regimes in laminar flow. Their results for the similarity regime, see Table 111, showed improved agreement with Martin's data for glycerine and rapeseed oil. T A B L E I11 VALUESOF tot

FOR

VARIOUSSIMILARITY SOLUTIONS Axisymmetric

Pr

Theory (78)

co 3000 1150 10 1 0.71

349,337"

Experiment (87)

Channel theory (83)

163 420 334

325" 210" 159"

Higher-order terms are omitted here.

They found that the similarity solutions could be extended as low as Pr = 0.4 so as to include all known gases, but that imaginary results were obtained for lower Pr numbers. This indicated that the similarity regime does not occur for liquid metals. Their solution for boundary layer flow was obtained by assuming equal thermal and momentum boundary layer thicknesses and eliminating the axial space coordinate between the governing equations of heat and momentum flow, Eqs. (lo)-( 13). They obtained a series solution to the resulting nonlinear, nonconstant coefficient ordinary differential equation and neglected terms higher than order one where the relative boundary layer thickness was the pertinent independent variable. Leslie and Martin felt that this approximation would be valid as long as the boundary layer thickness was less than 0.1. Some of their heat transfer results are shown later in Fig. 9. Finally, they considered the limiting condition which requires that the volume flow rate at the orifice must be a maximum (in order to have boundary layer flow) and found that boundary layer flow must cease when the Nusselt number becomes less than approximately 5.8 regardless of the Prandtl number. T h e first flow visualization for the open thermosyphon and also the first extensive treatment of the fully mixed turbulent regime was presented by Hasegawa et al. (51) in 1962 in Japanese and in 1963 in English (52) as a comprehensive experimental study of the open thermosyphon. Hasegawa et al. used several test cells both for flow

22

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visualization and heat transfer measurements. Most visualization experiments were conducted in a parallel-walled test cell on which Schlieren or shadowgraph techniques were used. A few observations were made in tubes with dye. They were able to measure heat transfer in the parallel plate apparatus, but did most of this work in two cylindrical cells. T h e first had a fixed length and electrical heaters with variable power input which allowed a nearly constant wall temperature profile to be effected. T h e second test cell consisted of a tube submerged in a stirred heated bath. This arrangement allowed easy variation of tube length. Each had a suitable reservoir and in spite of slightly different wall temperature profiles, comparable results were obtained. Various fluids were used. Finally, in all cases the base boundary condition appears to have been nearly one of constant temperature which was the same as the wall temperature. T h e flow visualizations of Hasegawa et al. confirmed the boundary layer regime and the steady laminar regime wherein the boundary layer extended throughout the entire test cell. Between the two regimes they observed an unsteady flow mechanism such as described before by Martin (86, 87) in which the flow tries to establish a boundary layer but is only temporarily successful. However the up-flow and down-flow conditions were not just restricted to the region near the opening but rather seemed to occur near the bottom and rise to the top. Their quantitative observations of fluctuation frequencies corroborated those of Martin. A stagnant region in the tube bottom was observed for very low to, as anticipated. They were also able to shed additional light on the problem of turbulence generation. They noted that wall turbulence spreads very slowly to the entire flow if it originates in a flow in the laminar boundary layer regime, but if it arises in the impeded regime it will soon trigger turbulence throughout the entire thermosyphon. These findings are in good agreement with those previously published by Martin (87). Additionally, these authors reported a third manner in which turbulence can arise-as a result of the fluid shear which has a maximum between the wall and core flow regions. If the Pr number is sufficiently high, this turbulence only affects the core flow initially; hence they reported that laminar boundary layer flow results are still valid. They remarked that this would not be true for lower Pr numbers, especially for liquid metals. Hasegawa et al. reported extensive experiments (see Fig. 7a) which showed the effect of Pr variation and L / a variation in the fully mixed turbulent regime. Martin’s (87)results are included as well as Lighthill’s predictions (80). Hasegawa et al. developed an empirical correlation for this fully mixed

ADVANCESIN THERMOSYPHON TECHNOLOGY

23

I

(b)

I -

2 3

4

5 6

3

z

g

-

0-

0

-I

-

FIG. 7. Experimental heat transfer results and correlation for open thermosyphon turbulent flow of water (52). Notes (a): - - - laminar flow prediction, Lighthill (80); turbulent flow prediction, Lighthill (80) (1) L / a = 20, (2) L / a = 50, (3) L / a = 100; _ _ _ _ Martin's (87) results for water, L / a = 47.5; (b) laminar flow prediction, Lighthill (80); - correlation due to Hasegawa et al. (52)

~

NU,

= tot

[I Z

if Z

(5)

5

=

-

exp(-

=

5720

--

5

-3/4)]/

t

715

Ot

( L / u ) Pr-'/*; ~/~

= 12; if 2 > 12, 5 = 2 ; ( I ) f = 12, (2) 5 = 40, (3) 5 = 63, (4) f = 95, 119, (6) f = 160. General notes: A L / a = 31.6, f = 39.2; 0 L / a = 50.9, 71.5; 0 L/u = 76.6, 5 = 119.5; 0 L / a = 96.0, 5 = 158.0.

< 12, 5

5

=

turbulent regime. T h e correlation and data is shown in Fig. 7b. T h e agreement is rather poor for t,, less than about lo4, but becomes reasonably good for larger values of to,. It should be noted that the sudden decrease (crevasse) in heat flux, mentioned before, occurs in this region for to, < lo4. For practical use, they recommended the form Nu,

=

but if

(t,t/750){1 - exp[-(5000/[)

t$'4]}

8 falls below 10, it is simply

and

6 = ( L / U ) ~ / ~(18) P~-~/~

set equal to 10.

24

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Finally, Hasegawa et al. presented several analytical considerations in which they confirmed Hartnett and Welsh’s findings, and applied Ostrach and Thornton’s work to an actual problem and gave an integral solution for free convection flow on a vertical flat plate where the thermal and momentum boundary layer thicknesses were not assumed equal. They thus found that the occurrence of equal thicknesses is realized at Pr = 0.278, but such a low value of the P r number for the above situation is inconsistent with the concept that the Pr number represents the ratio of momentum to thermal transfer effects. Instead, a value of about unity should be expected. Liu and Jew (81) were the first contributors to offer a mathematically complete solution to the full partial differential equations pertinent to open thermosyphon behavior. Their model, however, could not be directly applied to any specific experimental or practical cases but rather served to give a method to study the stagnation phenomena observed in certain cases. General studies were not carried out, but a specific case was presented showing clearly the stagnation phenomena involved. This work would also provide a suitable foundation for conducting an analytical study of thermal stability in thermosyphons. No subsequent related publications by the above authors has been found in the open literature. Martin and Lockwood (90) investigated the effect of entrance orifice shape by flow visualization techniques and heat transfer measurements. T h e transparent test cell was a glass tube 1 in. in diameter, 7B-in. long with an 8-in. square by 6-in. high reservoir. Electrical heating was used with no cooling coil being necessary. T h e heat flux measurements were carried out in an opaque test cell used before (86, 87) by Martin, however it was slightly modified. T h e reservoir dimensions and the cooling technique were thus probably different between these two test cells. T he authors found that for very viscous laminar flows, a hot annulus of fluid rose out of the thermosyphon into the reservoir as shown in Fig. 8. This annulus was penetrated by egg-shaped holes through which cool reservoir fluid entered the thermosyphon core. Though the number and location of such holes varied randomly, the total hole area appeared roughly constant. No mixing of the hot and cold streams was observed. This entrance mechanism was apparently restricted to the laminar impeded flow regime. For large Nu numbers, mixing occurred between the two streams at the orifice and hence some of the hot wall fluid was carried back down within the core. This mixing never extended more than +-in. below the orifice plane. Under turbulent flow conditions only the latter mechanism was

ADVANCESIN THERMOSYPHON TECHNOLOGY

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FIG. 8. Open thermosyphon inlet flow pattern. From Martin and Lockwood (90).

evident, but the mixing mechanism near the orifice continued to grow further into the tube with increasing heat flux. T h e effect of tilting was to allow the tube to eject the mixed core and to prevent its reappearance. T h e heat transfer results demonstrated that for laminar flow the rounded orifice (;-in. radius) generally gave better results than the sharp orifice, though the latter became a bit better for large to,. T h e authors suggested that the rounded orifice is initially superior because it does not cause mixing near the orifice, which the sharp-edged orifice, serving as a turbulence promoter, does and hence reduced heat transfer. For large to,, the effect of turbulence was judged to be good on the basis that the turbulence must have negligible adverse effect by causing core-boundary layer mixing while yet having a beneficial heat transfer effect within the boundary layer. For turbulent flow, the sharp-edged orifice was initially advantageous for heat transfer and the previous explanation was again cited. For larger to, the orifice shape did not affect the heat transfer due to the inevitable development of the fully mixed turbulent core. It must be noted that the concept of beneficial turbulence in the boundary layer which originates at the orifice, with no adverse coreboundary layer mixing, has not been directly verified (though it may well be correct) by any worker and in fact goes contrary to certain specific findings of Hasegawa. Unfortunately, the authors apparently did not make any visual checks on wall turbulence under these conditions. They also, it appears, conducted the flow visualization experiments with only the sharp-edged orifice, thus preventing a deeper understanding of this problem. Finally, Chu and Hammitt (19) presented in 1964 an open thermosyphon study in which they obtained a laminar boundary layer solution

26

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for arbitrary Pr numbers and experimental results for water and mercury as working fluids. Since the analytical work was done previously and more simply by Leslie and Martin (78),no details will be given except to mention that for Pr > 1 the limit on boundary layer flow was found to occur for lower values of the Nu number. Their experimental results for water fell somewhat above Martin’s (87) for the fully mixed turbulent regime; however, the thermocouple probe used was quite large relative to the tube diameter and if it was in use during the heat transfer experiments, sizeable errors could have resulted. T h e most recent open thermosyphon study is that due to Japikse and Winter (59, 63);advances in the laminar boundary solution, in the nature of laminar boundary layer transition, and in the correlation of the various turbulent heat transfer studies were reported. T h e laminar boundary layer problem was re-solved, using an integral method [Eqs. (lo)-( 15)] by assuming independent thermal and momentum boundary layer thicknesses and using both cubic velocity and cubic temperature profiles, which were found to be in better agreement with experimental findings. I t was pointed out that all such integral solutions are strictly valid only for the adiabatic base condition and that for the isothermal base problem with small L / a , large differences can be expected. Likewise, it is unreasonable to expect these solutions to apply for liquid metals where conduction effects can be significant. Figure 9 presents a comparison of the laminar boundary layer results of the study, those of other workers, and the experimental work of Martin (87). For fluids of large viscosity such as ethylene glycol (Pr effectively infinite), Lighthill’s solution shows up to 7 yo error whereas the others show up to 20% error with the exception of the study by Japikse and Winter. T h e data joined by bars is transitional data. For 9), Lighthill expected that his solution (Pr = a) water (4 Pr would be good to within about 10% at Pr = 2, but it is already high by about 17% for the higher Pr value of water. Clearly the results of Leslie and Martin are too high at Pr = 1 and should fall well below the water data. T h e results of the present study at Pr = 3 with the cubic temperature profile show reasonable agreement with the water data though of course the curve is a bit too high. T h e experimental data used for comparison is affected by the orifice shape, but only slightly (90). I n general it appears that a third-order profile not only fits temperature profile data well (59, 63) but gives good heat transfer predictions. Also the necessity of using p f 6 to model the boundary layer for Pr of order unity is evident. T he study also was extended for Pr < 1 with solutions down to Pr = 0.025 being obtained. Of interest was the fact that /3 ‘v S at

<

<

27

ADVANCES IN THERMOSYPMON TECHNOLOGY

I

4.5

.

,

,

.

l

l

l

l

l

l

5.5

5 0

.

.

I

.

l

6.0

‘og10 ‘ 0 ,

FIG. 9. Comparison of open thermosyphon laminar boundary layer solutions with experiment, revised from Japikse (59). Linked data points indicate incipient transition. _ _ Lighthill (80); - - - Leslie and Martin (78); Japikse (59). Experimental data, Martin (87); 0 ethylene glycol, 80 < Pr < 280; water, 6 < Pr < 10. ~

Pr = 1.0. As noted before, Hasegawa et al. (52) predicted that this should occur at Pr = 0.278. T h e heat transfer results, however, for liquid metals did not show any sort of agreement with the results of Fig. 13, as indeed no other laminar studies have. No doubt, the omission of conduction effects is significant. Two parametric studies were conducted using the previous analysis. I n the first case the effect of linear wall temperature variations was studied, and in the second case the effect of centrifugal (nonconstant) acceleration was examined. In a general sense an increase in either effect causes an increase in the basic driving potential; and thus an increase in heat transfer rates. T h e opposite is, of course, true for a decrease. Of particular importance, however, are the effects of decreasing wall temperature or weak acceleration fields. Such a decrease in the temperature or acceleration field causes a reduction in the driving potential and thus a flow retardation. Ultimately the boundary layer becomes too thick and the flow is blocked. Detailed graphical results are given in Japikse (59) for both heat transfer and boundary layer growth. I n addition, Table IV gives tabulated laminar heat transfer results which have been very useful for analyzing closed thermosyphons. Part of the problem concerning transition from laminar boundary

28

D. JAPIKSE TABLE IV

LAMINAR OPEN THERMOSYPHON BEHAVIOR"

Pr 00

100 10 3 1

bl -0.243 -0.246 -0.265 -0.295 -0.343

bz 0.172 0.161 0.0904 - 0.00823 -0.137

b, -0.414 - 0.406 -0.354 - 0.285 -0.204

a These equations represent the results of the study (3.7 < log tot < 6.75) linearized on a log-log plot using the values at log,, tot = 5.25 and 6.75. They are related by k ( t m c - to) = Nu& which can be used as a check to show that the fits are indeed quite good.

layer to turbulent flow (as opposed to transition from laminar impeded flow) was also resolved by noting that the pressure distribution in a thermosyphon is not constant but rather increases from the orifice to the base, so that the boundary layer experiences a favorable pressure gradient and the core an adverse one. Furthermore, an inflection point exists in the velocity profile virtually between the up- and down-flow so that the probable instability associated with it would be felt in either flow. It is well known that the occurrence of a profile inflection in an adverse pressure gradient can be highly conducive to transition. (Consider, for example, the classic stability analysis as in Schlichting (1057.1 I n order to assess the general effect of the pressure gradient, Japikse and Winter (63) sought an answer to the following question: Given one known transition point on a Gr, vs. Pr chart, under what restraint would it have to be bound if it were to trace out a transition curve ? They argued that the answer appears to be, in a very simplified form, along a locus where tot is approximately constant. Under such a condition the pressure gradient was found, while studying the results of the preceeding laminar boundary layer analysis, to scale on tot and show no significant further dependence on Pr. Similarly, the velocity profile was found to depend most strongly on tot although some secondary dependence on Pr was noted, but this is a smaller effect. Th u s since tot roughly fixes the profile in which the instability occurs as well as the pressure gradient which affects the magnification or attenuation of such instabilities,

ADVANCES IN THERMOSYPHON TECHNOLOGY

29

holding tot fixed should be approximately a sufficient condition to trace out a plausible transition curve. If in reality such a transition curve is unique, then, within the limits of the approximations, this restraint should also be a necessary condition. Obviously tot = const implies Grcrit cc Pr-I(L/a). All existing transition data, see Fig. 10, were

tog,,

Pr

FIG. 10. Open thermosyphon laminar to turbulent transition data. Revised from Japikse and Winter (63). 0 Data with strong evidence of transition from laminar boundary layer flow (63). 0 Data reported for transition from laminar boundary layer flow. A Data known or presumed to represent transition from laminar impeded flow to fully mixed turbulent flow. a, Mercury; b, air; c, water, d , light oil; e, ethylene glycol. B, Bayley et al. (6); F, Foster (34);J1, Jallouk (58); J 2 , Japikse (59); L, Lock (82); M , Martin (87).

examined by Japikse and Winter (63) for those cases which truly represent transition from a laminar boundary layer. They reported that the data fell in the band Grcrit = 106.68*0.5Pr-4/3. Several other data points (open circles) have been added which probably also represent boundary layer transition. With this data included, all but the points for Pr > 200 follow the trend Pr-I for boundary layer transition. T h e predicted trend is in qualitative agreement with what has been

30

D.

JAPIKSE

observed experimentally as far as the Gr cc Pr-l is concerned. T h e experimental data is not clear concerning L/a effects. Indeed, the above discussion is limited by the rather general approach taken and the fact that conventional stability studies were used as a guide, which only give the start of instability, not actual transition. Nonetheless, the combination of profile instability and adverse pressure gradient in the thermosyphon core was concluded to be a significant cause of the trend found in Fig. 10. T h e problem of describing turbulent impeded flow is also considered in the study by Japikse and Winter (63) and it was found to be quite perplexing. First of all, it is not entirely clear that the flow covered by this term is invariably truly turbulent since it may also include certain cases in which random measurements are obtained (and hence called turbulent) from a random generation process at the orifice and which subsequently yields seemingly random fluctuations throughout the flow; yet they often decay, not amplify, through the flow and hence the flow is stable. This has definitely been found to occur in the closed thermosyphon at the midtube exchange region. Also, Hasegawa noted that one can have turbulent core flow with laminar boundary layer flow the heat transfer for which follows the laminar trends. Regardless of the true nature of the fluctuations, a so-called turbulent impeded flow does exist which gives lower heat transfer results than either laminar or turbulent boundary layer flow. Invariably, the results of any one study have failed to agree with any other study and this discrepancy has been traced to a missing diameter dependency. Various observations led to the hypothesis that the heat transfer rate was dependent on diameter including (63): (1) the diameter appears as a meaningful parameter in heat transfer studies after L / d variations have already been considered (Figs. 2 and 13); (2) this turbulent impeded flow regime, as it exists in the open thermosyphon, was not found in the closed thermosyphon under equivalent conditions but rather a boundary layer regime was found (see next section) and the fundamental difference, the presence of an orifice, introduces only the diameter as a characteristic length; (3) the diameter must play a singular role as a limit or constriction on the size of eddies which can be formed anywhere in the system; (4) two authors, Foyle (35) and Foster (34), suspected or included the diameter as a meaningful parameter in their experiments (although subsequent workers found the experiments unclear in this matter, as noted earlier). Consequently, the following form was hypothesized for the correlation: Nu, = C,Ra,m(a/L)C2;

C,

=

C,(Pr, a), C,

=

C,(Pr, a).

(2)

ADVANCES IN TIIERMOSYPHON TECHNOLOGY

31

Various workers have found this form (but with C, and C, independent of radius) very suitable for their results though their correlations fail when applied to other studies with different tubes. Although C, might be a function of Pr, the fact that it is associated with a purely geometrical term might suggest an independence of Pr. Clearly, as a 4 00, C, 40 but of course C, cannot, in order to agree with the flat plate solution which is the limiting case. Now, the diameter must limit the size of eddies which can be formed anywhere in the system, hence as a+O, C, and C, 4const since there should be a point where the smallness of the tube diameter will always restrict the eddy size possible for entrainment; any further increase in entrainment rate is blocked as the radius is decreased.6 Th u s C, must be constant for small radius and also as a 4 CO, but may be a function of radius in between and C, must be a constant for small radius and approach zero as a -+ CO. T h e experimentally determined constants are shown in Fig. 11. T h e

2-

-0

Const. wall heat flux -A

Symbols Water

0 Ethylene glycol

10

2 0

30

40

50

60

Diameter ( i n )

FIG. 1 1 . Heat transfer correlation coeficients for turbulent impeded flow in the sharp-lipped, vertical, open thermosyphon. From Japikse and Winter (63). This implies some kind of local equilibrium, which Lighthill (80) chose to neglect in his turbulent study.

32

D.

JAPIKSE

predicted trends are obtained satisfactorily and, apparently, rather uniquely; the parameter C, appears to be a function only of radius. T h e choice of m = 0.25 for water and m = 0.3 for mercury was only used so as to be consistent with what has been used by most authors for those fluids. T h e data from Larsen and Hartnett (75) give an exception to the trends observed; it can only be concluded that the method they used to average the wall temperature distribution for their case of constant heat flux is only adequate to represent the gross characteristics as shown later in Fig. 13. Japikse and Winter noted that the data for the first four and last two points on the C, graph were taken directly from the original correlations. While this is convenient and avoided errors due to recomputation, the presence of terms such as for C, , for example, indicated that the value might have been slightly idealized in the original study and might give a small shift from what is shown in these figures. T h e point shown for Martin (D = 2-in.) was calculated from his data where the strongest reduction in heat transfer occurred for all L/a cases, i.e., however, C, drops to a lower value of 0.44 Ra = 106.5.At Ra = as can be expected from Fig. 2 or 6b. This exception found in Martin’s data was thought to be of extreme importance. T h e behavior expressed by Eq. (2) with C, and C, as function of radius is most likely caused by core-boundary layer entrainment. Indeed, this process generally begins before normal boundary layer transition would be expected, (see Fig. 10) and, too, it is possible to obtain a purely turbulent boundary layer flow without this degrading entrainment effect (e.g., ethylene glycol, Fig. 2 or 6a). As indicated in item 2 above, the inlet effects may be of considerable importance. I n fact, it appears that Martin’s studies were with a rounded inlet lip whereas the others were apparently with a sharp lip. Likewise, it was found for the inclined open thermosyphon, to be discussed later, that inclination influences the inlet process, and consequently the coreboundary layer entrainment process, with improvements in heat transfer. T hus Fig. 11 is probably restricted to sharp-edged inlets in vertical open thermosyphons. T he question of the appropriateness of using a dimensional parameter was also considered. T h e authors felt that the same good agreement found with Eq. (2) would probably also be obtained if the system to which nondimensional reasoning was applied were enlarged to include the entire orifice region, including at least part of the reservoir, and additional variables of which all have probably not yet been discovered. T h e radius of the orifice lip might be one such additional variable. I n fact, if the system were increased to include the entire reservoir, the thermosyphon would then be closed. Indeed, for the sake of completeness

ADVANCES IN TI-IERMOSYPHON TECHNOLOGY

33

it would have been advantageous and more accurate, although obviously more complicated, if the entire closed system had been considered in the various studies. As it is, it is very difficult to determine what kind of orifice existed in some studies to say nothing of the influence of reservoir and cooling coil size and shape in most of the studies.

C. THECIRCULAR OPENTHERMOSYPHON, LIQUIDMETALS T h e use of liquid metals in thermosyphons has attracted several research investigations due to the good thermal properties. However, it will be seen that the fluid mechanics of the open thermosyphon change considerably, largely offsetting some of the advantages of liquid metal usage in the open thermosyphon. Several aspects of liquid metal fluid flow in the open thermosyphon are apparent from the previous discussion: T h e existence of a laminar similarity flow was found to be mathematically impossible, and all laminar boundary layer analyses were found to be quite inadequate. Fortunately, several good studies are availzble which experimentally show some of these differences. T he first liquid-metal-related thermosyphon study is that due to Bayley ( 4 ) in which he analytically considered the turbulent free convection flat plate problem with particular attention to the low Pr range for liquid metals. He developed a theory to relate the nondimensional heat transfer coefficient to the boundary layer thicknesses that were determined from solutions to the heat and momentum equations. I t was necessary to introduce a variety of assumptions, and so a comparison with data is necessary for a realistic evaluation of his work. H e gave a comparison for air which showed very good agreement and in a later study (6) he was able to make a liquid metal comparison as shown in Fig. 12. T he results are generally quite good, and it will soon be evident that this flat plate solution serves as an upper limit on the amount of heat transfer which is possible for liquid metals in the vertical open thermosyphon. It was found that molecular conduction plays a dominant role in liquid metal heat transfer, and therefore the results for laminar and turbulent flow were expected to be nearly the same, as indeed was later found (see Fig. 12). Hartnett, Welsh, and Larsen (50) and Larsen and Hartnett (75) presented studies in 1958 and 1961 in which the use of mercury in the open thermosyphon under a constant wall heat flux condition was investigated. They obtained heat transfer results which were substantially lower than those for water for an equivalent Ra number. This reduction in heat transfer rate was apparently due to extensive turbulence making

D.

34

JAPIKSE

the adverse core-wall flow mixing effect found for water even more severe. Large variations in fluid and wall temperatures as well as a centerline temperature distribution which increased rapidly from the open to the

1.01 7.5

0.0

8.5

9.0

9.5

10.0

Log,oGr,

FIG. 12. Liquid metal open thermosyphon heat transfer results for large a / L (6).

-Turbulent analysis ( 4 ) .

closed end (even more severely than for water) served as evidence of the existence of strong turbulence. T h e latter was also cited as a cause of instability since dense cold fluid would be located above lighter hot fluid. Bayley, Milne, and Stoddart (6) conducted a series of experiments with mercury in short tubes with u/L ranging from about 0.4 to 1.6. They found substantial evidence to indicate boundary layer flow, both laminar and turbulent. T h e latter results showed agreement with a simple extrapolation of the laminar results. Transition apparently was caused by interference between the hot and cold flows which caused a temporary reduction in heat transfer followed by complete recovery as a turbulent boundary layer was established. T h e heat transfer results exceeded the laminar predictions by Leslie and Martin (78) by about 35 yo but showed reasonable agreement with the predictions for turbulent flow of Bayley (4). Centerline temperature profiles showed a constant value for each test and radial profiles agreed best with a third-order profile. Further mercury experiments for longer tubes ( u / L from about 0.08 to 0.5) were subsequently performed by Bayley and Czekanski (7). Their basic heat transfer results for constant wall temperature conditions showed two distinct modes of heat transfer each of which displayed a strong variation with u/L when plotted on a log,,Nu, vs. log,,Gr,Pr2 plot. This abscissa was chosen since viscosity is not included, the choice being

ADVANCES IN THERMOSYPNON TECHNOLOGY

35

consistent with the negligible viscosity of mercury.' T h e first mode of heat transfer yielded a unit slope variation which was attributed to a stagnant lower section which, for length variations alone, yielded a constant heat flux for varying length or, in other words, Nu, decreases linearly as the surface area increases linearly with length. For larger values of the abscissa, the heat transfer varies with the 0.3ths power of Gr, as was the case in Bayley's earlier study. T h e geometry effect, however, prevented comparison with theory but a comparison with Bayley's earlier work ( 4 ) for equal values of a / L (= 0.5) showed that the current values were some 17% lower than those of the previous experiment, thus indicating a further dependence on radius which was different in the two cases. Bayley and Czekanski noted that since the centerline temperature increased with distance from the open end for the second mode of heat transfer, the flow is presumably a turbulent impeded flow. Similar results for the case of constant heat flux showed complete agreement with the isothermal results for a / L = 0.167. However, reduced heat transfer rates were found for smaller a / L and agreement with the results of Hartnett et al. (50) was good for the single comparison which was possible. Martin (91) presented an analysis which gave relations for the lower limiting conditions under which free convection subsides and heat transfer is purely by conduction. This work is of primary interest for applications for liquid metals, but applies for other fluids too. T h e heat transfer rate was found to depend only on radius, being independent of length unless L / a < 1.8. Under the latter constraint the bottom boundary conditions had a significant effect. Chu and Hammitt (19) conducted heat transfer measurements for mercury in tubes with Lla = 26.6 and 44.5, the largest ratios yet tried. Consequently, their results are the lowest values recorded inasmuch as the usual dependence on L / a was again evident. Japikse and Winter (63) have also applied the concept of a radiusdependent heat transfer correlation to liquid metals, as indicated in the previous section. T h e same good agreement obtained for ordinary fluids was again found for impeded turbulent liquid metal flows, see Fig. 11. Th i s study also gave a compilation of all liquid metal (Hg) heat transfer results yet obtained, see Fig. 13.

' This statement is often made; in fact, the viscosity of most liquid metals is about the same as that for water. However, by comparison to the conductivity, it may play a negli-

gible part in thermal energy transfer.

D.

36

JAPIKSE

L/o

D = 6." I

-

6.1

0-

-I

.o -

3.0

Chu and Harnmitt D = 1.35"

I

I

I

I

4.0

5.0

6.0

7.0

Log,,lGr, Pr2)

FIG. 13. Liquid metal heat transfer results for the open thermosyphon. From Japikse and Winter (63). - - - - Ogale (98), rotating test of semiclosed thermosyphon, D = 0.71 in.; - -- flat plate turbulent boundary layer solution from Bayley ( 4 ) adapted for Lla = 0.677.

D. NONCIRCULAR OPEN THERMOSYPHONS T h e initial study of a thermosyphon with a noncircular cross section was conducted by Hasegawa et al. (51,52). Using an equivalent hydraulic radius, he compared his results for a parallel plate (rectangular cross section) thermosyphon with Lighthill's laminar analysis and obtained generally good agreement for boundary layer flow but somewhat higher results for the laminar impeded regime than those predicted previously by Lighthill. Lockwood and Martin (83) conducted an extensive analytical and experimental study for thermosyphons with circular, rectangular, triangular, and airfoil shaped cross sections under both laminar and turbulent flow conditions. Their analytical model, restricted to twodimensional laminar flow, closely followed Lighthill's (80) analysis and

ADVANCES IN THERMOSYPIION TECHNOLOGY

31

yielded heat transfer rates a bit higher than Lighthill’s predictions for the boundary layer regime and considerably higher for the similarity flow. T h e latter agreed particularly well with the trend found by Hasegawa et al. (52). For laminar flow, the circular thermosyphon gave the best heat transfer rate since the corners of the others caused greatly reduced flow rates. For laminar impeded flow particularly, strong reductions in heat transfer were found when sharp corners existed. For turbulent flow, which occurred earlier than in the round thermosyphons due to larger shear stresses in corners, circular and triangular sections provided the best heat transfer for low Ra numbers. When the Ra number was increased and fully mixed turbulent flow existed, the rectangular and air foil shaped sections gave much higher heat transfer. Since this mode is anticipated in actual turbine applications, results better than those of the circular thermosyphon can be expected. I t was found, however, that the use of a hydraulic radius at large Ra was not satisfactory.

E. CORIOLISAND INCLINATION EFFECTS During the early years of thermosyphon research Eckert and Jackson (29) and later Alcock (2) observed that the influence of Coriolis forces on a rotating thermosyphon could be approximated by studies with an inclined, static, open thermosyphon, giving an acceleration component normal to the tube axis as well as one along it. T h e primary effect of this normal component would be expected to be the shifting of hot and cold fluid streams. T h e hot and less dense fluid would tend toward the higher side whereas the cold denser fluid, flowing in the opposite direction, would tend toward the lower side. Whereas no formal studies concerning Coriolis effects have been made on rotating apparatuses, several have been made with inclined thermosyphons. T h e analytical investigation of Eckert and Jackson (29) for turbulent flow in the open thermosyphon provided information for formulating an expression for the ratio of Coriolis force to buoyancy force as follows: Coriolis force - 2um,,w 1 X/Y, = 0.71 -_ _ _ Pr.0*9 Po, buoyancy force ~,,w~/30~

(

1

’

but otherwise only expressions for maximum Reynolds number (used for the above relationship) and momentum thickness were obtained. T h e latter formed a basis for estimating maximum length to diameter ratios for open thermosyphons for turbulent boundary layer flow for which core-boundary layer interaction was assumed not to occur.

38

D.

JAPIKSE

Freche and Diaguila (36) conducted tests with a rotating test rig that used the open thermosyphon scheme. These results are shown in Fig. 2; there appears to be agreement with the static tests when one considers the appropriate inlet and L/a conditions. It must be noted, however, that the authors felt that the true results might well be a good deal higher than those shown due to measurement difficulties. This would be consistent with the effect of Coriolis forces as estimated from inclination tests, to be discussed shortly. At most, the inclusion of this data in Fig. 2 might suggest that static tests give a lower limit for rotating applications. Martin and Cresswell (88) and Martin (89) presented the earliest experimental studies on inclined open-thermosyphon behavior and included both laminar and turbulent heat transfer results over a wide range of Pr numbers. For glycerin and rapeseed oil (laminar flow, 700 3 Pr 3 20) the Nu, initially decreased for increasing 6' (inclination angle measured between the axis and the vertical) from 0" to about 6", see Fig. 14a,b. T h e Nu number then increased again for larger 8, exceeding the original Nu, value (for 6' = 0) at about 6' = 45". T h e initial trend was attributed to "mingling" of particles of hot and cold fluid causing reduced heat transfer much as the similarly destructive effect in the (vertical) impeded turbulent regime. For larger angles, considerable evidence was given to show that secondary pressure gradients yield a more stable situation in which the core is displaced somewhat and significant core enlargement is encountered. I n this manner the boundary layer thickness was reduced and the heat transfer was increased. For ethylene glycol, similar results were obtained, but the Nu, never exceeded 10G.25) due to the transition its original value at 0 = 0" (except for Ra, to turbulent flow and due to the smaller Pr apparently causing a much stronger initial reduction. For water, which yielded fully mixed turbulent flow from the outset and hence poor heat transfer for vertical tubes, large increases in Nu, were obtained even for low Ra numbers as shown in Fig. 14c,d. Martin suggested that the normal acceleration would cause reduced mixing and the eventual development of turbulent boundary layer flow, thus accounting for the large increase. For tan 6' > 0.4, the Nu, results tended to level off at a constant value. Finally, Martin noted that the static thermosyphon duplicated the Coriolis effect on the cool fluid only. He also reapplied Eckert and Jackson's (30) analysis to show that an angle of 6.5" duplicates the relative influence of Coriolis to centrifugal acceleration and a similar treatment of a different turbulence analysis yielding a value of 6' E 11'. These values serve to suggest a reasonable angle range for correct modeling of Coriolis effects.

<

ADVANCES IN THEKMOSYPHON TECHNOLOGY 30

r

(b)

39

log Ra,, Pr

I

6'

012 014 016 018

110

112

;.4

tan 8 45r ( d ) 401

/ - - *

Log Ra,

, Pr

7.75, 2

35 30

7.25, 3.5

25

3

20

6.75, 5

15

6.25, 6

10

5.75, 7.5

5 L

'0 0 . 2 0.4 0 . 6 0 . 8 1.0 1.2 1.4

FIG. 14. Heat transfer in inclined open thermosyphon. From Martin (89). (a) Extreme vertical tube; o leading edge, tan 0 = 0.1; x trailing heat transfer values for glycerin: edge, tan 0 = 1.26; - - - - vertical laminar analysis, Pr = co; -.- vertical laminar analysis, Pr = 1. (b) Effect of inclination on glycerin heat transfer tests: 0 uniform trailing edge temperature; 0 uniform leading edge temperature. (c) Extreme heat transfer values for water; vertical tube; o tan 8 > 0.4; - .- vertical laminar analysis, Pr = m; - - - _ vertical turbulent boundary layer, Nu, = 0.10 Ra:'3 and Nu, = 0.0325 Ra2I5. (d) Effect of inclination on water heat transfer tests: 0 uniform trailing edge temperature; 0 uniform leading edge temperature.

Leslie (79) published the only analytical treatment of the inclined thermosyphon; he used a laminar perturbation technique which of course is restricted to small angles. His analysis showed that tilting caused an increased heat transfer both for the (impeded) similarity solution and for the boundary layer flow solution. Since unstable effects, which caused the initial decrease in heat transfer rate with increasing angle in

40

D.

JAPIKSE

Martin’s experiments, were not considered, agreement is poor with Martin’s findings for small B values, but qualitatively correct as 6 increases. Hartnett et al. (50) and Larsen and Hartnett (75) conducted inclined thermosyphon experiments for both water and mercury. T h e water results (turbulent) agreed favorably with those of Martin (89) in all respects. T he mercury results showed the same tendency for heat transfer to increase with increasing 8, but the cause appears to be different since strong centerline temperature gradients indicated that the mixing effect apparently did not abate. For a given Ra number, water always yielded higher Nusselt numbers than mercury. I n concluding this section, it can be observed that the results of the inclined thermosyphon studies for water and liquid metals, the most likely liquids for turbine cooling, show increased heat transfer with inclination and therefore the effect of Coriolis forces probably should be to improve the results under rotating conditions over those for vertical static conditions. T h e data of Freche and Diaguila (see Fig. 2) for rotating conditions show that there apparently is no reduction under rotating conditions; and, considering their claim that the results should be considered conservative, this data suggests that rotation may well give higher heat transfer rates. Th u s there appears to be qualitative agreement between the findings of static inclined and rotating open thermosyphon tests.

III. Closed Thermosyphons

A. GENERAL BEHAVIOR T h e simple closed thermosyphon, as shown in Fig. 15 (and to which this section is solely directed), quickly gained popularity as problems with containment, chemical compatibility, and pressurization became apparent in various applications of the simple open thermosyphon. Only a few basic studies of the closed system have been performed, but together they rather clearly outline the fundamental performance of this thermosyphon system. I t has been found that the closed thermosyphon, when modeled carefully, can be treated as two simple open thermosyphons appropriately joined at the midtube exchange region. Not surprisingly, most (probably all) of the modes of flow found for the open thermosyphon have been found in the closed thermosyphon (the primary exception being that impeded turbulent flow is considerably delayed or reduced,

ADVANCES IN THERMOSYPHON TECHNOLOGY

41

I-d-I

FIG.15.

T h e single-phase closed thermosyphon.

allowing, in certain instances, the performance of the closed system to approach that of an equivalent open system, in spite of the seemingly higher thermal resistance). Th u s the primary problem concerned with the closed thermosyphon is that of modeling the exchange region or, in other words, finding To,land T,,,z in order to apply open thermosyphon - Tl,z and/or high Pr, it has been found that the results. For low increases, exchange mechanism is basically convective; as Tl,l or Pr is decreased, the convective process becomes less stable and eventually degenerates to an intensive mixing exchange process. I n the following chronological survey of the closed thermosyphon literature, these concepts come to light, as well as other views; the reader will probably find this short synopsis a convenient guide in considering various viewpoints.

B. THEVERTICAL CLOSEDTHEKNIOSYPHON Initial contributions to this topic shed little light on the fundamental 2 appear to have avoided the basic problem of finding To,l and To S and questions of internal flow patterns and the effect of tube geometry and however, in his fluid properties on heat transfer rates. Lighthill (84, analysis of the open thermosyphon first suggested that the closed system could be treated as two open systems joined together. H e predicted that, under turbulent flow conditions, extensive mixing of the two ramming boundary layers would yield a uniform core temperature in the thermosyphon. Hahnemann (46) conducted the first extensive set of experiments

42

D.

JAPIKSE

on the closed, static, single-component thermosyphon. He apparently made no attempt to examine any of the above-mentioned questions by systematic variation of controlling parameters. Unfortunately, detailed experimental data were not reported, thus preventing any further study or reexamination of his work. Lock (82) and Bayley and Lock (8) have reported the first comprehensive experimental and analytical study of the closed thermosyphon where emphasis was placed on the type of midtube exchange mechanism and the effect of Pr and L/a on heat transfer rate predictions. Their experimental program was carried out using a vertical opaque test cell of variable length on which electrical heating and water cooling was used. T h e electrical heaters were controlled independently to assure reasonably isothermal wall conditions. Tests were run using air, water ethylene glycol, and glycerin. Experimental results are presented in Fig. 16. It can be noted that many of the flow regimes found in the open thermosyphon are evident here. Fig. 16a shows clearly a laminar impeded 1.8r x

/'

Convection

/'

14-

/'

/'

If"

/'

I .o -

I,,

2

z

p

{ Fr 1

1.4-

rn 0

0

-0.61

I

-2 00.6.2-

\1

';

\

,

-0.2

1,: 3.:

'

$8

3.0 4.0

0

!

'lo

-0.2-

z

0

p

I.

0.2-

1.0 -

9

i

0.6-

'

I

I

5.0 6.0 7.0 8.0 9.0 log,, t,, ( 0 )

Pr = a3 Pr = I Mixing

-

-6.03.0 4.0 5.0 6.0 7.0 8.0 9.0

w o

+,,

(b)

FIG. 16. Heat transfer measurements and prediction in the closed thermosyphon (L,/L, = 1, L,/d = 7.5). From Bayley and Lock (8). Notes: x water, 1 < Pr < 10; ethylene glycol, 10 < Pr < 300; 0 glycerin, 20 < Pr < 20,000;- -.--air; -.- open thermosyphon analysis, Lighthill (80); __ closed thermosyphon predictions, Bayley and Lock (8).

regime for glycerin and a laminar boundary layer regime for all other fluids. Transition occurred at various values of to, as shown in Fig. 10,24, 25, 26 and 27. Transition was dependent on fluid properties and Lla. For air, the usual crevasse is again evident. A slight geometry effect, giving reduced heat transfer, not accounted for in the to, parameter was noted for long tubes, although it is not shown in Fig. 16. T h e heat

ADVANCES IN THERMOSYPHON TECHNOLOGY

43

transfer is less than for the open thermosyphon with the same temperature difference; and there is an optimal Pr number, the latter being contrary to the behavior of the open system. Before proceeding, a comment on the interpretation of transition and turbulent impeded flow is in order. While conducting his own study, this writer reviewed Lock’s data and was able to confirm the transition values given by Lock for water. However, the transition value for ethylene glycol and the values for the onset of impeded turbulent flow for either fluid were not clear. Considering Fig. 24 in particular (using Lock’s ethylene glycol data), this alternative viewpoint can be explained. Any such plot, with Nu, vs. t,, , is compiled with the properties used in these parameters evaluated at one given temperature, the current convention being the use of T l , l . This implies, however, that Tl,2 is now an independent parameter insofar as good property modeling is concerned. I t will be stressed later that the properties, particularly Pr, are extremely important for understanding the closed system, and it will are necessary for a good representation. be shown that both Tl,l and I t was found upon reexamination of the data in Fig. 24 that when a shift in the data occurred, there had invariably been a definite change in Tl,2 which was otherwise only slowly varying. In fact, the “scatter” in Fig. 24 can be traced, point by point, to increases and decreases in T l ,2except the last several points (high tct) which may finally be affected by impedence. This effect was generally apparent near the points where transition and the onset of impedence were reported to occur for all L / a cases, thus making it difficult to interpret such changes. Indeed, all the ethylene glycol heat transfer results showed reasonable agreement with what might be expected theoretically for laminar flow. Perhaps the clearest case is that of water for L / a = 15 in Fig. 27; in this case the indicated impeded flow does not show the typical leveling off of heat transfer at all. Nonetheless, his boundary layer transition values have at least order-of-magnitude agreement or better with the values from open thermosyphon studies as shown in Fig. 10. I n order to interpret these results Lock proposed three idealized exchange mechanisms as shown in Fig. 17. T h e first mechanism is called “mixing” and was suggested originally by Lighthill (80). I t presupposes a violent collision process of the opposing streams and as Lock noted it is hard to reconcile it with very viscous fluids. T h e second exchange mode, termed “convective,” allows flow streams to exist that would directly carry fluid from a boundary layer into the opposing core. T h e third exchange mechanism, “conduction,” requires each boundary layer to return to its own core, thus allowing only conduction across a limited interface. Lock further noted that he expected in

D.

44

Mixing

JAPIKSE

Convection

Conduction

FIG. 17. Ideal exchange mechanisms according

to

Lock (82).

practice that the actual coupling mechanism would probably contain elements of each of the idealized cases. Lock noted that the conduction mechanism, which would give the lowest rate of heat transfer, must be significant in the case of glycerin which showed the least heat transfer (though it would not necessarily be the only mechanism present). H e suggested that the exchange region can be thought of as a radially focusing jet which must be considered to be inherently unstable. T h e greatest stability would be expected for high Pr numbers as in the previous case, but for lower P r numbers he suspected that the jet would break down almost at the point of formation giving the possibility for two more cases. First it is possible that the two annuli of colliding fluids will yield total or near total, mixing as suggested before. Figure 16b shows the comparison of Lock’s analytical model based on total mixing and measured heat transfer results. T h e correlation appears to give strong verification to this exchange mode. For turbulent flow, Lighthill suggested that this would be the only exchange mechanism. Using Saunders’ (103) turbulent empirical correlation and the mixing mechanism, excellent agreement was reported with turbulent heat transfer data for shorter tubes. For longer tubes, the relatively thicker boundary layers probably caused significant interference and hence a larger mixing regime. Consequently, poor agreement with predictions based on the previous model was reported. Observations with a forced-coupling device (a fabricated device placed in the tube middle to force the convection pattern giving two streams u p and two down) further verified the mixing mechanism for turbulent flow in moderate length tubes. Lock’s third exchange mechanism was that of convection which would occur for large temperature differences and large Pr numbers. T h e hot and cold layers in the jet are presumed to pass through each other with

ADVANCES I N THERMOSYPHON TECHNOLOGY

45

little heat transfer between them. He suggested that secondary conduction between the streams would probably be inevitable and would reduce the efficiency of the convection mechanism. Experimental support for this mechanism was obtained from centerline temperature measurements, as demonstrated in Fig. 18. He noted that the centerline temper-

Distonce from cold end (in.)

FIG. 18. Ccntcrlinc temperature distribution. From Lock (82).

ature profile should be uniform if mixing occurred, but that in fact the temperature of the heated section was a good deal lower than the temperature in the cooled section, which would agree with the convection concept. T he forced coupling results for larger t,, showed compatibility with the nonforced results. Unfortunately, it must be noted that the analytical results based on a pure convection mechanism do not show good agreement with the experimental results as shown in Fig. 16b. Finally, Lock carried out a closed-form Karman-Pohlhausen integral solution to the laminar boundary layer flow and coupled these results (Fig. 16b) together assuming either pure mixing or pure convection as possible exchange mechanisms. T h e boundary layer solution was Pr, and the assumption was made that the tworestricted to 1 dimension form of the boundary layer equations is valid. T h e nonzero velocity outside the boundary layer was accounted for. Curvature was neglected in the basic equations and was accounted for only in the velocity profile via the use of a mean hydraulic diameter. I t can be seen from Fig. 16b that the mixing mechanism yields reasonable predictions for the ethylene glycol results but would fail for glycerin. T h e mixing mechanism plus boundary layer analysis also reflected quite well the change in heat transfer associated with changing the ratio of heated to cooled length (8). Furthermore, he predicted in his analysis optimum heat transfer when this ratio of heated to cooled length equals 0.772 or 0.537 for the mixing and convective coupling mechanisms, respectively. Unfortunately, he could not check this experimentally. Also,

<

46

D.

JAPIKSE

he suggested that since mixing represents a condition between the optimum of convection and the undesirable case of conduction, it should provide a useful approximation for all laminar flow conditions. Actually, further conclusions can be obtained from Lock’s data and this author (Japikse (59), Japikse and Winter (63))has carried out a new analysis of Lock’s heat transfer data8 by using the reported heat fluxes and wall temperatures to calculate equivalent centerline temperature distributions, with the aid of the writer’s open thermosyphon study presented before (63).T h e results, for laminar boundary layer cases only, are as follows:

> To,l , i.e., the effect of (1) T he vast majority of cases showed TOs2 convection was strongly evident. Pure convection (giving To,2= Tmc,l) was not found. (2) There were certain cases for which To,l ’v To,2and thus some evidence for a mixing mechanism was found. These results occurred most readily in long tubes (constricted flow) and/or for low-viscosity fluids with small or moderate temperature differences. Generally, however, one would expect this mechanism to occur for large temperature differences and thus outside disturbances which might influence flow stability, such as vibrations, might have been present to cause early breakdown. (3) A number of cases in the longest tubes showed To,l > To,2or the opposite of a convection profile. Presumably these are the results of the adverse effect of core-boundary layer interaction which should be strongest in long, thin tubes. However, as the temperature difference increased, this effect diminished and convective profiles were again found. (4) For the glycerin experiments, interesting results were obtained. In the top half, based on To,2- T l , z ,the results fell in the range 80 to, 2600. Hence boundary layer flow (by comparison to the findings of Lighthill (80) or Martin (86))was never obtained in this half of the tube but rather laminar impeded flow. T h u s Lock’s low heat transfer results for glycerin are understandable, and there is no reason to believe that there is an optimal Pr value for laminar boundary layer flow.

<

<

T h e second study of the closed thermosyphon is that due to Schenk and his students Dieperink and den Ouden (26,104).Their experiments were conducted in a noncircular closed thermosyphon using only air as a test fluid. T h e intention was to employ a channel cross section with T h e author is indebted to Professor G. S . H. Lock for his assistance on several occasions and for making his data available in a form suitable for the analysis.

ADVANCES IN THERMOSYPHON TECHNOLOGY

47

considerable depth, but strong end-wall effects required the inclusion of numerous glass partitions which effectively created a rectangular thermosyphon cross section, with, of course, imposed thermal boundary conditions only on the two outside walls. This apparatus permitted the use of interferograms and smoke visualization techniques, although the superposition of successive rectangular thermosyphon sections often rendered the interferograms meaningless. Their study was thoroughly executed and provided valuable information concerning flow patterns, (see Fig. 19), velocities, and temperature distributions (using a thermocouple probe).

FIG. 19. Flow visualization in the closed thermosyphon of rectangular cross section with air. From Dieperink and Ouden (26) Schenk et al. (104). (a, b) Particles illuminated with collimated light of various colors corresponding to depth; note the very feeble circulation in the top portion; (c, d) smoke visualization, note the convective exchange process and the virtually stagnant upper end; (e, f, g) smoke visualization, limited secondary circulation is evident in the isolated top portions (f, g) and bottom portion ( e ) . Photographs courtesy of Professor Schenk.

T h e flow patterns which the authors observed were of the following nature. T h e air in the bottom portion of the thermosyphon rises along the two heated metal wall plates until it reaches the insulated wall portion (see Fig. 20). This insulated section thermally isolates the top cold wall from the bottom warm wall. At this point one of the two currents of rising air penetrates into the upper part of the thermosyphon as it passes through the central exchange region. Simultaneously, a stream from the top descends into the lower part through the same central exchange region (see Fig. 20b,c). T h e warm air which enters the upper cooled part proceeds in a spiraling path with the direction of rotation

D.

48

25

JAPIKSE

rl

25°C

(0)

FIG. 20. Closed thermosyphon temperature distribution (a) and flow patterns (b, c , d, e) from Schenk et al. (104); mode of flow is presumed to be impeded laminar flow (62).

determined by the direction with which it enters the exchange region. After the turbulent air is cooled it returns into the bottom part of the thermosyphon where it recombines with other descending streams and proceeds in a spiral with inverse rotation (Fig. 20c,d). T h e new vortex or spiral is supplied by the other stream of rising warm air so that the flow becomes completely symmetrical (compare Fig. 20b,d). In order to study the influence of increased Tl,l , this temperature was increased to 35°C. I t was established that the new flow pattern became less distinct but occupied a greater portion of the thermosyphon. At higher Tl,l the flow continued to become more and more mixed up and toward Tl,l = 60" the flow occupied the entire thermosyphon. T h e authors noted that regions with very limited flow existed at the ends of their apparatus (but note, too, that some feeble circulation does exist, see Fig. 19) and hence these portions did not contribute to the overall heat transfer. Th u s they felt there were certain differences to the work of Bayley and Lock. Unfortunately, the authors failed to draw upon the open thermosyphon literature and thus did not attempt to consider the nature of flow regimes. This has been done in part by Japikse (62) and will be outlined here. First it must be noted that although the authors often referred to the air flow as being turbulent, it is not clear if the flow is merely irregular, say, in the core, due to the midtube exchange process or if the flow is truly turbulent throughout, as discussed previously for open thermosyphons. Either case is of course possible, but to expect complete transition seems a bit premature compared to transition data for air in the circular open or closed thermosyphons (Fig. 10). Now using their temperature data one can calculate to, = 36 for the cold end and to, = 78

ADVANCES I N THERMOSYPHON TECHNOLOGY

49

for the hot end. If one uses Table I11 and extrapolates the channel case to Pr = 0.71 assuming a Pr dependency similar to that for circular tubes, one obtains a value of to, = 77 for laminar similarity flow. It would thus appear that the authors’ initial experiment involves an impeded fully circulating flow (or a similarity flow) in the lower tube half, and a similarity flow with an adjoining stagnant region in the upper half, which accords well with what they reported. With larger temperature differences, the stagnant regions would be expected to diminish or disappear, again as reported. I t is interesting to note that a slight circulation does exist in the so-called “stagnant” region, as originally suggested by Lighthill, but unconfirmed until this time; see Fig. 19f. It is of course possible that turbulent flow does exist, but even then rather similar flow regimes (a turbulent similarity solution does of course exist (80)) should be expected still giving nearly stagnant end regions. T hus we note in passing that the study conducted by Schenk et al. is for very low to,, presumably laminar ( ?)impeded flow with and without stagnant end regions. No effort was made to develop an analytical joining prediction model; the conditions described for this join are only for air at low t o , . Another Dutch study of the closed thermosyphon is that due to 1 using water and liquid Everaarts (32) for the case where LJL, metals. Unfortunately, this reference was found too late to be obtained and reviewed in detail. A detailed analytical and experimental study of the closed thermosyphon with emphasis on the joining mechanism was conducted by Japikse, Jallouk, and Winter (58-60, 64). T h e experimental results were based on two studies conducted both (1) to visually determine the nature of the fluid flow and temperature distributions (59) and (2) to determine the corresponding heat transfer rates (58) under both vertical and inclined conditions with various thermal boundary conditions. l’hus a transparent test cell and an opaque test cell with electrical heating were designed and built. Each apparatus had the same internal dimensions, a 2-in. i.d. and a 16-in. length, and equal heated and cooled lengths. Flow visualization (64) and temperature field measurements (60) were made for fifteen experiments using water, ethylene glycol, a water-glycerin solution, glycerin, and alcohol, as test fluids with values of A T of 1.0, 3.0, and 9.0”C. For the opaque test cell, experiments were conducted using water as a test fluid with log t,, 8.3 and 1.9 Pr 6.1. Thu s trends observed in the 6.2 transparent tests could be extended and corresponding heat transfer rates could always be determined.

<

<

<

<

<

50

D.

JAPIKSE

T h e fundamental flow process, in the sense of basic stable laminar flow, that was observed to exist is illustrated in Fig. 21. This illustration

Flow tube cross section ( f l o w splits ot d’)

I

Partial section view

FIG. 21. Exchange region flow patterns in the closed thermosyphon for flow with high Pr or low AT. From Japikse et al. (59, 60, 64). Symbols: - flow near surface; - - - - internal flow; N flow dividing lines. Flow Characteristics: (a-a’) flow rising along the wall, into the center, then up the center; (b-b‘) flow down along the wall, into the center, then down the center; analogous to a-a’ but inverted; (c-c‘) flow up along the wall but diverted into a-a’ due to influence of b-b’; (d-d’) analogous to c-c’ but inverted; ( e ) vortex formed by counterflow streams (e.g., a-a’, and d-d). Notes: This flow has three streams up and three streams down. There are three vortices in each end.

shows that the midtube exchange mechanism involves the creation of flow streams from the boundary layers in the vicinity of the tube mid section. T h e flow streams slightly penetrate the opposite boundary layer and are subsequently bent into a horizontal direction so that they proceed radially toward the tube center. At the centerline they are diverted once again so that they continue in their original axial direction forming a cluster of warm (or cool) streams in the cool (or warm) tube halves. Fluid which came down the wall directly counter to a dominant flow

ADVANCES IN THERMOSYPHON TECHNOLOGY

51

stream is returned to its original tube half and can form a stable vortex as illustrated in Fig. 21. T h e complete flow pattern is basically stable laminar flow although a small rotational realignment of the flow streams was found after several hours. No significant mixing was observed although small fluid elements were sheared off into an alien region on occasion. T h e observations were made using dye injection and fish flakes. Figure 22 is a set of photographs showing elements of the general exchange mechanism displayed in Fig. 21. Parts a, b, c, and f are dye

FIG. 22. Photographs of flow patterns in the vertical closed thermosyphon (59,64). (a, b) Flow around and through a convective exchange flow tube; (c, f ) wall and core flow with an entrained vortex; (d, e ) fish flake time exposures showing convective streaking. Notes: (1) T h e dark circles are inlet fittings into the external, annular heat exchanger region, not into the test cell proper (tube); (2) the horizontal bands are flow dividers in the annulus only; o o tube center plane; I tube center line; v tube wall.

52

D.

JAPIKSE

traces, and it should be noted that two types of dye marks can be observed. First, regions of very low or zero velocity, which must occur between adjacent streams of opposite direction, tend to accumulate and hold the dye stagnant. Thus a partial shell or tube is shown through which the actual flow stream travels (a, b). Secondly, the dye will follow a basic flow stream as shown in the photographs (a, b, c). Pictures d and e are fish flake time exposures and show the general fluid motion; they seem to show that two flow streams merge to flow through a given opening, but these are actually just high velocity portions of a single flow stream being formed as was observed directly. These portions accelerate greatly so as to pass through the restricted flow tube. For the example shown in Fig. 21 only three streams up and three down were observed. However, as few as two up and two down and as many as approximately ten up and ten down were also observed. I t was found that the number of streams increased as Pr decreased or AT increased and the stability of the exchange process likewise deteriorated (see Japikse and Winter (60) for details). Even after flow streams could no longer be seen, evidence of convection still existed in the centerline temperature profile giving a clean step at the exchange region (see Fig. 23). T o be sure, this process which is convective in the fundamental sense is not independent of other exchange mechanisms. For conditions of high viscosity and/or low temperature differences, definite evidence was found of conduction by comparing the temperature found in the core on one side of the exchange region with what would be expected based on the thermal conditions of the opposite boundary layer. By the same method and by direct observation evidence was found of considerable mixing effects as Pr decreased or A T increased. T h e Prandtl number was found to be quite important for describing the flow stability. For cases where Pr > 90, the flow was stable. For 10 < Pr < 20, the convective nature was apparent, but the streams were unstable and shifted around every few minutes. When Pr was less than 10, the convective process was basically unstable and shifted continuously by itself. Increasing the temperature difference aggravated this condition until no streams could be seen. In this study the authors found an interesting similarity between the observed exchange region and the inlet region of the open thermosyphon. Such a study for the open system was performed by Martin and Lockwood (90) (see Fig. 8). For low heat fluxes and very viscous fluids, they reported (for the open thermosyphon) that the existing hot fluid annulus necked down to form a hot rising stream just after leaving the orifice. T h e cold fluid which had to flow down the core, and which came from the floor of the reservoir, pierced this hot annular cone of fluid and

ADVANCES IN THERMOSYPHON TECHNOLOGY

I

0

53

I

4

8

12

16

Distance from bottom (in.)

501 I

I I

Distance from bottom (in.) (b)

FIG.23. Wall and centerline temperature distributions in the closed thermosyphon with insulated ends. From Jallouk [(58), see also in Japikse et al. ( 6 4 ) ] .(a) Convective type centerline profile; (b) mixing-type profile showing the influence of the adverse core-boundary layer mixing. Note: The open points show the range of temperature fluctuations.

entered the core through egg-shaped holes. However, a fundamental difference exists in the stability of the two processes. Under the conditions just mentioned for the open thermosyphon study, the egg-shaped holes had a random variation in size and orientation. This is the opposite of

54

D.

JAPIKSE

the result in the closed system where only a slight shift was found after several hours. Thus the presence of the tube walls has a stabilizing effect on an otherwise similar process. It is also interesting to compare these results to those of Shetz and Eichorn (110) who conducted a study using a cylinder submerged vertically in an aquarium of water, the bottom half of the cylinder being heated ( T I )and the top half being cooled ( T2).They were able to observe the rising and falling external boundary layers using the tellurium dye technique and the Schlieren method. Certain experiments were conducted where Eichorn (personal communication) observed that the rising and falling boundary layers broke from the wall where the temperature step occurred and pierced each other, much as one can interlace the fingers of one hand between those of the other hand to represent this interaction. I n general, though, the process was quite unsteady. Indeed the similarities and stability trends between this case and the preceeding two are quite striking. T h e heat transfer data confirmed the concept of a convective mechanism at low temperature differences with secondary losses in that the measurements obtained were higher than what would be expected assuming pure mixing and lower than what would be anticipated under conditions of pure convection. On this basis too and based also on centerline temperature measurements, it was found that the change over from the fundamentally convective exchange process to a purely mixing mechanism becomes rather complete for larger temperature differences, in this particular case corresponding roughly to the point where transition to turbulence occurs (58). Comparison of the experimental laminar results, (by Jallouk) see Fig. 26, up to t,, = 10i.6 with the very similar case from Lock (82) shows some disagreement, which has been attributed to the differences in end thermal boundary conditions (64). Of particular interest now are the results for t,, > 10i.6 which correspond to turbulent flow. In interpreting his results, Lock claimed to have found both turbulent boundary layer and fully turbulent impeded flow. T h e latter appears to be a mistake, however, since the results do not show any significant reduction in heat transfer rate as one finds in the open thermosyphon under impeded flow conditions (also shown in Fig. 26) where the rate nearly flattens out, so to speak, and where a turbulent boundary layer flow is never ultimately obtained. Although the heat transfer rates in Fig. 26 do not show strong evidence of impeded flow, some such effects are surely present. For the visualized experiments, some definite core-boundary layer mixing was observed (but generally restricted to the outer parts of the boundary layer), and

ADVANCES IN THERMOSYPHON TECHNOLOGY

55

a considerable amount of churning and mixing were evident in the exchange region for conditions approaching and including transition to turbulence. Although these factors can well lead to impeded flow, a convective type centerline temperature profile still existed in the observation tests (very similar to Fig. 23a), so the extent of this interaction could not have been too significant. However, in the heat transfer apparatus with turbulent flow, the centerline temperature profile changed from a constant temperature mixing case to one where the temperature was high at the bottom (in the core) and low at the top (Fig. 23b), which could only be caused by adverse core-boundary layer mixing, the very cause of the low heat transfer rates in the open thermosyphon under impeded conditions. Th u s some evidence of the causes of impeded flow were found but no evidence of a strong effect was present as judged by comparing the heat transfer results against those corresponding to open thermosyphon behavior from Martin (87),Fig. 26. This absence of strong impedence, which does occur under the same conditions in the open system, is in all likelihood due to the strong difference in stability found in the otherwise rather equivalent orifice (open system) and midtube exchange region (closed system). As a consequence of this fundamental stability difference, Fig. 26 clearly shows that the performance of the closed system can approach, and perhaps eventually exceed, the performance of the open system-in spite of the inefficiency inherent in the midtube exchange region. T he problem of analyzing heat transfer rates in the closed thermosyphon was greatly simplified by seeking a method of joining two open thermosyphons, as opposed to solving the governing flow equations for the entire tube region. Th e preceeding experimental results strongly suggest that this is a valid approach. T h e fundamental objective of any joining model is to obtain a centerline temperature profile for the closed thermosyphon which is then sufficient to determine the heat transfer rate using results from the open thermosyphon studies. Since the assumption of a constant core temperature in each half of the closed thermosyphon (with adiabatic ends) has been experimentally shown to be reasonable for boundary layer flow, this modeling problem reduces to the determination of two temperatures, one for the lower core To,l and one for the upper core Hence two governing equations are necessary. For cases with isothermal end conditions, the same approach can apparently be followed since the centerline temperature profile is rather constant throughout either half (64). T h e first equation is obtained from an overall energy balance while the second equation is obtained from a local energy balance. I t was observed that the value for a single temperature for the thermo-

56

D. JAPIKSE

syphon core under pure mixing conditions which is determined from the overall energy balance (below), was generally quite close to the average for Lock’s data. T h u s the first of the calculated values of To,l and To,2 step of this convection model is to determine Tmixand then alter it so as to get the convective profiles To,l = Tmix- T and To,2= Tmix T , where T is calculated from the local energy balance. Since the heat flow into the closed thermosyphon must equal the heat flow out, Q1 = Q, . Th u s one can show that

+

~

~

1

~

~

1

/

~

1-~To,,) ~ ~= ~( ~ 2~ N 1~ 2 ~ / ~ 21 ) ( ~2 ~ ~~ 2 ~ 2 )1( ’ ,~T1,J , 12

by using the definitions for h and Nu. Now for any open thermosyphon process which can be represented by the equation Nu = Cl(a/L)C2RaC3

(2)

(which includes all open thermosyphon results reported) the equation becomes klC,,,Rac~((a/L)~~lL1 AT,

=

k2C,,,RaC3(a/L)~~2L, AT,

or

where C , is assumed the same for each tube half. Solving for Tmix = To,1 = T0.2

where

x, A , ,

and A, are given by

A,

= gPa3/va,

A,

=

u/L.

Equations (19) and (20), using AT = Tl,l - Tmix,define the pure mixing model for any process which fits the form of Eq. (2). This includes laminar flows, Table IV, turbulent boundary layer flows, Eqs. (16) and (17), and turbulent impeded flow, Fig. 11. This analysis is presented in a more general form than given in Japikse et al. (64) to permit more general application.

ADVANCES IN THERMOSYPHON TECHNOLOGY

57

Now Eqs. (2), (19)’ and (20) are combined to give the general closed thermosyphon mixing model equation

NU^

=

($1

C1,121-4c3

c2,1-c3

1-( 1 +X X

1+c3

tzi.

It may further be remarked that heat transfer rates based on Tl,l - To,l = Tl,l - Tmix showed excellent agreement with experimental results for 5 yo mixing cases (i.e., those cases-using Lock’s data -where the difference in centerline temperatures was less than 5 % of the overall temperature difference and hence these data were quite close to truly mixing data (64)). T h e average error in these heat transfer rates was about 3 yo as compared with 23 yowhen using Lock’s mixing method which is sensitive only to property conditions in the bottom tube half. T o obtain 7,the second parameter for the convection model, a simple local energy balance in the exchange region is used. Thus, thermal energy core 2 cc thermal energy boundary layer 1 M2cp2(T0,2-

Tref) =

tfilcDl(Tmc,l- Tref)

or using certain nondimensional parameters with Tref = T o , l ,

or

+

where = T,,,i, - T and To,2= Trnix T and 5 is a fraction which characterizes all losses due to conduction and mixing. Together with Eqs. (19) and (20), Eq. (22) defines the simple convection model. Unfortunately only the laminar flow cases have been solved in sufficient detail to evaluate the ni’s and t,, in Eq. (22). Using the boundary layer information given in Table IV,

where the tot’s are to be evaluated using Tnlix.T h e equations for Tmix and T now define the simple laminar boundary layer convection model which is inherently an approximate one. Clearly, it presupposes (1) that Tmixis close to the average of and To,2and (2) that in determining 7 the new temperature profile obtained will not drastically alter the overall

D. JAPIKSE

58

energy balance from which Tmix was obtained. Hence the simplified convection model is actually a blend between pure mixing and pure convection models. T h e determination of is relatively easy after a few comparisons with data have been made. Equation (21) for the mixing model can now be applied to several specific cases and compared to data. For laminar boundary layer flow, Eq. (21) becomes

NU^ and

= 0.984

(

lob'$'

~

:j1.256

t0.256

ct

x is obtained for Cl,l = lob'*',

C1,, =

1Ob'3',

C2= C, = M I - 1

For turbulent boundary layer flow using Nu,

=

=

0.256.

0.10Ra1i3 [Eq. (17)],

4.

and x is obtained using C, = 0.10, C, = 0, C, = For turbulent boundary layer flow using Nu, = 0.0325Ra2/5 [Eq. (16)],

using C, = 0.0325, C, = 0, and C, = 6 to obtain turbulent impeded flow using Fig. 11

x.

Finally, for

using values of C, , C, , and m from Fig. 11 and p. 32. Before examining experimental comparisons, it should be noted that Japikse et al. also derived a pure convection model for laminar boundary layer flow assuming that all thermal energy from one boundary layer was convected across to the other core without loss, i.e., To,l = T,,,, and To,2= T,,,, . Thus

Figure 24 shows a comparison between the various analytical models and the experimental data from Lock (82) for ethylene glycol at various values of Lla with L, = L, . Included in the case for L / a = 7.5 are the predictions from Lock's mixing and convection models which lumped

ADVANCES IN THERMOSYPHON TECHNOLOGY

59

FIG. 24. Closed thermosyphon experimental and analytical heat transfer rates for ethylene glycol, L J L , = 1 . - - - -, Pure convection, Lock (82); - - -, pure mixing, Lock (82). a, Simple laminar convection, Japikse (59), Eqs. (19), (20), (23); b, pure laminar convection, Japikse (57), Eq. (28); c, turbulent mixing, Eq. (25).

all fluid properties together at the lower wall temperature. Values of E E 0.6 in the simple convection model are quite successful for the shorter tubes, but a value of f 'v 0.3 is necessary for the longer tube. T h u s the effect of constriction has caused a change from nominally convective toward the mixing behavior. Turbulent mixing predictions are also included but generally are not too consistent. I t has been indicated earlier that the interpretation of transition was quite difficult

60

D. JAPIKSE

for this data. Also of note is the difference between the pure convection model and the simple convection model for [ = 1.0 which underscores the approximate nature of the simple convection model. T h e use of a convective exchange parameter such as 5 is really valuable only if it can be readily estimated based on previous experience. Knowing several values for 5 based on the L, = L, ethylene glycol results, it is interesting to try to estimate behavior for L, # L, . Figure 25 shows three cases for L, # L, ; the two cases for which - Simple laminar o

convection model, Japikse (591, Eqns. 19,20,23 Experiment, Lock ( 8 2 )

p

Loglofcl

FIG. 25. Closed thermosyphon experimental and analytical heat transfer rates for ethylene glycol, variable LJL, .

5 P 0.6 is valid do carry over as reasonable guesses from the L, = L, study. However, the case for which 5 = 0.85 is valid, involves L / a ratios between 15 and 22.5 and thus a 5 between 0.3 and 0.6 would have been expected. Evidently varying the LJL, ratio can have a stabilizing effect and estimating 5 was incorrect for this case. T h u s the simple convection model can be erroneous when applied outside the range of conditions from which 5 can be estimated based on comparison to experimental studies. As would be anticipated, the effect of viscosity on the convective structure is evident in the choice of f for different fluids. Figure 26 shows the closed thermosyphon results of Lock (82) and Jallouk (58) for water. For water, a low value of [ = 0.25 was found to be best for L/a = 7.5

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61

1.7-

I .6 - Open thermosyphon behavior, Martin (87)

1.5-

1.4-

-

1.3-

z

P Cm

2

+

Experiment, Lock (82) L / a = 7.5

model, Eqns. 19, 20, 2 3

wotc,

FIG.26. Closed thermosyphon experimental and analytical heat transfer rates for water, L J L , = 1.

and ( dropped to zero for L / u = 15.0, thus becoming pure mixing. T h u s the constriction effect is again apparent but it is, so far as the present data reveal, a uniform effect which shifts the entire heat transfer curve as opposed to leveling off as in the case with impeded flow in the open thermosyphon. These values just quoted for ( are based on comparison with Lock’s data which had a nearly isothermal base. For the adiabatic base data from Jallouk with L / a = 8.0, a value of 5 = 0.75 was found to be best as compared with the 0.3. T h e results of comparing Eqs. (25), (26), and (27), which are turbulent boundary layer flows according to the and 6 powers and impeded turbulent flow respectively, with Lock’s experimental data are shown in

62

D.

JAPIKSE

Fig. 27. An alternative turbulent boundary layer equation was suggested in Japikse et al. (64) but was based on an erroneous analysis. These results show that the convective nature of the flow is still evident for ++ ++ Experiment, HO , from Lock (821 Boundary layer and impeded transition, Lock

1.3 -

-1 1.5

0

6

1.2-

.2 z

11.3 9

o

-

m 0

1.4

1

1.0-

I

,,

I

0.9 -

0.8

e

i=o

/f

a

L / a = 15.0

LOqlOtct

FIG.27. Closed thermosyphon experimental and analytical heat transfer rates for water, LI/Lp= 1, with attention to modes of turbulent flow. a, Simple laminar convection model. Eqs. (19), (20), (23); b, +-power turbulent boundary layer and pure mixing exchange mode, Eq. (25); c, g-power turbulent boundary layer and mixing exchange mode, Eq. (26); d, impeded turbulent flow plus pure mixing exchange mode, Eq. (27).

the case where Lla = 7.5 and the results are 10 to 20% higher than the pure mixing equation predicts. For the L / a = 15 case, the Q-power is rather good; the experimental results still generally falling above what would be predicted for turbulent impeded flow using the known data from impeded open thermosyphons. A further comment on 5 < 0 should be included. Such values appear relevant when the very low data points in Fig. 24 and 26 at large L / a are considered, as suggested in Japikse et al. (60, 64). Such usage, although

ADVANCES IN THERMOSYPHON TECHNOLOGY

63

perhaps convenient, is not strictly correct. I n such cases the coreboundary layer interaction can cause cases for which > T o , 2 such , as Fig. 23b, which is equivalent to 4 < 0. However, f is a parameter which applies to the exchange process in the exchange region, see Fig. 15, while the core-boundary layer interaction occurs outside this region. Th us values of f < 0 may be convenient, but they are not rigorously consistent with the f definition. For practical engineering applications, these models should be helpful for predicting accurate heat transfer rates in many cases and at least sensible trends in others. T h e values and trends given for 5 should serve as an adequate guide for the applied engineer to choose a value of 4 for L/a 30 and applications when using fluids with Pr 3 1 and 0 with either adiabatic or isothermal base conditions. Finally, one additional closed thermosyphon study should be mentioned which was omitted from the chronological sequencing since the more recent studies are necessary for meaningful interpretation. Donaldson (28) reported in 1961 a rather unique study of a static closed thermosyphon problem. He was interested in the temperature distribution in a closed bore drilled in the New Zealand thermal regions for steam power production. He modeled the bore, both analytically and experimentally, with a closed vertical tube with a linear wall temperature profile where the bottom temperature is greater than that at the top. Donaldson’s analytical model is essentially a modification of Lighthill’s (open thermosyphon) laminar similarity solution where account is taken of the wall temperature variation. Th u s the solution appears quite similar t o that of Ostrach and Thornton (99). Unfortunately, no effort was made to relate the analytical model to a specific flow pattern or to consider the possibility of various flow regimes (e.g., similarity, impeded, or boundary layer) and he applied this model without any such qualifications. By examining his velocity profile it can be shown that for a given radius the velocity cannot change sign. Hence a flow model such as shown in Fig. 15 cannot be realized from his solution. In fact, his equations imply that he has modeled the closed thermosyphon as an open thermosyphon (as per Fig. 1) with the top closed. This leaves the unanswered question of what happens to the wall and core flows at that closed off end. Also, would the wall flow be expected to develop upward or downward ? Nonetheless his solution is of merit if applied correctly. H e solved for the ratio of axial temperature gradient to wall temperature gradient and obtained a curve that was equal to unity for H - Ttop)/~aLtOtal, i.e., very similar to tct] between 10 [ H = /3gu4(Tbottom and lo2 then dipped down smoothly to a low of about 0.6 for H between

<

<

64

D.

JAPIKSE

lo2and lo4.For H greater than lo4,the ratio tended smoothly up to unity again. His experimental studies were conducted using a 36-in. long tube of 1-in. i.d. A surrounding counter flow heat exchanger provided the linear (approximately) wall temperature distribution. He measured centerline and wall temperature distributions and two basic types were found as shown in Fig. 28. T h e results shown in Fig. 28 led Donaldson

52 5 -

c--9

42.5

0

0.2

0.4 0.6

0.8

To 1.0

Depth fraction

30

(b)

0

0.2

0.4

I

I

I

0.6

0.8

1.0

Depth fraction

FIG. 28. Temperature profiles in the closed therrnosyphon with quasi-linear wall temperature variation (28).

to assert that since the centerline temperature profile could have a negative slope in the middle, there must be a nonsymmetric flow. However, negative profile slopes have been found under suitable conditions in all closed thermosyphon studies (see Figs. 18,23) due to the convective exchange process and hence the conclusion seems unlikely.

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65

By using the slope of the distributions at the tube middle, the ratio mentioned in the previous paragraph was obtained. These results showed good agreement with the analytical solution for values of H less than 10, and greater than lo4 but dropped to less than zero in between. A better interpretation of these results could be obtained, following Japikse (59), if it is noted that the lower portion, i.e., H u p to about lo2, corresponds roughly to Lighthill’s laminar impeded regime; and for larger H , Donaldson’s work corresponds to boundary layer flow, either laminar or turbulent (actually probably of a fully mixed type if turbulent). Although his solution appears to be for an open thermosyphon model, it could be applied to either half of a closed thermosyphon model and the above-mentioned ratio results would still be nearly the same thus giving the expected good agreement for H < 10,. For H on the order of lo3, his solution fails and his experimental results could be explained via a laminar boundary layer regime (with a constant core temperature) which could yield a ratio value of zero. For larger W, since water (low viscosity) was used, turbulence no doubt sets in and, recalling Foster’s or Martin’s results applied to each half of Fig. 15, a roughly linear centerline temperature distribution can once again be obtained yielding ratio values of order 1. Thus his analytical model in this latter regime would correctly represent this gross characteristic (by coincidence) but would be expected to fail in almost all other details. Such a description as given in this paragraph is consistent with all centerline temperature profiles yet measured, but clearly a further study of this case would be desireable.

C. CORIOLISAND INCLINATION EFFECTS T h e first study which gave any information concerning rotation (particularly Coriolis) effects on closed thermosyphon performance is the Pametrada project as reported by Brown (14). He reported that heat transfer tests on a liquid metal rotating cylindrical closed thermosyphon gave heat transfer according to Nu cc (Gr Pr2)2/13

and that varying the thermosyphon geometry required only the additional inclusion of L l d and L J L , variables. T h e low exponent was noted as being considerably less than what commonly is found for freeconvection processes. Temperature variations across the thermosyphon were found and attributed to Coriolis forces but were not of a significant magnitude to affect applications.

66

D. JAPIKSE

Other references to the Pametrada research are now known but are not yet generally available (3, 15,21, 22, 113). Colclough et al. (22) have quoted a heat transfer correlation for the variable cross section (a, # a2), liquid metal closed thermosyphon of the form

NU^

= (Lz/L,)o.66

(29)

(Gr Pr2)1.7d+5.35

+ +

where L,is an effective length of thecooled endand LTisL, L, amidsection spacer; the definitions of the other terms seem to be the same as in this text. Of particular interest is the appearance of the diameter as a dimensional parameter (inches) in the correlating equation much as it appears in Eqs. (2) and (27). Evidently the necessity of using the diameter was known prior to being suggested by Japikse and Winter (63); unfortunately the earlier information was restricted and in fact the details are not available even today. T h e most significant difference between Eqs. (29) and (2) is the use of the diameter with the power of the Grashof number and the large magnitude of the power itself. T h e absence of a multiplicative numerical constant is strange, though possible. Ogale (98) investigated a rotating closed thermosyphon for turbine cooling with variable diameter and called the system a "semiclosed" thermosyphon. Ogale's liquid metal thermosyphon had a base with radius (or hydraulic radius) a, greater than the blade radius a, since he felt it was necessary to reduce the length of the cooling section to meet turbine space requirements while still providing sufficient area for heat transfer. He measured heat transfer rates for cylindrical and airfoil cross section thermosyphons; the results are shown in Fig. 29. Unfortunately,

[

go 00000

, I 0'

, , ,,,,,, o"o"oo , O , , ,,,,,, los

I 0'

, ,

-

Io8

Gr P r 2

FIG. 29. Heat transfer in the rotating, closed thermosyphon with variable crosssectional area. From Ogale (98). 0 NaK, blade shaped cross section; x NaK, circular shaped cross section; o Na, circular shaped cross section. Notes: N u and G r are based on diameter of the blade section, d, , and AT = TI - Tzmuch as for all other closed thermosyphon work but the properties are referenced to the average of T I and Tz. NaK of 22 %Na/78 $'(, K was used.

ADVANCES IN THERMOSYPHON TECHNOLOGY

67

he did not ascertain what the nature of the flow process was in this system, but he felt that it may be behaving more as an open than a closed thermosyphon. T h e actual nature of the flow path is rather speculative (64). T h e cylindrical results are also shown in Fig. 13 for comparison to liquid metal open thermosyphon results. If one considers the L/a value, the low D, value and the probable effect of rotation, the results are in general agreement but, of course, not precisely so. T h e heat transfer results were found to correlate as Nu

=

0.3325(Gr Pr2)0.1a

for circular section,

(30)

for airfoil section.

(31)

and Nu = 0.6038(Gr PrZ)O.O9

Clearly there is a substantial difference between Eq. (29) and Eq. (30) or (31). It is interesting to compare predictions from Eq. (27) using the data from Fig. 11 for H g (which of course is not strictly valid for NaK) with the test data reported by Ogale in his Table 5.5 as given here in Table V. NaK property values from Jackson (56) were employed. Clearly there is a need for improvement in the prediction but considering the nature of the assumptions and the provisional nature of Fig. 11, the agreement is not too bad. TABLE V COMPARISON OF HEATTRANSFER PREDICTION, EQ. (27), TESTDATAFROM OCALE (98),NaK CIRCULAR SECTION

TO

1000 2000 3000 4000 4500

500 1000 1500 1750 2500

125 173 216 271 267

32 39 48 49 48

0.396 0.526 0.598 0.593 0.651

0.340 0.581 0.724 0.805 0.896

T h e only static study conducted with an inclined closed thermosyphon (to model Coriolis forces) is that due to Japikse et al. (59, 64) and Jallouk ((58), see also in Japikse et al. (64)). T h e same visualization studies were made with the transparent apparatus (as previously described) tilted so that the centerline was inclined u p to 18" from the vertical, this angle being denoted 8. T h e observations were made for all

68

D.

JAPIKSE

fluids at temperature differences of 3" and 9" using fish flakes and dye. A basic behavior pattern was found in all experiments. Certain differences, attributable either to increasing A T or decreasing Pr, were also found and will be discussed later. Observations made at 0 = 1.5", 3", and 6" showed that the initial effect of inclination was to cause a gradual rearrangement of the convective exchange mechanism (as shown in Fig. 21) for the more viscous fluids and a gradual alignment of disorganized fluid streams for the less viscous fluids. I n cases where a steady convective exchange mechanism existed for 0 = 0", the first development at small inclinations was the thickening of a single flow stream (one passing from the lower boundary layer into the upper core) along the upper edge and an analogous but opposite effect on the lower edge. T h e thickened flow stream would penetrate the opposing boundary layer a bit further than for 0 = 0" and slant more gradually on over into the core. With larger inclinations the two preferred flow streams became larger, flowed faster, and penetrated the opposing boundary layers further. At 0 = 6" only two flow streams were left, regardless of Pr, and they penetrated the opposing boundary layers about I to 2 in. For the less-viscous fluids, two dominant flow streams would develop thus imparting a higher degree of order to these flows. Photographs of these effects can be found in Fig. 30a,b, and d. T he authors found that for still larger inclinations the two dominant streams cut further into their opposing boundary layer flows so that at 0 = 12" and 18" a continuous loop flow existed as shown in Fig. 30c,e, and f and Fig. 31 except occasionally for a thin wedge of counter flow near a tube end which was actually a remnant of the original opposing boundary layer. This wedge would disappear for certain cases of extreme inclination or temperature difference or low viscosity and was a negligible portion of the flow even when present (for 0 = 12" or 18"). At 18" the mass flow rates in the tube appeared to have increased by an order of magnitude over the corresponding perpendicular case. Certain regions of limited churning existed at the tube ends where the flow streams collided into the ends and at the tube middle where about two thirds of the wall flow from a given tube half would neck down to form the single stream (about one third of the opposing wall flow, see Fig. 31) that would rise (or fall) through the opposing tube half. These effects varied appreciably from case to case and are discussed further now. For small angles (8 = 6" in particular) at AT = 3", the pattern described earlier was observed to form for all fluids but with some occasional unsteadiness for the 35 yo glycerin/65 yo water and distilled water cases, caused by the opposing forces exerted by the boundary layer and flow stream. T h e basic pattern was, however, still maintained.

ADVANCESIN THERMOSYPHON TECHNOLOGY

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FIG. 30. Photographic flow observations in the inclined closed thermosyphon (59,64). (a, b, d) 0 = 6"; (c, e, f ) 0 = 12"or 18". Notes: ( 1 ) T h e dark circles are inlet fittings into the annular region, not into the test cell proper (tube). (2) T h e horizontal bands are flow dividers in the heat exchanger annuli which surround the test cell. (3) T h e shining streaks are due to glare: o o tube center plane; I tube center line; v tube wall.

I n all cases a vortex was observed in the tube middle (see dye trace photos, Fig. 30b) which generally presented minimal interference. However, for the 9" temperature difference cases with 35 yoglycerin165 % water and distilled water, serious interference was found between B = 6" and B = 11". I n these cases the two prevalent flow streams were actually

70

D.

---_FIG. 31.

JAPIKSE

Flow direction divider Direction of fluid flow

Flow pattern in the inclined closed thermosyphon (59, 64).

intermittent or spurting. This was caused by a cyclic process which apparently was started and perpetuated by core vortex interference. This interference would cause temporary reductions in the flow rate through the two flow streams which was quickly augmented by increased growth of the opposing boundary layer wedge, thus enhancing the flow reduction. With decreased flow, however, the dominant boundary layers experienced increased heat transfer and hence increased buoyancy forces which were soon large enough to overcome the factors causing flow reduction, one of which (the vortex interference) also tended to abate simply as the flow rates decreased. When the tube inclination was increased to % = 18" this phenomenon ceased and vortex interference was again slight (Fig. 30). Martin and Lockwood (90) observed similar flow patterns in the inclined open thermosyphon. They found that the fluid entered the thermosyphon at the lower edge, went to the bottom, then rose along the upper edge. They found a steady improvement in stability over the vertical case as the inlet mixing was reduced or eliminated with increased tilting. T h e heat transfer measurements, see Fig. 32, follow quite well what would be anticipated based on the observed flow patterns, although the region where vortex interference was found might have been expected to correspond to decreased heat transfer. Also, the results follow the same trend as observed in the open thermosyphon (88,89) (see Jallouk (58) for a numerical comparison between these cases). T h e open thermosyphon results (89) for ethylene glycol showed, however, an initial decrease for increasing 8 and then a subsequent reversal in trend giving a definite rise. Based on the observations for the water/glycerin solution, this might be expected in the closed thermosyphon as well but probably at lower Pr since the vortex interference was not observed for

ADVANCES IN THERMOSYPHON TECHNOLOGY

I

1

I

I

0.2

04

06

0.0

tan

71

I

1.0

8

FIG. 32. Heat transfer in the inclined closed thermosyphon for water. From Jallouk (58) and Winter [see in Japikse et al. (64)].

ethylene glycol. T h e measured temperature distributions were in complete agreement with the flow process observed. An additional study (117) conducted by Zysina-Molodjen et al. for the rotating closed thermosyphon gives both heat transfer rates and flow pictures. Unfortunately, greater clarity is necessary (61). Finally, 1eGrivks and GCnot (44) have also conducted rotating liquid metal closed thermosyphon tests, intended primarily for comparison to their two-phase thermosyphon study (discussed later) using a thermosyphon with a rectangular cross section. Their heat transfer results roughly followed the trend for a constant wall heat flux found by Bayley and Czekanski (7) (incorrectly referenced to (78)) for the static open thermosyphon. T h e agreement is quite interesting particularly since there is no particular reason for a constant-area closed thermosyphon to perform as an open one. In general, one would expect the closed thermosyphon to yield heat transfer results which are inferior to those of an open thermosyphon; perhaps the higher values found are due to the influence of the Coriolis force which can yield a more favorable flow pattern, as per Fig. 29. They found that the data followed the trend (Gr Pr2)o.2,the low-exponent value being rather similar to those found in other rotating experiments (14, 36, 98). Perhaps of greatest interest is the fact that the temperature distribution along the thermosyphon was rather flat, particularly as compared to the temperature distribution for their two-

72

D.

JAPIKSE

phase thermosyphon. This is particularly of interest from the standpoint of minimzing thermal stresses. Another interesting result is that the thermosyphon showed improved heat transfer capability, as evidenced by reduced blade temperatures, as the rotational speed increased up to about 3500rpm. From about 3500rpm to 6000rpm (the maximum obtained) no further improvement was observed. This latter behavior is rather surprising since the body force must continue to increase.

IV. Closed-Loop Thermosyphons

Two distinct virtues make the closed-loop thermosyphon profitable to study, although, oddly, it has received about the least attention. First, it is a very natural geometric configuration which can be found (or easily created) in many industrial situations. Secondly, it avoids the entry choking or mixing that occurs in the open thermosyphon, the complex midtube exchange process in the closed thermosyphon and, largely, the adverse core-boundary layer interaction common to both the open and closed thermosyphon. Thus, in addition to being convenient and common, the closed- loop thermosyphon should be capable of attaining much larger heat transfer rates. In principle, there is virtually no limit to the types of flow that could be obtained in a closed loop subject to various thermal, geometric, body force, and thermodynamic state conditions. Most cases so far considered have been simple single-phase, continuous, loop flows. An example of a more complex case would occur (see Fig. 33) if T I = T , < T , = T4 and the geometry in all sections was identical (and no rotation). I n this

FIG. 33. where

T h e simple closed-loop thermosyphon, circulation shown for the case

T2= T4> TI= T 3 .

ADVANCES IN THERMOSYPHON TECHNOLOGY

13

event each side of the loop would behave virtually independently as a simple closed thermosyphon with no appreciable circulation between the two sides. If, however, a significant change is made in the constraints, such as geometry or wall temperatures, then a strong loop circulation can be induced. Initially the strong loop flows will be considered with attention to other more complex forms later. All analyses to date of the loop flow have been simple one-dimensional force and energy balances (16, 17, 25,47,53, 72). Inasmuch as they are quite straightforward and simple, no elaboration is necessary. Apparently such analyses are quite adequate for many of the loop flows. Lapin (72) has conducted constant wall flux heating and cooling; he found very good agreement between his turbulent experimental results and a similar onedimensional constant flux analysis as shown in Fig. 34. T h e results that show some deviation are for laminar or transitional flow.

Gr PrZ

FIG. 34. Heat transfer in the simple closed-loop thermosyphon, constant flux Q , into loop; Q 2 = Q , out of loop. From Lapin (72).

=Q,

Another rather simple closed-loop thermosyphon which has been studied experimentally is shown in Fig. 35 Morris ((95,96), see also Davies and Morris (25),Humphreys et al. (55)) has performed a number of experiments on this configuration under the condition of rotation (0-300 rpm) using both water and glycerol. For a given temperature difference between the wall and coolant, increasing the speed of rotation gave strong increases in the heat transfer coefficient. An interesting comparison is formed if one models the radial limbs of the apparatus, along which the centripetal acceleration is felt, as copper conductors and computes the effective conductivity of the rotating thermosyphon. Morris showed that the closed loop at 300 rpm (Fig. 3 5 ) has an effective conductivity about 35 times that of copper for a 100°F overall temperature difference. T h e heat transfer results obtained were Nu Rad/Pr2

=

O . 1 5 0 A ~ ~ . ~ ~ ~for R ewater ~.~~

74

D.

I

JAPIKSE

Rotating seal

,Axis

of rotation

1

0 0

lnstrumentotion

600-4

FIG. 35. Rotating closed loop thermosyphon. From Morris (96).

and Nu Rad/Pr2 = 0 . 0 8 2 A ~ ~ . ~ 8 ~ Rfor e ~glycerin .~~

where Ac = Rw2/g, Ra, = (Ru2/3d3 A T/v2)Pr. T h e Reynolds number must be calculated from a simple momentum analysis, which Davies and Morris (25) found to be quite satisfactory. Strangely, no complementing one-dimensional heat transfer analysis was reported; it would have been interesting to compare results, even though the effects of rotation could be expected to cause some deviations since these effects are two or three dimensional. There is no doubt that the correlations given are sufficient to represent the data obtained, but it is not clear whether or not these correlations can be used for other geometries and flow conditions due to the form of the equations reported. Other geometries are doubtful since only one configuration was tested, and other flow conditions are questionable since Gr/Re or Gr/Re2, the usual mixed convection parameters, do not appear in the equations as a basic parameter (of course they could be rearranged to obtain any such parameter, but just what that parameter should be is not evident). T h e two examples considered so far have been rather simple loop flows. T h e process becomes more complex, as indicated at the beginning of this section, if one considers more complex operating conditions. For example, if a multichannel system is employed, the possibility exists that any one channel could have flow up or down depending on its thermal

ADVANCES IN THERMOSYPHON TECHNOLOGY

75

conditions and its relationship to all other channels. I n this event a mixed-convection flow results with pressure gradients imposed on some channels so as to cause flow in a direction opposite to that of natural convection flow. T h e importance of the mixed forced and free convection, with reverse flow regimes, was first referred to by Kunes (71) in a study of the multichannel (closed-loop) thermosyphon system for cooling transformer cores, and he was able to generate reverse flow regimes. This problem has received some analytical attention by Chato and Lawrence (17). Chato likewise conducted some experiments (16) and attempted to compare them with his analysis, assuming laminar flow. At very low heat fluxes he had fair success, and predicted flow conditions satisfactorily; but for moderate heat fluxes early transition to turbulence caused sizeable deviations. When body force and pressure effects are in opposition and of comparable orders of magnitude, one-dimensional analyses fail due to the strongly two- or three-dimensional nature of the resulting flows. One of the better known thermosyphon applications is the use of closed loop thermosyphons for cooling internal combustion engines (70, 111, 115). Such applications were made for automobiles prior to about 1945 (24) and for farm equipment up to at least 1955. T h e loop passes through the engine block, Fig. 36, where the fluid is heated and then is passed through a conventional radiator with fan to remove the heat prior to entering the engine block again. Upon inve~tigating,~ this writer learned that thermosyphon cooling was adequate up to compression ratios of about 5:l and that it was an economical and service-free system due to the absence of water pumps and thermostats (115). Later it was necessary in some applications to include a radiator shutter to maintain higher water temperatures to control oil-fuel dilution as vehicles (tractors in particular) were used more in cold weather. Evidently each company developed its own design procedures (no references to such work in the heat transfer literature are known); in the one case which this writer has been able to review, rational design procedures evolved over the years of usage. However, systematic heat transfer studies to determine the optimal design conditions apparently have never been made, and it seems that the range of applicability can be extended. I n any event, such cooling schemes should be quite appropriate for the numerous small internal combustion engines which are prevalent today in garden and leisure-time vehicles. This author is grateful to Mr. R. Deryl Miller, Manager of Service-Engines and Transmissions, Deere & Company, and his colleagues for their correspondence and informative notes on this application.

76

D.

JAPIKSE

FIG. 36. The closed-loop thermosyphon applied to cooling a John Deere Model M T Tractor; illustration courtesy of Deere & Company, Moline, Illinois.

Another early application for closed-loop thermosyphons is as a heat transfer device in nuclear reactors. Several investigations have been reported which are both experimental and analytical ( 4 7 , 5 3 , 5 6 ) . A recent paper (85) outlines the use of various single-phase thermosyphons as heat exchanger fins (another investigator also suggested the use of two-phase thermosyphons for the same applications during about the same time period (73)).T h e authors of the paper, Madejski and Mikielewicz, failed to draw on the very considerable thermosyphon literature relevent to their problem but made valuable contributions nonetheless. They used the Galerkin-Zhukhovitskii variational method to calculate

ADVANCES IN THERMOSYPHON TECHNOLOGY

77

the condition for the onset of circulation. Furthermore they constructed a recuperator using constant-diameter closed-loop thermosyphons as fins symmetrically installed so that the loop section in the hot gas was very similar to the loop section in the cold gas. They experimentally determined that such fins when filled with NaK eutectic have an apparent conductivity ranging from 50 to 100 times the thermal conductivity of the fluid, NaK, for the range of operating conditions tested. Values of about 150 to 250 were obtained using Hg. I t seems strange that the loops were not constructed in such a manner so as to induce a strong circulation, perhaps by geometry variations, as opposed to relying on thermal instabilities to initiate some degree of fluid circulation. T h e authors compared their results to a correlation for horizontal enclosed fluid spaces and found exceptional commonality. They concluded that the agreement was “curious.” Simple one-dimensional energy and force balances should be sufficient to determine ways of inducing stronger circulation. I t is not clear why two-phase systems were not considered for this application.

V. Two-Phase Thermosyphons

A. GENERAL BEHAVIOR T h e idea of operating a thermosyphon as a two-phase system is fundamentally appealing due to the high heat fluxes associated with the latent heats of evaporation and condensation, the much lower temperature gradients associated with these processes, and the reduced weight of such a system over a similar liquid system. For a given application, the higher heat fluxes usually imply a smaller system volume and smaller heat transfer areas when compared to other thermosyphon systems. Schmidt (107) capitalized on the high heat transfer and the low-temperature gradients and regulated the pressure in his open thermosyphon turbine blade so that boiling would occur at the free surface giving a very efficient heat sink. Cohen and Bayley (20) additionally took advantage of using small fillings with their rotating and static experiments and found that very good heat transfer could be obtained with as little as 1.5 yo volume filling of their system. Various investigators have considered systems in which differing filling quantities are employed; small filling quantities give performances based on evaporation and condensation with some type of return film flow whereas larger filling quantities additionally involve regions of fluid buildup in which boiling and convection are possible. Consequently, this section naturally

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divides into a portion on film flow with evaporation and condensation, or small filling quantities, and two-phase phenomena with large filling quantities. For completeness, a section on critical state operation is included.

B. TWO-PHASE PHENOMENA WITH SMALL FILLINGS Cohen and Bayley (20) conducted a series of experiments using hollow thermosyphon tubes in a rotating test rig which were filled with water in quantities ranging from 0 to 100 yo of the tube volume (at speeds up to 2000 rpm). They reported that the heat flow for fillings from 1.5 to 1 0 0 ~ owere not noticeably different, except that the larger fillings had a longer transient before the common steady state value was attained, due to the larger mass of coolant. T h e investigators were led to believe that the exchange process consisted of condensation in the base and a return film flow along the wall with evaporation. For fillings less than 1.5 yo,insufficient fluid for wetting the walls was the suspected cause of the reduced heat flow. For sufficiently large fillings, a liquid pool can be expected in the heated end, and the reported indifference of heat flow to percent filling would imply a heat transfer process in the pool similar to that in the film above. T h e investigators felt that a series of static experiments were necessary for further insights to this process. Once again the static tests confirmed the relatively indifferent nature of the thermosyphon to the amount of coolant present, past a small minimum value sufficient to ensure wetting of the walls with the liquid film which returned the condensed vapor from the cool end of the thermosyphon to the warm end where evaporation again occurred. T h e authors were able to relate the mechanical aspects of the film flow (gravitational and viscous forces) with the thermal aspects (evaporation and conduction through the film) in a manner which is essentially the reverse of Nusselt's condensation theory (but using heat flow/per unit area rather than the wall minus vapor temperature difference) to give

T he experimental results confirmed the parameters and slope given by this equation, but not the exact magnitude, most likely due to the breakup of the film into rivulets or drops, as the authors observed directly in a glass test cell. Even for a very clean surface, differences were found, probably due to film rippling. These irregularities are not totally surprising since most of the film in the evaporating end is, by the nature of

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this conjectured film process, in an unstable superheated state. Surface roughness or impurities could then be expected to cause local film breakdown leaving regions to be cooled only by conduction thus causing larger A T’s to be found in practice than those given by the film equation, as indeed were found. A consequence of the film theory just described is that as the heat flow rate increases, so must the film thickness and hence the quantity of fluid being circulated. If the initial mass of fluid in the thermosyphon is greater than that which is required, a liquid pool will form at the bottom of the heated end; when the initial mass is less than that required, the film will not continue to the end of the heated section and the danger of burn-out will arise. This actually occurred in one of the reported experiments, but fortunately not before the authors determined that the maximum heat flux obtainable with their system should be about 80,000 BTU/hr ft2. Actually this implies a film temperature drop greater than 100°F so the authors felt that the actual limitation to their experiments may well be due to the degree of superheat which the liquid film can withstand. I n concluding the present remarks concerning Cohen and Bayley’s work, it can be noted that they justifiably put great emphasis on the nature of the film which is the key to the mass circulation, the primary thermal resistance (i.e., conduction across it), and the ultimate limitation to the heat flow rate due to film breakdown. More recently Chato(18) has solvedthe integral momentum andenergy equations for the condensation/laminar film flow problem with a linearly varying acceleration field, as occurs in turbines. By using the method of successive approximations, he obtained results which showed that the condensate thickness and heat transfer approach limiting values with increasing radius and that heat transfer increases slightly with increasing I . He temperature difference if Pr > 1, whereas it decreases for Pr further considered the effect of surface shear due to the moving vapor and the effect of hydrostatic pressure change in the vapor phase. In each case it was found that significant influences on heat transfer could be obtained. On the other hand, whether or not his assumptions of laminar flow and similarity profiles are valid were not examined. No experimental data were given. For the cases just considered, the results are influenced by the choice of geometry in which films are likely to form. If, however, a thermosyphon is employed which has a condenser end of large cross section and reduced length (compared to the evaporator end), it is quite likely that the primary condensation will occur on the end wall perhaps without any film flow (in the condenser end). In this case, dropwise condensation needs to be considered in detail as GCnot and leGriv2s (42) have done.

<

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T h e forces on a single droplet include surface tension, dynamic pressure of the flux of the approaching vapor and the inertia force from the body force field. Th u s for droplet formation on a surface normal to the body force field the force balance gives (1

+ cos c)'

(1 - 4 cos C) w2RRd2- $(pv/pL) UV2&

-

(3u/pL) sin2 E = 0. (33)

Detachment of the drop from the surface is taken to occur when R, in this equation achieves a maximum. T h e solution for this value requires that the nondimensional group g(pv Uv2/2)2/apLw2Rbe less than dropwise condensation 2.95 x lop3. Thus, for w 2 R 2 508(pvUv2/2)2/apL will be occurring, and the authors conclude that for turbine cooling applications the conditions for dropwise condensation should generally be achieved when condensing on a surface normal to the body force. For parallel or inclined surfaces, it is quite easy to visualize film condensation with inferior heat exchange. I n a second study conducted by leGrivb and GCnot (44) a rotating, two-phase, NaK thermosyphon with a rectangular cross section was tested at rotational speeds u p to 6000 rpm. Their tests demonstrated that this system is quite adequate for blade cooling and the authors expressed their preference for this system over a similar single-phase closed system. They cited as major advantages the lighter weight and more favorable base temperatures facilitating easier base heat exchanger design. On the other hand, their study showed smoother temperature distributions for the single-phase system (advantageous for thermal stress minimization) and more extensive cooling. Presumably their tests can be taken to indicate that the Coriolis influence on the liquid film (two-phase system) is not severe, at least u p to 6000 rpm. This conclusion, although tentative, is reasonable since blade failure was not reported; unfortunately, temperature measurements around the circumference were not reported; these would have been very helpful in this regard. It would seem that this question would be amenable to analytical treatment which would be quite helpful, particularly since rotational speeds up to 50,000 rpm can be obtained in some turbines. One additional reference concerning this problem is Palanikuman (100) who studied the closed evaporative thermosyphon at various angles of inclination. T h e most recent, and most comprehensive, study of two-phase thermosyphons with small fillings is that due to Lee and Mital (77). They conducted an experimental study using a 1.06-in. i.d. by 54411. long (maximum) electrically heated, water-cooled thermosyphon using water and freon as test fluids and varied filling quantity, L,/L, , pressure (or Ts),and heat flux.

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I t was learned that the quantity of filling had very little influence on the heat transfer coefficient, h = q/( T , - T,) past a certain small filling quantity at a given T , (as Cohen and Bayley (20) reported). At very small filling quantities (say less than 5 % ) the effect of filling quantity was significant and caused variations in h,,,, by a factor as large as 2. T h e effect of decreasing L,/L, was to increase h,,, , much as was found in the single-phase thermosyphon. A limiting value of LJL, where this effect should reverse was anticipated but not obtained in the range of LJL, considered (0.8 to 2.0). Hence the advantage of larger condensation area is evident. T h e heat transfer coefficient was found to increase considerably with increasing mean pressure due to at least three factors: (1) since the mass flow for a given heat flux is nearly constant and density increases with pressure, a lower pressure drop (and hence lower A T ) is necessary for the same flux; (2) for larger P, , P, varies much more rapidly with T , for the fluids considered, hence requiring smaller A T’s at higher pressure; and ( 3 ) for lower pressure drops, a more favorable force balance exists on the condensate film permitting faster liquid return. A heat flux as large as 22,400 BTU/hr ft2 at 3.97 psia without reaching burnout was obtained. Some effect of heat flux on h was noticed, perhaps weaker at higher pressures. Water was found to give heat fluxes superior to those of freon due to better values of the latent heat of evaporation and thermal conductivity. T h e authors also considered the analytical problem of predicting the maximum heat transfer rate for a laminar film, constant wall temperature condensing section and constant heat flux evaporator section. Starting with a force balance on the falling film and including only the effects of gravity and fluid shear while neglecting the forces due to vapor pressure drop and momentum changes, an expression for the mass flow rate in the film was obtained. Then using a local energy balance in the condensing section between the latent heat of condensation and conduction through the film, plus an overall energy balance between the heat into the heated end and the heat removed from the condensing end the following equations for q and T , were obtained

where C= ~-~y2+~y4-&ydlny D

= y4 lny(31ny2 -

y

=

1

-

6/R.

g)

+ &y4+ y2($lny

-

$) -1-

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These equations were solved iteratively for q and T s with the properties evaluated at the film temperature ( T , T2)/2.Figure 37 shows results

+

FIG. 37. Heat transfer results for the two-phase closed thermosyphon with small fillings. From Lee and Mital (77).

of these equations compared with Lee and Mital’s experimental data. Due to the simplifying assumptions, errors as large as a factor of two exist, but the qualitative behavior is correct so far as trend, LJL, behavior (not shown), T , and working fluid are concerned. Finally, in concluding this section several applications for two-phase thermosyphons with small fillings will be mentioned. First, the application for precooling cryogenic equipment deserves comment and has been reported on several times in the past decade (10, 11,43, 69). T h e primary advantage of the two-phase thermosyphon for this application is its diode behavior, i.e., it transfers heat in one predominate direction over a fixed temperature range. For example, Gifford (43) d’iscusses a

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stainless steel tube operating with N , . For the geometry chosen one obtains an effective conductivity 16 times that of copper when operated in a normal manner with T I > T , , but when T I < T , the effective conductivity is only 1/400 that of copper; hence the conductivity in the normal manner of operation was calculated to be 6400 times the opposite direction. Using this diode type behavior several thermosyphons with different fluids can be used in series or parallel to cover a large cool-down temperature range by transferring heat to different reservoirs. Pumps and compressors are not needed and short cool-down times are possible. Gifford also discusses cryogenic applications in which the two-phase thermosyphon can be used as a thermal triode and for thermal damping. Although most of these papers have been application oriented, some have concentrated on the actual mechanics of the process. Bewilogua, Knoner, and Kappler (11) constructed a thermosyphon for cryogenic applications and found that the quantity of liquid filling, past a certain small initial amount, has no influence on heat transfer as the other studies have also shown. They discuss several actual application experiences and report cool-down times and refrigerant consumptions which are competitive with other precooling methods; the most notable advantage of the thermosyphon is its simplicity. Knoner (69)has reported results of a study on the heat transfer characteristics of cryogenic thermosyphons. H e found that pressure has a marked influence on heat transfer, higher pressure giving higher heat transfer (as Lee and Mital(77) also report). Perhaps his most important finding was that the performance of such systems is highly controlled simply by where the largest thermal resistance is. For large temperature differences, this occurred at the heat transfer surface (boiler end) since, due to the resulting Leidenfrost phenomena, the liquid nitrogen was not in contact with the heat transfer area. At low temperature differences, about 30°K and lower for N, , the thermal resistance of the condensation portion and the heat sink became important. Heat transfer coefficients were measured for H, , N, , and Ne for d T from 0 to 200°K and were found to be rather similar in form for these gases provided sufficient flow area existed to permit the varying flow rates. T h e effect of pressure on h was evidently not studied. Transient cool-down time was calculated with a very simple heat balance with reasonable accuracy. Finally a cryogenic two-phase closed-loop thermosyphon study by Bewilogua and Knoner (10) should be included here for completeness. They transferred 500 W for a AT = 8°K as compared to less than 20 W with the small filling evaporative/condensing thermosyphon (11). This was done over a horizontal distance of two meters, with larger distances being possible. In this case the highest thermal resistance occurred in the condenser. Good transient times and good thermal control were reported.

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Several other applications of these thermosyphons include (1) heat extraction from permafrost to provide solid arctic foundations and (2) heat extraction from the ocean to keep buoys from freezing in the winter. T h e permafrost application was first discussed by Long (84) (such thermosyphons have been termed “Long thermopiles”). He has discussed the construction and installation of them in great detail and has offered a few suggestions as to good operating fluid selection. Propane is perhaps the most economical, but butane and carbon dioxide were also considered. Propane has been found to operate efficiently from 32” to -60°F with pressures up to 60 psi. Such thermosyphons have been used at the Aurora and Glennallen communication sites built in 1960. Very good performance has been obtained. PHENOMENA WITH MODERATE FILLINGS C. TWO-PHASE T h e problem of understanding two-phase phenomena with large or moderate fillings is important not only to explain how the liquid pool in the heated end behaves, as referred to several times in the past section, but also as a basic operating mode in its own right (for example, with the heated end full, one need not be concerned about the influence of the Coriolis force on a film). Cohen and Bayley (20) conducted a set of static rig experiments with various diameter tubes and various fillings, in each case just filling the heated end. They found that the process was one of nucleate boiling and, due to the complexity of the process, sought a correlation based on dimensional analysis in the form

as opposed to pursuing an analytical analysis. Guided by some very interesting vapor bubble/drag experiments which suggested a nearly constant characteristic velocity, they took the velocity as constant and obtained a good graphical correlation based on k ATlpvLeVd vs. (~,’pvL,Vdz)(pLip,)(L,/d). Th u s A T was claimed to vary as the 0.75 power of Q. However, it appears that the data quite clearly follow the correlation

which differs somewhat from what Cohen and Bayley implied. At this point two observations can be made: (1) T h e heat exchange process

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which occurred in the liquid-filled tubes, or liquid pools in the previous section, gives heat fluxes characteristic of a boiling and not a convective nature and hence the indifference of the two-phase thermosyphon tubes to the amount of coolant previously reported is not greatly surprising. (2) On the other hand, the fact that A T cc q4/3 in the one case and AT cc q0.5s in the other case should have shown up as some influence on the total heat flow which was reported unaffected by the filling quantity-perhaps the effect was too small to be detected by the methods employed. Another interesting characteristic is that the maximum heat flux was found by Cohen and Bayley to be diameter dependent: about 40,000 BT U / hr ft2 in a 0.25-in. tube and about 92,000 BTU/ht ft2 in a 0.50-in. tube at atmospheric pressure (at 190 psi the first value was reported to rise to 73,500 BTU/h r ft". They were thus led to believe that the limitation on heat transfer rate was due to flow restrictions resulting from large vapor-bubble movement in a restricted tube, giving, for large fluxes, regions where the liquid is completely displaced and thus no cooling. Since only the vapor motion is thus significant a new dimensional analysis suggests

and, in fact, a plot of Qmax/pVLcVd2 versusL,(pL/o)1'2gave a unique result, roughly hyperbolic in shape, this shape suggesting that Qmaxwill not go e8). to zero and the limiting value is reasonably large (Qmax/pvLcVid2 For unrestricted passages, film boiling is still a final limit, but vapor blockage is likely to occur first in thermosyphon problems. A second study of the two-phase system with moderate filling quantities was reported by Bayley and Bell (5). They studied the heat flow with evaporation and condensation in a rotating thermosyphon with 25, 50, 75, and 100% of the heated length filled with mercury. T h e results which they obtained (see Fig. 38), were quite interesting and both increases and decreases in heat transfer were obtained for changes in rotational speed and tube filling; yet most of these changes were quite consistent with the known heat transfer theories. By assuming that boiling occurred in the liquid pool and evaporation occurred in the film above the pool, the calculations (described in detail in Bayley and Bell ( 5 ) ) showed good agreement with data, except at very large fillings (essentially full). At these largest fillings Bayley and Bell were able to demonstrate that the thermosyphon wall temperature probably lay below the saturation temperature of the mercury, thus prohibiting evaporation; and the strong gradients present evidently cause the large heat fluxes. We can

D.

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Calculated behavior assuming 2 phase phenomena (for the full tube case, behavior believed to be single phase)

0

0

112

I/4 Fraction

3/4

I

of heated length filled

FIG. 38. Heat flow in the two-phase, rotating closed thermosyphon. Plotted from Bayley and Bell (5).

therefore conclude, generally along with, but somewhat in contradiction to, the authors: (1) Operation with a partially filled heated end with evaporation is not necessarily superior to a filled heated end without evaporation; in fact, the data suggests the opposite to be generally so. (2) Operation with small filling quantities (in this case one-quarter full) is generally as good as a filled heated end and obviously weighs less. (3) Reasonably good agreement with the concepts presented throughout this section was obtained by comparison to data, except as noted in (4). (4) T h e previous finding that heat flow is independent of percent filling does not hold, though at least part of the reason for this is accounted for in the authors’ analysis: Hg, being heavier, would have larger variations in hydrostatic pressure and hence saturation temperature which of course depends on the quantity of Hg introduced. T h e effect should be stronger with increasing rotation speed, as Fig. 38 shows. Thus apparently the significance of mass filling is accounted for by the computations, but nonetheless it is now an important parameter for determining heat transfer. Another factor may be the different relations

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which govern AT, q for the liquid pool and film portions, as mentioned previously. Finally, this section can be concluded by noting that Bayley and Bell’s lOOyo filling case without evaporation is quite similar to Schmidts’ (107) scheme for cooling turbine rotors where, by suitably regulating the hydrostatic head, he maintained a pure liquid phase in the blade passages and permitted evaporation to occur only at the outer free surface where it served as an excellent heat sink. Another advantage of his scheme was that the “pure liquid phase” was often allowed to operate very near the thermodynamic critical point. T h e most recent study of a two-phase thermosyphon with large filling quantities is that due to Larkin (74). H e used electrically heated, water cooled thermosyphons withL,/L, varying from 2: 1 to 1 :2 with 0.75-, 1.0-, and 1.25-in. nominal internal diameters for measuring heat transfer characterisitcs. Other tests with a glass thermosyphon permitted fluid flow observations. Tests with air flow through water in a tube provided further information about the two-phase behavior. It was not possible in this study to develop precise relationships for the boiling and condensing heat transfer coefficients, but considerable knowledge of the mechanics of the processes was gained. Larkin observed the processes in the glass test cell with Freon 11 to be rather similar to the type of flow described by Cohen and Bayley (20): condensation, return film flow into the heated end with heat conduction through and evaporation from the film in that end. A boiling pool in the bottom may accompany such a film flow process. I n the glass tube the condensation was generally dropwise and irregular. T h e film was rippled (after about 2-in., Re ‘v 30, see Fig. 39) and Larkin noted that this effect is not accounted for in most laminar film flow analyses. T h e film was observed to flow about half way into the heated end before breaking up into rivulets. However, when using water as a test fluid the above film flow did not cool the heated end in any notable manner. Keeping the heat flux constant, he increased T ; as the density correspondingly increased it was observed that the level of the two-phase mixture subsided proportionally. Upon falling below the end of the condenser the condensate film did not continue to wet the tube cell but rather began boiling immediately. Some of the liquid was thrown off, the rest gave the appearance of Leidenfrost boiling. T h e temperature under this condition corresponded to the dryout condition in the brass heat transfer test cell. T he heat flux, as measured in the opaque test cell, was found to increase with decreasing fluid depth up to some optimum value. Larkin cited an example for an 18-in. heated length and a 36-in. cooled length with the 2-in. diameter tube using water for which the optimum filling

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FIG. 39. Rippled film condensate flow in the two-phase closed thermosyphon (74); photograph courtesy of Dr. Larkin.

was 9-in. At low operating temperatures the heat transfer parameter,

Q/(T,- T2), showed some dependence on heat flux. T h e optimum performance was limited by dryout. So far as these tests could show, no general advantage was found by using a flow divider to separate the liquid condensate flow from the vapor flow. Dryout could occur by several different means. First, dryout would occur if the two-phase mixture did not reach high enough in the tube and the condensate film did not reach low enough. Secondly, even when the tube was adequately filled, vapor blockage was a cause of dryout, much as dryout occurs in conventional two-phase flows. Thirdly, dryout could occur at the bottom of the tube if conditions were such to cause all the fluid to be held in the condensate film. Larkin noted that his observations were made for constant heat flux and that other conditions, such as a constant wall temperature, might well give different behavior. This may

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well be an important factor for dryout conditions. T h e author noted that dryout could occur by reducing the filling quantity or increasing the heat flux; for the 1-in. and Ii-in. tubes only the first type was possible due to the limited power supply but for the 3-in. tube a critical flux existed regardless of filling quantity. Another limitation on performance was observed when the bulk temperature (for water) was below about 120°F. In this case boiling was intermittent due to erratic nucleation releasing energy stored as superheat with a slug of water being thrown up the tube, as shown clearly in the photographs, Fig. 40. Noisy and violent operation was observed as the power was increased.

FIG.40. Slug-boiling in the two-phase thermosyphon (74); photograph courtesy of Dr. Larkin.

Larkin found that his boiling heat transfer coefficients were generally four times larger than those of Cohen and Bayley [Eq. (35)]. H e concluded the analysis was suspect due to the assumption of constant bubble velocity. T h e assumed constant velocity of 0.7 ft/sec was in particularly bad agreement with values measured in a thermosyphon of as much as 15 ft/sec. Filling quantity had little effect on heat transfer except near the dryout condition. Likewise, the heat flux generally had no significant effect on the boiling heat transfer coefficient. Perhaps the most significant effect on the condensation heat transfer coefficients is the amount of liquid which is carried over from the boiler end of the tube. T h e presence of such liquid in the condenser end obviously has a negative influence on the condensation process. An inactive length between the two ends was found to be quite helpful in controlling the liquid carryover. As indicated previously, carryover decreases as the temperature increases, and hence density decreases.

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T h e optimum filling depth gave much higher coefficients than predicted by the simple laminar flow Nusselt condensation theory. T h e selection of the optimum filling depth was very important for obtaining large condensation coefficients. Finally, tests were conducted by flowing air through a tube of water to model the two-phase pool behavior. It was found that the wetted length to static depth ratio did not vary indefinitely as the air flow was increased, so perhaps a constant quantity of working fluid can have a broad performance range where strong carryover to the condenser can be avoided. These tests showed no significant dependence on tube inclination up to 60" from the vertical suggesting that inclined thermosyphons with large fillings will probably behave much as vertical ones. T h e latter observation was also confirmed in the transparent thermosyphon. Before finishing this section mention should be made of a major application of this type of thermosyphon. For many years bakers' ovens have been heated by steam tubes which are simply inclined, two-phase thermosyphons with large fillings. They were studied by Moorhouse (94) due to a number of failures causing property and human damage. He reported various thermal and metallurgical observations and perhaps most significantly noted that the steam tube in many cases was probably operating under a condition of vapor dryout due to a nearly horizontal positioning of the short boiler end of the tube. A bent tube with a more nearly vertical boiler section was suggested. His tests were conducted with a tube filled with a nearly critical volume of water so as to assure two-phase operation as close to the critical temperature as possible. In applied practice, however, he found numerous deviations from this principle.

D. CRITICAL STATEOPERATION Obviously the preceding parts of this section were concerned with subcritical operation in that they involved a change of phase, which was seen to be very advantageous from a heat transfer standpoint. Another advantageous thermodynamic state at which to operate is the critical point, or at least in the vicinity thereof. At the critical point, the dynamic viscosity p takes on a minimum value and the coefficient of volumetric expansion /3 as well as the specific heat C , take on large local maximum values (theoretically approaching infinity), thus creating ideal conditions for extremely high rates of convective heat transfer. A thermosyphon operating at the critical point can have an apparent thermal conductivity up to lo4 times that of an equivalent copper bar; but it is restricted to a small temperature range near the critical temperature for

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effective operation, and it is difficult to maintain the system stably at this point. Schmidt ((108), Schmidt et al. (106))utilized these advantages in his turbine cooling program reported on in other work (107, 109) as well. Subsequent work by Hahne (45) investigated this effect further using different fluids, various phases, and variable inclination. Unfortunately, the disadvantages associated with critical state thermosyphon operation have apparently outweighed the advantages and little new has appeared in recent years, although a large literature has developed concerning critical state processes in general.

VI. Turbine Applications A. THERMOSYPHONS FOR TURBINE COOLING For more than 30 years the possible use of thermosyphons (54, 107) for cooling turbine blades has been recognized, and at least five different attempts (14,36,37,39-41, 107) have been made to construct turbines cooled by this concept and there is evidence of growing interest by engine manufacturers today (44, 112). Several strong advantages can be cited for the use of thermosyphons including: (1) Attainment of high convective heat transfer coefficients without the necessity of additional pumping equipment; (2) a broad selection of coolants (for closed systems) and; (3) no aerodynamic penalties for dumping air into the gas path such as occurs for film and transpiration cooling (65). T h e primary disadvantages appear to be: (1) possible large thermal stresses due to the large temperature differences necessary to conduct large fluxes across a blade wall, characteristic of the next-generation, maximum-temperature blades; (2) the necessity to transfer the heat from the thermosyphon passages to a secondary heat exchanger, generally with less area than the blade itself; and (3) limited research background to support meaningful applications. On this basis, a strong case can be built for thermosyphon blade cooling. T h e most probable limitation on future applications will likely be the stress question (items 2 and 3 should be readily resolvable), but there is definitely a lack of sufficient evidence to ascertain if this really is or is not a major problem. I n any event, it must be weighed against the large gains to be had by not dumping large quantities of air directly into

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the gas path. At worst, it might be necessary to accept a somewhat shorter blade-life in order to get a high-performance blade for hightemperature operation (say above 2800"F), but such problems should not be encountered for lower-temperature blades. Clearly, further studies to place these qualitative concepts on an acceptable quantitative basis would be highly desireable. T h e preceding sections have shown that quite a variety of types of systems are available, but the stringent demands on a good blade cooling scheme quickly narrow the field. T h e desirable, in fact, necessary attributes include: maximum possible heat transfer (this is not to imply that overcooling is recommended but rather that the system should function as efficiently as possible), minimum weight, dependable operation, and, as a result of the geometry used and temperatures obtained, an acceptable stress distribution and an efficient secondary heat exchange process. Furthermore, additional constraints are often imposed by particular manufacturers in order to ensure system or philosophy compatability. It is true, on the one hand, that one could conceivably find turbine applications for almost any one of the thermosyphons discussed, but, on the other hand, the various industrial requirements and the higher turbine inlet temperatures for aircraft gas turbines of the near future greatly limit the fields to which the preceding schemes can be applied. A discussion of the thermal problems which occur with today's turbine blades has recently been given (65). As will soon be seen, the open thermosyphon (see Table VI for a survey of turbine tests) apparently has nearly run its course and is no longer seriously considered for quite a variety of reasons including system vibrations, containment, limited heat transfer, etc. T h e simple closed system, using liquid metals, has been shown to have adequate heat transfer capability for many lower temperature applications (see Table VII for summary of rotating tests). T h e closed system would probably use a liquid metal if a liquid were selected, or very high pressures if a gas were employed. T h e former might have some containment difficulties and bad stresses due to the large mass in a strong force field; the latter would require excessive pressures and hence stresses. T h e use of metals solely in their liquid state is strongly resisted by aircraft manufactures due to weight considerations, but might find future marine applications. Both systems would be inadequate for trailing-edge cooling. One possible application for the open thermosyphon might still be worth considering. Gas turbines are receiving increasing use for emergency and peak-load electric power generation. Current development efforts are aimed at using the hot turbine exhaust for steam production and subsequent additional power production.

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Actually, this is quite similar to early suggestions for the use of the steam produced in Schmidt’s (107)blade cooling scheme (or Friedrich’s (37) more recent version). It would thus be very interesting to have a cycle study conducted to see if one can profitably remove large quantities of heat in the turbine via steam production (perhaps beyond what is necessary merely for blade cooling) for subsequent electrical power production, as opposed to using only the exhaust gases-or a combination of the two. T h e closed-loop thermosyphon and the two-phase thermosyphon apparently have very good future prospects. T h e closed-loop system can yield very high heat transfer due to the unidirectional flow or, in other words, the absence of core-boundary layer interactions and orifice or midtube exchange processes common to the simple open and closed thermosyphon. T h e high heat transfer rates in the two-phase thermosyphon of course follow directly from the high latent heats of evaporation and condensation. Naturally, certain problems exist with each of these methods; these will be brought out in the following discussion and historical review of thermosyphon turbine applications.

B. REVIEWOF THERMOSYPHON COOLEDTURBINES T h e first attempt at thermosyphon blade cooling was Schmidt’s (207,109) turbine cooled by open thermosyphons using water near the thermodynamic critical point with the heat being removed by evaporation at the blade root (see Table VI). Although Robinson (101) had difficulty repeating the tests presumably due to worn or damaged equipment, Schmidt’s tests (107)and later those of Friedrich (37) demonstrated that the system was quite adequate from a heat transfer standpoint but subject to large vibrations apparently caused by the loose fluid at the blade root. Freche and Diaguila (36) successfully avoided vibrations by using a single-phase (liquid) root requiring forced water flow past the thermosyphon passages. Unfortunately, their system had water leakage and, as in the other cases, also required additional pumping equipment. Consequently, the emphasis among turbine workers was changed entirely to closed systems. T h e only known turbine actually built using the simple closed thermosyphon is the Pametrada project (3, 13, 14, 21,22, 98) which used NaK eutectic with forced convection water for root cooling. Such root cooling is perhaps acceptable for marine applications for which the project was intended, but probably should be replaced by air cooling for aero applications. T h e project apparently was successful from the standpoint of rotor cooling, and the liquid metal leakage reported for the first

TABLE VI

THERMOSYPHON COOLEDTURBINES Worker, organization, and reference

Date and country

Fluid

Heat removal

TIT" (max.)

Estimated average blade temperature

hours run

Remarks

OPEN THERMOSYPHON (Schmidt's System) Schmidt (107)

194C45 Germany

Water

Evaporation 2190°F (1200°C)

Robinson 1950 NGTE (101) England

Water

Evaporation 1200°F (650°C)

Freche and Diaguila NACA (36)

Water

Bulk water flow past

1950 U.S.A.

1740°F (950°C)

1120-1470°F (600-800°C)

-

350°F

-

60

Captured German turbine, required considerable repair; vibration permitted attaining only 61 yo of design speed (19,000 rpm design); corrosion breakthrough and considerable oxide blockage. Apparently successor to above.

-

Limited circulation in T E holes due to inadequate sizing with resulting higher temperatures; coolant leakage; no vibration; pumping power = 2 yo turbine power at design conditions.

(176°C)

T. S. opening

Friedrich (37)

1962 Germany

Water

Evaporation 1920°F (1050°C)

Pioneer thermosyphon work; water used near critical point; blade passage pressures to 300 atm; tests limited by war.

1215

Strong vibrations due to flow instability in the ring of water at the blade root.

U '

* 9

zx

cn

m

Ivanov et al. (55c)

1966 U.S.S.R.

Water

Bulk water flow past T. S. opening

750-1000°C

191-269°C

-

Very similar to turbine of Freche and Diaguila (36). See Appendix.

CLOSED THERMOSYPHON Brown ( 1 4 ) Pametrada see also: Colclough et al. (22); Ogale (98)

1957 Britain

g

NaK Water 2200°F Eutectic forced (1200°C) convection

700-810°F (370-430°C)

400 8 at max. temp.

Ref. (98) reports these run times and actual usage of the system in a turbine. Ref. (98) gives rotating test data and design conditions liquid metal leakage overcome in phase 11.

9

5 !?

2

2

M

w

z

CLOSED LOOP THERMOSYPHON Gabel et al. Continental Aviation

(39-41)

a

1968 U.S.A.

5

Water Fuel forced 2450°F vapor convection max. at 8OOOpsi (1340°C) (liquid)

T I T ; turbine inlet temporature-the

1500°F max. 790°C

21 Some blade failures due to poor tip cap above welds, casting errors (walls too thin), 2100"F, and fuel seal leaks; no catastrophic 155 failures and remedies appear feasible. total T i p cap failures did not cause shut down.

gas temperature from the combustion chamber into the turbine.

;

2 8 3r 0

0 4

TABLE VII: ROTATING THERMOSYPHON EXPERIMENTS W

Q\

Worker and/or organization, and reference

Type of study

Principle heat transfer results found

Applicability to turbine cooling

Brown, Pametrada Project (14) England

NaK, closed thermosyphon H,O as secondary coolant

Nu a (GrPr2)2/13 constant not given LID and Lh/Lcalso significant. Effect of Coriolis force noticeable but not significant.

Results were sufficiently encouraging to build an entire rotor based on this study (see Table VI). Further information becoming available in references (21, 22, 98).

Ogale, University of Delft (98) Holland

NaK, closed, singlephase thermosyphon H,O as secondary coolant, 5004500 rpm

Used a variable area closed thermosyphon (so-called semiclosed) but effect of area change not studied. for circular section: Nu = 0.3325 (Gr Pr2)13/100 for blade-shaped section: Nu = 0.6038 (Gr Pr2)9/100

Worker was sufficiently satisfied to design a turbine to operate at:

Cohen and Bayley (20); Bayley and Bell (5) England

Two-phase closed thermosyphon Evaporation and condensation of H,O and Hg, 500-2000 rpm

T,,, = 1200°C T~~~~~~~ = 7 0 0 ~ Apparently not yet built.

* zx

Demonstrated large heat fluxes obtainable with this system and showed effects of filling quantity and rotational speed on heat transfer (see Section V).

Investigators concluded the system was attractive for blade cooling and other workers have pursued this approach.

IeGrivhs and GCnot One- and two-phase L’ONERA and closed thermosyphon using NaK SNECMA (42,44) France Air as secondary coolant Speeds up to 6000 rpm

Studied heat transfer and temperature distributions subject to various rotational speeds.

Demonstrated the adequacy of either system, particularly two-phase operation, to turbine cooling.

Usltov and Tseitlin, Central BoilerTurbine Institute (1126) U.S.S.R.

Suspected conduction plays significant role in heat transfer. Noted influence of gravity and centripetal acceleration.

Low rpm, no turbine conclusions reached (see Appendix).

Na, closed thermosyphon, 0-710 rpm

P cl

8

ADVANCES I N THERMOSYPHON TECHNOLOGY

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turbine build was evidently overcome in the second build. Rotating rig studies (24,44,98)have also demonstrated the adequacy of this technique and have been reviewed in previous sections, but it should be noted that thick (or air-cooled) trailing edges are often necessary with the simple closed thermosyphon. One good possibility for blade cooling is the use of evaporating and condensing thermosyphons which have received some attention (5, 20,42,44, I12), though it has not, as yet, been tested in an actual turbine. Although very high heat transfer rates are obtainable, it appears that large stresses with circular passages (67),possible lack of uniformity in cooling due primarily to Coriolis forces on the film (20), and the necessity for thick, or uncooled, trailing edges (20,42, 98), may hamper the development of this version. One recent rotating rig test ( 4 4 , see Table VII, has demonstrated successful two-phase operation up to 6000 rpm using a thermosyphon with a rectangular cross section, as reviewed earlier. Also, the high heat flux at the leading edge, which is imposed on a very narrow passage, may yield a very hot leading edge and possibly cause film boiling, as suggested by Ainley ( I ) . One such study seems to have illustrated this problem (67, 112). This study was first reported by Sucio (112) of General Electric Corporation. T h e blade which he illustrated was the first thermosyphon (two-phase) blade which had an air-cooled root and an air-cooled trailing edge, thus circumventing the thick trailing-edge problem. A very promising system for blade cooling is the closed loop thermosyphon which Gabel et al. (39-41) have successfully employed. One of the strongest attributes of this system was shown by Eckert and Jackson in 1950 (29) when they suggested unidirectional flowing channels joined in parallel to a primary bidirectional open thermosyphon channel for cooling thin trailing edges (although bidirectional channels are not necessary). Gabel and Tabbey also realized a weight savings by employing for the first time a vapor, or gas, thermosyphon which, operating at about 1000 psi, gave quite adequate heat transfer capability. Unfortunately, they used fuel cooling of the root which required additional pumping as opposed to an air-cooled root. Also, many manufacturers find fuel cooling undesireable for aircraft applications, though this method may be more attractive for land usage, since additional safety equipment, if necessary, can be added more easily. Fuel cooling for thermosyphons is reviewed by Edwards (31). One additional static study (72) of this system has shown the large heat transfer capabilities which can be obtained in a closed-loop thermosyphon especially with liquid metals. Other loop systems which are of interest are a variety of schemes suggested by Jackson and Livingood (57) and their colleagues at NACA

98

D.

JAPIKSE

in 1951. These systems generally involved the entire rotor radius to develop large pressure differences and apparently did not receive further attention. Finally a particular blade cooling application with some possibility for the future may arise in the cooling of supersonic turbines for advanced lift fans (see Yaffee (116)). I n this case the turbine blades are mounted directly on the perimeter of the fan blades. Heat picked up from cooling the turbine blades can probably be rejected to the fan air via the fan blades by means of various thermosyphon designs. Such an application is reminiscent of early British and American patent applications as reported in Ainley ( I ) and Davies and Morris (24). It might very well afford a much simpler and more efficient design.

VII. Future Research T h e need for considerable future research is readily apparent if one compares closely the table of contents of this review, which by the breadth or brevity of each section reveals how much attention has been given to it, and Table I, which shows what type of applications are merited. There is a clear discrepancy between current knowledge and future needs. T h e shortest major section, that pertaining to closed-loop or mixed-convection thermosyphons (single or two phase) accounts for more than half the entries in Table I. T h e next shortest, the two-phase section, again pertains to a very large portion of the entries. T h e industrial value of two-phase thermosyphons and various types of closed loop thermosyphons has been made evident in the previous sections: They constitute inexpensive, reliable means of transporting heat under a very wide range of conditions. A number of research projects along these lines are underway in Europe and North America; hopefully, both universities and industry will open channels of communication to guide future work to useful applications. This dialogue has often been missing in the past. Appendix : Current Contributions

I n recent months several additional references have appeared which contribute greatly to our understanding of practical thermosyphon applications and to a better theoretical understanding of certain types of thermosyphon systems.

ADVANCES IN THERMOSYPHON TECHNOLOGY

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A. OPENTHERMOSYPHONS Recently Gosman et al. (43a) have published the first finite-difference solution for the full laminar elliptic equations governing heat and momentum transfer in the open thermosyphon (as shown in Fig. 1). Th i s study merits careful consideration inasmuch as it is clear that the large body of thermosyphon data available makes this problem a logical choice for testing elliptical internal flow computational methods. T h e numerical procedures used have been discussed by these authors in a number of publications and they will not be reviewed here; rather attention will be focused on their prediction of the physical phenomena. Generally speaking the authors presented a very fine report on the detailed and careful computations which they performed. They report solutions over the following range of parameters: 2 x lo2 < tot < lo6,

1

< Pr < lo4,

5

< L/a < 45.

Their results for L / a = 45 and Pr = 100 are in excellent agreement with the data of Martin (86,87) for rapeseed oil and glycerin. They are slightly lower than Lighthill’s (80) predictions for boundary layer flow and about 20 yo higher than his predictions for similarity and nonsimilarity flow. They confirmed the validity of the boundary layer approximations over the range of L / a examined. They reconfirmed the fact that their is no significant effect of Prandtl number for values greater than 100. Upon comparing adiabatic and isothermal base results, they found no meaningful difference in overall heat transfer; but a small ring vortex was found for the isothermal base. There are, however, two results reported which require very close attention. First, the authors note that their solutions are of a continuous nature when plotted on a Nu vs. to, graph in contrast to the two curves obtained by Lighthill. His solutions are firstly for similarity and nonsimilarity flow and secondly for boundary layer flow. I n fact Lighthill’s solutions caused him to speculate that perhaps the flow would be unsteady in the region where his two curves come together but do not intersect (see Fig. 3). Experience bore out his expectation as shown by the studies of Martin (86, 87) and to a lesser extent Hasegawa et nl. (52).Consequently the continuous nature may be aesthetically pleasing but is not more valid than Lighthill’s results. Naturally this study would not be expected to reveal the unsteady nature of the flow; however, the authors should be careful of conclusions made in such a region, and hence their claim for a different variation of heat transfer (from that of Lighthill) with distance from the orifice in this regime is on unsure grounds. Secondly, their examination of property variations (which is a major

100

D.

JAPIKSE

asset to recommend such finite-difference studies) requires close scrutiny. Several cases were computed with both completely variable properties and with constant properties, the wall temperature being selected as the appropriate reference. T h e investigators concluded that their was no significant difference in the computed overall Nusselt numbers even though the viscosity varied by as much as 320% in one case. Variable property solutions yielded only slightly lower Nusselt numbers than the constant property solutions. It is interesting to note, however, that in each case the constant property solution is as good or better when compared to the actual data than the variable property solution, although the differences are nearly within the scatter band. It seems strange that the authors chose only these cases with rapeseed oil as a test for property variation effect when their conclusion was long held to be so anyways; perhaps a better choice would have been air in the nonsimilarity regime which appears to be strongly property dependent (see Section 1,b). A second aspect of their property variation study is the manner in which Nusselt number varies with Prandtl number. Their results for the similarity regime are qualitatively correct and quantitatively appear to be in better agreement with the data than Lighthill’s predictions. Their results in the so-called boundary layer regime are qualitatively and quantitatively strange, see Fig. 41. For a flow of this general type it is very strange to find the Nusselt number decreasing with increasing Prandtl number and in fact no experimental thermosyphon studies have

Rape seed oil x-

-t- _ _ _ _ _ _ _ _ _ _ _ _ _ - -

c _ _ _ _ _ _ _

x-

/-

Glycerin+

Ethylene glycol

0

Martin, exp.

+

Lla = 7.5

0 Lla = 15.0 + Lla = 47.5 Boundary loyer regime tot = lo5

4 Transitional

FIG.41. Comparison of Gosman’s et al. (43a) elliptic open thermosyphon analysis and Japikse’s (59) integral boundary layer analysis with Martin’s (86, 87) experimental results for the steady flow laminar open thermosyphon.

ADVANCES IN THEI~MOSYPHON TECHNOLOGY

101

reported such behavior to date. T h e included datalo illustrates this as does Fig. 9. T h e authors cite certain distinctions in elliptic vs. integral boundary layer velocity profiles to defend their conclusions. They note that the velocity profile was not found to have a constant core value as claimed to exist in integral boundary- layer studies (but their velocity near the wall was consistent with boundary-layer profiles). Rather, their profile was generally similar to those typical of similarity solutions. T h e temperature profiles did show a constant core region typical of the boundary layer assumption (and of course not typical of the similarity solutions). Since the difference in core velocity profile was found, the authors suggested that their N u versus Pr variation is plausible due to the qualitative similarity of their velocity profiles to the various similarity solutions which have the same Nu versus Pr variation. Of course, this argument requires that we ignore the discrepancy in temperature profiles in their argument; the argument is most specious. Finally, they claim that this type of velocity profile was found for to, as large lo6 and Pr as low as unity. They did not clearly state what range of L/a was used although hopefully it should not be important. In their conclusions (b) and (c), however, they seem to suggest that large Grashof numbers and short tubes were used leaving us unsure as to just how general their distinctive velocity profiles are. Further attention appears necessary for the variable property solutions. Ivanov et nl. (553, 55c) have conducted rotating turbine tests of an open thermosyphon configuration which is very similar to the work of Freche and Diaguila (36). They too experienced difficulty in computing the characteristics of the mixing chamber, or reservoir, at the open thermosyphon orifice due to elevated thermal and hydraulic resistances there. Consequently they employed a convection coeficient that is defined as the amount of liquid forced into the mixing chamber to the amount of liquid circulating in the blade passages to help correlate their heat transfer results.

B. CLOSEDTHERMOSYPHONS First, Schneider ( 2 0 9 ~ has ) outlined a new closed, liquid metal, thermosyphon cooled turbine blade in a recent patent. Perhaps the most novel aspect of the patent is that the inventor located his heat removal system completely inside the confines of the thermosyphon. That is, lo T h e data points from Martin for Pr N 8 are taken from his study with water for L / a = 7.5 and 15.0 where the effect of using different L/a data is of course insignificant. These cases appear to be for good steady flow experiments without any evidence of strange inlet effects or core-boundary layer interactions.

D.

102

JAPIKSE

he used a set of parallel tubes passing through the closed thermosyphon through which he passed compressor air to remove heat from the thermosyphon. T h e air was then discharged through an air-cooled blade trailing edge. His scheme provides for a compact, efficient heat removal system. A meaningful contribution to our understanding of the liquid metal vertical closed thermosyphon has been made by Pucci and Gerretsen ( 1 0 0 ~ ) .They have measured heat transfer rates and temperature distributions for two thermosyphons of L i d = 5.63 with diameters of 10.6 and 26.1 mm. They considered both the case of constant wall temperature and constant wall heat flux on the heated end. Their results using water were in reasonable agreement with those of Bayley and Lock ( 8 ) and Everaarts (32), although a systematic comparison based on L l d and d was not made. T h e constant wall temperature and constant heat flux data for water showed nearly equal Nusselt numbers. Using N a K and Hg, they presented data over the range 2.3 log(GrPr2) 6.2, including some data from Fransen (35a) at high GrPr2 which heretofore had not appeared in the open literature. It is not clear if both sets of data are for the same L / d . I n general, the data for constant wall heat flux fell distinctly below the constant wall temperature data. Thus the authors speculated that conduction effects must be significant inasmuch as the two cases have distinctly different temperature distributions. This writer has considered the accuracy of Eq. (27) for Nu, and the Hg data from Fig. 11 (which, of course, is approximate for the NaK case) against the present data of Pucci and Gerretsen. Rather good agreement (&25%) for Eq. (27) was obtained with the NaK data in the small diameter tube, rough agreement (- 10to -SOY0) for NaK in the large diameter tube and H g in the small diameter tube, and poor agreement (low by more than 1 0 0 ~ ofor ) Hg in the large diameter tube. Thus we can see that further research is necessary before we will have realistic prediction tools for closed, liquid metal thermosyphons, either vertical static, as shown here, or rotating (see Table V). Indeed, a deliberate study of the open system to systematically obtain data such as was pieced together to yield Fig. 11 would be desirable. At the very least, future investigators hopefully will study their closed systems both closed and open with a reservoir in order to make possible predictions over a wide range of geometries. Other studies of the liquid metal closed thermosyphon have been reported in the Soviet literature, although apparently not all of them are yet available. Evidently Romanov ( I O l a ) has executed a study similar in nature to that of Bayley and Lock (8) as early as 1956 (see

<

<

ADVANCES IN THERMOSYPHON TECHNOLOGY

103

(112a) and (112b) for reference to this work). Likewise Captieu and Musse (15a) have reported theoretical and experimental studies of this . another study (Ivanov (55a)), a thermosystem (see also ( 2 1 2 ~ ) )In dynamic study of various operating fluids is reported. I n other Soviet works, a variety of rotating closed thermosyphon studies are reported. Belichenko et al. (9a) are reported (see (112b)) to have measured the overall effects of heat transfer in rotating, liquid sodium, closed thermosyphons. A similar study was reported by . and Tseitlin (112b) have Slobodyanyuk and Omelyuk ( 1 1 0 ~ ) Uskov reported results for rotating, sodium filled, closed thermosyphons for speeds from 0 to 710 rpm. Their wall temperature distributions were similar to those of leGriv&sand GCnot (44) (see Section 111, C) in that they became very flat as the rotational speed increased. T h e authors suggested that molecular conduction may well be important in this case. They correlated their results in the form

where G r is based on g, not w2R, and hence both the earth’s acceleration and centripetal acceleration were included. Surprisingly, w 2 R was used rather than a Grashof number based on w2R. Good correlation was claimed for ~ ~ R 1 9 . 8u 1p to 700 and 3 Ltotal’d 20. Finally Tkachenko (122a) has reported another study of flow visualization in the rotating closed thermosyphon. Tellurium dye techniques were employed. Results are given for a channel, presumably of rectan= 4,L = 72 mm, R = 50 cm gular cross section, for whichLtotnl’desuiv, that is rotating at w = 200 and 360 rpm. He reported that the flows are not symmetrical between the cold and hot ends. He stated that, at 200 rpm, the flow rose to the heated end through the core and then down along the two sides. T h e forward side was stabilized by the Coriolis force, whereas the trailing side was destabilized by it. Intensive eddy generation was observed at the top end and to a lesser extent at the bottom end. A single circulating eddy was observed in the cold end. At higher rotational speed, w 3 360 rpm, extensive eddy motion was observed throughout the thermosyphon. There are clearly many points of similarity between all the qualitative trends listed here and results found by Japikse et al. (59, 64) in an inclined closed thermosyphon. A careful comparison cannot be made, though, due to a lack of technical data from which to, for each section and t,, can be computed for Tkachenko’s cases. Lacking this data, it is not even possible to estimate what modes of flow should be expected in

<

<

104

D.

JAPIKSE

each tube section. For example, the lack of end-to-end symmetry might not be at all surprising as found in other cases by Japikse (62). This study (222a) evidently is a part of the study reported by ZysinaMolodjen et al. (227). Some further details are given there, particularly that the heat transfer rates are believed to be governed by Nu

K

(GrPr)0.5

(37)

in the rotating system. This large exponent stands in sharp contrast to many values reported in Section 111, C of about 0.2. Perhaps this issue will be clarified in a forthcoming article by Zysina-Molodjen and Tkachenko ( 2 2 6 ~ ) .

C. CLOSEDLOOPTHERMOSYPHONS A recent comprehensive evaluation of thermosyphon turbine blade cooling has been reported by Stappenbeck and Moskowitz (220b). They considered several closed loop (multichannel) thermosyphon systems which used liquid metals, high-pressure air and high-pressure steam as coolant fluids. Both air and fuel were considered for heat removal from the thermosyphon. Most of the systems employed a rotating heat exchanger fixed to the turbine shaft with the thermosyphon coolant fluid piped between the blade and the heat exchanger to form the complete loop. They evaluated the relative merits of these systems and compared them to the merits of both ordinary and modulated transpiration cooled blades. T h e comparison was based on fabrication, performance, and durability of the entire system. In all cases the thermosyphon systems out ranked the transpiration schemes. However, the heat exchanger design is complex and raises the absolute level of complexity and possible maintenance problems regardless of relative merits. Hence the sponsoring organization decided against any further development programs. This specific program was intended for small gas turbines. It appears that the primary deficiency of this study was insufficient consideration of the problem of removing heat from the thermosyphon systems. Perhaps the previously mentioned root or internal heat exchangers plus trailing edge air discharge would obviate the need for the large rotating heat exchangers and render the systems practical. I n any event, other types of gas turbines, such as large marine turbines, might provide a practical place to apply some of these ideas. Closing in a different vein, the study reported by Walters (123a) raises the prospect that our understanding of closed loop thermosyphons may soon be put to use for designing geothermal power producing systems. T h e loop may be as much as 15,000 ft deep with a driving

ADVANCES IN TI~ERMOSYPHON TECHNOLOGY

105

temperature difference of 600°F. Under steady operation, it would simply function as a closed loop thermosyphon and extract thermal energy from deep in the Earth's crust and bring it to a surface power plant.

ACKNOWLEDGMENTS T h e author is particularly grateful to his graduate committee and to Professor E. R. F. Winter, his major professor at Purdue University, for their guidance during the early years of the author's work on this subject and specifically for their suggestion and encouragement to prepare this review. Special thanks is also given to Professors B. W. Martin, F. J. Bayley, G. S. H. Lock, and J. Schenk and Dr. B. Larkin and Mr. R. Deryl Miller who provided original data and/or photographs for use in preparing this article. Many other workers are also to be thanked for their correspondence and suggestions relevant to their own topics. Finally, my greatest appreciation is to my wife, Ellen, who has assisted in typing and translating and who has been most understanding.

NOMENCLATURE tube radius, inside ordinate intercept value of certain linear correlations property groupings specific heat (at constant pressure) tube diameter, inside gravitational acceleration convective heat transfer coefficient thermal conductivity tube length of the open thermosyphon or half-length of the closed thermosyphon latent heat of evaporation mass flow rate, non-dimensional =

V

V X

X Y a

B

A?lC,/kL mass flow rate, dimensional slope values of certain linear correlations pressure heat flux heat flow radius, nondimensional = Ria radius, dimensional gas constant temperature, dimensional, "C temperature difference velocity in x direction, nondimensional = u2U/mL velocity in x direction, dimensional

8

E

0 h P Y

5 P 0 7

X w

velocity in r direction, nondimensional = u V / a velocity in r direction, dimensional axial coordinate, nondimensional measured from bottom = X / L axial coordinate, dimensional measured from bottom distance from the wall, dimensicnal thermal diffusivity = k / p C , coefficient of thermal expansion or the (complementary) thermal boundary layer thickness from tube centerline; depending on specific application momentum boundary layer thickness from tube centerline (complementary thickness) contact angle angle of inclination parameter for turbulent similarity viscosity, dynamic viscosity, kinematic convection effectiveness parameter density surface tension centerline temperature defect mixing temperature parameter angular velocity

D.

JAPIKSE

NONDIMENSIONAL GROUPS Gr Pr Ra Nus

NU^ t tot tc t

C

Grashof number = bgd Ta3/v2 Prandtl number = pC,/k Rayleigh number = Gr Pr Nusselt number (used for open sys-~T , ) ~ L tem) = ha/k = Q / 2 5 7 ( ~ Nusselt number (used for closed system) = hd/k = Q/.(TI.l - T,,,)kL Pga4(T1 - T)/vaL - To)/vaL BgdYTi - T A / v d

f d e ct max mc min Ot S

1 0 1,2

SUBSCRIPTS a b

based on radius condition at the base

L V

centerline condition film droplet or diameter see definition of Le closed thermosyphon maximum mixing cup (temperature) minimum open thermosyphon saturation condition at wall condition at orifice on the centerline denotes bottom or top half, respectively, of closed thermosyphon; when paired subscripts are used the second number always denotes the tube half liquid vapor

REFERENCES 1. D . G. Ainley, Proc. Inst. Mech. Eng. (London) 169 (53), 1075 (1955). [Discussion of reference 20.1 2. J. F. Alcock, Proc. Znst. Mech. Eng. A . S . M . E . (London) Conf. Proc. 378 (1951). [Discussion of reference 1071. 3. T. A. Andvig, Rotating Rig Experiments on the Effectiveness of the Closed Thermosyphon System for the Cooling of Turbine Rotor-Blades. Pametrada Contract Rep. No. C108 (1956). 4. F. J. Bayley, Proc. Inst. Mech. Eng. (London) 169 (20), 361-370 (1955). 5. F. J. Bayley and N. Bell, Engineering (London) 183, (47/48), 3 W 3 0 2 (1957). 6. F. J. Bayley, P. A. Milne, and D. E. Stoddart, Proc. R o y . S o t . , Ser. A 265, 97 (1961). 7. F. J. Bayley and J. Czekanski, J. Mech. Eng. Sci. 5(4), 295-302 (1963). 8. F. J. Bayley and G. S. H. Lock, J. Heat Transfer 87, 3&40 (1965). 9. F. J. Bayley and B. W. Martin, Proc. Znst. Mech. Eng. 185(18), 219-227 (1971). 9a. N. P. Belichenko, K. M. Pepov, and A. G. Romanov, Tr. Tsent. 51 (1964). 10. L. Bewilogua and R. Knoner, Cryogenics 2, 46-47 (1961). 11. L. Bewilogua, R. Knoner, and G. Kappler, Cryogenics 34-35, Feb. (1966). 12. R. C. Biggs and J. W. Stachiewicz, Proc. Int. Heat Transfer Conf., 4th, ParisVersailles Paper NC3.1 (1970). 13. T. W. F. Brown, Proc. Inst. Mech. Eng. (London) 169(153), 1075 (1955). [Discussion of reference 20.1 14. T. W. F. Brown, C I M A C Congr. Zurich pp. 443-484 (1957). 15. A. H. 0. Brown, Sodium and NaK. The Natural Convection Cycle with Reference to Turbine Blade Cooling. Brit. Siddeley Engines Ltd., Ref. G N 4257 (1961). 15a. M. Captieu and S. Musse, Theoretical and Experimental Study of Closed Thermosyphons with Liquid Metals, Heat and Mass Transfer, 1, Energiya Press, Moscow, 1968.

ADVANCES IN THERMOSYPHON TECHNOLOGY 107 J, C. Chato, J. Heat Transfer 85(4), 339-345 (1963). J. C. Chato and W. T. Lawrence, Develop. Heat Transfer pp. 371-388 (1964). J. C. Chato, J. Eng. Power 87, 355-360 (1965). P. T. Chu and F. G. Hammitt, A S M E Winter Ann. Meet., New York Paper 64 WA/HT 51 (1964). 20. H. Cohen and F. J. Bayley, Proc. Inst. Mech. Eng. (London) 169(53), 1063 (1955). 21. C. D. Colclough, Measurement of Rotor-Blade Temperature in a Liquid-Cooled Gas Turbine. B.S.R.A. Rep. N.S. 2, Mar. Eng. Rep No. 2 (1963). 22. C. Colclough, P. Barden, and R. Darling, Tests on the Single-Stage Liquid-cooled High-Temperature Gas Turbine with the Mark I1 Rotor. B.S.R.A. Res.Rep N.S.4 (1963) 23. J. G. Collier, B. E. Boyce, A. S. deForge Dedman, and R. Khanna, Natural Convection through Narrow Vertical Unheated Annuli at High Gas Pressures. Proc. Int. Heat Transfer Conf., 4th, Paris- VersaillesPaper NC 2.6 (1970). 24. T. H. Davies and W. D. Morris, Innd. Eng. Dig. 26(11/12), 87-91 (1965). 25. T. H . Davies and W. D. Morris, Proc. Int. Heat Transfer Conf., 3rd, Chicago 2, 172-181 (1966). 26. G. W. Dieperink and C. den Ouden, Stroming En Temperatuurverdeling in Een Gesloten Thermosyphon. Intern. Rep., Laboratorium voor Technische Natuurkunde, Lorentzweg, Delft, Netherlands (1968). 27. B. A. Dmitriev, I m . Akad. Nauk S S S R , Otd. Tekh. Nauk 5, 103-107 (1956). [D.S.I.R. Transl. Conts. L. Russ. Period., Sept. 1956.90.64.1 28. I. G. Donaldson, Aust. J. Phys. 14(4), 529-539 (1961). 29. E. R. G. Eckert and T. Jackson, Analytical Investigation of Flow and Heat Transfer in Coolant Passages of Free-Convection Liquid Cooled Turbines. Nat. Adv. Comm. Aeronaut. NACA 50d25 (1950). 30. E. Eckert and T. W. Jackson, Analysis of Turbulent Free-Convection Boundary Layer on Flat Plate. Nat. Adv. Comm. Aeronaut. NACA T C 2207 (1950). 31. J. P. Edwards, Liquid and Vapour Cooling Systems for Gas Turbines. A.R.C. C.P. No. 1127 (1969). 32. D. H. Everaarts, Free Convection Heat Transfer with Mercury and Water in a Closed Thermosyphon. Technische Hogeschool Delft, Report 10.M.006, Delft (1 967). 33. H. Ellerbrock, Proc. Heat Transfer Conf. London Sect. V (1951). 34. C. V. Foster, Heat Transfer by Free Convection to Fluids Contained in Vertical Tubes. Ph.D. Thesis, Univ. of Delaware, Newark, Delaware, 1953. 35. R. T. Foyle, Proc. Inst. Mech. Eng. A.S.M.E. (London) Conf. Proc. 382 (1951). [Discussion of reference 107.1 35a. J. W. M. Fransen, Metingen aan de Met Kwik Gevulde Thermosyphon Meetopstelling, Technische Hogeschool Delft, Rept. 10.H.060, Delft (1969). 36. J. C. Freche and A. J. Diaguila, Heat Transfer and Operating Characteristics of Aluminum Forced-Convection and Stainless-Steel Natural-Convection WaterCooled Single-Stage Turbines. Nat Adv. Comm. Aeronaut. NACA R M E5OdO 3a (1950). 37. R. Friedrich, Brennst.- Waerme-Kraft 14, 368-373 (1962). 38. P. Fries, Int. J. Heat Mass Transfer 13, 1503-1504 (1970). 39. R. M. Gabel, S A E Int. Automot. Eng. Congr., Detroit No. 690034 (1969). 40. R. M. Gabel and J. Tabbey, Advancement of High Temperature Turbine TechCooled Axial Flow Turbine. nology for Small Gas Turbine Engines-Fluid USAAVLABS Tech. Rep. 68-65, AD 686312 (1969). 16. 17. 18. 19.

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41. R. M. Gabel, S. L. Moskowitz, and T. E. Schober, S A E (SOC. Automot. Eng.) 77(5), 65-72 (1969). 42. J. GCnot and E. leGrivks, Proc. Znt. Heat Transfer Conf., 4th, Paris-Versailles Paper H E 2.2 (1970). 43. W. E. Gifford, Advan. Cryog. Eng. 7, 551 (1962). 43a. A. D. Gosman, F. C. Lockwood, and D. G. Tatchell, Znt. J. Heat Transfer 14, 10 (1971). 44. E. 1eGrivks and J. Gknot, Refroidissement des Aubes de Turbines par Mbtaux Liquides. Off. Nat. Etud. Rech. Aerosp. T.P. Na 872 (1970). 45. E. W. P. Hahne, Znt. J. Heat Mass Transfer 8, 481-497 (1965). 46. H. Hahnemann, Stationary Rig Experiments on the Heat Extracting Power of Closed Thermosyphon Cooling Holes. Min. Supply (Gt. Brit.), Nat. Gas Turbine Estab., Memorandum m. 128 (1951). 47. D. C. Hamilton, F. E. Lynch, and L. D. Palmer, T h e Nature of the Flow of Ordinary Fluids in a Thermal Convection Harp. Oak Ridge Nat. Lab. ORNL 1624 (1954). 48. F. G. Hammitt, Heat and Mass Transfer in Closed, Vertical, Cylindrical Vessels with Internal Heat Sources for Homogeneous Nuclear Reactors. Ph.D. Thesis, Univ. of Michigan, Sch. of Nucl. Eng., Ann Arbor, Michigan, 1957. 49. J. P. Hartnett and W. E. Welsh, Trans. A S M E 79(7), 1551-1557 (1957). 50. J. P. Hartnett, W. E. Welsh, and F. W. Larsen, Chem. Eng. Progr., Symp. Ser. 55(23), 85-91 (1958). 51. S. Hasegawa, K. Yamagata, and K. Nishikawa, Nippon Kikai Gakkai Rombunshu 28, N o . 191, 930-960 (1962). Mech. Eng.) 52. S. Hasegawa, K. Nishikawa, and K. Yamagata, Bull. J S M E ( l a p . SOC. 6, No. 22, 93Ck960 (1963). 53. S. K. Hellman, G. Habetler, and H. Babrov, Use of Numerical Analysis in Transient Solution of Two-Dimensional Heat Transfer Problems with Natural and Forced Convection. Amer. SOC.Mech. Eng. ASME Paper No. 54-SA-53 (1954). [Also Trans. A S M E 78, 1155-1161 (1956).] 54. H. Holzwarth, Die Entwicklung der Holzwarth-Gasturbine. Holzwarth-Gasturbinen, Gmbh Muellheim-Ruhr (1938). 55. J. F. Humphreys, H. Barrow, and W. D. Morris, J. Heat Transfer 92, 58G586 ( 1970). 55a. V. L. Ivanov, Ispol’zovanie vody kak promteplonositelya v avtonomnoi sisteme okhlazhdeniya rabochei lopatki gazovoi turbiny, Izvestiya Vysshikh Uchebnykh Zavedenii, Mashinostroenie no. 3, 77-80 (1967). 55b. V. L. Ivanov and Yu. D. Lapin, Teploobmen v usloviyakh svobodnoi konvektsii na uchastki kanala s mestnym soprotivleniem, Izvestiya Vysshikh Uchebnykh Zavedenii, Mashinostroenie no. 10, 70-72 (1966). 55c. V. L. Ivanov, E. A. Manushin, and Yu. D. Lapin, Zzv. Vuz. Aviats. Tekhn. 9(2), 143-150 (1966) [Sou. Aeron. 87-91 (1966)l. 56. C . B. Jackson, ed., “Liquid Metal Handbook-Sodium (NaK), Supplement.” Atomic Energy Commission and Dept. of Navy, Washington, D.C. (1955). 57. T. W. Jackson and J. N. B. Livingood, Analytical Investigation of Two Liquid Cooling Systems for Turbine Blades. Nat. Adv. Comm. Aeronaut. NACA RM E51FO (1951). 58. P. A. Jallouk, Experimental Investigation of Heat Transfer in a Closed Thermosyphon. M.S. Thesis, Purdue Univ., Lafayette, Indiana, 1969. 59. D. Japikse, Heat Transfer in Open and Closed Thermosyphons. Ph.D. Thesis, Purdue Univ., Lafayette, Indiana, 1969.

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109

60. D. Japikse and E. R. F. Winter, Proc. Int. Heat Transfer Conf., 4th, Paris-Versailles Paper N C 2.9 (1970). 61. D. Japikse, Proc. Znt. Heat Transfer Conf., 4t/z, Paris-Versailles 10, 163-164 (1970). 62. D. Japikse, Proc. Int. Heat Transfer Conf., 4th, Paris-Versailles 10, 146-147 (1970). 63. D. Japikse and E. R. F. Winter, Int. I. Heat Mass Transfer 14(4), 427-441 (1971). 64. D. Japikse, P. A. Jallouk, and E. 11. F. Winter, Int. J. Heat Mass Transfer 14(7), 869-887 (1971). 65. D. Japikse, Proc. Znst. Mech. Eng. (London) 185(18), 228 (1971). [Discussion of reference 9.1 66. G. H. Johnston, Permafrost and Foundations. Nat. Res. Counc., Can. Bld. Dig. CBD 64 (1965). 67. A. Kaufman, Analytical Study of Cooled Turbine Blades Considering Combined Steady-State and Transient Conditions. Nat. Aeronaut. Space Admin. NASA T M X 1951 (1970). 68. G. A. Kemeny and E. V. Somers, J . Heat Transfer 84, 339-346 (1962). 69. R. Knoner, Cryog. Eng. Conf., Tokyo and Kyoto, 244-245, April (1967). 70. J. T. Kulhavy, Winter Meet. Amer. SOC. Agr. Eng., New Orleans Paper No. 64-637 (1 964). 71. J. J. Kunes, Power App. Syst. 39, 973-977 (1958). 72. Y. D. Lapin, Therm. Eng. ( U S S R ) 16(9), 94-97 (1969). 73. B. S. Larkin, Heat Transfer in a Two-Phase Thermosyphon Tube. Nat. Res. Counc., Can., Div. Mech. Eng., Quart. Bull. Div. Mech. Eng. Nat. Aeronaut. Estab., Ottawa, No. 3, pp. 45-53 (1967). 74. B. S. Larkin, 84th Ann. Mtg. Eng. Inst. Canada, Ottawa, Paper No. 70-CSME-6, EIC-70-1046, Sept., 1970. Also, to be published in Tram. CSME. 75. F. W . Larsen and J. P. Hartnett, 1. Heat Transfer 83, 87-93 (1961). 76. Y. Lee, Preservation of Permafrost by Means of Two-Phased Closed Thermosyphon. Univ. of Ottawa, Ann. Rep. D R B 951 1-96 (1969). 77. Y. Lee and U. Mital, Proc. Znt. Symp. Two-Phase Syst., Technion City, Haifa (1971). 78. F. M. Leslie and B. W. Martin, /. Mech. Eng. 1(2), 184-193 (1959). 79. F. M. Leslie, J . Fluid Mech. 7, 115-127 (1959). 80. M. J. Lighthill, Quart. J. Mech. AppZ. Math. 6, Part 4, 398-439 (1953). 81. V. C. Liu and H. Jew, 2. Angew. Math. Mech. 42, No. IOjll, 431-437 (1962). 82. G. S. H. Lock, Heat Transfer Studics of the Closed Thermosyphon. Ph.D. Thesis, Univ. of Durham, Appl. Sci., Durham, England, 1962. 83. F. C. Lockwood and B. W. Martin, I . Mech. Eng. Sci. 6(4), 379-393 (1964). 84. E. L. Long, Proc. Pernmfrost Znt. Conf., pp. 487-491 (1963). 85. J. Madejski and J. Mikielewicz, Znt. /. Heat Muss Transfer 14, 357-363 (1971). 86. B. W. Martin, Brit. J. Appl. Phys. 5(3), 91-95 (1954). 87. B. W. Martin, Proc. Roy. SOC., Ser. A 230, 502-530 (1955). 88. B. W. Martin and D . J. Cresswell, Engineer 204, 926-930 (1957). 89. B. W. Martin, Proc. Znt. Mech. Eng. (London) 173(32), 761-771 (1959). 90. B. W. Martin and F. C. Lockwood, /. Fluid Mech. 19(2), 246-256 (1963). 91. B. W. Martin, Int. J. Heat Mass Transfer 8 , 19-25 (1965). 92. B. W . Martin, Proc. Roy. SOC.,Ser. A 301, 327-341 (1967). 93. R. Mitchell and V. A. Ogale, Gas Turbine Blade Cooling-Retrospect and Prospect. Amer. SOC.Mech. Eng. ASME Paper No. 67-WA/GT-9 (1967). 94. W. E. Moorhouse, Proc. Inst. Mech. Eng. (London) 174(16), 561-574 (1960). 95. W. D. Morris, Heat Transfer Charactcristics of a Rotating Thermosyphon. P1i.D. Thesis, Univ. of Wales, Swansea, 1964.

110

D.

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96. W. D. Morris, Some Observations on the Heat Transfer Characteristics of a Rotating Mixed Convection Thermosyphon. Aeron. Res. Counc. C.P. No. 1115 (1970). 97. C. Moussez and A. Mihail, Entropie, 102-109 (1966). 98. V. A. Ogale, On the Application of the Semi-closed Thermosyphon System to Gas Turbine Blade Cooling. Dr. Thesis, Technological Univ. of Delft, Delft, Netherlands, 1968. 99. S. Ostrach and P. R. Thornton, On the Stagnation of Natural-Convection Flows in Closed End Tubes. Amer. SOC.Mech. Eng. ASME Paper No. 57-SA-2 (1957). 100. P. Palanikuman, The Effect of Tube Inclination on Heat Transfer in the Closed Evaporative Thermosyphon. M.S. Thesis, Univ. of Durham, Durham, England, 1960. 100a. P. F. Pucci and J. C. R. Gerretsen, ASME Gas Turbine and Fluids Eng. Conf. Products Show, San Francisco, Paper No. 72-GT-36 (1972). 101. A. F. Robinson, Mechanical Performance of the T 3 Water Cooled Turbine. N G T E Memo. M83 (1950). 101a. A. G. Romanov, Investigation of Heat Transfer in a Blind Channel under Conditions of Natural Convection, Zzv. A N S S S R , OTN, no. 6 (1956). 102. S. K. Runcorn, Phil. Trans. R o y . SOC.London, Ser. A 258, 228-251 (1965). 103. 0. A. Saunders, Proc. Roy. SOC.,Ser. A 172, 55 (1939). 104. J. Schenk, G. W. Dieperink, and C. den Ouden, A p p l . Sci. Res. 20, 246-250 (1969). 105. H. Schlichting, “Boundary Layer Theory,” 4th Ed., p. 409. McGraw-Hill, New York, 1962. 106. E. Schmidt, E. Eckert, and U. Grigull, Heat Transfer by Liquids Near the Critical State, Translation of Captured Enemy Document. Air Doc. Div., Wright Field, Dayton (1946). 107. E. Schmidt, Proc. Znst. Mech. Eng. A . S . M . E . (London) Conf. Proc. 361-363 (1951). 108. E. Schmidt, Znt. /. H e a t Mass Transfer 1(6), 92-101 (1960). 109. E. Schmidt, Int. Develop Heat Transfer, Proc. Heat Transfer Conf., Univ. of Colorado, XXIX-XL (1963). 109a. W. F. Schneider, Liquid-Metal-Filled Rotor Blade. U. S. Pat. No. 3, 623, 825 (Nov. 30, 1971). 110. A. Shetz and R. Eichhorn, J. Fluid Mech. 18, Part 2, 167-176 (1964). 110a. L. I. Slobodyanyuk and V. A. Omelyuk, Zzv. Vy‘yssh. Ucheb. Zaved., Energ. 3, 61-66 2 4; AD-622466; Oct. (1965)l. (1965) [Transl. as FTD-TT-65-1370/1 1lob. A. Stappenbeck and S. Moskowitz, Small Axial Turbine Stage Cooling Investigation. USAAVLABS Tech. Rep. 71-74 (1971). 111. A. A. Stone, “Farm Tractors.” Wiley, New York, 1932. 112. S. N. Sucio, A G A R D Conf. Proc. Paper No. 15, pp. 21-25 (1970). 112a. G . M. Tkachenko, Izv. Vuz. Energ. 6 , 61-65 (1970). Also, Heat Transfer-Sov. Res. 2(6), 48-52 (1970). 112b. I. B. Uskov and L. M. Tseitlin, Teploenergetika 17(1), 56-58 (1970). Also: Therm. Eng. 17(1), 80-83 (1970). 113. L. F. Walker, Further Investigation into the Effectiveness of the Closed Thermosyphon System using Aerofoil-Shaped Blades. Pametrada Rep. No. 160, British Ship Res. Ass. (1957). 113a. S. Walters, Mech. Eng. 94(3), 32-33 (1972). 114. D. Wilkie and S. A. Fisher, Znt. Develop. Heat Transfer, Proc. Heat Transfer Conf., Univ. of Colorado, pp. 995-1002 and pp. D271-274 (1963). 115. W. H. Worthington, SOC. Automot. Eng., Farm, Constr., Znd. Machinery Meet., Milwaukee No. 660584 (1966).

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116. M. L. Yaffee, Awiat. Week Space Technol. 95, 68-69 (1971). 116a. L. M. Zysina-Molodjen and G. M. Tkachenko, Heat Transfer-Sou. Res. 3(5), 142 (1971). 117. L. M. Zysina-Molodjen, R. M. Jablonic, M. P. Poljak, I. B. Uskov, and G. M. Tkachenko, Proc. Znt. Heat Transfer Conf., 4th, Paris-Versailles Paper NC 4.7 (1970).

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Heat Transfer to Flowing Gas-Solid Mixtures CREIGHTON A. DEPEW University of Washington, Seattle, Washington

AND

TED J. KRAMER Boeing Company, Seattle, Washington

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Observations and Heat Transfer Correlations . . 111. Fluid Mechanics of Suspensions . . . . . . . . . . . . . . A. Experimental Techniques . . . . . . . . . . . . . . . B. Flow Model Development . . . . . . . . . . . . . . . C. Measured Flow Properties and Correlations . . . . . . . IV. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . V. Concluding Remarks . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

. . .

. .

113 116 134 136 145 156 167 175 176

I77

I. Introduction As heat transfer media, both gases and liquids have some serious deficiencies which lead t o design limitations or extraordinary considerations to overcome the deficiencies when high heat fluxes are required. Liquids generally have better coolant properties than gases because of their higher density and thermal conductivity, but unless high pressure is used, change of phase may occur which may cause instability and or burnout. Temperatures are necessarily low to avoid boiling when liquids are used, but gases can be carried to high temperatures. Gases, however, have low thermal capacity and low thermal transport properties 113

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CREIGHTON A. DEPEWAND TEDJ. KRAMER

which give rise to low heat transfer coefficients in typical applications. When the careful and organized laboratory experiments of Farbar and Morley ( I ) showed that it was possible to overcome these deficiencies by the addition of solid particles to the gas stream, interest in gas-solids suspensions as a basic heat transfer medium was renewed. Nuclear reactor cooling offered a particularly interesting new application since moderate-to-high heat fluxes and temperatures are involved in power production, but Farbar and Morley’s work was originally suggested by the process of extracting waste heat from flue gas laden with aluminasilica catalyst: a process which had been used in the petroleum industry for a number of years. Though energy recovery from the catalyst regeneration process was widely practiced, the heat transfer process had not been systematically studied previously. T h e recovered heat was used to generate steam by passing the mixture of excess air, combustion products, and carried-over catalyst through the tubes of a low-pressure boiler. Steam generation was not the only means used to recover energy from the exothermic regeneration stage, since the mixture was also used successfully to power gas turbines directly. Augmentation of heat transfer by the addition of solids potentially offered many desirable improvements to conventional cooling of reactors. T h e same heat transfer rates could be obtained at a lower system pressure, and costs could be reduced since less heat transfer area would be required by using higher power densities. Due to the higher volumetric heat capacity of suspensions, less flow area would be required. Increased cycle efficiency could be realized by higher reactor coolant temperatures. For the same heat transfer rate, the pumping power would be reduced because of lower friction factors and higher heat content per unit volume. Additionally, the lower void fraction for coolant passages would result in economy of neutrons, and lower channel temperatures resulting from the improved cooling effectiveness would permit a wider choice of cladding and fuel materials. T h e influence of suspended particles on the transport characteristics of the suspension, as they are manifested at the wall, has been widely discussed by numerous authors. An obvious point of agreement is that the solid phase increases the volumetric thermal capacity of the flow which in turn increases the heat transfer rate for the same temperature conditions. Beyond this elementary view there are numerous explanations for the phenomena which are encountered in the experiments. There is generally a consensus that direct heat transfer from the wall to the solid phase is negligible due to small time and area of contact between the particles and the wall. A consequence of this view of the heat transfer mechanism is that the temperature of the solids lags the temperature

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115

of the gas. Assuming that the solids and gas temperatures have equilibrated prior to the onset of heating, the gas temperature increases more rapidly than the solids temperature until some balance of temperature potential and heating rate between the wall and the gas and the gas and solids is achieved. It is reasonable to expect that at some distance from the thermal entrance, the flow will become fully developed thermally with similar temperature profiles persisting if the thermodynamic and transport properties are reasonably constant. T h e transfer of heat to the solid particles depends on the particle geometry, physical properties, and motion relative to the gas, whereas the net transfer of heat to the gas depends on the loss of heat to the solids versus direct gain from the wall through the sublayer, the buffer region, and the turbulent core. Alteration of the laminar or turbulent structure of these regions may profoundly affect the rate of transfer to the suspension, and the fluid dynamics of the flow is a basic facet of the convective process. Solids can alter convection in several ways: (1) penetration of solids through the buffer layer and into the laminar sublayer would cause a disturbance and thinning of that layer and a reduction in its resistance to heat transfer; (2) the presence of solids may cause a damping of the convective eddies and a consequent reduction in turbulent transport of energy; ( 3 ) on the other hand, slip between the particle and gas may enhance the turbulent mixing of the carrier gas; and (4) radial motion of the solids would promote the exchange of energy between the laminar sublayer and the turbulent core. T h e rate of heat transfer to the suspension clearly depends on the fluid dynamic properties of the flow, and the transport processes are governed by complicated interactions between: (1) fluid and particle, (2) particle and particle, and ( 3 ) the mixture and the flow boundary. Local mean fluid dynamic properties which control the energy and momentum transport include particle velocity, gas velocity, and local particle flux. T o these mean properties could be added the microscale properties such as fluctuating velocity components of each phase and the turbulent diffusivities. T h e technical review which follows is presented in three parts. Experimental heat transfer results are reviewed in Section 11. T h e discussion includes correlations of heat transfer data and discussion of correlating parameters which are based on theoretical models of the energy transfer process. Most of the experimental research has been conducted in systems having vertically oriented, heated or cooled sections, but the asymmetry caused by a transverse gravity force when the flow is horizontal has led to some interesting results and hypotheses about the nature of the particle motions in this orientation.

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T h e fluid dynamic properties of the suspension flow system are fundamental to the heat transfer process, and any advances in understanding and analysis of the thermal system must be based on progress in the description of the isothermal system. Therefore, Section I11 contains new information which is based on experiments on the point mean properties of isothermal flow of a suspension of glass spheres in air. A universal velocity law for the suspension as a continuum is derived from the fundamental equations, and the constants for the law are derived from the experimental observations. Values of the eddy diffusivity for momentum are also derived from the data and presented. T h e final part, Section IV, contains a discussion of theoretical analyses of the problem. Previous attempts are discussed and criticized, and although the present analysis is based on the data of Section I1 and the solution is by numerical computation of the energy equation in finitedifference form, the results contain a quantitative accuracy not found in more purely theoretical models.

II. Experimental Observations and Heat Transfer Correlations Farbar and Morley’s (I) data showed that the heat transfer coefficient was increased by as much as 2.6 when alumina-silica catalyst was added to air. T he enhanced heat transfer process was demonstrated in an isothermal, vertical tube by adding solids u p to a solids-to-gas loading ratio (on a weight basis) of 13.5 while maintaining a constant air rate. Two of the physical features of the system should be noted: ( I ) T h e test section was 48 diameters long (17.5-mm i.d.), and (2) the catalyst powder had a range of sizes with a considerable fraction of fines95 yo was less than 200p and 50 yo was less than Sop. T h e data showed that the heat transfer coefficient increased relatively more at the lower Reynolds number (13,500) than at the higher Reynolds number (27,000) and that the coefficient showed little improvement when the solids loading ratio was less than unity. Farbar and Morley (1) were able to correlate their results at solid loading ratios above unity with an equation which resembles the Dittus-Boelter equation for single phase flow: Nu,

=

0.14

(1)

Although the increase in heat transfer coefficient is impressive, it should be pointed out that Nu, is less for a suspension than for the equivalent fluid having the density of the suspension. I t was not altogether clear that the addition of solids was an economically justifiable technique

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117

for improved heat transfer, for if suspension were to compete with other heat transfer augmentation schemes, it would have to be shown that operating costs, e.g., pumping power, were consistent with the gains. T h e early laboratory work led to a large-scale effort by the Babcock and Wilcox Company (2-5) under the sponsorship of the U.S. Atomic Energy Commission. T h e purpose was to assess, through demonstration, the feasibility and desirability of using gas-solids suspension as primary reactor coolant. T h e solid phase was graphite with an arithmetic mean in the “as received condition.” T h e mean size was reduced size of 2 . 0 1 ~ to less than 0.711. due to attrition after 46 h r of circulation time. Nitrogen, helium, and Freon-I4 were used as suspension gases. T h e initial series of tests in vertical circular tubes with electrical heating resulted in the correlation equation Nu,

=

0.02(Reg Pr)’.’ ( M C

+ l)0.45.

(2)

Turbulence promoters in the form of spiral-shaped strips inserted in the full length of the heated section greatly enhanced the heat transfer performance. I t has been postulated that the spiral strips created a rotational motion which forced the solids to the wall by centrifugal force where a scouring action took place. A variety of pitch lengths and tube sizes were involved in the study, but no single correlation of the results for all conditions was possible. Using their heat transfer and pumping power data, they were able to show very positive benefits. For example, with a 13.5-mm tube and 102-mm pitch turbulence promoter the pumping power which will maintain the same heat transfer coefficient as in the clear gas can be reduced by a factor of 10 without promoters and 25 with promoters. For helium gas, the power is reduced by factors of 5 and 15, respectively. One of the reports which resulted from this study cited several properties of the system which point up the potential of gas-solids systems as coolants: (1) heat transfer coefficients can be increased by factors up to 8. (2) heat transfer can be increased by factors u p to 25. ( 3 ) low neutron cross section. (4) compatibility of graphite with most gases and materials used in gas-cooled reactors. ( 5 ) potential scavenges of fission, corrosion, and mass transfer products. (6) ease in varying effective density and heat capacity by variation of solids loading. T h e advantages were clearly demonstrated for graphite suspensions over

118

CREIGHTON A. DEPEWAND TEDJ. KRAMER

conventional gas-cooling, but two other points were demonstrated through the course of the study. T h e system was a closed loop which operated continuously when needed, and 125 kW power was continuously transferred to the suspension and subsequently removed in shell and tube water heat exchangers giving a heat flux of at least 870 kW/m2. Although the system was constructed from conventional equipment and materials, no detectable corrosion, erosion, or wear was found in 135 hr of operation. T h e system could be stopped and started with ease, even though the loading was quite heavy at times, up to 176 kg/m3. Performance on a sizable scale was demonstrated, and the reports concluded that further development should proceed to full-scale application. T h e Babcock and Wilcox results were reevaluated by a group at the Franklin Institute (6) who conducted a careful criticism of the experiments and data reduction. They concluded that a number of similar equations could be used to correlate the data, and Pfeffer, Rossetti, and Lieblein (7) have pointed out that all of the correlations presented give essentially the same representation of the data. Further experiments with graphite powder having an average size of 2p in helium and nitrogen were conducted at the U.S. Bureau of Mines (8, 9 ) in a closed system. Their objectives were also to determine the role of such suspensions in nuclear reactor cooling. T h e initial series of tests was conducted at low apparent Reynolds numbers (800-6000). These tests were at considerably lower flow rates than the previous Babcock and Wilcox experiments. T h e Bureau of Mines test loop utilized induction heaters which brought the suspension to nominally 500°F. Heat transfer results were reported for the water cooler. T h e first heat exchanger used was made of finned tubing where water passed through the center tube and the suspension passed between the longitudinal fins in the annulus. During the first 22 hr of operation the finned annulus became almost completely plugged with graphite. T h e cooler was subsequently replaced with a water jacketed counter-flow heat exchanger and operated for 200 hr without noticeable abrasion. However, heat transfer performance had degraded during the operation by the deposition of carbon on the cold heat exchange surfaces. Two methods were investigated for removal of the deposition. Large graphite particles were effective in scouring the surfaces, but they broke up after several hours of operation. T h e addition of large alumina-silicate particles was likewise effective in scouring the surface, but severe pump impeller erosion would have prevented continuous circulation for an extended period. In the subsequent tests, turbulence promoters were used as in the Babcock and Wilcox experiments. Deposition on the cold walls continued, but the absence of graphite on the turbulators

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

119

themselves and the greater deposition on the walls near the heat exchanger inlet led the authors to conclude that a temperature gradient was the major driving force for deposition. Babcock and Wilcox had likewise previously had similar difficulty in their coolers, but the phenomenon occurred only with the lightest gas, helium. T h e researchers at the Bureau of Mines also constructed a “high velocity system” which was capable of producing Reynolds numbers in the range from 6600-16,000. This horizontal test loop was constructed from 16-mm copper tubing with large radius bends to eliminate abrupt changes in direction. I t appears from the schematic diagram that no flow straighteners or distributors were used, and it is likely that the flow was stratified through most of its cycle. T h e purpose of the high-velocity loop was to eliminate deposition, but the sought-for relief was not found. T h e authors report that only limited improvement in heat transfer characteristics was found. They recommend that the flow must be maintained in the highly turbulent range to minimize build-up, but the higher pumping costs offset the advantage in heat transfer. I n a subsequent investigation at the U.S. Bureau of Mines, Bluman et al. (Ya) report that heat transfer coefficients with 30p glass particles in a mixture of carbon dioxide and nitrogen were not significantly higher than the clear gas values, over Reynolds numbers from 20,000 to 35,000 and loadings up to 0.7. Rosenecker, Coates, and Lucas (5%) investigated the pumping power relationship for glass spheres in the 1 8 - 4 0 ~range also in a mixture of carbon dioxide and nitrogen. Gorbis and Bakhtiozin (10) also studied graphite suspensions in air. T h e experiments covered a rather wide range of variables: tube sizes from 12 to 33 mm, particle size from 0.15 to 2.08 mm. T h e heat transfer measurements were made by cooling the preheated mixture by passing cold water counter-current to the suspension flow in an annulus. T h e higher heat transfer performance which was previously determined in the Babcock and Wilcox study was confirmed even though the particles were generally many times larger. T h e authors recommend a correlation equation Nus Nu,

-- -

1

+ 6.3

CM ,

(3)

where Re, = Vpdpg/pgand V , is the particle terminal velocity. A group at Urals Polytechnic Institute (11)also performed experiments with graphite suspensions having a mean particle size of approximately lop in atmospheric pressure air. Heat transfer results are reported for a vertical cooler with cooling water flowing in the annulus jacket, concurrent with suspension. A large range of gas-phase Reynolds

CREIGHTON A. DEPEWAND TEDJ. KRAMER

120

numbers and solids loading ratios was covered, and the data are correlated in terms of the clear gas Nusselt number Nu, : region of low concentration: 0.5

< M < 25, Nu,/Nu,

=

570 1

< Re, < 72,900

+ 5.35 Rego.3M0.78;

(4)

region of average concentration: 25

< M < 60, Nu,/Nu,

=

8000 < Re, 1

+ 0.0246

< 23,000 M2.45;

region of high concentration: 60

< M < 110, Nu,/Nu,

=

5,600 < Re, 1

< 12,000

+ 2.26 Repo.3M1.33;

region of maximum heat exchange: 100 < M

< 150,

5000

Nu,/Nug

-

< Re, < 7000 (7)

80.

These investigators concluded from their work that with the same power requirements, the quantity of heat transmitted, and consequently the heat exchanger's effectiveness (all other things being equal) is several times higher than when using air alone. Where M < 15, the relative effectiveness is equal to or even less than unity. T h e authors did not report any operational difficulties such as deposition on the wall or erosion of components. Sukomel, Tsvetkov, and Kerimov (12) at the Moscow Power Institute, conducted experiments which were similar to those by Babcock and Wilcox. Graphite particles and aluminum oxide particles were suspended in horizontal flows of helium, nitrogen, and air in an electrically heated test section. They were able to correlate their data at x / D = 72 for 3 < M < 40 by the equation N U ~ ~ / N= U 0.55 ~ , Co.'Mn, where n

=

0.33 for 65p particles and n

=

(8)

0.25 for 130-29Op particles.

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

121

Heat transfer was reduced for M < 3. They conclude that the thermal entrance region is greatly extended for dust-laden gas. A large amount of industrial data on fluid catalytic cracking units was correlated by Danziger (13) of the M.W. Kellogg Company. T h e correlation covers a wide range of operating conditions: Reynolds numbers from 178 to 25,400, solids loading ratios from 2 to 446, and tube i.d. from 17.5 to 38.0 mm. Danziger was careful to reject the data that did not satisfy a consistency check, and he also performed an approximate calculation showing that the difference between the solids and gas temperature was within the overall error in the data themselves. His results verified the 0.45 power of the loading ratio which had been derived by Farbar and Morley. T h e correlation for catalyst suspensions of about 50p particle size in vertical transport is Nu,

=

0.0784 Re:.66

(9)

T h e average deviation from the equation is 4.3 yo for the Kellogg data, and Farbar and Morley’s data had an average deviation of 9.2%. T h e Babcock and Wilcox correlation for graphite did not agree with the catalyst data, probably due to the large differences between particle size and mass. Danziger also showed that a correlation derived by Wen and Miller (14) was inadequate for predicting values of the suspension heat transfer coefficient h, . T h e correlation does not contain an influence due to gas mass velocity variation, and this appears to be a fundamental shortcoming since the experimental data demonstrate a rather strong dependency. Danziger’s equation appears to be adequate for design for catalyst of 50p average size in vertical isothermal transport lines, and it should be useful for engineering work. As the efforts to develop the art of heat transfer with gaseous suspensions of graphite were proceeding, two other groups were conducting basic research on the characteristics of gas-solids flows, Professor S. L. So0 at Princeton University under the aegis of the U S . Office of Naval Research’s Project Squid was investigating a variety of two-phase flow problems (15, 16). Some of these studies will be discussed in Section 111. While associated with the Princeton group, C. L. Tien, as part of his doctoral dissertation, carried out the first analysis of the heat transfer problem starting with the fundamental laws (17). This analysis will be discussed in Section IV. During the same time period the studies at the University of California under Professor Leonard Farbar were being continued on an experimental and analytical basis. A new system to study the effect of particle size was constructed. T h e test section was heated by condensing carbon tetra-

122

CREIGHTON A. DEPEWAND TEDJ. KRAMER

chloride on the outside of a vertical glass tube which was 17-mm i.d. by 806-mm long. Solid spherical glass particles of uniform size were entrained in an air stream with loading ratios up to 11 on a mass basis. T h e study by Farbar and Depew (18) showed that the smallest of the four sizes used, 30p, was the most effective in increasing the heat transfer coefficient. Heat transfer coefficients based on the logarithmic mean temperature were increased by as much as 300% at Reynolds numbers of 15,300 and 26,500 with the 30p size particle, but they were essentially constant when 2001. particles were added. T h e study clearly showed that heat transfer was strongly dependent on particle size, and the authors suggest several contributing phenomena: (1) prolonged thermal entry length, (2) larger internal thermal resistance of larger particles, and (3) reduction of turbulent eddy transport for light loadings but tendency toward a pluglike flow at high loadings. T o test the influence of the tube length and to assess the constancy of the heat transfer coefficient which is required for the legitimate use of the logarithmic temperature difference, a heat transfer tube with uniform flux was constructed by Depew and Farbar (19, 20). T h e wall temperature of the 18-mm i.d. tube was measured along its length, and local heat transfer coefficients were calculated on the basis of the local equilibrium suspension bulk temperature. Virtually the same influence of particle size as was previously found (18) persisted in the latter investigation; the local Nusselt number was almost unaffected by 200p glass particles up to loadings of 7. Furthermore, the variation of h, with distance was only slightly altered by 200p particles. On the other hand, 30p particles prolonged the thermal entry length to the extent that it was found to be over three times as long as with clear air at a loading ratio of 1.17. An analysis of the system was also performed, and this will be discussed in Section IV. Continuing this progression of experiments, Tien, who had joined the faculty at University of California, Berkeley, and Victor Quan, a graduate student, used Depew’s apparatus which is described in (21) to measure the heat transfer response with lead particles in air. They repeated Depew’s experiments with glass particles at Re = 15,000 and 30,000 in order to make a direct comparison with the lead powder suspension. T h e conclusions of Depew and Farbar (20) were substantiated, and further observations were possible. Tien and Quan (21) found that the suspensions with lead always produced Nusselt numbers which were less than with air alone and which were significantly lower than with glass particles. Partially at least, the explanation lies in the comparison of the thermodynamic properties of glass and lead. Lead has a density about four times greater than glass, but the heat capacity is only about one-seventh that of glass.

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

I23

Therefore the thermal capacity for equal size particles is lower (about 60%) for lead. Particle dynamics and particle-particle interaction have been studied by Richard L. Peskin as part of the Project Squid effort at Princeton University under the leadership of Professor S. L. So0 (presently at the University of Illinois). Peskin's interest in suspensions continued at Rutgers University, and many of his papers are discussed in Section 111. More recently Briller and Peskin (22) performed heat transfer measurements in a system where very high Reynolds numbers were obtainable. T h e horizontal heated and cooled test sections were 7.8-cm i.d. by approximately 3 m in length. Heating was supplied by radianttype electrical heaters which were controlled to provide near isothermal tube wall conditions, and the cooling section was surrounded by five isothermal water jackets. Experiments were performed in air at a nominal Reynolds number of 130,000 with glass spheres of 28, 84, and 147p diameter. Solids loading ratios for the small size were less than 2, but loadings up to 8 were obtained with the larger sizes. Special probes were inserted into the flow to determine the air and solids temperatures, and although the system instrumentation was inadequate for accurate quantitative conclusions, the authors observed that there was no significant improvement in the heat transfer coefficient at Re = 130,000 within the range of the experimental variables. They conclude that the major advantage of a gas-solids suspension compared to pure air as a heat transfer medium at high Reynolds numbers is its higher heat capacity. T h e Rutgers University gas-solids flow loop was modified to eliminate the inadequacies previously cited, and further experimental work was performed. A new vertical section (3.17-cm i.d. x 366-cm long) was electrically (ac) heated. T h e test section was preceded by a straight entrance section which was 57 diameters in length. Wall thermocouples were mounted at 12 longitudinal stations, 90" apart. Experiments were conducted over a range of Reynolds numbers for 20,000 to 40,000 with glass particles 85 and l50p in diameter. Even though solids loadings were as high as 6, Briller (23) reported no change in the Nusselt number at a distance of 29 diameters from the inlet. Briller and Peskin (24, 25) carried out an analysis of suspension flow heat transfer in an attempt to develop a parameter which could reasonably account for enhancement of the heat transfer coefficient under certain conditions of particle size and loading. Three parameters are utilized. T h e first is the laminar sublayer thickness 6, for the clear gas flow evaluated at the same Reynolds number as the suspension flow based on the gas properties. Th e second parameter is the number density of particles np which is evaluated by assuming a uniform

124

CREIGHTON A. DEPEWAND TEDJ. KRAMER

distribution of particles and no slip between the phases. T h e third parameter is the penetration depth, which is an estimate of the degree of disturbance which is caused by the individual particles to the laminar sublayer. By reasoning that np1/3 is a measure of the average particle spacing and other dimensional considerations, the correlation is presented in the form Nu,/Nug

=

function(nr 8,PD).

(10)

T h e first two parameters can be readily calculated from measured variables of the flow, but the penetration depth requires four assumptions and consideration of the particle dynamics in order to relate it to the primary flow variables. These four assumptions are the following: (1) the component of fluid turbulent intensity transverse to the wall at the edge of the boundary layer is proportional to the average axial gas velocity, (2) the transverse slip velocity can be described by the steady-state drag law for spheres for particle Reynolds numbers less than 100, (3) the transverse slip at the edge of the boundary layer is proportional to the transverse turbulent intensity which is related directly to the average axial gas velocity, and (4)the particle penetration into the laminar sublayer results in the same fractional reduction in its transverse slip velocity for all experiments and that the constant proportionality implied in (3) above is the same for all experiments. Finally, PD is related to properties of the flow by

K5 is constant by assumption (4)and is chosen as unity for convenience. T h e experimental results of a large number of investigators representing a wide range of system variables was used in the development of a correlation equation. T h e results are shown in Fig. 1 along with a mean curve which is represented by NuS/Nug= 0.0195(nE'3 81PD)o.6.

(12)

T h e experimental curves are identified by Briller and Peskin (23-25). Figure 1 does demonstrate a certain grouping of the data in a way that appears physically reasonable, i.e., enhancement of heat transfer is expected with an increase of any of the terms in the abscissa. Significant increases in the ratio occur more generally for values of ng/36,PD > lo3; for values of ni/361PD< lo3 the Nusselt number ratio appears to be little affected. I t should be noted that n2/36,PD can be related to the

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

FIG. 1.

I25

Correlation of experimental results using Briller’s parameter.

primary flow variables using Briller’s (23, 25) relationships such that 6,PD

7~:’~

=

39.2 M2’3(p,/pp)1’3(D/d)0.3

(13)

T h e strongest dependence is seen to be on the solids loading ratio with considerable influence due to Dld. T h e density ratio is fairly constant for all of the experiments represented, and the dependence on Re, is weak. I t is apparent that the correlation does not explain the constancy of the Nusselt number ratio for the author’s own experiments. It is also interesting to note that substitution of Eq. (13) back into the correlation (12) produces an equation which is dependent on The value 0.4 compares favorably with the exponent 0.45 of Farbar and Morley (1) and Danziger (13). Brandon (26, 27) in collaboration with Thomas, concluded that a critical-size particle should exist which would interact with the continuous phase turbulence to enhance the heat transfer through the highly resistive laminar boundary layer. Brandon argued that small particles which travel with the turbulent eddies do not aid the convective process but merely add to the heat capacity of the mixture. T h e higher thermal capacity per unit volume produces a higher heat transfer coefficient, but h, is not improved more than the comparable continuum fluid based on weighted average properties. Moreover, Brandon continued, particles with low inertia which follow the turbulent fluctuations would not demonstrate a particle size influence on heat transfer. He

126

CREIGHTON A. DEPEWAND TEDJ. KRAMER

theorizes that interaction with the small, intense eddies near the wall is essential for significant influence on the heat transfer process from the wall to the suspension. Particles which are very much larger than these small eddies would be little affected by these eddies due to the overwhelming inertia of the individual particle. Thomas postulated that the Kolmogoroff microscale 7 is the appropriate characteristic dimension for the smallest influential eddy size. T h e Kolmogoroff microscale in homogeneous turbulence is related to the energy production per unit volume. Although pipe flow turbulence is decidedly nonhomogeneous, it is known that the small scales are locally isotropic regardless of whether the large scales are anisotropic or not, and the very small scales are chiefly governed by viscous forces and the energy transferral from the large eddies (31).Brandon suggests that 7 should be evaluated for the eddies’ maximum turbulent dissipation. Laufer (32) has determined that the maximum dissipation is approximately ten times the average dissipation which can be related to the flow variables through the Blasius friction factor law. T h e resulting relationship is d/q = ( d / D )Re11/16.

(14)

Reasoning that for the most effective particle-fluid interaction to take place, the mass of the most effective fluid particle should be the same as the mass of the solid particle, another ratio is produced

Brandon has also considered the effectiveness .of the optimum-size solid particle by proposing that the “stopping distance’’ can be used as a scaling factor. I n a manner which is similar to Briller’s derivation of penetration depth, the ratio of the fluctuating transverse component of particle velocity to the mean axial velocity is assumed to be proportional to Re” (where n is a constant to be determined by experiment). T h e relaxation time which is derived by Davies (33) for deposition on the wall is taken to be the time of flight. By incorporating the previous two relationships, the stopping distance ds is given by

If one assumes that the resistance to heat transfer can be lumped into a stagnant film of thickness A x , where A x = k/h, the ratio can be written as d/Ax cc

Pr1I3 Re(0.42+n).

(17)

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

127

Experiments were conducted in a steam-heated pipe with suspensions of glass microspheres in water to assess the validity of the foregoing heuristic arguments. Figure 2 shows the incremental increase in h versus d/r, for vertical and horizontal orientations of the tube. Figure 3 shows the results of other investigations ( I , 18, 20, 21, 28, 29, 30) of gas-solids heat transfer plotted as h,/h, versus d/q, and Fig. 4 shows

.

V

0 V 0

A

d micnms 9 , microns 20-860 16.0

170 32-370

15.4 l o 20.2

50

15.1 l o 23.4 10.6 to 27

20-100

14.8

-

10

WLUUE FRACTION SOLIDS

-

8-

6-

4-

0.I

I

\

.\

1.0

10

MEAN PARTICLE DIAMETER = KOLMOGOROFF MICROSCALE

0

V

I I

$-+dl116

FIG. 2. Effect of particle size and system characteristics on the convective heat transfer to dilute water suspensions.

the results of experiments performed by Brandon and Grizzle (27) for heating of suspensions of glass microspheres in air. All three of these figures substantiate the hypothesis that there is a critical ratio of the particle diameter to Kolmogoroff microscale for which the convective processes of suspensions are enhanced. But as Brandon and Grizzle point out, the single point in the 25-mm tube does not confirm that the theory has general quantitative application.

I28

CREIGHTON A. DEPEWAND TEDJ. KRAMER

d. mlcmm 7. microns -

o

m

15 - 2 5

0

00-600

53-71

0ROT.7. HlBY AND MULLER (19%

30-200 15-22 A m-200 16-26 .i7 30-200 15-24

' 0

REFERENCE

FAQBAR AND MORLEY (19571 FARBAR AND DEPEW (19631 DEPEW AND FARBAR (19631

TIEN AND W A N 11962)

90-1oOo

10-55

JEPSON. POLL AND SMITH I1963

40-120

24-61

WlLKlNSON AND NORMAN (19671

2.

'

2.

J

I

I.

h , , , , , h.,,

I.

I.

I.

I.

0, 1.0

' 'f- " 10

-

l 0

MEAN PARTICLE DIAMETER KOLMOGOROFF MICROSCALE

FIG. 3. Effect of particle size and system characteristics o n the convective heat transfer to dilute gas-solid suspensions.

Heat transfer studies at the Technical and Physical Research Center of the Socialist Republic of Romania under the direction of Ion Curievici (34-36) have produced data which verify previous results and with which the authors develop new correlations. One investigation (34) was devoted to the influence of swirling suspension flow created by a twisted tape in the pipe upstream of the heated section. T h e heat transfer coefficient for the swirling flow was as much as 2.42 times as large as the suspension heat transfer coefficient without the turbulence promoter as in the Babcock and Wilcox work. Alumina-silica catalyst, predominantly in the range 60-100p, in air was used in the vertical, isothermal Re 25,200 and I M 10 for the tube. I n the range, 17,170 turbulator and system used, the authors correlate the ratio of the

<

<

< <

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES I

I

I

I

I

I

129

I

v 7 0 u 79,0007 0 p 42,000 o 701.1 34.0007 0 p 30,000 5 0 p 30,000-

4\ I

//

A

~r\

x 1 O O p 30,000 Dz25rnm 0 7 0 pll 1 6,000-0--

1.0

0

1

2

3

4

Mean Particle Diameter = d 11/16 Kolrnogoroff Microscale DRe

FIG. 4. Heat flux enhancement in turbulent air suspensions of glass microspheres.

suspension swirl flow coefficient to the clear gas swirl flow to within 10 yo by the equation hsv/h,

=

41,930 Re-1.076(1

+ CM)o.662.

(18)

Another flow loop was similar in many respects to the setup used by Babcock and Wilcox with continuous circulation of a graphite suspension through an electrically heated tube in series with four water coolers. Large graphite particles were reduced by attrition to an average of 3p with a maximum of 5p. Local heat transfer coefficients were based on the calculated bulk average temperature in the manner of Depew and Farbar (20),and thermal equilibrium between the two phases probably prevailed for the length of the tube due to the very small particle size. Curievici et al. found that for the range of Reynolds numbers covered, 11,500 Re 43,000, the thermal entrance region length increased from 12 to 30 diameters and that the asymptotic value of the heat transfer coefficient was greater than for pure gas when M > 5. They also found, in agreement with previous work, that a minimum value in (Nu,/Nu,), was obtained at about M = 3. They were able to correlate their data with an equation the form of which was suggested by Hawes et al. (37)

<

<

Nus/Nug = 0.43(1

+ CM)0.765(1 + 0.6 e-0.085@/D 1-

(19)

I n a subsequent report (36), Curievici and Dinulescu surveyed a number of experimental investigations (2, 5 , 20, 12, 27, 20, 21, 35, 37)

130

CREIGHTON A. DEPEWAND TEDJ. KRAMER

and derived new constants for the above correlation which more accurately represented the data for d < l o p and M > 4.

T h e experimental measurements by Chu (38) were used in the calculation of values of the eddy diffusivity for heat as is described in Section IV of this report, but the experimental results merit presentation and some discussion here. T h e system developed by Chu was similar in design to the ones previously used by Depew (29), Rajpaul (39), and Wahi (40). Two test sections having uniform heat flux were used: 25.4-mm i.d. and 50.8-mm i.d. by 4.23-m and 4.33-m long, respectively. Solid glass microspheres were added to the air stream from a weighing hopper at a controlled rate, and they were returned to a storage hopper by a double effect cyclone separator. Each heated section was preceded by a hydrodynamic calming section and temperature measurements were made on the outside tube wall at 16 axial locations instrumented with two thermocouples and at three locations with four equally spaced thermocouples. T h e local heat transfer coefficient and Nusselt number were based on the average suspension temperature as in Depew (19). Figure 5 shows some typical results in terms of the ratio NuJNu,

-

16

l

l

r

$

~

-

3-

30p-

62"

. .______

200P-.-

.

-

151rnrn tubel

= 0 6

1

0

1

1

1

1

1

1

1

1

1

1

1

1

2

3

4

5

6

1

2

3

4

5

6

0.8 06 0

Mass Loading Ratio (M)

FIG. 5. Nusselt number ratio based on suspension mean temperature vs. solids loading ratio.

~

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

131

versus M at xID = 50. M was limited by an apparent stratification near the heat section outlet as manifested in a nonsymmetrical tube wall temperature difference at high loadings. Whereas with air alone or light loading one side of the tube would normally run hotter than the other side by less than 1°C due to nonuniform wall thickness, at higher loading the normally hotter wall would become cooler by several degrees. Van Zoonen (41) observed stratification of the flow in a vertical glass riser, and it is believed that the cooler wall was being cooled preferentially by an accumulation of solids. T h e location of this stratification moved down the tube (upstream) with increased loading, and sometimes negative axial wall temperature gradients were observed. Figure 5 shows that in almost all cases Nu, was lower than Nu,. As was discussed previously in this section, a temperature potential that is based on the mean gas temperature would give a more realistic heat transfer coefficient, and Nu, were recomputed using Danziger’s suggested method of estimating the heat exchange between the gas and solid phases. These results are displayed in Fig. 6 for the 200p size. Results 51 rnrn tube

RQ:30,000

0

1

2

3

4

5

6

Mass Loading Ratio (M)

FIG.6. Nusselt number ratio based on gas mean temperature vs. solids loading ratio.

for the 30 and 62p are not shown since the values are imperceptibly different from those in Fig. 5. A comparison of Figs. 5 and 6 shows that Nu, is larger than Nu, for both tube sizes when Re > 18,500. When Re = 12,000 in the larger tube, Nu, is essentially the same for both calculations, but Nu, is significantly altered for the 25.4-mm case. T h e reason for the difference in behavior for the same Reynolds number is that the residence time in the larger tube is greater, and both phases are much closer to the same temperature in this case. Since it was assumed in the calculation that h,d/kg = 2, the minimum theoretical value, the values of Nu, which are based on Tgmare the largest possible values. T h e difference in temperature between the two phases was on the order of 10°C for the cases shown. This is on the order of 10% of the total temperature potential. According to Fig. 5, the largest decreases in Nu,/Nug occur for 30 and 62p particles. T h e ratio increases in most of the cases monotonically with increasing Re for the 30 and

132

CREIGHTON A. DEPEWAND TEDJ. KRAMER

62p size in contrast with 200p particles. For the larger tube size, Nu, decreases with increasing Re in an orderly manner; however, the 200p size shows a mixed trend. For the 30p-25.4-mm tube combination and the 62p-50.8-mm tube case, a certain similarity exists. Various authors have suggested that d / D is an important correlation parameter. A similarity exists, but if we compare the cases at Re = 15,000, at M = 1, Nu,/Nu, is 0.85 for 62p and 0.62 for 30p. This difference may in part be due to a higher relative particle-fluid interaction in the slower moving stream. Kramer (42) has shown that the influence of fluid turbulence on the particle motion is at a higher level when the mean velocity of the carrying fluid is lower; consequently the convective process may be enhanced more for suspensions at lower mean velocities. Chu’s study (38)contains discussion of attempts to use Boothroyd’s dimensional analysis (43, #4), an equation of the form Nu, = AReB, and Briller’s parameter to correlate the data. These techniques were inadequate in representing the diverse trends shown in Fig. 5, and it is clear that there is much room for progress in the understanding and correlation of these data. T h e first efforts to examine quantitatively the effects of stratification on heat transfer in horizontal flows was presented in a report by Depew and Cramer (45). A longer and more adequately instrumented tube was used by Bowen (46) to confirm and extend Cramer’s (47) results. Earlier observations of stratification were made by Thomas (48) in isothermal horizontal flow experiments. Thomas correlated the minimum transport velocity (which is defined as the mean velocity required to prevent the accumulation of a layer of stationary or sliding particles on the bottom of a horizontal tube) with the fluid and system properties. He suggested that the major factor affecting the vertical distribution of suspended solids in a flowing horizontal stream is the ratio of the particle settling velocity to the friction velocity. A value of 0.2 was chosen as the point of division between suspension flow and sliding bed flow. Depew and Cramer showed that the same criteria could not be applied to the diathermal system, and they concluded that it was unlikely that the criterion proposed by Thomas is adequate. Some of Bowen’s extended results are shown in Figs. 7 and 8. These curves are representative of the results of the investigation which used separate tubes for measuring pressure drop and heat transfer. T h e pressure drop system was made of glass tubing 17.7-mm i.d. by 7.30-m long with pressure taps at 0.6-m intervals, and the uniform heat flux tube was 19 mm by 3.05-m long. Four thermocouples at each of ten axial locations measured the wall temperature at the top, bottom, and both sides. Bowen’s experimental program covered a range of Re from 10,000 to

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

0

I

I

I

I

I

I

1

2

3

4

5

6

133

Mass Loading Ratio (M)

FIG.7. Nusselt number ratio in horizontal stratified flow vs. solids loading ratio.

1 ' " " " ' " ' l 3

1

FIG.8. Pressure drop ratio in horizontal stratified flow vs. solids loading ratio.

30,000 and included three sizes of glass microspheres: 30, 62, and 200p. Only a portion of the results are shown in Figs. 7 and 8. T h e asymptotic Nusselt numbers were circumferential1y uniform for the 2001. particle o the bottom of the suspension, but the values were higher by 1 0 0 ~ at tube when 30p particles were used. Nu, based on the side wall temperatures were intermediate to the values at the top and bottom. While the differences between the top and bottom results were much smaller for the 62 than the 30p size, higher Nu, were observed for the top side than for either of the other two sizes. T h e heat transfer results suggest that the suspension particle distribution became more uniform

134

CREIGHTON A. DEPEWAND TEDJ. KRAMER

with increasing particle size. This trend is contradictory to Thomas’ (48) theory of sedimentation, but an explanation for the phenomenon can be found in the observations of Adam (49). I n these studies, particles were observed to bounce from wall to wall as they were transported along the pipe. Th is mechanism is substantiated by the pressure gradient curves shown in Fig. 8 where the significantly higher pressure drop for the 200p size can be attributed to direct momentum transfer from the solid phase to the tube wall. This same bouncing-type flow was seen in the context of another suspension heat transfer study. Min and Chao (50, 51) observed with high speed photography that large glass particles were apt to bounce off the wall in nearly rectangular trajectories. Their study was directed to demonstrating that a transverse fluctuating electric field could enhance the heat transfer rate in a suspension flow. T h e enhancement depends on the ability of the particles to transport heat and acquire heat directly from the wall as they are driven in transverse motion by the interaction of their electrostatic charge and the electric field. T h e ability of the particles to carry a charge, the charging uniformity, and the frequency and strength of the applied field are important variables, and Min observed that the heat transfer rate was increased up to about 60% for a potential of 10 kV and frequencies of less than 10 Hz. T he enhancement was reduced at high Reynolds numbers and at solids loading ratios greater than unity. El-Saiedi (52) extended Min’s study by further experimentation and improved analysis, and he confirmed Min’s findings.

III. Fluid Mechanics of Suspensions T h e analogous nature of momentum and energy transport in single phase flows is well known. T h e possibility of extending this analogy to encompass gas-solid suspension flows serves as sufficient impetus to prompt an investigation of the fluid mechanics of suspensions; since if such an analogy were to exist, an understanding of the mechanics of momentum transport would serve as a starting point for the development of a model for energy transport. A defense for the assumption that the analogies observed in singlephase flows can be extended to suspensions rests with the fact that the analogy between interphase transport of momentum and heat has already been established (e.g., the analogy between the drag coefficient and heat transfer coefficient for a sphere in flow stream). Furthermore, transport of both quantities by single phase interaction (i.e., between

HEAT TRANSFER TO FLOWING GAS-SOLIDMIXTURES

135

fluid eddies or by particle collision) will be predominantly by the continuous phase since particle collisions are relatively infrequent. Hence, if the flow field disturbance caused by the presence of the dispersed phase is accepted to have a similar influence on both transport processes, the existence of the analogy is possible. Historically, the motivation for studying suspension fluid mechanics has been varied. Early investigations such as those performed by Gasterstaedt (53) and Rose and Barnacle (54) were prompted by an interest in pneumatic conveyance of solids and the design of ducting and pumping requirement for conveyance systems. Later, the possibility of using suspended particles to enhance the heat transfer characteristics of gases led to investigations such as the one performed at Babcock and Wilcox Company (2-4). More recently, the presence of a solid phase in rocket motor exhausts, transport of rock particles from highspeed drilling operations, deposition of debris from nuclear explosions, and the diffusion of solid emissions by the atmosphere have all motivated investigations of suspension mechanics. These investigations can be classified into two broad categories: analytical and experimental. Analytical studies have dealt mainly with development of models which are capable of expressing observed phenomena in mathematical form. Two approaches have been employed: (1) the study of single particle dynamics and the extension to multiple particle systems; (2) modification of continuum mechanics models. T h e dynamics of single particles in a turbulent flow field was investigated by Tchen (55), So0 (56), Chao (57), and So0 and Peskin (58). These studies dealt mainly with obtaining a description of the diffusion of particles by fluid turbulence. Though important scientific contributions, these studies did not provide sufficient information to describe the flow of suspensions in channels which probably represents the most important case from the standpoint of heat transfer systems. Much of the recent analytical work has been directed toward modifying the conservation equations of continuum mechanics to describe suspension flows. T h e conservation equations and the attendant boundary conditions alone do not represent a complete flow description. I n addition, the flux relationship expressing mass and momentum fluxes as functions of field variables must also be known. Definition of these relationships relies heavily on experimental observations in exactly the same manner that experimental observation fills the gaps in the semiempirical single phase model for turbulent flow. For this reason, discussion of analysis and experiment cannot be completely divorced. Most experimental investigations unfortunately have not been performed in sufficient depth or breadth to promote the evaluation of a generalized

136

CREIGHTON A. DEPEWAND TEDJ. KRAMER

model describing suspension flows. There are several reasons for the inadequate amount of experimentally observed data. First, introduction of a solid phase doubles the number of field variables that must be measured, and increases the number of flow conditions by a power of two. For example, a single-phase experiment involving three Reynolds numbers and three channel diameters would require measurements into nine flow conditions. A similar experiment for two-phase flow could have as additional parameters three particle diameters and three loading densities which would expand the number of flow conditions to 81. A second reason for the scarcity of experimental data is the difficulty most investigators have encountered in measuring flow variables. Impacting particles damage and clog measuring probes inserted in the flow. Remote sensing devices such as cameras and laser Doppler velometers appear to be attractive alternatives. However, practical considerations, such as the time required to reduce data from a statistically significant sample of measurements, limit the usefulness of camera techniques for measuring particle velocity or density. Peskin at Rutgers University and Brandon at Clemson University are currently developing laser anemometers for use in suspension flows. Such instruments provide an accurate means of measuring particle velocity, but techniques have not yet been perfected that allow the laser to be used to measure particle mass flux. A survey of recent advances in the study of gas-solids suspension flows will be presented in this section. T h e presentation is divided into three parts. First, experimental techniques for measuring suspension flow properties are described, and their merits and limitations are discussed. Techniques for measuring gas and particulate phase velocities, mass flux, and turbulence are described. A summary of measured suspension characteristics is then presented and discussed. I n the third and final part, the problem of developing a descriptive flow model from the conservation equations is presented. Several simplifying assumptions are offered, and their influence on the flow description is discussed.

A. EXPERIMENTAL TECHNIQUES Unique measuring techniques have evolved from experimental studies of gas-solid suspension flows. Since high-speed suspension flows either erode or completely destroy most instrumentation, fragile instruments such as hot wire anemometers or thin film transducers cannot be used. T h e hostile environment presented in suspension flows has caused many investigators to turn to remote sensing techniques.

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

137

Such techniques generally employ optics and have been successfully used to measure the velocity and density of the particulate phase. Remote sensing instruments for measuring properties of the gaseous phase are promising, but they are still in the developmental stages. A survey of special techniques which have been developed to measure the properties of suspension flows follows.

1. Particle Velocity Measurements High-speed photography has been used by several investigators including Hinkle (59), Doig and Roper (60), and McCarthy and Olson (61). Photographic data while being relatively easy to obtain, involves lengthy and tedious data reduction procedures. T h e particle velocity can be calculated from high-speed motion picture recorders by the following equation:

v, = mL/de,

(21)

where L is the distance the particle image travels, m the linear magnification factor, and LIB the elapsed time. T h e accuracy of data obtained with high-speed motion picture cameras depends on the accuracy to which the magnification factor m is known and the accuracy of the measurement of L from the film record. T h e elapsed time dB is accurately known from timing marks placed on the film by most cameras. I n the case of dilute suspensions very often there are not enough particles passing through the viewing field to provide a statistically significant sample of measured data on the film. Conversely, when the suspension is very dense, discrimination of individual particles can be difficult or impossible. A streaking camera was used by Kramer (42) to measure particle velocities in air containing 62 and 200p particles. Particle velocities ranged from 7 to 40 ft/sec. T h e camera, which is shown schematically in Fig. 9, was a Beckman-Whitley high-speed drum camera with the framing mechanism removed. Camera optics consisted of a magnification system and a transfer system. Th e magnification system contained two 35-mm camera lenses mounted on the ends of a telescoping tube to provide linear magnification of 7.25. Transfer optics consisted of a 0.020-in. wide vertical slit, a shutter system, and a first surface mirror. A real image was formed at the plane of the slit and was transferred to the film at a one-to-one magnification factor. Particle paths appeared as diagonal streaks on the exposed film as shown in Fig. 10. Velocity was obtained from the film record by measuring the angle between the streak and a line perpendicular to the edge of the film.

138

CREIGHTON A. DEPEWAND TEDJ. KRAMER

CAMERP, DRUM SUSPENSION

/

COLLIMATED LIGHT BEAM BEAM LIGHT

i

l

l

ILLUMINATED VOLUME

FIG. 9. Particle velocity measurement using streaking camera.

A third technique for measuring particle velocity is the optical cross-correlation technique developed by Kramer (42). A typical test setup is shown in Fig. 11. T h e optical cross-correlation technique employes photomultiplier tubes instead of a film to record the passage of particles through the viewing volume. Magnified particle images are formed on a plate having two vertical slits cut on a line parallel to the direction of flow. Slit separation is on the order of 75 to 100 mm, and a photomultiplier tube is placed behind each slit. The images of particles having predominant components of velocity in the direction of flow appear at first one slit and then the second. T h e time delay AT between the output signals of the photomultiplier tubes is inversely proportional to the particle velocity. T h e signals from the photomultiplier tubes are amplified and input to a correlation function computer. T h e computer output is the cross-correlation function

T h e delay time T at which the functionf(7) is maximum represents the mean time required for a particle image to move the distance between the slits. A strip chart record of a typical cross-correlation function is shown in Fig. 12. Mean particle velocity can be calculated from the optimum delay time, the magnification factor, and the distance between

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES 139

FIG. 10. Typical streaking camera record.

the slits. T h e main advantage of the cross-correlation technique is the ease with which the average particle velocity can be obtained from the measured data. I n addition, the photomultiplier tubes are sensitive to low levels of illumination for which it would be impossible to obtain high-speed film records. A light interruption technique of measuring particle velocities was

CREIGHTON A. DEPEWAND TEDJ. KRAMER

140

A CORRELATION COMRITE:

SUSPENSION FLOW TUBE

RESISTANCES

I In

t

22,000 STRIP CHART RECORDER

1o.ooo

RESISTOR

n

VARIABLE RESISTOR

FIG. I 1. Particle velocity using cross correlation.

DELAY TIME

FIG. 12. Strip chart record of cross-correlation function.

used by Eichhorn et al. (62). This technique is in principle identical to the optical cross-correlation method discussed above except a measuring probe must be placed in the flow stream. T h e probe consists of two fiber optic rods which are oriented with respect to the flow as shown in Fig. 13. A beam of light is directed into the fiber optic rods. Particles traveling parallel to the flow axis interrupt the light entering the rods causing the photomultiplier tubes located at the other end of

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

141

FIG.13. Schematic of light interruption technique for measuring particle velocity.

the rods to produce signals. T h e particle velocity is proportional to the ratio of the separation distance and the elapsed time. One disadvantage of the light interruption technique is the inability of the fiber optic rods to withstand the erosive flow environment. Applicability of the technique to measuring velocities of particles smaller than loop is doubtful since the percent reduction in light energy entering the rods due to passage of a particle would be extremely small. Doppler-shifted light signals back-scattered by suspended particles have been successfully used to measure the velocity micron-size particles. Morse et al. (63) developed a Doppler shift velometer using an He-Ne laser light source and a Fabry-Perot interferometer frequency filter. A schematic of the instrument and test setup is shown in Fig. 14. T h e intensity of particle-scattered light varies with the angle p from the axis of the incident beam. Although the intensity of forward ( p = 180') scattered light is maximum, the frequency shift due to particle motion is zero. Back-scattered light, while having lower intensity, exhibits the greatest frequency shift and therefore is used to measure velocity. T h e Fabry-Perot interferometer consists of two circular 97 yo reflecting mirrors, flat to approximately hj200. T h e mirror separation is driven by three stacks of piezoelectric crystals. Voltage across the crystals is linearly proportional to mirror separation. Significant light transmission through the interferometer occurs only at integer values (modes) of (24 cos # ) / A where 4 is the mirror separation, h is the wave length of light, # is the angle between wave-front normal and mirror

142

CREIGHTON A. DEPEW AND TEDJ. KRAMER

FIG.14. Schematic of Laser-Doppler velometer.

normal. T h e interferometer is first calibrated with nonshifted light to determine two reference mode locations as a function of mirror separation (or voltage). During operation, the interferometer is scanned between the two reference modes by applying a sawtooth voltage across the piezoelectric crystals. T h e Doppler shift in back-scattered light entering the interferometer will cause the transmission modes to shift and occur at some new mirror separation A*. Since mirror separation is proportional to applied voltage, the frequency shift and the particle velocity can be determined.

2. Gas Phase Velocity Measurements T h e measurement of gas phase velocity in the presence of suspended particles is relatively easy if the particles are larger than loop and the flow velocity is below 30 m/sec. A classical Pitot tube can be used if the mouth of the tube is flattened to prevent particles from being ingested. Stainless steel impact probes were used by Kramer (42) to measure total pressure in suspensions of 2001. particles in which the gas velocity ranged from 4.6 to 40 mjsec. When particle velocities exceeded 30 m/sec, the probes were eroded by impacting particles and

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

143

in some cases complete erosion of probe walls occurred within 20 min. Although smaller particles do not erode the probes as rapidly, considerable difficulty can be experienced in measuring gas velocity in suspensions containing particles smaller than 30p. T h e mouth of the impact tube must be made small to prevent ingestion. As a consequence, a manometer cannot be used to measure pressure differential because response time becomes prohibitively large due to restricted mass flow. This problem can be circumvented by replacing the manometer with a constant-volume pressure transducer. Several types are available which convert pressure-induced deflection of a thin diaphragm into a voltage signal. Several additional problems arise when the dimensions of the probe mouth are decreased. First, the small opening is very susceptible to clogging by the fine dust which is always present in suspension flows. Second, the mouth of the probe can be completely peened shut by the thousands of impacting particles. Both of these problems were experienced by Kramer (42), prompting the search for an improved probe design. I t was found that total pressure probes drawn from fine borosilicate glass tubing could withstand the erosiveness of the flow and could last longer than their stainless steel counterparts. T h e 0.75-mm diameter probes were drawn from 2-mm diameter tubing, and the mouths were fire polished and ground with a fine abrasive to produce the desired capture area. Dussourd (64) conducted an extensive study of instrumentation for measuring gas stagnation pressure and velocity in particle-laden flows. T h e following three requirements were identified as being necessary for a successful design:

(1) T o prevent clogging of the probe and pressure lines, the rear of the probe must be vented (or the capture area of the probe can be decreased to preclude particle ingestion). (2) T h e probe must be made sufficiently small to minimize upstream aerodynamic interaction. (3) T h e internal pressure tap should be situated close to the mouth of the tube to minimize the effects of particle-gas interactions within the tube. A number of vented stagnation probes were tested by Dussourd. Satisfactory results were obtained for measurements made in an airstream laden with water droplets.

3. Particle Mass Flux and Density Measurements Measurement of particle flux in suspension flows involves counting the number of particles crossing a control plane per unit of time. A

144

CREIGHTON A. DEPEWAND TEDJ. KRAMER

number of techniques for obtaining this information have been developed. So0 and Regalbuto (65) measured particle flux by capturing particles with a tube placed in the stream. Air was scavenged through a vent line at the rear of the tube to minimize the effect of the probe on the upstream flow. Trotter (66) used a similar type of capture tube in his study of gas-solids flows in vertical tubes. So0 et al. (67) developed an electrostatic probe to measure particle flux distributions in particle-laden flows. Operation of the probe is based on the fact that solid particles can take on a static charge when flowing though a channel. Assuming that the magnitude of charge increases with particle size, a probe inserted in the flow stream will receive a charge at a rate dependent on particle mass flux. Soo’s electrostatic probe consisted of a &-in. diameter stainless steel sphere mounted on a stainless steel shaft. T h e sphere and lead wire were insulated from the shaft by a glass sleeve. Current flowing between the probe and ground was assumed to be proportional to the particle mass flux. Further studies of electrostatic charge effects were carried out by Min and Chao (51) and El-Saiedi (52). Kramer (42) developed a remote technique for measuring particle flux using collimated light to illuminate the flow stream. Light dispersed by particles was focused on a photomultiplier tube whose signals were tallied with an electronic counter. Figure 15 shows an

FIG. 15. Particle counter signal.

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

145

oscilloscope trace of a typical signal. A major problem with optical sensing techniques for measuring particle flux in dense particle clouds is the spatial variation in light intensity due to attenuation by the cloud. It is necessary to first measure the amount of attenuation and calibrate the particle counting system accordingly. A more direct approach to obtaining particle density information was taken by Doig and Roper (60) who used a still camera and a 3 - p e c flash to record particle concentrations. This technique worked very well for the dilute suspension of large particles (350-756~)investigated by Doig and Roper but data reduction would be an overwhelming task for dense suspensions of small particles. An accurate definition of the optical depth of field is also difficult to obtain but necessary for interpretation of the photographic records.

B. FLOWMODELDEVELOPMENT T h e equations of mass conservation and momentum for two-phase mixtures are presented with clarity in Soo’s book (15). T h e purpose of the following analysis is to develop from these equations the mathematical expressions for those field properties of the flow which are directly measurable. Seven assumptions are necessary for the development of the model:

(1)

instantaneous changes in the air density are negligible

(2) instantaneous changes in particle density are small enough that triple velocity-density correlations can be neglected (3) steady flow exists (4) the flow field is symmetrical and fully developed (5) there is no swirl (6) there is no net radial flow (7) the only body force acting on the suspension is gravity which acts in the axial direction. When these assumptions are incorporated in the momentum equation, the component forms can be written for the radial and axial directions (due to the assumptions, the angular component becomes unimportant)

146

CREIGHTON A. DEPEWAND TEDJ. KRAMER

Th e definition of the mass fraction piips and the species flux ji = - V,) enable the mass conservation equation to be separated into particle and fluid terms, and the particle phase conservation can be written

pi( Vi

2(rjpl.)/ar= 0.

(25)

Eleven unknowns are present in Eqs. (23)-(25). Flux relations are required to relate mass and momentum fluxes to the field variables. T h e previous conditions along with incompressibility and the assumption that the suspension acts like a Newtonian fluid reduces the molecular stress tensor components to the following form: Y,, = 0, Y,, = 0, Y,, = -ps(8V,,/8r) and by analogy to single phase flow, it will be assumed that the Reynolds stresses are related to the mixture velocity gradient by the mixture “eddy viscosity’’ clvl .

T h e eddy viscosity is, of course, not a physical characteristic of the mixture, but its value depends on the field characteristics, and it must be calculated from experimental measurements. Solutions are sought in the later sections for the mean flow properties in the turbulent core, and an assumption which helps to make the analysis possible is the neglect of the momentum transfer on a molecular scale. If this assumption, along with Eq. (26), is incorporated into Eqs. (23) and (24), they become

I n considering the distribution of particles in a suspension, both steady diffusion forces and turbulent diffusion forces must be considered. T h e former can be described with the aid of the Nernst-Einstein equation for diffusion in liquids which introduces the concept of “mobility” A, and the latter can be described by a turbulent diffusivity Dhti which is analogous to a Fick’s law coefficient. T h e mass conservation equation for the particle phase becomes

Integration of Eq. (27) shows that the static pressure gradient is a

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

147

function of axial position only for fully developed flow. Therefore, i3p 'ax = dp,/dx and Eqs. (28) and (29) constitute a system of equations with three unknowns: pp , Vsz and p , . Solution of these equations depends on the development of expressions which relate cM and DLY to the field variables. Before proceeding with the solution of the fundamental equations, it is well to examine the influences on the particle distribution and the conditions under which a uniform density can be expected. Equation (29) expresses the balance of mass flux as produced by the diffusion driving forces and the turbulent diffusivity of particles. For the case where the diffusion driving forces are zero, a solution to Eq. (29) is dw,/dr = 0 or w , = const. Thus, unless a net diffusion driving force exists within the flow field, the particle phase will be uniformly distributed. T h e radial forces which may exist to give rise to a nonuniform distribution are: (1) the force due to an electric field on a charged particle (2) the force due to the static pressure gradient in the radial direction (3) the lift force on a particle in shear flow. Forces (2) and (3) act inwardly toward the centerline while the charged particle force acts outwardly. T h e charge on a suspended particle and its magnitude can only be estimated, but some charge undoubtedly exists due to friction with the air and interactions with the tube wall. So0 et al. (67) have reported values of charge per unit mass for 52p glass spheres ranging from 0.4 to 0.5 x C/kg. Under these conditions a charged particle in a 6-mm radius tube carrying a suspension with a mass loading of 5 would experience a maximum electrostatic force of about 1.75 x lop4 N/kg. T h e pressure gradient force can be estimated using Laufer's work (32) for values of turbulent intensity. N kg for a 6-mm radius This force is calculated t o be about 2 x tube and an air Reynolds number of 30,000 (pressure gradient is proportional to the gradient of the radial component of turbulent intensity). T h e structure of the turbulent flow stream would, of course, be altered by the presence of a large number of particles, but the conclusion that the pressure gradient force predominates is probably valid for light loadings. Analysis of the lift force in a shear flow is even more obscure than the others, but Eichhorn and Small (68) have developed an expression for the specific lift force in Poiseuille flow. Using their equation, it was found that in the vicinity of the wall the lift force is an order of magnitude less than the pressure gradient force for 62p particles but about the same for the 200p size. Based on the above estimates of force strength, it will be assumed

148

CREIGHTON A. DEPEWAND TEDJ. KRAMER

that the predominant diffusion force is due to the radial pressure gradient, and an expression for the particle concentration distribution in terms of the turbulent particle diffusivity will be developed. Assuming that Stoke’s law provides adequate description of the drag force on the particle when considering the steady driving force, the “mobility” of particles in air can be written as A

=

(30)

1/3.rrdp.

Substitution into the mass conservation equation yields

Integrating once and noting that the constant of integration must be zero gives

and further integration yields

Thus, the particle concentration ratio depends on the coefficient dzpso1/18pand the ratio of the diffusion force per unit mass to the

turbulent diffusivity of particles in air. Some representative values of the coefficient are 9.84 for 200p and 0.943 for 62p glass particles. It can be noted that if the function

were independent of particle size, large particle suspensions would have larger concentration gradients. An expression for the radial pressure gradient force can be obtained with the aid of Laufer’s (32) work, and substitution gives WP w

PO

=

exp

(-

1r

2 x 10-4pgd2(Vx)0 9PR

O

dr.

(34)

Pg

This expression shows that the concentration is a maximum at the tube center, and particle distribution is dependent on the turbulent diffusivity of particles in air Dk;. An approximate expression can be developed

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

149

which relates 0;;to the turbulent diffusivity for air alone. T h e analysis is based on the instantaneous equation of motion in the radial direction = F(U& -

dU&/dt

U&),

(35)

and the assumption that the radial relative velocity is small such that Stokes' drag law provides an adequate description. Uip is the instantaneous value of a fluctuating-random varying velocity, and the solution of Eq. (35) requires some description of it. It will be assumed that the Lagrangian velocity of air is represented by a single turbulent frequency y and U&. = A sin{y(t

-t to)},

(36)

the velocity encountered by a particle, is

Ui,= A sin{yt, + Pot},

(37)

where Po = 2 ~ { [ Uir)2]1/2 ( + V,,}/Ao and A, is the characteristic wavelength of fluctuation of a fluid particle. Upon integration of Eq. (35) and calculation of the average value,

For long diffusion times, the coefficient of eddy diffusion is defined as -

(t) Dg =

r2 ~

2t

,

( U')Z A, ~

=

(39)

and the last two terms of Eq. (38) are small compared to unity. Thus, the ratio of particle and air diffusivity can be expressed as

D;;

F2

-- -

P

I

F2

g

ALP

+ Poz A,,

T h e Lagrangian integral time scale A,, integral length scale by

.

, is related to the Lagrangian

~

A,,

=

4g/[(~Jz11~z?

(41)

and the particle time scale is ALP

= AL,/([(11/2

+

VS1iP,J.

(42)

CREIGHTON A. DEPEWAND TEDJ. KRAMER

150

Substitution yields

where S,

=

1

+ T/sli,,,/[(U~J2]1/2

K

and

-

=

2[(U;r)2]1/2/F/1Lg

Note that SRaccounts for the particle slip in the axial direction, and with S , = 1.0 comparison can be made to Peskins' work (69) where no mean slip was allowed. Figure 16 shows the relationship between diffusivity ratio and K while Fig. 17 shows the influence of S , on the

MEAN S L I P VELOCITY ASSUMED TO BE ZERO

',

\ 5.0 -

K

10-

a5 -

0.I 0

I

a2

I 04

I

06

I

aa

1

10

DlFFUSlVlTY RATIO, DZ/$

FIG. 16. Relationship between the diffusivity ratio and the parameter K.

diffusivity ratio. T h e results of Fig. 17 indicate that increased particle slip results in a decreased diffusivity ratio. T h e variation of the diffusivity ratio across the flow channel is shown in Fig. 18 for a particular set of conditions occurring in the experimental investigations by Kramer (42). T h e diffusivity reaches a maximum where the slip velocity is zero ( y / R = 0.16). Particle concentration profiles were calculated by Kramer (42) from Eq. (34) using experimentally measured data. T h e results

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

151

DIFFUSIVITY RATIO , D~/D;''

FIG. 17. Influence of slip parameter on diffusivity ratio.

Re = 24,500 R 025 in d = 62p 2

1 OD

I 01

I

I

I

I

02

03

04

05

DIMENSIONLESS DISTANCE , y/2R

FIG.18. Predicted variation of diffusivity ratio across flow channel.

are presented in Fig. 19. Fair agreement between calculated and measured particle concentration is shown. T h e turbulent diffusivity of air only Dp) was considered to be a free variable in the calculations.

152

CREIGHTON A. DEPEWAND TEDJ. KRAMER 10-

0.9 -

3 0 I-

2

ae-

M:

07-

2

Re = 24,500 R 0.25 in d = 62p

0 0.6k-

Fz w

M:

a5-

Y

00

a4-

EXPERIMENT CALCULATED

5 0.1

0.2

0.3

0.4

0.5

0

DIMENSIONLESS DISTANCE , y/2R

FIG. 19. Comparison of predicted and measured particle concentration ratios.

Although Eq. (34) shows that the radial concentration of particles can be nonuniform under conditions of low turbulent diffusivity and high mean velocity, the solution of the suspension momentum equation is developed for uniform distribution. Justification for this assumption is based on experimental results which show that, for a large range of conditions, the assumption is good. Additionally, the accuracy of the particle diffusivity approximation outlined above and the lack of knowledge about DLt’ do not warrant a more rigorous approach. T h e shear stress due to molecular momentum transport can be neglected, and it can be shown that the Reynolds stresses due to the cross-correlation terms, V&.Vbz and VL,Vg,, can be neglected when compared to V&.V,& and Vb,Vkr . Consequently, the suspension turbulent shear stress can be written as the linear combination of the stress due to particles and the stress due to the carrier gas

Also, following Prandtl’s development, the suspension turbulent shear stress can be related to the mixture velocity gradient using a mixture mixing length. T h e model requires the following assumptions: (1) T h e random components of particle velocity, Vbx and Vkr , are approximately equal.

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

153

(2) Only drag and gravity forces act on the particle in the axial direction. ( 3 ) Particle collisions can be neglected. T h e mean free path for momentum transfer is thus dependent on particle interaction with the mean air flow field. (4) T h e flow field is two dimensional. With these assumptions, the equation of motion for a particle which is initially moving in plane 1 at a velocity (VPJl and which moves radially a distance 1, to plane 2 is mP

d

VPx)1

+Gxl

=

m,F

[(v,x)l + 2 d, vgxT

-

(VPX)l -

GX] - mP g, (45)

where it has been assumed that the velocity profiles are approximately linear over the distance lg . Vbx is the change in particle velocity due to a change in the air velocity 1, dVg,.dx. If the motion in plane 1 is steady, (dV,,/dt), = 0 and where

Also if T is the time required by the particle to move a distance 1, at a velocity of Vhp, Eq. (45) can be integrated for the initial condition Vbr = 0 when t = 0 to read

Using the first assumption above and A, becomes

=

-(dVgZ/dr)-l, Eq. (47)

I t is interesting to note that for very large F , T approaches A, . Equation (48) constitutes a relationship between the characteristic response time T and the air velocity gradient (A,)-l as a function of the momentum transfer relaxation time F-l. If FA, < 0.368, FT < 1.0. For the range of variables in most experiments, this condition is satisfied. Expansion of the exponential and elimination of the higher-order terms results in an approximate solution hvF m ( F T ) 2 / 2 !

(49)

154

CREIGHTON A. DEPEWAND TEDJ . KRAMER

T h e particle mixing length is obtained as follows:

and the particle eddy diffusivity becomes

T h e effective Reynolds stress is dVgX dr

1%

+p dr

e 1l , PS

and the eddy diffusivity for the suspension becomes

dV,, dr

Equation (48) shows that large particles (which have small values of F ) have large T . Hence, the Reynolds stress for suspensions should decrease with increasing particle size. Further, since X i 1 increases in moving away from the centerline to the wall, T-1 becomes larger and the particle phase contribution to the effective Reynolds stress should increase. Introducing the eddy diffusivity for particles and air and rearranging Eq. (54) gives

Since dVp,/dr (dVgz/dr)-' rn hv/T,

Eq. (56) can be written

With K , defined such that I, I, is related to y by

=

Is

K,I, and K , defined such that I, =

K2K,y.

=

K,y, (57)

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

155

T h e momentum equation (28) can be written for the turbulent core where molecular momentum transfer can be neglected

For a given particle drag law and air velocity gradient, it is possible to calculate the ratio A,/T with Eq. (48). As a first approximation it is not unreasonable to assume that the air velocity is undisturbed for dilute suspensions; however, the experimental data can be used for determination of the continuous phase velocity gradient. For two extreme cases, solutions can be found:

(1) large particles and large gradients I / T 2m F/2hv;

(2) small particles and small gradients

T 2 hv2. For this latter case, K , becomes a function of only the particle distribution, and it is constant if pp is uniform. Integration of (58) with the assumption that the density Kl and K , are constant, yields

+

Vs-l-= Iny+

+ 7,

(59)

where

and -q = const. One objective of Kramer's research (42) was to determine the constants Kl and K 2 . An estimate of K , can be made without resort to experiment by calculating K , with A, the clear air value and T from Eq.(48). However, the remaining unknown Kl must be calculated from the experimentally determined values of 4 and the estimated values of K , . T h e results of these calculations are presented in Section II1,C. T h e friction factor, which is a readily verifiable parameter, follows from the definition, f = 2Yw/psVs,, f-1/2

=

(1

+ M ) l / ,{ 1.625 + log[Rcf

ll2

(1

+ M ) 3 / 2 ]- 1.7954 + 0.7077). (60)

156

CREIGHTON A. DEPEWAND TEDJ. KRAMER

C. MEASURED FLOWPROPERTIES A N D CORRELATIONS Over the years, a number of experimental investigations have been conducted with the purpose of adding to the general knowledge of the characteristics of suspension flows. These investigations, while contributing to the understanding of certain aspects of suspension flow mechanics, were restricted in scope and did not present a complete picture. Many of the investigations dealt with a single facet of the problem such as the pressure drop studies of Rose and Barnacle (54) and Mehta (70), the concentration measurements of So0 et al. (67), and the turbulence diffusion measurements of Kada and Hanratty (71). Other experimental investigations such as those of Doig and Roper (60) and McCarthy and Olson (61) provided a more complete picture of suspension flow characteristics for a limited range of conditions. Doig and Roper measured velocity profiles of both gaseous and particulate phases as well as particle density profiles and static pressure gradients. Measurements were made in a vertical 43-mm diameter tube carrying suspensions of 300 and 750p diameter glass spheres. Air Reynolds numbers ranged from 12 x loJ to 44 x lo3 and particle mass loading ratios varied between 0 and 5.0. McCarthy and Olson obtained measurements similar to Doig’s and Roper’s for gas Reynolds numbers between lo5 and lo6 and particle mass loading ratios less than 1.0. Measurements were taken in a horizontal 25-mm diameter tube carrying suspensions of air and 60p glass spheres. T h e experimental study by Kramer (42) was the first to provide enough information for the determination of suspension eddy diffusivity from measured point mean flow properties. Particle and air velocity profiles, particle mass flux profiles, and static pressure gradient were measured in vertical circular ducts of 12.7-, 19.1-, and 25.4-mm diameters. Gas Reynolds numbers ranged from 5670 to 50,000 and suspensions of air and 62 and 20Op particles were studied with particle loading ratios ranging from 0 to 5. T h e experimental flow system utilized high pressure plant air at a pressure of about 4.4 atm. Air entering the system passed through a regulator and fluid and particulate filters. T h e air then passed through a laminar flow element, used to measure the gross air flow rate, and into a plenum chamber. After leaving the plenum chamber, the air passed under a feed tank where solid particles were added at the throat of a controllable converging-diverging section which made the flow symmetrical and enhanced flow development. T h e bulk of the solid particles were removed from the flow and returned to a storage hopper by a double-effect cyclone separator. T h e remainder was separated by

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

157

a cloth filter before the air exhausted to the room. T h e gross particle mass flow rate was determined by continuously monitoring the weight of the solids feed tank which was mounted on a platform scale. T h e scale arm was restrained by a transducer which was monitored by a strip chart recorder. T h e recorder response adjusted to 200pV/kg, producing a sensitivity to weight change of 0.05 kg. Flow rates determined in this way were accurate to a few percent. T h e vertical channels which served as test sections were fabricated from 3-m lengths of precision bore borosilicate glass tubing with static pressure taps abrasively drilled through the tube walls. Local velocity was obtained directly by imposing the total pressure from a traversing pitot tube and static pressure from an adjacent wall tap on opposite sides of a capacitance pressure transducer. Pitot tubes were fabricated as discussed in Section II1,A. T h e existence of a radial static pressure gradient introduced a systematic error of about 3 % in the dynamic pressure data. T h e “optical cross-correlation” technique and streaking camera were used to measure particle velocities. Particle flux was determined by counting the number of particles passing through a control volume per unit time as described in II1,A. Static pressure gradients were measured at wall taps spaced at 0.254-m intervals. T h e pressure differential between taps was detected with a capacitance transducer. Analytical solutions for universal suspension velocity profiles were presented in II1,B. These profiles were developed with the assumption that the suspension density is uniform, and it is well to examine this assumption in the light of measurements. Figures 20 and 21 are samples

”r “._ . C.

In

3

m

..

1

‘

Re=24,500 d = 62p R = 0.5in

ai

02

A

1

03

04

1

05

RELATIVE DISTANCE FROM WALL, y/2R

FIG.20. Suspension density profiles-62p particles,

I58

CREIGHTON A. DEPEWAND TEDJ. KRAMER Re = 24,500 d = 200p R = 051n

I

0

I 0.1

I

I

I

I

02

03

04

0.5

RELATIVE DISTANCE FROM WALL, y/2R

FIG. 21.

Suspension density profiles-2OOp

particles.

of the density distribution data. For all cases, the 62p particle density was a maximum at the centerline. This was also true for the 20011. size at low Reynolds numbers, but as the Reynolds number increased, the distribution flattened. At Re = 18,000 the profiles were almost flat for all loadings, and at Re = 24,500, as shown in Fig. 21, the concentration was higher at the wall. Particularly at the low loadings, the variation in concentration was modest, and the assumption of uniform density is warranted for the purposes of analysis. T h e analysis of suspension flows shows that the mass average local suspension velocity profile is logarithmic, if the particle density is uniform, such that the velocity V,+ is given by Vs+ = 4 lnys+

+ 'I,

(61)

where and Vs+= V,/(T~/~~)'/~

ys+ = y( V*/va)(1

+ M).

Values of V,+ and ys+ were calculated from the measurement of air and particle velocity, particle flux, and wall pressure gradient. T h e results are plotted in Figs. 22 and 23. I n preparing the universal velocity plots, it was observed that the data for the 62p size scattered considerably for flows having greatly different values of the suspension Froude number. Further investigation showed that plots of In V,+ vs. In F r were straight lines dependent only on M , and that the slope of the lines thus plotted was equal to O.O4(M - 2). Figure 23 shows that the scatter was reduced to an acceptable level by including the Froude

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

0 Re. 5.670 R=Q25in.

0

A Re-0.100

LI

R=Q25in. Re*11.900 R*Q25In. v Re = 18.600 R=QWin. 0 Re=18.400 R=Q3751n.

159

Re = 24.500 R *amin. Re = 24.500 R = 0.25 in.

0

0

5

0

FIG. 22.

I00

500

lpoo

5poo

Universal velocity profiles-2OOp particles.

+>"

5t

0

5

I

I

I

I

0

100

500

I.Oo0

I

5,000

FIG. 23. Universal velocity profiles-62p particles.

number dependence in Vs+.T h e data in the velocity plots are subject to experimental error in both coordinates, and the classical method of fitting straight line functions to data is inadequate. Instead, the confluence analysis of Wald (72) was used to calculate the constants in Eq. (61), and these are presented in Figs. 24 and 25 along with the

160

CREIGHTON A. DEPEWAND TEDJ. KRAMER

0

62p PARTICLE

SUSPENSION o 200p PARTICLE SUSPENSION

cn

I

I

.o

0

I

I

I

2.0

3.0

I

I 4.0

5.0

MASS LOADING RATIO, M

FIG.24. Influence of mass loading ratio on the slope of the logarithmic velocity profile.

SUSPENSION

PARTICLE SUSPENSION

95% CONFIDENCE LIMITS ON EACH POINT

0

I

I

I

1

I

I

10

20

30

40

50

MASS LOADING RATIO. M

FIG. 25. Influence of mass loading ratio on the constant 7.

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

161

95% confidence limits. It must be remembered that the dependent variables for the two particle sizes are different and that a direct comparison is not possible. T h e universal velocity profiles lead directly to an expression for the friction factor as in the case of single phase flow. Using Eq. (61) and the definition of the friction factor

f = 2 ~ w / P s ~,~ I n it can be easily shown that the smooth pipe resistance low is f-lI2 =

p( 1 + M)Il2{ 1.6254 log[Ref112(1 + M ) 3 / 2 ] 1.7954 + 0.7077},

where

p=1

for

d

=

200p

p

for

d

=

62p.

= Fy0.04(M-2)

(62)

Comparison was made with the previous experiments of Depew (29) and McCarthy and Olson (61) using values of and 7 from Figs. 24 and 25. Figure 26 shows good agreement with the data even though

+

0 McCARTHY [61] d =62p Re > lo5 DEPEWC191 d = K O / L Re= 13,500 DEPEWCI91 d: 30p Rez13.500 o DEPEWC191 d = 3 0 p Re=27,400

v o

EQUATION (62)

0

10

20

30

40

50

MASS LOADING RATIO, M

FIG.26.

Comparison of experimental and theoretical friction factors.

the particle size and Reynolds numbers were outside the range of the present experiments.

I62

CREIGHTON A. DEPEWAND TEDJ. KRAMER

T h e mean gas phase velocity distribution depends on Kl as shown in the previous section. These values are not easily obtained from the experimental results, but the range of Kl and K , can be ascertained from the suspension velocity profiles. T h e analysis of the preceding section showed that the slope 4 is given by )

but K , was also shown to be related to the properties of the flow

Assuming that the air velocity profile can be approximated by

vg= Vg0(y/R)1/6.3, the average velocity gradient over the turbulent core is given by IjXv

=

1.726 Vg,/R.

(65)

Using Eq. (47) it is possible to estimate the range of Av/T and to calculate the corresponding values of K , (assuming pp is uniform).

I '

0.7

0

I

2

3

4

MASS LOADING RATIO, M

FIG.27. K 2 vs. mass loading ratio.

5

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

163

T h e results of the calculation are shown in Fig. 27, and K , , calculated from Eq. (63) using mean values of K 2 is shown in Fig. 28.

-

1.2

@K,

-

08

-

t ao

21) MASS LOADING RATIO,

4.0

M

FIG. 28. ,!3K, vs. mass loading ratio.

Suspension eddy viscosity is related to mean flow variables by the expression YS EM = (66) p s dVsz/dr ’ T h e variable Y , is the local suspension shear stress and is related to the static pressure gradient and suspension density by the axial equation of momentum conservation

Point measurements of suspension velocity and density along with measured static pressure gradients can be used in the above expression to calculate suspension eddy viscosity. Figure 29 presents dimensionless eddy viscosity profiles calculated from Kramer’s (42)point measurements made in circular flow channels. T h e general trend is for dimensionless eddy viscosity to increase with mass loading ratio and for the maximum value to shift toward the channel centerline with increasing mass loading ratio. These trends arise from the characteristic flattening of suspension velocity profiles as mass loading ratio is increased. Table I summarizes suspension eddy viscosities calculated from Kramer’s measurements.

TABLE I SUSPENSION DIMENSIONLESS EDDYVISCOSITY Particle Air Channel Mass diameter Reynolds diameter loading (p) number (in.) ratio 200

YID 0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

5,670

0.50

1 2 3 4 5

0.0259 0.0368 0.0458 0.0524 0.0566

0.1108 0.1574 0.1693 0.1624 0.1672

0.1543 0.2290 0.2508 0.2417 0.2627

0.1853 0.2451 0.2954 0.3270 0.3444

0.1768 0.2075 0.2764 0.3725 0.3691

0.1580 0.1956 0.2671 0.3694 0.3956

0.1203 0.1758 0.2521 0.3511 0.4479

0.0702 0.1250 0.2142 0.3622 0.5783

0.0308 0.0698 0.1660 0.5276 -

10,100

0.50

1 2 3 4 5

0.0192 0.0265 0.0337 0.0411 0.0482

0.1467 0.1963 0.2316 0.2617 0.2871

0.1422 0.1841 0.2159 0.2419 0.2625

0.1433 0.1882 0.2241 0.2532 0.2758

0.1461 0.2019 0.2485 0.2888 0.3218

0.1456 0.1975 0.2430 0.2864 0.3330

0.1401 0.1829 0.2232 0.2645 0.3201

0.1311 0.1816 0.2359 0.2906 0.3614

0.0698 0.1121 0.1869 0.2960 0.4680

1 2 3 4 5

0.0167 0.0235 0.0323 0.0431 0.0562

0.1400 0.2081 0.2699 0.3261 0.3781

0.1404 0.2024 0.2602 0.3140 0.3655

0.1468 0.2241 0.3101 0.3987 0.4948

0.1657 0.2845 0.4478 0.6511 0.9219

0.1733 0.2996 0.4349 0.5937 0.7109

0.1752 0.2848 0.3506 0.4108 0.4345

0.1895 0.3191 0.3890 0.4413 0.4737

0.0748 0.1366 0.2409 0.3986 0.8957

1 2 3 4 5

0.0228 0.0337 0.0440 0.0545 0.0651

0.0845 0.1137 0.1289 0.1683 0.2137

0.0952 0.1316 0.1543 0.2060 0.2735

0.0970 0.1248 0.1624 0.2317 0.3096

0.0903 0.1235 0.1735 0.2795 0.4058

0.0818 0.1367 0.2747 0.4791 1.0531

0.0766 0.1357 0.7527 2.0920 5.8645

0.0691 0.1245 0.7488 2.7775

0.0553 0.0979 0.5996 2.8938

-

-

1 2 3 4 5

0.0215 0.0322 0.0426 0.0523 0.0613

0.0669 0.0959 0.1235 0.1482 0.1721

0.0966 0.1297 0.1616 0.1920 0.2213

0.0980 0.1344 0.1723 0.2122 0.2518

0.1004 0.1481 0.2011 0.2644 0.3317

0.1162 0.1840 0.2551 0.3608 0.4933

0.1480 0.2623 0.3649 0.5658 0.8848

0.1831 0.3758 0.5905 0.9199 1.4918

0.0867 0.1435 0.2455 0.3157 0.4112

11,900

18,600

18,400

0.50

1.0

0.75

m +d

U

5

m Xl

200

62

18,600

0.50

1 2 3 4 5

0.0179 0.0254 0.0334 0.0420 0.0513

0.1594 0.2392 0.3077 0.3701 0.4207

0.1391 0.2027 0.2608 0.3126 0.3581

0.1212 0.1716 0.2188 0.261 1 0.2986

0.1118 0.1554 0.1955 0.2315 0.2625

0.1178 0.1620 0.2040 0.2392 0.2709

0.1364 0.1870 0.2333 0.2673 0.300

0.1421 0.1971 0.2309 0.2547 0.2743

0.0648 0.0917 0.1080 0.1254 0.1380

24,500

1.o

1 2 3 4 5

0.0242 0.0350 0.0441 0.0520 0.0601

0.0948 0.1158 0.1234 0.1403 0.1530

0.1190 0.1561 0.1763 0.2078 0.2280

0.1240 0.1669 0.2233 0.2712 0.2925

0.1243 0.1699 0.2491 0.3316 0.3931

0.1205 0.1593 0.2446 0.3932 0.6343

0.1129 0.1458 0.2584 0.4531 0.9795

0.0952 0.1220 0.2375 0.4412 1.3386

0.0526 0.0845 0.1007 0.3951 1.2165

24,500

0.50

1 2 3 4 5

0.0200 0.0295 0.0392 0.0490 0.0588

0.1478 0.2382 0.3282 0.41 87 0.5077

0.1255 0.1927 0.2499 0.2991 0.3445

0.1193 0.1626 0.1990 0.2266 0.2493

0.1168 0.1440 0.1709 0.1898 0.2029

0.1315 0.1688 0.2075 0.2341 0.2498

0.1575 0.2459 0.3486 0.4326 0.4813

0.1430 0.2381 0.3567 0.4638 0.5455

0.0554 0.0850 0.1210 0.1574 0.1991

10,100

0.50

1 2 3 4 5

0.0127 0.0141 0.01 74 0.021 1 0.0255

0.0633 0.0697 0.1021 0.1 354 0.2020

0.0866 0.1242 0.1846 0.2479 0.4945

0.1100 0.2036 0.2528 0.3104 0.5508

0.1352 0.3310 0.3176 0.3413 0.43 16

0.1595 0.5775 0.4625 0.4080 0.4398

0.1756 0.9101 0.7200 0.4819 0.4535

0.1664 0.4486 0.41 18 0.4482 0.3738

0.1084 0.1360 0.1334 0.2934 0.2216

1 2 3 4 5

0.01 77 0.0235 0.0287 0.0355 0.0414

0.0409 0.0450 0.0544 0.0759 0.1117

0.0500 0.0582 0.0765 0.1109 0.2127

0.0735 0.0798 0.1262 0.1834 0.3387

0.0734 0.1 182 0.2191 0.3153 0.4452

0.0555 0.1609 0.2799 0.4735 0.5552

0.0853 0.2016 0.3681 0.7769 0.6451

0.0940 0.2221 0.6247 1.1506 0.7759

0.0666 0.1579

1 2 3 4 5

0.01 3 1 0.0137 0.0161 0.0186 0.0210

0.0795 0.0842 0.1115 0.1346 0.1556

0.1042 0.1856 0.2776 0.3326 0.3447

0.1003 0.1821 0.2684 0.3090 0.2997

0.0830 0.1242 0.1781 0.2026 0.2014

0.0940 0.1481 0.2081 0.2263 0.2306

0.1346 0.2468 0.3383 0.3409 0.3748

0.1515 0.2494 0.3630 0.4612 0.6034

0.1343 0.1550 0.2645 1.1601 -

11,900

62

11,900

1.o

0.50

-

-

(continued)

TABLE I (continued)

Particle Air Channel Mass diameter Reynolds diameter loading (p) number (in.) ratio 18,600

1.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.192 0.0264 0.0317 0.0377 0.0422

0.0450 0.0544 0.0794 0.0928 0.1004

0.0526 0.0656 0.0962 0.1139 0.1388

0.0732 0.1092 0.1220 0.1737 0.2180

0.1004 0.2342 0.2424 0.3659 0.3701

0.1156 0.3468 0.8688 0.9071 0.9461

0.1307 0.4988 3.0772 7.9151

0.1458 1.9706 2.0551

-

0.0798 1.7908 2.1932 5.7805 -

0.0176 0.021 1 0.0351 0.0287 0.0317

0.0413 0.0424 0.0486 0.0561 0.0641

0.0671 0.0835 0.1119 0.1389 0.1765

0.0781 0.1247 0.2076 0.2875 0.3973

0.0761 0.1480 0.3000 0.4493 0.5184

0.0681 0.1525 0.4250 0.6982 0.5816

0.0573 0.1394 0.4481 0.6674 0.4359

0.0539 0.1727 0.6586 0.9684 0.5282

0.0662 0.6771 2.4637 -

0.0816 0.0971 0.1189 0.1383 0.1548

0.0986 0.1250 0.1662 0.2085 0.3246

0.1158 0.1595 0.2391 0.3027 0.4678

0.1259 0.2008 0.3677 0.4636 0.4690

0.1367 0.2560 0.4388 0.5965 0.6331

0.1435 0.3169 0.4273 0.6193 1.0182

0.1367 0.3363 0.5413 0.7967 1.8623

0.1088 0.3343

5

0.0145 0.0161 0.0157 0.0137 0.0103

1 2 3 4 5

0.01 84 0.0246 0.0314 0.0426 0.0448

0.0595 0.07 11 0.0836 0.0383 0.1276

0.0638 0.0873 0.1076 0.0942 0.2719

0.0455 0.1023 0.1401 0.1308 0.5815

0.0695 0.1275 0.2927 0.2627 1.6593

0.1584 0.8889 1.5704 -

0.0754 0.1840 1.4412 -

0.0751 0.2508 -

-

-

0.0787 0.5773 -

1 2 3 4 5

0.0130 0.0170 0.0226 0.0296 0.0381

0.0898 0.1098 0.1313 0.1483 0.1641

0.0938 0.1270 0.1570 0.1719 0.1776

0.111 0.1551 0.1944 0.2140 0.2181

0.1350 0.1926 0.2460 0.2773 0.2899

0.1504 0.2454 0.2460 0.3605 0.3676

0.1495 0.3050 0.4620 0.4872 0.4667

0.1358 0.2963 0.5300 0.6596 0.6810

0.0883 0.1968 0.4610 0.3973 -

1 2 3 4 5

18,400

0.75

1 2 3 4 5

18,600

24,500

62

24,500

0.50

1.00

0.50

YID

1 2 3 4

-

-

-

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

167

0.8-

-

0.7-

Re = 24,000 R: 051n

>

F

w‘

I

0.6-

>

t v)

8IL,

0.5 -

>

0.4 -

W v,

0.3 -

& n v, W

_J

6

02-

v,

z

W

0.1

-

D 0

0I

I 0.2

I

I

I

03

0.4

0.5

DIMENSIONLESS DISTANCE , y/2R

FIG. 29. Eddy viscosity variation over the flow channel.

IV. Analysis

Researchers have used many techniques in the analysis of gas-solids suspension heat transfer. Boothroyd’s work (43, 44) is an exarnple of dimensional analysis; Brandon (26) and Briller (23) have derived “figures of merit” based on the mechanism of interaction between the particles and the fluid; the analogy between wall shear stress and wall heat transfer is quite often used as exemplified in recent analytical work by Gorbis (73). These techniques have all been more or less successful over a limited range of variables. They involve assumptions about the fundamental mechanisms that require a high degree of understanding of those same mechanisms. Presumably, a surer path would be to integrate the fundamental partial differential equations which express conservation of mass, the force-momentum balance, and the conservation of energy. Tien (17, 74) derived the energy relationships and discussed the statistical nature of the flow fluid. These equations were much simplified in order to produce a set which was mathematically tractable. Some of the more important assumptions are as follows:

(1) Fluid properties are constant. (2) Radiation effects are negligible.

168

CREIGHTON A. DEPEWAND TEDJ. KRAMER

(3) T h e temperature of an individual particle is essentially uniform. (4) T h e slip velocity based on the mean particle and mean fluid (5)

(6) (7) (8)

velocities is zero. Solid particles are uniformly distributed over the pipe cross section. T h e particles have negligible effect on both the mean and fluctuating components of the continuous phase motion. Viscous dissipation is negligible. T h e particle eddy mixing term is negligible compared to the fluid eddy mixing term.

T h e equations which express the balance of energy transfer for each phase thus produced are

T h e solution for the isothermal wall boundary condition is made possible by assuming that

where An are eigenvalues. This condition is satisfied most readily by small particles. Tien notes that the ordinary differential equation which results from the separation of variables is identical to the equation which governs the single phase flow case. Under these conditions, the asymptotic Nusselt number is the same for both cases, single and two phase, when the Reynolds number is evaluated for the clear gas properties. T h e influence of the particles occurs in the prolongation of the thermal entrance region. T h e axial local heat transfer coefficients depend on the heat capacity loading ratio M C and enhancement of the average heat transfer coefficient is due to delay in the development of similar temperature profiles which are manifested in terms of uniform h,, . Tien computed average values for the heat transfer coefficient for Farbar and Morley’s experimental conditions, and these compare well with the data for very light loadings. Agreement for very dilute mixtures is anticipated since the prerequisite assumptions on the basic equations are possibly valid for this condition.

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

169

A more critical test of the theory is made when h, for theory and experiment are compared. Depew’s experiments were conducted with a tube having uniform heat flux to facilitate determination of h, ; therefore it was necessary to solve Tien’s equations for this wallboundary condition. T h e solution proceeds along the lines suggested by Sparrow, Hallman, and Siege1 (75) for single phase flow where the dimensionless temperature function is considered to consist of two additive parts: 0, is the fully developed solution that is attained far down the pipe from the onset of heating, and 0, is the deviation from similar profiles which accounts for entrance effects during the development region. Separation of the variables takes place by assuming a solution in product form, and ordinary differential equations for 0, and 0, for the axial and radial directions result. As in Tien’s case, an assumption regarding the magnitude of the particle and fluid properties was necessary in order to reduce the problem to a Sturm-Liouville system. For the system of glass particles transported in air, it is required that Nu,dP Bn2(V/Vm)lo5, where B, are eigenvalues. Depew (19) has shown that this is only approximately valid for 30p spheres and Re < 30,000 for the first eigenvalue and that it is invalid for 200p spheres away from the pipe wall. T h e above condition can also be interpreted as requiring that both phases must be at about the same temperature. A solution for Nu, is obtained within the limitations of the gbove restrictions, and the local Nusselt number is given as

>>

NuS(x) =

[Nug(X+)]-l

+

1 ’ $(@gm - @sm),

’

where

and where Nu,e is to b evaluated at the pseudolocation X+. T h term is the difference between the dimensionless mean temperatures for the gas phase and for the suspension far from the inlet, and it was shown that if the temperature profiles for each phase are similar

(egm O,,),

T h e factor ib?,~,/(ib?~c~+ ib?PcP)zis zero at both high and low mass loading ratios, and it reaches a maximum at 1.26 for glass spheres. T h e effect is to decrease Nu, to a minimum in the region 0.5 < M < 1.5.

170

CREIGHTON A. DEPEWAND TEDJ. KRAMER

For 200p spheres the decrease is 18 yo,but for 30p particles the decrease is only 1 yo. T h e decrease which is anticipated for the 200p size was not experimentally confirmed, but a large decrease was found for the 30p size even though it was not predicted by the analysis. T h e values of Nu, for the analysis and experiment are based on the difference between the suspension average temperature and the wall temperature. Since it is likely that heat is transferred directly to the continuous phase and subsequently to the solid phase, a more physically realistic potential would be based on the mixed mean gas temperature. Also, since the solids are at a temperature which is lower than the suspension mean, the gas must be at a mean temperature which is higher than the mixture. T h e consequence of this relationship is that the heat transfer coefficient based on the suspension mean is always less than that which is based on the gas mean, and part of the decrease in h, that is seen experimentally can be attributed to its definition. However, it is interesting to note that-at the experimentally observed minimum h, for the 30p size-the maximum heat transfer coefficient that could exist based on Tgmis 1 4 % below the value for air alone. This maximum is determined by assuming that none of the heat from the wall is transferred to the particles. It seems certain that a change in the basic convection mechanism has taken place, and one or more of the assumptions used in the analysis apparently is invalid even at this modest loading rate. Tien invoked several assumptions in the formulation of his model which were plausible for very dilute mixtures and which were necessitated by an almost complete lack of quantitative information about the fluid dynamics of the mixture. Very little data on the fluid dynamics of suspended particle was available, and in no case had a complete characterization of the system been made by an experimenter. Furthermore, no theory of the structure of the turbulence of suspensions had been proposed, as is the case today. Research by Peskin et al., as summarized in Peskin (24), has produced a large number of significant observations, but as yet the influence of solid particles on pipe flow turbulence is not quantitatively determinable. Peskin has shown that the turbulence characteristics are altered by particles such that energy is dissipated at a higher rate at higher wave numbers and that the eddy diffusivity for momentum is reduced. T h e existing analyses are sufficient in number and accuracy to conclude that the clear gas properties will not suffice in the analysis of the heat transfer problem and that realistic fluid dynamic properties must be used. T h e work of Kramer (42) which has been introduced in the previous section produced a characterization of the flow of glass microspheres in air which is

HEATTRANSFER TO FLOWING GAS-SOLIDMIXTURES

171

adequate if the suspension is treated as a continuum. Local mean values of particle velocity, air velocity, and particle flux were determined for 12.7-, 19.0-, and 25.4-mm i.d. tubes for 62 and 200p particles. T h e experiments covered a range of Reynolds numbers from 5670 to 24,500. A primary objective of Kramer’s work was to make those measurements of the suspension flow which were necessary to carry out the analysis of the diathermal system. T h e analysis carried out by Chu (38) is based on the assumptions that the suspension can be regarded as a continuum and that the flow is fully developed. T h e assumptions are as follows: (1) (2)

(3)

(4)

(5) (6) (7)

T h e gas-solid suspension flow can be treated as a single phase continuum. T h e suspension flow field is symmetrical and fully developed. T h e fully developed hydrodynamic field is not affected by the temperature field, and the temperature field is fully developed. T h e heat flux at the pipe wall is uniform. Viscous dissipation is negligible. Axial diffusion of heat is small compared to radial diffusion. Heat transfer by radiation is negligible.

Under these conditions the energy equation for turbulent pipe flow is

a

pscsv, aT, = -1 a (rk, arTs ax r

a~

~

-p

ycsrm).

I n this expression ps is the suspension density which can be calculated from ps = p g

t- fippsolv

(73)

for all but very high loading ratios. c, is the mass average heat capacity for the suspension cs =

(pgcg

+ pso1npvcp)/ps

(74)

*

T h e suspension thermal conductivity k, is calculated using Orr’s (76) equation 2k, k, - 2X(k, - Kp) (75) ks = k g 2kg k , X ( k , - kp)

[

+

+ +

1.

where X is the volume fraction of the solid phase X = n p v p . T h e boundary conditions for the system being considered are

K, aT,/ar I ~ ==9, ~ ~aT,/ar

= 0.

(76)

172

CREIGHTON A. DEPEWAND TEDJ. KRAMER

Since the flow is fully developed, the axial temperature gradient can be related to the wall heat flux by an energy balance:

Also, since the suspension is to be treated as a continuum, it is reasonable to introduce the eddy diffusivity for heat eH Vs'Ts' =

-cH

aTs/ar.

T h e energy equation is reduced to a second-order ordinary differential equation by the introduction of the above relationships

It has been assumed in previous analyses (17,19) that V , is unchanged from the clear gas values by the addition of the particulate phase and that n, is uniform. These assumptions, undoubtedly, are among the weakest of the lot, and a major objective of Kramer's (42) research was to obviate these limitations. Kramer's results include nP(r),ps(r), and V8(r);however, it was necessary to use tabulated values in a finitedifference form of the equations since the results are not expressible in analytic form. Th e eddy diffusivity for heat in turbulent suspension flows has also been assumed to be uninfluenced by solids even though other turbulent properties have been shown to change with solids addition. I n view of the lack of experimental evidence and theoretical prediction regarding the effect of solids addition on e H , one of the primary objectives of Chu's work (38) was to determine the relationships between eH and eM . Kramer's tabulation of cM for a vertical system of 62 and 200p glass spheres suspended in air over a range of Re from 5,670 to 24,500 was available, and Chu's heat transfer research program was planned to duplicate the largest tube size (25.4-mm i.d.) used by Kramer and to include a 50.8-mm i.d. tube which could be used to test the analytic technique. T h e solution of the energy equation was carried out in finite difference form with eH = e M . T h e ratio of eddy diffusivities 01 is the remaining unknown. It is well known that a is close to unity for gases and that it takes on values of 0.2 to 0.4 for liquid metals, but 01 is unknown for suspension flows. It was therefore decided to carry out the solution with 01 as a parameter, and to pick the value of 01 which gave the best agreement with the experimental results.

HEAT TRANSFER TO FLOWING GAS-SOLIDMIXTURES

173

T h e derivatives are approximated for a variable grid size (subscripts are omitted without ambiguity)

A

= (ri

-

ri-#

-

ri)

+

-Y

~

(rj ) ~- rjPl).

With these approximations, the energy equation becomes

(83)

where r

+ dksdr f

kS

+

ks

-

G

~

d&H pScS

f

dPs

f

CSEH

pSeS

dCS cs 7 + 7 &H PS

.

P ~ C ~ C H

These equations for each of the nodal points along with the boundary conditions form a set of simultaneous equations which can be written in matrix form and solved by matrix inversion. T h e Gauss-Jordan elimination method was used, and the computation was performed on a CDC 6400 computer. With the temperature profiles obtained from the numerical analysis, the suspension mean temperature could be calculated from

0

T h e local Nusselt number is defined as

CREIGHTON A. DEPEW AND TEDJ. KRAMER

174

Nusselt numbers for clear air were calculated using the universal velocity profile and Sleicher’s eddy diffusivity model (77). The grid spacing was nonuniform with a higher density of nodal points near the wall where the flow field variables and the temperature are changing most rapidly. T h e difference approximations generally become more accurate as the number of grid points is increased, but the required computer storage and computation time increases prohibitively. T h e adequate grid spacing was determined by making trial runs with 50, 75, and 100 nodal points. I t was found that 50 points gave asymptotic Nusselt numbers that were within 5 % of Sleicher’s values and that there was negligible difference among the results with 75 points, 100 points, and Sleicher’s curve. I t was therefore concluded that an adequate solution could be obtained with 75 points. Solutions were obtained for 62 and 200p particles in air in 25.4-mm, 19.0-mm, and 12.7-mm tubes with 01 as a parameter. Reynolds numbers were chosen on the basis of available heat transfer results for these tube and particle sizes. T h e values of 01 which gave agreement with the experiments at x / D = 50 are shown in Fig. 30. Data on the 25.4-mm 10

-

-

,-. -8

-s;<

\ ‘ \

1

1

I I I I I

‘A

I

I

I

I-

-

-

-

-

2

4

(r

0’ -

3

1

?

3

=0 r

U 1,

3

-

Particle Reynolds Size Number (rnrn) (micron)

-

254

Tube Size

- A 254 - x 254 254

0 254 -

127 127 I 190 I 001 01 02

62 62 62 200 200 200 200 200 I

I

04

18,600 18,600 24,500 18,600 24,500 18,000 I 1 I Ill 06 0810

1 20

I

I

40

FIG.30. Eddy diffusivity ratio vs. mass loading ratio.

I

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

175

tube were available from the experimental phase of Chu’s study, and Rajpaul’s (39) data were used for the 19.0-mm tube. Kramer’s data on a 12.7-mm tube were used in conjunction with Rajpaul’s data on a 9.5-mm tube since there were the closest available in tube size for the particles being considered. T h e eddy diffusivity ratio is shown as a function of solids loading ratio from I to 4. T h e results for 200p particles are grouped with minor scatter about a straight line which represents the points quite well. For 62p particles at Re = 24,500 a line parallel to and above the line for 200p particles represents the points quite well, while the values for Re = 11,800and 18,600are less-well correlated by the line. T h e most obvious conclusion from Fig. 30 is that the eddy diffusivity for heat is less than the eddy diffusivity for momentum. This seems to indicate that the mechanism for thermal energy transfer by turbulent motion of the particles is weaker than for momentum transfer. I t is also clear that the ratio is always greater for the smaller particles. This is perhaps a manifestation of the larger surface to volume ratio for the smaller particles which have a faster response to changes in environment and temperature.

V. Concluding Remarks

Heat transfer to gas-solid suspension flow is far from being a unified subject. Data by various investigators indicate that heat transfer can be augmented by the addition of solid particles to a gas stream, and the results for systems of small (micron-size) particles can be correlated to within an order of magnitude. But results for larger particles indicate that reductions in heat transfer, below what is expected from increased heat capacity, have been encountered. T h e theory of suspension flows has been approached from a fundamental point of view, but when the results are complete, they apply only to very dilute concentrations and to very small particles which follow Stokes’ drag law. In this study, the suspension has been treated as a continuum, and a universal logarithmic velocity variation was predicted for that model. Values of the suspension velocity which were derived from point mean observations of particle velocity, air velocity, and particle flux form a family of curves which include a range of Reynolds numbers and which depend on the mass loading ratio. T h e constants for the logarithmic velocity profiles were calculated using confluence analysis. Friction factors which are based on the experimentally derived constants were compared with friction factors of other investigations with good agreement.

176

CREIGHTON A. DEPEW AND TEDJ. KRAMER

I n the heat transfer study, the energy equation for the suspension as a continuum was solved in finite-difference form using the analogy between suspension eddy momentum diffusivity and the eddy diffusivity for heat. T h e appropriate ratio of diffusivities was determined through a parametric study, and the results show that the diffusivity for heat is always lower than the diffusivity for momentum in the cases studied. Progress has been made in the quantitative description of suspension flows, but much research remains to be done.

ACKNOWLEDGMENTS The part of the authors' research which was conducted at the University of Washington was supported by the National Science Foundation under Grant GK-1134, by the Mechanical Engineering Department, and by the College of Engineering at the University of Washington. The authors would also like to gratefully acknowledge the contribution of Dr. Nan-Cheng Chu.

NOMENCLATURE A B* C

surface area eigenvalue specific heat ratio, cp/cg specific heat C D tube diameter D(t) turbulent diffusivity particle diameter d stopping distance ds signal voltage or see Eq. (83) E F time constant for momentum transfer .F diffusion driving force friction factor f see Eq. (83) G acceleration due to gravity g heat transfer coefficient h mass flux j see Eq. (43) K Kl defined by Eq. (57) K2 defined by Eq. (57) k thermal conductivity distance particle image travels, L Eq. (21) mixing length for momentum 1 transfer M solid to gas loading ratio mass flow rate & I

magnification factor or particle mass A? particle mobility in continuous phase Nu Nusselt number number density n PD penetration depth Prandtl number Pr pressure P heat flux 4 tube radius R Reynolds number Re radial distance r see Eq. (43) SR temperature, except in Section 111 T where T is a characteristic time time t fluctuating Lagrangian velocity U' fluctuating gas phase encountered U by a particle time-averaged velocity V V+ dimensionless velocity, V/V* randomly varying component of V' velocity V* friction velocity, [ Yw/ps]1/2 volume V defined with Eq. (70) X'

m

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES distance in flow direction distance from the wall dimensionless distance from the M)/va wall, yV*(l eddy diffusivity ratio, eH/cM defined with Eq. (62) see Eq. (37) defined with Eq. (69) defined with Eq. (68) turbulence frequency mirror separation or see Eq. (82) laminar sublayer thickness for clear gas eddy diffusivity for heat eddy diffusivity for momentum constant defined by Eq. (59) or Kolmogoroff microscale dimensionless temperature time Lagrangian integral time scale eigenvalue wavelength inverse of the absolute value of the gas phase velocity gradient Lagrangian integral length scale Eulerian integral length scale

+

p Y

p

Y

+ I/J

w

177

viscosity or microns dynamic viscosity mass density shear stress constant defined by Eq. (49) angular coordinate concentration

SUBSCRIPTS a g i j m 0

P S

sol V W

X

air gas node node area mean value tube centerline particle suspension pertaining to solid material vortex generators as used in Eq. (18) wall value local

SUPERSCRIPTS (t)

turbulent

REFERENCES 1. L. Farbar and M. J. Morley, Ind. Eng. Chem. 49, 1143 (1957). 2. Babcock and Wilcox Co., Gas-Suspension Coolant Project. Final Rep. BAW-1159 (1959). 3. Babcock and Wilcox Co., Gas-Suspension Task 111. Final Rep. BAW-1201 (1960). 4. Babcock and Wilcox Co., Supplement to the Gas-Suspension Task 111. Final Rep. BAW-1207 (1960). 5. D. Schuderberg, R. Whitelaw, and R. Carlson, Nucleonics 19, 67 (1961). 6. G. P. Wachtell, J. P. Waggener, and W. H. Steigelmann, U. S. At. Energy Comm. Rep. No. NVO-9672 (1961). I . R. Pfeffer, S. Rossetti, and S. Lieblein, N A S A Tech. Note NASA TN D-3603 (1966). 8. W.T.Abel, J. P. O'Leary, D. E. Bluman, and J. P. McGee, U . S. Bur. Mines, Rep. Invest. RI-6255 (1963). 9. W. T. Abel, D. E. Bluman, and J. P. O'Leary, Amer. SOC.Mech. Eng. ASMEPaper NO.63-WA-210 (1963). 9a. D. E. Bluman, A. F. Galli, N. H. Coates, J. D. Spencer, and C. N. Rosenecker, U . S. Bur. Mines, Rep. Inwest. RI-7057(1967). 9b. C. N. Rosenecker, N. H. Coates, and H. G. Lucas, U.S. Bur. Mines, Rep. Inwest. RI-7019 (1967). 10. 2. R. Gorbis and R. A. Bakhtiozin, Sov. J . A t . Energy 12, 402 (1962).

178

CREIGHTON A. DEPEW AND TEDJ. KRAMER

1 1 . V. S. Nosov and N. I. Syromyatnikov, Teploenergetika 12, 84 (1965). [Also Therm. Eng. ( U S S R ) 12, 106 (1965).] 12. A. S . Sukomel, F. F. Tsvetkov, and R. V. Kerimov, Teploenergetika 14, 77 (1967). [Also Therm. Eng. ( U S S R ) 14, 116 (1967).] 13. W. J. Danziger, Ind. Eng. Chem., Process Des. Develop. 2, 269 (1963). 14. C. Y. Wen and E. N. Miller, Ind. Eng. Chem. 53, 51 (1961). 15. S. L. SOO, “Fluid Dynamics of Multiphase Systems.” Ginn (Blaisdell), Boston, Massachusetts, 1967. 16. S. L. Soo, in “Advanced Heat Transfer” (B. T. Chao, ed.), p. 415. Univ. of Illinois Press, Urbana, Illinois, 1969. 17. C. L. Tien, J. Heat Transfer 83, 183 (1961). 18. L. Farbar and C. A. Depew, Ind. Eng. Chem., Fundam. 2, 130 (1963). 19. C. A. Depew, Heat Transfer to Flowing Gas-Solids Mixtures in a Vertical Circular Duct. Ph.D. Thesis, Univ. of California, Berkeley, California, 1960. [Also UCRL 9280 (1 960).] 20. C. A. Depew and L. Farbar, J. Heat Transfer 85, 164 (1963). 21. C. L. Tien and V. Quan, Arner. SOC. Mech. Eng. ASME Paper No. 62-HT-15 (1962). 22. R. Briller and R. L. Peskin, J. Heat Transfer 90, 464 (1968). 23. R. Briller, An Analytic and Experimental Study of the Mechanisms of Gas Solid Suspension Heat Transfer and Friction Factor. Ph.D. Thesis, Rutgers Univ., New Brunswick, New Jersey, 1969. 24. R. L. Peskin, Basic Studies in Gas-Solid Suspension. Final Rep. to U. S. At. Energy Cornm., Tech. Rep. No. 133-MAE-F (NYO 2930-15), Rutgers Univ., New Brunswick, New Jersey (1966). 25. R. Briller and R. L. Peskin, in “Augmentation of Convective Heat Transfer” (A. E. Bergles and R. L. Webb, eds.), p. 124. Amer. SOC.Mech. Eng., New York, 1970. 26. C. A. Brandon, An Investigation of the Interaction of Solid Particles with Fluids in Turbulent Flow. Ph.D. Thesis, Univ. of Tennessee, Knoxville, Tennessee, 1968. [Also ORNL-TM-2124 (1968).] 27. C. A. Brandon and T. A. Grizzle, Proc. Int. Symp. Two-Phase Syst., Technion City, Ha;f., Paper No. 4-8 (1971). 28. W. Brotz, J. W. Hiby, and K. G . Muller, Chem.-Ing.-Tech. 30, 138 (1958). 29. G. Jepson, A. Poll, and W. Smith, Trans. Inst. Chem. Eng. 41, 207 (1963). 30. G. T. Wilkinson and J. R. Norman, Trans. Inst. Chem. Eng. 45, 314 (1967). 31. C. C. Lin, in “Turbulent Flows and Heat Transfer” (C. C. Lin, ed.), p. 196. Princeton Univ. Press, Princeton, New Jersey, 1969. 32. J. Laufer, N A C A (Nut. Adv. Cornm. Aeronaut.), Rep. 1174 (1954). 33. C. N. Davies, Proc. Roy. Soc., Ser. A 289, 235 (1966). 34. I. Curievici, D. Peretz, C. Capatu, and A. Andreescu, BuZ. Inst. Politeh. Iasi 16(20), Sect. I1 (1970). 35. I. Curievici, D. Peretqand H. Dinulescu, Bul. Inst. Politeh. Iasi 16(20),Sect. IV (1970). 36. I. Curievici and H. Dinulescu, “Local Heat Transfer by Flowing Gas-Solids SUSpensions in Tubes.” Tech. Phys. Res. Center, Iasi, Romania, 1970. 37. R. I. Hawes, E. Holland, G. I. Kirby, and P. R. Waller, U . K . At. Energy Auth., Reactor Group, Rep. AEEW-R 244 (1964). 38. N. C . Chu, Turbulent Heat Transfer of Gas-Solid Two-Phase Flow in Circular Tubes. Ph.D. Thesis, Univ. of Washington, Seattle, Washington, 1971. 39. V. K. Rajpaul, Tube Size and Particle Size Effects on Heat Transfer to Flowing Gas-Solid Mixtures with Low Solid Loading Ratios. M.S. Thesis, Univ. of Washington, Seattle, Washington, 1963.

HEATTRANSFER TO FLOWING GAS-SOLID MIXTURES

179

40. M. K. Wahi, Effect of Particle Size and Test Section Orientation on Heat Transfer to Flowing Gas-Solid Mixtures. Univ. of Washington, Seattle, Washington, 1966. 41. D. van Zoonen, Proc. Symp. Interaction Fluids Particles, London, p. 64 (1962). 42. T. J. Kramer, Mean Flow Characteristics of Flowing Gas-Solid Suspensions. Ph.D. Thesis, Univ. of Washington, Seattle, Washington, 1970. 43. R. G. Boothroyd, Amer. SOC.Mech. Eng. ASME Paper No. 68-MH-11 (1968). 44. R. G. Boothroyd, Appl. Sci. Res. 21, 98 (1969). 45. C. A. Depew and E. R. Cramer, J. Heat Transfer 92, 77 (1970). 46. W. Bowen, Heat Transfer to Stratified Suspension Flows with 30, 62 and 200 Micron Particles. M.S. Thesis, Univ. of Washington, Seattle, Washington, 1969. 47. E. R. Cramer, Heat Transfer to Stratified Suspension Flow. M.S. Thesis, Univ. of Washington, Seattle, Washington, 1968. 48. D. G. Thomas, AIChE J. 8, 373 (1962). 49. 0. Adam, Chem. Eng. Tech. 29, 151 (1957). 50. K. Min, Particle Transport and Heat Transfer in Gas-Solid Suspension Flow Under the Influence of An Electric Field. Ph.D. Thesis, Univ. of Illinois, Urbana, Illinois, 1965. 51. K. Min and B. T. Chao, Nucl. Sci. Eng. 26, 523 (1966). 52. A. F. I. El-Saiedi, The Effect of Electric Fields on Heat Transfer in Gas-Solid Suspensions Flowing in Rectangular Channels. Ph.D. Thesis, Univ. of Illinois, Urbana, Illinois, 1969. 53. J. Gasterstaedt, V D I (Ver. Deut. I g . ) Forschungsarb. No. 265 (1924). 54. H. E. Rose and H. E. Barnacle, Engineer 203, 898 (1957). 55. C. M. Tchen , “Mean Value and Correlation Problems Connected with the Motion of Small Particles Suspended in a Turbulent Fluid.” Nijhoff, The Hague, 1947. [Dissertation, Univ. of Delft.] 56. S. L. Soo, Chem. Eng. Sci. 5, 57 (1956). 57. B. T. Chao, Oesterr. Ing.-Arch. 18, 7 (1964). 58. S. L. So0 and R. L. Peshin, Statistical Distribution of Solid-Phase in Two-Phase Turbulent Motion. Proj. SQUID Tech. Rep. PR-80-R, Princeton Univ., Princeton, New Jersey (1958). 59. B. L. Hinkle, Acceleration of Particles and Pressure Drop Encountered in Pneumatic Conveying. Ph.D. Thesis, Georgia Inst. of Technol., Atlanta, Georgia, 1953. 60. I. D. Doig and C. H. Roper, Ind. Eng. Chem., Fundam. 6 , 247 (1967). 61. H. E. McCarthy and J. H. Olson, Ind. Eng. Chem., Fundam. 7 , 471 (1968). 62. R. Eichhorn, R. Shanny, and U. Navon, Determination of the Solid-Phase Velocity in a Turbulent Gas-Solids Pipe Flow. Proj. SQUID Tech. Rep. PR-107-P, Princeton Univ., Princeton, New Jersey (1964). 63. H. L. Morse et al., J. Spacecr. Rockets 6, 264 (1969). 64. J. L. Dussourd, A Theoretical and Experimental Investigation of a Deceleration Probe for Measurement of Several Properties of a Droplet-Laden Air Stream. Sc.D. Thesis, Massachusetts Inst. of Technol., Cambridge, Massachusetts, 1954. 65. S. L. So0 and J. A. Regalbuto, Can. 1. Chem. Eng. 38, 160 (1960). 66. D. P. Trotter, Ann. Chem. Eng. Symp. Dyn. Multiphase Syst., 28th, Amer. Chem. SOC., Univ. of Delaware (1961). 67. S. L. Soo, G. J. Trezek, R. C. Cimick, and G. F. Hohnstreiter, Ind. Eng. Chem. 3, 98 (1964). 68. R. Eichhorn and S. Small, J. Fluid Mech. 20, 513 (1964). 69. R. L. Peskin, Proc. Int. Symp. Stochastic Hydraul., Univ. of Pittsburgh (1971). 70. N. C. Mehta, Ind. Eng. Chem. 49, 986 (1957).

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CREIGHTON A. DEPEWAND TEDJ. KRAMER

71. H. Kada and T. J. Hanratty, AIChE J. 6 , 624 (1960). 72. A. Wald, Ann. Math. Statist. 11, 284 (1940). 73. Z. R. Gorbis, Proc. Int. Heat Transfer Conf., 4th, Paris-Versailles, Paper No. CT-2.3 (1970). 74. C. L. Tien, Turbulent Processes in Two-Phase Turbulent Flow. Ph.D. Thesis, Princeton Univ., Princeton, New Jersey, 1959. 75. E. M. Sparrow, T. M. Hallman, and R. S. Siegel, App. Sci. Res. 7 , 37 (1957). 76. C. Orr, Jr., “Particulate Technology.” Macmillan, New York, 1966. 77. C. A. Sleicher, Jr. and M. Tribus, J. Heat Transfer 79, 789 (1957).

Condensation Heat Transfer HERMAN MERTE. JR

.

Department of Mechanical Engineering. University of Michigan. Ann Arbor. Michigan

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . I1 Nucleation . . . . . . . . . . . . . . . . . . . . . . . A Surface Tension . . . . . . . . . . . . . . . . . . . . B. Latent Heat . . . . . . . . . . . . . . . . . . . . . C. Equilibrium Across a Curved Surface . . . . . . . . . D Bulk Phase Nucleation . . . . . . . . . . . . . . . E. Nucleation on Solid Phase . . . . . . . . . . . . . I11. Liquid-Vapor Interface Phenomena . . . . . . . . . . . IV . Bulk Condensation Rates . . . . . . . . . . . . . . . A Condensation on Drops . . . . . . . . . . . . . . B. Condensation in Liquid Bulk . . . . . . . . . . . . V Surface Condensation Rates . . . . . . . . . . . . . . A . Prediction of Mode . . . . . . . . . . . . . . . . B . Dropwise Condensation . . . . . . . . . . . . . . C . Film Condensation . . . . . . . . . . . . . . . . VI . Mixtures . . . . . . . . . . . . . . . . . . . . . . . . VII . Similarities between Boiling and Condensation . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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181 183 184 190 192 199 203 215 222 223 226 221 221 231 244 264 266 261 268

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I Introduction

Condensation is the change in phase from the vapor state to the liquid or solid state. I t can be considered as taking place either within a bulk material or on a cooled surface. and is accompanied by simultaneous heat and mass transfer . Condensation plays a significant role in the heat rejection parts of the Rankine power generation cycle and the vapor compression refrigeration cycle. which generally involve single components (pure substances) . Dehumidification in air conditioning and the production of liquified 181

182

HERMANMERTE,JR.

petroleum gases, liquid nitrogen, and liquid oxygen are examples in which condensation in mixtures take place. Condensation is initiated by a nucleation process, either in a bulk vapor or on a solid surface. Fog and clouds are droplets which have grown from nuclei in a gas-vapor mixture. T h e energy release or latent heat removal with condensation in clouds, under certain conditions, serves as the driving force for the severe thunder storms with which we are all familiar. Cryopumping and vacuum depositing of metals are examples of condensation in which the vapor-solid phase change takes place. Condensation takes place during the collapse phase of the cavitation process, and in the boiling of subcooled liquids. T h e development of the heat pipe, which can provide an almost infinite effective thermal conductivity, depends on the high rate of heat transfer possible with condensation. After consideration of the processes of nucleation, a moderately detailed examination of the liquid-vapor interface, and a brief discussion of bulk condensation, the majority of effort in this writing will be devoted to condensation on solid surfaces since this has the greatest engineering significance. Condensation on a cooled solid surface occurs in one of two ways; film or dropwise condensation. I n film condensation the liquid condensate forms a continuous film which covers the surface, and takes place when the liquid “wets” the surface. This film flows over the surface under the action of gravity or other body forces, surface rotation, and/or shear stresses due to vapor flow. Heat transfer to the solid surface takes place through the film, which forms the greatest part of the thermal resistance. I n dropwise condensation the vapor impinges on the cool wall, decreasing its energy and thereby liquifying, and forming drops which grow by direct condensation of vapor on the drops and by coalescence with neighboring droplets until the drops are swept off the surface by the action of gravity or other body forces, surface rotation, and/or shear stresses due to vapor flow. As the drops move they coalesce with other droplets in their path, sweeping a portion of the surface clean so that condensation can begin anew. T h e details of dropwise condensation are not completely understood, but it is known to take place under circumstances where the liquid does not “wet” the surface. Dropwise condensation of steam has heat transfer coefficients over 10 times as large as film condensation ( I ) . However, it has been difficult to sustain dropwise condensation commercially for long periods of time, so conservative designs must be based on operation with film condensation. Efforts are continuing- on means to predict the heat transfer coefficient

CONDENSATION HEATTRANSFER

183

for condensation more accurately, and to increase the heat transfer coefficient. This will result in reduced condenser sizes and would increase significantly the attractiveness of condensers in applications such as steam cars. T h e increase in the heat transfer coefficient will result if dropwise condensation can be sustained, or if the liquid film can be reduced in thickness in the case of film condensation. Condensation heat transfer can be studied from either the macroscopic or microscopic point of view, as is the case in other physical phenomena. T h e ultimate objective for the engineer is to understand the phenomena so it can be described, and hence its behavior predicted for design purposes. T h e macroscopic viewpoint considers matter as a continuum, and requires the introduction of certain phenomenological laws, the macroscopic transport equations, in addition to the conservation equations of mass, energy, and momentum, in order to deal with rate processes. On a microscopic level, dealing with molecules, atoms, and subatomic particles, only the conservation principles need be introduced. However, because of limitations in describing these particles and in relating the large numbers of particles which constitute a reasonable size system one resorts to concepts of statistical and quantum mechanics. Both points of view, macroscopic and microscopic, will be introduced here as appropriate to aid in understanding the phenomena of condensation.

II. Nucleation Nucleation involves a nucleus-a starting point for growth. Nucleation and subsequent condensation can occur only if energy of the vapor is removed or reduced such that the temperature is sufficiently below the saturation temperature. This can be considered in one of two categories, depending on how the energy removal takes place: bulk nucleation and solid surface nucleation. Bulk nucleation occurs within the bulk of the vapor away from solid boundaries, and takes place either as homogeneous nucleation in the absence of any other phases such as entrained foreign particles, or as heterogeneous nucleation on foreign particles entrained in the vapor, such as dust (2). Bulk nucleation of a pure vapor can occur only by an adiabatic expansion, such that a transformation of energy from internal energy to other forms takes place. T h e vapor then becomes supersaturated until nuclei spontaneously form. Such an expansion in a nozzle produces a so-called condensation shock, and can occur within the low-pressure stage of a steam turbine, or in expansions to very low pressures as in space. T h e strange particles observed outside of the

184

HERMAN MERTE,J R .

spacecraft by the astronauts in the Appollo flights are believed to be solid condensed vapors. T h e Wilson cloud chamber, used to detect cosmic radiation, is an example of a nonflow adiabatic expansion to produce a supersaturated vapor. T h e path of the charged particles is made visible by the condensation trail left behind. I n a mixture of vapor and “noncondensable” gases, such as water vapor and air, bulk nucleation can also take place by expansion of the mixture, as is the case with pure vapors mentioned above. Meteorologically, the clouds in the center of a low-pressure region are in part a consequence of this expansion; one rarely observes clouds in the center of a high-pressure area. T h e temperature of a mixture can also be reduced by a mixing process, by introducing a cold constituent to the mixture, T he condensable component is then cooled below its saturation condition by a combination of mass and thermal diffusion and turbulent mixing. T h e consequences of mixing of this type on a meteorological scale, the meeting of a “warm” and “cold” front, are quite familiar. I n addition, the cooling and bulk nucleation of both pure vapors and mixtures can occur by the radiant emission of energy to the cooler boundaries for certain substances. Nucleation on solid surfaces can be placed in the category of heterogeneous nucleation, where nucleation takes place on a foreign substance, or homogeneous nucleation, where the solid is the same substance as the vapor. I n this case, however, as contrasted to bulk nucleation, the energy removal from the vapor can take place by heat transfer through the solid phase in addition to the vapor phase. Nucleation results in interfaces between phases and substances, and thus requires that certain interfacial phenomena be examined in some detail first. Extensive treatments of this subject are found in Adam (3) and Davies and Rideal ( 4 ) . A. SURFACE TENSION A commonly observed property of liquid surfaces is that they tend to contract spontaneously to the smallest possible surface area. Surfaces of minimum area are described at any point ( 3 ) by 1

1

const

where R, and R, are the principal radii at a point. T h e driving forces for this minimum area are intermolecular, as illustrated in Fig. 1. Liquids are distinguished from solids by the freedom of the molecules to move, and from gases by the larger cohesive forces between

CONDENSATION HEATTRANSFER 0

0

185

0 0

0

0

0

0

FIG. 1. Intermolecular forces at a surface.

molecules which inhibit their freedom of motion and hold them close together. I n the interior of the liquid in Fig. 1, each molecule is subject to the same attractive forces Fi in all directions. At the surface, the tangential molecular forces F , cancel out, but because of the greater intermolecular spacing in the vapor the outward forces do not balance the inward ones. Hence, every surface molecule is subjected to a strong inward attraction Fi perpendicular to the surface. This force causes inward movement of the molecules until the maximum number are inside with a minimum on the outside, resulting in a minimum surface area. That the liquid surface contracts spontaneously shows that thermodynamic “free energy” is associated with it. Any system that can undergo a spontaneous process possesses free energy and can produce work. Converse to the spontaneous contraction of a surface above, the extending of a surface requires that work be done in order to bring additional molecules from the interior to the surface. This results in an increase in surface energy, the surface free energy. Surface tension has been characterized (3) as a mathematical device introduced to simplify calculations. A hypothetical tension is substituted, acting in all directions parallel to the surface and equal to the surface free energy. One should be careful, however, and not think of surface tension as arising from a sort of stretched membrane at the surface since the physical basis for the tensile stress is different, and erroneous conclusions may be drawn. This is particularly true with regard to the vapor pressure about a curved surface, which will be considered shortly. Another view of the source of surface tension is presented in Davies and Rideal ( 4 ) . Referring again to Fig. 1, equilibrium between the molecules at the interface and in the bulk requires that the free energies in both places be the same. If the forces Fi and F , are identical, the free energies will be unequal, and molecules at the surface will be attracted to the interior, depleting somewhat the molecules at the surface. This

186

HERMANMERTE,JR.

increases the intermolecular spacing at the surface, resulting in added attractive forces between molecules in the surface which reduce their tendency to escape from the surface. T h e added attractive force at the interface thus constitutes the so-called surface tension phenomena. Several important consequences arise from the phenomenon of surface free energy. First, a “flat” liquid-vapor interface is an unnatural state of affairs, and exists only because of the overwhelming magnitude of other forces compared to the surface energy-mainly gravity. For a very small drop, the surface tension is large compared to the force of gravity, and it assumes a nearly spherical shape if it is nonwetting with the surface on which it rests. A large nonwetting liquid drop on the other hand will become spherical only under very low gravity. A liquid is wetting on a solid surface when the adhesive forces between the liquid and solid are greater than among the liquid particles themselves, and conversely. A parameter describing the relation between the body or acceleration forces and the capillary or surface-tension forces is the Bond number, given by B,

= F,/F, =

MaIuL

= pL2a/o.

(2)

L is some characteristic length of the system under consideration. T h e Bond number is useful for describing the conditions under which a wetting liquid will be settled at one end of a container under very low gravity fields, as in space (5). T h e second consequence of surface free energy is that a pressure difference exists across a curved interface, with the pressure being smaller on the convex side. Consider the general surface A in Fig. 2, described by radii of curvature R, and R, with origins at 0, and 0,. T h e displacement of surface A by a distance 6 to A‘ parallel to itself requires the expenditure of work, since the surface area is increased. This work is supplied mechanically by the pressure difference (PI- P2).Evaluating the work in Fig. 2 results in the fundamental equation of capillarity (3):

7

-+-. R2

PI - P 2 = a ( R l

For a spherical surface R, = R,

=

PI - P2

R and =

2o/R.

Measurements show that for the great majority of substances, the surface tension decreases as temperature increases. Kelvin (given in Adam (3)) showed that it follows that there must be an absorption of heat as a liquid surface is extended isothermally. By careful application

CONDENSATION HEATTRANSFER

187

FIG.2. Pressure difference across a curved interface.

of the energy equation to a liquid-vapor interface being extended isothermally, it can be shown that the total surface energy is given by eS =

G

+ qs = u

-

Tda/dT,

(5)

where 9s = -T dUldT.

(6)

qHis the latent heat of the surface, since it represents the amount of heat that must be added to the surface to maintain its temperature during an isothermal expansion. T h e total energy eB of the surface (per unit area) then consists of two parts: the surface free energy u and the latent heat -T daldT. E~ is the difference between the total energy per unit area of the molecules in the surface and the energy of the same number of molecules in the interior of the liquid. Heat is absorbed in extending a surface because molecules must be brought from the interior against the inward attractive force in order to form the new surface. T h e inward attraction tends to retard the motion of the particles as they leave the interior, so that the temperature of the surface layers would be lower than that of the interior were heat not supplied from outside. T o compare the magnitudes of the surface free energy and the surface latent heat Eq. ( 5 ) is written as

188

HERMANMERTE,JR.

Some values of the right-hand side of Eq. (7) are listed in Table I, and show that the surface latent heat is of the same order as the surface free energy. For quantities of liquid of reasonable size, where the number of molecules in the interior of the liquid is large compared to the number TABLE I COMPARISON OF THE SURFACE FREEENERGY AND SURFACE LATENTHEAT ~

Liquid

(Ed4

Temperature (OF)

80

0.82

I20 77 200

3.63 0.64 1.28

- 320

1.82

Ethyl Alcohol n-Hexane H2O H2O Liquid Nitrogen

-

1

in the surface, the contribution of the surface latent heat can be neglected. However, when considering small agglomerations of molecules which could serve as embryos for the onset of the nucleation under the appropriate conditions, the surface latent heat should be taken into account. Increasing the pressure of the vapor or a gas over the liquid surface decreases the surface tension by bringing a larger number of molecules closer to the surface on the vapor side. T h e attraction of these molecules neutralizes to some extent the inward attraction on the liquid surface molecules, hence decreasing the surface tension. For a pure substance, this decreases to zero at the critical pressure and temperature. T h e surface tension at the interface between two nonmixing liquids likewise will be smaller than the surface tension of either liquid in equilibrium with its own vapor or a gas because of the reduced net inward attraction on the molecules at the interface. If two liquids B and C, initially in contact with A (Fig. 3a) are brought into contact with each other the interfacial energy changes from uBA uca in Fig. 3a to uBCin Fig. 3b. Substance A in Fig. 3 may be air, a vacuum, or any other defined material. T he work required to separate them again is called the work of adhesion (6),as presented in Adam (3):

+

wBC

=

uBA

+

uCA

- uBC

*

(8)

As is obvious from Eq. (8), the work of adhesion is defined in terms of substance A (vapor, vacuum, etc.) and must be so understood. When

CONDENSATION HEATTRANSFER

189

uBb

OBC uCA

ib )

(0)

FIG. 3.

Work of adhesion.

+

the work of adhesion WBc < (oBA oCA), uBC> 0 and an interface between B and C is formed. If the interfacial tension uBC is zero or negative, an interface between them cannot exist since the work of adhesion is equal to or greater than the energy (uBA ucA)required to form the two interfaces. If the liquid B is the same as liquid C, then the work of adhesion is termed the work of cohesion, and uBA= ucA and uBc = 0, and Eq. (8) gives WBB = 2 u B A . (9)

+

If in Eq. (8)

3 UBA

~ C A

+

UBC

,

then the energy will be decreased by B spreading on C, and liquid B will spread spontaneously on C. T h e difference between the left and right side of Eq. (10) is defined as the spreading coefficient S (3)

For S 2 0, B will spread on C, while for S < 0, B will not spread, but form a drop. If substrate C in Fig. 3 is a solid, Eqs. (8) and (1 1) also apply as the work of adhesion and the spreading coefficient between liquid B and solid C. I n addition, a particle of liquid resting on a solid surface will form an angle of contact 6 at the triple interface, as in Fig. 4. Resolving the horizontal components in Fig. 4,at equilibrium

Combining Eq. (12) with Eq. (8),

I90

HERMAN MERTE,JH. A

C (%id)

FIG. 4.

Contact angles,

Generally speaking, a liquid is considered nonwetting on a surface if d > 90" and wetting if d < 90". ucAand oBC in Eq. (12) are difficult to measure directly, but measurement of the contact angle d permits a convenient determination of the adhesion work, Eq. (13), which is related to the solid properties by Eq. (8). A procedure is presented (7) using drops on inclined planes which permits the straightforward calculation of the liquid-solid surface tension, uBc above. Data on the work of adhesion, spreading coefficient, contact angle, and surface tension between various liquids and fluorinated polymers are presented in Fox and Zisman (8),along with a review of the rigorous solid-liquid interfacial relationships. These properties are of importance in the dropwise condensation process.

B. LATENT HEAT T h e latent heat of evaporation, like surface energy, is a phenomena involving intermolecular forces, and a relation might be expected between them. This will be demonstrated on an order of magnitude basis. T h e total surface energy per unit area in Eq. ( 5 ) , at the liquid surface, can be expressed ( 9 ) as eS =

u(z - z') n',

(14)

where u is the mutual potential energy of two neighboring molecules of mean spacing I, z is the number of nearest neighbors within liquid bulk, z' is the number of nearest neighbors at liquid surface, and n' is the number of surface particles per unit area (-1/r2). T o evaporate the liquid at the surface requires that the potential energy between the surface particles and those beneath must be overcome. T h e energy to accomplish this is the latent heat of evaporation, and can be expressed on a unit volume basis as

CONDENSATION HEATTRANSFER

191

where n is the number of particles per unit volume (-l/r3). T h e factor 1/2 takes into account the fact that only the attractive forces of particles on the underside of the surface are effective. Dividing Eq. (14) by Eq. (15) and substituting the approximate values for n’ and n:

-€8_

%

- 2(z

-

z

z’)

r w

Y

w 10-8 cm.

T h e intermolecular spacing of most liquids is on the order of lo-* cm, and Eq. (16) has been verified for almost all liquids. Consider water at 100°C, e.g., eS a w 60 dynes/cm (erg/cm2)

fi;g

-

330 cal/cm3 = 1.32 x 1O1O erg/cm3.

Th us

r

es/hTg = 60/(1.32 -

I

x lolo) = 0.45 x

cm.

Another manifestation of surface tension is the ability of pure liquids to withstand great tensions. T o estimate the tensile stress required to rupture a liquid, use is made of Eq. (9), which gives the minimum energy required to separate a liquid over area A at constant temperature as 2oA since this is the amount by which the surface energy increases. It can be considered sufficient to rupture the liquid if the molecules are separated by an additional distance on the order of the molecular spacing r . T h e work of separation is then the product of this distance and the maximum tensile force (negative pressure x area) that the liquid can withstand without separating, i.e., PmaxAr zz 2uA

or P,,,

R5

2a/r.

Again, for water at 100°C, Pma,

w (2 x 60)/10-R N

1.2 x

1O1O

dyne/cm2

10,000 atm

Th i s is the theoretical upper limit for the process Th a t cavitation on solid surfaces occurs at much consequence not only of the different intermolecular liquid and solid, but also of the influence of foreign entrapped or dissolved gases and oil films.

of bulk cavitation. lower values is a forces between the substances such as

192

HERMAN MERTE,JR.

ACROSSA CURVEDSURFACE C. EQUILIBRIUM T h e growth of a nucleus requires a nonequilibrium condition. T o determine the required nonequilibrium condition, it is necessary first to determine the equilibrium condition. Consider a spherical drop of pure liquid of radius r at pressure P , and temperature T , somehow suspended inside an enclosure, as shown in Fig. 5. Surrounding the drop is a pure vapor of the same substance

FIG. 5. Vapor-liquid drop equilibrium.

as the liquid, at pressure P, and temperature T,. Initially, take the vapor at P, to be in equilibrium with a liquid at T, , having a flat interface. T h e vapor is therefore saturated at P, . Also, take the drop to be in thermal equilibrium with the liquid. Th u s Ti

=

Tv = T s .

(19)

From Eq. (4) which relates the pressures across a curved liquid-vapor interface, P, - Pv = 2uIr. Then P, > P,, and the liquid is compressed with respect to the “normal” saturation pressure corresponding to the temperature (or subcooled with respect to the “normal” saturation temperature corresponding to the liquid pressure). T h e presence of a curved interface influences equilibrium conditions in a liquid-vapor system, as was shown by Thomson (10). T o determine the necessary conditions for thermodynamic equilibrium, the development will follow that of Frenkel (9).

CONDENSATION HEATTRANSFER

193

T h e total free energy (or Gibbs free energy, or thermodynamic potential) of the system in Fig. 5 is, for T , = T, = T ,

where n, , n, are the number of moles of vapor and liquid, and g, , g, give the Gibbs free energy of vapor and liquid per mole (chemical potential for a pure substance), respectively. It should be noted that the surface energy is included in the last term of Eq. (21). P,in Eq. (21) is given from Eq. (20) as PI = P v -1 ~ U / Y = P v

+ AP,

(22)

where AP = 2a/r. For thermodynamic equilibrium, at which state the system free energy is at a minimum, dG(P, T

=

const)

=

0

nv dgv

=

+ gv dnv + nl dgi + gi dni .

(23)

If this occurs at a state where the system free energy is a minimum, we have a state of stable equilibrium. If this occurs at a maximum free energy, a state of metastable equilibrium exists. A test will be conducted below to examine which form exists. Since nv n, = const

+

dn,

=

-dnl.

(24)

Also, at constant pressure and temperature, dgv = 0,

dgl = 0.

(25)

Substituting Eqs. (24) and (25) into Eq. (23), the requirement for equilibrium is given by the result gv(Pv, T ) = gl(p1, T ) ,

or from Eq. (22) gv(P")

= gdPv

+ 4.

Making a Taylor series expansion on the right side of Eq. (27),

Employing the thermodynamic relation v1 = (ag,/aP), ;

(26) (27)

HERMAN MERTE,JR.

194

By comparing Eqs. (26) and (29)

As Y .+

00, Eq.

(29) gives

gv(Pv)= gl(Pv). For a given value of T, , Eq. (31) is satisfied at a certain pressure P, , the “normal” vapor pressure. But for any value of r other than Y = co the value of P, necessary to satisfy Eq. (29) will be different, depending on r . T o determine this P, , differentiate Eq. (29), holding T = const dgv - dgl

=

2avl d(l/r).

(32)

For constant temperature the following thermodynamic relations apply; dgv

=

vV dP,

= ~1 dP

(33)

2avl d(l/r).

(34)

dgi

Substituting Eq. (33) into Eq. (32), (vV- 211) dP

=

>

If nv nl , valid far from the critical state, and considering the vapor to approximate ideal gas behavior such that vV = RTv/Pv,

(35)

RT, dPv/Pv = 2av1 d(l/r).

(36)

Eq. (34) reduces to: Integrating Eq. (36) between the limits r

= CO,

Pv = Ps

and

r = r*,

Pv = Pv

ln(Pv/Ps) = 2avl/r*RTv

(37)

Pv/Ps= exp(2avl/r*RTv).

(38)

or This gives the fractional increase in vapor pressure around a liquid drop of radius r*, above its “normal” vapor pressure (for Y = 00). This might be termed the supersaturation pressure ratio. For a liquid-vapor interface other than flat, then thermodynamic equilibrium requires that the vapor be supersaturated at a pressure greater than the saturation pressure corresponding to the temperature, shown as points A and B, respectively, in Fig. 6. Table I1 gives values of this pressure increase for water at 68°F for a variety of droplet diameters. One notes that the drop diameter

CONDENSATION HEATTRANSFER ,Super

195

saturoted Vopor

Vopor- Pressure Curve ( r = m )

Liquid

---------pz

P ps

- --- - - - - --

I I I

I

I I

T FIG. 6.

Vapor-liquid drop equilibrium states.

TABLE I1 EFFECT OF DIAMETER OF DROPOF WATER ON ITS VAPOR PRESSURE AT 68°F

Diameter (in.)

Approximate number of molecules

Supersaturation PvlPs

10-2 10-4 10-6 10-7

2.1 x 1017 2.7 x 10" 270,000 270

1.000009 1.00086 1.0905 2.38

must be quite small before the supersaturation pressure ratio becomes of apparent significance. If the supersaturation in a vapor surrounding a drop is less than that given by Eq. (38) for the given drop size, the droplet will evaporate since its effective vapor pressure is higher than that of the surroundings. Conversely, if the supersaturation in a vapor surrounding a drop is greater than that given by Eq. (38), condensation will take place on the drop since its effective vapor pressure is lower than that of the surroundings. A well-known example of this effect in operation is that water vapor will not condense in a dust- and ion-free atmosphere until its vapor pressure exceeds the saturation point by a considerable amount. From Table 11, condensation to a droplet 1.0 pin. in diameter requires a 9 % increase in vapor pressure. A sphere of this size contains about 270,000 molecules of H,O, and the probability of so many coming together spontaneously to form a drop of this size is quite small. Therefore, nuclei

196

HERMANMERTE,JR.

of some type providing a smaller curvature must be present if condensation is to occur anywhere near the ordinary saturation vapor pressures. Equation (38) assumes that is constant with curvature, but for very small sizes this is no longer valid. Based on arguments from statistical mechanics the following relation between surface tension and droplet radius has been proposed (11) (T

where urnis the surface tension for a flat surface and 6 is a length between 0.25 and 0.6 of the molecular or atomic radius in the liquid state. Equation (38) gives the increase in vapor pressure above the normal saturation pressure necessary for equilibrium between the vapor and a liquid drop of radii Y * . I t can also be viewed in another way. Solving Eq. (37) for r*,

For a given vapor temperature T, (hence a given P,) and a supersaturated vapor at vapor pressure P,, a drop of radius r* is in a state of metastable equilibrium. If, due to random fluctuations, the drop becomes slightly greater than Y * , condensation will take place on it. If the drop becomes slightly smaller than Y * , it will evaporate completely. Y * can thus be considered as the critical radius which can subsequently serve as a nucleus for condensation. That a drop of radius Y* is in a state of metastable equilibrium can be verified as follows: T h e total free energy of the system in Fig. 5 was given by Eq. (21). From Eq. (30) it can be shown that

T h e last term of Eq. (41) is the contribution of the surface free energy. Equation (21) can now be written as

Let Go correspond to the state in Fig. 5 where only vapor is present (the drop is completely evaporated). Then GO

=

(nv

+ nl)gv(Pv

9

T).

(43)

CONDENSATION HEATTRANSFER

197

T h e difference between Eqs. (42) and (43) is the free energy change associated with the formation of a drop of radius r, and gives AG

=

At equilibrium, r Eq. (44) becomes

G - Go = n , ( g l - gv)(Pv,T )

=

+ 4nr2a.

(44)

r* in Eq. (29), and noting that n, = +7rr3/zll,

+

AG = 4na[- $(r3/r*) r 2 ] .

(45)

O n substituting Eq. (40) into Eq. (45) AG

=

4xa

(-

r 3 RT

-_

3 av1

P, Ps

In -

+

r2).

Equation (46) is plotted in Fig. 7 as a function of r for the cases where

AG

FIG.7.

I

p v

<

PS -

(Superheated o r saturated)

Free energy of formation of a liquid drop.

<

1 (superheated or saturated), and where P,/Ps > 1 (supersaturated). A maximum in dG, designated as AG,,, , occurs only for P, P, > 1 at r = r*, and is given by

P,,'P,

where r* is given by Eq. (40). AG in Eqs. (45)and (46) is the free energy of formation of a drop of radius r . Since spontaneous processes are those associated with a decrease in the free energy, it is obvious from Fig. 7 that droplets will grow spontaneously only in a supersaturated vapor, that droplets of radius r < r* will not grow but will evaporate, and that drops will grow, serving as nuclei for condensation, when r > r*. Equation (40) holds true only for a pure substance. T h e presence of a monomolecular surface film of foreign substances (as is always present

198

HERMAN MERTE,JR.

in the atmosphere) has a considerable influence and requires a more complex analysis (3, 4). A supersaturated vapor, A in Fig. 6, is at a pressure greater than the saturation pressure P, corresponding to the temperature T , at B. It can also be considered as a vapor cooled to a temperature T,, below the saturation temperature T , corresponding to the vapor pressure P,, , point C in Fig. 6. T h e degree of subcooling T , - T,, corresponding to equilibrium with a droplet of radius r*, can be evaluated from Eq. (38) combined with the Clausius-Clapyron equation, which relates the normal saturation temperature and pressure:

>

For v, vl, and assuming the vapor behaves as an ideal gas, Eq. (35), Eq. (48) becomes dP P

--

hf, dT R T2'

(49)

Assuming h,, remains approximately constant over a small interval, Eq. (49) is integrated over the limits p

=

Pv,

T

=

T,

( C in Fig. 6)

,

T

=

Tv

(B in Fig. 6)

and p = p,

Equating Eqs. (50) and (37) and rearranging gives Ts - Tv = 2ovlT,/hf,r*.

(51)

This gives the subcooling of the vapor below the saturation temperature T , necessary for equilibrium with a droplet of radius r*. T h e actual state of the liquid within the droplet is also subcooled, or compressed, as indicated by point D in Fig. 6. From Eqs. (22) and (37) Pl

=

+ -20p.

Ps exp(2avl/r*RT)

CONDENSATION HEATTRANSFER

199

D. BULKPHASENUCLEATION For nucleation occurring in the bulk of a vapor away from solid surfaces, two cases can be distinguished, with and without the presence of foreign particles which can themselves act as nuclei. Growth of a drop on a foreign particle also requires a supersaturated vapor condition as specified by Eq. (38), except here the shape of the particle has considerable bearing on r * . T h e intermolecular forces between the particle and the liquid can be expected to modify the equilibrium conditions. An important source of condensation nuclei are salt particles from the sea, and considerable effort has been expended to study the significant parameters in the production of these nuclei (2). T h e effectiveness of AgI smoke in promoting atmospheric precipitation is believed to be due to its action as initial nucleation centers for ice crystals, which have a lattice spacing very similar to that of AgI ( 4 ) . For pure substances in which no detectable foreign nuclei were present, the maximum supersaturation ratio attained experimentally, as defined by the pressure ratio in Eq. (38), was 8 in a nonflow device (12) and up to 100 in a nozzle (13). Figure 8 compares the supersaturation limit line attained for nitrogen vapor with the normal vapor pressure curve, and Fig. 9 shows the variation of supersaturation ratio as a function of the vapor temperature. If foreign nuclei of some unknown but fixed size r* can be considered as present and limiting the degree of supersaturation, then the behavior in Fig. 9 appears to follow the trend predicted by Eq. (38). For a pure vapor with no foreign nuclei present, condensation nuclei must come from the vapor phase itself. Liquid embryos, or agglomerations of molecules, can form spontaneously even in a thermodynamically stable system owing to local fluctuations. From Eq. (45) these embryos are thus at a higher potential or free energy level, and may reach a level sufficient to become a nucleus for condensation, the maximum point in Fig. 7, if the necessary degree of supersaturation is present. Equation (46) shows that a maximum in AG does not exist with a saturated vapor. With a supersaturated vapor, AG,,, can be considered as a potential barrier which plays a role similar to that of the activation energy in chemical reactions ( 9 ) , and must therefore be a part of any analysis predicting the conditions under which nucleation takes place. T h e classical liquid-drop model of steady state nucleation of a pure substance is reviewed in Hill et al. (14). It is based on the description ( 9 ) of a supersaturated vapor as a dilute solution of substances (liquid embryos of various size) in a vapor as the solvent. These liquid embryos form as a result of spontaneous and random density fluctuations due to

200

HERMAN MERTE,JR. 1.0,

0.005

’

I

1

I

TEMPERATURE

I

I

1

I

T , OR

FIG. 8. Supersaturation limit in a nozzle with nitrogen. From Goglia and Van Wylen (13).

collisions between molecules. With equilibrium the sizes of these embryos is given by a Boltzmann-type distribution, Eq. (53), Nr = C exp(--dG,,,/kT).

(53)

With supersaturation a quasi-equilibrium condition is assumed to exist, in which embryos reaching the critical size become condensation nuclei and are immediately removed from the distribution. T h e rate at which these particles are replaced is computed from the rate at which embryos having one molecule less than the critical number gain a single molecule, using results of the kinetic theory. T h e resulting rate of formation of growing nuclei per unit volume is given (14) as

T he critical radius r* is related to the supersaturation pressure ratio

20 1

CONDENSATION TEMPERATURE Tc, "R

FIG. 9. Supersaturation ratio versus condensation temperature with nitrogen. From Goglia and Van Wylen (13).

I

I

3

5

7

SUPERSATURATION PRESSURE RATIO, Pv/Ps

FIG. 10. Rate of formation of condensation nuclei in water vapor. From Hill et al. (14).

HERMAN MERTE,JR.

202

P,/P,by Eq. (40).T h e larger the supersaturation (with smaller r*) the greater is J in Eq. (54).J is plotted for water vapor in Fig. 10 as a function of the supersaturation pressure ratio. T h e curve is so steep at low values of J that it is possible to define a critical supersaturation ratio beyond which condensation occurs spontaneously. A number of measurements have been made by observing the condensation upon expansion in a cylinder (14,and these are compared in Table I11 with the values TABLE I11

SUPERSATURATION PRESSURE RATIO^

(I

Vapor

Temperature (OK)

Water Water Methanol Ethanol I-Propanol Isopropyl alcohol n-Butyl alcohol Nitromethane Ethyl acetate

275.2 261.0 270.0 273.0 270.0 265.0 270.0 252.0 242.0

Measured

4.2 f 0.1 5.0 3.0 2.3 3.0 2.8 4.6 6.0 8.6-12.3

Calculated

4.2 5.0 1.8 2.3 3.2 2.9 4.5 6.2 10.4

From Volmer and Flood (15).

calculated from Eq. (54).I n nearly all cases the observed ratios agree well with the calculations. Using the limiting supersaturation pressure ratio of 4.2 for water vapor in air at 2.2"C from Table 111, the critical radius calculated from Eq. (40)is 6.58.T h e liquid nucleus just large enough to continue to grow thus contains about 40 molecules of water ( 4 ) . Contours of constant nucleation rate are plotted in Fig. 11 for water vapor from Eq. (54).Superimposing the isentropic expansion permits an estimate of whether and how soon condensation may be expected during an expansion in a turbine nozzle. T h e similarity in form between Figs. 11 and 8 is interesting, although they apply to different gases. More recent data on nucleation of water vapor in a nozzle follows the prediction of the classical model (16).However, with NH, the nucleation rate follows the Lothe-Pound equation, which predicts nucleation rates higher than the classical model by 1012-1018. T h e Lothe-Pound equation (17) is based on a quantum-statistical model, and corrects for the effect of size on the surface tension. T h e classical theory considers that the surface tension is constant. A comprehensive review of nucleation theory is given by Feder et al. (18) and presents a new kinetic treatment

CONDENSATION HEATTRANSFER

203

Io2

5 v)

a I

w

n

3 v) v)

W

n a

10

500

I

I

700

900

I

1100

TEMPERATURE - O R

FIG. 11. Nucleation rate for water vapor. From Hill et al. (14).

that accounts for the latent heat. T h e irreversible thermodynamics of nonisothermal nucleation also is discussed.

E. NUCLEATION ON SOLIDPHASE Nucleation on a solid surface proceeds to one of two cases: dropwise or film condensation. Nucleation as such does not exist with established

204

HERMAN MERTE,JR.

film condensation, only a continuous absorption or emission of molecules. This will be considered in the next section. Dropwise condensation is classified as a nucleation phenomenon (19), where active sites for drop formation are microscopic pits, scratches, and solid particles. A discussion of dropwise condensation must include other aspects related to the nucleation phenomena; the droplet population and size distribution, the role of promoters in the regulation of droplet population, and the mechanism of droplet removal. These will be considered below. I t has been well established that higher heat transfer rates arise with dropwise condensation than with film condensation, of which Fig. 12 shows one example. A number of different models

k

l CUPRIC O L E A T E )

-

AT OF

FIG. 12. Comparison of dropwise and film condensation of steam at atmospheric pressure on vertical copper surface. From Welch and Westwater (20).

of the dropwise condensation phenomenon have been proposed to describe and account for the large heat transfer rates. They differ essentially as to the relative importance of condensation directly on the drops and that, if any, on the spaces between the drops. T h e salient features of several of these models are represented in Fig. 13 and discussed below.

1. Fatica and Katz (21) This model considers that drop growth occurs primarily by condensation on the drop, with the latent heat transported to the solid surface

CONDENSATION HEATTRANSFER Conduction in \ Drop

205

SuDersa turated

Adsorbed surface layer

Fotico ond K o t z (Ref. 21)

Eucken (Ref. 24)

Thin liauid

Emrnons (Ref. 27)

Jokob (Ref. 281

FIG. 13. Several models for dropwise condensation.

by conduction through the drop. A two-dimensional flux plot is used to compute the rate of heat transfer. Assuming a uniform drop size and arbitrary fractional coverage of the surface, it was possible to compute an average heat flux. Because of the relatively low thermal conductivity of the liquid the heat flux is low except near the intersection of the solid wall and the edge of the drop. I t has been suggested (22) that the heat transfer rate through the drop would be enhanced considerably if internal circulation within the drops takes place because of variations in surface tension about the surface with temperature. A detailed finite difference solution of this circulation in a hemispherical drop indicates that the contribution of the circulation is insignificant (23).

2. Eucken (24) I n addition to condensation directly on the drop, it was proposed that a monomolecular layer forms between the drops. Adjacent to the drop this layer is saturated while farther away it is supersaturated. T h e concentration gradient in the layer associated with the difference in the degree of saturation serves as the driving force for surface diffusion of the molecules toward the drop. Eucken (24) also used the kinetic theory of gases to calculate the heat transfer by condensation of steam at atmospheric pressure if all molecules striking the cooling surface were immediately condensed. T h e

206

HERMAN MERTE,JR.

result was a flux of q/A = 72 x lo6 BTUjhr ft2, about 260 times greater than the maximum observed. According to this only about 0.4% of the vapor molecules striking the solid are retained as liquid.

3. Emmons (27) This mechanism proposes that the vapor molecules striking the solid bare surfaces are reevaporated, but at the temperature of the surface and hence subcooled, and then followed by recondensation onto the droplets. T h e bare cooling surface between drops are thus blanketed by supersaturated vapor. Emmons (27) also considers that the rapid condensation on a drop causes a local reduction in pressure which sets up violent local eddy currents between drops.

4. Jakob (28) Jakob (28) suggested that dropwise condensation results from the fracture of a thin film of condensate, which completely covers the solid surface, into drops after the film had grown to some critical thickness, after which the process repeated itself. Observation of dropwise condensation under a microscope appeared to support this (20). I t was noted that drops large enough to be visible (0.01 mm) grew primarily by coalescing with other drops, after which the bare metal beneath is exposed, as noted by a lustrous appearance. T h e luster disappears very quickly, indicating that the condensation was building up again. It was concluded that heat transfer occurs primarily between the drops, which are “fed” by the fracture of the thin liquid film. Subsequent works have indicated that the film-fracture mechanism does not take place. Using the optical technique of measuring the change in ellipticity of polarized light upon reflection from a transparent thin film, as manifested by a change in intensity, the thickness of the film can be computed, if one exists (19). T h e intensities are recorded against time, as shown in Fig. 14. T h e upper part of the figure served as a test of the technique. T h e alcohol was injected on the surface and formed a continuous film on the gold surface. From optical theory the intensity of the reflected elliptically polarized light should increase if a film is present, as is the case toward the right. Each cycle corresponds to a On the lower part of Fig. 14 the heater change in thickness of 2500 within the cooled surface was turned off so that condensation could begin. If a thin film formed between drops in well-established dropwise condensation, it should also form on a bare surface before drops begin to form. This is seen not to be the case; the intensity did not increase as it would were a film present, but rather decreased because of scattering

a.

CONDENSATION HEATTRANSFER

207

on surface

Time

-

( a ) Calibration test. Gold surface

with alcohol

-

t Irn

Test area momentarily cleared

1

1

Surface heater shut o f f Drop formation

of the reflected light by the drops. It was concluded that any film, if present, could not be more than one monolayer thick, and therefore no net condensation takes place between the drops. T h e conclusion of the above is that condensation occurs only at certain sites on the surface, and dropwise condensation may be classified as a nucleation phenomena in a fashion similar to nucleate boiling. This has been supported by high-speed photographs of dropwise condensation taken under very high magnification (29), where it was observed that drops formed repeatedly at fixed sites, both natural and artificial, on the surface. It was postulated that liquid trapped in pits on the surface are the nucleation sites. If this is the case, then the presence of the second phase aids the change of phase process considerably in a fashion similar to nucleation in boiling. It has also been observed that grain boundaries of metals serve as nucleating sites for dropwise condensation (30). Despite the observations cited above, that no net condensation takes place between drops, more recent studies using an interference microscope (31) indicate the presence of liquid films between drops, which reach a critical thickness of about 0.63 p before the film fractures into

208

HERMAN MERTE,JR.

0,P: 1600-

19mm

0 , P= 39mm

~ 5 0 0 , 0 0 0

0 370,000 1400-

FIG. 15. Heat flux during dropwise condensation of water on a horizontal surface. From McCormick and Westwater (33).

CONDENSATION HEATTRANSFER

209

T h e effect of variability in population density no doubt accounts for some of the large differences in experimental data in the literature for similar conditions, an example of which is shown in Fig. 16. It is obviously not possible to designate any one result as the “correct” one.

Calculated coefficient f o r a condensation

.

0

100

200

300

400

Heat Flux-(Btu/ft2-hr)

500

600

x

FIG. 16. Summary of published results of heat transfer coefficient for dropwise condensation nucleation cavities. From Tanner et al. (48).

One can surmise from Fig. 15 that there must exist an upper limit on the population density, beyond which complete coalescence would take place between droplets, i.e., the surface would become covered with a film of liquid, called film condensation. I t was noted in Fig. 12 that the heat flux decreases with film condensation. This is analogous to the upper limit in heat flux with nucleate boiling, called the peak heat flux, beyond which the heat flux decreases as AT increases until the surface is blanketed with a vapor film-called film boiling. I n both cases it is a cessation of the distinct nucleation process that gives rise to the formation of a film, and the accompanying decrease in performance; in both cases it is the nucleation process that gives rise to the high rates of heat transfer. From Fig. 15, one can also surmise that for most effective performance it is desirable to maintain as high a population density as possible with dropwise condensation, but not too high, lest film condensation take place. Experimentally it has been found that dropwise condensation of

210

HERMANMERTE,JR.

steam on metal (except for the noble metals, cf. Umur and Griffith (19)), occurs only with the use of “promoters.” A number of promoters are listed in Table IV. TABLE IV

PROMOTERS FOR STEAM Ref. Stearic acid Montanic acid Dioctadecyl disulfide Benzyl mercaptan Oleic acid Dimethyl-polysiloxane (silicon oil KF-96) Dibenzyl disulfide Montan wax Dodecane (ethanethic) silane

(34) (34) (34-36) (29) (34 (34) (34) (35,36)

The detailed role that the promoters play has yet to be clarified. They do form a coating on the surface, and it was noted that the effectiveness changed little with variations in thickness of the coating between 0.2 and 11 equivalent monolayers, on a relative scale (34). Thicknesses less than this did not produce dropwise condensation, while those greater seemed to introduce an additional thermal resistance. It was also observed that both abrasion and oxidation of the surface tended to cause the surface to revert to film condensation. These were explained on the basis that gross roughening assists wetting by capillarity, hence increasing the number of potential nucleating sites. Surfaces with oxide layers required relatively more promoter to make the surfaces hydrophobic (nonwetting). I t appears that a promoter tends to eliminate some of the many natural nucleating sites that are present on any real surface by perhaps filling in some of the cavities, or by changing the solid-liquidvapor surface energy relationships such that a critical size drastically different from that given by Eq. (40) becomes necessary for nucleation to take place. Since filmwise condensation is the more common mode which takes place, to increase heat transfer rates requires the establishment of dropwise condensation on a reliable basis and hence the reduction in number of nucleating sites such that coalescence does not take place. Using a variety of particles ranging in size from 1-100 p (1 p = cm = 40 pin.) and resting on a cooled surface (29), the nucleating ability of these particles were observed on the basis of their composition

CONDENSATION HEATTRANSFER

21 1

and size ranges. Table V lists the various materials used in the order of their nucleating ability with water vapor, from the best on down. Also TABLE V

RELATIVEORDEROF NLJCLEATION ABILITYOF PARTICLES~ Particle size Particle Sodium chloride Platinum Glass Aluminum oxide Starch Bone charcoal Silver iodide Titanium dioxide Graphite Mercury Teflon Coconut charcoal Pyrolytic graphite a

(PI

540 1-100 2-40 2-50 9-35 3-33 2-24 10-25 7-44 4-70 6-43 11-29 5-50

Nuclation ability

Net heat of adsorption (kcal/mole)

Excellent Very good Very good Very good Very good Good Poor Poor Very poor Very poor Very poor None None

25.5 (soluble) 5.5 2.7 1.6 0.5 0.5 0.4 0.03 0.05 0 0 -0.85 - 2.0

From McCormick and Westwater (29).

listed is the net heat of adsorption for each substance, and almost a direct relationship exists between nucleating ability and the net heat of adsorption, which is the difference between the heat of adsorption and the latent heat of condensation. Adsorption of vapor on the particle is the first step in the nucleation of drops on the particle and, from Table V, takes place when the heat of adsorption is greater than the latent heat of condensation. T h e greater the difference, the more easily nucleation takes place. Assuming for the moment that the condensation taking place on a surface will be dropwise, it is of interest to predict the wall temperatures at which condensation nucleation may be expected to occur. T h e analysis is similar to that used to predict the surface superheat for incipient nucleate boiling (37) and assumes that nucleation will begin at a cavity of radius r e , in Fig. 17. T h e analysis is also similar to that for predicting the supersaturation necessary for condensation on particles (29). A cavity of radius rc exists, Fig. 17, and is considered filled with liquid to give a hemispherical protrusion of height r c . T , is the saturation temperature corresponding to the normal vapor pressure, and T, indicates the vapor temperature profile in contact with a cooled surface at

HERMAN MERTE,JR.

212

cavity

cooling surface

’ y,r*

FIG. 17. Model for determining effective nucleating site.

temperatures T , . As had already been indicated, in order for condensation to occur on a curved interface, the local vapor subcooling must be greater than a minimum amount given by Eq. (51). I n order for the cavity of radius Y, in Fig. 17 to act as a nucleating site, i.e., for the drop to grow, the vapor temperature at distance Y, from the wall must have a subcooling greater than that given by Eq. (51) for Y * = r C . Letting T, = T,, , Eq. (51) gives

T,,

=

Ts (1 -

s). fgr*

(55)

T,., is plotted in Fig. 17. T h e embryo will grow (i.e., nucleation will occur) when the vapor temperature T, at y = Y, equals T,., at r* = Y, . Assume that near the wall the vapor profile is linear over a sufficient range, and can be represented in terms of a boundary layer thickness 6, bulk vapor temperature TVm, and wall temperature T , by

If the wall temperature is T , as indicated in Fig. 17, the cavity cannot

CONDENSATION HEATTRANSFER

21 3

serve as a nucleation site since the temperature T, at y = r0 is greater than the critical temperature given by T,, at r* = rc . If, however, T, is TWias indicated in Fig. 17 such that T, is equal to T,, (point X), then the site is just at the incipient nucleating point for condensation. T o solve for the wall temperature at this state, set T,, in Eq. (55) for r* = rC equal to T, in Eq. (56) for y = rC . T h e result is AT,i

=

-)Arc

(ATsup4-

S ~

6 - rc '

(57)

where

This gives the minimum value of AT, (= Tvm- T,) at which the cavity of radius rc may become an active condensation nucleating site. Note in Fig. 17 that the boundary layer temperature profile also intersects T,., at a smaller value of r*, designated as Y . If A T , is given but the cavity sizes cover the entire spectrum, the range of cavity sizes which can become active is given by the solution of Eq. (57) for r , :

where

For saturated vapor conditions 8, = 0. For T, = TWiin Fig. 17, the two roots of Eq. (58) would correspond to points X and Y , and within this range the local vapor temperature T , is lower than the critical temperature T,., , hence cavities in this range can become nucleation sites. If the value of the terms within the square brackets of Eq. (58) is negative, then no site can become active. If a wide range of cavity sizes is present on a surface, then the criteria for incipient condensation is given setting the discriminate of Eq. (58) equal to zero and solving

If the vapor is saturated (ATsup= 0), then Eq. (59) reduces to ATwi = 4A/S.

(60)

HERMANMERTE,JR.

214

Equation (58) is plotted in Fig. 18, along with data for dropwise condensation of water on a horizontal copper surface promoted with benzyl

cn 2

70

-

60

-

50NUCLEATION

0

a

uI I

POINTS

0 0

40-

L U

a W

00

I-

g

30-

5 0 >

k

p

20-

0

I

\

NUCLEATION

ATw%

FIG. 18. Effective size range of condensation nucleation cavities. From McCormick and Westwater (29).

mercaptan (29). T h e data points fall within the plotted size range, even though it was difficult to do better than estimate a value for the boundary layer thickness 6 in the experiments. An analysis to determine the critical radius for subsequent growth of

CONDENSATION HEATTRANSFER

21 5

the outer surface of a liquid film covering a protrusion on a surface gives the same result as Eq. (51) for homogeneous nucleation (38). HI. Liquid-Vapor Interface Phenomena

Once nucleation has taken place (the embryo has become a nucleus and growth begins), subsequent condensation takes place from the vapor onto the liquid. T h e dynamics of the phenomena at the liquidvapor interface will now be considered. T h e first question raised might be that regarding the nature of the interface-how well defined is it, sharp or diffuse? Strong evidence exists that the change in density from liquid to vapor is very abrupt, the transition layer being only 1-2 molecules thick. This is shown most clearly by the nature of light reflected from the surface. From Fresnel’s law of reflection, if the transition between air and a medium of refraction index n is abrupt, the light is plane polarized if the angle of incidence is at the Brewster’s angle, or tan-ln. But if the transition is gradual the light will be elliptically polarized (3). This test is so sensitive that it will detect layers of the order of one molecule thick. An experiment by Rayleigh (cited in Adam ( 3 ) )with water indicated that a water surface had a transition about one molecule thick. A related question is the temperature of the interface. If the interface is not sharply defined but consists of a transition zone, the concept of an interface temperature loses its meaning. However, in light of the results above, it seems reasonable that a temperature can be assigned to each phase at the interface, and the zone of uncertainty is quite small. I n fact, the classical concept of temperature itself loses meaning on the molecular scale. T o avoid the complication of a curved interface, consider the condensation of a pure vapor on a plane interface, as represented in Fig. 19. T h e process is first examined on the basis of a continuum or macroscopic viewpoint. With thermodynamic equilibrium, in the notation on the upper part of Fig. 19, Ti = Tii

=

Tvi = Tv = Tv,

=

Ts ,

(61)

and no changes will take place in the absence of any driving forces. With nonequilibrium, temperature differences will exist in the system, in T , and T, . To explore the temperatures at the interface, the temperature of the liquid and the vapor at the interface must be defined. If a submicrominiature temperature measuring device is placed in the liquid, and its temperature distribution explored as close to the interface as

HERMAN MERTE,J R .

216

,

Interface

A/

Solid

Wall

~L1j+ Distance of 1-2 mean-free paths in vapor

e-

-+X FIG. 19. Continuum representation of liquid-vapor interface.

possible, the extrapolation of this temperature to the interface is defined as the liquid interfacial temperature Tli. I n the vapor also, the extrapolation of the vapor temperature to the interface is the vapor interfacial temperature, Tvi . This is extrapolated over the distance of one to two mean free paths to avoid the problem of defining temperature in terms of a non-Maxwellian velocity distribution which may exist in the immediate vicinity of the interface (39). Because of inability to detect the differences between Tli and Tvi, and in the absence of any known relation between them, non-equilibrium processes treated on a continuum basis consider that Tii =

Tvi

= T,,

(62)

although there exists no “a priori” basis for this. For condensation to take place, the latent heat must be removed, by conduction in one or both phases. If the liquid is stationary, the interface will move to the right in Fig. 19. A first law control volume analysis on the interface gives (40) plhfg dXi/dt = Kl(dTl/d~)i- rZV(dTV/dx)i. (63) T h e rate of mass transfer between the two phases may be described in terms of dX,/dT. T h e first term on the right side of Eq. (63) represents

CONDENSATION HEATTRANSFER

21 7

conduction heat transfer in the liquid phase, and the second term is conduction heat transfer on the vapor side, with the difference being the net energy removal of latent heat-the left side of Eq. ( 6 3 ) . With a density difference between the two phases, bulk motion in the vapor takes place, expressed by

A driving force is necessary for the vapor motion to take place. With condensation, the vapor concentration at the interface is reduced by the net removal of the vapor to a value below that for equilibrium conditions. A pressure depression thus exists, as represented on the lower part of Fig. 19, Pli < Pvi.T h e influence of hydrostatic heads are neglected here. T o achieve this reduced pressure requires that the liquid interface temperature TIi be below the equilibrium saturation temperature T , , corresponding to the vapor pressure P, . T h e vapor interface temperature Tvi may or may not be the same as T , . I t depends upon the velocity distribution of the vapor molecules at the interface, which consists of molecules coming from the bulk vapor region on the right, molecules reflected from the interface, and molecules emitted from the liquid. If the net sum of these furnishes the same Maxwellian velocity distribution present at temperature T , , then

Schrage (39) provides an argument showing that when the entire vapor region is at the saturation temperature (i.e., T, = T,), then Tvi = T , . However, with superheated bulk vapor, it would be likely that interactions between the three groups of molecules would result in a nonMaxwellian velocity distribution, and one would not expect that Eq. (65) would hold. T he net conclusion of the above discussion is that an interfacial temperature difference ( TVi- T,,), sometimes called a “temperature jump,” must exist across a liquid-vapor interface, even with a continuum treatment of the process. T h e calculation of the magnitude of this temperature jump, however, requires the introduction of microscopic considerations. One might be tempted to include the temperature difference ( Tvi - TI,)in the macroscopic formulation of the energy interactions at the interface, given as Eq. (63). It would not be correct, however, since the energy transfers in the liquid and vapor are already accounted for in the defined temperature gradients at the interface. T h e formulation of the problem, however, requires an additional relation

21 8

HERMAN MERTE,JR.

between Tli and T v i . In seeking such a relation one might view the difference as a thermal manifestation of a molecular resistance, particularly when dealing with a series of thermal resistances as in film condensation. T he difference (TIi - Tvi) was not detected until much work on condensation of liquid metals had taken place (e.g., see Sukhatme and Rohsenow (41), Kroger and Rohsenow (42), and Barry and Balzhiser (43)). Its detection depends on the magnitude of the interfacial “resistance” compared to other thermal resistances. If the liquid film in the upper part of Fig. 19 has a relatively high thermal resistance, then

(T1i - Tw)

> (Tvi

-

Tli).

(66)

T h e right-hand side might be negligibly small, and no error exists with taking Tvi m TIi M T,.This accounts for the success of the Nusselt theory of film condensation (44) for many fluids like water. T o derive an expression for the interfacial temperature difference it will be assumed that the bulk vapor is at the saturation temperature so that Tvi = T , . T h e presentation follows that of Schrage (39) and Wilhelm (45). T h e interphase mass transfer is viewed as a difference between the rate of arrival of molecules from the vapor space toward the interface and the rate of departure of molecules from the surface of the liquid into the vapor space. When condensation takes place, the arrival rate exceeds the departure rate; with evaporation the opposite occurs; with equilibrium they are equal. From the kinetic theory of gases, assuming a Maxwellian velocity distribution for the vapor emitted from a liquid surface at T,,, and considering this vapor to behave as an ideal gas at pressure P,, , the vapor pressure corresponding to T I ,, the absolute rate of evaporation of a liquid (as into a high vacuum) at TIi is given by

, and is attained with the evaporation of a pure liquid into a high vacuum. T h e evaporation coefficient U, corrects the evaporative mass flux for effects associated with polyatomic molecules and the equilibration of their internal degrees of freedom in passing from the initial to the activated states on evaporation. I t has been shown theoretically (46)that spherically symmetrical molecules such as CCL, and monatomic molecules have values of ae max = 1, while unsymmetrical ones have values 0 ue max 1. Measured values for liquid metals also give a, max M 1. ue max is the maximum value of an evaporation coefficient u,

<

<

CONDENSATION HEATTRANSFER

219

When condensation and evaporation occur simultaneously u, may be reduced considerably below u, max because of interactions between the evaporating flux and the condensing flux. ue may be considered as a correction factor for the combined efiects mentioned above plus the possible departure of the velocity distribution of evaporating molecules from the equilibrium lhlaxwellian distribution. Thus, 0

< ue <

u e max

,

and the mass flux of evaporating molecules is given by (69)

By a similar analysis, the mass flux of vapor molecules condensing on a liquid surface is given by

% = ,,rPv A

(+-j1'2, M 27rRTv

where u, is the fraction of incident molecules that actually condense; the remaining fraction ( I - u,) is reflected and contributes nothing to the net flux. r arises from the analysis because of the influence of the bulk vapor velocity V , moving toward the condensing plane in Fig. 19, and is given by

r = exp(-#2) + # ~ l / ~+( lerf +), where

V , is given by

#

=

Vv/(2RT/M)1/2.

w/A

= pvVv

.

(71) (72) (73)

For w,see Eq. (75) below. U, is called the condensation coefficient, and for mathematical simplicity is often taken as U c = U e = U,

(74)

although no physical justification exists for this assumption. Each coefficient is associated with a different mechanism-the capture by and escape from a strong intermolecular force (45). T h e net mass flux at the interface for a pure substance is then given by the difference between Eqs. (70) and (69),

HERMAN MERTE,JR.

220

With thermodynamic equilibrium conditions Eq. (74) is strictly true, and r = 0. Then P, = P l i , T, = Tli, and w / A = 0. If Eq. (74) is assumed valid under nonequilibrium conditions, then Eq. (75) becomes

This shows that, for any given fluid and mass transfer rate, the interfacial temperature drop ( T , - Tli) increases as pressure P, is decreased. Examples of calculations of this are shown in Table VI, taken from Sukhatme and Rohsenow (42), and based on an assumed temperature drop of 5°F across the liquid film. Table VI demonstrates why the TABLE VI

TEMPERATURE DROPAT LIQUID-VAPORINTERFACE^ Assumed value of Fluid

U

1.o 0.04

Water Mercury Sodium a

1.o 0.1 1 .o

(Tv - Tlj) ( O F ) , for Pv

=

760mm 100mm 10mm

0.003 0.1 0.3 5.9 0.6

0.01 0.7 1.6 29.6 2.8

0.1 4.8 10.0

Mass flux from Nusselt's theory (lbmlhr ftz) 1.9

1115

-

18.0

250

From Sukhatme and Rohsenow (41).

presence of this interfacial temperature drop with fluids other than liquid metals has not been observed. T h e heat flux corresponding to complete condensation of steam at atmospheric pressure, using the mass flux of steam calculated from Eq. (70) with r = 1 and uc = 1 , resulted in q/A = 72 x lo6 BTU/hr ft2 (26). T h e maximum heat flux observed with condensation of steam has been about 252,000 BTUihr ft2, so if Eq. (74) is taken as valid, the condensation coefficient is about 0.04, which was stated as agreeing with the values of others (26). Recent measurements of the condensation coefficient of water with net condensation, using a transient technique, gave values of u ranging over 0.01 u 0.2, and with net evaporation in the range 0.02 o 0.8, which varied with time in a decreasing fashion (47). An interferometer was used to obtain an approximation of the transient temperature at the liquid-vapor interface. Th e asymptotic values with net condensation appeared to be 0.01, and with net evaporation was 0.02, in about

< <

< <

CONDENSATION HEATTRANSFER

22 1

0.2 sec. T h e relationship between these transient meas1.irements and steady ones has not yet been established. Other recent results with steam at atmospheric pressure indicate that higher values of the condensation coefficient are possible: u 3 0.08 (48), u >, 0.45 (49),u FZ 1.0 (50). Table VII, taken from Mills and Seban (49), shows other values for steam. TABLE VII PREVIOUS EXPERIMENTAL VALUES FOR

THE

CONDENSATION COEFFICIENT OF WATER"

Date

Temperature ("C)

Alty

1931

18-60

Alty and Nicoll Alty Alty and Mackay Pruger

1931 1933 1935 1940

18-60 -8-$4 15

Investigation

u

Nature of the experiment

GROUP 1

Hammecke and Kappler 1953 Hammecke and Kappler 1955 Delaney et al. 1964

0.006-0.016

? 0-43

evaporation from a suspended drop same 0.01-0.02 same 0.04 same 0.036 evaporation from a 0.02 horizontal surface same 0.045 same 0.100 0.0415-0.0265 same

0

0.42

100

20

GROUP 2 Hickman

1954

Nabavian and Bromley

1963

10-50

Jamieson

1965

0-70

Berman

1961

?

a

0.35-1.0 0.35 near 1.0

evaporation from a tensimeter jet film condensation on a fluted tube condensation on a tensimeter jet film condensation on a horizontal cylinder

From Mills and Seban (49).

For both water and liquid metals condensing on a vertical surface, there appears to be considerable evidence that a direct relationship exists between the condensation coefficient and pressure, as shown in Fig. 20. Three different liquid metals as well as water are represented over a wide range of pressures, from nine independent investigations. T h e asymptotic value of u = 1.0 for low pressures appears to support the description, given earlier, that the condensation coefficient is a correction factor for interaction between molecules.

222

n

HERMANMERTE,JR.

n

,A

L

0 K .K

0-

0 No

A Ha

1c 0.01 0.001

0

A K-

0

O K

+No

0.01

0.I

.o

Ps, ATMOSPHERES

FIG. 20. Condensation coefficient of potassium versus saturation pressure. From Kroger and Rohsenow (42).

An analysis which takes into account the influence of vapor subcooling in a thin zone adjacent to the liquid-vapor interface (on the order of ten mean free paths) results in a calculation of the condensation coefficient u = I for much of the data, rather than u decreasing as pressure increases as in Fig. 20 (51).

IV. Bulk Condensation Rates Devices such as steam desuperheaters, spray condensers, and direct contact steam-water heaters provide examples where phase changes of pure substances take place in a bulk continuous phase, i.e., away from solid surfaces. T h e rate at which the phase change takes place is of concern in order to design these devices properly. T h e growth of hailstones, rain drops, and snow are examples of bulk condensation of mixtures in which one of the components is relatively inert or noncondensable. T h e conditions under which these processes arise and the rate at which they occur are of great interest to meteorologists (52).

CONDENSATION HEATTRANSFER

223

Bulk condensation can be classified in two categories, depending on whether the continuous or primary phase is liquid or vapor. Condensation in a supersaturated vapor or on injected subcooled liquid drops would take place as the growth of dispersed liquid drops, while a vapor injected into a subcooled liquid bulk would result in collapse of the vapor bubble because of condensation. T h e complete governing equations are identical for both cases, consisting of the equations of conservation of mass, momentum, and energy for the region within and outside of the drop or bubble. T h e growth of the drop is much slower than the collapse of a vapor bubble for the same initial temperature difference. This, combined with the much lower density of the vapor moving medium for the drop case as compared to the liquid for the collapsing bubble, means that momentum effects can be neglected in the case of the condensation on a drop. These cases are therefore considered independently. Except perhaps for liquid metals, the influence of any interfacial temperature drops can be ignored.

A. CONDENSATION ON DROPS Two different cases of condensation on drops will be considered depending on whether the sink for latent heat removal lies outside of, or within the liquid drop. 1. Supersaturated Vapor If, as a result of expansion of a pure vapor below the saturation temperature nucleation takes place, and the expansion continues, further growth of the drop occurs by the transfer of heat to the surroundings, and continues as long as the surroundings are supersaturated. It should be recognized that the rate of condensation will depend on the rate at which nuclei are spontaneously formed from the supersaturated vapor as well as the rate of condensation on the nuclei. T h e rate of formation of nuclei was considered earlier so only the latter process will be considered here. T h e physical situation is represented in Fig. 21. T h e ambient temperature T,,(t) is related to the pressure Pv(t)in some form such as given in Fig. 11. Also, TVm< T , , where T , is the saturation temperature corresponding to the pressure P,. T h e surface temperature is assumed to be at the saturation temperature T , . This neglects the effect of curvature on the vapor pressure exerted by the liquid, and also assumes that the process is limited by thermal diffusion in the vapor rather than by mass diffusion. T h e transient conduction in the liquid is represented quite simply in differential form (40) and the relation

224

HERMANMERTE,JR.

T

Supersaturated Vapor

Su bcooled LiquidSuperheated Vapor Tv,>Ts (@pp,)

FIG. 21. Dropwise bulk condensation.

coupling the liquid and vapor region is given by Eq. (63). However, the differential form of the energy equation for the vapor becomes quite complex because of the vapor motion and expansion taking place simultaneously. A simplified solution of this problem has made (53) by considering the adiabatic expansion of a sphere of vapor of radius Rv(t),in Fig. 21, containing a liquid drop at the center, and writing the integral form of the first law of thermodynamics for the system. Th e assumption is made that the drop is always at a uniform temperature, i.e., the liquid has a large thermal conductivity, and that no relative motion occurs between the drop and its surrounding vapor, i.e., no slip. A perturbation procedure is then used to solve for the drop size as a function of time. Another theoretical solution to a problem similar to the above has also been made (54). Measurements of the growth rates of droplets of water, methanol, and ethanol from supersaturated vapors were made. By comparing the calculated growth rates with the theoretical ones, the condensation coefficients were computed. If relative motion exists between the drop and the surrounding vapor, and if the primary resistance is in the continuous phase, provided that the growth rate is not too large, one can use relations for heat and mass transfer obtained for steady conditions for the unsteady processes. This

CONDENSATION

WEAT

TRANSFER

225

has been done for calculating the growth of spherical hailstones (52), where the relations used were Nu

=

2.00

+ 0.60Pr1/3Re1/2

(77)

Sh

=

2.00

+ 0.60S~l/~Rel/~

(78)

for heat transfer, and for mass transfer, where Sh is the Sherwood number, giving the mass diffusion coefficient, and Sc is the Schmidt number, giving the ratio of momentum to mass diffusivities. 2. Subcooled Liquid Drop

If subcooled liquid drops are sprayed into a bulk vapor held at a constant pressure P, , but superheated to temperature Tvm, condensation may or may not initially occur on the surface of the drop, depending on the relative rates of heat transfer in the liquid and in the vapor at the surface of the drop. T h e physical representation is shown in the lower part of Fig. 21. If the superheat remains constant, the drop will obviously evaporate, ultimately. However, if the drop is moving with the vapor in an adiabatic system, then the superheat will be reduced. Depending on the relative amounts of liquid and vapor present, net condensation may or may not occur. For a constant system pressure, the solution of this problem is somewhat simpler than the one described previously, shown in the upper part of Fig. 21, in that the surface temperature of the drop, T, , is constant. If the bulk vapor is saturated to begin with ( TVm= T,) then only the liquid domain of the temperature field remains to be solved. This is given by

with initial and boundary conditions T(r,0 ) = To,

T(0,t )

=

finite,

T ( R ,t ) = T , (>T,).

(80)

T h e unknowns T(r, t ) and R(t)are coupled by the energy equation at the boundary:

with R(0) = R , .

226

HERMANMERTE,J R .

Even with the many simplifications in this system of equations, a closed form solution is mathematically complex. A spray-type condenser, described by Eqs. (79)-(82), has potential in the automotive application of a steam Rankine cycle in that a small size of condenser is possible. T h e size of condensers necessary has been a great handicap to the development of a compact closed-water cycle steam power plant. I t is necessary, of course, to have a source of subcooled water available, but this could be achieved with the use of the present heat exchanger (radiator ) on a partial by-pass basis as illustrated in Fig. 22. SPRAY CONDENSER

SPRAY PUMP

FIG. 22. Application of a spray-type condenser to Rankine-cycle power plant.

B. CONDENSATION IN LIQUID BULK That the process of condensation of vapor injected into a large subcooled bulk of liquid is dynamic is obvious to anyone who has been in the vicinity of steam being bubbled into cold water. Whether a vapor bubble grows or collapses in a bulk liquid depends primarily on whether the liquid is superheated or subcooled with respect to the saturation temperature corresponding to the pressure within the bubble. T h e literature is quite extensive in the treatment of bubble dynamics arising with boiling and cavitation, and it appears impractical to cover the area here. A comprehensive review covering an earlier period is given in

CONDENSATION HEATTRANSFER

227

Zuber (55). Florschuetz and Chao (56) have formulated the general problem, and obtained analytic solutions for the inertia dominated and thermal dominated cases, where the energy and momentum equations are respectively neglected. Because of the nonlinearity of the general problem, it was necessary to solve the intermediate case, where both inertial and thermal effects are present, by numerical means. Good agreement is presented with experiments conducted to eliminate the relative motion induced by buoyant forces. Where buoyant forces were not eliminated, rather severe discrepancies arose (57). V. Surface Condensation Rates

Prediction of heat transfer rates with film condensation has been quite successful when compared with dropwise condensation. Both will be considered in this section. I t would be desirable, however, first to be able to predict a priori the prevailing mode of condensation, dropwise or filmwise.

A. PREDICTION OF MODE As mentioned previously, it has been experimentally observed that steam will condense dropwise on metal surfaces only with the use of promoters, except where noble metals are used. This would seem to indicate that the criteria for the onset of film or dropwise condensation should depend on surface energy relationships. A criteria has been developed (8,58) based on Eqs. (8) and (1 1). Figure 23(a-c) represents the progressive changes in surface free energy as a solid is brought into contact with a liquid, beginning with a liquid in contact with its saturated vapor and a solid in a vacuum. From Eq. (8), the work of adhesion is given by Wl,

= Ulv

+

uc

- u1c

.

(83)

T h e work of adhesion between the solid and the saturated vapor is defined as r E ,

wvc= u,

- (3°C

=TE.

(84)

sometimes referred to as the equilibrium film pressure (59 or the equilibrium spreading pressure (8). Substituting G~ from Eq. (84) into Eq. (83), the work of adhesion becomes

rEis

WlC = ulv

+ + c%C

*E

-

.

(85)

HERMAN MERTE,JR.

228

vapor

C vocuum

K c s o l i d

Cc

+

C*v

Cvc

+

C'pv

Uf C

(b)

(0)

(C)

FIG. 23. Interface surface free energy.

then the surface free energy in the final state, Fig. 23c, will be lower than the surface free energy in the initial state, Fig. 23a, and the liquid will spread spontaneously on the solid. As with Eq. (1 l), a spreading coefficient is thus defined as

s=

UVC

+

T E - (ulV

f

ulC)*

(87)

Thus, from Eqs. (86) and (87), the criteria for film and dropwise condensation is given by: S 30 S

<0

spreading-filmwisecondensation non-spreading-dropwise condensation.

In order to express the solid surface energies in terms of the contact angle 8, Eq. (12) is rewritten for the drop in Fig. 4 as

Substitution of Eq. (88) into Eq. (87) gives

s = T E + ulv(cose - I).

(89)

CONDENSATION HEATTRANSFER

229

I t has been shown that on low energy surfaces such as polytetrafluoroethylene (PTFE), the equilibrium film pressure rE is negligible (8).T h e spreading coefficient S , Eq. (89), then becomes

T h e spreading coefficient is thus a function of the liquid-vapor surface free energy and the contact angle of the liquid on the particular solid surface. I n principle, dropwise condensation should take place on any low-energy surface for which the contact angle is greater than zero, since alr in Eq. (90) is a positive quantity, and also would be independent of alv (except insofar as it influences the contact angle 0). T h a t this is not a sufficient criteria has been determined experimentally (59). T h e concept of a critical surface tension uCrappears to have been successful in describing the spreading behavior of a great variety of liquids on various surfaces (60),and in determining the onset of dropwise condensation on a limited number of surfaces (58, 61). By measurement of the contact angle of a series of homologous liquids on different surfaces, it was observed that for a given surface, the plot of cos 0 vs. alv yielded an approximately straight line, as with samples shown in Fig. 24 (60). T h e critical surface tension ucr for a given solid 10

I

\

0 8 -

\

I

I

\

\

'\

\

\\ \\

06-

\

\

*\ *\\

\

KEI-F

*\ \

cn

\ TEFLON

Lc

0 0

04-

PLAT I NUM WITH PERFLUOROBUTYRIC ACID MONOLAYER

02-

PLATINUM WITH PERFLUOROIAURIC ACID MONOLAYER

\

\

\

o\ O\

\

3

00

5

10

15

a,,-DY

20

25

30

NE /CM

FIG. 24. Definition of critical surface tension. From Shafrin and Zisman (60).

230

HERMANMERTE,JR.

is then defined as the surface tension at which the liquid completely wets the surface (i.e., 0 = 0), and is determined by extrapolating the data to cos 0 = 1. T h e linear relationship was also found to be approximately valid for a variety of nonhomologous liquids, although the intercept giving the critical surface tension is less well defined (60). Nevertheless, the critical surface tension is a useful parameter since it is a characteristic of the solid only (which would include a surface layer, if present). Dropwise condensation (nonspreading) will take place when the liquid-vapor surface tension ulv is greater than the critical surface tension of the solid (and its surface layer). T h e critical surface tension of some solids are listed in Table VIII. Table I X lists the liquidvapor surface tensions for several different liquids, and the type of condensation taking place on TFE coated surfaces. I t is noted that filmwise condensation takes place when the liquid-vapor surface tension is on the order of the critical surface tension for Teflon from Table VIII, uCr= 18 dynes/cm. Most metallic surfaces are so-called high-energy TABLE VIII CRITICAL SURFACE TENSIONS OF SOME SOLIDS~ Solid

TFE (Teflon) Polyethylene Kel-F Polystyrene Polyvinyl chloride Nylon Platinum with perfluorobutyric acid monolayer Platinum with perfluorolauric acid monolayer a

18 31 31 33 39

46 10

6

From Shafrin and Zisman (60).

surfaces, but the critical surface tensions of these does not yet appear to have been investigated. Water has one of the highest surface tensions among the more common liquids, and the fact that film condensation occurs with most metals indicates that the critical surface tension of these metals is above the liquid-vapor surface tension of water. T h e effect of dropwise condensation promoters then is to form a layer which reduces the effective critical surface tension below that of the liquid, Nonspreading of certain liquids on some high energy surfaces such as

CONDENSATION HEATTRANSFER

23 1

TABLE IX

CONDENSATION ON TETRAFLUOROETHYLFNE SURFACE^ Substance

T (“C)

Ethylene glycol Nitrobenzene Aniline Water CCLI Benzene Methanol

120 110 110 100 76.8 80 64.7

olv( T

38.1 33.3 33.7 60.8 20.2 20.5 18.9

)

Mode of condensation Dropwise Dropwise Dropwise Dropwise Fi 1mwise Filmwise Filmwise

From Davies and Ponter (58).

platinum takes place even with no promoters present, and is attributed to the liquids having the property of being “autophobic,” or unable to spread on its own adsorbed film (62).

B. DROPWISE CONDENSATION I n attempting to predict or correlate heat transfer rates with dropwise condensation on solid surfaces, or in evaluating experimental results, a number of possible complicating factors should be kept in mind. These include the effects of noncondensable gases, promoter, surface thermal properties, and the droplet removal mechanism. Each will be discussed in turn.

1. Noncondensables

It is important that noncondensables be removed to a high degree for the best performance and reproducibility of condensation (63). This factor will be dealt with in the discussion on condensation of mixtures.

2. Promoters T h e promoters used to produce dropwise condensation can introduce variables. For stable operation, the entire system must be saturated with promoter, not just the condensing surface (64). Cupric oleate in steam in concentrations between 3-50 ppm resulted in good dropwise condensation on copper, stainless steel, inconel, and copper-nickel, and had an effective continuous life of at least 10,000 hr (20). However, in a case

232

HERMANMERTE,JR.

where the operation was intermittant on a daily basis, with exposure of the surface to air in between, the effectiveness decreased by one-half over a ten day period. T he type of promoter also can have an influence. Four different promoters were used in one study (36), keeping all other quantities constant. A maximum difference of 50% in the heat transfer coefficient resulted between the various promoters, all with dropwise condensation. It was also observed in this work that if cooling of the surface continued while the surface was exposed to air, such that water vapor in the air would be deposited on the surface, then the promoter remained effective upon starting up again. However, if the surface was permitted to dry up in contact with air, a breakdown in effectiveness of the promoter occurred, accompanied by a discoloration of the surface. I n another case (29), the surface was baked for several hours between condensing tests without exposing the surface to air. Upon condensing on the surface afterward, the number of nucleating sites was reduced by one half. T h e baking process most likely removed liquid that had become trapped in microscopic pits. By flooding the entire surface with liquid before condensing again, the original number of sites was reactivated. One might expect that with continuous operation over a long period of time all potential sites would become activated by a random process of “seeding” from adjacent active sites. Dropwise condensation was obtained with no promoter by coating the outside of a $-in. O.D. admiralty brass horizontal tube with a 0.0001-in. film of PTFE (polytetrafluorethylene) (65). At 7-in. Hg of steam pressure at q/A = 13,000 BTU,’hr ft2, the heat transfer coefficient increased from h = 2500 BTU, hr ft2 O F with film condensation for the uncoated tube to h = 6500 BTU/hr ft2 O F with dropwise condensation for the coated tube. T h e latter value includes the additional thermal resistance of the coating. T h e advantage of a stable coating over promoters to produce dropwise condensation is obvious: the additional operation of maintaining promoter concentration is eliminated and the vapor is not contaminated.

3. Eflect of Surface Thermal Properties It has been experimentally demonstrated that the thermal properties of the condensing surface exert considerable influence on dropwise condensation (64). Saturated steam at atmospheric pressure was condensed on the underside of a horizontal surface with three different metals: stainless steel, zinc, and copper. T o eliminate differences due to surface effects, each material was plated with 0.005 in. thick gold,

CONDENSATION HEATTRANSFER

233

each promoted by oleic acid. T h e heat transfer coefficient is defined in the standard way by

It was observed that in all cases the heat transfer coefficient h was independent of the temperature difference (Tv- Tw)and thus also independent of the heat flux. This could be a consequence of the particular character of a gold surface. T h e results are tabulated in Table X, with h given in BTU/hr ft2 OF. It is noted that for a 20-fold increase in the thermal conductivity of the TABLE X DROPWISE CONDENSATION OF STEAM ON DIFFERENT MATERIALS, ATMOSPHERIC PRESSURE kw

Material

Stainless steel Zinc Copper

(BTU/hr ft

10 63 220

O F )

h Gold plated (64) 2,000 4.500 10,500

h Bare (34)

8.000 42,000

substrate material, the condensing coefficient increases fivefold, both for the gold plated surface (64) and the bare surface (34). T h e absolute differences between the plated and bare surfaces are indicative that surface properties as well as substrate thermal properties have an important effect. T h e effect of substrate thermal properties is postulated as resulting from the temperature variation along the metal surface between the drop center, where the surface is effectively insulated by the drop itself, and the triple interface at the edge of the drop where the majority of heat transfer takes place (64). This is also implied in models for dropwise condensation proposed much earlier (21,22). An analytical basis for the influence of the substrate was obtained by solving the twodimensional conduction problem for the substrate, assuming that the drops effectively insulate the surface (66).I t was found that for stainless steel as the condensing surface about 84% of the total resistance was that occurring in the plate itself due to heat flux nonuniformity over the surface, while for copper this was 20 yo.

234

HERMANMERTE,JR.

4. Droplet Removal It was shown in Fig. 15 that for a given A T , the heat flux depended on the population density of the drops. These experiments were conducted at short times after admission of vapor to the test vessel so that the mechanism of drop removal was not a factor, and at low pressure so that the growth rates were low. One might anticipate that as the drops continue to grow in size the population density would change, and would influence the heat flux. That this is indeed the case is shown in Fig. 25,

0

50

I00

I50

21 3

RECIPROCAL DISTANCE TO NEAREST NEIGHBOR, CM-'

FIG.25. Effect of nearby drops on drop growth rate. P (water vapor) From McCormick and Westwater (33).

=

19 mm Hg.

where the effect of nearby drops on the growth rate is presented. T h e ordinate D2/t arises from the following: T he relation for spherical phase growth in an infinite medium where the process is governed by heat transfer, due to Scriven (67),as applied to the case of condensation (33) is

where D is the drop diameter, and t is time. For a given subcooling, A T , = ( T , - Tm), the left side of Eq. (92) should remain constant. This is not the case, as seen in Fig. 25; for a given subcooling the coefficient for the growth equation depends on the nearest-neighbor distance, the smaller the spacing the lower is the growth rate. This effect is attributed to competition between drops for the same vapor.

235

CONDENSATION HEATTRANSFER

All of the results in Fig. 25 plus others at a different pressure were correlated by a single expression (33), given by G = -0[ 2 t

+

I"

(93)

(0.79 Ts A T ) pv '

where G is a function of nearest-neighbor spacing only, and is plotted in Fig. 26. As the spacing between drops decreases the growth rate, and

0

P = 19 MM

I

0

50

I

1

100

I50

2

RECIPROCAL DISTANCE TO NEAREST NEIGHBORS, CM-'

FIG.26. Correlation of growth rates of condensation water drops on copper promoted by benzyl mercaptan. From McCormick and Westwater (33).

hence heat transfer rate, associated with each drop decreases. Complicating the process further is the variation in drop sizes and coalescence when contact between adjacent drops occurs. Figure 27 shows an example of measured drop size distributions for several population densities, and is compared with the maximum theoret-

HERMAN MERTE,JR.

236

12,50C

I

I0,OOC

V

2n W

a

d n W

+ W

750C

I

4

n LL

0

m

a 0

n

MAXIMUM THEORETICAL NUMBER, SQUARE PACKING

500C

U

0

OP OF POPULATION, 17,800/CM2

n W m

I 3 z

250C

C

00

160

240

320

400

4

D, DROP DIAMETER, MICRONS

FIG. 21. Average condensation water drop distribution for maximum diameters between 500-3000 microns. From McCormick and Westwater (33).

ical one. Each point represents drops counted in the size range * 2 0 p diameter. Note that as population density increases, it is chiefly in the small sizes because coalescence then takes place with the larger sizes. Any realistic model for representing dropwise condensation must take into account the population density, size distribution, and the related mechanism for removal of the droplet from the surface, which will influence the maximum possible size. T h e influence of the size distribution and maximum droplet size was studied theoretically by means of a computer model (68). I t is assumed that condensation occurs only on the drops, and the rate is limited by conduction through the drop. It is similar t o the work in Fatica and Katz (21) except that now a

CONDENSATION HEATTRANSFER

231

random distribution of sizes and various states of crowding are permitted. T h e drops grow by condensation and coalescence when contact between adjacent drops occur, and the drops are removed when some arbitrary maximum size is reached. T h e solution, giving a mean heat transfer coefficient as functions of site population density and maximum drop size, was carried out for the condensation of water vapor at atmospheric pressure, using a digital computer and a Monte Carlo technique. T h e result is shown in Fig. 28, in which R is the maximum permitted radius 2

I -

y

- - - - - - - - - --

10-1

I

2

3

4

5

6

7

8

9

log N - sites/,.,2

FIG.28. Theoretical curves from a computer model relating heat transfer coefficient to nucleation site density and maximum drop size. From Gose et al. (68).

of a drop. Included is the range of heat transfer coefficients observed in experiments with steam at atmospheric pressure condensing on vertical surfaces. For large nucleation site densities, the heat transfer coefficient is dependent only on the maximum permitted radius. For the small cm) and site densities greater than lo8 sites/cm2, maximum radius ( the predicted heat transfer coefficients are at least ten times greater than have been actually observed. This implies that extremely large heat transfer coefficients could be obtained if a mechanism for efficient removal of these drops from a condensor surface could be developed (68). T o yield the heat transfer rates obtained in dropwise condensation experiments, Fig. 28 indicates that, according to the model, the nucleation site density must be greater than 5 x lo4 sites/cm2.

238

HERMANMERTE,JR.

5. Correlations I n view of the above complexities it is surprising that any success in correlating experimental data is possible. Care must be exercised in applying particular correlations derived on the basis of experimental work to see that conditions of similarity are met. Using dimensional analysis to obtain the dimensionless parameters and experimectal data for the functional relationship, the following correlations were obtained describing dropwise condensation of steam on vertical surfaces (69): Nu

1.6 x 10-4(Re)-0.*4(Pr)1/3(IIk)l.l6

=

for 8 x

< Re < 3.3 x

(94)

and Nu

=

2.5 x 10-6(Re)-1.57 (Pr)1/3(IIk)lJ6 for

3.3 x

< Re < 1.8 x

(95)

where

All physical properties are evaluated at the saturation temperature. Equations (94) and (95) are compared with independent data for steam in Fig. 29, with reasonable agreement at higher values of d T,corresponding to larger values of Re. These equations are also compared in Fig. 30 with data for dropwise condensation of ethylene glycol at various pressures on a copper surface promoted with oleic acid. Because of the deviation at low values of Re, a modification to these equations has been suggested (32) as: Nu

=

1.46 x 10-6(Re)-1.63(Pr)l/z(IIk)l.16.

(96)

This covers a range of Pr from 1.65 to 23.6, and 17k varies from 7.8 x to 2.65 x Equation (96) is plotted in Fig. 31 and compared with data for both steam and ethylene glycol. Fatica and Katz (21) derived a quantitative expression for the heat transfer coefficient with dropwise condensation based on the heat transfer being conduction limited through the drop, and with uniform drop size

CONDENSATION HEATTRANSFER 239

240

HERMANMERTE,JR.

I

o3

lo2

-I

10

Re

FIG. 31. Improved correlation for dropwise condensation on vertical surfaces. From Peterson and Westwater (32).

and uniform distribution. T h e application of the equation, however, requires the experimental determination of a number of quantities; the fractional area covered by the drops, the receding, average and advancing contact angles, and the overall heat transfer coefficient through the bare areas between the drops. T h e necessity for this latter quantity results in limited usefulness of the equation. Another analysis for dropwise condensation had been begun (70) where the heat transfer is also conduction limited by the drop, but it requires additional work. A further analysis considers the conduction through the drops ( 3 4 , but takes into account the effect of surface tension on the temperature on either side of the liquid-vapor interface, plus the interfacial temperature difference. A distribution of drop sizes is assumed and adjusted to fit experimental data of heat transfer rates by means of an arbitrary coefficient. Five parameters must be specified, which may make the specific application of the correlation difficult. T h e various parameters are quite lengthy, and the reader is referred to LeFevre and Rose (35). An analysis of dropwise condensation of liquid metals assumes that

24 1

CONDENSATION HEATTRANSFER

the unwetted portions of the heat transfer surface do not participate in the heat and mass transfer (72). T h e temperature distributions in the drop and wall are calculated with an analog computer, assuming that the heat of condensation is given up at the drop surface. T h e solution appears to require at least one empirical constant for a system, and a mean drop diameter and the fractionalcoverage of the surface by the drops. T h e assumption of a quasi-steady temperature distribution in a condensation drop permits an analytic solution for the drop growth rate, given (19) by

odd

where P, are Legendre polynomials of the first kind. Comparison with a particular set of experimental data is shown in Fig. 32. T h e slow rate 1800 SATURATED WATER VAPOR Tv = 70" F ro = 5 MICRONS

1600

T, -Ts

0 014"

I400

1200

N

9

1000

0

cc 0 I N

,

800

0

600

400

200

0

0

I

I

2

4

6

1

I

0

10

12

TIME, MINUTES

FIG. 32. Comparison of steam condensation drop growth with quasi-steady predictions. From Umur and Griffith (19).

242

HERMAN MERTE,JR.

of growth might be noted, and is a consequence of the very low pressure used; 0.363 Ibfjin2. Whether such good comparison would arise under more dynamic conditions is open to question. Using Eq. (97) and a model similar to that described in Ivanovskii et al. (71)) an expression was derived for the overall heat transfer coefficient with dropwise condensation (66):

rj are the radii of the active or growing drops, rk are the radii of the nonactive drops, and hi is the interfacial heat transfer coefficient at the surface of the drop. As a result of their experimental observations Sugawara and Katsuta (31) conclude that the heat transfer in condensation takes place between as well as on the drops, forming a thin film which then fractures

t

0' 0

5

0

LARGE SURFACE

0

S M A L L SURFACE

10 ATo C

FIG. 33. Comparison of calculated and experimental values of condensation heat transfer coefficient. From Sugawara and Katsuta (31).

243

CONDENSATION HEATTRANSFER

to form drops when a critical thickness is reached. T o predict the rate of growth of this film, hence the heat transfer rate, requires the solution of two coupled one-dimensional transient conduction equations; one for the thin film and the other for the semiinfinite vapor domain. T h e sample calculations, shown in Fig. 33, appear to consider only the heat transfer between drops. When compared with experiments the ratio of hexp/hcomputed decreases as A T increases, most likely because of the decrease of average area of contact between the vapor and the condensing metal surface, i.e., the bubbles appear to act as thermal insulators in this model. An improvement might be to combine the heat transfer between the drops and the heat transfer through the drops, as in Fatica and Katz (21) or McCormick and Baer (70), and then prorate the contribution of each to the total heat flux on the basis of a fractional coverage of the surface area.

6 . EfJect of Vapor Velocity I n one research the velocity of condensing steam, at atmospheric pressure, was varied as it swept past the vertical surface, copper promoted with cupric oleate (72). T h e results are shown in Fig. 34, and appear to indicate that there exists some critical velocity. T h e velocity is the mean between inlet and exit, and varies across the plate because of the con-

LL

NI

I-

LL I

a

I \ 3 + m h

0 al Jz

0

I

2

3

VAPOR VELOCITY

4

5

6

7

8

FT/SEC

FIG.34. Effect of mean vapor velocity on dropwise condensation. From Roblee et al. (72).

244

HERMAN MERTE,JR.

densation taking place. From the behavior of the heat transfer coefficient, decreasing as d T , increases, it appears that noncondensables were present and exerted considerable influence. T h e noncondensables were measured and given as 0.0036 yo, a seemingly insignificant amount. However, because of the condensing process, the migration of the vapor molecules toward the condensing surface will “drag” gas molecules along and cause them to become concentrated at the surface, as has been shown experimentally (36),resulting in an increased thermal resistance. I n another experimental work with a vertical surface (48) it was observed that steam velocity had a marked effect on A T at high heat fluxes, but very little effect at low fluxes. An interesting experimental study was conducted to determine the influence of centrifugal acceleration on dropwise condensation (73). A 6-in. vertical copper cylinder, on which dropwise condensation of steam took place on the outside, was rotated to produce 48 g. Cooling water was circulated on the inside of the cylinder, and because of instrumentation limitations it was possible to measure only the overall heat transfer coefficient, which decreased as acceleration increased. If the inside single phase coefficient remains constant or increases with acceleration, as appears reasonable, then the outside dropwise condensation coefficient must decrease with increasing acceleration. It is believed that the shearing action of the vapor on the drops inhibited their early removal by centrifugal action and elongated them such as to effectively further insulate the surface.

C. FILMCONDENSATION 1. Classical Derivation T h e theory of laminar film condensation on vertical and inclined surfaces was first formulated by Nusselt (44). Improvements have been made over the years, but excepting the behavior with liquid metals, the original theory has been reasonably successful. T h e formulation of this pioneer work will be reviewed briefly here since it serves as the basis for many subsequent theoretical efforts. Figure 35 shows the model and notation followed. T h e downward flow of the film under the action of gravity is laminar, and all fluid properties are considered constant. If the vapor is stagnant or has low velocities, shear stresses at the liquid-vapor interface may be reasonably neglected, and if the film is presumed to be thin, then momentum effects in the liquid may also be neglected. Thus, as far as forces are

CONDENSATION HEATTRANSFER

245

concerned it is necessary only to consider equilibrium between the gravity and viscous forces in the liquid film. Applying the momentum equation to the control volume dx shown shaded in Fig. 35, (8 - Y)(Pl - Pv) g sin 0

= Pl(dU/dY),

(99)

Saturated Stagnant Vapor

Liquid

Film

x FIG.35. Physical model for film condensation.

Integrating Eq. (99) from y = 0 to y = 6, with boundary conditions u = 0 at y = 0, gives the velocity profile in the film at any position x

depends on x only insofar as 6 depends on x. T h e mean velocity in the film is given by

u

and the velocity at the liquid-vapor interface is u(6) = I.%. Neglecting the convective heat transfer in the liquid film, which would take place in the x direction, only conduction in the y direction remains, described by d2T/dyz= 0

(102)

246

HERMANMERTE,JR.

with boundary conditions T ( y = 0) = T , and T ( y = 6) = T , . It is here that the interfacial temperature drop is neglected, in considering that T ( y = S) = T , . Using Fourier’s equation which relates the heat flux to the temperature distribution, q/A = -kdT/dy.

(103)

Equation (102) gives a linear temperature distribution which can be put in the form Q / A= h(Ts - Tw)/6.

(104)

T h e heat transfer coefficient is defined by

h=

4lA

TS - Tw

-

ki

6

*

To obtain the as yet unknown thickness S(x), the conservation of mass principle is applied, which states that any increase in the mass flow of the film must come from condensation at the liquid-vapor interface, which is related to the heat transfer rate. Let the condensate film flow rate per unit width of the wall be

then, from the energy equation

Substituting a from Eq. (101) into Eq. (107), a differential equation in 6 results,

Assuming that condensation begins at x integrates to

= 0,

6

=0

and Eq. (108)

Substituting Eq. (109) into Eq. (105) gives the local heat transfer coefficient =

[

h 3 p 1 ( p 1 - pv) g sin Ohfg 4(Ts - T,) PIX

1

1/4.

(1 10)

CONDENSATION HEATTRANSFER

247

One notes that the heat transfer coefficient decreases as x increases because of thickening of the liquid film. T h e mean heat transfer coefficient over a length L is obtained by integrating Eq. (1 10)

Defining a Nusselt number, Eq. (111) is expressed in dimensionless form as

Defining a Reynolds number of the liquid film by

where wL is given by Eq. (106) at length x = L. Neglecting the subcooling in the liquid film, the energy equation gives, from Eqs. (105) and (107), W,hfg

=

h,(T, - T,)L.

(1 14)

Substituting Eq. (1 14) into Eq. (1 13) for w, , and then Eq. (1 11) into Eq. (1 13) for h , , taking 0 = 90°, and for the state far removed from the critical point of the fluid such that p1 pv , we obtain for a vertical plate

>

where u is the kinematic viscosity. Since ( ~ : / g ) ’ / ~has units of length, the left-hand side of Eq. (1 15) could be interpreted as a Nusselt number.

2 . Laminar Film Condensation T h e accomplishments in laminar film condensation since the work of Nusselt will be subdivided into four groups, depending on the mechanism by which the liquid condensate film is set in motion and removed. T h e thickness of the film governs the heat transfer rate and this is in turn dependent on the driving force for liquid motion. T h e four driving forces are gravity body forces, forced convection or vapor shearing action, rotational or centrifugal forces, and miscellaneous (magnetic, electrostatic, capillary) forces.

248

HERMAN MERTE,JR.

a. Gravity Body Force. T h e first modification to the Nusselt theory was to apply a correction to take into account the heat capacity of the liquid film (74). T h e heat transfer at the wall is greater than the energy from the latent heat by the subcooling of the liquid film that takes place, it being saturated only at the liquid-vapor interface. An approximation for the correction was developed and consists of replacing h,, in Eqs. (107)-( 112) by hi, as follows:

( + 0.4

hf, = hfg 1

ho

This form of the correction applies for values of ( C , d T/h,,) up to about 3. This same form has been applied to account for superheating effects in film boiling (74). An approximate analysis using an integral procedure, removing the

9-5-; -4 1.6 X

F:

= 100, ALSO FOR NO ACCELERATION TERMS

a 0 OI

04

12

08

16

2.0

C,AT hb

FIG. 36. Local heat transfer from solution of complete boundary layer equation. From Sparrow and Gregg (76).

CONDENSATION HEATTRANSFER

249

restriction of neglecting the convection heat transfer ( 7 3 , resulted in h,, in Eqs. (107)-( 112) being replaced by

This is listed as being valid for ( C , A T/h,,) up to one. T h e problem of film condensation on a vertical surface was formulated in terms of the boundary layer theory, and includes the influence of both convection and momentum in the liquid, but still neglects shear at the liquid-vapor interface (76). T h e results are presented in Fig. 36, which show that for Pr > 10 liquid momentum (acceleration terms) can be neglected. T h e momentum effects arise because of the necessity for accelerating the vapor in a direction parallel to the wall as it changes phase to liquid. Even for Pr = 1, Fig. 36 indicates that the error due to neglecting acceleration will be less than 5 % . For small Pr numbers, such as with liquid metals, the momentum of the vapor becomes important, as noted in Fig. 37. Also included is Nusselt's theory (neglecting

1.0

0.9

0.6

0 5

0 0001

I

0 01

0 001

CpAT ~

hf,

FIG. 37. Local heat transfer for low Pr numbers from solution of complete boundary layer equation. From Sparrow and Gregg (76).

01

250

HERMANMERTE,JR.

momentum of the liquid film) in which the ordinate has a value of one, the dimensionless form of Eq. (110). In a further work involving a boundary-layer-type solution to the same problem, the shear forces at the liquid-vapor interface are taken into full account (77). These result in induced motion in the vapor, which must come a t the expense of momentum in the liquid. The liquid thus has a lower velocity when shear is acting, and the heat transfer coefficient should be somewhat smaller. That this indeed is the case for Pr = 1 is seen in the solutions plotted in Fig. 38, and in Fig. 39 for all 1.151

. 1.10 -

WITH INTERFACIAL SHEAR J ~ L

Y

~

_ _ _ _ _ _ WITHOUT INTERFACIAL SHEAR

(Ref. 7 6 )

/

-

I .05 .05.

I00

0.95

/

I

I

I , , ,

I

I

I

, , I .

C A T hf,

FIG. 38. The effect of interfacial shear stress on heat transfer, Pr condensation on vertical surface. From Koh et al. (77).

> 1. Laminar film

of the low Pr number cases. For Pr > 10, interfacial shear can be neglected; and even for Pr = 1, the effect is small, T h e dashed lines in both Figs. 38 and 39 correspond to the same results seen in Figs. 36 and 37, respectively. T h e ordinate = 1 in Fig. 39 again corresponds to the Nusselt theory, so considerable error would arise in applying it to liquid metals. An independent solution was obtained for this same case, where momentum, convection and interfacial shear effects are retained, but using a perturbation scheme on a modified integral boundary layer equation (78). T h e results appear similar to those in Koh et al. (77), except are presented as average values. Figure 40 presents results from

CONDENSATION HEAT TRANSFER

25 1

1.1 NUSSELT THEORY 7

1.0

0.9 -

0.8-

0.7

-

0 0 .. 6 6 --

-WITH

INTERFACIAL SHEAR INTERFACIAL SHEAR (Ref. 76)

_ _ _ _ WITHOUT

0.4L 0.0001

0.001

0.01

c,,nT hf,

FIG. 39. Effect of interfacial shear stress on heat transfer, liquid metal range. From Koh et al. (77).

Chen (78), and for the ordinate value of one is the same as Eq. (112). I n Fig. 41 comparison is made with the theoretical results neglecting interfacial shear (76) and with experimental data for condensing mercury (79).It appears that further work, experimental as well as analytical, is needed to eliminate discrepancies. It is possible that thermal interfacial resistances are in part responsible for these discrepancies. An approximate working equation, valid within 1 yo for the results plotted in Fig. 40, is given (78) as

where h , is the mean heat transfer coefficient, and hNusseltis the mean heat transfer coefficient as computed from the simple Nusselt theory Eq. (1 1l), and

0 .I

HERMANMERTE,JR.

252

0.5 MERCURY DATA MlSRA 8 BONILLA REF. 79

_____ _ _ - -- - _ _ _ _ -

FOR T, SEE FIG. 4 0

____-

_ _ - -__ -_ _ _ _-__

0.4

- -- - --0.3 0.02

____

0.1

0.2 0.3

0 5 0.7 1.0

2.0 3 0

5.0 7.0

10

k,AT

__ Plhf,

FIG. 41.

Comparison with data and theories for liquid metals. From Chen (78).

253

CONDENSATION HEATTRANSFER

<

<

T h e ranges of applicability of Eq. (1 18) are given by 2, y 20, and for liquids with Pr > 1 or Pr < 0.05. Boundary-layer-type analyses were also applied to film condensation on vertical surfaces to solve the problem for variable physical properties of the condensate, notably viscosity (80), and for the case where the plate is nonisothermal (81). A similarity solution also was obtained for film condensation on a flat plate where the gravity field varied in a direction parallel to the plate (82). Interfacial shear is neglected, and the gravity field must vary in a special way, given by ).(g

=

c&

(120)

where C, and C, are constants. Another solution was obtained where the gravity field varied linearly (83). Although experimental data for film condensation on horizontal tubes are quite extensive, that for vertical surfaces is relatively scarce. Four different sources reporting data obtained with steam are mentioned in Spencer and Ibele (84), and which give heat transfer coefficients varying from 22 to 53% above that predicted by the Nusselt equation. More recent results with steam, both saturated and superheated ( 8 4 , are plotted in Fig. 42. T h e data points on the lower part of the figure x DATA OF SHEA 8 KRASE o.30t

0.50

o.60

t

-

0.40 -

NZl"

0.30-

HEAT METER DATA ONLY SATURATED VAPOR ONLY 0 Ts.,-Tw = 16.1' F ,T ,, -T, = 13.5O F 0 0

r

NUSSELT EQUATION, Eq (115)

Tsot-Tw = 13.0°F A Tso~-T,,II.O0F

-2c)

I

~rlx-

0.250.20

-

0.15

-

0.101

40

60

80 100

150

200

I

300

400 5 0 0 6 0 0 7(

FIG.42. Experimental data for condensation of saturated and superheated steam on (84).

a vertical surface. From Spencer and Ibele

254

HERMAN MERTE,JR.

represent superheats varying between 0 and 100°F. These results are considerably below that predicted by the Nusselt relation, and a distinct change in some mechanism occurs, as manifested by a knee in the data. Perhaps a departure something like a transition to turbulence occurs. Also interesting is the fact that at low Reynolds number the heat transfer coefficient remains constant. The heat flux measured with film condensation of saturated n-butyl alcohol vapor on a vertical surface (85) is plotted against the condensate film temperature drop in Fig. 43, and compares well with the Nusselt

10

m

-

e

'0

NO INTERFACIAL RESISTANCE -

x

NUSSELT ANALYSIS

N.-

c

L

.

6

c

3 + m

(T,-T,)

-OF

FIG. 43. Heat flux versus temperature difference for film condensation of saturated n-butyl alcohol on a vertical surface. From Sleger and Seban (85).

theory. A number of different computed curves are included which take into account different interfacial resistances associated with various values of the condensation coefficient. One can conclude from Fig. 43 only that the condensation coefficient lies in the range 0.25 < (T < 1.00. It was found in this same research (85) that noncondensables were being created continuously and it was felt that some of the low values of condensation coefficients reported earlier for this class of material may be due to the presence of noncondensable gases.

CONDENSATION HEATTRANSFER

255

Data obtained with film condensation of saturated potassium vapor on a vertical surface (42) was used, in conjunction with the Nusselt equation ( 1 1 1 ) for determining the liquid film temperature drop, to compute condensation coefficients from Eq. (76). These were shown earlier in Fig. 20 and compared well with the data of others. I n another study of film condensation of liquid metals, the condensation heat transfer coefficient is computed from the kinetic theory, assuming u = 1 and neglecting the thermal resistance of the liquid film (71). Reasonably good comparison is shown with experimental data obtained for lithium, sodium, potassium, and mercury. Experimental data for vertical tubes, which should be similar in behavior to vertical flat plates if the tube diameter is much larger than the film thickness, are shown in Fig. 44 and compared with the Nusselt I

.

I

I

I

1

I

I

0

Y = 1.88.X-1/3 (Ref. I)

X

oxx

m

>

I "I5t -

-

Tube Dia. x 2 2 MM 0 28 M M

'.

Fluid 2-Propanol Methanol 2 Propanol

1-Butanol

0.10 200

300

400

500

600

700 800 900

FIG. 44. Variation of condensing film coefficient with Re for vertical tubes. From Selin (148).

equation ( 1 15). T h e comparison is better if the coefficient is changed from 1.47 to 1.88 as recommended in McAdams ( I ) . Laminar film condensation on the underside of horizontal and inclined surfaces is treated analytically and experimentally in Gerstmann and Griffith (86),and an approximate solution is given for film condensation

HERMANMERTE,JR.

256

on top of a horizontal cooled surface of finite width in terms of the liquid film thickness at the edge (87). This is similar, phenomenologically, to film boiling on the underside of a horizontal surface of finite width, treated in Lewis et al. (88). Analyses of transient laminar film condensation on vertical surfaces have been made (89,90)for the case where momentum effects and interfacial drag are neglected. I n one case (89)the transient arises by suddenly decreasing the wall temperature below the saturation temperature, while in the other, transients in both wall temperature and in the gravity field are considered. Nusselt (44) obtained an average film condensation coefficient for the outside of a horizontal pipe by integrating Eq. (1 10) about the periphery of the tube. T h e result is, for p1 pv

>

h,

= 0.725

[ ki3pi2ghfg AT PlDO

]1/4

’

where Do is the outer diameter of the tube and AT = T , - T, , the temperature drop across the liquid film. Placing Eq. (121) in dimensionless form gives, for horizontal tubes,

where the Reynolds number is given by Eq. (113), in which the mass flow rate of condensate per unit length is used. Experimental data for the horizontal tube are compared with Eq. (122) i n Fig. 45. It is noted that the data are correlated better if the coefficient in Eq. (122) is changed from 1.51 to 1.27. Film condensation on the outside of inclined tubes is predicted reasonably well if the body force in Eq. (121) is multiplied by cos a, where a is the angle of the tube to the horizontal (91),

Equation (123) is compared with experimental data in Fig. 46. A more realistic model for film condensation on inclined tubes is claimed by considering two zones (92);the upper portion of the tube, which may be treated with the Nusselt-type model and where surface tension effects are neglected, and the bottom part of the tube, where the flow is affected by surface tension and may be laminar or turbulent. Boundary-layer-type equations for laminar film condensation on

CONDENSATION HEATTRANSFER

I

I

0.2 20

0

Tube dia. Fluid 2 8 rnm Methanol

A 0.

28rnrn 2-Propanol A 28 rnrn I-Butonol v 42.1rnm Methanol + 42.1 rnrn 2-Propanol 42.1 rnm I-Butano! 40 30

257

*-

P

50

60

70

80 90 100

X = R e = 4 w,+,

FIG.45. Condensing heat transfer data with horizontal tubes compared. From Selin (148).

a

Degrees

FIG.46. Variation of condensing film coefficient with tube inclination. From Selin (148).

single and vertical banks of horizontal tubes have been solved and are reported in Chen (93).

b. Forced Convection. In cases considered now, it is the shear stresses at the liquid-vapor interface that causes the liquid film flow to take place. A study of forced convection condensation over a flat plate using a

258

HERMANMERTE,JR.

boundary-layer type of analysis takes into account the presence of noncondensable gases and interfacial thermal resistance (94).A related work considers the additional effect of superheating the vapor (95). I n both of these a liquid and a vapor gas boundary layer are used, with continuity of shear at the interface serving as the liquid driving force, neglecting the momentum transfer associated with the condensing vapor. T h e combination of body-force and vapor-forced convection induced motion of the liquid film over a flat plate is treated approximately by an integral method (96).An analytical and experimental study was made of film condensation of a saturated vapor moving parallel to a subcooled liquid film with no heat transfer to the solid surface (97). Thus, once the liquid has reached saturation no further condensation will take place. T h e analysis accounts for turbulence in the liquid film, interfacial shear stresses, and assumes that the interface is smooth although experiments show this not to be the case. Comparison of the measurements with the computed wall temperatures and film thicknesses appear reasonably good. An analytic solution, based on the Nusselt assumptions, has been obtained for laminar film condensation of a vapor flowing perpendicular to a horizontal cylinder (98). Nusselt (44) solved the problem of forced convection condensation flow up and down the inside of vertical tubes, using shear stresses at the liquid-vapor interface computed from normal pressure drop data for tubes. This assumes that the film thickness is small compared to the tube diameter. T h e use of the usual friction factor in determining interfacial shear with forced convection condensation has been under discussion (99). Condensation inside horizontal tubes is somewhat more complicated than that inside vertical tubes because of the tendency of the condensate to fill the bottom of the tube at the lower flow rates. According to Jakob (26), if this effect is neglected Nusselt’s theory of condensation on the outside of horizontal tubes should apply. A recent work presented a method for computing the local heat transfer coefficient about the tube (100).Near the top the coefficient is independent of the amount of liquid and Nusselt’s theory applies. Near the bottom the analogy between heat and momentum transfer is used to determine the heat transfer coefficient as with a single phase. Pressure drop in a tube with condensation is treated as pressure drop due to the uncondensed vapor (102).As velocity increases, some recovery of pressure from the axial momentum of the vapor should occur, and is taken into consideration in Silver and Wallis (102) and Ginwala (103). Vapor momentum becomes an important means for obtaining liquid film driving forces in zero gravity, and studies have presented the criteria

CONDENSATION HEATTRANSFER

259

necessary for stable operation (104). An experiment with nonwetting mercury condensing in a tapered horizontal tube at standard and short time zero gravity showed that the effect of gravity was negligible (105). Many correlations of nondimensional types with empirical coefficients have been proposed for condensation in horizontal tubes. Data have been correlated by a single-phase type flow equation of the form (106).

where the Reynolds number is based on the liquid mass velocity equivalent of the mixture of vapor and liquid. Data of condensing methanol and Freon-12 were correlated by the following (107), giving the axial local values of the heat transfer coefficient; for 1000 < (T)($)1/2 DGv

and for 20,000

< 20,000.

(-)(y <

< DGv

100,000

PI

where G, = pvV, , the mass velocity of the vapor. Equations (125) and (126) are plotted in Fig. 47, along with other recent data (108). Reynolds analogy between heat and momentum transfer was applied to the condensation of saturated vapor in a horizontal tube (109). T h e result is an expression for the local heat transfer coefficient given by hzlhL

=

(PL/PP,

(127)

where pa: is the average density of the local liquid-vapor mixture and pL is the liquid density. T h e single phase exit heat transfer coefficient

hL is computed from a general turbulent correlation, given as NU = 0.021Re0.8Pro.4(Prl/Prw)1/4.

(128)

Comparison of Eq. (127) with experiments for steam are shown in Fig. 48.

HERMANMERTE,JR.

260

An empirical correlation which was successful in drawing together within 15 yo the experimental data for the condensation of refrigerants I

1000

I -

I

I I

I

I

I l l l l

I

I

I

I

I I 1 1 1 1

Akers and Rosson A Altman, Staub and Norris 0 Chen T h e present data

I

o

I

-

... ..+

-

-

-

-

-

0.0

-

--

-

-

I

I

I 1

I

I I I I I

I

I

I

1

1

1

1

1

1

-

FIG.47. Correlation of condensation with forced convection in horizontal tubes From Bae et al. (108). 12

+

o

P=

0

P=~ ~ K G K M '

KGC IM'

10- x P = 100KGICM'

0

EQUATION (127)

x

0

0.1

0.2

0.3

0.4

0.5

0.6

07

xx x x

0.8

X

0.9

X

1.0

L O C A L VAPOR Q U A L I T Y - X

FIG.48. Local heat transfer coefficient with condensation of steam. From Ananiev et al. (109).

CONDENSATION HEATTRANSFER

26 1

R-11, 12, 21, 22, 113, 114 obtained in four different laboratories, in both vertical and horizontal orientations, is given (110) as

where Reeg.

+

= Re,(~~/~1)(pl/p~)~ Re1 .~

.

(130)

T h e Nusselt and equivalent Reynolds numbers are the mean between inlet and outlet, while the Re, and Re, used to compute the equivalent Reynolds number are determined as if the vapor and liquid each respectively occupied the tube alone. Further experimental correlations and analytical work dealing with condensation within horizontal tubes are presented in Boyko and Krushilin (111), Rufer and Kezios (112), and Soliman et al. (113). c. Rotating Condensation. By replacing the gravity body force by centrifugal acceleration it is possible to increase the condensation heat transfer coefficient significantly since the liquid film thickness can be decreased. It also provides a mechanism for obtaining liquid film motion under zero gravity. T h e theoretical prediction for condensation on a rotating disc (114) shows that the condensate film thickness is uniform over the entire disc and inversely proportional to the square root of the rotational speed. Experimental results (115) fall about 25 yo below the predicted values. With film condensation on the outside of a horizontal tube rotating about its own axis, the heat transfer coefficient first increased with rotation, then decreased (116). T h e initial increase occurred because of the net thinning of the liquid film, referred to above, owing to centrifugal forces. T h e decrease occurs approximately when the axisymmetric waves in the condensate film degenerate into drops. Interfacial drag on the drops then tends to retard their removal from the surface, resulting in a decrease in the heat transfer coefficient. I n two other experimental works, with rotating vertical cylinders, the film condensation coefficients increased with rotation (73, 117). Other works involving condensation on rotating systems are found in Bromley et al. (118) and Sparrow and Hartnett (119). d. Miscellaneous Liquid Film Removal. Means other than gravity and centrifugal acceleration for removal of the liquid film in condensation have been considered in order to either improve the performance of film condensation, or to provide for its steady operation in zero gravity. I n Singer (120) an electromagnetic field replaces the gravity body forces, but can be used only with liquid metals. For non-conducting fluids, electrostatic fields have been successful in improving performance.

262

HERMANMERTE,JR.

Increases in the heat transfer coefficient by up to threefold were obtained with a dc field (124, and u p to tenfold with an alternating electric field (122). Another method for condensed liquid film removal is by suction through a porous cooled plate (123, 124). Surface tension or capillary forces can also be effective for liquid removal under zero gravity (103) by placing a wicking material in the vicinity of the cooling surface. An attempt to improve film condensation on the outside of a vertical tube was made by the use of transverse vibrations of the tube (125). Previous experiments show that beyond a critical condition, below which the effect of vibration is negligible, the condensation heat transfer coefficient increases continually to about 55 yo above the stationary value. Observation of the condensate film on the tube showed that the liquid was not thrown from the tube, by the vibration, as might be expected, but rather moved from one side of the tube to the other. It was believed that this motion contributed to greater mixing in the film, and hence the improved heat transfer. 3. Turbulent Film Condensation T he laminar character of the condensate film constitutes one of the most basic assumptions of Nusselt’s theory. Since the mass flow rate increases with x on a vertical surface, one might expect that at a given Reynolds number the film becomes turbulent and Eqs. (1 10) and (1 12) no longer apply. T h e Prandtl analogy for pipe flow was used to compute the heat transfer with a turbulent condensate film (126). Based on this a mean empirical equation is suggested for turbulent condensation on a vertical plate:

Th e ReL’ is defined differently here than by Eq. (1 13), as =3= PI

W T s-T w P Plhg

*

Equation (131) is plotted in Fig. 49, taken from Grober et al. (126), along with the Nusselt equation, and includes a wide variety of data. T h e critical transition between laminar and turbulent film flow takes place at ReL’ = 350. An analysis similar to the above was conducted to cover wider ranges of Prandtl numbers (127, 128). T h e results are compared with liquid metals in Fig. 50.

2/3

1/3

k,p, 9

(T*-T,)L

5/ 3

PI h,,

FIG.49.

Heat transfer with condensation on a vertical surface. From Grober et al. (126).

DUKLER'S ANALYSIS (REF 128) ANALYSIS (REF. 127) Pr = 10

.-

0 -

N4

v

42 0.10

0.02 200

I000

10,000

IOC )OO

Re,

FIG. 50. Turbulent condensation on vertical surfaces for various Prandtl numbers. From Lee (127).

264

HERMANMERTE,JR. VI. Mixtures

T h e condensation of mixtures of vapors may include gases which are noncondensable under the conditions present. Mixtures with components of complete mutual solubility of the liquids will behave approximately as simple substances, and the film theories discussed earlier can be used. However, the dew-point temperature of the mixture is used instead of the saturation temperature at the interface. T h e dew-point temperature depends upon the composition of the vapor, and the mol fraction of the condensed liquid must come from a composition diagram of the constituents. T h e problem has been theoretically solved for condensation of a binary mixture on a vertical plate under the action of gravity (129). When one of the components in the gaseous mixture is relatively insoluble in the liquid and does not itself condense, the process is classified as condensation in the presence of a noncondensing gas, and appears to have received the greatest attention in the literature. Discrepancies in the computation of the condensation coefficient using early experimental results have been attributed to the presence of noncondensable gases (38). Figure 51 shows the relative influence of a noncondensable gas on the steam side heat transfer coefficient. T h e minimum inert gas concentration was less than 1 ppm. T h e sensitivity of the heat transfer coefficient to changes in the amount of nitrogen associated with the vapor increases with heat flux and as the system pressure is lowered (130). This effect of heat flux is due to the increasing concentration of the noncondensable gas resulting from the convective flow associated with the condensation of the vapor. With no forced convection, an equilibrium condition is established in which the rate of removal of the gas by diffusion and natural convection away from the condensing surface equals the rate at which it is brought to the interface. T h e resulting increased gas concentration at the interface reduces the partial pressure of the vapor, and hence the vapor condensation temperature. With film condensation the temperature drop across the condensate film is thus smaller, as is the associated heat flux. T h e increasing concentration of the noncondensable gas as the condensing surface is approached is demonstrated well in the experimental results shown in Fig. 52 (63). The influence of the location of a continuously operating vent on the solid condensing surface subcooling is shown as a function of different heat flux levels. As the vent is brought closer to the surface, the subcooling necessary to maintain the heat flux is reduced, since the noncondensable gas concentration near the surface is reduced. Other experimental work has shown that condensation in the presence of noncondensables can be predicted by diffusion analysis (131).

CONDENSATION HEATTRANSFER

265

10 STEAM PRESSURE 0 8 i n Hg I . 5 t n Hg

---___

7

'\"

301n Hg NUMBERS GIVE HEAT FLUX RANGE

23,000-26,000

oi

0

I

0 5 10 NITROGEN CONTENT OF STEAM-VOLUME %

FIG. 51. Effect of noncondensable gas on the steam-side heat transfer coefficient. From Tanner et al. (130).

Several analytical and approximate solutions of forced convection condensation with noncondensables (94, 132) show that if the presence of noncondensable cannot be avoided, its detrimental effects can be reduced by forced convection of the vapor, which reduces the local concentration. A recent analytical work includes both gravity and forced convection effects and results in the same conclusion (133). A number of experimental studies with noncondensables under gravity and forced convection conditions provide empirical correlations for both heat and mass transfer (134-137). Other references dealing with noncondensable gases are cited (33, 34, 36, 48, 102, 138-147).

266

HERMAN MERTE,JR.

FIG. 52. Effect of vent position on the steam-to-surface temperature difference for fixed venting rate. From Citakoglu and Rose (63).

VII. Similarities between Boiling and Condensation Based on the discussion of the condensation process it may be observed that many similarities exist between the boiling and condensation phenomena; (a) Nucleate boiling and dropwise condensation are both nucleation governed phenomena. I n nucleate boiling the vapor bubbles appear to effectively disrupt the boundary layer, and a similar process most likely occurs with dropwise condensation, provided that the drops can be removed before growing too large. (b) For a given d T , the heat flux increases as the number of nucleating sites increase, both for nucleate boiling and dropwise condensation. (c) With nucleate boiling, as d T increases the number of active sites increase until the population density becomes so great that coalescence takes place, giving rise to the peak or maximum heat flux phenomena. It has not yet been demonstrated that a corresponding peak heat flux exists with dropwise condensation on a steady basis as AT is increased. With nucleate boiling the buoyant forces are large compared to the surface forces which hold the vapor bubble on the surface, thus the departure of the bubbles from the heating surface takes place with facility.

CONDENSATION HEATTRANSFER

267

On the other hand, with dropwise condensation the surface forces tending to keep the drop in place are large relative to the gravity forces which might tend to remove it from the cooling surface. It might be anticipated that with a positive droplet removal mechanism (e.g., centrifugal force), corresponding to the buoyant forces on vapor bubbles, that a peak flux will be reached with dropwise condensation as AT increases, beyond which coalescence between the drops will increase. (d) With nucleate boiling, when A T has been increased sufficiently to produce complete coalescence between vapor bubbles the process of film boiling takes place. Likewise with dropwise condensation, it might be anticipated that when the AT is increased such that complete coalescence occurs between adjacent drops, film condensation will be taking place. I n practice, film condensation is much more apt to occur than dropwise condensation unless special “promoters” are used, as discussed earlier, because of the lack of a sufficiently effective droplet removal mechanism. NOMENCLATURE Unless defined otherwise locally, the following apply: a

A BLl

c,

D

f F g

G h

hpg

J

k L m

M n No Nr Nu

P

acceleration area, or defined below Eq. (57) Bond number, Eq. (2) specific heat diameter fractional area covered by drops force acceleration due to gravity, Gibbs free energy per mole total free energy, mass velocity heat transfer coefficient latent heat of vaporization rate of formation of condensation nuclei, Eq. (54) thermal conductivity, Boltzmann constant length mass per molecule mass, molecular weight number of moles Avogadros’ number distribution of embryos of radius r , Eq. (53) Nusselt number pressure

Pr 4 4s

r, R R r* Re

S sc Sh t

T U V

W

W 01,

0

P> Y Y

6 €8

r

“E (5

Prandtl number heat transfer rate latent heat of surface, Eq. (6) radius universal gas constant critical radius Reynolds number spreading coefficient, Eqs. (11),(87) Schmidt number Sherwood number time temperature velocity specific volume mass flow rate work angles defined parameters specific heat ratio film thickness total surface energy, Eq. ( 5 ) Eq. (71) equilibrium film pressure, Eq. (84) surface tension, condensation coefficient

HERMANMERTE,JR.

268 p p P

density dynamic viscosity kinematic viscosity

SUBSCRIPTS

e

i 1 L m S V

c cr

condensed, solid critical

vm W

emitted liquid-vapor interface, incipient liquid based on length L mean “normal” saturation vapor bulk vapor wall

REFERENCES 1. W. McAdams, “Heat Transmission,” 3rd Ed., Ch. 13. McGraw-Hill, New York, 1954. 2. W. D. Garrett, J. Geophys. Res. 73, 5145-5150 (1968). 3. N. K. Adam, “The Physics and Chemistry of Surfaces,” 3rd Ed. Dover, New York, 1968. 4. J. T. Davies and E. K. Rideal, “Interfacial Phenomena.” Academic Press, New York, 1963. 5. E. W. Otto, Symp. Eff. “Zero Gravity” Fluid Dyn. Heat Transfer, AIChE 55th Nat. Meet., Houston, Tex. Part I, Preprint 17a (1965). 6. A. Dupre, “Theorie Mechanique de la Chaleur.” Gauthier-Villars, Paris, 1869. 7. G. Macdougall and C. Ockrent, Proc. Roy. SOC.,Ser. A 180, 151 (1942). 8. H. W. Fox and W. A. Zisman, J. Colloid Sci. 7, 109 (1952). 9. J. Frenkel, “Kinetic Theory of Liquids.” Oxford Univ. Press, London and New York, 1946. 10. W. Thomson, London, Edinburgh Dublin Phil. Mag. J. Sci. 42, 488 (1871). 11. R. C. Tolman, J. Chem. Phys. 17, 333-337 (1949). 12. C. T. R. Wilson, Proc. Roy. Soc., London 61, 240-243 (1897). 13. G. L. Goglia and G. J. VanWylen, J. Heat Transfer 83, 27-32 (1961). 14. P. G. Hill, H. Witting, and E. P. Demetri, J. Heat Transfer 85, 303-317 (1963). 15. M. Volmer and H. Flood, Z . Phys. Chem., Abt. A 170, 273 (1934). 16. H. L. Jaeger, E. D. Willson, P. G. Hill, and K. C. Russell, J. Chem. Phys. 51, 5380-5388 (1969). 17. J. Lothe and G. M. Pound, J. Chem. Phys. 45, 63Ck634 (1966). 18. J. Feder, K. C. Russell, J . Lothe, and G. M. Pound, Advan. Phys. 15, 111-178 (1966). 19. A. Umur and P. Griffith, J. Heat Transfer 87, 275-282 (1965). 20. J. F. Welch and J. W. Westwater, Znt. Develop. Heat Transfer, Proc. Heat Transfer Conf., Univ. of Colorado, 1961, Part 11, pp. 302-309 (1961). 21. N. Fatica and D. L. Katz, Chem. Eng. Progr. 45, 661-674 (1949). 22. L. Trefethan, Drop Condensation and the Possible Importance of Circulation Within Drops Caused by Surface Tension Variation. General Electric Co., G.E.L. Rep. 58GL47 (1958). [Cited in reference 25.1 23. J. J. Lorenz and B. B. Mikic, J. Heat Transfer 92, 46-52 (1970). 24. A. Eucken, Naturwissenschaften 25, 309 (1937). [Cited in reference 26.1 25. W. M. Rohsenow and H. Choi, “Heat, Mass and Momentum Transfer.” PrenticeHall, Englewood Cliffs, New Jersey, 1961. 26. M. Jakob, “Heat Transfer.” Wiley, New York, 1949.

CONDENSATION HEATTRANSFER 27. 28. 29. 30.

269

H. Emmons, Trans. Amer. Znst. Chem. Eng. 35, 109 (1939). M. Jakob, Mech. Eng. 58, 729 (1936). J. L. McCormick and J. W. Westwater, Chem. Eng. Sci. 20, 1021 (1965). U. Grigull, Inst. fur Tech. Thermodyn., Tech. Hochsch., Munich, personal communication (1968). 31. S. Sugawara and K. Katsuta, Proc. Znt. Heat Transfer Conf., 3rd, Chicago 2, 354-361 (1966). 32. A. C. Peterson and J. W. Westwater, Chem. Eng. Progr., Symp. Ser. 62, 135 (1966). 33. J. L. McCormickand J. W. Westwater, Chem. Eng. Progr., Symp. Ser. 62, 120(1966). 34. D. W. Tanner, D. Pope, C. J. Potter, and D. West, Int. J. Heat Mass Transfer 8, 427-436 (1965). 35. E. J. LeFevre and J. W. Rose, Proc. Znt. Heat Transfer Conf., 3rd,Chicago 2,362-375 (1966). 36. E. J. LeFevre and J. W. Rose, Znt. J. Heat Mass Transfer 8, 1117-1133 (1965). 37. A. E. Bergles and W. M. Rohsenow, J. Heat Transfer 86, 365 (1964). 38. E. Ruckenstein and H. Metiu, Chem. Eng. Sci. 20, 173 (1965). 39. R. W. Schrage, “A Theoretical Study of Interphase Mass Transfer.” Columbia Univ. Press, New York, 1953. 40. V. S. Arpaci, “Conduction Heat Transfer.” Addison-Wesley, Reading, Massachusetts, 1966. 41. S. P. Sukhatme and W. M. Rohsenow, J. Heat Transfer 88, 19-28 (1966). 42. D. G. Kroger and W. M. Rohsenow, Znt. J. Heat Mass Transfer 10, 1891-1894 (1967). 43. R. E. Barry and R. E. Balzhiser, Proc. Znt. Heat Transfer Cmf., 3rd, Chicago 2, 318 (1966). 44. W. Nusselt, 2. Ver. Deut. Zng. 60, 541-569 (1916). 45. D. J. Wilhelm, Condensation of Metal Vapors: Mercury and the Kinetic Theory of Condensation. Argonne Nat. Lab. Rep. ANL-6948 (1964). 46. E. M. Mortenson and H. Eyring, J . Phys. Chem. 64, 846 (1960). 3rd, Chicago 2, 47. R. K. M. Johnstone and W. Smith, Proc. Znt. Heat Transfer Conf., 348-353 (1966). 48. D. W. Tanner, D. Pope, C. J. Potter, and D. West, Znt. J. Heat Mass Transfer 8, 419-426 (1965). 49. A. F. Mills and R. A. Seban, Znt. J. Heat Mass Transfer 10, 1815 (1967). 50. H. Wenzel, Znt. J. Heat Mass Transfer 12, 125-126 (1969). 51. E. D. Fedorovich and W. M. Rohsenow, Int. J. Heat Mass Transfer 12, 1525-1529 (1969). 52. R. List, J. Atmos. Sci. 20, 189-197 (1963). 53. R. Puzyrewski, Znt. J. Heat Mass Transfer 10, 1717-1726 (1967). 54. H. Wakeshima and K. Takata, Jap. J. Appl. Phys. 2, 792 (1963). 55. N. Zuber, Appl. Mech. Rev. 17, 663 (1964). 56. L. W. Florschuetz and B. T. Chao, J. Heat Transfer 87, 209-220 (1965). 57. H. C. Hewitt and J. D. Parker, J . Heat Tvansfer 90, 22-26 (1968). 58. G. A. Davies and A. B. Ponter, Int. J. Heat Mass Transfer 11, 375 (1968). 59. G. A. Davies, W. Mojtehedi, and A. B. Ponter, Int. J. Heat Mass Transfer 14, 709-713 (1971). 60. E. G. Shafrin and W. A. Zisman, J. Phys. Chem. 64, 519 (1960). 61. P. G. Kosky, Int. J. Heat Mass Transfer 11, 374 (1968). 62. E. F. Hare and W. A. Zisman, J. Phys. Chem. 59, 335 (1955). 63. E. Citakoglu and J. W. Rose, Znt. J. Heat Mass Transfer 11, 523-537 (1968).

270

HERMAN MERTE,JR.

64. P. Griffith and M. S. Lee, Znt. J. Heat Mass Transfer 10, 697-707 (1967). 65. A. R. Brown and M. A. Thomas, Proc. Znt. Heat Transfer Conf., 3rd, Chicago 2, 300 (1 966). 66. B. B. Mikic, Znt. J. Heat Mass Transfer 12, 1311-1323 (1969). 67. L. E. Scriven, Chem. Eng. Sci. 10, 1 (1959). 68. E. E. Gose, A. N. Mucciardi, and E. Baer, Znt. J. Heat Mass Transfer 10, 15-22 (1 967). 69. V. P. Isachenko, Teploenergetika 9, 81-85 (1962). 70. J. L. McCormick and E. Baer, J. Colloid Sci. 18, 208 (1963). 71. M. N. Ivanovskii, V. I. Subbotin, and Y . V. Milovanov, Therm. Eng. 14, 114-122 (1967). 72. L. H. S. Roblee, J. T. O’Bara, and E. S. Killian, Dropwise Condensation of Steam at Atmospheric and Above Atmospheric Pressures. ONR Tech. Rep. No. 1 , Contr. Nonr 3357(02) (1966). 73. D. C. P. Birt, J. J. Brunt, J . T. Shelton, and R. G. H. Watson, Trans. Znst. Chem. Eng. 37, 289-296 (1959). 74. L. A. Bromley, Znd. Eng. Chem. 44, 2966 (1952). 75. W. M. Rohsenow, Trans. A S M E 78, 1645-1648 (1956). 76. E. M. Sparrow and J. L. Gregg, J. Heat Transfer 81, 13-23 (1959). 77. J. C. Y . Koh, E. M. Sparrow, and J. P. Hartnett, Int. J. Heat Mass Transfer 2, 69-82 (1961). 78. M. M. Chen, J. Heat Transfer 83, 48-54 (1961). 79. B. Misra and C. F. Bonilla, Chem. Eng. Progr., Symp. Ser. 52, 7-21 (1956). 80. G. Poots and R. G. Miles, Znt. J. Heat Mass Transfer 10, 1677-1692 (1967). 81. K. T. Yang, J. Appl. Mech. 33, 203-205 (1966). 82. S. A. Matin, A Similarity Solution for the Condensate Film on a Vertical Flat Plate in Presence of Variable Gravity Field. Amer. SOC.Mech. Eng. ASME Paper 65-AV42 (1965). 83. J. C. Chato, J. Eng. Power 87, 355-360 (1965). 84. D. L. Spencer and W. E. Ibele, Proc. Znt. Heat Transfer Conf., 3rd, Chicago 2, 337 (1966). 85. L. Slegers and R. A. Seban, Int. J. Heat Mass Transfer 12, 237-239 (1969). 86. J. Gerstmann and P. Griffith, Int. J. Heat Mass Transfer 10, 567-580 (1967). 87. G. Leppert and B. Nimmo, J. Heat Transfer 90, 178-179 (1968). 88. E. W. Lewis, J. A. Clark, and H. Merte, Jr., Boiling of Liquid Nitrogen in Reduced Gravity Fields with Subcooling. Dep. of Mech. Eng., Univ. of Michigan, ORA Rep. 07461-20-T (1967). 89. E. M. Sparrow and R. Siege], J . Appl. Mech. 26, 12Ck121 (1959). 90. P. M. Chung, ASME-AZChE Heat Transfer Conf. Exhibit, Houston, Tex. A S M E Paper 62-HT-23 (1962). 91. K. E. Hassan and M. Jakob, Trans. A S M E 80, 887 (1958). 92. A. G. Sheynkman and V. N. Linetskiy, Heat Transfer-Sov. Res. 1, 9Ck97 (1969). 93. M. M. Chen, J. Heat Transfer 83, 55-60 (1961). 94. E. M. Sparrow, W. J. Minkowycz, and M. Saddy, Znt. J. Heat Mass Transfer 10, 1829-1 845 (1967). 95. W. J. Minkowycz and E. M. Sparrow, Znt. J. Heat Mass Transfer 12, 147-154 (1969). 96. H. R. Jacobs, Znt. J. Heat Mass Transfer 9, 637-648 (1966). 97. J. H . Linehan, M. Petrick, and M. M. El-Wakil, ASME-AZChE Nut. Heat Transfer Conf., Ilth, Minneapolis, AIChE Preprint 3 (1969). 98. V. E. Denny and A. F. Mills, 1.Heat Transfer 91, 495-501 (1969).

CONDENSATION HEATTRANSFER

27 1

99. I. G. Shekriladze and V. I. Gomelauri, Int. J. Heat Mass Transfer 13, 942-943 (1970). 100. H. F. Rosson and J. A. Myers, Chem. Eng. Progr., Symp. Ser. 61, 190-199 (1965). 101. E. J. Hoffman, Pressure Drop in Condensation. Amer. SOC.Mech. Eng. ASME Paper 68-WA/HT-28 (1968). 102. R. S. Silver and G. B. Wallis, Proc. Inst. Mech. Eng. 180, Part 1, 36-40, (1965-1966). 103. K. Ginwala, Engineering Study of Vapor Cycle Cooling Equipment for ZeroGravity Environment. WADD Tech. Rep. 60-776 (1961). 104. R. T. Lancet, P. Abramson, and R. P. Forslund, Symp. Fluid Mech. Heat Transfer Low Gravitational Conditions Lockheed Res. Lab., Palo Alto, California (1965). 105. J. A. Albers and R. P. Macosko, Condensation Pressure Drop of Non-Wetting Mercury in a Uniformly Tapered Tube in 1 - g and Zero-Gravity Environments. N A S A Tech. Note NASA TN D-3185 (1966). 106. W. W. Akers, H. A. Deans, and 0. K. Crosser, Chem. Eng. Progr. 54 (lo), 89-90 (1958). 107. W. W. Akers and H. F. Rosson, Chem. Eng. Progr., Symp. Ser. 56, 145-154 (1960). 108. S. Bae, J. S. Maulbetsch, and W. M. Rohsenow, Refrigerant Forced-Convection Condensation Inside Horizontal Tubes. Dep. of Mech. Eng., MIT, Cambridge, Massachusetts, Rep. No. DSR 79760-59 (1968). 109. E. P. Ananiev, L. D. Boyko, and G. N. Kruzhilin, Int. Develop. Heat Transfer, Proc. Heat Transfer Conf., Univ. of Colorado, 1961, Part 11, p. 290 (1961). 110. A. Cavallini and R. Zecchin, Proc. Int. Congr. Rejrig., 13th, Washington, D . C . Paper 2.29 (1971). 1 1 1 . L. D. Boyko and G. N. Kruzhilin, Int. J. Heat Mass Transfer 10, 361-373 (1967). 112. C. E. Rufer and S. P. Kezios, J. Heat Transfer 88, 265-275 (1966). 113. M. Soliman, J. R. Schuster, and P. J. Berenson, J. Heat Transfer 90,267-276 (1968). 114. E. M. Sparrow and J. L. Gregg, Trans. A S M E 81, 113 (1959). 115. S. S. Nandapurkar and K. 0. Beatty, Chem. Eng. Progr. Symp. Ser. 56, 129 ( 1960). 116. R. M. Singer and G. W. Preckshot, Heat Transfer Fluid Mech. Inst. Paper No. 14, pp. 205-221 (1963). 117. A. A. Nicol and M. Gacesa, J. Heat Transfer 92, 144-152 (1970). 1 1 8. L. A. Bromley, R. F. Humphreys, and W. Murray, Condensation on and Evaporation from Radially Grooved Rotating Disks. Amer. SOC.Mech. Eng. ASME Paper 65-HT-26 (1965). 119. E. M. Sparrow and J. P. Hartnett, J. Heat Transfer 83, 101-102 (1961). 120. R. M. Singer, Laminar Film Condensation in the Pressence of an Electromagnetic Field. Amer. SOC.Mech. Eng. ASME Paper 64-WA/HT-47 (1964). 121. H. R. Velkoff and J. H. Miller, J . Heat Transfer 87, 197-201 (1965). 122. R. E. Holmes and A. J. Chapman, J. Heat Transfer 92, 616-620 (1970). 123. N. A. Frankel and S. G. Bankoff, J. Heat Transfer 87, 95-102 (1965). 124. J. W. Yang, J. Heat Transfer 92, 252-256 (1970). 125. J. C. Dent, Int. J. Heat Mass Transfer 12, 991-996 (1969). 126. H. Grober, S. Erk, and U. Grigull, “Fundamentals of Heat Transfer.” McGrawHill, New York, 1961. 127. J. Lee, AIChE J. 10, 540-544 (1964). 128. A. E. Dukler, Chem. Eng. Progr. Symp. Ser. 56, 1 (1960). 129. E. M. Sparrow and E. Marschall, /. Heat Transfer 91, 205 (1969). 130. D. W. Tanner, D. Pope, C. J. Potter, and D. West, Znt. J. Heat Mass Transfer 11, 181-190 (1968).

272

HERMANMERTE,JR.

D. G. Kroger and W. M. Rohsenow, Int. J. Heat Mass Transfer 11, 15-26 (1968). J. W. Rose, Int. J. Heat Mass Transfer 12, 233-237 (1969). V. E. Denny, A. F. Mills, and V. J. Jusionis, J. Heat Transfer 93, 297-304 (1971). W. W. Akers, S. H. Davis, Jr., and J. E. Crawford, ASME-AIChE Nut. Heat Transfer Conf., 3rd, Storrs, Conn. AIChE Preprint 113 (1959). 135. P. D. Lebedev, A. M. Baklastov, and Z. F. Sergazin, Int. J. Heat Mass Transfer 12, 833-842 (1969). 136. Y. M. Vizel’ and I. L. Mostinskiy, Heat Transfer-Sou. Res. 1, 97-105 (1969). 137. C. L. Henderson and J. M. Marchello, J. Heat Transfer 91, 447-450 (1969). 138. H. Hampson, Int. Develop. Heat Transfer, Proc. Heat Transfer Conf., Univ. of Colorado, 1961, Part I , pp. 310-318 (1961). 139. S. J. Meisenburg, R. M. Boarts, and W. L. Badger, Trans. Amer. Inst. Chem. Eng. 31, 622-638 (1935). 140. E. M. Sparrow and S. H . Lin, J. Heat Transfer 86, 430-436 (1964). 141. P. M. Brdlik, I. A. Kozhinov, and N. G. Petrov, J. Eng. Phys. ( U S S R ) 8, 164-166 (1965). 142. E. Marschall, Kaltetech. Klimatisierung 19, 241-245 (1967). 143. S. E. Sadek, Ind. Eng. Chem., Fundam. 7, 321-324 (1968). 144. F. Stern and F. Votta, Jr., AIChE J. 14,928-933 (1968). 145. Y. Taitel and A. Tamir, Int. J. Heat Mass Transfer 12, 1157-1169 (1969). 146. M. N. Ozisik and D. Hughes, Nucl. Sci. Eng. 35, 384-393 (1969). 147. K. I. Chang and D. L. Spencer, Int. J. Heat Mass Transfer 14, 502-505 (1971). 148. G. Selin, Int. Develop. Heat Transfer, Proc. Heat Transfer Conf., Univ. of Colorado, 1961, Part 11, p. 279 (1961).

131. 132. 133. 134.

Natural Convection Flows and Stability .

B GEBHART Sibley School of Mechanical and Aerospace Engineering. Upson Hall. Cornell University. Ithaca. New York

I. I1 111. IV

Introduction

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

213 215 280 282 A . Flat Vertical Surfaces . . . . . . . . . . . . . . . . . 283 B. The Power-Law Variation . . . . . . . . . . . . . . . 286 C . The Exponential Variation . . . . . . . . . . . . . . . 289 D . Convection in Stratified Media . . . . . . . . . . . . . 290 E . Viscous Dissipation . . . . . . . . . . . . . . . . . . 292 F. Additional Effects . . . . . . . . . . . . . . . . . . . 293 G . Summary Concerning Flow Adjacent to Vertical Surfaces . 295 H Convection over Inclined and Horizontal Surfaces . . . . . 296 I . Natural Convection Plumes . . . . . . . . . . . . . . 291 J . Line Source Plumes . . . . . . . . . . . . . . . . . 299 K . Axisymmetric Flows . . . . . . . . . . . . . . . . . 301 V . Combined Buoyancy Mechanisms . . . . . . . . . . . . . 303 VI . Flow Transients . . . . . . . . . . . . . . . . . . . . . 310 VII . Instability and Transition of Laminar Flows . . . . . . . . . 321 VIII . Instability in Plumes . . . . . . . . . . . . . . . . . . . 335 IX . General Aspects of Instability . . . . . . . . . . . . . . . 339 X . Separating Flows . . . . . . . . . . . . . . . . . . . . 342 References . . . . . . . . . . . . . . . . . . . . . . . 346

. The Relevant Equations . . . . . . . . . . . . . . . . . Boundary-Layer Simplifications . . . . . . . . . . . . . . . Steady Laminar Boundary-Layer Flows . . . . . . . . . . .

.

.

I Introduction I n the world around us. we continually encounter transport processes in fluids wherein the motion is driven simply by the interaction of a difference in density with the gravitational field . T h e motion-driving buoyancy effect may be due to a temperature-caused density difference. as in atmospheric and oceanic circulations. in our heated and cooled 273

274

B. GEBHART

enclosures, in,the cooling oil in a power transformer, or in the air current arising from a cooling object. T h e density difference may also be due to a varying composition or phase of a fluid, as in most air rising, in ocean circulations due to differences in salinity or in suspended particulate matter, or in a mixture of liquid and vapor in a steam generator or processing device. All such occurrences are very similar and are termed “natural convection.” They are quite different from transport processes driven by a forcing condition such as an imposed motion or a fan or pump. T h e principle differences are that in natural convection little is known a priori about the resulting flow; the flow and temperature fields are invariably completely coupled and must be considered together; and the flows are relatively weak, i.e., the velocities are always relatively small with the momentum and viscous effects of the same order. T h e total group of such processes is subdivided into two useful classifications. Flows arising in a very extensive medium due to a given inhomogeneity in density or to imposed temperature or energy conditions at a location or at a surface are called external natural convection. Those without a surface, such as plumes, thermals, and jets, are called free boundary flows. Flows arising in a body of fluid contained in a cavity or completely bounded by surfaces are internal natural convection. This classification is not controversial. However, the terminology is. One body of usage calls the external case “free” convection, another uses this term for both cases. There is some sentiment that neither term, natural of free, is ever appropriate and that something more descriptive, such as “buoyancy convection” or “gravity-induced convection”, should be used. This divergence of usage clearly stems from the absence of any obviously suitable term, and my only position in this matter is the tentative choice of a single term for all processes. I n recent years, we have seen a very rapid increase of the intensity of research in the field of natural convection. Some of this effort represents a shift in interest from several conventional fields of fluid mechanics of greatly diminished importance. However, most of the increased intensity is due to enhanced concerns in science and in technology concerning buoyancy-induced motions in the atmosphere, in bodies of water, in quasi-solid bodies such as the earth, and in devices and process equipment. T h e intensifying desire to have detailed and quantitative information already has resulted in an appreciable expansion of knowledge in areas little considered only a few years ago. I n this review I will attempt to draw together some recent information scattered through technical journals and will try to fashion a reasonable combination and summary of what is known in several important areas.

NATURAL CONVECTION FLOWS AND

STABILITY

275

I must be limited and the chosen limitations are of laminar and transition flows, of external and free boundary flows, and of boundary layer regimes. A consequence of this choice is that the direct applications of the knowledge is to flows of the small physical scale found mostly in technological applications, in micrometeorology, and in small circulations generally. T h e particular problems I will consider will be a collection among those which seem to be of greatest interest and immediate importance. It is inevitable, therefore, that our own past work will occupy a prominent place. Before considering particular known things, the general equations of natural convection flow are set forth. Then the approximations are introduced in terms of their physical significance, to yield, when they are all invoked, the conventional equations of analysis. T h e boundarylayer simplifications follow, for two-dimensional vertical flows. T h e equations for vertical axisymmetric flows are also given. Then, for steady laminar boundary layers arising from temperatureinduced buoyancy, known solutions for vertical and axisymmetric flows are reviewed, including plumes. Combined mass transport and thermally induced buoyancy is then treated. Consideration is next given to laminar transients. T h e next segment of the review concerns how laminar flows become unstable and how disturbances amplify to lead the flow toward transition. Vertical flows adjacent to surfaces and in plumes are considered. T h e nature of transition and turbulent flow are treated, as limited present knowledge permits. Then the general question of instability in buoyancyinduced flows is treated. Th e interesting and important question of separation in natural convection flow is reviewed and distinctions are drawn relative to forced flow separation. 11. The Relevant Equations

T h e governing equations for natural convection in Newtonian fluids stem from those used to analyze momentum and thermal transport generally but necessarily include the body force. T h e continuity, force-momentum, energy, and mass transport equations are written below: aplar w (pv)= 0; (1)

+ p

DVIDT

pcP DtlDr = W

DCIDr

+ p V‘V $- &pW(0 V ) ; k Wt + /3T DplDr + q‘” + p@;

= pg - Wp

=

*

V * D WC.

*

(2) (3) (4)

276

B. GEBHART

T h e body force per unit volume is written as pg, where p is local density and g is the gravitional force per unit mass. T h e usual approximation is made for bulk viscosity, and the viscosity p is taken as constant for brevity. T h e volumetric coefficient of thermal expansion is p, and k and D are the thermal and mass diffusion conductivities, taken as variable here. T h e distributed energy source 9”’ is in addition to viscous dissipation p@, where @ is the usual dissipation function. T h e relation for C, the concentration of a single diffusing species, is based upon the assumptions that the concentration is low and that no chemical reaction occurs. Finally, no force-producing electrical or magnetic effects are included. T h e complexity and coupling inherent in natural-convection processes are apparent in this set of equations. Motion results because the local densityp in Eq. (2) is variable due to variable t and/or C. Neglecting the density effects of chemical species diffusion, the variable density in Eqs. (1) and (2) is known only by taking into account the “temperature” equation. This equation, in turn, inevitably involves velocity. Thus, the distributions of p, p , V, and t in space (x,y , x), and perhaps also in time T, must be found simultaneously from five scalar equations and one property relation. I n spite of these difficulties, the study of natural-convection flows via the governing equations has produced a large amount of information of importance in understanding and in predicting transport behavior. Most of this information comes from the simpler forms of the governing equations which are applicable in most physical circumstances. We will consider various approximations (and their limitations) which can be made to simplify the equations. T h e principal complexities in the above equations result primarily from the possible variation of the transport properties p, k, and D on the one hand and of the density p on the other. These are separate aspects for the vast majority of interesting problems. Since p, k, and D are dependent primarily upon temperature, an important variation occurs only in processes involving large temperature differences. This is either true or not true, and this question will be ignored here. These properties are taken as constant. However, the density variation must always be taken into account, to provide motion. Nevertheless, density differences may be approximated, for processes not involving large temperature differences, in a way which greatly simplifies the equations. T h e question of density variation is considered here particularly in relation to external flows, that is, for flows of limited extent occurring in a much more extensive ambient medium of the same material. Such external flows are the principal concern of this review. Consider the external flow which results if an object at to is placed in an

NATURAL CONVECTION FLOWS AND STABILITY

277

extensive and quiescent medium, at, for example, a lower temperature. T h e fluid near the surface is heated, becomes lighter, and rises. This fluid is shed at the top of the object as a wake, which perhaps rises as a plume. Fluid from the extensive medium continually flows into the convection region near the surface to replace the rising material. If the surface is essentially vertical, as in this example, and the coordinate x is taken positive upward in the vertical direction, then the only term of the body force in Eq. (2) above is pg appearing in the x component of that equation. As a result, the u component of V will be the principal velocity. T h e direction y is taken normal to the surface. Note that from hydrostatic considerations the pressure gradient ap/ax in the remote ambient fluid at x is -pmg, where pmis the local density at large y . T h e impetus to the motion is the difference in Eq. (2) between the local body force -pg and the local external pressure gradient -pmg. This difference g(p, - p ) is the buoyancy force. Since p = p(p, T) in general for a single phase fluid, this is how the coupling occurs between the equations. This density difference and the effect of density in Eq. (1) are simplified as follows through what is called the Boussinesq approximation. T h e density difference at a given x, ( p m - p ) , is written in terms of the local temperature and pressure differences as Pm

-P

+ ... + (+la~)t (P, -PI + ... + ( W a t a p ) ( L t ) ( p , - p ) +

= (apiat), ( t m - t )

-

***.

(5)

T h e local pressure decrease (p, - p) arises because of the momentum increase in entrainment to the convection velocity level Uc and because of any viscous forces. For moderate and small Prandtl numbers these effects are comparable and Pm -

P

=

0(Uc2).

T h e convection kinetic energy UO2at x arises primarily through the buoyancy force g(p, - p ) acting through the distance x. These are equated, for moderate viscous effects:

uc2= O[gX(Pm

- P)1 = O ( p ,

-

P).

(6)

Thus LIP is known in terms of dp and Eq. ( 5 ) may be thus modified, I t should be noted that a different procedure is necessary for very high Prandtl number fluids. Then the convection velocity U , is taken from the balance of conduction and convection and is of order k/pc,x. T h e relative magnitude of the various terms in pm - p may be most

278

B. GEBHART

simply evaluated for an ideal gas ( p = pRT and p = l / T ) . Recall that the effects of p and T on p are greater for a gas than for liquids generally. Thus, with Eq. ( 6 ) , Eq. ( 5 ) becomes

+ additional cross terms, where the first term is the complete expansion in A t , the second is the first pressure term (all the rest are zero) and the third is the first cross term. T h e quantity R, = g p x / R is very small for gases at the terrestrial intensity of gravity. Also R, = p A t (= At/T for gases) is very small in many flows of interest, that is A t lip (= T for gases). Thus for both R, and R, 1 we may approximate the density difference as the first term of the temperature effect as follows:

<

pm - P

=z

<

pP(t - tm).

(7)

This approximation is even better for liquids, except near extreme states. T h e convection velocity becomes

u, = [gPx(t, - trn)]1/2.

(8)

This is an estimate of the maximum value since it was made neglecting viscous forces. This estimate is too large for fluids of high viscosity, i.e., of large Prandtl number. T h e effect of variable density in the continuity equation is next assessed. T h e ratio of the two components of the term a(pu)/ax is evaluated as follows

The variation of p may thus be interpreted as the combination of the variation of ( p a - p ) with x and of an x dependence of pa (i.e., density stratification in the remote fluid). T h e stratification effect is written in terms of a scaling height or exponential rate factor H as H = ( - FI )ap, Pw x

-1

.

This height is yRT/g for a gas adiabatically stratified, where y is the specific-heat ratio. Otherwise, H is found from some other prescription,

NATURAL CONVECTION

FLOWS AND STABILITY

279

such as tm(x). H' is, analogously, the scale height for the variation of R, = P A t . For x H and x H' for R, 1, density variations may be ignored in the continuity equation. This result, combined with the approximation of pm - p in Eq. ( 7 ) , is the Boussinesq approximation in natural-convection flows. However, further simplifications are permissible for many processes of great interest. These occur in the energy equation and concern the magnitude of the viscous-dissipation effect p@ and the pressure term PT DplDr. T h e nonlinear viscous dissipation term p@ in Eq. (3) is often of negligible effect. T h e condition for this is determined by comparing it with the conduction term

<

<

<

T h e quantity gP/c, becomes important for large g and/or very low temperature. For liquids of large Prandtl number, the effect is appreciable even for gP/c, very small. T h e magnitude of the pressure term is next assessed. T h e pressure which appears is the actual local static pressure. I t is convenient to dissociate this into two components, the local pressure in the distant medium, i.e., the local hydrostatic, P , ~(= p , above), and p , the component associated with motion. Thus, p = p , p , and p , 0. T h e ratio of the pressure and convection term is thus two quantities as follows:

<

+

and

T h e term in p , is thus usually very small. However, the condition for R, , is not satisfied when R, 1 in many neglecting p,, , i.e., PTRo flows of interest. For an ideal gas which is approximately adiabatically R, the term in p , may be stratified, R, = O(x/HR,). Then for x / H neglected. Since the orders of the viscous dissipation and p,l terms are similar it is interesting to compare them directly:

<

<<

<

280

B. GEBHART

<

For gases and R, 1 viscous dissipation is a smaller effect. For liquids having large Prandtl numbers, they may be comparable or the viscous dissipation effect may even be much larger. p , and invoking the approximation of Eq. (7) has Writing p as p , the following result for the combination of the body force and pressure gradient terms in the Navier-Stokes equation. Assume g acts in the negative x direction.

+

-pg

- 09 = -pg

+ p m g - VPm

= gpP(t - tm) - VPm

.

(9)

It might appear that the term V p , is small. It will be seen later that it may be omitted for some flows, but it is not always unimportant. T h e set of equations of motion embodying the Boussinesq approximations are written below for constant transport properties p, k, and D.T h e relative magnitudes of several possibly small terms are written in parentheses. T h e coupling, which is characteristic of natural convection, is seen in the presence of ( t - t m )in the force-momentum balance:

V *V

+ (Dp/Dr)(R2)= 0 ;

(10)

DCIDr

(13)

=

DV2C.

This set of equations is adequate for almost any need in continuum flow when the body force is purely gravitational. I n rotating systems, such as the oceans and the atmosphere, a coriolis force also arises. I n the presence of electric and magnetic fields and of electrical charges and currents, other forces also arise. Note that the buoyancy term in Eq. (1 1 ) is written as though x were parallel to g. If it is not, there will be additional components. Then V p , usually emerges as an important quantity. III. Boundary-Layer Simplifications Experimental and analytical investigations of natural convection were carried out very early in the study of heat transfer. From early measurements it was evident that the heat transfer rate from a surface depended upon the temperature difference to a higher power than the first,

NATURAL CONVECTION FLOWS AND STABILITY

28 1

indicating that the surface coefficient depends not merely upon geometry and fluid properties but also upon the driving force, the temperature difference. T h e analysis of Lorenz in 1881 for a vertical surface at uniform temperature in an extensive fluid at rest assumed that the flow in the convection layer is primarily parallel to the surface. A force balance including only the effects of buoyancy and fluid shear resulted in an expression for the thickness of the convection layer. By considering conduction across this slab, an expression for the surface coefficient was found. T h e average convection coefficient, combined in the Nusselt number, was found to be

where Ra is the Rayleigh number. This very simple result is consistent with similarity arguments. However, it was almost 30 years after the origination of Prandtl’s laminar boundary-layer concept for forced flows that the applicability of the concept was first investigated for natural-convection flows. The measurements of Schmidt and Beckmann in 1930, in a flow around a heated vertical plate in air, indicated that the boundary region may be thin compared with the height of the surface and that, as a result, the components of gradients parallel to the surface are relatively small. T h e result is the boundary-layer approximation of the general equations. T he boundary-layer equations may also be obtained from the general relations, Eqs. (10)-(13) by order of magnitude analysis. For twodimensional flow in Cartesian coordinates, the stream function #&x, y) and temperature function +g = ( t - t m ) / ( t n- t m ) may be expanded in ) G-l, where terms of the small parameter ~ ( x =

G cc [gpx3(tn - t , ) / ~ ~ ] = l / ~( G I - ~ ) ~ / ~ . Terms are separated in orders of E and the terms of order EO, in +(x, y ) and in #(x,y), written in terms of velocity components u = 4, and -u = #z and angle of inclination 8, are

-au+ - =avo ax

ay

(15)

282

B. GEBHART 0

= gp(t

-

t m )sin 0

+ gp*(C - C,)

sin 0 - -1 ap, ; P aY ~

(17)

I n the above forms the transport properties p, k, and D may be variable. I n Eq. (18) the included viscous dissipation term is neglected for g/3x/cp 1 and the pressure term may be neglected for R, R, 1. The mass diffusion equation is obtained by similar arguments for C small where Gr, is defined in terms of ( C , - C,). T h e added buoyancy term in Eq. (16) results from mass diffusion, where p* = - l / p ( + / X ) . A term c"' is included in Eq. (19) for a species source due to chemical reaction. These relations allow for inclination angle of the x axis of B from the vertical. This form is convenient for inclined surfaces. For B small, the motion pressure term in Eq. (16) is negligible, but for 0 near 90" it must be retained. For vertical axisymmetric flows (as would arise, for example, around a long vertical cylinder) similar approximations can be made. T h e resulting steady-flow boundary-layer equations, in terms of the vertical coordinate x and radial distance y from the axis of symmetry, are

<

< <

at

u-+v, ax

2t 1a at = ff-y r , dy y ay dy

(

1

where viscous dissipation, the pressure term, property variations, and mass diffusion have been omitted. T h e considerations leading to all the foregoing boundary-layer equations proceeded as though a surface of prescribed temperature to were present. However, these relations are also valid for a plume rising from an energy source and for a buoyant jet issuing from an opening or slit. IV. Steady Laminar Boundary-Layer Flows

T h e above boundary-layer equations admit of many interesting and valuable similarity solutions and these collectively clarify many aspects

NATURAL CONVECTION FLOWS AND STABILITY

283

of natural convection flows. This section considers solutions known for the pure natural convection circumstance, i.e., for flows without any forcing condition other than buoyancy. T h e remote condition away from the induced flow field is taken as quiescent, no motion. These are problems with homogeneous boundary conditions in similarity form, quantities go to zero as y increases. Consideration of mixed convection is thus not included here. Many solutions have been found since the earliest one given by E. Pohlhausen in 1930 for a vertical flat isothermal surface. Succeeding pages of this section will review these solutions for vertical surfaces and plumes, for inclined and horizontal surfaces, and for axisymmetric vertical plumes and flows adjacent to surfaces. I n most of the summary given here the methods of Ref. ( I ) will be used to permit concise arguments. Flows induced by either thermal and mass diffusion separately are included. T h e following section considers flows wherein both effects occur simultaneously, but with different diffusion characteristics.

A. FLATVERTICAL SURFACES Consider a surface at y = 0 extending from x = 0 upward, maintained at t o , perhaps with an assigned level of concentration C, of a diffusing species at the surface, and located in an extensive medium at t , and C , . We shall take properties p, K, and D as constant. Each of to and C, as well as C , and t , may be specified by some physical consideration to be functions of x,e.g., temperature stratification in the atmosphere. T h e x dependencies of these quantities are written in terms of functions d, e, j , and r as follows: to - t , = d ( x ) , (23) C,

-

C,

=

e(x),

(24)

where t , and C , are reference values to be determined later. T h e temperature and concentration distributions in the convection layer are written as

A similarity variable 7 and a generalized stream function are postulated:

284

B. GEBHART

We seek functions b, c, d, and/or e (also j and r for stratification) such that +, C, and f depend only on 7. Substituting Eqs. (23)-(28) into Eqs. (l5)-( 19) for constant transport properties p, k, and D yields, after some rearrangement,

where the primes indicate derivatives with respect to 7. T h e definitions at y = 0 (7 = 0), the behavior as y -+ a (7 -+ a),and the conditions at the wall result in the following boundary conditions for the above equations, independent of x and y : f'(0)

=f ( 0 ) =

1 - 4(0) = 1 - C(0) = ~ ' ( c o = ) $(a) = C(OO) = 0.

(32)

For similarity to result (i.e., thatf, 4, and/or C be functions only of 7)) the functions b, c, d, e, j, and r must be such that x disappears from the equations. This requires that

5 b = c,

(334 momentum, thermal, and mass convection

~

CbX = c, b2

d g-Pcb3

~~

u2

gp*

cb3 v 2

cd,/bd b2c2 v2 d cD

d

pK1

(33b)

KlC,

thermal buoyancy

(334

K 2 = K2C, cb3

mass-diffusion buoyancy

(334

C,

nonuniform to

(334

b2c2 d

viscous dissipation

(33f)

=

=z

=

---K3= KSC,

NATURAL CONVECTION FLOWS AND STABILITY cj,/bd = C,

temperature stratification

ce,lbe = C,

nonuniform C,

cr,/be

c g bd cz,

=

diffusing-species stratification

C, C

PT - - = /3T - K4 = PT K,C,,

pressure term

q "_ _1 _l _ - Fl(d

distributed-energy source

bd

b2d Pr k

1 c'!!

--b2e v - F,(d

chemical reaction,

where a number of the quantities C, , C , ,..., C,, must be constants, depending upon the physical effects to be included. I n addition, the 7 functions F, and F, , if nonzero in the convection region, must go to zero sufficiently fast with increasing 7 to give boundary-layer flow. Also, the boundary conditions peculiar to natural convection, written in Eq. (32) in terms off, 4, and C as functions of 7, must be satisfied. We must determine the kinds of arbitrary specifications of some or all of the functions d, e,j, and r which result in b and c values for which the quantities C , , C , ,..., C,, are constants. At first sight, it may appear unlikely that any result is possible, given the very large number of relations that b and c must simultaneously satisfy. However, there is at least one realistic case known for every physical effect included above. Clearly, including a smaller number of physical effects should increase the latitude and perhaps also increase the number of cases. Thus, the best way to proceed is to take first the simplest kind of problem and then add effects one at a time. This procedure is followed. T h e remainder of this section considers first flows over a vertical surface for which the effects of stratification, viscous dissipation, distributed heat and species sources, the pressure term in the energy equation, and surface mass addition may be omitted. Then these effects are added one by one. Among the constants C , through C,, only the first two conditions, C , and C , , and either the third (C,) or the fourth (C4), arising from buoyancy, need be met in all cases. T h e fifth (C,) and eighth (C,) conditions relate to nonuniform surface conditions to and C, , that is d and e. T h e seventh (C,) and ninth (C,) are for stratification of t , and C , in the distant medium. Note also that the conditions for a purely thermaldiffusion process in terms of d(x) and j ( x ) are identical to those for a purely mass-diffusion process in terms of e(x) and r ( x ) . Thus, any d and/or j functions which give similarity for thermal transport will also give similarity for mass diffusion when e and/or r are taken as the same

286

B. GEBHART

functions. Therefore, it is not necessary to consider further any flow driven entirely by mass-diffusion-induced buoyancy. Joint buoyancy effects are analyzed in a later section. T he simplest circumstance is an isothermal surface (at x 3 0, y = 0, x positive upward for to > t m )in an unstratified medium. This involves C,, C,, and C,. However, this circumstance is included in d(x) = to - t , . Therefore, consider d = d(x). T h e conditions are Cx/b

=

C, ,

cb,/b2

=

C, ,

d/cb3 = C , ,

cd,/bd

=

C,

.

(34)

T h e conditions may all simultaneously be satisfied (2, 3) by d of either power law or exponential form in x. They are also satisfied ( 4 ) if d is a power of a linear function of x, i.e., ( q x). Thus,

+

d ( x ) = Memx.

(33))

T h e limitations are, see ( I ) ,N and M greater than zero, q 2 0, and m # 0.

B. THEPOWER-LAW VARIATION For this variation of t o , n may have any value; n = 0 is the isothermal surface, n < 0 is singular in to at x = 0 for q = 0, and n > 0 results in to - t , = 0 at x = 0 if q = 0. For the power-law case, the values of b and c are

c = 4(Grx/4)1/4=

G.

T h e resulting differential equations and boundary conditions for any n and for q 3 0 are

f”’+ 4 + (n + 3)fj’’ - (2n + 2 ) ( f ’ ) 2

=

0

(364

=

0

(36b)

1 - 4(0) = f’(a3) = +(a) = 0.

(364

4‘‘ + Pr[(n + 3)f4’ - 4nf’4] f(0)

= f’(0) =

T he power-law-dependence circumstance has a number of interesting aspects. These are best seen in terms of the local surface heat flux at x

NATURAL CONVECTION FLOWS AND STABILITY

287

and the total amount of energy convected by the boundary region flow at x, Q(x). These are calculated for unstratified media as

T h e first relation shows that a surface temperature variation (to- t m )of n = 6 results in a uniform surface heat flux (at y = 0), as might occur from an electrically heated surface. T h e second shows that for any choice of n, the amount of energy convected in the boundary region is not zero at the leading edge for q # 0. Therefore, n = 6 and q = 0 are the uniform-surface-flux case treated through the use of specialized parameters (5). This circumstance is a special case of the general powerlaw formulation. T h e assigned surface heat flux q" is related to the temperature-difference parameter N by Eq. (37) as follows:

N

=

(4"/k[-4'(0)1)4/5 (4v2/gp)1/5.

T h e nonzero value of Q(x) in Eq. (38) for q > 0 indicates that what is taken to be the leading edge of the vertical surface (at x = 0) must be preceded by a heat source. T o be consistent with the present treatment, this source must be merely an extension of the surface to x = -q. T h u s the simpler form to - t , = Nx" includes all the possibilities. Taking q = 0, the general values of b and c are b

=

(l/x) (gPx3 d / 4 ~ ~ )= ' / (~l / x ) (Gr,/4)lI4 = G/4x,

c =

G.

(39)

Another aspect of the power-law variation (Nx") may be seen by calculating the boundary-layer thickness 6 and velocity component u at x = 0. Only for -1 < n <: 1 are these quantities zero. An additional consideration is apparent. Equation (38) indicates that for Q(x) to be constant or increasing with x,n >, -0.6. T h e value n = -0.6 amounts to a line source at the leading edge and an adiabatic condition at y = 0, i.e., q"(x) = 0 for x > 0 [because d'(0) = 0 for n = -0.61. For n < -0.6, the line source is infinite in strength and q"(x) < 0 for all x. T h u s the limit of physical realism for the power-law case is -0.6 n <1 and q = 0.

<

288

B. GEBHART

The heat transfer characteristics of a surface at x >, 0 and y calculated in terms of the local heat flux as Q"(x) =

-k(at/ay,)

=

=

[-+'(O)] k db.

0 are (40)

The local heat transfer coefficient is given in terms of the local Nusselt number and Grashof number as NU,

h x

= LL =

k

[-+'(O)]

xb

=

[-c'(0)l

.\/z (Gr,)1/4

= F(n, Pr)

(Grz)1/4.

The function -+'(O), or F(n, Pr) is found by solving the two point boundary problem posed in Eq. (36). The first solution was given by Pohlhausen in 1930 for an isothermal surface and Pr = 0.733. Solutions for various Prandtl numbers were given in Schuh (6) and Ostrach (7). Values of F(0, Pr) are listed in Table I TABLE I VALUES OF F(n,Pr) FOR AN ISOTHERMAL SURFACE, ADAPTED FROM EDE (U), UNIFORM FLUXSURFACE (n = i),PREPARED BY DR. C. A. HIEBER

AND FOR A

0 0.01 0.72 0.733 1.O 2.0 2.5

5.0 6.7 7.0 10 102 103 104

co

0.600(Pr)1/2 0.0570 0.357

0.71l(Pr)1/2 0.0669 0.410

0.401 0.507 0.616 0.675 0.829 0.754 0.826 1.55 2.80 5.01 0.503(Pr)1'4

0.931 1.74

for Prandtl numbers over a wide range and also for the Prandtl number both asymptotically large and small, as determined in LeFevre (8) and Kuiken (9, lo). Values of F(0, Pr) are plotted in Fig. 1. T h e asymptotic limits are seen to be very good approximations, except in a narrow central range of Pr. Values of F(n, Pr) are also shown in Table I for the uniform surface flux condition n = +.

NATURAL CONVECTION FLOWS AND STABILITY

289

10

I

-h -4; 10-1

-$I? I

10-2

10-3 40-4

10-2

1

102

1o4

Pr

FIG. 1. Heat transfer characteristics for laminar natural convection from an isothermal, vertical surface.

C. THEEXPONENTIAL VARIATION For this variation of the assigned temperature difference (to- t m ) , Eq. (35b), the values of b and c and the differential equations are b

=

m ( g / 3 4 4 ~ ~ m= ~ )m(Grm/4)1/4, ~/~

+

f”’ 4 + f f ” - 2( f ’ ) 2

4’’ + Pr(f4‘

c = 4(Gr,/4)1/4

=0

- 4f’4) = 0.

(424 (42b)

T h e parameter m is seen not to appear in the equations. T h e range m > 0 is an exponential increase of to , m < 0 is a decay, and m = 0 is excluded by the analysis. This category of solutions is discussed in Gebhart and Mollendorf (12) and the m < 0 circumstance is shown to be always physically unrealistic due to implications with regard to flow at negative x. T h e local Nusselt number for the exponential distribution is found to be Nu, = h/mk = {[-4’(0)]/21/2}(GI-,#/*. (43) is a (Note that h is dependent on x through d . ) T h e quantity [-+’(O)] function of Prandtl number, and it is plotted together with the drag parameterf”(0) in Fig. 2. Two related peculiarities of the exponential similarity circumstance are that 6 and Q(x), as defined generally in Eq. (38), are not zero at x= 0. This is because the m > 0 condition implies an exponential decay of positive to - t , to zero along a surface extending in the negative

290

B. GEBHART 40 20 10

05

02

05

10

20

50

20

10

50

100

200

Pr

FIG. 2. Heat transfer and drag quantities for an exponential surface-temperature distribution.

direction. Therefore, at x = 0 there are both momentum and heat flows in the boundary region. T h e remainder of this leading-edge effect at x = L is shown to be small (12) if mL is large compared to 1.0.

D. CONVECTION IN STRATIFIED MEDIA If the temperature t , in the remote medium is dependent upon x, the function j(x) = t , - t, which applies must satisfy Eq. (33g) for similarity to result. A stable condition of stratification, (i3pm/ax) 0, applies if pm = p,(t,), an approximation in the foregoing treatment. This requires merely that j , 2 0, i.e., that t , increases with x if

<

dp/dt

< 0.

For p m = pm(p, , T,) the condition on t , for adiabatic stratification is the following lower limit of change, where S is entropy, dT,/dx

= jx

2 (dT/dx)s = --pg(aT/ap), , =

-g/c,

,

small for liquids

for an ideal gas.

(44)

(45)

Taking the condition as j , >, 0 for liquids, t , must be constant or increase with x. T h e value of j , from Eq. (33g)

j,

=

(bd/c) C, = C, $N xn-1

and

C, iMmemx 2 0

indicates that C, 2 0 for the power law variation (since N is positive) and that C,m 3 0 for the exponential (since M is). This again amounts to C, 3 0 since m < 0 are not physically realistic flows. T h e j variations are j = t , - tr = (C7/4n)Nx" and t C , Memx. (46)

NATURAL CONVECTION FLOWSAND STABILITY

29 1

T h u s the permissible j variations are multiples of the d = to - t , variations. T h e circumstance n < 0 is unreasonable since t , is then necessarily negative around x = 0 for any choice of t , . Then since to - tI. = d + j

=

Nx"[l

+ (C7/4n)],

(47)

where C,/n > 0, there are no solutions with stratification for an isothermal surface. Finally, for q =/- 0, in ( q x ) ~ n, < 0 is permissible and Eq. (47) shows that to may be constant, for -4n = C, > 0. Such cases were considered in Cheesewright (13), but q # 0 has been shown to result in ambiguities at the leading edge. For either the power law or exponential variation of j , the value of C, is chosen on the basis of the amount of stratification. T h e additional term -C,f' appears in Eq. (36b) or (42b). T h e stability condition on degrcc of temperature stratification is more generous with gases since they would cool with decompression in rising. T h u s a decreasing t , with x is shown by Eq. (45) to be permissible. T h e limiting rate of decrease is linear in x:

+

j

=

t,

~

fI. =

(48)

-(g/c,).

Considering only the power law variation j

=

C,Nxn/4n2 - ( g / c , ) x ,

(49)

the simplest similarity circumstance is for n = 1, where C,N,4, the degree of stratification, or rate of change of t , with x, must be equal to or greater than -g,'c, (or -C, 4gn c , N ) . For C, negative, i.e., for dt,'dx < 0, the limit of n is sccn below:

<

j,

=

(-C,) Nxn-l/4 < g/c,

or

xn-l

< (g/c,)

[4/N(-C7)].

For n < 1, there is a lower limit on x for stability; thus n may not be less than 1 .O. For n > 1, there is an upper limit of x and n > 1 may be used for flow from x = 0 to x = L whereI, is given as Ln-1

5 4 g / r p N (-C,).

Therefore, solutions for stable stratification are found for C , < 0 where n > 1 when the above limit on L is observed. With n = 1 there is no limit. Recall that n 3 1 produces peculiarities at the leading edge. For C, > 0, dt,'dx > 0 and all n 3 0 are reasonable. However, n < 0 results in t , being unbounded around x = 0.

B. GEBHART

292

T h e limiting condition on j, adiabatic stratification, is seen from Eq. (49) to be n = 1, (C,,’4) -g/c,N, and j = -gx/c, . If N is taken as g/c, (C, = -4), then Eq. (47) indicates that to = t , = Constant. Thus, there is a similarity solution for a uniform temperature surface in an adiabatically stratified gas. Reference is made here to the particularly simple result given in Gill (14). I t is a one-dimensional solution (in y ) which applies for a doubly infinite surface (-00 < x < 00) when d and j, are constant. T h e applicability of these conditions to a physically reasonable flow circumstance is not clear. T h e solutions for ( t - t m )and u, written in current notation, are 1

+ = ( t - tm)/(to- tm) u = (a$ d2/vjZ)l/2 e-ylp

=

e-y/?’

cos(y/p)

sin(y/p),

where

p4 = 4vm/gpj,,

01

=

k/pc,

.

E. VISCOUSDISSIPATION Inclusion of this effect, in addition to the foregoing ones of variable to and t,, requires the constancy of C,, C,, C,, C,, C,, and the additional quantity C, , Eq. (33f). It has been shown (15) that neither the isothermal nor the uniform flux surfaces have similarity. However, perturbation solutions were given for a Prandtl number range of lop2 to lo4. It is shown that the parameter gpx/c, arises as the measure of the importance of the viscous-dissipation effect. T h e range of these results was extended to infinite Prandtl number (16). More recently it has been shown (12) that, although similarity is not found for the power-law variation of (to- t,), it does result for the exponential variation to - t , = Memx. For this variation, Pr K,C, is evaluated to be 4 Pr(gp/mc,). T h e additional term + 4 Pr(g/3/mcp)(f”)2 appears in the energy equation for the exponential case, Eq. (42b). T h u s another parameter, gp/mc,, arises in addition to the Prandtl number. Solutions were given for the Prandtl number range of 0.72 to lo2 for the dissipation parameter 4gp/mc, (m > 0) from 0 to 2.0. T h e effect on drag and heat transfer in terms off”(0) and +’(O), normalized by their values for no viscous dissipation,f;(O) and +,’(O), are shown in Fig. 3. It is seen that the larger values of 4gp/mc, result in an appreciable reduction of heat transfer. Such a circumstance would arise, for example, in applications at very high g, such as in turbine blade cooling. Heat transfer information may be obtained from Fig. 3 by combining

NATURAL CONVECTION FLOWS AND STABILITY 1.06 I

293

I

4.00

090

-

080

-

d'(0) d'(0)o

'

0.70 00

I

I

I

05

40

1.5

2 .o

4gP/mcp

FIG. 3.

Effect of viscous dissipation on heat transfer and drag.

this information with that on Fig. 2 for no viscous dissipation, to determine +'(O). This value may then be used in Eq. (43).

F. ADDITIONAL EFFECTS T h e pressure term PTDpIDr in the energy equation for gases has the residual form shown in Eq. (18) for the approximations appropriate to most natural-convection flows and to boundary-layer behavior. This term gives rise to the similarity consideration, Eq. (33j) which requires that pTcjbd be independent of x. In general PT is a complicated function of x and y in the flow field. However, for ideal gases PT = 1. T h e combination clbd is independent of x for d = to - t , = Nxn where

294

B. GEBHART

n = 1 . T h e additional term in Eq. (36b) is then -(4g/Ncp)f’. T h u s an additional parameter arises. This result may also be used with stratification w herej is given by Eq. (49) for n = 1 for C, either positive or negative. T he energy input that causes the rising convection layers may include, in some circumstances, a distributed source of thermal energy accounted for by g”’(x,y). This could arise, for example, through a temperatureor concentration-controlled chemical reaction or through a thermalradiation mechanism. T h e additional condition for a similar solution, Eq. (33k) must be satisfied. This condition means that q”’(x,y ) must be b2dFl(q), where Fl(q) goes to zero sufficiently fast with q. This form is reasonable since both the temperature and the concentration distributions will depend solely on q. However, the additional x dependence may not be useful. For n = Q, g’” = q”’(q). T he last physical effect to be considered in this section for the vertical-surface geometry is that which would arise if a fluid of the same nature as that involved in the convection process were being added (or removed) at a relatively small velocity Z,I at the surface. This could occur with a porous plate and would be interesting in some applications. Th e difference from the previous formulation, Eqs. (29)-(31), is not in the physical effects but rather in the boundary conditions, Eq. (32). T he conditionf(0) = 0 results from the condition 71 = 0 at an impermeable boundary at y = 0. For material addition, v would have the value vo(x), which may be variable with x and either positive or negative. In terms of the formulation of similarity variables, Eq. (28), the new boundary condition replacingf(0) = 0 becomes

.f(O)

= -%(x)/%!

.

(50)

Similarity results only forf(0) = 0 or constant. Therefore, vo(x)oc c, and the required x dependence of the imposed v,, is

and

zlo(x) cc x ( ” - ~ ) / ~

for

d

=

Nx“

q,(x) cc ernxI4

for d

=

Memx.

Thus a choice of n or m fixes the distribution of vo in x. T h e power law case has been discussed (17) and a perturbation solution given (18) for n = 0 and no constant. An integral method analysis has also been given (19). There is another interesting solution (20) for suction related to boundary-layer flow. Considering a uniform suction velocity vn , one finds a solution for which u(x,y ) = u ( y ) ,v(x, y ) = -vn , and O(x, y ) = O(y). This solution would apply for a doubly infinite surface or for a

NATURAL CONVECTION FLOWSAND STABILITY

295

half-infinite surface (x > 0) and at large x for on uniform. T h is solution applies for to - t , constant. T h e temperature and velocity distributions, where LY is the thermal diffusivity, are

9 = ex~[-(un/a)~l and

G. SUMMARY CONCERNING FLOWADJACENT TO VERTICAL SURFACES T h e foregoing paragraphs of this section established the appropriate boundary-layer equations and set forth the requirements for similarity for two-dimensional steady laminar flow in Cartesian coordinates, Circumstances leading to similar solutions were delineated, including all the physical effects for flows in which a temperature difference is the motive effect for motion. T h e requirements for the similarity of chemicalspecies diffusion processes at small C are analogous to those for thermal diffusion, except for some effects present only in the latter process. Therefore, the similar flows above for thermal diffusion may be translated into equivalent forms ( d = e a n d j = Y) for chemical-species diffusion processes, if the normal velocity at the surface can be neglected. I n addition, the simplicity of the energy and species diffusion equations under the present formulation permit treatment of the two processes simultaneously in the same flow. This is done in the next section. A number of additional results, related to the kinds of problems considered in this section, will be mentioned. T h e wake from a vertical surface of finite height has been considered (21). T h e above solutions apply strictly for surfaces which are semiinfinite in the x direction. x L. Results for height L are simply taken as those for the portion 0 Using asymptotic expansions, the surface boundary-layer solutions were matched to the distant plume behavior. A vertical surface having uniform surface flux from x = 0 to L and adiabatic thereafter has also been studied (22). An experimental study (23) considered flat vertical surfaces having slanted leading edges, i.e., at an angle other than 90" with the x direction. Measurements, using the technique of Baker ( 2 4 , indicated that heat transfer behavior depended only upon the distance directly upstream to the leading edge; there were no appreciable lateral effects. However, our studies and use of the method of Baker (24) suggest caution. Apparently appreciable bubble formation is a necessary concomitant of the electrolytic process involved. Non-Newtonian fluid behavior has been considered for natural convection flows adjacent to a vertical

< <

296

B. GEBHART

surface. Similarity of several classes of non-Newtonian behavior is discussed in Acrivos (25) and Na and Hansen (26). H. CONVECTION OVER INCLINED AND HORIZONTAL SURFACES For many interesting and practical flows, the surface is curved or is inclined to the local direction of the gravity field. T h e buoyancy force has a component in both the tangential and the normal directions as indicated in Eqs. (16) and (17). Surfaces may be flat or curved. They may be curved in the direction of flow and transverse thereto. There have also been a number of analyses of vertical wedges and cones using approximations which are justifiable for small surface inclinations from the vertical. Cones and two-dimensional closed bottom shapes were considered in Merk and Prins (27) and Braun et al. (28). Configurations were found which give similarity. However, both studies ignored the normal component of buoyancy force and the entire motion pressure field. These approximations are admissible only for small angles from the vertical. Cone flows were again treated in Hering (29) and Hering and Grosh (30). Measurements (31) of heat transfer rates on inclined plates give some support to the approximations of such analysis. These studies amount to a consideration of how a reduced and perhaps varying tangential buoyancy force affects transport and some also show the effect of geometry-determined flow divergence. They do not relate to the perhaps more interesting effect, that of the modification of the pressure field by the normal component of buoyancy. This becomes a very important consideration at large inclination and is very interesting. For any strictly horizontal surface, the tangential component of buoyancy is zero. Th u s flow in the x direction is generated entirely by the gradient in p , , Eqs. (16) and (17) for 0 = 90". Clearly, ap,/ax must be negative and this arises for heated surfaces facing upward, as determined experimentally (32). T h e first analysis (33), showing similarity for a flat plate with a single leading edge, i.e., for a semiinfinite flat plate, led to the opposite conclusion because of a sign mistake. This was corrected (34), similarity was shown for power law variations of to - t , and extensive calculations were carried out [34, 35). Analysis proceeds in terms of the following values of b(x) and c(x), where Gr, = gpx3(t, - t,)/v2, as before: b(x) = (l/x )(G1 -~ /5= )~G/x, /~

C(X) =

The variation of p , is

pm = (5v2p/x2)G4P(7).

5vG.

(54

NATURAL CONVECTION FLOWSAND STABILITY

297

Solutions for the isothermal and uniform heat flux surface conditions ( n = 0 and *) were given (36) for Prandtl numbers between 0.1 and 100. T h e heat transfer relations are hx/k = 0.3904 (GrJ115(Pr)l14,

n

=0

(524

0.5013 (Gr2)1/5(Pr)lI4,

n

=1 3’

(52b)

hx/k

=

Experiments, using interferometry, and for n = 0, were in agreement with both the calculated temperature distributions and heat transfer.

I. NATURAL CONVECTION PLUMES I n many configurations in technology and in nature we encounter masses of fluid rising freely in an extensive mediumbecause of a buoyancy effect. Solar-heated surfaces warm the adjacent air, which may rise in motions of small physical extent, as seen over a road or field, or on the grand physical scale apparent in the flows which cause the ascending columnar shapes that form clouds over a sun-heated prairie. T h e total meteorological process is a buoyancy-controlled motion. Such circulations occur also in the effluents from chimneys and stacks. In liquid media, such as the oceans, differences in salinity, temperature, and suspended particulate matter give rise to circulations. T h e discharge of either chemical, thermal, or particulate pollutants into air or water produces buoyancy-affected diffusion and dispersion mechanisms. Such processes have received relatively little attention. However, in most recent years interest has increased and a considerable amount of detailed analytical and experimental study has provided a beginning of understanding of the physical mechanisms involved. Both laminar and turbulent transport mechanisms have received some consideration. This section presents some of the information available for steady laminar processes for flows caused exclusively by an energy or thermal addition. Such a steady input in a concentrated region of an extensive fluid medium results in a continuously rising region of fluid above the energy source as a plume and is thus differentiated both from the flow resulting from a concentrated impulse thermal input, which results in what is conventionally called a thermal, and from a continuous jet of buoyant material, as from a chimney, called a buoyant jet, or forced plume. I n analyzing the steady plume flow which results in an extensive medium from a steady and purely thermal input, one is led to the simplest idealizations-the infinitely long line source and the point source of thermal energy. I n practice, the two circumstances may be

298

B. GEBHART

realized by a long horizontal heated wire and by a small heat source. An interferogram of a plume formed above a long electrically heated fine wire in atmospheric air is reproduced in Fig. 4. T h e fringes are,

FIG.4. An interferogram of a plume rising in air at 1 atm from a horizontal wire of 6-in. length. One fringe is approximately 7.25"F; q' = 54 BTU/hr ft length.

approximately, isothermal regions corresponding to different temperatures in the air rising above the energy source. Counting fringes in from a uniform t , in the distant medium, we see that the temperature of the midplane material of the plume decreases with height x above the source as energy is diffused outward. This is the characteristic of all such flows initiated from a source. Buoyancy tends to increase velocity, and the temperature difference driving the flow diffuses into the adjacent material. Some of the important questions about such a flow are concerned with its extent and with the nature of the velocity and temperature fields. One of the first studies of plumes (37) described flow for both point- and line-source plumes. There have subsequently been many studies of laminar plumes. Numerical solutions were given (6) at P r = 0.7 for both point- and line-source plumes. Closed-form solutions for various values of Pr and measurements above a simulated line source are found in Yih (38,39) and Rouse et al. (40).T h e plume above a line source was

NATURAL CONVECTION FLOWS AND STABILITY

299

again considered (41). A line source in a gas of variable properties was analyzed in an approximate way for Pr = 3, (42).T h e line-source plume in the limit Pr + 00 has been analyzed (43). Closed-form and numerical solutions were obtained for both configurations (44). T h e results of experimental studies for the line-source plume will be discussed briefly after the analysis is presented.

J. LINESOURCE PLUMES T h e analysis of plume flows is based on the natural-convection approximations discussed above, with the assumption that boundarylayer approximations are also valid. T h e plume of Fig. 4 makes this approximation a t least initially plausible. Past terminology and procedures in line-source plume analysis have often been indirect and somewhat obscure. T h e formulation used here (45) treats the plume as merely another particular boundary condition. T h e coordinates and the general definitions and formulations are the same as before, Eqs. (23)-(28), resulting in the same conditions, Eqs. (29)-(31), for similarity. However, some of the boundary conditions are different from those in Eq. (32). T h e plume midplane (at y = 0) is a plane of symmetry and also is an adiabatic and zero-fluid-shear surface. Therefore, the apparent boundary conditions are

d’(0) = f ” ( 0 )

=f(0) =

1

-

d(0) = +(m) = f ’ ( O O )

=

0.

We must determine which flow, among all those included in the general formulation above, is the one satisfying known conditions for a line-source plume such as that shown in Fig. 4. T h e condition in this plume is that the convected energy 2Q(x) = Q must be the same at all x > 0. This quantity was calculated in Eq. (38) and the value of n was found to be -3. Thus, ( t o - t m )decays as x-3/5 as follows: to

where I

=

t

- m

=

Nx-315

= ( Q4/45gSpp2p2~,414)1/5 x-~/

~,

(534

Srf ‘5, dy. T h e vertical component of velocity is given by u

=

( g S p Q / ~ ~ p ~ / ‘ pf’‘(7) ~ ~x1I5. I)~/~

(53b)

Thus u is predicted to increase without limit. T h e differential equations for n = -$, neglecting stratification and viscous dissipation, are f”‘

+ ( L 5 Z ) f f ” - (*)f’2 + 4

0

(544

4‘’ + (15g)Pr(f4)’ = 0.

(54b)

z

300

B. GEBHART

Note that Eq. (54b) is a perfect differential. These equations have closedform solutions for Pr = $, +,2, and 3. T he boundary conditions above are too numerous. I n addition, Eq. (38) with n = -3 might be used as a boundary condition. Past literature differs not only in formulation but also in how to impose boundary conditions. It was shown (45)that the following conditions are independent and that they are the optimum choice for numerical calculations: #’(O)

=

1 - #(O)

= f ” ( 0 )= f ( 0 ) = f ’ ( C O ) =

0.

(54c)

Solutions for Prandtl numbers in the range 10-2-102 were given in the above reference, and values of f ’ ( O ) , of I , of 7 where f ’ , f ’ ( O ) = 0.01 (q8), and of 7 where c,b = 0.01 (q8,) appear in Table 11. TABLE I1 PLUMEPARAMETERS Pr I f’(0) 76

7%

0.01

0.1 1.545 0.9751 0.8408 14.6 9.3 11.0

0.1 0.623 0.661 8 4.1 3.9

1 .o 0.527 0.6265 3.8 3.2

2.0 0.378 0.5590 3.7 2.2

6.7 10.0 100.0 0.204 0.164 0.4480 0.4139 0.2505 4.1 4.3 5.6 1.2 1 .o 0.4

T he above solution is for thermal plumes. Species diffusion may also cause these flows, and the results are the same for processes which may be formulated as in the previous sections. Mention is also made here of other thermal-energy input conditions which would result in plume flows having similarity solutions. T h e energy input might occur along the plane at y = 0 for x >, 0, as with an upward-facing-slit radiation source at x = 0. T h e value of n in ( t o - t m ) = Nxn for any such plume would be found either from the expression for q”(x), Eq. (37), or from Q(x), Eq. (38). For example, for q”(x) increasing linearly with x, the value of n is seen to be 1. T h e differential equations would be (36) and the boundary condition c,b’(O) = 0 would be changed to account for the flux condition. I n recent years there has been some experimental study of laminar plumes rising above long electrically heated wires of small diameter. Measurements of velocity and temperature distributions were made in air above a wire having a length-to-diameter ratio ( L / D ) of 3330 (46). Temperatures were measured (47) in air above a wire of LID = 250. Temperatures were determined (48) in an interferometric study of a plume, as in Fig. 4, in a light silicone oil (Pr = 6.7) above a wire of 0.005-in. diameter and 6-in. length, L / D = 1200. T h e data of all three investigations indicated plume temperatures significantly below the

NATURAL CONVECTION FLOWS AND STABILITY

30 1

predictions of analysis. T h e generalized midplane temperatures from all three studies at different heating rates are shown in Fig. 5, plotted against the local Grashof number in the plumc. T h e results have the correct slope but are uniformly low by approximately 15 %.

. *

I

r:

0.001

’

I

I

I

I

I

Schlieren studies of plumes in air and in water, in a direction normal to the plane of the plume, clearly show large end effects, i.e., necking of the plume, an I , D effect. Schorr and Gebhart (48) have shown that allowing for finite wire size and nonsimilar flow near the wire by finding a “virtual” line source does not remove the systematic disagreement between theory results and existing data. Also, a “conduction” end effect is shown to be inadequate as an explanation of the temperature deficiency, compared with two-dimensional theory.

K. AXISYMMETRIC FLOWS Such plume flows arise from concentrated energy sources, and the idealization is a point source, often realized in practice by a small flame or heated sphere. Most plumes found in nature are more closely approximated by the axisymmctric form. T h e initiating material for such a plume often gathers, through a thermal instability mechanism, into a relatively compact form in the early part of its ascent. Even plumes arising from a line source of finite length will eventually modify, while rising, to an axisymmetric form.

302

B. GEBHART

T h e boundary-layer equations for steady laminar axisymmetric flow, Eqs. (20)-(22), are in terms of x, the vertical distance from the origin of the flow, and y, the distance out, normal to the axis of symmetry. Employing a Stokes stream function of the following form satisfies the continuity condition: and

yu = v#,

yv = - v & .

Employing the following transformation, the force-momentum and energy balances are written in terms of the similarity variable 7 : 7 = ( Y / x )[gPx3(t0 - tm)/~211/4 =( Y / W ~ J ~ ’ ~ #(x, , Y) = x f ( 4

+ Pr(f4)’

(rl$’)’

= 0.

(554

(55c)

A similar indirectness concerning boundary conditions is found in the literature for this case, as for the plane plume flow. T h e sufficient and simplest boundary conditions are f ( 0 ) = $’(O)

= f ’ ( 0 )=

1

-

$(O)

=

0,

f ’ ( c o ) bounded.

(55d)

T h e above formulation still leaves unknown the exact relation between to - t , and x. T h e thermal energy convected in the plume at any x, Q(x), is independent of x. Calculation of this quantity, as done above for the plane-flow case, leading to Eq. (38), indicates that to - t , cc l/x. Thus the temperature field decays in magnitude inversely with x as it diffuses outward. T h e velocity level, however, remains constant with x, but the velocity region increases in extent. T h e exact relation between center-line temperature and the energy content of the plume Q is given by to - t ,

=

hQ/2~kx

where H isO.759,0.687,0.667,0.625,0.561, and 0.500forPr = lop2, 0.7, 1 , 2, 10, and 00, respectively. T h e first consideration of this plume (6) gave numerical results for Pr = 0.7. Closed-form solutions were given (38) for Pr = 1 and 2. T h e same problem was formulated (49) in connection with a study of natural convection from spheres. Numerical solutions for Pr = 0.01, 0.7, and 10 and also closed-form solutions for Pr = 1 and 2 are given in Fujii (44). T h e closed form solutions are repeated in Brand and Lahey (50). Boundary-layer flows over vertical cylinders are governed by the

NATURAL CONVECTION FLOWS AND STABILITY

303

axisymmetric form of the equations, Eqs. (20) to (22). T h e physical requirement for boundary-layer behavior is that the thickness 6 be small compared with the distance x from the beginning of the flow. A similarity solution of the equations was found (51, 52) for a vertical cylinder, of diameter D,in a uniform ambient fluid at t , for a surface temperature to increasing linearly with x, i.e., to - t , = Nx. T h e equations are transformed by replacing y by the new variable Y defined in y / D = er. T h e resulting equations and boundary conditions for a temperature variable and stream function contain the Prandtl number and a temperature-gradient Grashof number Gr’ = g/3D4N/v2.Numerical solutions for Pr = 0.733, 1.0, 10, and 100 for various values of Gr’ indicate that the local heat transfer characteristics of the collection of results are accurately represented by hD/k

=

+ 7Pr)]1/4.

1.058 (Gr’)li4[Pr2/(4

(56)

T h e local convection coefficient is seen not to vary with x for a given circumstance. This is in agreement with various experimental observations for small-diameter vertical cylinders. A solution for a vertical “needle” has also been given (53). For vertical cylinders, it is valuable to know the conditions for which the transversely curved surface may be considered flat, so that the results for vertical plates apply. Clearly, when the boundary-layer thickness 6 is very small compared with D, the effect of transverse curvature becomes small. T h e boundary-layer equations for an isothermal flat surface were perturbed with the transverse-curvature effect (54). Calculations for Prandtl numbers 0.72 and 1.0 indicate that the flat-surface solution is not in error by more than 57” in heat-transfer rate for vertical cylinders of sufficiently large diameter such that DiL 3 35/(Gr)1/4, where the Grashof number is based upon L. V. Combined Buoyancy Mechanisms

T h e preceding section considered flow for which thermal diffusion provided the buoyancy effect. I t was noted that conditions for similarity in the equations were the same for mass diffusion, in the formulation which applies for a very low concentration level C of the diffusing species. This is often the practical circumstance in the atmosphere, in bodies of water, in plant communities, and in many applications of technology. An added complexity of such flows is the equivalent velocity at a surface providing the diffusing materials, Eq. (50). Similarity is found

304

B. GEBHART

for some conditions when this is taken into account. For plume flows this is not a consideration. It may be shown (55) that, even for flow generated by diffusion from surfaces, this velocity effect is very small at low concentration levels. Neglecting it amounts to ignoring a term of order dp/p, as for the Boussinesq approximation itself. T h e formulation above is adequate for either thermal or mass diffusion alone. T h e former problem becomes the latter when the Prandtl number is replaced by the Schmidt number Sc = .ID, p by p* and to - t , by C, - C , . Similarity is found for mass diffusion with a power-law variation (C, - C,) = Nxn and the mass diffusion equation in terms of C = (C - C,)/(C, - C,) is C”

+ Sc[(n + 3)fC’- 4nf’Cl 1

-

0

(574

C(0) = C(C0) = 0.

(57b)

=

These relations with Eq. (36a), where 4 becomes C, are governing. However, the combined buoyancy mechanisms are to be considered here and Eqs. (29), (30) and (31) are to be taken simultaneously. T h e parameters of flow and transport will be Pr, Sc, and a Grashof number for each driving mechanism. There have been past studies of such flows adjacent to flat vertical surfaces. T h e collection (56-58) used integral methods or neglected important terms in the equations. Studies ( 5 9 4 2 ) assumed binary diffusion mechanisms, which lead to direct numerical integration. Using a nonsimilar form of zl,,(x) some numerical results were presented (63).Studies of behavior at large Pr and Sc and for Sc Pr has led ( 6 4 4 6 ) to indications of asymptotic transport behavior. T h e purpose here is to review a recent analysis of combined mechanism flows, (55,67), using the approximations of Eqs. (57) and the often reasonable approximation that zl,(x) = 0. It is shown that many atmospheric processes, as well as those in bodies of water, inevitably involve combined effects in which the two buoyancy force terms in Eq. (16) [/3*(C- C,) and P ( t - t,)] are of the same order. It is shown that in air (Pr = 0.7) the practical range of Sc is 0.1 to 10. For water (Pr = 7.0), the range of Sc considered was 1.0 to 500. Pr and Sc are not assumed in general equal. Similarity results for both vertical and horizontal flows for power law variations. T h e simplest formulation is for both to - t , and C, - C, varying in the same way with x, i.e., the same value of n. T h e uniform condition (n = 0) was considered for both geometries and vertical plumes (n = -0.6) were also analyzed.

<

NATURAL CONVECTION FLOWS AND STABILITY

305

Considering vertical flows first, values of b(x) and c(x) in Eq. (28) must be determined which include the two components of buoyancy force. T h e following combination accomplishes this and gives similarity: b(x)

G/4x,

=

c(x) =

G,

where G

=

[ i ( PGr,,t

+ Q Gr,,c)]1/4,

(58)

P and Q are constants, and Gr,,t

= gpX3(t0

-

tm)/v2

and

Gre,c = g/3*x3(C0- Cm)/v2.

T h e ratio Gr,,,/Grz,t is taken as N . We note that the effects of multiple diffusing species may be treated in an extension of this formulation. Flow on to the leading edge, x increasing upward, is ensured for P = Q if Grz, Gr,,c > 0. For Pr = Sc, P must equal Q and N > - I. However, for Pr # Sc the condition on N is not always this stringent. Then GrzSt Grz,, may be negative and still result in net positive buoyancy force and flow over all or most of the boundary region. This condition results for Sc > Pr for Grz,, < 0 and for Pr > Sc for Grz,t < 0. T h e combination in Eq. (58) is made positive by a suitable choice of P and Q. With the definitions of Eq. (58) the thermal and mass diffusion equations remain the same as before, Eqs. (36b) and (57a). T h e forcemomentum balance is still (36a), with the buoyancy force replaced by (4 + N C ) / ( P+ Q N ) . T h e boundary conditions for the surface (n = 0) and plume (n = -0.6) are as follows:

+

+

+

f ‘ ( 0 )= f ( O )

=f‘(OO)

=

1 -+(O)

=

1

-

C(0) = +(a) = C(w) = 0

and f ” ( 0 )= f ( 0 ) = f ‘ ( c o )

= +’(O) =

C’(0) = 1 - +(O)

=

1 - C(0) = 0.

T h e later conditions are in the form shown (45) to be optimum for plumes and Eqs. (36b) and (57a) may be integrated once in 7 to lower the order of the problem. T h e transport parameters, Nu,, and Nu,,, (or the Sherwood number) are found, assuming Gr,,t > 0, as Nu,,t

= h,,,x/K =

Nu,,,

=

{[-+’(0)]/21/2}( P

+ QN)1/4(Gr,,t)1/4

= {[-C’(0)l/21/z} [I(P

+QN)/N

(I Gr3C.C W4.

(594 (59b)

Velocities are found in similar form. Numerical calculations for Pr # Sc, are very demanding; the set of

B. GEBHART

306

coupled equations for f,4,and C i s a seventh-order two-point boundaryvalue problem. Solutions were determined for air, Pr = 0.7, for Sc = 0.1, 0.5, 0.94 (CO,), 5, and 10 and for water, Pr = 7.0, for Sc = 1, 100, and 500. Values of N were -0.5, 0.5, 1, and 2, i.e., for effects opposing and for various ratios of aiding effects. The results of these calculations will be briefly summarized here. Figures 6 and 7 show the velocity distribution u through the boundary S c = O . l Pr.0.7

0.5

0.4

ux -

A

0.3

4u

0.2

0. I

0

3

1

2

3

4

5

6

FIG. 6. Flow adjacent to a vertical surface, the effect on flow velocity of varying Schmidt numbers at given Prandtl number, for N = -0.5 (dashed) and 1 (solid).

region for N = -0.5 and 1.0 for both the n = 0 and n = -0.6 flows. Results are shown for Pr = 0.7 and 7.0 for various values of the Schmidt number. They are plotted in terms of Gr,,l and, therefore, show the added effect of mass diffusion. Considering Pr = 0.7, these results indicate that the Schmidt number effect upon the temperature and velocity levels is reversed for N = -0.5 and 1.0, for both flows. These results follow from a very complicated interaction of the two buoyancy effects, through velocity, to the diffusion

NATURALCONVECTION FLOWS AND

n = 315

0.8

* ux -

307

STABILITY

0.6

4v

0.4

0.2

0

- ~ -....... -

0

I

3

2

4

~~

~..~...

5

K f i Y

4

FIG. 7. Plume flows, the effect on flow velocity of varying Schmidt number at given Prandtl number, for N = -0.5 (dashed) and 1 (solid).

mechanisms. For aiding effects, decreasing Schmidt number increases the velocity level and its extent, thus thinning the temperature region. For opposed effects, velocities are much lower and less for decreasing Schmidt number. Thus, the temperature region becomes thicker. Another interesting result is the large distortion of the velocity field caused by decreasing values of N. Negative values of velocity are predicted in the outer boundary region for smaller values of Sc for N negative. This results from the incoming flow first experiencing the negative buoyancy effect of the thicker species diffusion layer. T h e flow is eventually drawn upward by the combined action of thermally caused buoyancy and the shear that results therefrom. These large effects upon the velocity distribution would be expected to have very large effects on the stability of such laminar flows. I n particular the very major distortions shown to result for Pr f Sc and opposing effects should drastically reduce stability limits for such flows. Note, parenthetically, that the roles of Pr and Sc may simply be interchanged in interpreting the results.

B. GEBHART

308

T h e effects of Schmidt number on the heat and species transfer parameters of Eqs. (59a) and (59b) are shown as the solid curves in Figs. 8 and 9 for Pr = 0.7 in terms of the parameter N . Results for 0.6

1

-I0

sc=oI

-05

,I

1

0

0 5

I

10

15

I

1

1

2 0

25

30

(N + I)

FIG. 8. For flow adjacent to a vertical surface, values of the heat transfer parameter at various Schmidt numbers at a Prandtl number of 0.7, as a function of the relative value ( N ) of the two buoyancy mechanisms. __ Present theory with P = Q = 1 ; - .. - Present theory with P = (SC)'/~,Q = (Pr)*lz; - - - - integral ' method.

Pr = 7.0 are similar. T h e values shown at N = 0 are for thermal buoyancy alone. T h e results for N # 0 are very interesting. First, around N = 0, heat transfer is not strongly affected by species diffusion although the mass transfer parameter becomes very large. This is because the flow is induced almost entirely by thermal buoyancy and produces a very effective species diffusion mechanism at very low concentration. For opposed effects ( N < 0), both diffusion parameters fall off rapidly. T h e values shown as broken lines for Sc > Pr and N negative were obtained by choosing P and Q in Eq. ( 5 8 ) different from 1.0. This is discussed subsequently. For aiding effects ( N > 0), heat transfer is continuously improved by the added impetus to the flow, whereas the species diffusion parameters fall toward the values which would pertain for flows driven primarily by species diffusion caused buoyancy. Thus, as N + co the heat transfer parameters become infinite (as for species diffusion at N = 0) while species diffusion parameters go to the asympotic values which pertain for each particular

NATURAL CONVECTION FLOWS AND STABILITY

309

1.8 1.6 1.4 -

s c = 10.0

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

I

-1.0

-0.5

I

I

0

0.5

I

1.0

I

1.5

>o.I

1

4

I

2.0

2.5

3.0

(N+ I)

For flow adjacent to a vertical surface, values of the species transfer parameter various Schmidt numbers at a Prandtl number of 0.7, as a function of the relative value ( N ) of the two buoyancy mechanisms. __ Present theory with P = Q = 1 ; - .. - present theory with P = ( S C ) ~ /Q~ ,= (Pr)lI2; - - - - integral . method.

FIG.9.

at

Schmidt number. These characteristics are found for each Prandtl number. T h e foregoing results, calculated from the appropriate complete equations, imply conditions under which integral and other approximate methods of analysis become unrealistic in predictions concerning either the flow or the transport parameters. T h e results show the ways in which approximate analyses are inaccurate for N much different from zero, particularly for N increasing negative, and for the Schmidt number appreciably different from the Prandtl number. Assumed forms of temperature, concentration, and velocity fields should include not only inversions of boundary region velocity levels, as a function of Schmidt number as N changes sign, but also of boundary region thicknesses. T h e relative thickness of the velocity and temperature region is not simply a function of Pr, or of Sc. Similarly, the combined buoyancy effect for N < 0 greatly alters the velocity distribution and even produces locally negative values for Sc # Pr. T h e assumptions employed in integral method analysis are appropriate for Pr = Sc. However, such

310

B. GEBHART

combined buoyancy effect processes (Le = l), are the same, in terms of the generalization employed herein, for any N , as those arising from a single buoyancy mechanism. Thus, in its most appropriate application, integral analysis is not necessary. T h e limitations of such analysis may be best seen by comparing the resulting predictions of heat and species transport rates with those calculated here. T h e dashed curves in Figs. 8 and 9 compared to the solid ones and their calculated extension (with P # Q # 1) show that the disagreement always increases as the Schmidt number diverges from the Prandtl number. Errors become large for Sc < Pr. This is particularly Pr > 1 when the true for species diffusion, Fig. 9, except for Sc velocity layer is much thicker than both diffusion layers. For Sc < Pr, errors in the species transfer parameter may be 35 yo.T h e approximate result also becomes unreliable as N becomes sufficiently large negative that the opposing byoyancy effect is appreciable. T h e trends of the two results become quite different. These results are a guide to the limitations of more approximate methods in flows with combined buoyancy mechanisms. A similar analysis has been carried out (67)for flow over a semiinfinite ) the same except horizontal surface. T h e formulation of b(x) and ~ ( xare that 5 and the fifth root of Gr, appear as in Eq. (51a). T h e presence of the pressure field results in an eighth-order two-point boundary value problem. Numerical calculations were for Pr = 0.7, Sc = 0.1, 0.5, 0.7, 1.0, 5.0, and 10, for N = -0.5, 0.5, 1.0 and 2. Very similar effects were found both in boundary region distributions and in both thermal and species transport.

<

VI. Flow Transients

Natural convection flow transients arise from transients in thermal conditions which in turn cause changing buoyancy forces. T h e thermal changes that have been considered in some of the analyses of transient behavior have been changes in surface temperature or in surface heat flux to the adjacent fluid. T h e assumed condition has invariably been a step in boundary condition from a no-flow condition. Such step changes are unrealistic. What happens is most actual circumstances is that a change occurs in thermal input to a element of finite thermal capacity, C” per unit of surface area. T h e surface temperature to(%,T ) then changes in a coupled relation between the element thermal capacity and the conductive and convective transport rate to the adjacent fluid. Thus, if an

NATURAL CONVECTION FLOWS AND STABILITY

31 1

input flux of qtt(x, T ) per unit surface area is imposed, the appropriate surface condition is Q” = C ” ( a t o / a T ) - k(at/ay),=o

.

(60)

Conduction internal to the element in the x direction is assumed negligible and in the y direction is assumed complete, as for thin surfaces. Since such surfaces, electrically heated, are both practical and a good experimental configuration, this condition is used in the present discussion. Considering qr’(x, T ) to be a simple uniform step input q“, we may analyse transient flow response. Clearly there are different regimes of such flow. For example, at very short times, i.e., T < xi U , ,the presence of a leading edge is not felt downstream of x. Therefore, in the equations the normal velocity component ZI and all x derivatives are zero. Thus the problem is linear and u( y , T ) may be easily found, even for more general transient boundary conditions. This region is called the “conduction transient regime” and has been extensively analyzed (68-71). One may estimate from such results the distance D(T)of the propagation of the leading edge effect as

T h e experimental results (72, 73), give somewhat larger values of D . Nevertheless, this regime arises early in all processes and even dominates a major portion of those for which C” is small compared to the energy absorption capability of the adjacent fluid. This characteristic is indicated by small values of the parameter Q:

Q

~c C ” l p c p 8 ~c ~ s l ~ 3 l c

where p and cp are fluid properties and 6 is the characteristic thickness of the convection layer. Q is also written above as the ratio of the time constant for the transient in the element to that in the fluid. For Q 1, the conduction transient regime dominates. Another limiting regime arises when Q is considerably larger than this limit. For this condition q” is mainly consumed during the transient in heating the element. Convection begins and grows weakly, the starting time is controlled by c“ and is long. T h e transient momentum and energy effects in the fluid are negligible. Temperature increases are controlled by the storage requirement of the surface and the convection process is approximately quasi static. T h e analysis of this regime is also relatively simple, at least with some approximations. An indication of when the two limiting regimes occur in actual

<

312

B. GEBHART

processes is interesting. Considering the dependence of Q above, it is clear that c" might be less variable over the range of circumstances of practical interest than pcP , 6 is not as highly variable. For liquids and gases at atmospheric pressure, the quantity pcp differs by a factor of about lo3. For values of c" of practical interest, gases are usually in the quasi-static regime. Transients with conventional liquids are sometimes in the conduction transient regime. However, gases a t high density, with small c", and liquids with large c", reach into the intermediate region from the two extremes, respectively. Thus, an analysis of the intermediate region is of practical importance. T h e optimum analysis of transients would result if conditions of selfsimilarity were found to apply to the boundary layer equations with the transient terms and at/& included, i.e., if x, y , and T went over into a single variable ~ ( xy,, T), resulting in ordinary differential equations. Conditions on t,(x, T ) were enumerated ( 4 ) for which this occurred. These forms are interesting but quantitatively do not seem to be of practical importance. An integral analysis (74) for a surface temperature step leads to a solution by the method of characteristics. However, unreasonable predictions resulted for any process for which more realistic boundary conditions, such as Eq. (60), apply. An alternative procedure is the numerical integration of the transient boundary-layer equations for conditions of interest. Some such solutions are found in the literature. As always, many parameters arise, and the results are very special and do not lead to conceptual simplicity. Some years ago an approximate analysis appeared (75, 76) that retained all of the principal physical features of such transient flows and predicted transient response in terms of a few parameters. T h e experimental results which have subsequently accumulated support the predictions and the appropriateness of the parameters, over the range from conduction transients to quasi-static processes. T h e analysis seems very approximate, but its measure of success, coupled with the relevance of its form to other flow problems such as laminar instability, perhaps make it worth presentation here. T he boundary region adjacent to the first length L above the leading edge of a flat vertical surface with thermal capacity C" was treated by a "double" integral method. Integrals were taken over the instantaneous convection region thicknesses and over the height x = 0 to L. Transient terms for both momentum and thermal storage were retained. T h e local instantaneous surface temperature to - t , , the maximum velocity and the boundary region thickness are generalized as $, X and Y , in terms of the final steady-state values. Various integrals and other combinations of these quantities arise.

NATURAL CONVECTION FLOWS AND STABILITY

313

These in turn are expressed in terms of instantaneous averages (over x) $, _ _ X,Y , times distribution weighting factors. T h e approximation arises in evaluating these weighting factors. They are functions of a thermal capacity factor, such as Q, of time, of Prandtl number, and of location. Inspection of known transient and steady-state results suggested that these factors are actually most strongly Prandtl number dependent, through the effect of this parameter on the forms and relative thicknesses of the temperature and velocity boundary regions. Therefore, the weighting factors were evaluated from accurately known steady-state flows, for various Prandtl numbers over the range from to lo3. T h e results of the analysis are differential expressions for the time and Y as functions of generalized time T , in dependent quantities $, terms of a thermal capacity parameter Q and Prandtl number dependent constants a, S , U , W , M , and b. T h e instantaneous average value of 4, i.e., $, is the increase in average surface temperature, expressed as a fraction of the final steady state value. For example, for a step in input flux q", the value of $ goes from zero initially to 1.O in steady state. An important practical question is whether or not $ exceeds 1.0, that is overshoots, during the transient. T h e equations, initial conditions, and definitions of generalized time T and thermal capacity parameter Q, follow:

x

S$F - U(X/F) - d ( X F ) / d T - WFX2 = 0

(62b)

at

T

=

(ai-/L2)[b Gr* Pr]2/5 w 01i-/6~

(63)

Q

=

(c"/pcsLM)[b Gr*

(64)

w c"/pcs6.

T h e ratio q"/qL = f ( ~ relates ) the instantaneous input flux to that at infinite time, which then becomes the uniform steady state surface flux. Time T is seen to be a Fourier number. T h e flux Grashof number, G r * = g p x 4 q ~ / k v 2is, a special one particularly useful for flow resulting from a uniform flux boundary condition. T h e Prandtl-number-dependent constants are listed in Table I11 for the range 10-2-103. T h e element temperature response $ has the following dependence:

B. GEBHART

3 14

Although several conditions of f ( 7 ) have been treated, I will consider results only for a step in input flux, i.e.,f(T) = 1. T h e many experiments used this condition. TABLE I11

VALUESOF UPON

THE PRANDTL-NUMBER-DEPENDENT CONSTANTS BASED STEADY-STATE DISTRIBUTIONS FOR AN ISOTHERMAL SURFACE

Pr

a

S=(U+W)

u

0.01 0.72 1 .o 5 10 100 1000

0.1844 0.2000 0.1971 0.1936 0.1894 0.1924 0.1905

9.082 16.13 17.67 33.40 41.69 126.9 263.2

1.165 9.242 10.97 26.78 35.29 121.4 258.6

W 7.918 6.886 6.704 6.616 6.398 5.523 4.677

M

b x lo4

1.88 1.79 1.79 1.78 1.77 1.77 1.76

1.408 40.25 71.15 75.97 81.43 137.1 118.4

Numerical solutions of Eq. (62) for an element of zero thermal capacity (Q = 0) indicated no temperature overshoot, i.e., $ 1, and a negligible Prandtl number effect on response in terms of T. Thus,

<

4 = $(T)

for Q = 0.

Calculations for Q # 0 over the range from to 10 indicated that again the Prandtl number effect was negligible if T was expressed as T/Q. We have, therefore,

4=

(65)

4(%?9Q).

These calculations also showed the limits of the quasi-static and conduction transient response regimes. For Q 3 1, the calculated response is quasistatic and actually becomes a function only of TIQ. I n the present formulation this is equivalent to neglecting rates of change in momentum and thermal energy content in the fluid layer, but retaining the feature of changing storage in the element itself. This solution is called For Q 0.1 the calculated response follows that of pure conduction, accounting for all effects, until has risen to approximately 0.8. Then adjusts rapidly to the asymptote = 1 .O, while the one-dimensional conduction solution continues to increase without bound. Thus, the Q 1, to the level intermediate regime occupies only the region 0.1 of sophistication of this theory. Calculated curves are shown in Fig. 10, the intermediate range is seen to be of small extent.

4

<

4,.

4

4

4

< <

NATURAL CONVECTION FLOWS AND STABILITY I

nL

315

P = 10.1

x

/Q=Ol

7

-

Q=C'

x

n =7 48

+

0 =0 029

387

-

0 0 = 0 0014

0

I

I

I

I

1

I

I

2

3

4

5

6

7

T/ 0

FIG. 10. Calculated temperature response and comparison with measured response in the two extreme regimes.

T h e interest in such transients, along with the desire to test the theory, has resulted in a number of experimental determinations of transient response characteristics. T h e results cover the range of Q from0.009 to 3 1, T h e fluids used were water and gases. Element temperature response was determined in all cases, and by sensors as different as resistance thermometers, infrared detectors, and interferometers. T h e interferometric determinations of response were carried out by averaging instantaneous interferograms like those shown in Fig. 1 1. A summary of test conditions, transient times, instrumentation, etc. for the five investigations compared with theory are given in Table IV. T h e results of this comparison are given briefly in Figs. 10,12, and 13. I n the first figure, experimentally determined response is shown for the asymptotic regimes. There is little systematic diagreement and all differences are thought to be within experimental error. These results are a test of the theory particularly in the prediction of the lower limit of Q for quasi-static processes and in the dominance of the conduction transient mechanism of early response for low-Q processes. T h e comparison shown in Figs. 12 and 13 is for the intermediate region and near quasi-static regime. This is a more critical test of the theory. Again the agreement is within experimental accuracy. One concludes

316

B. GEBHART

FIG. 1 1 . Interferograms at various times during the convection transient at Q = 0.2516, Gr = 3.98 = lo*, p = 17.1 atm. Field normal to the foil enlarged by a factor of six with anamorphic lens system.

that the averaging procedure of the theory very closely, and perhaps unexpectedly, matches, over the whole range of response, the average characteristics of flows which proceed from early one-dimensional conduction transient mechanisms, through the propagating leading edge effect, through the final convection transient to steady state.

TABLE IV SUMMARY OF

References

Points on Figs. 12 and 13

NATURAL CONVECTION TRANSIENTS STUDIED

TO

DATE

Fluid

Q

Gr

Gr*

Height of foil (in.)

Water

0.0014

9.8 x lo6

5.3 x lo8

6.5

Water Water

0.0087 0.029

9.3 x 107 9.4 x lo8

1.0 x 1010 1.5 x loll

3.5 3

5.0 x 105

7.7 x 106

1.5

47.5

233

2.48

2.2 x 106

4.5 x 107

2.5

5.0

225

0.71

4.6 x lo6

8.7

lo7

3.0

1.75

198

0.1614

6.94 x los

5.75 x 1Olo

6.0

0.2516

3.98 x lo9

5.11

10l1

3.9

0.2549

3.58 x lo8

2.53 x 1Ol0

6.2

0.508 1

1.71 x 108

9.99 x 109

0.5203

1.37 x lo9

1.010 1.016

Air 1 atm Air 1 atm Air 1 atm

30.8

X

Time to (sec)

45 4.7 1.50

NZ

17.4 atm NZ 17.1 atm NZ 9.27 atm NZ 3.93 atm NZ 6.87 atm

(to -

tm)m

4 = 0.90 (OF)

Instrumentation

1.2

Interferometer

38 114

Infrared detector

4.80

> Z

X

8.21

6.9

22.9

1.351 x 10l1

3.5

75.5

5.38 x 107

2.36 x 109

7.4

57.3

9.29 x 107

4.66 x 109

6.8

74.1

7&

U

30.6

NZ

1.54 atm Na 1.80 atm

Resistance thermometer

Interferometer

B. GEBHART

318

1.0

\I.

0.5

-

I 2 3 4

Q Q Q Q

= 0.1614 = 0.2516

=0.2549 =0.5081

+ 0

0

5 0.0.5203

0 0

I

2

3

4

T/Q

FIG.12. Measured transients in Nz gas compared with calculated responses, the convection transient regime.

1.0 -

Q = 1.013

Q = 0.5142

4 Q = 0.5081 0 5 Q = 0.5203 6 Q = 1.010 A 7 Q = 1.016 A

0

0

I

I

I

I

I

2

3

4

T/Q

FIG.13. Measured transients in N, gas compared with calculated responses, for conditions near quasistatic.

NATURAL CONVECTION FLOWS AND STABILITY

319

FIG. 14. Interferograms at various times during the convection transient Q” 29.4 BTU/hr ft2, Q = 0.16, Gr = 4.30 X lO’O, p = 17.97atm.

=

3 20

B. GEBHART

This collection of comparisons of predictions and experimental results certainly validate the analysis as applied to estimating average element temperature response. All of the results indicate that the early concern over the possibility of temperature overshoot during transients was unfounded. T h e other predictions, concerning velocity levels and region thicknesses, have not been tested. This consideration has been of laminar boundary layer regime flows. A more recent experimental study (73) concerned much more vigorous transients in pressurized nitrogen. Transient response was determined for Grashof numbers, based on average steady-state temperature difference, of approximately 4.3 x 1O1O and 2.5 x loll. The values of Q were 0.16 and 0.25. Such vigorous flows would be turbulent at large

1.67 Sec.

2.33 Sec.

2.00 Sec.

2.12 Sec

3.00 Sec.

FIG. 15. Interferograms at various times during the convection transient q" B T U / h r ft2, Q = 0.25, Gr = 2.52 x lo", p = 17.97 atm.

=

269

NATURAL CONVECTION FLOWSAND STABILITY

321

distances from the leading edge in steady state. T h e principal purposes of this study were to see if delayed transition could cause the temperature to overshoot that of the final turbulent flow condition and to determine whether or not a rapidly developing transient flow would be subject to important delays in transition. Th e immediate results of the observations were a possible positive answer to the first question and a definite positive one to the second. We also found that the temperature response in the laminar part of the transient was in accord with the theories discussed above. Leading edge effect propagation rates were again faster than predicted. However, the interferometric records are much more interesting than that. Figs. 14 and 15 show a sequence of frames for each of the two transients. For the weaker one we see transition and relaminarization, for the other just transition. But the characteristics that they have in common, and the surprising observation, is that the disturbances which grow to the turbulent bursts and transition appear to be derived from the propagating leading edge effect. Further, these disturbances are principally a single harmonic and the frequency is very close to that which would be most preferred by the unusual filtering mechanism predicted by linear stability analysis of steady laminar flows. This brings us to the question of laminar stability, to the coupling between temperature and velocity disturbances which occurs in natural convection flows, and to the importance of element and fluid thermal capacity coupling in disturbance propagation. T h e next section considers stability, disturbance growth, and what is known of transition for such flows. T h e relations between the special characteristics of natural convection transients discussed above are related to these latter questions.

VII. Instability and Transition of Laminar Flows

Many configurations in fluids subject to a buoyancy force are known to be unstable. Instabilities lead first to laminar disturbances or circulations and then to turbulence. This section will review recent findings concerning these mechanisms for flows adjacent to surfaces. T h e following two sections concern plumes and other flows. Most natural-convection processes found in nature occur on such a large scale and are of such long duration that the detailed transport mechanism is largely turbulent. However, for the fluids and for the scale of size important in technology, as well as in the intimate details of such flows in nature, one more typically encounters either laminar flow, a stably stratified fluid, unstable laminar flow, a flow in transition, or

322

B. GEBHART

a newly turbulent flow. Often all these regimes may be found in the same geometry or in different regions of a particular natural-convection field. Conditions in practice are often appropriate for the body of fluid to become unstable to imposed (and ever-present) disturbances. Two different kinds of instability arise. One kind results from the tendency to motion present in a stratified medium in which heavier fluid overlays lighter fluid. This is called thermal instability. T h e other kind of instability arises in a laminar flow when a balance of buoyancy, pressure, and viscous forces may contribute net energy to a disturbance, causing it to grow as it is convected along. This is called hydrodynamic instability. I n transport processes generally, the rate of transport depends very strongly on the regime of flow. T h e importance of natural convection has led to intensified study of laminar instability and of its consequences in determining flow regime. We are interesting in the sequence of mechanisms whereby a laminar flow is converted to turbulence. Consideration is first given to vertical surfaces bounded by an extensive region of fluid at rest. This idealization is also appropriate in internal flow circumstances if the convection layers are thin compared with the dimensions and spacings of the bounding surfaces. One of the earliest studies of laminar-turbulent natural-convection flow over a vertical surface (82) inferred the presence of turbulence from measured heat transfer characteristics. Interferograms of the same flow configuration (83) suggested for the first time that the advent of turbulence in such flow was, in all likelihood, the amplification of initially small disturbances. T h e disturbances were seen to amplify in twodimensional form, initially as a pure sinusoidal disturbance and later as a more complicated wave. Several qualitative hot-wire measurements of amplified disturbances in air and in water over a vertical isothermal plate were later reported (84, 85). T he early observations suggested that the sequence of processes whereby an initially laminar flow progresses toward a turbulent circumstance may be similar, at least in broad outline, to those which appear to apply in forced flow. Over the previous two decades, a number of quantitative analytical and experimental studies had indicated that a laminar boundary layer over a plate, placed in a uniform stream, became unstable to disturbances. These amplified to produce a circumstance in which three-dimensional, or spanwise, effects became important. These effects, in turn, were found to cause a condition (a “shear layer”) under which concentrated turbulent bursts are produced. These bursts, of whatever origin, are known to consume the remaining laminar flow, in a transition region, to produce an eventually completely turbulent

NATURAL CONVECTION FLOWS AND STABILITY

323

flow. See Stuart (86) for a convenient recent discussion of this sequence of processes. There are reasons to believe that these mechanisms are also in operation in the early part of the natural-convection process. However, there are a number of compelling reasons, from both analytical considerations and experimental observations, to expect that later stages of the process are different from, and far more complicated than, the forced-flow analog. For example, the disturbance velocities and wavelengths of importance are very different, the formation of a “shear layer” is not so straightforward, and the growth of any bursts would involve additional mechanisms and energy sources. Nevertheless, the picture of the initiation of the whole process of the conversion of a laminar flow to a turbulent flow must be the same for both flows. T h e laminar flow must become unstable and amplify various frequency components of a disturbance present in the flow. These disturbances are the cause of later breakdown. T h e apparent importance of this initial mechanism led to the formulation (87) of linear stability theory for such flows. T h e fundamental assumption of this theory is that the basic laminar flow, e.g., over a vertical surface as indicated by u, v , and t - t , = 8, is perturbed by small two dimensional velocity and temperature disturbances of similar form. These disturbances are decomposed into a periodic series representation. A typical periodic component (of frequency f)of the disturbance u“, v”, p”, and 6 is superimposed on the base flow to determine its behavior. Is the base flow stable, neutrally stable, or unstable for this particular disturbance ? In principle, this question is asked for all frequencies in order to find stability limits for the laminar flow. Figures 18, 19, and 20 (see pp.329,330and 331)show interferograms of convected disturbances rising adjacent to a thin, electrically heated foil. Amplification and damping of such controlled disturbances are clearly seen. Given a disturbance of frequency f superimposed on the laminar base flow, does this flow contribute or remove energy from the disturbance as it is convected along ? T h e sum of buoyancy, momentum, and viscous effects determines the result. T h e question is asked by postulating such a disturbance, putting it into the complete two-dimensional forcemomentum and energy equations governing the flow, and asking whether it is damped or amplified and at what rate. T h e stability equations set forth in Plapp (87), the Orr-Sommerfield equations, are in terms of the velocity and temperature disturbance amplitudes. Since these two disturbances are coupled through buoyancy, and since pressure disturbances are admitted, the resulting set of equations are of sixth order. It may be considered a fourth-order problem

324

B. GEBHART

when buoyancy coupling and any thermal coupling between the surface and the flow are neglected, but it becomes sixth order when thermal effects are admitted. T h e complexity of natural convection flow profiles impeded early attempts (87,88)neglecting thermal effects, to treat the stability problem analytically. There are some exceptions for special or extreme flows, see, e.g., Gill and Davey (89), Pera and Gebhart (90), and Hiber and Gebhart ( 9 I , 9 2 ) .However, the adequate treatment of the more common flows considered here awaited sophisticated numerical techniques and greater computer capability. Then the direct and reliable method became numerical integration of the equations. T h e first integration of the uncoupled equations (93),for Pr = 0.733, estimated stability limits. However, special numerical techniques (94) were necessary to make it practical to treat the sixth-order problem. T h e neutral stability results for air and water ( 9 4 , P r = 0.733 and 6.7, indicated that the inclusion of the effect of coupling between velocity and temperature disturbances, through buoyancy, may have a very large effect on predicted laminar stability limits. These results, strangely, suggested very low stability limits, in terms of local Grashof number, compared to previous experimental observations of the occurrence of turbulence. An additional thermal coupling effect, between the flow and the surface which generates it, arises as a relation involving their relative thermal capacity. I t is analogous to the mechanism considered above for transient flows, in terms of the parameter Q. Temperature disturbance propagation may be modified by interaction with the surface material and thus, through buoyancy coupling, affect the velocity disturbances. For example, a massive surface of high relative thermal conductivity would completely damp fluid temperature disturbances at the interface. Admission of this effect (95) amounts to a more complicated boundary condition. Again, this coupling was found to have an important influence on predicted stability limits under some flow conditions. This later result is important in a number of ways. First, predicted stability limits are dependent on this thermal coupling over the range of a parameter similar to Q. Many actual surfaces are sufficiently low in thermal capacity, relative to liquids, that it is of practical importance. Further, the design of our experimental program to study actual stability limits and disturbance growth rates inevitably led to designs in which relative thermal capacity Q varied widely between gases and liquids. T h e optimum apparatus became thin, electrically heated foils in pressurized gases and in the dielectric liquid silicone. T h e thin foils were necessary in order to obtain the very short test times necessary to ensure

NATURAL CONVECTION FLOWS AND STABILITY

325

quiescence in media of reasonable extent. Long tests lead to circulations and stratification in the distant medium. From the results concerning transient times discussed in the previous section, it was apparent that one could carry out complete experiments in steady flows in a few minutes, with thin, electrically heated foils. As a result, the stability limits and disturbance growth rate calculations for flows adjacent to vertical surfaces were carried out for a surface having a uniform (in x) time average surface heat flux q“ and for various conditions of thermal capacity coupling between the surface material and the adjacent fluid. T h e formulation of the linear problem of twodimensional disturbance growth in these terms is given below. T h e basic laminar flow, subject to disturbances, is formulated in the manner of (5) for a uniform heat flux surface condition. T h e usual stream function $, and the temperature difference are written as functions f and q5 in terms of similarity variable q as follows:

where the convection velocity, boundary region thickness and conduction estimate of the temperature difference are

T h e base flow distributionsf(q) and +(T) are obtained from two coupled ordinary differential equations, the parameter being the Prandtl number. T h e equations and boundary conditions are fiif - 3

4” + Pr(44’f f(0) = f’(0) = f ’( 00)

+=0

(674

- +f ’) = 0

(67b)

f ~+ 4 f f ” -

= +( co) = +’(O)

+ 1 = 0.

(674

T h e disturbance temperature t’ and stream function $’ are postulated as follows: =

( 5 v ~ x ) 1 / ~2 ( 7exp[i(Sx )

-

/&)I

(684

where fl is taken real (&) and becomes the disturbance frequency and 8

326

B. GEBHART

is complex, the real part BR being the wave number and the imaginary part B, the spatial exponential amplification rate of the disturbance. T h e physical quantities B and are generalized as

6

p

=

ps/u,,

a = 66.

These disturbances are substituted into the complete time-dependent flow equations. T h e base flow f, is assumed to be one-dimensional in the disturbance equations, i.e., v is taken as zero, and x derivatives of u and are neglected. T h e resulting Orr-Sommerfeld and energy equations for the disturbance amplitude distributions @(v) and s(y), in terms o f f , 4, the parameters G* and Pr and the eigenvalues a and /3, are

+

+

(f’- p/a)(@’’ - a2@)-f”@

=

(I/iaG*)(@”” - 2a2@“

(f’- p/a) s - +‘@

=

(1 /;a Pr G*)(s” - 2 s ) .

+ a4@ + s’)

(69a) (69b)

These stability equations were derived for the special uniform flux formulation but are the same as those which would result from the definition of as ( t - t , ) / d ( x ) (where 7 = 8 in this circumstance) and the formulation in terms of G. G* above then becomes G and the base flow equations are (36). Thus the plume has the same stability equations as will be seen in the following section. T h e boundary conditions in this notation for Eqs. (69a) and (69b) are

+

@(O)

=

@‘(O)

=

@(a)= @’(a= ) s(c0)

=

0

(694

and s(0) = (i/pQG*3//“) ~ ’ ( 0=) ( z ’ / Q * s’(O), ~~)

(694

where, as before,

Q

=

Pr(c”/pc,)( g & ’ ’ / k ~ ~ ) ~ / / “ .

Th e limiting characteristics of the last boundary conditions are s(0) = 0 for Q very large, i.e., temperature disturbances damped at the wall, and s’(0) = 0 for Q = 0, no thermal capacity at the wall to damp temperature disturbances. T h e above formulation has been investigated to determine G*&) for aI = 0, i.e., for conditions of neutral stability, and to determine the disturbance amplification characteristics (for q < 0). Such calculations have been carried out (95) for small and moderate Prandtl numbers. Results for gases and for Pr = 6.7 are discussed below. T h e effect of disturbance coupling through buoyancy is seen in the

327

NATURAL CONVECTION FLOWS AND STABILITY

neutral stability curves of Fig. 16, for the Prandtl number of air. T h e dashed curve was calculated (96) from Eq. (69a) above alone (s = 0), 0.14 I

t

I

0.12

I

I

I

-

010 -

w

1 d = 0.003

0.08-

P 0.06 -

0.04 -

I

Uncoupled solution

0.02 -

O V

0

1

20

40

60

80

100

120

140

I

160

I

180

I

200

G*

FIG. 16. Neutral stability conditions for a uniform flux base flow and Prandtl of 0.733, from Knowles and Gebhart (95).

with the first four conditions of Eq. (69c). T h e admission of buoyancy coupling between disturbances, but with complete disturbance damping at the surface, results in the curve labeled Q* = co [s(O) = 01. Buoyancy coupling destabilizes the flow in the low-frequency range. However, this will be shown later to be of no appreciable importance in actual transition mechanisms for this Prandtl number. T h e effect of surface thermal capacity coupling, s(0) # 0, is seen to cause a further destabilization at lower frequencies. Flows arising from a uniform surface temperature condition have very similar neutral stability characteristics. This is also true for a Prandtl number of 6.7, the value for water and for a light silicone oil used in experiments. For Pr = 6.7, the effects of buoyancy coupling are very large, reducing predicted levels of G* for neutral stability by a factor of about 10. Only the coupled curve is shown in Fig. 17. T h e uncoupled one lies in the vicinity of G* = 500. Results with different levels of surface coupling Q are similar, but slightly displaced for the two extreme surface conditions s(0) = 0 and s’(0) = 0. T h e results of Fig. 17 are for s’(0) = 0, i.e.,

Q

=

0.

' B. GEBHART

328

0 12

1 i

1 300

G*

FIG. 17. Curve of neutral stability for a Prandtl number of 6.7 and with s'(0) = 0 (100). Points are from the experiment of Knowles and Gebhart (98).

T h e neutral stability predictions for these two Prandtl number levels were experimentally verified by experiments in pressurized nitrogen and in 0.65 centistoke silicone oil (97-99). T h e method of disturbance observation in both circumstances was interferometric, with disturbances of controlled frequency and amplitude introduced by a vibrating ribbon. Demonstration interferograms are shown in Figs. 18, 19, and 20. Experimentally estimated neutral stability points shown in Figs. 17 and 21 support the predictions of linear theory, within our experimental accuracy. T h e next question of interest is how disturbances amplify as they are convected along in an unstable laminar flow. Detailed calculations (92, 100) have resulted in the two extensive stability planes of Figs. 22 and 23, for Prandtl numbers of 0.733 and 6.7. T h e neutral curve is shown as zero, followed by contours of equal downstream amplitude. T h e numbers on the contours are A = -J a , dG*/4, the integral being taken along constant physical frequency paths from the neutral curve. T h e quantity e A represents the local disturbance amplitude divided by that of the same disturbance (of the same frequency) as it crossed the neutral curve. T h e dashed lines are paths of the propagation of constant physical frequency. Predicted disturbance amplitude

NATURAL CONVECTION FLOWS AND STABILITY

329

FIG. 18. The convection of amplifying disturbances in pressurized N, gas. Note temperature disturbance propagation through the foil, to the right side, as a result of thermal capacity coupling.

330

B. GEBHART

FIG.19. Damped disturbance in pressurized N, gas.

distributions and growth rates in G* have been experimentally verified

(98, 99). These two stability planes show surprising characteristics. For each fluid only a narrow band of frequency is highly amplified by the flow. Note that the first frequency to be unstable along the surface (G*) does not experience important amplification. This is in sharp contrast, for

FIG.20. Disturbances in silicone oil (Pr = 6.7), first damped, then amplified after the location of neutral stability.

B. GEBHART

332

0’9 0.8

c 1 i A

data Polyrneropoulos 8 Gebhart (1967)

01

60

00

I

100

I

I

120

140

I

160

I

180

200

I

G*

FIG.21. Comparison of the experimental estimates of neutral stability from Polymeropoulos and Gebhart (97) with calculations from Knowles and Gebhart (95). The conditions of the experiments were approximately Q* = 0.04.

example, to disturbance amplification in forced flow boundary layers. The implication of these results is that such natural convection flows filter complicated disturbances for certain frequencies. All the experimental evidence we have confirms that this filtering mechanism operates and that its consequences vitally affect latter processes. Over the years there have been a number of observations of highly amplified disturbances which disrupt the laminar flow regime, in circumstances in which no “controlled” disturbances were introduced. These are so-called “natural transitions.” A number of such observations in air and in water, for which frequencies could be determined, are shown as points on Figs. 22 and 23. All points lie near the frequency path of greatest amplification. Thus, even transition appears to be controlled by this filtering mechanism. A review of the whole collection of such observations strongly suggests that the appearance of large oscillations and of transition are linked to G* (or G), actually to A. Values of A = 6 and A = 10 are suggested (91) as the locations of large disturbances and of transition for both Pr = 0.7 and 6.7.

NATURAL CONVECTION FLOWSAND STABILITY

,

0.141

I

0

333

I

200

000 I

1000

600

400

G*

FIG. 22. Stability plane for Pr = 0.733, showing amplitude ratio contours in the unstable region, s(0) = 0. The circled data point is from Eckert and Soehngen (83) and the crosses are from Polymeropoulos and Gebhart (97).

1

0

I

200

600

400

000

I

1000

G*

FIG. 23. Stability plane for Pr = 6.7, showing amplitude ratio contours in the unstable region, s(0) = 0. The crossed data points are from Knowles and Gebhart (98).

334

B. GEBHART

These special stability characteristics have been found only for flows adjacent to surfaces. Different behavior is predicted for plumes and for other flows as well. T h e study of vigorous transients (73) referred to in the previous section unexpectedly provided additional support for the above conclusions concerning the importance of disturbance filtering. T h e observed local formation of turbulent bursts corresponded approximately to the calculated instantaneous location of the propagating leading edge effect. T h e observed dominant frequency of the two-dimensional disturbance preceding the bursts, seen in Figs. 14 and 15, along with the instantaneous local Grashof number, located the burst in p, G* coordinates. These locations fell very close to the most amplified frequency curve of Fig. 22, but at much higher G*, in the range from 1000 to 2000. This implies that the leading edge disturbance, whose detailed characteristics are presently unknown, was filtered by the flow. T h e delay to higher G* is characteristic of the delay of amplification in transients reported in Gunness and Gebhart (101). Another curious observation was of relaminarization, after the consequences of the leading edge effect had passed. This may be characteristic of natural convection flows in very quiet surroundings, such as these were. There is no continuing source of disturbances until downstream circulations, turbulence, or stratification have had an opportunity to feed back to the upstream regions. This collection of predictions and observations of the stability, disturbance amplification, and transition characteristics indicate the special properties of these natural convection flows. T h e flows are weak and the coupling of disturbances through buoyancy and with the surface are often very important. Conditions in the remote fluid also affect these processes in ways not now understood. For completeness, attention is drawn to extensive calculations (91, 92) of neutral stability conditions to very large values of G* and at both large and small Prandtl numbers, over the range from oils to liquid metals. Filtering characteristics appear to be the same. For large Prandtl numbers, neutral stability curves are seen (91) to scale approximately in powers of the Prandtl number. T h e nature of instability limits were determined as the Prandtl number becomes very large (92). Two modes of instability are found, the one associated with the inner (temperature) region is found to be dominant. T he indicated question at this point in the development of knowledge concerning disturbance growth to turbulent bursts is, how are twodimensional disturbances related to three-dimensional effects ? Now we only know that the flow is initially more stable to the latter disturbances.

NATURAL CONVECTION FLOWSAND STABILITY

335

However, some things about three-dimensional disturbances are in evidence for flows adjacent to horizontal surfaces and in mechanisms related to what has sometimes been called “flow separation” in natural flows. This is discussed in a later section. VIII. Instability in Plumes Flow adjacent to a vertical surface is seen to have some very surprising instability characteristics, for example, in the very sharp frequency filtering and in the large Prandtl number dependence of coupling. Several other kinds of buoyancy induced flows have also been analyzed and the plume flow considered below has been shown (90) to have very different characteristics. Consider a plume arising from a line source, Fig. 4. It was shown that to - t , = Nxn where n = -0.6. The coupled flow temperature 4 and stream function f are determined in q by Eq. (54). Two-dimensional disturbances are postulated for this flow in the same form as before, Eq. (68). T h e stability equation remains exactly Eq. (69), the functions f and are now the plume solutions. G* in Eq. (69) is replaced by the G relevant to a plume, G = 4(Gr,/4)lI4. All generalizations are the same as before. However, there are differences in some of the boundary conditions the eigenfunctions @(q) and s(7) must satisfy. T h e following ones are the same as before: @’(+XI) = @(&oo) = s(*co) = 0. (714 T h e other necessary three conditions admit the possibility of motion at q = 0 and also that disturbances on the two sides of the plume may be symmetric about q = 0, or nonsymmetric. T h e extreme of nonsymmetry is entirely asymmetric. T h e boundary conditions at q = 0 of symmetric and of asymmetric disturbances, one set to be used with those above, are @(O)

=

@”(0)= s’(0)

@’(0)= P ( 0 ) = s(0)

=

0

=

0.

T h e plume flow has nonzero inviscid asymptotes, at large G, of instability in both disturbance modes. T h e vertical surface flows apparently do not. T h e asymptotic values for Pr = 0.7 were = 0.7088 and = 1.3847, respectively. Thus the asymmetric mode appears less stable and is the only one considered hereafter. Asymptotic values were then found over the Prandtl number range from lop2to lo4.

B. GEBHART

336

Neutral stability limits were then determined for the uncoupled mode 0) and for coupled disturbances for Pr = 0.7 to yield the curves of Fig. 24. T h e first values of G for instability are very low, an order of

(s =

I .4 I .2 I .o

0.0 a

06 0.4

02 0

FIG.24. Computed neutral stability curves. Coupled and uncoupled flow. Asymmetric disturbances. u is the Prandtl number.

magnitude less than those shown in Fig. 21 for flows adjacent to surfaces. Both curves approach the asymptote at large G. Coupling has a large effect at low G, it may be expected to decrease uniformly as G increases. T h e short segments of neutral stability curves for other Prandtl numbers, uncoupled, show a small effect of this parameter in the range considered. The paths which disturbances follow as convected along at constant frequency are indicated on Fig. 25. T h e particular frequencies shown relate to an experiment to be discussed subsequently. This is very different behavior than for vertical flows adjacent to surfaces. T h e base flow filters for all frequencies below a certain limit. However, all frequencies are eventually stable, i.e., all eventually would emerge again into the stable region. Of course this does not happen in the actual flow. Nonlinear mechanisms enter first, for some of the frequencies present. Experiments were carried out to test these stability predictions. A 6-in. long horizontal wire of 0.005-in. in diameter was electrically heated in atmospheric air. T h e plume rose in the field of a Mach-Zehnder interferometer. T h e interferometer sensitivity was 7.25" per fringe for a two-dimensional field, 6-in. wide. Adjustment was made to the infinite fringe and each fringe represents an isothermal contour.

NATURALCONVECTION FLOWS AND STABILITY

337

Figure 4 is an interferogram of an unperturbed plume. It clearly shows the extent of the thermal boundary region. T h e steadiness indicates the quiet surroundings in the test section.

0.5

P

0. I

0.05

FIG.25. Computed neutral stability curves. Coupled and uncoupled flow. Asymmetric disturbances. u = 0.7. -.-.-*- Constant frequency contours for air at test conditions. u is the Prandtl number.

Since for a Prandtl number of 0.7 the velocity and the thermal boundary regions are of almost equal extent, the region seen is essentially the whole plume. T h e rectangular grid is a check of optical distortion and serves as a frame of reference for distance measurements. T h e vertical distance between the lines is 3 in. and the horizontal distance is in. Controlled disturbances were introduced with the vibrator seen near the plume source in Fig. 26. This figure shows a sequence of plumes in air perturbed with controlled sinusoidal oscillations at different frequencies. Low-frequency disturbances are strongly amplified and after a few oscillations the laminar base flow is completely transformed. As the frequency is increased, the amplification rate of the disturbances appears to be less, a longer distance is apparently required to disrupt the flow. Disturbances of yet higher frequency were very difficult to visually

4

B. GEBHART

338

5.1 Hz

7.0 Hz

FIG.26. Plumes perturbed with sinusoidal disturbances at several frequencies. Air at atmospheric conditions. Q = 58.6 BTU/hr ft, wire length = 6 in., wire diameter = 0.005 in.

NATURAL CONVECTION FLOWS AND STABILITY

339

detect downstream. A hot-wire anemometer was used for higherfrequency disturbances. Disturbances with frequencies higher than about 12 Hz were not detected downstream. These observations are in very good agreement with the predictions of Fig. 25. T h e discrepancy between 12 and 15 Hz is not unreasonable. T h e introduced disturbance is perhaps not of perfect asymmetric form and may not quickly become so. I t was not possible to determine experimentally the conditions of neutral stability. T h e unstable region extends to very low values of G and, for our test conditions in air, this is at very small x (around 0.1 in.). I t was not possible to introduce disturbances at even smaller x. I n any case, boundary layer approximations are in severe doubt at such low values of Gand little importance should be ascribed to the stability results there.

IX. General Aspects of Instability T h e above results of studies of instability and two-dimensional disturbance growth, combined with others concerning yet additional configurations of buoyancy-induced flows, are very interesting when looked at collectively. T h e previous discussion concerned flows adjacent to both uniform flux and isothermal vertical surfaces and in plumes above a line source. I n addition to these flows those adjacent to horizontal and slightly inclined surfaces (Z02), in axisymmetric plumes (103), and in nonbuoyant and buoyant jets (103) have been studied for stability and for disturbance growth characteristics. All of these results are collectively considered here with respect to how disturbances amplify in the different flows, as they are convected downstream. T h e way disturbances propagate on a p, G stability plane is determined. Various flows have very different characteristics. Consider any flow, taking x as the distance from the beginning of the flow to the downstream location of interest. I n treating stability, the frequency of a given disturbance f (Is = 2 4 ) , its local wavelength X (4, = 27ilX) and its exponential growth rate --olI are generalized in terms of the flow region thickness S(x) and its vigor, as indicated by buoyancy generated velocity level U,(x), as follows: 6 /3

=

k,x/G,

U,

=

k2uG2/x

=@/U, = /?k,~~/k,u = Gk~p

x2/G3

(72) (73)

B. GEBHART

340

where G = G ( x )is the kind of Grashof number relevant to the particular flow circumstance. A somewhat different formulation is used for a jet where G becomes the Reynolds number, which does not vary with x. T h e nature of the variation of G with x, as well as the values of the constants k, depend on flow geometry. Linear stability calculations proceed by finding rates of amplification in x (-aI) for chosen frequencies fl (or wavelength aR)and flow locations G. T h e result in each circumstance is a stability plane p, G, shown schematically in Fig. 27. If consists of a stable region (aI> 0) and unstable one ( a , < 0) separated by the neutral curve (aI= 0). Such

\

\ \ \ \

\

P

STABLE REGION

\ NEUTRAL CURVE

/

I

,6

UNSTABLE REGION

0

G FIG. 27. Typical stability plane for buoyancy induced flows, showing downstream paths of a disturbance of a given frequency for different kinds of flow. Numbers are related to flow configuration as follows: (1) isothermal vertical surface; (2) uniform heat flux vertical surface; (3) horizontal and slightly inclined surfaces and disks; (4) plane plume; ( 5 ) axisymmetric plume; (6) jet, nonbuoyant and slightly buoyant.

planes have been generated for flows adjacent to vertical, horizontal, and slightly inclined surfaces, in plane and axisymmetric plumes, and in buoyant and nonbuoyant jets and they are all of the general form shown in Fig. 27.

NATURAL CONVECTION FLOWS AND STABILITY

34 1

T h e thing that is remarkably different among this collection of flows is how disturbances of various given physical frequencies fl are amplified as they propagate downstream in the flow. From Eq. (73) the quantity /3G3/x2is seen to be constant for any given frequency and is proportional to 0. T h e x2 may be converted to G(x) for each flow to yield the following /3, G relations along paths of constant physical frequency. T h e constants C, , C, ,..., C, depend on fl. (1)

(2) (3)

(4) (5) (6)

/3G1l3 = C, , isothermal vertical surface. /3G1/, = C, , uniform heat flux vertical surface. /3G-l/, = C, , horizontal or slightly inclined surfaces with a leading edge (also horizontal disk flows). /3G-l/, = C,, a plane plume. /3G-l = C, , an axisymmetric plume. G = R = C, , a jet, nonbuoyant or slightly buoyant.

These paths have strikingly different characteristics on the /3, G plane, Fig. 27. For vertical surfaces, the paths continue to penetrate more deeply into the highly amplified region of the stability diagram as they are convected downstream. Detailed behavior of aI in the unstable region indicates that only a very narrow band of frequencies is highly amplified, as we have already seen. Th u s such flows filter a complicated disturbance for certain frequencies. However, the parabolic forms ( 3 ) , (4),and ( 5 ) cross the unstable region; the upper branch of the neutral curve is known to be bounded for most of the flows considered. Therefore, any given frequency over a very broad band is unstable, but only over a range of G, i.e., of downstream region x. T h e same is seen to be true of jets (6). T h e implications of this are very interesting since initially small two-dimensional disturbances are thought to be the origin of the later (in x) more complicated disturbances which disrupt a laminar flow and convert it to turbulence. This occurs as disturbance amplitude increases and leads to other and nonlinear effects. T h e results in Fig. 27 suggest that flows adjacent to vertical surfaces are more inevitably unstable. However, the free boundary flows (4),( 5 ) , and (6), along with (3) in which the buoyancy force is normal to the flow direction, are eventually stable in the linear range of amplitude to all disturbances. This group thus seems more stable. However, experimental studies show that all these latter flows are actually much less stable. That is, transition, or appreciable nonlinear effects, occur at much smaller G than for flows (1) and (2). Thus we know that other mechanisms become important much more quickly (in x) for flows (3), (4),( 5 ) , and (6). These mechanisms presumably quickly

342

B. GEBHART

dominate because of the absence of a stabilizing surface in these free boundary flows. For horizontal flow the secondary mechanism is perhaps associated with a thermal instability mode which arises due to unstable stratification (102). These results and comparisons suggest differences which might be valuable guides in finding explanations of later events in the transition processes.

X. Separating Flows I n forced flows the phenomenon of flow separation is frequently encountered and some of the principal mechanisms of steady separation in flows with important viscous effects are understood. For boundary layer flows over bluff bodies, e.g., the external pressure gradient consideration provides a basis for understanding. Past years have seen some study of transport and flow characteristics for buoyancy-driven flows around bluff bodies, which by simple and sometimes misleading analogy to forced flows, would be expected to separate. There has been some tendency to interpret the results of the few experiments and observations bearing on this question in the terminology and mechanisms of forced flow separation. However, the basic mechanism one might intuitively expect to be the genesis of separation is completely different. Consider, for example, a large heated horizontal cylinder in an extensive quiescent medium. Convection layers form and they flow around the surface. It might be expected that these layers would tend to separate on the top half of the cylinder. But the tendency to separate would be produced by the buoyancy force which operates only internal to the convection layers, not by an agency external thereto. T h e buoyancy component away from the surface seems to be the probable cause, if separation should occur. Another fundamental difference arises for any natural convection flow. What material would be found in the separated region ? In forced flow it is the mixture of the vortex-layer fluid and free-stream fluid induced by the interaction of vortices normal to the flow direction with the more distant moving stream. External momentum drives the mixing process. I n the natural convection flow there is no apparent external mechanism to produce this flow induction, or a back flow. One might assume the formation of vortex systems parallel to the cylinder axis to accomplish this, and one might also suppose that reliance on the analogy to forced flow separation is based on an unformulated idea of some such mechanism. We decided to look into this question. Since the Grashof number is an

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estimate of the magnitude of buoyancy forces, a large horizontal steam heated cylinder was used in water. Grashof number to 1O1O were obtained. One-micron particles of pliolite provided visualization of the flow field. Preliminary observations showed nothing which looked like separation, there was no chaotic wake region of mixing. More careful visualization indicated that the laminar layers from the two sides of the cylinder joined smoothly against the surface, to rise as a single wake. One might accept any local back flow as sufficient evidence of separation. There was no appreciable back flow. However, an apparently periodic but very small secondary flow was seen in the spanwise direction. Further experiments were carried out on an inverted U-shaped surface, 21-in. high, in water. This geometry gave much higher Grashof numbers and flow velocities on the upper part of the surface. I n steady state no separation or back flow was found. However, in the starting transient both effects were seen immediately following the time calculated for the arrival of the leading edge effect. Th is time was calculated as indicated in the above dicussion of transients. These observations (104) indicated either that separation attends much higher Grashof numbers or that the buoyancy force away from the surface might operate in more subtle ways. At this same time we were investigating natural convection flows above long horizontal and near-horizontal flat surfaces with a leading edge. T h e strictly horizontal problem was treated by Stewartson (33) and Gill et nl. (34),as discussed above. T h e slightly inclined one is treated by Pera and Gebhart (36). Relatively simple solutions may be obtained for a heated surface facing upward. T h e principal interest was to assess the validity of these solutions interferometrically. Agreement was sufficiently close to warrant the consideration of the linear stability characteristics of such flows to two-dimensional disturbances. Calculations, as well as experiments with controlled disturbances, were carried out (102). T h e details of these calculations and experiments are not of interest here, inasmuch as they relate to the boundary region portion of the flow and to the question of two-dimensional disturbance propagation therein. T h e relevant questions concern the mechanism and meaning of separation as observed in these experiments. T h e flow was generated above a thick, electrically heated and approximately isothermal plate of 17-in. length, normal to the leading edge, in air. Independent of Grashof number, of plate inclination up to 12", and of conditions further downstream, the boundary region flow appeared to separate in a very complicated way some distance downstream of the leading edge of the heated plate. T h e interferometer indicated temperatures higher than ambient in

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the apparently separated region. However, the interferometer integrates over the width of the region and thus did not give any information about the spanwise structure of the flow either before, during, or after this thickening of the flow region. However, smoke filaments introduced into the attached laminar boundary layer indicated the formation of longitudinal rolls just before thickening, and the enlargement and persistence of these rolls during and immediately after. T h e process was not as regular as perhaps suggested by speaking of rolls, but the secondary flow was ordered in this manner. T h e rolls persisted after apparent separation. This flow configuration may be seen in Fig. 28. T h e leading edge of

FIG. 28. Flow of air above a heated horizontal surface with a leading edge: (a) no introduced disturbance; (b) controlled upstream disturbance of 1.7 Hz.

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the surface is at the right. Smoke was introduced upstream into the attached boundary region at a central spanwise location. T h e rolls are seen best in the bottom picture. What apparently happens is that the longitudinal rolls are inducing an inflow of ambient fluid to mix with the boundary region material. This provides the rapid thickening we ordinarily associate with, and might call, separation. There are a number of interesting aspects of this question, interpreted as above. We have carried out calculations of the stability characteristics of two-dimensional disturbances in these flows, in the linear theory as discussed in preceding sections for other flows. This flow is found to be very quickly unstable and the calculated amplification rates are very high. A buoyant flow over a warm surface facing upward has an additional instability mechanism, compared to flow adjacent to vertical surfaces. There is unfavorable stratification. This perhaps accounts for the early instability and high amplification rates of disturbances. It might also account for the early and powerful effect of a three-dimensional feature which becomes the observed longitudinal rolls. This is curiously similar to the observations in Husar and Sparrow (105) and Sparrow and Husar (106), using an electrochemical technique to visualize laminar disturbance mechanisms for flow adjacent to horizontal and inclined surfaces in water. These observations suggested that longitudinal vortices are the first mode of laminar instability in such flows. Additional study (107) suggested that this instability mode dominates for inclinations greater than about 14" from the vertical. Our calculations and observations would suggest that two-dimensional disturbances may have preceded and, even perhaps caused, the longitudinal vortices. T h e bottom photograph of Fig. 28 shows a much more ordered flow. This one resulted with the introduction of a controlled disturbance, by a vibrating ribbon upstream. T h e frequency of 1.7 Hz is near that predicted by calculations to be the most unstable one for these conditions. Perhaps the additional mechanisms were thus offered the most favorable circumstances for rapid and ordered disturbance growth. This would support the surmise of the previous paragraph. These various observations, taken together, suggest that natural convection flows which have an appreciable component of buoyancy force away from the flow-generating surface do not separate in the conventional sense. No back flow or vortex system normal to the twodimensional flow were found. However, the normal component of buoyancy force may quickly develop secondary longitudinal vortices, which in turn induce ambient fluid and mixing. This mechanism may account for rapid growth in the thickness of such buoyant layers.

B. GEBHART ACKNOWLEDGMENTS The writer wishes to acknowledge the continuing support of the National Science Foundation, currently under grant NSF 18529, for his own research and that of his students included in the above account. He wishes also to thank all of his students and colleagues who have contributed to this continuing study.

REFERENCES 1. 2. 3. 4. 5. 6.

B. Gebhart, “Heat Transfer,” 2nd Ed. McGraw-Hill, New York, 1971. M. Finston, 2.Angew. Math. Phys. 7, 527 (1956). E. M. Sparrow and J. L. Gregg, Trans. A S M E 80, 379 (1958). K. T. Yang, J. Appl. Mech. 27, 230 (1960). E. M. Sparrow and J. L. Gregg, Trans. A S M E 78, 435 (1956). H. Schuh, Boundary Layers. Sect. B.6, Brit. Min. of Supply, Ger. Doc. Cent., Ref. 3220T (1948). 7. S. Ostrach, N A C A (Nut. Adv. Comm. Aeronaut.), Rep. 1111 (1953). 8. E. J. LeFevre, Appl. Mech., Proc. Int. Congr., 9th, Brussels 4, 168 (1956). 9. H. K. Kuiken, J. Eng. Math. 2, 355 (1968). 10. H. K. Kuiken, J. Fluid Mech. 37, 785 (1969). 11. A. J. Ede, Advan. Heat Transfer 4, 1-64 (1967). 12. B. Gebhart and J. Mollendorf, J. Fluid Mech. 38, 97 (1969). 13. R. Cheesewright, Int. 1.Heat Mass Transfer 10, 1847 (1967). 14. A. E. Gill, J. Fluid Mech. 26, 515 (1966). 15. B. Gebhart, J. Fluid Mech. 14, 225 (1962). 16. S. Roy, Int. J. Heat Mass Transfer 12, 239 (1969). 17. R. Eichhorn, 1.Heat Transfer 82, 260 (1960). 18. E. M. Sparrow and R. D. Cess, J . Heat Transfer 83, 387 (1961). Mech. Eng.) 6, 223 (1963). 19. I. Mabuchi, Bull. J S M E (Jap. SOC. 20. S. F. Shen, personal communication. Grad. Sch. of Aerosp. Eng., Cornell Univ., Ithaca, New York, 1969. 21. K. T. Yang, J. Appl. Mech. 31, 131 (1964). 22. H. S. Takhar, J. Fluid Mech. 34, 81 (1968). 23. E. M. Sparrow and R. B. Husar, Int. J. Heat Mass Transfer 12, 365 (1969). 24. D. J. Baker, J. Fluid Mech. 26, 573 (1966). 25. A. Acrivos, AIChE J. 6, 584 (1960). 26. T. Y. Na and A. G. Hansen, Int. J. Heat Mass Transfer 9,261 (1966) 27 H. Merte, Jr., and J. A. Prins, Appl. Sci. Res., Sect. A 4 , Parts 1-111, 11, 195, 207 (1953-1 954). 28. W. H. Braun, S. Ostrach, and J. E. Heighway, Int. J. Heat Mass Transfer 2, 121 (1961). 29. R. G. Hering, Int. J. Heat Mass Transfer 8, 1333 (1965). 30. R. G. Hering and R. J. Grosh, Int. J. Heat Mass Transfer 5, 1059 (1962). 31. B. R. Rich, Trans. A S M E 75, 489 (1953). 32. J. Fishenden and 0. A. Saunders, Engineering (London) 130, 193 (1930). 33. K. Stewartson, 2.Angew. Math. Phys. 9a, 276 (1958). 34. W. N. Gill, D. W. Zeh, and E. Del Casal, Z . Angew. Math. Phys. 16, 539 (1965). 35. A. Rotem and L. Claassen, J. Fluid Mech. 39, 173 (1969). 36. L. Pera and B. Gebhart, Int. J. Heat Mass Tran:fer (to be published).

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37. Y. B. Zeldovich, Zh. Eksp. Teor. Fiz. 7, 1463 (1937). 38. C. S. Yih, Proc. U.S. Nut. Congr. Appl. Mech., lst, Illinois Institute of Technology, Chicago, p. 941 (1951). 39. C. S. Yih, Trans. Amer. Geophys. Union 33, 669 (1952). 40. H. Rouse, C . S. Yih, and H. W. Huniphreys, Tellus 4, 201 (1952). 41. I. G. Sevruk, J. Appl. Math. Mech. ( U S S R ) 22, 807 (1958). 42. L. J. Crane, Z. Angew. Math. Phys. 10, 453 (1959). 43. D. B. Spalding and R. G. Cruddace, Znt. J. Heat Muss Transfer 3, 55 (1961). 44. T. Fujii, Znt. J. Heat Mass Transfer 6, 597 (1963). 45. B. Gebhart, L. Pera, and A. W. Schorr, Znt. J. Heat Mass Transfer 13, 161 (1970). 46. K. Brodowicz and W. T. Kierkus, Znt. J. Heat Mass Transfer 9, 81 (1966). 47. R. J. Forstrom and E. M. Sparrow, Znt. J. Heat Mass Transfer 10, 321 (1967). 48. A. W. Schorr and B. Gebhart, Znt. J . Heat Mass Transfer 13, 557 (1970). 49. J. J. Mahony, Proc. Roy. Soc., Ser. A 238, 412 (1957). 50. R. S. Brand and F. J. Lahey, J. Fluid Mech. 29, 305 (1967). 51. K. Millsaps and K. Pohlhausen, J. Aeronaut. Sci. 23, 381 (1956). 52. K. Millsaps and K. Pohlhausen, J . Aeronaut. Sci. 25, 357 (1958). 53. M. Van Dyke, “Free Convection from a Vertical Needle,” Sedov Anniv. Vol. Moscow, 1967. 54. E. M. Sparrow and J. L. Gregg, Trans. A S M E 78, 1823 (1956). 55. B. Gebhart and L. Pera, Znt. J. Heat Mass Transfer 14, 2025 (1971). 56. E. V. Somers, J. Appl. Mech. 23, 295-301 (1956). 57. W. G. Mathers, A. J. Madden, and E. L. Piret, Znd. Eng. Chem. 49, 961-968 (1957). 58. W. R. Wilcox, Chem. Eng. Sci. 13, 113-119 (1961). 59. W. N. Gill, E. Del Casal, and D. W. Zeh, Znt. J. Heat Mass Transfer 8, 1131-1151 (1965). 60. R. L. Lowell and J. A. Adams, AZAA J. 5 , 1360-1361 (1967). 61. J. A. Adams and R. L. Lowell, Int. J. Heat Mass Transfer 1 1 , 1215-1224 (1968). 62. J. L. Manganaro and 0. T. Hanna, AZChE J. 16, 204-211 (1970). 63. D. V. Cardner and J. D. Hellums, Znd. Eng. Chem., Fundam. 6, 376-380 (1967). 64. E. N. Lightfoot, Chem. Eng. Sci. 23, 931 (1968). 65. D. A. Saville and S. W. Churchill, AZChE J. 16, 268-273 (1970). 66. D. A. Saville and S. W. Churchill, J. Fluid Mech. 29, 391-399 (1967). 67. L. Pera and B. Gebhart, Znt. J. Heat Muss Transfer 15, 269-278 (1972). 68. J. A. Schetz and R. Eichhorn, J . Heat Transfer 85, 334 (1962). 69. E. R. Menold and K. T. Yang, J. Appl. Mech. 29, 124 (1962). 70. B. Gebhart, J. Heat Transfer 85, 184 (1963). 71. R. J. Goldstein and D. G. Briggs, J. Heat Transfer 86, 460 (1964). 72. B. Gebhart and R. P. Dring, J . Heat Transfer 89, 274 (1967). 73. J. C . Mollendorf and B. Gebhart, J. Heat Transfer 92, 628 (1970). 74. R. Siegel, Trans. A S M E 80, 347 (1958). 75. B. Gebhart, J. Heat Transfer 83, 61 (1961). 76. B. Gebhart, J. Heat Transfer 85, 10 (1963). 77. R. J. Goldstein and E. Eckert, Znt. J. Heat Mass Transfer 1, 208 (1960). 78. J. H. Martin, An Experimental Study of Unsteady State Natural Convection from Vertical Surfaces. M.S. Thesis, Cornell Univ., Ithaca, New York, 1961. 79. H. Lurie and H. A. Johnson, J. Heat Transfer 84, 217 (1962). 80. B. Gebhart and D. E. Adams, J. Heat Transfer 85, 25 (1963). 81. B. Gebhart, R. P. Dring, and C. E. Polymeropoulos, J. Heat Transfer 89, 53 (1967). 82. 0. A. Saunders, Proc. Roy. SOC.,Ser. A 157, 278 (1936).

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83. E. Eckert and E. Soehngen, General Discussion on Heat Transfer. IME & ASME, 321 (1951). 84. P. Colak-Antic, Jahrb. WGLR p. 172 (1964). 85. P. Colak-Antic, Sitzungsber. Heidelberg. Akad. Wiss., Math.-Naturwiss. Kl. Jahrgang 1962/64, p. 315 (1964). 86. J. T. Stuart, Appl. Mech. Rev. 18, 523 (1965). 87. J. E. Plapp, Ph.D. Thesis, California Inst. of Technol., Pasadena, California, 1957; also see J. E. Plapp, J. Aeronaut. Sci. 24, 318 (1957). 88. A. A. Szewcyzk, Int. J. Heat Mass Transfer 5, 903 (1962). 89. A. E. Gill, and A. Davey, 1.Fluid Mech. 35, 775 (1969). 90. L. Pera and B. Gebhart, Int. J. Heat Mass Transfer 14, 975-984 (1971). 91. C. A. Hieber and B. Gebhart, J. Fluid Mech. 48, 625-646 (1971). 92. C. A. Hieber and B. Gebhart, J . Fluid Mech. 49, 577-592 (1971). 93. E. F. Kurtz and S. H. Crandall, J. Math. Phys. (Cambridge, Mass.) 41,264 (1962) 94. P. R. Nachtsheim, Stability of Free-Convection Boundary-Layer Flows. N A S A Tech. Note N A S A TN D-2089 (1963). 95. C. P. Knowles and B. Gebhart, J . Fluid Mech. 34, 657 (1968). 96. C. E. Polymeropoulos and B. Gebhart, A I A A J . 4, No. 11, 2066 (1966). 97. C. E. Polymeropoulos and B. Gebhart, J. Fluid Mech. 30, 225 (1967). 98. C. P. Knowles and B. Gebhart, in “Progress in Heat and Mass Transfer” (E. R. G. Eckert and T. F. Irvine, Jr., eds.), Vol. 2, p. 99. Pergamon, London, 1969. 99. R. P. Dring and B. Gebhart, J. Fluid Mech. 35, 447 (1969). 100. R. P. Dring and B. Gebhart, J. Fluid Mech. 34, 551 (1968). 101. R. C. Gunness, Jr. and B. Gebhart, Phys. Fluids 12, 1968 (1969). 102. L. Pera and B. Gebhart, Int. J. Heat Muss Transfer (to be published). 103. J. C. Mollendorf, The Effect of Thermal Buoyancy on the Hydrodynamic Stability of a Round Laminar Vertical Jet. Ph.D. Thesis, Cornell Univ., Ithaca, New York, 1971. 104. L. Pera and B. Gebhart, Int. J. Heat Mass Transfer accepted for publication (1972). 105. R. B. Husar and E. M. Sparrow, Int. J. Heat Mass Transfer 11, 1206 (1968). 106. E. M. Sparrow and R. B Husar, J Fluid Mech. 37, 251 (1969). 107. J. R. Lloyd and E. M. Sparrow, J. Fluid Mech. 42, 465 (1969).

Cryogenic Insulation Heat Transfer . .

C L TEN Department of Mechanical Engineering. University of California. Berkeley. California

AND

. .

G R CUNNINGTON Lockheed Pa10 Alto Research Laboratory. Pa10 Alto. Culifornia

I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1 Cryogenic Insulation . . . . . . . . . . . . . . . . . . . A. General Considerations . . . . . . . . . . . . . . . . B . Types of Cryogenic Insulation . . . . . . . . . . . . . 111. Fundamental Heat Transfer Processes . . . . . . . . . . . . A. Gas Conduction . . . . . . . . . . . . . . . . . . . B . Solid Conduction . . . . . . . . . . . . . . . . . . . C . Radiation . . . . . . . . . . . . . . . . . . . . . . IV . Evacuated Powder and Fiber Insulation . . . . . . . . . . . A . Physical and Optical Properties . . . . . . . . . . . . . B . Conduction Heat Transfer . . . . . . . . . . . . . . . C . Radiation . . . . . . . . . . . . . . . . . . . . . . D . Total Heat Transfer . . . . . . . . . . . . . . . . . . V . Evacuated Multilayer Insulation . . . . . . . . . . . . . . A. Thermophysical Properties of Reflective Shields and Spacers B. Normal Heat Transfer . . . . . . . . . . . . . . . . . C . Lateral Heat Transfer . . . . . . . . . . . . . . . . . VI . Test Methods . . . . . . . . . . . . . . . . . . . . . . A. Boil-Off Calorimetry . . . . . . . . . . . . . . . . . B. Electrical-Input Method . . . . . . . . . . . . . . . . C . Indirect Methods . . . . . . . . . . . . . . . . . . . VII . Applications . . . . . . . . . . . . . . . . . . . . . . . A . Nonevacuated Insulation . . . . . . . . . . . . . . . . B. Evacuated Powder and Fiber Insulation . . . . . . . . . C . Evacuated Multilayer Insulation . . . . . . . . . . . . Symbols . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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350 352 352 353 356 356 359 361 365 366 368 377 319 381 382 389 394 399 400 403 404 405 401 408 408 413 414

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C. L. TIENAND G. R. CUNNINGTON I. Introduction

Thermal insulation has long been a subject of great importance to heat transfer engineers and was indeed one of the major concerns in the early development of heat transfer technology. It is interesting to note that one of the very first textbooks on heat transfer has the title of “Elements of Heat Transfer and Insulation” ( I ) . In the last few decades, however, a multitude of new heat transfer research and developments have moved the subject of insulation heat transfer from its earlier prominence to a stagnant obscurity. Thermal insulation had become a classical subject that was considered by many as already well developed and of concern only to the manufacturing and design engineers (2). I n the meantime, developments in many new emerging technologies have extended considerably the ordinary temperature range of operation, and have presented a great number of formidable engineering problems at the extreme temperature limits. One major problem has been the application of effective thermal insulation at extreme temperatures. Consequently, the past few years have registered an intensive surge of renewed interest in thermal insulation, particularly for high-temperature and cryogenic applications. Despite common basic features of insulation, such as the use of multiple radiation shields, fibrous materials or powders, high-temperature insulation (say, for the approximate range of 500 to 2500°K) differs in many fundamental aspects from cryogenic insulation (say, for temperatures below 100°K). T h e different temperature ranges dictate the use of different insulation materials and methods that in turn result in fundamentally different thermal-property characteristics as well as transport phenomena. For instance, the multishield (or multilayer) insulation concept is employed in both high-temperature and cryogenic insulation, but the insulation material, the arrangement, and most important of all, the detailed heat transfer characteristics are quite different. Furthermore, cryogenic insulation is normally operated under the evacuated condition (i.e., moderate or high vacuum), while hightemperature insulation often encounters oxidizing or reducing atmospheres. Surface oxidation and material sublimation is indeed a major problem in high-temperature insulation but not in cryogenic insulation. Cryogenic insulation is unique in many ways, when compared to insulations for applications in other temperature ranges. From the practical viewpoint, it has played and is continuing to play a most prominent role in the field of cryogenics ( 3 , 4 ) . T h e importance of insulation in cryogenics is easily realized by noting that the heat of vaporization of cryogenic liquids as well as the specific heats of matter at

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cryogenic temperatures are much smaller than the corresponding ones at room temperature, and it takes little inflow of heat from outside to boil off the cryogenic liquids or to raise the system temperature. In fact, in many instances, the development of a better cryogenic insulation has constituted a giant step in the growth of cryogenic science and technology. T h e study of cryogenic phenomena became possible only after the discovery of Dewar flasks (or simply dewars) in 1893. T h e development of low-cost porous (foam, fiber, or powder) insulation for transportation and storage has been primarily responsible for the largescale use of liquified gases in industry. I n the last ten years, the advances in evacuated insulation, especially the evacuated multilayer insulation, have contributed enormously to the rapid development of rocketry and space exploration programs, liquid helium technology, superconducting technology, and many others. On the fundamental side, cryogenic insulation has presented a score of new and unique heat transfer problems that have consistently puzzled and troubled heat transfer engineers and have provided great challenges to researchers. I n addition to the complex internal geometry involved in cryogenic insulation medium, unique material behaviors at cryogenic temperatures originate heat transfer processes and mechanisms that are uncommon to the conventional thinking and analysis of heat transfer phenomena at moderate temperatures. Moreover, the relatively young field of cryogenics provides ample opportunity for further fundamental research in cryogenic insulation heat transfer. Indeed, a sharp contrast exists today between the importance and wide acceptance of newly developed cryogenic insulation and the lack of fundamental understanding in its heat transfer processes. I n view of its importance and the need for better understanding, several monographs and review articles on heat transfer in cryogenic insulation have been made available in the past few years (5-8). T h e rapid recent developments, however, point to the need of a more comprehensive and updated treatment of the subject. T h e purpose of the present article is twofold: first, to provide the fundamental information necessary for the design and evaluation of the thermal performance of cryogenic insulation, and secondly, to review and assess the existing contributions in the literature for future research in this area. T h e scope of the present article is limited to those aspects that have direct relevance to the understanding of heat transfer processes in cryogenic insulation. Many other important topics that could affect the thermal performance of cryogenic insulation will not be treated here. These include, for instance, manufacturing of insulation materials, mechanical supports and penetrations, evacuation and purging processes,

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test standards, systems design, etc. Discussion on these topics can be found in the literature (2-6, 9 ) . T h e present article is subdivided into several sections. T h e next section (Section 11) is devoted to the introduction of the general aspects of cryogenic insulation. Section I11 presents a general discussion of the heat transfer processes relevant to the thermal performance of cryogenic insulation. Considerations of specific types of cryogenic insulation are presented in the next two sections: Section IV dealing with evacuated powder and fibrous insulation and Section V with evacuated multilayer insulation. Finally in Sections VI and VII, test methods and applications of cryogenic insulation to a number of physical systems are discussed.

II. Cryogenic Insulation T h e present section is concerned with the general background information on various aspects regarding cryogenic insulation and serves as a basis for discussions in subsequent sections.

A. GENERAL CONSIDERATIONS Thermal insulation refers to either a single homogeneous material or a mixture of materials in a composite structure that is designed to reduce heat flow between their boundary surfaces. T h e choice of thermal insulation material and structure for a particular application depends on the required thermal effectiveness of the insulation as well as many other factors such as economy, weight, volume, convenience, ruggedness, etc. Due to the extreme-temperature conditions and the required ultrahigh thermal effectiveness, cryogenic insulation normally exists in a composite form with combinations of materials of desired thermal and mechanical properties. Various types of cryogenic insulation will be introduced after this subsection. T h e inhomogeneous composite structure of cryogenic insulation makes the heat transfer analysis a rather complicated problem. Consideration must be given to the complex interactions of various heat transfer mechanisms in an inhomogeneous medium and in some cases, to the highly anisotropic (i.e., dependent of the heat-flow direction) behavior. Heat transfer through cryogenic insulation usually consists of the simultaneous action of the following mechanisms: solid conduction through the insulation materials and between individual insulation components across areas of contact, gas (or residue gas under vacuum conditions) conduction in void spaces within the composite structure,

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and radiation across these void spaces and through some of the insulation components. Because of the complex interactions of various mechanisms, it is practically useful and convenient to define an “apparent” (or “effective” or “equivalent”) thermal conductivity to characterize the thermal effectiveness of the insulation. Consider, for simplicity, the onedimensional case involving a slab insulation. T h e apparent thermal conductivity k, is defined by the following relation:

where Q is the total heat flow through insulation, A the heat-flow crosssectional area, T , and T,the boundary temperatures, and L the insulation thickness. For an anisotropic insulation, there will be more than one apparent thermal conductivity. In steady-state operation, the apparent thermal conductivity is the important parameter for the evaluation of thermal effectiveness. T h e product of conductivity times density becomes of major importance in certain spacecraft applications because of launch and injection weight consideration. For instance, to design for a specific heat flux q (W/m2), the insulation weight w (kg/m2) is determined as follows:

where p is density. T o minimize the insulation system weight requires the product pk, to be a minimum. Under unsteady conditions such as during cool-downs, warm-ups, and boundary temperature transients, the volumetric heat capacity (i.e., the product of density and specific heat) of the insulation is also an important parameter. T h e greater the volumetric heat capacity, the longer the temperature response time will be and the larger the heat change that will be required for a change of insulation temperature.

B. TYPES OF CRYOGENIC INSULATION There exist two general classes of cryogenic insulations: the unevacuated and evacuated. T h e unevacuated insulations are porous materials such as solid foams, powders, and fibers in which the interstitial spaces are filled with gas at atmospheric pressure. T h e porous structure serves to reduce solid (and in certain instances gaseous) conduction as well as radiative transfer between boundary surfaces. While the thermal effectiveness of porous insulations is relatively poor as a result of gas conduction, they are widely used in less-demanding cryogenic insulation

3 54

C. L. TIENAND G. R. CUNNINGTON

systems on account of their low installation cost. For systems operating at temperatures below the liquid-oxygen temperature, the insulations normally used are the evacuated ones. T h e evacuated insulations can be subdivided into three major types: simple high-vacuum insulation, evacuated porous insulation, and evacuated multilayer insulation. In particular, the last two types constitute the major recent advances in this area and will receive primary attention in the presentation here. Figure 1 illustrates schematically various types of cryogenic insulation. Their respective thermal effectiveness is shown in Fig. 2. SIMPLE VACUUM

POROUS EVACUATED OR UNEVACUATED

I ‘I

WARM CRINKLED OR METAL COATED PLASTIC FILM

FIG. 1.

Various types of cryogenic insulations.

T h e simple high-vacuum insulation system is composed of a wellevacuated space bounded by highly reflective walls. T h e classical Dewar flask and the Thermos bottle both are of this type. It is structurally and conceptually the simplest, has the least weight and least heat capacity, but is not very effective thermally because of the direct radiative exchange between two bounding surfaces. Conceptually, the evacuated multilayer insulations are a direct extension of the simple high-vacuum insulation system by merely placing many radiation shields in between two boundaries. They normally consist of a laminated assembly of numerous thin (-0.15-3 mils) plastic films coated on one or both sides by a thin vapor-deposited of high reflectance metal, usually aluminum or gold. layer (-400 T he plastic films are employed instead of solid metal films because of their high mechanical strength, low density, and low thermal conductivity. These radiation shields (-10-50 per cm) are separated from each

a)

CRYOGENIC INSULATION HEATTRANSFER THERMAL CONDUCTIVITY x BULK DENSITY (mW-gm/cm4 6

t o*

MULT ILAYER INSULATIONS

to

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EVACUATED POWDERS, FIBERS 0

EVACUATED’ OPAClFl ED POWDERS

EVACUATED c

NON-EVACUATED POWDERS, FIBERS FOAM ETC

-

MULTILAY E R

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-1

1

EFFECTIVE THERMAL CONDUCTIVITY (mW/cm OK)

FIG.2. Apparent thermal conductivity of cryogenic insulation. Notes: a, open cell foams; b, effective thermal conductivity dependent upon optical properties, boundary surfaces.

-

MULTILAYER INSULATIONS

EVACUATED POWDERS AND FIBERS

NON-EVACUATED POWDERS, FOAMS FIBERS, ETC

-

356

C. L. TIENAND G. R. CUNNINGTON

however, the most expensive and difficult to install, and highly anisotropic with the lateral (i.e., parallel to the lamination) apparent thermal conductivity three to six orders of magnitude greater than that normal to lamination. Such a large disparity in directional thermal resistance presents a serious thermal design problem for systems where structural members and plumbing lines penetrate the insulation and provide lateral heat leaks at their junctures. T he evacuated porous insulations are essentially the same as the unevacuated insulations except that they are operated under highvacuum conditions. In comparison with the evacuated multilayer insulations, they are isotropic, simpler to install, less expensive, and have roughly the same heat capacity, but their apparent thermal conductivity is one or two orders of magnitude higher than the normal apparent conductivity of the evacuated multilayer insulations. T h e unsurpassed thermal effectiveness of the evacuated multilayer insulation, however, may soon be challenged as a result of recent developments in evacuated porous insulations by using packed beds of hollow dielectric spheres (-loop in diameter) coated with highly reflective films (see Section IV).

In. Fundamental Heat Transfer Processes Since the primary function of thermal insulation is to reduce heat flow through insulation, it is important to understand the various heat transfer processes responsible for this heat flow. These processes include conduction in the solid and gas phases and radiation exchange between surfaces, and in general they interact with each other in a complex manner. Free convection within the voids is always negligible even for the unevacuated insulations, since the characteristic length of the voids is so small cm or less) that the product of the Prandtl and Grashof numbers is much less than the critical value (-lo3) for the onset of convection (10). Under certain conditions, however, they can be effectively decoupled and considered separately. T h e purpose of the present section is to review and to discuss the general fundamental aspects of each individual process in order to provide a basis for subsequent detailed consideration of heat transfer in a particular type of insulation.

A. GAS CONDUCTION T he prominence of gas-conduction contribution is clearly demonstrated by the difference in thermal conductivity between unevacuated

CRYOGENIC INSULATION HEATTRANSFER

357

and evacuated insulations. T h e degree of vacuum required for desired insulation effectiveness is an important design problem for evacuated insulation, and can only be established by a careful consideration of gas conduction. Even for a highly evacuated insulation or insulations in a high-vacuum environment such as in outer space, outgassing and gas entrainment inside the insulation may render the gas-conduction contribution significant. Heat conduction in gases is normally considered in separate molecular regimes (12): namely, free-molecule (Kn > lo), transition (10 > K n > O.l), temperature-jump (slip) (0.1 > Kn > 0.01) and continuum (Kn < O.Ol), where Kn is the Knudsen number (Kn = ZIL, I the mean free path of molecular collisions, and L the characteriztic length of the gas layer (e.g., the vacuum spacing)). T h e various regimes have been under extensive investigations in the field of rarefied gas dynamics ( Z Z ) , but these studies are mostly restricted to linearized 1, where T , and T , are temperatures problems, i.e., [(TJT,) - l)] of the two bounding surfaces. In extending these results to heat transfer calculations for cryogenic insulations, however, care must be exercised since the boundary temperatures are often quite different, rendering the linearization condition invalid. A general discussion on gas-conduction calculation in cryogenic insulations was first given by Corruccini (12). T h e following contains a brief discussion of the current status. T o characterize the mode of gas conduction, the mean free path of molecular collisions must be known. T h e mean free path can be obtained from kinetic theory (13) and a convenient relation in terms of macroscopic properties is given as (12)

<

1 = 8.6(p/p)(T/M)l/',

(3)

where 1 is in centimeters, p is viscosity in poise, p pressure in millimeters mercury, T temperature in degrees Kelvin, and M molecular weight. At one micron pressure and 76"K, I = 0.87 cm for air, 1.8 cm for hydrogen, and 3.2 cm for helium. Under free-molecule conditions (Kn > lo), the conductive heat flux for parallel plates, coaxial cylinders, and concentric spheres can be estimated by Knudsen's formula (10, 12) 4FM

QFM/Ai =

1

+ I)/(Y

OI[(Y

-

l)l(no/8nMT)"2~(T2 - Ti),

(4)

where A is the area, y the specific heat ratio of the gas, R" the molar gas constant, T = ( T , + T,)/2, a: the overall thermal accommodation coefficient 01=

[%

+4

al%

1

-

4 4/4l'

(5)

358

C. L. TIENAND G. R. CUNNINGTON

and subscripts 1 and 2 refer to the inner and outer surfaces respectively. T h e thermal accommodation coefficient is a measure of the efficiency of thermal energy interchange that occurs when a gas molecule collides with the surface. It may vary between 1 (complete accommodation, diffuse reemission) and 0 (specular reemission). Its exact value depends upon the kind of gas molecule, the surface temperature, and most importantly the exact condition of the surface. At the present state of knowledge, 01 can only be regarded as an empirical parameter that must be determined from measurements. Values of N under various conditions have been reported by many investigators (11-15). It may be indicated that, in general, 01 increases with decreasing cleanliness, heavier gas molecules, and decreasing temperature (except for the interaction of helium gas and clean tungsten filament). One notable feature of the free-molecule conduction is that the conductive heat flux is independent of the gas-layer thickness L (or vacuum gap spacing). This is analogous to the radiative transport between two surfaces separated by a nonradiating medium. T h e parameter L, however, is important here in defining the free-molecule regime, i.e., Z'L > 10. For spacing on the order of 1 cm, the vacuum required for free-molecule conduction is about 10-3-10-4 torr (1 torr = 1 mm Hg). For the same pressure level, if n shields of identical surface accommodation characteristics are separately spaced in the gap region, 1). I n other the conductive heat flux will be reduced by a factor ( n words, the same vacuum insulation effectiveness can be achieved with lesser vacuum requirement when more shields are used. Thus the shielding concept in insulation applies to residual gas conduction as well as to radiation. Conduction shielding, however, is often overlooked because in many situations either the natural surrounding is at such a high vacuum (e.g., outer space) or it is very convenient to reduce the gas pressure to such a level that gas conduction is negligibly small compared to radiation. Gas conduction in the transition and slip regimes is a rather complicated subject and has been under numerous recent investigations (11). For practical calculations for parallel plates, coaxial cylinders, and concentric spheres, it is recommended that the following simple interpolation formula be used (11)

+

where qc is the continuum heat flux. For instance, for a plane layer, qc can be written from the simple kinetic theory (13) as,

CRYOGENIC INSULATION HEAT TRANSFER

3 59

K can be obtained through simple manipulation of Eqs. (3), (4), (6), and (7). Equation (6) indicates that under the same temperature and pressure conditions (therefore, the same I), free-molecule conduction gives the maximum heat flux. This should not be confused with the fact that gas conduction does decrease as pressure goes down. Gas conduction in complex geometries such as powder and fiber insulations defies any rigorous quantitative description and semiempirical representation becomes necessary. This will be discussed further in Section IV,B.

c, is the constant-volume specific heat, and the constant

B. SOLIDCONDUCTION Conductive heat transfer through solid components of the insulation often constitutes the predominent mode of heat transfer in porous and multilayer insulations. To reduce or to eliminate the solid-conduction contribution is thus a major objective in the design of thermal insulation. Unfortunately, solid conduction cannot be easily reduced without affecting other heat transfer modes, particularly radiation, as well as many other mechanical and structural considerations. An excellent example of this dilemma is the Dewar flask, in which solid conduction is eliminated at the expense of structure integrity and relatively poor thermal effectiveness due to direct radiative exchange. For zero-gravity applications, both conduction and radiation can probably be reduced through the use of loosely packed, floating particles. I n general, however, reduction of solid conduction must be achieved by increasing thermal resistances to conductive heat flow. T h e logical way is, of course, to increase the effective length of heat flow paths and to decrease the effective flow cross-sectional area. This is best accomplished by creating tortuous and constricted conduction paths through the use of finely divided solid elements (particles, fibers, foams, screens, etc.) so that constriction resistances to heat flow are formed throughout the insulation. A brief discussion of the thermal constriction (or contact) resistance (16, 17) is given below in order to demonstrate the various physical parameters involved. Thermal constriction resistance originates from the constriction of heat flow near the contact (or connecting) regions. When the radius rc of the contact spot is much smaller than the radius of the unconstricted flow tube, as is usually the case, the constriction resistance is given as

360

C. L. TIENAND G. R. CUNNINGTON

which was first suggested by Holm in terms of electrical contact resistance and has been widely quoted. T h e radius r0 depends on the mechanical properties of the contacting material as well as the geometry of the contact. For the simple case of the elastic deformation resulting from two identical spheres in contact, the radius is defined by the wellknown Hertz formula:

where p is Poisson's ratio, E Young's modulus, F force acting on the spheres, and rs sphere radius. From the above simple consideration, it will suffice to say that solid conduction in porous or multilayer insulations is a function of the thermal, mechanical, and geometric properties of the solid elements and the force acting on the insulation. T h e dependence of temperature is implicit in the thermal and mechanical properties. Further consideration of solid conduction in specific types of insulation will be given in Sections IV and V. Among the inherent physical properties of the solid elements, thermal conductivity still exerts the major influence on thermal constriction resistance as indicated in Eqs. (8) and (9). It should be emphasized that in the cryogenic temperature range the thermal conductivity of solids exhibits a strong dependence of temperature as well as molecular structure and purity (18, 19). Shown in Fig. 4 are low-temperature thermal

'K

HIGH PURITY COPPER

E

3 105 E

/SINGLE

I

CRYSTAL ALUMINUM

V

3

99% DRAWN COMMERCiAL ALUMINUM

'1 0

I

lV 0

FIG. 4.

/--

-yEXTRUDEO AMORPHOUS CARBON

I 50

I

I

I

I00 150 200 TEMPERATURE (OK)

I 250

300

Thermal conductivities of low temperature solids.

CRYOGENIC INSULATION HEATTRANSFER

361

conductivities of some representative solids. In theory, the thermalenergy transport in solids is due primarily to two major mechanisms: mechanical interaction between molecules (i.e., lattice vibrations) and translation of free conduction electrons. Because lattice vibrations can be treated as phonons, thermal transport in solids can be regarded as energy transport in phonon and electron gases, and the mean-free-path concept in the kinetic theory of molecular gases is directly applicable here (13). T h e free-electron contribution dominates in the energy transport in metals and the phonon contribution is predominant in dielectric solids, whereas in very impure metals or in disordered metals, the phonon contribution may be comparable with the free-electron contribution. T h e disordered dielectrics with no free electrons and considerable lattice imperfection are the poorest solid conductors of heat, and consequently most porous or multilayer insulations are made of materials such as glass or polymeric plastics. Poor solid conductors, however, are also poor reflectors of radiation and are relatively ineffective for radiation shielding. A remedy is the use of metal particles dispersed in a dielectric medium as is evidenced in the opacified powders and in the aluminized plastic shields in multilayer insulations. Another significant characteristic of thermal conduction in cryogenic solids is the increase in the mean free path of phonons and electrons T-I and I, T-'-P3, as the temperature decreases. Approximately, I, and at room temperatures I,, m 10-lo2 for crystalline dielectrics while I, M lo2 A for pure metals (13, 18). At cryogenic temperatures, these mean free paths are indeed comparable to the characteristic dimensions of the dielectric powders, fibers, and metallic coatings commonly used in cryogenic insulations. Under these conditions, many of the phonon or electron free paths will be shortened as a result of termination at the boundary surface of the solid elements, and the thermal conductivity is expected to be less than that of the bulk solid. T h e significance of this size effect on thermal conductivity has been quantitatively demonstrated for thin metallic films at cryogenic temperatures (20).

a

-

-

C. RADIATION Despite the fact that radiant energy involved at cryogenic temperatures is much smaller than that at room or high temperatures, radiation is still a major mode of heat transfer in cryogenic insulations. T h e importance of the radiation mode is manifest in the large difference (see Fig. 2) between the thermal effectiveness of ordinary

362

C. L. TIENAND G. R. CUNNINGTON

porous insulations and that of opacified-powder and multilayer insulations, in which radiative exchange is greatly reduced by radiation shielding. Radiative transfer refers to the transport of energy by electromagnetic waves and attenuation of radiation takes place in the forms of reflection, absorption, and scattering, all of which are essential elements in the heat transfer process in high-performance cryogenic insulations. T h e theoretical basis for radiative transfer calculations rests on the concept of local thermodynamic equilibrium (21,22). T h e equilibrium radiation within a uniform-temperature enclosure is called blackbody radiation and is described by Planck's law: IbA =

2hcO2 n2h5[exp(hc,/nXkT)- 11 '

where IbAis the spectral blackbody intensity, h the Planck constant (h = 6.6256 x lop2' erg sec), c,, speed of light in vacuum (co = 2.9979 x 1O1O cm/sec), n the refractive index of the nonabsorbing (i.e., extinction index K = 0) medium in the enclosure, X wavelength, k the Boltzmann constant (k = 1.38054 x erg/"K), and T temperature. Attempts to generalize Eq. (10) for absorbing media have is related met with limited success (23). T h e blackbody intensity IbA to the blackbody emissive power ebAby ebA= TI,,,. For a given temperature T , the maximum radiation intensity occurs at wavelength ,A when nX,T

=

2897.6 p°K.

(11)

This equation is known as Wien's displacement law. One limiting expression of Planck's law is found useful in many approximate calculations. This is Wien's distribution:

which closely approximates the energy content given by Planck's law over most of the spectral range. Where the approximation fails to give close correspondence, there remains only a small portion of the total radiation energy. Specifically, Wien's distribution has been employed to arrive at simple expressions for radiation of metallic surfaces at cryogenic temperatures (24). T he Stefan-Boltzmann law for the total blackbody emissive power can be obtained by integration of Eq. (10): eb =

nIb

=

n2(2n5k4/15h3c,2) T 4 = n2uT4,

(13)

CRYOGENIC INSLJLATION HEATTRANSFER

363

where G is the Stefan-Boltzmann constant (u = 5.6697 x erg’cm2 OK4 sec). T h e significant feature of cryogenic radiation lies in its long wavelengths, as indicated by Eq. (11). Consider, for instance, blackbody radiation at 10°K. Most energy is contained in the spectral range from 100 to lOOOp, which is indeed of the same order of magnitude or higher than the characteristic dimensions of solid elements and voids in highperformance cryogenic insulations. T h e long-wavelength radiation results in a reconsideration of many radiation phenomena, which can be legitimately neglected at room or high temperatures but become increasingly important with the lowering of temperature. Considerable attention has been given to these phenomena recently (25-27). These include the modification of Planck’s law for blackbody radiation in a small cavity (26), the enhancement of radiative transfer at close spacings (27), and the anomalous skin effect of radiation dissipation in metals (25). There exist also other better-known effects of longwavelength radiation, such as thc increase in specularity of reflected radiation from rough surfaces (28), and the electromagnetic scattering effect from curved bodies (29).A careful assessment of these phenomena and their associated effects is often necessary in the consideration of heat transfer through cryogenic insulations. Further discussion of these long-wavelength effects will be given in Sections I V and V, when applicable. Radiation characteristics of matter are usually presented in macroscopic terms such as emissivity, absorptivity, and reflectivity. For particulate media, the scattering coefficient must be included to take into account the electromagnetic field scattering, which combines the effects of reflection, diffraction, and refraction. I n ideal situations such as those with solid bodies of optically clean and smooth surfaces, the macroscopic radiation properties can be related to the more fundamental optical parameters, i.e., the optical constants n = n - in’ where n is the complex refractive index, n the refractive index, and n’ the extinction index. T h e optical constants are still macroscopic parameters, but can be further expressed in terms of microscopic parameters of the radiating medium. An outstanding example of the interrelationships among all these parameters is the radiation behavior of strong absorbing media at the long-wavelength limit, given by the well-known Hagen-Rubens relation (21, 22, 25):

where en,+, anA, and pnA are spectral normal emissivity, absorptivity, and

364

C. L. TIENAND G. R. CUNNINGTON

reflectivity, respectively. T h e parameter uo is the dc electrical conductivity, and from simple kinetic theory it can be written as u, = neezT/m,

,

(15)

where n, is the electron number density, e electron charge, T electron relaxation time (or roughly the average period between collisions), and me electron mass. T h e temperature dependence of oo is implicit in n, and T , but an explicit expression for the dependence on temperature as well as other molecular parameters is available from a rigorous treatment of electron-transport processes (30). For metals at low T-5. temperatures (T the Debye temperature), a, I t is demonstrated above that the spectral radiation properties of solid matter are complicated functions of wavelength, material constants, and temperature, not even to mention the effects due to impurities and dislocations on the surface and in the bulk. But, at cryogenic temperatures, further complications might occur. T h e Hagen-Rubens relation is a limiting expression (i.e., at long wavelength) of a general theory called the Drude free-electron theory, which is valid for pure metals when the electron mean free path I, is small compared to the penetration depth of the radiation field. With longer I , at low temperatures, the Drude theory must be modified to include the penetration-depth effect (i.e., the skin effect). This is considered in the anomalous-skin-effect theory, which has been employed recently to predict emissivities of metals at cryogenic temperature (31, 32). At cryogenic temperatures, there exists also an appreciable size effect on emissivity when I, is of the same order of magnitude as the characteristic length of metal elements. Theoretical and experimental studies on emissivity of thin metal films at cryogenic temperatures have been reported (33-35). Another unknown factor is the magnitude of quantum mechanical effect on the radiation of cryogenic solids (25). T h e role of these various effects in the calculations of cryogenic-insulation heat transfer will be considered in most details in Section IV and V. Knowing the radiation property constitutes only the beginning of the problem of calculating radiative transfer. Between two planar radiation shields separated by a nonparticipating medium or vacuum, the radiative transfer on a spectral basis is given by (21, 22)

-

<<

qA

+

(ehAi - e b ~ 2 ) ( l / c h ~ i 1 / c h ~ 2-

l)-'?

(16)

where approximation has been made regarding the use of spectral hemispherical emissivity ehA . For radiative transfer through a dispersive medium such as porous insulations or multilayer insulations with

CRYOGENIC INSULATION HEATTRANSFER

365

spacers, however, calculations must be based on the rather complicated solutions of the equation of radiative transfer (21). Moreover, the radiation properties of the dispersive medium are given in terms of the scattering coefficient y A,the absorption coefficient K,, , and the extinction coefficient (= y A K,,), and to determine these coefficients in a general way is a problem of considerable difficulty from both theoretical and experimental viewpoints (22). For a single particle of simple shapes (e.g., spheres and cylinders), some solutions for these coefficients are available from the theory of electromagnetic wave scattering (29). When particles are closely spaced or packed, multiple scattering renders the problem untractable. More discussions on radiative transfer through dispersive media will be given in the following two sections.

+

IV. Evacuated Powder and Fiber Insulation As indicated previsously, this class of insulation typically has a thermal conductivity one to two orders of magnitude greater than multilayer systems. T h e major heat transfer mechanisms are conduction through the solid phase and radiation. Conduction through residual gas is comparatively insignificant except for poor vacuum conditions ( p > 10-2Torr), and under these conditions the enhancement of thermal contact conductance due to the presence of an interstitial gas must also be taken into account in addition to residual gas conduction (36). Any attempt to analyze the heat transfer characteristics of powder and fiber insulations faces immediately two formidable problems, i.e., the complex, irregular geometry of interfaces and pores and the coupling of solid conduction and radiation processes. T h e complex geometry problem is normally handled through the use of appropriately defined effective thermophysical properties in the heat transfer analysis. T h e effective properties can be regarded as the average over a volume element consisting of numerous interfaces and tortuous voids, and they are functions of inherent material properties as well as geometric parameters characteristic of the porous medium such as porosity, pore size, etc. T h e coupling problem must be considered when the interactions between conduction and. radiation are strong. I n many situations, however, the total heat transport can be justifiably approximated as the sum of the individual processes (Section IV, D). This section presents first a brief discussion of the physical and optical properties of the powder and fiber insulations that are important to the heat transfer processes and then the analytical methods used to predict solid-to-solid conduction and radiation heat flow as well as the total

366

C. L. TIENAND G. R. CUNNINGTON

heat transfer in the evacuated powder and fiber cryogenic insulation. Consideration is given to both opacified and nonopacified systems. In the former case, the basic elements (powders or fibers) of insulation are either coated with highly reflective thin metallic films or include interdispensed reflective or absorbing particles to provide effective radiation attenuation.

A. PHYSICAL AND OPTICALPROPERTIES T he material properties which are of importance to the heat flow in cryogenic powder and fiber insulations are those which influence the conduction (i.e., thermal conductivity, density, elastic modulus, and Poisson’s ratio) and radiation (complex index of refraction-extinction coefficient) processes. Unfortunately, data on these properties at low temperature are lacking for many of the materials used for cryogenic systems such as perlite, calcium silicate, and polymers. While some data are available for materials such as silica and glasses and for opacifiers such as aluminum or carbon, it should be noted that these are only for bulk materials. Because of the very small sizes of basic elements used for porous insulations, thermal size effects due to the cutoff of the phonon or electron mean free paths may become significant at very low temperatures as indicated in Section 111, B. There has been little study to date of these effects t o achieve an accurate assessment, however, and they are typically neglected in heat flow analysis for this class of insulation. Thermal conductivity data as a function of temperature for typical insulation materials are shown in Fig. 4.Data on other physical properties at low temperature are very limited, and in general room temperature values must be used. Typical room temperature data are given in Table I. TABLE I PHYSICAL PROPERTIES OF SOMECRYOGENIC INSULATION MATERIALS AT 300°K

Materials Borosilicate glasses Silica Carbon Aluminum Copper A1,0, . SiO,

Thermal conductivity (mW/cm OK) 1.1 1.5 8.6 1.9 3.8 2.6

x 10 x 10 x 10 x 103 x 103 x 1G

Density (gm/cm3) 2.5 2.2 1.4 2.77 8.95 3.2

Elastic modu1us (N/m2) 6 7 3 7 12 10

x 1Olo x 1010 x 109 x 10’0 x 10’0 x 10’0

Poisson’s ratio 0.22 0.16 -

0.33 0.36 0.2

CRYOGENIC INSULATION HEATTRANSFER

3 67

A similar problem exists in regard to data of the complex refractive index at low temperatures for commonly used insulation materials. Spectral transmission data have been reported for some materials (Fig. 5 ) , but they do not cover the far-infrared region, 20-200p, which is 1.0

I I I I -SILICA AEROGEL 3 ~ 1 6 ~ r n T H l C K N € S S [ R e f . 3 P E R L I T E , 3 x 1 6 ~ TmH I C K N E S S [ R e f . 31 S I L I C A P O W D E R . 3x10-4rnTHICKNESS [Ref. S I L I C A F I B E R S . t x ID-' ~ T H I C K N E S S , D E N S I T Y = 400 K g / m 3

-----0.8

z

0 m

'" I 0.6 m

z

u k -I

0.4

q rK b-

0

w

a 0.2

0 0

5

10 WAVELENGTH

20

15

25

(PL)

FIG.5. Spectral transmission of insulation materials.

particularly applicable to the low-temperature problem. Also, these data have most commonly been used to obtain a qualitative assessment of transmittance rather than to compute index of refraction and extinction coefficients for quantitative evaluation of the radiation process. Additionally, some absorption and scattering coefficient data have been presented for fibrous materials such as silica, carbon (38), and a polyurethane foam. However, neither low-temperature nor far-infrared data were obtained for these materials so their usefulness for cryogenic applications is mainly to indicate an order of magnitude rather than absolute values. Typical values for several porous materials are presented in Table I1 which illustrates the magnitudes that might be expected for these coefficients. Reported values for total extinction coefficients of several opacified powders are given in Fig. 6. By comparison with Table I1 it is seen that these values for opacified systems are an order of magnitude greater than those for the low-density silica or polyurethane materials and are nearly equal to those of carbon fibers and the high density borosilicate glass.

368

C. L. TIENAND G. R. CUNNINGTON I

5

I

I

10

15

METAL V O L U M E CONCENTRATION x lo3

FIG. 6. Total extinction coefficients of mixed powders.

B. CONDUCTION HEATTRANSFER T h e heat flow by conduction across evacuated powder and fiber insulations may be considered as due solely to that occurring by solid-tosolid contact between the particles or fibers of the medium. Conduction through any residual gas phase may be neglected at pressures less than 10-2-10--3Torr, the pressure being a function of the void size and temperature level. Typically, heat flow as a function of gas pressure decreases until this range of pressure is reached. Further reduction of pressure below this range does not produce a significant decrease in total heat transfer since the gas conduction is now much smaller than radiation and solid conduction. Data of this type are in good agreement with analytical considerations of the effective thermal conductivity of a gas contained in a cellular volume of dimensions less than the mean free path of the gas. T h e thermal conductivity of a gas at ordinary temperature and pressure conditions can be expressed in terms of density p and mean free path I as k, K Cpcvl, (17) where is the mean molecular speed. For a free or unrestricted gas, the

CRYOGENIC INSULATION HEAT TRANSFER

369

TABLE I1

TOTAL A~SORPTION AND SCATTERING COEFFICIENTS FOR SEVERAL INSULATIONMATERIALS

Material

Borosilicate glass fibers

Density (kg/m3)

200

Diameter

(4 1 x 10-6

1.5 x

Silica fibers

50

Silica fibers

50

I x 10-5

Carbon fibers

65

I x 10-5

Polyurethane foam

35

Random pore size

Source temperature (OK)

Absorption coefficient (m-7

Scattering coefficient b-l)

500 650 800 lo00 1700 500 650 800 lo00 500 650 800 lo00 775 923 1123 1273 500

1300 1100 1100 700 600 <200 <200 < 100 < 100 <200 <200 <1 0 0 < 100 400 200 200 400 200

26000 27000 28000 3100 2500 3300 5000 710 740 3800 5700 730 760 38500 26000 18500 20000 2850

thermal conductivity is independent of pressure since density varies directly with pressure while the mean free path is inversely proportional to pressure, as indicated in Eq. (3). For a contained or restricted gas, however, the dimensions of the volume, or void in an insulation, may become much smaller than the gaseous mean free path at low pressures, resulting in free-molecule conduction (Section 111, A). When complex void geometries are involved such as in powder and fiber insulations, semiempirical formulas for gas conductivity must be used for different conduction regimes. A convenient semiempirical technique is based on the use of an effective mean free path (39), which takes into account both the molecule-to-molecule and molecule-to-wall collisions,

where L is a characteristic dimension of the void. T h e effective gas thermal conductivity at reduced pressures then is

C. L. TIENAND G. R. CUNNINGTON

370

where a is the accommodation coefficient introduced in Section 111, B. Considering the heat flow to be by parallel gas and solid paths, an effective thermal conductivity of the insulation due to the gas phase is then kgt = (1 - 6,)

kg’.

(19)

It is recommended (40) that L

= 0.67Dp/6, for

powders

and

L

=

0.78Df/6, for fibers, (20)

where D, and D, are the equivalent diameters for the media and 6, is the fractional volume of solids. Some typical solid fractions for particles and fibers are given in Table 111. T h e wide ranges shown for particles and fibers are due to the range of particle sizes in a single lot of material and compression or densification techniques for fiber systems. TABLE I11

TYPICAL SOLIDFRACTIONS FOR POWDER AND FIBER INSULATIONS Material-Packing Spheres-Compact, hexagonal Spheres-Simple cubic Agglomerate angular particles (i.e., perlite or colloidal silica)-Random Angular single particle-Random Fibers-Random

Solid fraction

0.74 0.52 0.02/0.2 0.410.6 O.Olj0.2

Examining the data from Fig. 7, the effective thermal conductivity of the insulation is relatively unaffected by pressure in the near atmospheric pressure end of the scale because the gaseous thermal conductivity is 1. As pressure is reduced, the mean independent of pressure for L free path approaches the void size L and there is a distinct reduction in gas thermal conductivity which is reflected in the decrease in insulation conductivity. Finally, at low pressures a further reduction in pressure does not change the insulation conductivity since the heat flow is due only to solid conduction and radiation. When the insulation temperature decreases, correspondingly lower gas pressures must be achieved to eliminate the effect of gas conduction since the mean free path is proportional to temperature. Moreover, in this case, radiation and conduction processes are smaller, so that a small gas conduction effect may be important to the total heat flow. m and D, from Typical values of D,range from 1 x to 1 x

>

CRYOGENIC INSULATION HEATTRANSFER

'

I

E

E >e

@ @

l -

-

-

@

@

I

I

I

I

I

37 1

I

COLLOIDAL SILICA IN N 2 . 3 0 0 " / 7 7 " K . D p = l O - a r n [ref 31 COLLOIDAL S I L I C A IN H e , 77'/20"K, O , ~ l O ~ * r n [ref 31 PERLITE I N N?. 3 0 0 ~ 7 7 D~ ~ ~I .I 5 1 05 m , P = 8 5 K g / r n 3 [ r e f 51 GLASS FIBERS I N N 2 , 3 0 0 ' / 7 7 " K . D p i 1 x 1 0 - 6 r n , P = 1 6 K g / r n 3

T

1 x 10-8 to 1 x m for cryogenic insulations. Based on Eq. (18) pressure dependent effective gaseous thermal conductivity values for helium and nitrogen as residual gases are shown in Fig. 8 for representative diameters. For the very small particle sizes (colloidal), gas conductivity at a pressure of 10-2Torr is two orders of magnitude less than the .-

a Y

,ol

I----FIBERS ----FIBERSAT

IO-'rn

AT T = Z O O * K . ' N ~

L'=o.780f,'

T = 50°K,He} 8, POWDER I C O L L O I D A L I AT T = 2 0 0 ° K , N p

PRESSURE

(Torr)

FIG. 8. Effect of pressure and void size on gaseous thermal conductivity.

372

C. L. TIENAND G. 9. CUNNINGTON

total conductivity of the best opacified powder or fiber systems (see Fig. 2). T h e larger values of L ; re typical of the higher conductivity fiber insulations, and at 10-2Torr the gas conduction is of the same order of magnitude as the total heat flow measured under good vacuum conditions. I n order to effectively eliminate the gas effect in this case the pressure should be reduced to 10-4Torr or less. It should be noted that Fig. 8 represents an ideal se assuming all one-size particles. I n the real case insulations are comyxed of a range of particle and pore sizes and perhaps shapes, and an averagi: or mean size must be used for computations. T h e so-called enhancement of conductance by the residual gas is due to the heat transfer through the gas in the gap between the adjacent surfaces of the solid particles as opposed to the true solid-to-solid contact conductance. I n a semiempirical manner, this heat flow may be ascribed to a conductivity ki given by (36)

where

For air or nitrogen and 01 M 1, there follows g w 21. T h e total heat transport from the gas phase is then considered as a conductivity k,, which is k,t = (1 - 6,) K‘, 6,k; .

+

T h e solid fraction terms relate equivalent areas for solid and gas in the direction of heat flow. For an estimate of the importance of this process at low pressures, let 01 M 1 and 6, M 0.5, then conductivity due to this process in powder insulation for 1 L is approximately one half of that for the gas conductivity given by Eq. (18), which results in a total effective gas conductivity of 0.75kg’.For small values of 6, and L 1 (near atmospheric pressure) the total gas conductivity is slightly greater than kg’. I n the near atmospheric pressure range, experimental thermal conductivity data for small 6, fiber insulations (40) show a gas conductivity which is greater than predicted by Eq. (19). Little study effort has been devoted to this problem, but this discrepancy may be due in part to the process given by Eq. (21). For evacuated cryogenic porous insulations, conduction through the solid phase may represent a major portion of the total heat transfer, and this process becomes increasingly important as temperatures are

>>

>

CRYOGENIC INSULATION HEATTRANSFER

373

lowered because of the fourth-power temperature dependence in the radiation process. Materials which have a discontinuous solid phase such as loose fill powders or unbonded fibers minimize the solid conduction because of the large thermal resistance at each particle-to-particle interface. For solid particles, the resistance of the region at a distance from the contact is neglected. In the case of spherical particles which are lightly loaded, the radius of the contact area is much smaller than the sphere radius and this contact represents the overriding resistance. However, for porous or thin-walled particles, a second resistance may be important; but no analysis of this problem has been developed for application to insulation. This second resistance has been considered for fibrous materials in the case where an effective distance between fiber junctions may be assumed. Although the contact resistance problem has received much attention, little success has been realized in the development of an analytical model for the general problem of conduction transport in the low-density powder and fiber systems. This is due to several factors such as the random orientation of particles and fibers and the range of sizes and shapes in the typical insulation materials. Also, some particulate materials are agglomerates of smaller particles which in addition to having a tortuous internal void structure possess many internal contact points, which presents an extremely complex analytical problem. While a number of investigators have developed models for the conduction transport in fibrous and powder insulations, these generally are based either on a semiempirical treatment of experimental data for a specific set of conditions or an assumption of packing arrangements and contact geometries (41). T h e simplest system to analyze is a cubic packing of uniform size spherical particles of radius I , . By assuming that each layer of particles is isothermal perpendicular to the direction of heat flow, the heat transfer may be represented by a group of parallel resistances each of which is a series of resistances corresponding to the number of contact points in the heat flow direction (Fig. 9). Each contact resistance is given by Eqs. (8) and (9), and the material properties are independent of temperature so that the compressive force F is the only variable in the series path. For a large number of spheres in the heat flow direction, the number of contacts is equal to the number of spheres; and each series resistance is Rs

NtRc,

1

where Nt is the number of spheres per unit length (1/2r8). Correspondingly, the number of parallel paths is equal to the number of spheres

374

C. L. TIENAND G. R. CUNNINGTON

per unit area normal to the heat flow N , , and the equivalent resistance per unit area is Re,

=

(Nt/Na)Re

with the heat flow across a layer of thickness t 4 = Na ATINtRtt.

(23)

Now N , = N,/Nt, where N , is the number of spheres per unit volume ( N , = 66,/8rr,3) and N , = 66,/4rr,2. For cubic packing, 6, = i ~ , ' 6 . By substitution of Eqs. (8) and (9) and the values for N , and Nt into Eq. (23)the heat flow is 4=

0.9O8Ks[(1 - p2)/E]'/3( F ) 1 / AT 3 (Ys)2/3 t

Next consider two limiting cases of compressive force. First, the condition is of an externally applied compressive load such that the contact force is independent of the force due to sphere weight. T h e force on each contact is now uniform through the thickness and F

= p/Na =

4nrS2p/66,

Substitution of this into Eq. (24) results in

and the contact conduction is independent of sphere size. For the case of zero external load, the force on each contact is equal to the weight of the spheres above it. Therefore, the contact resistances are not all equal and decrease with increasing depth from the uppermost surface. T h e weight of each sphere is p,V,, and the series resistance becomes n=t/2r,

R,'

==

a-'(w)-l/s

n-l/3, n=l

where

n is the contact number from x = 0, and w is weight per sphere. T h e summation term appears like the Riemann zeta function and is actually different (42), but it can be effectively approximated by an integral,

CRYOGENIC INSULATION HEATTRANSFER

315

especially when t/2r, is large. Considering N , resistances in series, the heat flow is

Through integration, an approximation of the series for large values of n is #nil3 where no = tj2r, , and the heat flux is given by

and, the solid conduction is again independent of sphere size. T h e effective solid conductivity is expressed as

When neither the external load nor the weight govern the contact force, the heat flow is given in terms of both pressure force F' and weight force w as q = a N a LI T / C (F' nz0-113. (27)

+

T o present an example of the conduction heat transfer analysis, consider recent data (43) on the heat flow through a horizontal layer of metallized hollow glass spheres under high-vacuum conditions (Fig. 9). 70

I

-

I

I

I

I

0

3 8 p SPHERES, t =3.94mm

A

4 3 p SPHERES, t =5.84rnm 7 3 p SPHERES, t =5.84mm

0

L

N

E 50-

\

-3 4 0 -

1

I

/'-

W A R M BOUNDARY T E M P E R A T U R E

( O K )

FIG. 9. Heat flux through horizontal layers of differing sizes of metallized hollow glass spheres.

376

C. L. TIENAND G. R. CUNNINGTON

For the 5.84 mm thickness of 73p and 45p spheres (average sphere size) the measured heat flows were nearly equal, which at low temperatures when conduction is predominant agree with the predicted size independence of Eq. (26) for spheres of like true density and material properties. Similarly, for the 3.94 mm thickness of 38p spheres the heat flow is greater by a factor of approximately 1.3, which corresponds to the t-2/3 dependence of Eq. (26) for horizontal layers with zero external load. It should be emphasized that many powder-type insulations are composed of a range of particle sizes and shapes, and it is extremely difficult to characterize the system. T h e number of contact areas in a unit volume vary spatially because of variations in size, shape, and agglomeration or bridging as well as with compressive loading. Because of the assumptions necessary to achieve a solution, the formulas are useful in a qualitative sense; but they do not permit accurate heat flow calculations on a material unless constants have been evaluated from prior heat transfer experiments. A similar problem exists in describing the conduction process for fibrous systems because of the random arrangement of fibers in an actual insulation. For a symmetrical array of uniform size fibers, an expression for the conductive heat flux has been given as (44) k S 4 m2n A T " = r ln[12.5(E/F)2/3 r 4 / 3 ] + Lf '

where n is the number of fiber contacts per unit area, L, the distance between fiber junctures, and F the load on the fiber juncture. Another expression developed for a specific packing geometry that illustrates the effect of the individual fiber mat parameters is (45)

where

and C is the compressive pressure on the fiber mat. T h e usefulness of the preceding formulas for quantitative calculation is indeed restricted due to their highly semiempirical nature as well as many unknown quantities in the system. I n view of the many complicated factors in fibrous or powder insulations such as particle size distributions, packing arrangements, and contact areas, it is not feasible to predict quantitatively the conduction heat flow in such materials. Although this

CRYOGENIC INSULATION HEAT TRANSFER

377

is not a significant problem for higher temperature systems because the radiative process is often the overriding heat transfer mode, at very low temperatures conduction becomes the predominant mechanism and prediction of performance cannot be made without experimentation. I n order to evaluate the conduction process, investigators typically measure heat flux in terms of several temperature conditions and compute a radiative heat flux, which can be done with acceptable accuracy. Conduction is then estimated by subtracting this from the total measured value, assuming that the interactions among various heat transfer modes are weak and their individual contributions are linearly superposable. T h e conduction process is then examined in terms of temperature, material properties, and compression loading. This permits engineering heat transfer calculations for a specific insulation in terms of loads, temperatures, and gas pressures. It does not, however, lead to methods suitable for the computation of the performance of other materials or different size and shape configurations.

C. RADIATION Heat flow by radiation has as its sources the boundaries and the volume of material contained within these boundaries. For optically thick insulation systems, i.e., large extinction coefficient and thickness, the radiosity of the boundaries may be neglected, and the radiation process depends only on the optical properties of the medium between the boundaries. This occurs in most high-density insulations and the opacified ones. I n the case of transparent or optically thin conditions (small extinction coefficient or small distances between boundaries), the radiative properties of the boundaries must be included in the overall heat flow problem, and this applies to the nonopacified, lowdensity powder, and fiber materials in small thicknesses. Other important considerations to the radiation problem include, first, whether the system can be adequately treated by assuming spectrally gray radiation properties, and secondly, the specular-diffuse character of the boundaries for the optically thin case. Rigorous studies for absorbing and isotropically scattering planar media of thickness L resulted in expressions for the radiative heat flow in the form (21,46) n2u(T,4 -

qr

=

(3/4)7

+ (li.1) +

T24) (1/4 -1

where the optical thickness 7 3 PL, the extinction coefficient p

= (K

+ y),

378

C. L. TIENAND G. R. CUNNINGTON

K is the absorption coefficient, y the scattering coefficient, and E the boundary emissivity. I n the opaque limit, T + co, Eq. (30) becomes

qr = 4n2u(T,4 - T , 4 ) / 3 ~ .

(31)

Equation (31) can also be obtained by regarding the radiation process as a diffusion process with an effective radiation conductivity given by

Kr

=

(32)

16n2aT3/3P.

Strictly speaking, the concept of radiation conductivity is valid only for optically thick systems. For an anisotropic scattering medium such as in porous insulation systems, exact radiative transfer analysis becomes prohibitively complex. Most existing analyses for radiation through porous insulations are based on the two-flux model, assuming the existence of only two discrete fluxes, one forward and one backward (47-49). Equation (30) also 1. For applies well to this case in the optically thin situation since 7 optically thick systems, however, the result is given by

<

kr

=

~uT~f / ( 2yb). K

(33)

where y,, is the backward scattering coefficient. It would be highly desirable for actual applications to relate /3, K , and yb to the properties of porous media, such as particle size, surface reflectance of particles, and void fraction. Some attempts have been made but only with very limited success (49, 50). T h e above treatment generally considers gray conditions and does not include the temperature-dependent radiative properties. However, the fundamental formulations are such that spectral and temperature dependent functions should be included. For cryogenic applications, the boundaries are typically highly reflective surfaces such as aluminum which may be considered gray for long-wavelength radiation (> 5p), and the temperature dependence of the reflectance or emittance could be incorporated. No attempt has been made to include spectral or temperature dependence of the dielectric insulation materials since no investigations of these properties have been made for the cryogenic application. T h e effect of optical thickness on radiative heat transfer is illustrated qualitatively by the data (50) on powder insulations between boundary temperatures of 300 and 76°K. Thermal conductivity values were obtained for a 2.5-cm thickness of plain and metal-flake opacified silica powders, and the influence of boundary emittance on measured conductivity values is shown in Table IV. For the opacified powders,

CRYOGENIC INXJLATION HEATTRANSFER

379

is very large compared to the boundary emittance term, and this approaches the opaque limit given by Eq. (32) or (33). However, for the plain powders the boundary properties have a strong influence on the radiative transfer, Eq. (31).

T

D. TOTAL HEATTRANSFER To define the total heat transfer through a powder or fiber insulation, the effect of interaction between the simultaneously acting radiation and conduction processes must first be examined. Once approximate limits in terms of temperature, optical properties, and the solid conduction have been defined, the form of the appropriate solution for total heat transfer can be established. For a gray conducting, absorbing, and isotropically scattering medium between two parallel walls, the problem of the interaction of radiation and conduction has been investigated extensively (21,51, 52). Despite the assumption of a continuous homogeneous, isotropic medium in the analysis, the result should be a reasonable approximation of the powder and fiber systems, including those opacified by additives, since the particle dimensions are generally much less than the system thickness. With regard to the anisotropic scattering effect, some improvement in the analytical model can be achieved through the use of the two-flux model (48,49,53), but it still contains the basic problem of how to estimate the back-scattering coefficient. T h e following discussion is based on the analysis for the isotropic case (52). T h e dimensionless parameters that characterize the importance of the interact ion between radiation and conduction processes include the ratio of heat flow by conduction and radiation N , , optical thickness r , and ratio of scattering to extinction coefficients W : N,

kP/4uT13,

r

Pt,

w =

y/P,

(34)

where h is the thermal conductivity. In porous insulation calculations, k should include contributions due to gas and solid phases. T h e general solution for total heat transfer for combined radiation and conduction w < 1 is given by (46) 4= (‘)T

+

(2) k p ~ , ( i- e) 4-n 2 0 ~ ~ 4 (1 e4) (3+ ( 1 ($1 + (1 - c2)!3]

[

Nc .-I- 2t1;?’3i

(35)

+ [( Nc/03)f 2t2/3

where 8 = T 2 / T ,and n is the effective index of refraction for the porous medium (54).

C. L. TIENAND G. R. CUNNINGTON

3 80

For low-density, fiber or powder insulations, the extinction coefficient

,8 is in the range of 5 to 500 cm-l, w > 0.7, t from 2 to 10 cm, and k typically on the order of 2 x mW/cm OK. For cryogenic applications, T I from 100 to 300"K, N , is in the range of 0.01 to 20, T from 10 to 5000, and w is close to unity. As T becomes very large (> 500), the second and third terms in the denominator of Eq. ( 3 5 ) are negligible and the expression for the total heat flow becomes 4=

m,(i

-

e) + 4n2uTl4(1 -

e4)

(36)

37

It is shown that conduction and radiation contributions can be effectively separated. This is the case for opacified systems where ,8 > 1 and t > m. For nonopacified materials with high emittance boundaries, E ---t 1, the heat flux is also described by Eq. (36). If N , 1 and for low emittance boundaries, E GZ 0.03, the heat flux becomes

<

4=-

k q i - e)

+ % + ( 1 / d +-( 1 k 2 ) - 1 n2u~14(1

e4)

(37)

Computed values of the total heat fluxes for a nonopacified and an opacifi-.l insulation are shown in Fig. 10 as a function of gas pressure.

loZ--.-

I

N

----

5

2

L

1

, arad1095xio-1mw/cmZ QtotDl FOR NON-OPACIFIED, € 0 5003, Prod = 0 4 9 x l d ' m W / c m 2 atotal OPACIFIED, Orad= 0.1 x 1 0 - I m w / c m 2 Qtota1

E 10' I

X 3 J

I

FOR NdN-OPaCiFIED, cb = I

I

-

-

-

-

-I 4

I

The computations are performed for t = 2.5 x m, n = 1.1, = 3.9 x 10-5m, T I = 300"K, 6' = 0.067. For the nonopacified material, T = 75 and boundary emittances are 0.03 and 1.O. For the opacimW/cm OK, fied system, T = 750. T h e solid conductivity term is 2 x and 6, = 0.1. I n the case of the opacified system, the radiation heat flux is 1 x 10-2 mW/cm2and the solid conduction heat flux is 2 x mW/cm2. Heat flux from gas conduction increases from 2 x 10-3mW/cm2 at

L

CRYOGENIC INSULATION HEATTRANSFER

38 I

10-4Torr to 1.1 x lo2mW/cm2 at near atmospheric pressure. I n the case of the nonopacified system, the radiation heat fluxes are 4.9 x mW/cm2 and 9.5 x mW/cm2 for boundary emittances of 0.03 and 1.0, respectively. At high gas pressures the total heat flux is not significantly dependent upon radiation properties because gas conduction is the overriding process. For very small values of T, such as a very low density insulation, the radiation effect is more pronounced; and it does become a major contributor to heat transfer for high emittance boundaries. T h e importance of the radiation process is most significant at high vacuum conditions where reductions of 1 to 10 times can be achieved by use of opacifiers and low emittance boundaries (Table IV). However, TABLE IV EFFECTOF BOUNDARY EMITTANCE IN EVACUATED POWDER INSULATION .

k, (mW/cm OK) Insulation Cab-o-sil H-5 Syloid Microcel T-5 20% Al" Cab-o-sil 66% Al" Cab-o-sil

+ +

a

Eb =

0.02

Eb =

0.0135 0.0077 0.0061 0.0044 0.0023

0.8

0.0298 0.0128 0.0095 0.0058 0.0024

Weight percent of 44p aluminum powder.

in a real insulation the addition of opacifiers increases the solid conduction so that there is an optimum point for the amount of material added in order to achieve the minimum total heat flux for a system (50).

V. Evacuated Multilayer Insulation

T h e increasing demand for compact, lightweight, and highly effective thermal insulation for cryogenic applications has resulted in an impressive development and wide acceptance of evacuated cryogenic multilayer insulation systems. Great strides have been made in the last few years in understanding the complex heat transfer mechanisms involved. Reliable working formulas and computation techniques for heat transfer calculations are now available for thermal design and performance evaluation.

382

C. L. TIENAND G. R. CUNNINGTON

A. THERMOPHYSICAL PROPERTIES OF REFLECTIVE SHIELDS AND SPACERS T he importance of the thermophysical properties of insulation has been clearly demonstrated in the preceding section. In particular, advances in evacuated multilayer insulation have been made largely through the development of materials with optimal thermophysical properties. Furthermore, for reliable thermal design and performance evaluation which is crucial to all high-performance insulations, it is important to have at hand the correct values of thermophysical properties of the medium involved. This is especially the case in multilayer insulation since the basic elements of insulation (reflective shields and spacers) were developed not too long ago and relatively little information is available in the literature regarding its thermophysical properties. T h e primary thermophysical properties of the reflective shields and spacers that affect in a direct way the insulation heat transfer calculation are the thermal conductivity, specific heat, and radiation properties of the shields and the spacers. These have received considerable attention in many recent investigations.

I . Thermal Properties of Reflective Shields T he thermal conductivity of the reflective shield consists of contributions from the plastic film and the metal coating. For standard reflective shields such as &mil aluminized Mylar, the plastic contribution is small as compared to that due to metal. (Table'V shows typical values for the thermal properties of various substrate materials.) T h e thermal conductivity of metals at cryogenic temperatures has long been a subject of intensive studies, but most of these studies are concerned with bulk metals. I n multilayer insulation, however, thin metal films are involved, and there might exist a substantial size effect in the heat transfer mechanism. T h e transport coefficients of thin films, such as electrical and thermal conductivities, are indeed expected to be less than those of the bulk metal, for some of the electron free paths will be shortened due to termination at the boundary surface. Since as the temperature decreases, phonon excitation (i.e., lattice vibration) will be reduced and the electron mean free path for the electron-phonon interaction will increase, the size effect is more pronounced at cryogenic temperatures. T h e method of calculating the thermal conductivity of thin metallic films and wires at cryogenic temperatures has been presented in detail in a recent paper (20). Only a very brief account will be given here.

TABLE V

THERMOPHYSICAL PROPERTIES OF TYPICAL SUBSTRATE MATERIALS FOR REFLECTIVE SHIELDSAT Rooni TEMPERATURE

Film

Mylar (polyethylene terephthalate) Kapton (polyimide) Tedlar (PolYTnl fluoride) FEP Teflon

Thermal conductivity

Specific heat

( x lo6, m)

(kg/m3)

( x lo5, OK-')

(W/MK)

(j/gk)

3.81

1384

2.7

0.151

1.318

Slow burn

420

7.62

1495

3.6

0.155

1.088

Self-extinguishing

670

F+I

1578

5.0

-

-

Slow burn

380

2>

Flammability (ASTM D-635)

Maximum usable temperature

?

Density

Thermal expansion coefficient

Minimum thickness

("K)

2

r

s 2 z

12.7

5v 12.7

2131

8.3-1 0.4

0.209

1.172

Self-extinguishing

480

m z

w

co

w

C. L. TIENAND G. R. CUNNINGTON

384

T he thermal conductivity of thin metallic films and wires can be expressed in the following functional form: k/kb = f(d/lk

,$')

(38)

where k , is the bulk thermal conductivity, d the characteristic length (thickness for films and diameter for wires), lk the electron mean free path for thermal conductivity, and p the probability of specular reflections of electrons at the boundary. Experimental evidence indicates that the reflection of electrons is primarily diffuse or p is near zero. T h e calculation of Zk is rather involved, but for pure gold or aluminum at room temperature, Zk is of the order of 400 8 (Z.?), and increases with T-3 (20). Since for cryogenic reflective shields, d w 400 8, d is much smaller 1, than lk in the cryogenic temperature range. For p = 0 and d/lk Eq. (38) assumes the explicit form

<

(39)

k/kb = $(d/Zk) In(lk/d).

T he thermal conductivity size effect is most revealingly shown in Fig. 1I, where the experimental data of the effective thermal conductivity

-Y 0.12 .E.

ALUMINUM FILM

3 0.10-

0

1

(l~d/>~of~ ncm'

EXPERIMENTAL DATA (COSTON AND V L l E T l

-THEORETICAL

>-

t

>

RESULTS

; 0.08V 3

n

z

8 006_I

0

a

5.

5+

0.04-

/

p=o

W

00

>

t, 0.02-

0

0

0

W

LL LL

W

0

0

I

I

I

I

50

I00

I50

200

25

of a double-aluminized Mylar-Dexiglas multilayer insulation system (55) are presented along with the theoretical results. T h e name effective thermal conductivity was used here because the data were reduced

CRYOGENIC INSULATION MEAT TRANSFER

385

from heat conduction measurements by assuming the linear temperature dependence of thermal conductivity. This assumption is indeed reasonable as demonstrated in both the theoretical and experimental results. I t should be also pointed out that the lateral radiation contribution (see Section V, C) has been neglected in the data interpretation presented here and this can be justified through a complete lateral conductionradiation interaction analysis (56). T h e totally different trends in temperature dependence between the bulk and the thin film results shows clearly the importance of the thermal-conductivity size effect in actual applications even in the temperature range near room temperature. T h e specific heat of reflective shields is largely due to the contribution of the plastic substrate material. For instance, the metal coating contributes roughly one percent of the total specific heat for a doubly mil). aluminized (400 thick) Mylar sheet

a

(a

2. Radiation Properties of Reflective Shields T h e size effect is also expected in the radiation properties of thin metallic films as the absorption and emission of radiation of metals has its origin in electron motions. This has been briefly mentioned in Section 111, C. I n addition to the similar-size effect as in the conductivity case which is characterized by the ratio of two characteristic lengths, i.e., film thickness and electron mean free path, another effect comes into play in the radiation properties of metals as a result of one additional characteristic length, the field penetration depth. When radiation (i.e., a batch of electromagnetic waves) impinges on a metallic surface, the actual state within the metal is one in which the amplitude of the electric field decays with distance into the metal. When the electron mean free path becomes large as compared to the penetration depth of the decaying field, such as is the case at cryogenic temperatures, the electrons experience some effects due to the spatial variation of this field. These effects which are neglected in the ordinary free-electron theory of absorption, are given consideration in the ASE (anomalous-skin-effect) theory. Figure 12 (31) shows the predictions and measurements of the total normal emissivity of copper at cryogenic temperatures. It is evident that the ASE theory represents a considerable improvement over the DSE theory in the prediction, but experimental data scatter and deviate in a substantial degree from predictions. T h e major cause for the data scatter and deviation from the theory is due to the surface impurities and imperfections, since as indicated in Domoto et al. (32) some limited data obtained at room temperature under extremely stringent conditions

386

C. L. TIENAND G. R. CUNNINGTON I

o',

0

I

I

I

I

I00 200 T E M P E R A T U R E (OK)

I

I

300

FIG. 12. Total normal emissivity for copper at cryogenic temperatures. [Data sources referred to in Fig. 12 can be found, fully cited, in Ref. (31).]

of purity and surface finish do agree with the ASE prediction. Simple approximate formulas for the prediction of the spectral and total emissivity of metals at low temperatures have been reported (31,53),and further extension of the ASE theory for transition metals has been made (32). For practice, particularly in the thermal performance calculation for multilayer insulation where there exists a substantial difference in temperature among various shields, it is desirable to express total hemispherical emissivity of the shield in a simple function of temperature. This is often approximated by (58) E

=

aTb,

(40)

where constants a and b are prescribed on the basis of experimental information. For low-emittance metals, b w 0.67, while a may vary considerably in the range to lop4 depending on the surface condition. For thin metallic films (e.g., aluminum on Mylar) at cryogenic temperatures, the calculation of radiation properties must incorporate both the size effect and the anomalous skin effect. Such a calculation has been made (34,35) and a parallel experimental study has also been

CRYOGENIC INSULATION HEATTRANSFER

387

carried out (33).As shown in Fig. 13, for the class of thin metallic films (about 400 thick aluminum or gold) actually used in multilayer

a

0. 0 4004 GOLD ON KAPTON v 400$ GOLD ON COPPER

$ GOLD ON MYLAR 5 2 0 A GOLD ON COPPER 0 5 5 0 ! GOLD ON MYLAR 0 7 2 0 A GOLD ON MYLAR b 2 5 0 0 GOLD ON COPPER 4000 GOLD ON COPPER

A 470 0 W

V

z a

+

I-

k

A A

0

5

0

W

.bQ

J

a

0 a

g 0.0 a

-

v)

5 W

r

J

a I-

0

+

THEORETICAL

_ _ _ _ _ 400;

ASE

600d ASE ASE BULK

0.oc

I

/

I

FIG. 13. Total hemi-spherical emissivity of gold films at cryogenic temperatures.

insulation, there exists an appreciable effect of film thickness on the emissivity. T h e thin film emissivity increases with the decrease of thickness, and the effect becomes more pronounced at lower temperatures. the substrate material For the range of film thicknesses (> 400 does not seem to exert any influence. Th e discrepancy between theory and measurements is due again to the imperfect condition of the films used for the measurements. T h e films used, however, are equivalent to those commercially available. For reflective shields with metallic layers coated on only one side, such as the widely used singly aluminized Mylar, the plastic-side emissivity of the shield must also be determined. This requires information regarding infrared radiation characteristics of thin plastic films. Radiation in plastics, which consists of randomly oriented long-chained

a),

C. L. TIENAND G. R. CUNNINGTON

388

polymeric molecules, has its origin in the excitation of radical bonds in the large molecule and can be described by a field of randomly distributed excitation centers. As a result, the radiation spectrum of plastics, somewhat analogous to that of radiating gases, contains a larger number of resonances, especially in the infrared (Fig. 14).These sharp spectral

v)

01 2

I

I 4

I

I

6

I

I

I

1

I

8 10 WAVELENGTH

1

12

(pl

I

“A

14

I

I I 18 22 26

FIG. 14. Spectral normal reflectance of plastic films on aluminum.

variations would result in cumbersome computations if exact nongray calculations are to be made. For engineering calculations, a simple analytical technique, based on the concept of band-averaged optical constants, has been developed recently to calculate radiation of thin plastic films (59). Shown in Table VI are the band-averaged optical TABLE VI BAND-AVERAGED OPTICAL CONSTANTS OF MYLARAND BAND-AVERAGED REFLECTANCE OF MIL MYLARON ALUMINUM Reflectance Wavelength range (PI

4.0 6.8 9.8 14.3 21.0

to 6.8 to 9.8 to 14.3 to 21.0 to 100.0

Refractive index

Absorption index

Theoretical

Experimental

1.805 3.648 2.191 1.565 1.955

0.00327 0.01763 0.01700 0.01 114 0.02215

0.936 0.651 0.757 0.932 0.800

0.88 0.40 0.75 0.91 -

CRYOGENIC INSULATION HEAT TRANSFER

389

constants of Mylar as well as the reflectance of Mylar on aluminum. I t is found that, for singly aluminized Mylar (&mil thick), the plastic-side emissivity (about 0.4) is one order of magnitude greater than that on the metal side.

3 . Thermal Properties of Spacer Materials

As mentioned previously in Section 11, some multilayer insulation systems do not use any spacers, and the radiation shields are separated from each other simply by crinkling or embossing the shields. For those systems with spacers, the commonly used spacer materials are shown in Table I1 along with their absorption and scattering coefficients. It should be emphasized that these properties as well as effective solidphase thermal conductivity are strongly dependent on the density or the pressure imposed on the material. Consequently, their thermal properties are difficult to estimate without any experimental information under simulated conditions. T h e general dependence of the effective solid-phase thermal conductivity upon various relevant parameters follows that already discussed in Section IV, B. B. NORMALHEAT TRANSFER Heat transfer in the direction normal to the layers often constitutes the major criterion in the thermal design and performance evaluation of multilayer insulation. T h e heat transfer mechanism involved is influenced by a large number of system parameters. T h e primary system parameters consist of the layer density (including thicknesses of insulation blankets and spacers, and the imposed pressure), radiation properties of the reflective shields and thermal properties (conduction, absorption, and scattering) of the spacer material. T h e calculation of normal heat transfer in multilayer insulation as a whole is always built on the calculation for a basic segment consisting of two neighboring shields, across which combined radiation and conduction (i.e., conduction through spacers or contact conduction in case of no spacers) takes place. For the spacer materials (including void) and thicknesses utilized in typical multilayer insulation systems, the optical thickness is very small compared to one, and the radiation and conduction contributions can be calculated separately (53). Furthermore, in the calculation of radiation contribution, the spacer effect can be neglected. T h e conduction contribution depends little on the thermal conductivity of the spacer layer but primarily on the interface contact conductance,

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C. L. TIENAND G. R. CUNNINGTON

which is in turn strongly influenced by the contact pressure and deformation statistics (60, 61). Earlier treatments of normal heat transfer across evacuated cryogenic multilayer insulation (5-7) have been a direct extension of the two-shield case to the multilayer system. This analytical approach stays closely with the discrete physical system, but the resultant computations, even with improved numerical techniques (62), are cumbersome and the analysis is not flexible in its applications to a variety of physical situations. A continuum model has been suggested (63) which assumes the multilayer system as a continuous homogeneous medium and seeks a local equivalent thermal conductivity characterizing both conduction and radiation transport. Results obtained from this simple continuous model have been shown in good agreement with experimental data (63) as well as exact numerical solution (57). Further refinement of the continuous model has also been reported recently (57). A brief discussion of this model is given in the following. Consider a single segment of a multilayer composite consisting of one reflective shield and one spacer layer. T h e shield may be metallized on both surfaces or on one surface only. For the latter condition, different optical properties must be considered for each surface. T h e spacer layer may be a second material or simply a void, as in the case of the integral spacer systems. T h e general considerations and assumptions are as follows: (a) T h e shield layer is regarded as having negligible thermal resistance and, therefore, of uniform temperature. T h e assumption of negligible resistance is justified as the conductivity of the plastic film is at least three orders of magnitude greater than the equivalent conductivity of the multilayer system. (b) Radiation tunneling effect is negligible across the spacer or void layer. Typical spacer layer dimensions h are 2.5 x 10-3 cm or greater so tunneling is not a consideration above 20°K (27, 64-66). (c) Residual gas conduction is negligible as the systems are considered to be at a pressure of Torr or less. (d) For the spacer materials and thicknesses utilized in multilayer insulation systems, the optical thickness is very small, i.e., T 1 where T = ( K y)h, K is absorption coefficient, and y is scattering coefficient. Thus, the radiation and conduction contributions are separable as shown by Wang and Tien (52). T h e heat flux between adjacent shields may be expressed as

<

+

where k is the effective thermal conductivity for the spacer layer. As the

CRYOGENIC INSULATION HEATTRANSFER emittance of the metal surface is on the order of lop2and T reduces to

39 1

< 1, Eq. (41)

I n Eq. (42), the surface radiation contribution is obtained under the assumption of diffuse and gray surfaces, i.e., E is the total hemispherical emittance. I n general, the directional and the polarization effects are negligible (67). For relatively small temperature differences between the shields, such as occur in multilayer insulation, the nongray surface effect is small and may be neglected (68). For the insulation systems using spacers of refractive index n > 1, the blackbody emissive power of the surface becomes eb, =

n%T14,

(43)

and E = nE where 2 is the total hemispherical emittance of metal to vacuum (68). For the case of a shield having a single metallized surface, the emittance of the one surface is evaluated as that of a dielectric film over a metal substrate (59). Thus,

I n the conduction term, the conductivity coefficient k is dependent upon the interface contact conductance and not the conductivity of the spacer layer itself. Thus, k = Hh,", where H is a conductance, and N , is the number of interfaces, i.e., two for a separate spacer and one for a r integral spacer system. In general, the interface contact conductance follows a dimensionless relation (60) such that

where c is a constant depending upon surface conditions, k , is the thermal conductivity of the spacer material, p is pressure, E is a deformation parameter, and d is a correlation constant. I n the general case, the deformation and thermal parameters are temperature dependent. Since these properties are not easily obtainable for most systems, they are lumped into a coefficient C. Fletcher et al. (61) have reported d to be approximately 0.5 for materials such as fibrous papers or net. T h e solid conductivity term is now defined as k

= Cp"h/Nc

392

C. L. TIENAND G. R. CUNNINGTON

T he equation for the heat flux between two layers in the multilayer insulation becomes qz-

Cpdh (TI - T,) Nc

+

212,

+ 2, n3u(T14

-

€1

TS4).

(45)

When a system is composed of many shields and the temperature difference between two adjacent shields is small compared to the total difference, the heat transfer across each segment, composed of two neighboring shields, may be expressed by differentials rather than differences. This implies a continuum approximation to a discrete system such as the multilayer. T h e approximation approaches the exact case where the number of layers is large and the temperature drop = 2, and x = N ( h t) between two layers is very small. For ( N is the number of segments composed of one shield and one spacer, and t the shield thickness),

+

q=--

and for h

Cpdh AT Nc h

n3ah

A(T4)

'(7)'T

>t

where the bracketed term is the local equivalent thermal conductivity characterizing both conduction and radiation transport. For illustration, consider steady one-dimensional heat transfer across a multilayer insulation with the boundary conditions x = 0, T = T, , and x = N,(h t ) , T = T , , where N , refers to the total number of layers and subscripts H and C denote the temperature of the exterior surfaces. Let the temperature-dependent properties be approximated by a simple power function, i.e., C = a,Tbl and 2 = a2Tb2,cf. Eq. (40). T h e heat flux can be readily obtained as

+

(48)

T o apply the heat transfer equation t o multilayer insulations, which typically have a complex structure, the values of a, and b, are approximated from experimental data regarding the effect of pressure or layer

CRYOGENIC INSULATION HEATTRANSFER

393

density on heat flux. Data of shield emittance as a function of temperature are used to evaluate a, and b, (6, w 3). T h e index of refraction n is 1 for insulation not using a continuous spacer layer and n = 1.14 for thin fibrous paper-type spacers (54). T h e applicability of the solution for multilayer insulation heat transfer, given by Eq. (48), for engineering design calculations is examined by comparison with experimental heat-flux data for three types of insulation systems being considered for aerospace applications. Good agreement has been indicated in all three cases ( 5 4 , and Fig. 15 shows the trend of these data.

-2o I

I I

I

I

I I I l l I 10 I02 COMPRESSIVE PRESSURE ( N /cm2) I l l

I

I l l

I o3

FIG. 15. Heat flux as a function of applied compressive pressure for three multilayer insulations. ( 1 ) Crinkled, single-aluminized Mylar; (2) double-aluminized MylarTissuglas paper spacer; (3) double-aluminized Mylar, two layers of silk net for spacers.

From a more fundamental viewpoint, a number of recent investigations have been concerned with certain basic radiation phenomena that may affect significantly on a gross level the normal heat transfer in evacuated cryogenic multilayer insulation. T h e specific problems involved are the effect of nongray radiation of reflective shields (68), the increase of radiative transfer by wave interference and radiation tunneling due to small spacings between the shields (64,65),and the non-Planckian nature of blackbody radiation in small enclosures (66). T h e nongray effect is important generally but the small-spacing and non-Planckian effect becomes pronounced only for highly compressed multilayer insulation in the liquid-helium temperature range.

394

C. L. TIENAND G. R. CUNNINGTON

C. LATERAL HEATTRANSFER I n actual applications, heat flows inside the multilayer insulation system are seldom one dimensional in the normal direction. T h e complex geometry of the insulated system as well as penetrations by mechanical supports and plumbing often results in multidimensional heat paths and leaks. It would be a formidable problem to calculate the multidimensional heat transfer in such a highly anisotropic discrete medium as multilayer insulation. A logical starting point seems to be the extension of the continuous model in the case of normal heat transfer to consider the multilayer insulation as a continuous, homogeneous but anisotropic medium with prescribed normal and lateral effective thermal conductivities. T h e definition of these effective conductivities may not be an easy matter, however, since these conductivities, in contrast with the one-dimensional normal or lateral case, must include the interaction effect between the normal and lateral heat transfer. No analysis or calculation of this nature has yet been reported. Even the one-dimensional lateral heat transfer has not been easy for analysis and understanding. Until very recently, the lateral heat transfer had been regarded as governed solely by heat conduction in the thin metallized film on the plastic. Recent evidence (56, 69, 7 4 , however, reveals that multiple reflections (i.e., lateral radiation tunneling) along two conducting films could affect the lateral heat transfer in a significant manner, especially when spacers are not used. T h e use of highly scattering fibrous spacers such as Tissuglas and Dexiglas reduces the lateral radiation contribution, but it takes a few layers of them to eliminate effectively this contribution (72). Analysis of the lateral heat transfer in evacuated multilayer insulation has been presented for both cases with or without spacers (58, 72, 73). Simple approximate formulas are now available for use in design and performance evaluation. Numerical calculations based on simplified nodal techniques ( 4 ) have also been reported and are in good agreement with approximate analytical solutions. Experimental measurements of the effective lateral thermal conductivity have been shown to compare favorable with analytical predictions (72). T h e analytical problem is interesting in several aspects and deserves some detailed consideration. For the case of two radiating but nonconducting plates, the problem is analogous to that of infrared “light pipes” and has been analyzed in detail (75). Of particular fundamental interest radiation conduction interaction phenomenon here is the strong analytical resemblance to the unidimensional gaseous radiation-conduction interaction (54, which

CRYOGENIC INSULATION HEATTRANSFER

395

has been already discussed extensively in conjunction with normal heat transfer through multilayer insulations. T h e physical system under investigation consists of two parallel conducting and radiating plates of finite length but infinite width with ends maintained at temperature T I and T 2 , respectively (see Fig. 16).

p z”:ig’///”i 7 FIG. 16. Schematic of the physical system for lateral heat transfer.

T h e two plates are separated by a nonabsorbing dielectric with a refractive index of unity and the spacing h is small compared to the plate length L, but large compared to the characteristic wavelength of radiation so that anomalous small-spacing effects can be neglected (64). Boundaries 1 and 2 are opaque, and 3 and 4 are gray and externally insulated. Since L> h, radiation effects due to boundaries 1 and 2 can be effectively neglected, and lateral heat transfer is governed only by the radiation and conduction of plates 3 and 4. T h e intensity of radiation within the medium is I-1 in the forward direction and I - in the backward direction. T h e governing equation for the heat flow per unit width q, at any section x is simply dy/dx

=0

(or q = qr

+ yC

=

const),

(49)

where qr and qo denote the radiation and conduction contribution, respectively. T h e conduction contribution is

+

where k is the thermal conductivity at the mean temperature ( T , T2)/2, t the plate thickness, and y = x,h.T h e radiation contribution can be formulated as (75)

where 0 is the Stefan-Boltzmann constant, T the transmittance of radiation from one section to the other due to multiple reflection, and

396

C. L. TIENAND G. R. CUNNINGTON

Lo = Lib. Equations (49)-(51) combined with the boundary conditions y=O,

T = TI,

y=L,,

T = T2

(52)

constitute the complete analytical description of the problem. In order to understand the analytical behavior and functional dependence of the lateral conduction-radiation interaction, it is intended to seek an approximate analytical solution. Basic to the present approach is the approximation of T( y ) by an exponential function, i.e., (53)

T ( Y )= c a y ,

where a is a constant depending on the surface reflectance. T h e numerical solution of ~ ( yindicates ) this to be a valid approximation, and accordingly a can be determined by simply matching with the solution (75). I t is interesting that Eq. (53) is identical to the expression of transmittance for a radiating gas. Moreover, the exponential approximation allows the use of the kernel-substitution technique to reduce the integrodifferential equation to a differential one (21). Nondimensionalizing and differentiating twice Eqs. (49)-(5 1) gives

where N is a radiation-conduction parameter defined by N = (2n2aTI3h2/kta),

I an optical length parameter, I = aL,, 6' = TIT,, and E = x,'L. Incorporated with the boundary conditions, O(0) = 1 and Q(1) = 8, as well as the definition of effective lateral thermal conductivity: ke

f

gLl[t(T,- T2.K

(55)

Eq. (54) can be integrated to yield

The governing equation, although simplified by the use of the exponential approximation and the kernel-substitution technique, is still too complex to allow an exact analytical solution. Limiting solutions, however, can be obtained for various regions of physical interest such as conduction-predominant ( N l), optically thick ( I l), and optically thin (1 1). T h e parameter c employed to characterize

<

<

>

CRYOGENIC INSULATION HEATTRANSFER unidimensional gaseous radiation-conduction be used here, i.e., c =

1jlvP ( 2

< l),

c =

397

interaction (53) can also

ljlv ( 2 >

1).

(57)

T h e solid curve in Fig. 17 which separates approximately the radiationpredominant and conduction-predominant region was obtained by setting the interaction parameter unity.

FIG.17. Limiting regions for combined lateral radiation and conduction in multilayer insulations.

> >

Consider the optically thick limit (I 1) which applies to most cases 1). By applying the optically in multilayer insulations because of (Llh 1) to Eq. (56) and noting that @(O) and O’(1) are of the thick limit (1 order 1/Z and hence negligible, there follows

>

(kelk) = 1

+ “1 + e m + 02).

(58)

This implies that in the optically thick limit, radiation can be treated as a parallel mode to conduction. Formulas for other limits have also been reported (56).

398

C. L. TIENAND G. R. CUNNINGTON

Spacer effects on lateral heat transfer come into the analytical framework through the spacer refractive index n (the absorption index being assumed to be negligibly small as indicated in Table 11), and the transmittance constant a. I t has been shown (73) that a % 1.59h. Another important concern in the assessment of spacer effects is the fact that in practice, the spacer and the radiation shields are loosely packed to minimize the thermal contact and reduce the heat transfer in the normal direction. T hi s results in a nonhomogeneous case where the gap spacing is partly filled by the spacer and partly by vacuum. Such a system can be approximated by a homogeneous spacer of extinction coefficient /3h,ih, where h, is the thickness of the spacer(s). This approximation is exact for radiative equilibrium provided the reflection and the refraction suffered by the ray in the vacuum-spacer interfaces are neglected. This is also exact for the case of radiation-conduction interaction for a purely absorbing or scattering medium (22). Since most spacer materials used in practice have very large scattering coefficients compared to their absorption coefficients, this is a valid approximation. Figure 18 shows experimental results of the effective lateral thermal conductivity at different temperatures (70, 72). Measurements were > t

5

I

-

I

I

I

VACUUM GAP

0 +

v

S I N G L E LAYER DEXIGLAS D E N S E L Y PACKEO O E X I G L A S

0

MEASURED S H I E L D CONDUCTIVITY

3

-

z 0;

vo i

x

W

'

-

-

zoz3 r~u c

22 2 -

-

W a ,

G1 s w

2

-

I -

I-

u

LL W

h

0

I

I

I

I

made in the temperature range of 120 to 250°K for doubly aluminized Mylar sheets with and without Dexiglas spacers. While the use of spacers reduces lateral radiation transport, there are ways to reduce lateral heat conduction. Th e most effective technique is

CRYOGENIC INSULATION HEATTRANSFER

399

the selective slitting of insulation blanket to increase resistance in the direction of heat flow. ‘The slitting results in a two-dimensional heat conduction problem that is tractable (76), but the resulting two-dimensional lateral conduction-radiation interaction becomes extremely complicated. More studies of the slitting effect on lateral conduction and conduction-radiation interaction are needed. Finally, it should be noted that the above discussion and analysis assume that the lateral and normal heat transfer are uncoupled. This is a valid approximation as long as the normal heat transfer is much smaller than the lateral heat transfer, i.e., only a very weak interaction between them exists. With various means of reducing the lateral heat transfer such as using spacers and slitting, a stronger interaction might be resulted and the combined lateral and normal heat transfer must be analyzed. Unfortunately, the analysis will become so complicated that a reasonable solution of the analytical problem does not seem to be feasible at this time.

VI. Test Methods

A variety of test methods and types of apparatus have been used to measure thermal properties of cryogenic insulations. Although thermal conductivity is the most widely measured property, others which have been studied in some detail are specific heat, infrared radiation properties and outgassing characteristics of insulation materials (5, 33, 77, 78). T h e discussion in this section will be limited to thermal conductivity test methods which have general applicability to evacuated powders, fibers, foams, and multilayer insulations. Specific experimental details are contained in the cited references. T h e thermal conductivity test methods may be divided into three general categories (9, 79) which are defined in terms of the methods used to measure the amount of heat transferred through the specimen. These are boil-off calorimetry, electrical-input methods and indirect methods which use another property of the material or another solid material of known thermal conductivity for computation of specimen heat transfer. Boil-off calorimetry and electrical input are the most widely used methods, and they have been applied to all types of cryogenic insulations. T h e category of indirect methods which includes transient and heat-flow-meter procedures, has had more limited application than either of the other methods, and in general, does not have the accuracy of the others.

400

C. L. TIENAND G. R. CUNNINGTON

A. BOIL-OFF CALORIMETRY

As the name implies, the amount of thermal energy passing through the test specimen is determined from a measurement of the volume of gas vaporized from a fluid of known latent heat of vaporization at a constant temperature and pressure. T h e reservoir for this liquid forms the cold boundary of the specimen (Fig. 19), and a surface controlled at a higher temperature is located at the opposite specimen boundary to provide the driving potential. This warmer surface is generally heated either electrically or by a thermostated fluid bath. Cylindrical BOIL-OFF MEASUREMENT

SP

RY

GUARD SPECIMEN -WARM ~ ~ U N D A R Y

.GUARD

SPECIMEN WARM BOUNDARY

(C)

FIG. 19. Typical boil-off calorimeter configurations. (a) Guarded flat plate boil-off measurement; (b) double-guarded cylinder boil-off measurement; (c) tank calorimeter.

CRYOGENIC INSULATION HEATTRANSFER

40 1

and flat-plate specimen geometries are those most commonly used. Spherical specimens have also been used, but it is difficult to obtain a uniform density for powders or to apply a layered insulation to this configuration. If the liquid reservoir is not totally enclosed by the test material, it is necessary to provide guard reservoirs adjacent to the measuring surface area so that only thermal energy from the specimen enters the measuring reservoir. T h e guard is filled with the same liquid and maintained at nearly the same temperature as the measuring fluid. Th e cold boundary temperatures are those attainable with a range of fluids typically from LH, at 20°K to butane at 273°K. An early version of a calorimeter using the flat specimen geometry is the Wilkes device covered by ASTM C 420-62T (80). T h e specimen is in the form of a disk approximately 25 cm in dimater. T h e measuring reservoir diameter is one-half of the specimen diameter, and it is surrounded by a guard vessel having an outer diameter equal to the specimen size. This guard also surrounds the measuring vessel fill and vent lines. T h e edges of the test specimen are additionally guarded with a loose-fill insulation to reduce heat transfer from the surroundings. A larger version of this calorimeter was constructed for testing of insulation system for LH,-fueled hypersonic aircraft (81).T h e apparatus would accommodate a 1.5-m diameter specimen up to 15-cm thick. T h e hot surface plate could attain a maximum temperature of 81 1°K (heated by radiant lamps), and the calorimeter fluid was LH, (20°K). Higher hot boundary temperatures have been achieved with a smaller device using LN, as the calorimeter fluid (82) for studying insulations applicable to cryogenically fueled entry vehicles. A further improverncnt in the flat-plate type of device is the so-called “double guarded” calorimeter (83). As filling of the guard reservoir disturbs the equilibrium conditions, which is more significant for LH, because of its very low density, it is desirable to increase the time period between fills to a maximum. T o accomplish this a second guard vessel, filled with LN, , is placed exterior to the inner guard (filled with LH,) in order to reduce the extraneous heat transfer into this vessel. A larger version of this type of apparatus used for multilayer insulation testing is shown schematically in Fig. 20 (58). T h e large guard diameter is desirable because of the thermal anisotropy of multilayer insulations. T h e heater plate supports the insulation specimen and is movable so that thickness can be varied without removing the specimen or disturbing the temyerature and vacuum conditions. This plate is also supported on a load cell for measuring the compressive pressure applied to the specimen. Cylindrical calorimeters (3) may be used for powder, fiber, foam, or

402

C. L. TIENAND G. R. CUNNINGTON ,rGUARD

VESSEL FILL AND VENT LINES ILL AND VENT LINES

FIG.20.

Double-guarded flat-plate calorimeter schematic.

multilayer insulations. This type of apparatus (Fig. 21) employs a cylindrical measuring vessel guarded either at the top only or at both ends. T h e specimen is placed in the annulus formed by the cryogen vessels and an outer controllable temperature cylinder or the vacuum enclosure. For porous materials, the lower guard may be eliminated. However, for multilayer materials which are very anisotropic, a lower guard reservoir is used to minimize two-dimensional heat transfer at the measuring area (84). I n operating the boil-off calorimeter, care must be taken to minimize any extraneous heat transfer to the measuring reservoir such as through vent and fill lines. This is normally accomplished by thermally grounding these tubes to the guard vessel. I n order to prevent condensation of vapor in the measuring section and its vent line, the temperature of the guard must be maintained slightly above the boiling temperature of the fluid in the measuring reservoir. This is done by maintaining the guard at a pressure 1 to 2 Torr above that of the test vessel. T h e rate at which the calorimetric fluid is vaporized is generally measured by a wet type of flow meter or a thermal mass flow meter (58). Constant reservoir pressures are maintained by use of pressure controlling valves referenced to a constant pressure sink. An additional consideration in the testing of multilayer insulations, which applies t o all methods, is the problem of two-dimensional heat transfer within the insulation. I n order to assure one-dimensional conditions at the measuring area, the guard width to specimen thickness ratio must be sufficiently large. T h e minimum acceptable value is dependent upon the thermal conductivity normal to the layers and the ratio of normal to parallel conductivities (84-86), as well as the edge boundary

--

CRYOGENIC INSULATION HEATTRANSFER

PRESSURE CONTROLLER

I

THERMAL MASS FLOW METER

403

PRESSURE CONTROLLER

FLOW METER--]

VACUUM PUMP VACUUM CONTAINER HOT BOUNDARY

MEASURING RESERVOIR

FIG. 21. Cylindrical guarded boil-off calorimeter.

temperature. For an edge temperature at the average of the hot and cold boundary temperatures, the ratio of guard width to thickness should be greater than fifteen.

B. ELECTRICAL-INPUT METHOD For this method the insulation heat flux is determined from a measurement of the electrical energy which is dissipated thermally in a resistive load uniformly distributed over the measuring area of the test specimen.

404

C. L. TIENAND G. R. CUNNINGTON

Flat-plate or cylindrical geometries may be used for all types of insulations. T h e heated surface is located at the hot boundary of the specimen and a cryogenic fluid is used for the heat sink. No boil-off measurements are made so the cryogen reservoirs are not guarded. A guard heater is used to assure one-dimensional heat transfer in the insulation measuring area and that all of the measuring heater power is transferred only through the insulation. T h e classical example of the flat-plate type of apparatus is the guarded hot plate, A S T M C177-63 (80). For this method identical samples of the test material are placed at both surfaces of the guard-main heater plate arrangement, and heat sink plates which are at a lower temperature are located at the exterior surface of each specimen. T h e edges of this are insulated with a loosestack of sink-specimen-heaters-specimen-sink fill material to reduce heat transfer to the surroundings. T h e guard heater power is adjusted to maintain a temperature balance between main and guard heater surface plates. Measuring heater power is assumed to be equally divided between both specimens. Examples of the use of this apparatus can be found elsewhere (79, 78). Rilultilayer insulation tests have been conducted using a guarded cylinder method (88) wherein the insulation is wrapped on a cylindrical tube consisting of a central measuring heater with guard heaters at each end. T h e exterior surface of the insulation is in contact with or radiatively coupled to a cooled cylindrical heat sink.

C. INDIRECT METHODS Transient methods have been applied to multilayer (89) and powder materials ( 9 ) to measure insulation thermal diffusivity. T h e thermal conductivity is computed using known or estimated specific heat data and specimen density or weight. By this approach thermal conductivity can be studied for small temperature differences over a wide range of boundary temperatures and with shorter measurement times than achievable by steady-state methods. Comparative methods are also useful for some materials. I n this case the specimen is placed between a heat source (heated electrically or by a fluid bath) and a material of known thermal conductivity which forms a heat flow meter at the cold boundary (90).This approach is not considered satisfactory for multilayer insulation because of the requirement for measurement of very small heat rates, and standard materials are not available in the low-conductivity W/cm OK for calibration. I n general the indirect range, X < methods suffer from a poorer accuracy than the boil-off or electricalinput methods.

CRYOGENIC INSULATION HEATTRANSFER

405

VII. Applications

T o date, the largest application of cryogenic insulation has been for the storage and transportation of cryogenic fluids on the earth. T h e liquid cryogens handled in large quantities are liquefied natural gas (LNG), oxygen, nitrogen, and hydrogen, and to a lesser extent helium. Boiling temperatures of these fluids at atmospheric pressure range from 120 to 4.2"K. Capacities of storage and transportation vessels are from a few liters to greater than 100-million liters, the latter being associated with the L N G industry. Oceangoing L N G tankers in the 100-million-liter capacity range are in service, and storage tanks at L N G distribution and peak-load-shaving plants for utilities are of comparable size (91, 92). Both stationary and portable vessels in the size range of a few liters to thousands of liters are used for storage of cryogens in laboratories and manufacturing processing facilities such as in the electronics industry. Associated with these installations are insulated plumbing lines which range in size from small diameter transfer lines a meter or two long to large diameter pipe lines whose lengths are measured in kilometers (93,94). Insulation used for this broad category include vacuum; both evacuated and nonevacuated foams, fibers, and powders; evacuated spaces containing cooled radiation shields andlor multilayer insulation; and wood or cork board. Potential applications in the aircraft and aerospace industries have been the impetus for a major portion of the recent advances in insulation technology for cryogenic storage and handling (95). These have been dictated by stringent requirements for minimum heat transfer and weight with maximum reliability for long unattended periods of use. T h e largest aerospace vehicle using insulated cryogenic tankage for propulsion is the Saturn vehicle used for the U.S. Manned Space Program (96). Other applications include oxygen and hydrogen storage for fuel cells and life-support systems (such as the Apollo program), cryogenically cooled scientific space experiments, and cryogenic tankage for future space programs such as the reusable space shuttle orbiter and booster and the Modular Nuclear Vehicle for deep-space missions. Typical aircraft applications have been on-board oxygen storage for crew systems, and conceptual study programs for development of LO, LH, and liquid methane fueled hypersonic aircraft. Finally, a wide variety of insulated devices have been used for developmental studies, such as LN,-cooled electrical transmission lines (97) and scientific investigations in the laboratory. Examples of these are in the fields of cryogenic properties of materials, superconducting devices,

406

C. L. TIENAND G. R. CUNNINGTON

and high-energy physics. Often very sophisticated insulation schemes have been required because of the desirability of attaining extremely low temperatures (below 4°K) and the high cost of a cryogen such as helium or neon. Much use has been made of cooled shields and outer guard vessels containing a less expensive cryogen in conjunction with evacuated insulations. Many factors enter into the selection of an insulation system for a specific application. For large storage vessels, the cost, maintainability, and reliability are prime considerations. Minimum total cost per unit quantity of stored or transported cryogen may dictate the type of insulation system to be used for a specific vessel. For LNG or liquid nitrogen, satisfactory thermal insulation is typically achieved by using materials such as evacuated powders, granular materials, or unevacuated foams. As an example, a large LNG tanker ship currently in service has a relatively high boil-off rate (91), but this vaporized liquid is used to fuel boilers for propulsion of the ship which is an important factor in the economics of the choice of the optimum insulation system for this particular application. In most aerospace cases installed cost is not as important a consideration, and minimum heat flux for long-term storage in space or weight, including cryogen boil-off, is the prime factor. For launch or booster systems, the storage time requirements are for a few hours, and less efficient insulations may be acceptable. Such is the case of the Saturn vehicle (96, 98), which uses a foamed insulation. On the other hand, deep-space missions require storage times measured in months, and great care must be taken to assure minimum heat flux into the storage vessel at minimum weight. Multilayer insulations, sometimes used in conjunction with cooled intermediate shields, are prime candidates for these applications. Because of the minimum heat flux requirement, the gas pressure within the multilayer becomes an important consideration, and factors such as outgassing from the insulation materials and the venting of these products must be considered. Excessive outgassing with ineffective venting degrades the performance because of the higher interstitial gas pressure within the system which increases the gas conduction (99). Also, the outgassing products may cryodeposit on reflective surfaces which will increase the radiative heat transfer (100). Brief descriptions of applications of several types of insulation to ground and space cryogenic systems are discussed in the following paragraphs. Major emphasis is placed on evacuated foams, powder, and fiber, and multilayer materials as these are more pertinent in terms of heat transfer as discussed in the preceding sections. Multilayer insulations are of particular interest because of their thermal anisotropy and sensi-

CRYOGENIC INSULATION HEAT TRANSFER

407

tivity to compressive loading, which require the use of special design techniques to achieve optimum performance.

A. NONEVACUATED INSULATION Nonevacuated powder, fiber, and foam insulations are used extensively in the construction of large L N G storage tanks. Polyurethane foam, 22- to 33-cm thick and mounted internally in a concrete tank, has been used for a 110 million liter storage facility (50). Reported performance is a boil-off of 0.06% of tank volume per day which corresponds to a heat flux density of approximately 15 W/m2. Expanded Perlite has been used as the insulation in double-walled large L N G storage tanks (102) as well as for smaller dewars. A problem encountered when using a loose-fill material such as Perlite or silica is the tendency for the material to settle and compact with the thermal contraction and expansion of the inner tank wall during filling and emptying of the vessel. This results in the formation of voids in the upper regions of the insulation space which increases the heat transfer. One method used to overcome this problem is to place a layer of a resilient fiberglass blanket material between the inner wall and the powder or granular fill (102). T h e fiber blanket acts as a spring to take up the dimensional changes and prevents settling of the insulation. Perlite also has been used in conjunction with wood and glass fiber mixtures to provide a satisfactory thermal insulation for a large storage tank (203). In this application Perlite was used for roof and floor insulation with slightly compressed fiberglass blocks in the wall annulus; the compression being sufficient to retain the fibrous insulation upon contraction of the inner wall. T h e S-IVB stage of the Saturn I presents an example of the use of a nonevacuated system for short-term liquid-hydrogen storage. T h e system selected for this application was a low-density reinforced polyurethane foam bonded to the internal wall of the LH, propellant tank (96).A barrier layer was applied to the liquid side of the insulation to retard permeation of hydrogen into the foam. Test data showed that the presence of hydrogen in the foam increased its effective thermal conductivity by a factor of 2 to 3 over the normal value for this foam. T h e insulation system was designed to provide effective cryogen storage through ascent and for 4.5 hr in orbit. An internal foam was selected to prevent air liquefaction during ground hold, provide a higher temperature surface for bonding to the tank wall and minimize tank contraction during filling. T h e performance of this type of system has

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been demonstrated by the successful flights of this vehicle in the Apollo program.

B. EVACUATED POWDER AND FIBERINSULATION Powder insulations such as expanded Perlite, silica, and carbon or charcoal have been used in vacuum jacketed cryogenic liquid storage vessels and transfer lines. Filling the vacuum space with a powder or fiber material reduced the heat transfer over that of a simple vacuum jacketed system because of the radiation attenuation as discussed in Section IV. A further improvement in thermal performance is realized by opacifying the insulation through the addition of metallic flakes or powders (101). Also, the use of very small particles reduces the vacuum requirement as the effective void size becomes significantly smaller than the mean free path of the gas at higher pressures which reduces gas phase conduction as discussed in Section 111. Settling can be a problem with evacuated powders, and one method used to overcome this problem is by placing a reservoir at the uppermost point of the tank so that excess material will flow from this volume into any voids resulting from settling. This method has been used on a large transportable 6000-liter LH, dewar (104) having a boil-off rate reported to be about 1.5yo of the tank capacity per day. A thickness of 30 cm of Perlite fills the vacuum space of the horizontal, cylindrical tank, and because of the very small pore size and long path length, the evacuation of this powder system to a satisfactory pressure level generally requires several days of pumping. Also, the finely divided material has a large surface area, and consequently, moisture desorption can be a problem. An interesting example of the use of fibrous material is found in (14) which describes a program for the development of a permanently evacuated vacuum panel insulation. Several glass fiber paper-type materials were investigated for filling the inner space. T h e vacuum jacket was flexible so the filler material had to support an atmospheric pressure compressive loading. T h e heat transfer data indicate an effective W/cm "K which is nearly five times thermal conductivity of 3 x greater than the value for the evacuated fiber when not subjected to the external compressive load.

C. EVACUATED MULTILAYER INSULATION Extensive usage has been made of this very low thermal conductivity insulation in the field of cryogenics, particularly for aerospace applica-

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tions. From a standpoint of heat transfer analysis, it presents a complex system because of the multidimensional heat transfer considerations brought on by the large degree of thermal anisotropy, Section V. A number of analytical and experimental studies have been conducted to determine optimum methods for terminating the multilayer at penetrations or edges exposed to a different thermal environment (7,105). T h e sensitivity of thermal performance to variations in a compressive load which may be applied to this type of insulation presents another application design problem (57).Relatively small changes in compressive load, such as 10-100 N/m2 can increase the heat transfer by a factor of two or more as shown in Fig. 15 for three types of multilayer systems. T h e increase in heat flux is due to the increased solid conduction which may be correlated with pressure as shown by the equations fitting the experimental data. Variations in compressive loading of this order of magnitude can readily result from application of the insulation around a sharp corner or over small-radius surfaces. Another design problem area is in the attainment and maintenance of a very low gas pressure within the insulation. All of these design problems contribute to a large uncertainty in the thermal performance of a given system when installed on a tank or component, and it is obvious that in order to achieve the highest thermal efficiency of this system with a predictable degree of certainty careful design and fabrication procedures must be followed. Typically, uncertainty factors of two to four are used to predict installed insulation performance based upon calorimeter test data of the multilayer material. However, even with this large uncertainty the performance of multilayer insulated systems is significantly better than the opacified powders. Multilayer insulations have been used for very efficient ground storage dewars for liquid hydrogen and helium. In this type of application the problem of maintaining an adequate vacuum for very long periods of time is not important as the insulation space may be pumped periodically. I n many cases, however, the devices are transportable and the inner tank must be supported in a manner which will not excessively degrade thermal performance and yet can withstand the loads encountered in highway, rail, or air transportation. An example of the application of multilayer insulation to a moderate-sized storage vessel is that of a 3000-liter dewar for storage of slush hydrogen in a laboratory test facility. T h e cryogen container of this vessel was a vertical cylinder approximately 1.4-m i.d. by 6.7-m high with hemispherical ends. T h e inner tank was supported to the bottom of the outer shell through tubular fiberglass struts and a cone attached to the outer shell with a sliding cylindrical supporting surface at the inner tank to accommodate thermal

410

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contractions and expansions of the vessel. All external surfaces of the inner vessel, the fiberglass support struts and cone, and all intervessel plumbing lines were insulated with three overlapping multilayer insulation blankets. Total thickness of the combined multilayer blankets was approximately 2.54 cm. T h e multilayer blankets were prefabricated in oversized gores and polar caps for the inner vessel heads, and in oversized rectangular sections for the cylindrical shell. These blanket sections were then trimmed to fit at close tolerance butt joints during installation. Each multilayer blanket section consists of seventeen 0.15-mil double-aluminized crinkled Mylar radiation shields separated by sixteen Tissuglas spacers. During facrication, all blanket sections were covered on both faces with Dacron mesh to improve handling characteristics. T h e cylinder blanket sections were further reinforced in the vertical direction with Dacron ribbon, 1.27-cm wide, spaced on approximately 20.3-cm centers and sewn to the mesh net. T h e composite sections were then fastened together with molded nylon button retainers spaced approximately 15.2 cm on center. During installation, the cylinder blankets were suspended from the support cone at the top of the Dewar using the Dacron ribbons. T h e lower gore and polar cap blankets were subsequently attached to the cylinder blankets with aluminized Mylar tape and Dacron thread. Adjacent cylinder and gore sections were attached at the butt joints using Teflon tabs and aluminized Mylar tape. Measured equilibrium heat transfer to the cryogen Dewar was 17.5 W. Approximately 35 % was attributed to supports, 15% to plumbing and the remainder to the insulation. Laboratory data on insulation heat transfer were used and degraded by a factor of 2 to account for joints between blankets and buttons supporting the layer in each blanket. T h e heat flux density for this insulation system was computed to be 0.51 W/m2 which for the test temperature conditions corresponded to an insulation effective thermal conductivity of 5.0 x lo-' W/cm "K. This is compared to the laboratory test value of 2.5 x lop7W cm "K for the same boundary temperature conditions. A further improvement on the evacuated, multilayer insulated storage vessel concept has been achieved by incorporating into the insulation space a shield cooled by gas vaporized from the cryogen. Dewars of this type for liquid helium storage have shown significant reductions in storage losses for both liquid helium and nitrogen (106).Test data shown a factor of 16 reduction in cryogen loss for L H e and a factor of 3 for LH, . A cooled shield in conjunction with evacuated multilayer insulation has also been incorporated into a solid cryogen refrigerator used to provide one year of cooling for an experiment on a spacecraft. This unit uses two solid cryogens, and the shield is thermally attached to the tank

CRYOGENIC INSULATION HEATTRANSFER

41 1

containing the cryogen of higher sublimation temperature, in this case carbon dioxide (107). Argon was used to provide cooling for an infrared sensor, and the carbon dioxide container intercepted the thermal energy from the structural members supporting the tank assemblies. As shown by Fig. 22, a 5-cm thickness of a multilayer insulation composed of alter-

FIG. 22. Solid cryogen refrigerator with multilayer insulation.

nate layers of +-mil double-aluminized Mylar and a glass fiber paper, a total of 150 layers, surrounds the two cryogen tanks, the upper one for carbon dioxide and the lower one for the argon. The copper shroud is attached to upper tank and encloses the argon tank at the interior surface of the insulation. A gold-plated, floating radiation shield (not shown in the figure) was suspended between the shroud and argon tank.

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C. L. TIENAND G. R. CUNNINGTON

T h e thermal conductivity of the insulation as installed on this small W/cm OK which is nearly a factor system was estimated to be 5 x of 2 greater than laboratory data for tests under one-dimensional heat transfer conditions, 2.6 x lo-’ W/cm OK. A major problem in the application of multilayer insulation to small storage vessels is the contouring to insulate effectively the ends of the container. I n some cases this is done by cutting gores for each layer and then overlaying and closing each gore with aluminized Mylar tape. This is a very time-consuming procedure, and if not done with care the insulation performance can be severely degraded by openings at joints

FIG. 23.

Multilayer insulations in shingled pattern.

CRYOGENIC INSULATION HEATTRANSFER

413

or local compression due to excessive overlaps. One method that has been used to simplify insulation of curved surfaces is to apply a “shingle” type of multilayer in modules (108).A partially insulated 1.5-m diameter tank is shown in Fig. 23 for illustration for this concept. Layers of a crinkled 4-mil single-aluminized Mylar were bonded to a Dacron fabric sublayer and a Dacron net outer layer. T h e Mylar material is cut in shingle patterns and bonded at each edge to the Dacron fabric and net. T h e completed modules are then attached to the tank with adhesive or nylon fasteners. T h e effective thermal conductivity of the shingle method was determined to be approximately 1.1 x lop6 W/cm OK compared to a blanket type insulation value of 5 x lo-’ W/cm OK.

ACKNOWLEDGMENTS T h e authors wish to acknowledge the support of National Science Foundation through Grant No. N S F GK-30557 (to C L T ) and Lockheed Independent Research Program (to GRC). We wish to thank Mr. C. K. Chan at the University of California at Berkeley for having carefully read the manuscript and for many valuable suggestions.

SYMBOLS A CO

C”

C

D,> DP d

E eb

F h I K Kn

k ka , ke

L 1 M m

area speed of light in vacuum specific heat at constant volume mean molecular speed equivalent fiber, power diameter thickness Young’s modulus blackbody emissive powder force Planck‘s constant, shield spacing intensity constant defined in Eq. ( 6 ) Knudsen number thermal conductivity or Boltzmann constant apparent, effective thermal conductivity length or thickness mean free path or optical length molecular weight mass

N

radiation-conduction eter

Na Nc

NtINv

param-

conduction-radiation parameter Nt , Nv number of spheres per length, per volume n number density or refractive index R = n -in’ complex refractive index pressure P heat flow Q heat flux 4 continuum, free-molecule heat qC 9 QFM flux thermal resistance R gas constant RQ radius Y temperature T thickness t weight W coordinate along heat flow X direction Y xlh OL accommodation coefficient

C. L. TIENAND G. R. CUNNINGTON

414

E

extinction coefficient ratio of specific heats or scattering - coefficient fractional volume of solids TI TI emissivity

6

XlL

B Y

8,

e K

h Y P 0 00

T

relaxation time, optical thickness, transmittance

SUBSCRIPTS

absorption coefficient wavelength viscosity or Poisson’s ratio density Stefan-Boltzmann constant dc electrical conductivity

blackbody or bulk or boundary contact o r conduction electron gas normal phonon radiation solid or sphere

REFERENCES 1. M. Jakob and G. A. Hawkins, “Elements of Heat Transfer and Insulation,” 1st Ed. Wiley, New York, 1942 (2nd Ed., 1950). 2. J. F. Malloy, “Thermal Insulation.” Van Nostrand-Reinhold, Princeton, New Jersey, 1969. 3. R. B. Scott, “Cryogenic Engineering,” Ch. 6. Van Nostrand, New York, 1959. 4. R. H. Kropschot, B. W. Birmingham, and D. B. Mann, Technology of Liquid Helium, Nut. Bur. Stand. ( U . S . ) , Monogr. 111, 170-191 (1968). 5. M. G. Kaganer, “Thermal Insulation in Cryogenic Engineering” (Engl. transl. from Russ.), Isr. Program Sci. Transl., Jerusalem, 1969. 6. P. E. Glaser et al., Thermal Insulation Systems-A Survey. N A S A Spec. Publ. NASA SP-5027 (1967). 7. R. P. Caren and G. R. Cunnington, Chem. Eng. Progr., Symp. Ser. 87,67-81 (1968). 8. C. L. Tien, Cryog. Technol. 7, 157-163 (1971). 9. R. H. Kropschot, Advan. Cryog. Eng. 16, 104-108 (1971). 10. A. V. Luikov, “Heat and Mass Transfer in Capillary-Porous Bodies.” Pergamon, Oxford, 1964. 11. G. S. Springer, Advan. Heat Transfer 7, 163-218 (1971). 12. R. J. Corruccini, Advan. Cryog. Eng. 3, 353 (1958); Vacuum 7/8,19 (1957-1958). 13. C. L. Tien and J. H. Lienhard, “Statistical Thermodynamics,” Ch. 11. Holt, New York, 1971. 14. D. E. Klett and R. K. Ivey, Advan. Cryog. Eng. 14, 217 (1969). 15. F. C. Hurlbut, Particle Surface Interactions in Hyper-velocity Flight, an Annotated Bibliography. Memo R M 4885 PR. Rand Corp., Santa Monica, California (1967). 16. M. G. Cooper, B. B. Mikic, and M . M. Yovanovich, Int. J. Heat Mass Transfer 12, 279 (1969). 17. E. Fried, in “Thermal Conductivity’’ (R. P. Tye, ed.), Vol. 2, pp. 253-274. Academic Press, New York, 1969. 18. P. G. Klemens, in “Thermal Conductivity’’ (R. P. Tye, ed.), Vol. 1 , pp. 1-68. Academic Press, New York, 1969. 19. D. S. Dillard and K. D. Timmerhaus, Chem. Eng. Progr., Symp. Ser. 87,1-20 (1968). 20. C . L. Tien, B. F. Armaly, and P. S. Jagannathan, in “Thermal Conductivity’’ (C. Y. H o and R. E. Taylor, eds.), pp. 13-19. Plenum, New York, 1969.

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21. E. M. Sparrow and R. D. Cess, “Radiation Heat Transfer.” Brooks/Cole, Belmont, California, 1970. 22. H. C. Hottel and A. F. Sarofim, “Radiative Transfer.” McGraw-Hill, New York, 1967. 23. C. Fragstein, Ann. Phys. (Leipzig) 7, 63 (1950). 24. R. E. Rolling and C. L. Tien, in “Thermophysics of Spacecraft and Planetary Bodies” (G. B. Heller, ed.), Progress in Astronautics and Aeronautics, Voi. 20, pp. 677-693. Academic Press, New York, 1967. 25. C. L. Tien and E. G. Cravalho, Chem. Eng. Progr., Symp. Ser. 87, 56-66 (1968). 26. K. M. Case and S. C. Chiu, Phys. Rev. A I , 1 170 (1970). 27. E. G. Cravalho, C. L. Tien, and R. P. Caren, J. Heat Transfer 89, 351 (1967). 28. P. Beckman and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces.” Macmillan, New York, 1963. 29. H. C. van de Hulst, “Light Scattering by Small Particles.” Wiley, New York, 1957. 30. A. H. Wilson, “The Theory of Metals.” Cambridge Univ. Press, London and New York, 1953. 31. G. A. Domoto, R. F. Boehm, and C. L. Tien, Advan. Cryog. Eng. 14, 230 (1969). 32. M. C. Jones, D. C. Palmer, and C. L. Tien, J. Opt. Soc. Amer. 62, 353 (1972). 33. G. R. Cunnington, G. A. Bell, B. F. Armaly, and C. L. Tien, J. Spacecr. Rockets 7, 1496 (1970). 34. B. F. Armaly and C. L. Tien, Proc. Int. Heat Transfer Conf., 4th, Paris 3, R1.l (1970). 35. C. H. Forsberg and G. A. Domoto, J. Heat Transfer accepted for publication (1972). 36. C. L. Johnson and D. J. Hollwegen, Non-Evacuated Cryogenic Thermal Insulation Systems. Air Foce Mater. Lab. ML-7DR-64-260 (1964). 37. V. J. Johnson, ed., A Compendium of the Properties of Materials at Low Temperature. WADD T R 60-56 (1960). 38. G. R. Cunnington et al., Performance of Multilayer Insulation Systems for Temperatures to 700°K. N A S A Contract. Rep, NASA CR-907 (1967). 39. J. D. Verschoor and P. Greebler, Trans. A S M E 74, 961-968 (1952). 40. R. R. Pettyjohn, Proc. 7th Thermal Conductivity Conference. Nut. Bur. Stard. ( U . S.), Spec. Publ. 302, 729-736 (1967). 41. A. V. Luikov et al., Int. J , Heat Muss Transfer 1 1 , 117 (1968). 42. M. Abramowitz and I. A. Stegun, eds., “Handbook of Mathematical Functions.” NBS AMS-55. U. S. Nat. Bur. Stard., Washington, D.C. 43. G. R. Cunnington and C . L. Tien, Heat Transfer in Microsphere Cryogenic Insulation. Paper C-1, Cryogenic Engineering Conference, Boulder, Colorado ( 1 972). 44. H. M. Strong, F. P. Bundy, and M. P. Bovenkirk, J. Appl. Phys. 31, 39 (1960). 45. D. I. Wang, “Aerodynamically Heated Structures.” Prentice-Hall, Englewood Cliffs, New Jcrsey, 1962. 46. L. S. Wang and C. L. Tien, Int. J . Heat Mass Transfer 10, 1327 (1967). 47. B. K. Larkin and S. W.Churchil1, AIChE J . 5, 467 (1959). 48. J. C. Chen and S. W. Churchill, AIChE J. 9, 35 (1963). 49. J. D. Klein, Symposium on Thermal Radiation of Solids. N A S A Spec. Publ. NASA SP-55, 73-81 (1965). 50. B. J. Hunter, R. H. Kropschot, J. E. Schrodt, and M. M. Fulk, Advan. Cryog. Eng. 5, 146 (1960). 5 1. R. Viskanta, J. Heat Transfer 87C, 143 ( 1965). 52. L. S. Wang and C. L. Tien, Proc. Int. Heat Transfer Conf., 3rd, Chicago 5 , 190 (1966).

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C. L. TIENAND G. R. CUNNINGTON

53. J. B. Berggaum and R. A. Seban, J. Heat Transfer 93C, 236 (1971). 54. R. P. Caren, J. Heat Transfer 91C, 154 (1969). 55. R. M. Coston and G. C . Vliet, in “Thermophysics of Spacecraft and Planetary Bodies” (G. B. Heller, ed.), Progress in Astronautics and Aeronautics, Vol. 20, pp. 909-923. Academic Press, New York, 1967. 56. C. L. Tien, P. S. Jagannathan, and B. F. Armaly, AIAA J. 7, 1806 (1969). 57. G. A. Domoto and C. H. Forsberg, An Exact Numerical Solution to the OneDimensional Multilayer Insulation Problem. ASME Paper No. 70-HT/SpT-28. Amer. SOC.Mech. Eng., New York, 1970. 58. G. R. Cunnington, C. W. Keller, and G. A. Bell, Thermal Performance of Multi Layer Insulations. NASA Contract Rep. NASA CR-72605 (1971). 59. C. L. Tien, C. K. Chan, and G. R. Cunnington, J. Heat Transfer 94C, 41 (1972). 60. C. L. Tien, Proc. 7th Thermal Conductivity Conference. N u t . BUY.Stand. ( U . S.), Spec. Publ. 302, 755-759 (1968). 61. L. S. Fletcher, P. A. Smuda, and D. A. Gyorog, AIAA J. 7, 1302 (1969). 62. R. K. MacGregor, J. H. Pogson, and D. J. Russell, in “Heat Transfer and Space Thermal Control” (J. W. Lucas, ed.), Progress in Astronautics and Aeronautics, Vol. 24, pp. 473-486, Academic Press, New York, 1971. 63. G. R. Cunnington and C. L. Tien, in “Thermophysics: Applications to Thermal Design of Spacecraft” (J. T. Bevans, ed.), Progress in Astronautics and Aeronautics, Vol. 23, pp. 11 1-126. Academic Press, New York, 1970. 64. R. F. Boehm and C. L. Tien, J. Heat Transfer 92C, 405 (1970). 65. G. A. Domoto, R. F. Boehm, and C. L. Tien, J. Heat Transfer 92C, 412 (1970). 66. R. P. Caren, J . Heat Transfer 94C, 295 (1972). 67. R. V. Dunkle, Symposium on Thermal Radiation of Solids. NASA Spec. Publ. NASA SP-55, 39-44 (1965). 68. G. A. Domoto and C. L. Tien, J. Heat Transfer 92C, 394 (1970). 69. P. S. Jagannathan and C. L. Tien, J. Spacecr. Rockets 8, 416 (1971). 70. P. S. Jagannathan and C. L. Tien, Advan. Cryog. Eng. 16, 138 (1971). 71. J. G. Androulakis, Effective Thermal Conductivity Parallel to the Laminations of Multilayer Insulation. AIAA Paper No. 70-846. Amer. Inst. Aeronaut. Astronaut., New York (1970). 72. C. L. Tien, P. S. Jagannathan, and C. K. Chan, Lateral Heat Transfer in Cryogenic Multilayer Insulation. Paper C-3, Cryogenic Engineering Conference, Boulder, Colorado (I 972). 73. P. S. Jagannathan, Lateral Heat Transfer in Multilayer Insulation. Ph. D. Dissertation in Mechanical Engineering, University of California, Berkeley, California, 1971. 74. J. H. Pogson and R. K. MacGregor, in “Heat Transfer and Spacecraft Thermal Control” (J. W. Lucas, ed.), Progress in Astronautics and Aeronautics, Vol. 24, pp. 473-486, Academic Press, New York, 1971. 75. D. K. Edwards and R. D. Tobin, J. Heat Transfer 89C, 132 (1967). 76. J. H. Pogson and R. K. MacGregor, in “Heat Transfer and Spacecraft Thermal Control” (J. W. Lucas, ed.), Progress in Astronautics and Aeronautics, Vol. 24, pp. 487-501, Academic Press, New York, 1971. 77. P. F. Dickson and M. C. Jones, N u t . Bur. Stand. ( U . S.), Tech. Note 348 (1966). 78. A. P. M. Glassford, J. Spacecr. Rockets 7 , 1464 (1970). 79. Thermal Conductivity Measurements of Insulating Materials at Cryogenic Temperatures. Amer. SOC.Test. Muter., Spec. Tech. Publ. 411 (1967). 80. Book ASTM Stand. Part 14 (1 964). 81. C. L. Johnson, in “Thermophysics of Spacecraft and Planetary Bodies” (G. B.

CRYOGENIC INSULATION HEATTRANSFER

417

Heller, ed.), Progress in Astronautics and Aeronautics, Vol. 20, p. 849. Academic Press, New York, 1967. 82. J. M. Ryan et al., Lightweight Thermal Protection Development, Vol. I1 Air Force Mater Lab. AFML-TR-65-26 (1 965). 83. I. A. Black et al., Advan. Cryog. Eng. 5, 181 (1960). 84. A. S. Gilcrest and J. L. Fick, Thermal Problems Related to the Utilization of Highly Anisotropic Multilayer Insulation Systems. AIAA Paper No. 65-120. Amer. Inst. Aeronaut. Astronaut., New York (1965). 85. G. R. Cunnington and C. A. Zierman, Proc. Conf. Therm. Conductivity, 5th, Denver p-IV-c. Univ. o f Denver, Coll. of Eng. (1965). 86. L. B. Golovanov, Proc. Int. Cryog. Eng. Conf., 2nd p. 117. ILIFFE Second Tech. Publ., Surrey, England (1968). 87. S. J. Babjack et al., Planetary Vehicle Thermal Insulation Systems. Final Rep., JPL Contract 951537. Gen. Elec. Co., Rep. DJN: 68SD4266 (1968). 88. D. V. Hale and G. D. Reny, Advan. Cryog. Eng. 15, 324 (1970). 89. M. B. Hammond, Jr., Advan. Cryog. Eng. 16, 143 (1971). 90. R. M. Christensen et al., Advan. Cryog. Eng. 5, 171 (1960). 91. Anonymous, LNG: Low temperature giant of the seventies. Cryog. Ind. Gases 5(1), 33 (1970). 92. P. G. Prater, Cryog. Znd. Gases 5 ( 5 ) , 15 (1970). 93. A. T. Jeffs, Znt. Cryog. Eng. Conf.,2nd p . 73. ILIFFE Second Tech. Publ., Surrey, England (1968). 94. D. H. Tantam and J. Robb, Proc. Znt. Cryog. Eng. Conf., 2nd p. 147. ILIFFE Second Tech. Publ., Surrey, England (1968). 95. R. W. Vance, Proc. Int. Cryog. Eng. Conf.,2nd p. 43. ILIFFE Second Tech. Publ., Surrey, England (1968). 96. D. L. Dearing, Advan. Cryog. Eng. 11, 89 (1966). 97. G. R. Fox and J. T. Bernstein, Mech. Eng. 92(8), 7 (1970). 98. F. E. Mack and M. E. Smith, Advan. Cryog. Eng. 16, 118 (1971). Znt. Cryog. Conf., 1st p . 34. Heywood-Temple Ind. 99. R. S. Mikhalchenko et al., PYOC. Publ., London (1 967). 100. R. P. Caren, A. S. Gilcrest, and C. A. Zierman, Advan. Cryog. Eng. 9, 457 (1964). 101. Y. A. Selcukogh, Proc. Znt. Cryog. Eng. Conf., 2nd p. 131. ILIFFE Second Tech. Pub., Surrey, England (1968). 102. C. C. Hanke, Cryog. Technol. 5, 213 (1969). 103. D. R. Hauser, Cryog. Ind. Gases Sept./Oct., p. 19 (1970). 104. V. E. Isakson, C. D. Holben, and C. V. Fogelberg, Advan. Cryog. Eng. 3,232 (1958). 105. D. 0. Murray, Advan. Cryog. Eng. 13, 680 (1968). 106. P. J. Murto, Advan. Cryog. Eng. 7, 291 (1962). 107. R. P. Caren and R. M. Coston, Advan. Cryog. Eng. 13, 450 (1968). 108. R. T. Parmley, D. R. Elgin, and R. M. Coston, Advan. Cryog. Eng. 11, 16 (1966).

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Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed.

A Abel, W. T., 118(8, 9), 177 Abramowitz, M., 374(42), 415 Abramson, P., 259(104), 271 Acrivos, A,, 296, 346 A d a m , N . K., 184, 185(3), 186, 188, 189(3), 198(3), 215, 268 Adam, O., 134, 179 Adams, D. E., 317(80), 347 Adams, J. A., 304(60, 61), 347 Ainley, D. G., 97, 98, 106 Akers, W. W., 259(106, 107), 265(134), 271, 272 Albers, J. A., 259(105), 271 Alcock, J. F., 37, 106 Ananier, E. P., 259(109), 260, 271 Andreescu, A., 128(34), 178 Androulakis, J. G., 394(71), 416 Andvig, T. A., 3(3), 66(3), 93(3), I06 Armaly, B. F., 361(20), 364(33, 34), 382(20), 384(20), 385(56), 386(34), 387 (33), 394(56), 397(56), 399(33), 414, 415, 416 Arpaci, V. S., 216(40), 223(40), 269

B Babjack, S. J., 417 Babrov, H., 73(53), 76(53), I08 Badger, W. L., 265(139), 272 Bae, S., 259(108), 260, 271 Baer, E., 236(68), 237(68), 240(70), 243,270 Baker, D. J., 295, 346 Bakhtiozin, R. A., 119, 129(10), 177

Baklastov, A. M., 265(135), 272 Balzhiser, R. E., 218, 269 Bankoff, S. G., 262(123), 271 Barden, P., 3(22), 66(22), 93(22), 95(22), 96(22), 107 Barnacle, H. E., 135, 156, I79 Barrow, H., 73(55), 108 Barry, R. E., 218, 269 Bayley, F. J., 3(9, 20), 29, 33, 34, 35, 36,42, 45(8), 71, 77, 78, 84, 85, 86, 87, 96, 97(5, 20), 102, 106, 107 Beattp, K . O., 261(115), 271 Beckman, P., 363(28), 415 Belichenko, N. P., 103, 106 Bell, G. A , , 364(33), 386(58), 387(33), 394(58), 399(33), 401(58), 402(58), 41.5, 416 Bell, N., 85, 86, 96, 97(5), 106 Berenson, P. J., 261( 113), 271 Berggaum, J. B., 379(53), 386(53), 389(53), 394(53), 397(53), 416 Bergles, A. E., 21 1(37), 269 Bernstein, J. T., 405(97), 417 Bewilogua, L., 3(10, 1t), 82(10, I l ) , 83, I06 Biggs, R. C . , 7, 106 Birmingham, B. W., 350(4), 352(4), 394(4), 414 Birt, D. C. P., 244(73), 261(73), 270 Black, I. A., 401(83), 417 Bluman, D. E., 118(8, 9), 119, I77 Boarts, R. M., 265(139), 272 Boehm, R. F., 364(31), 385(31), 386(31), 390(64, 65), 393(64, 65), 395(64), 415, 416 Bonilla, C . F., 251(79), 270

419

420

AUTHORINDEX

Boothroyd, R. G., 132, 167, I79 Bovenkirk, M. P., 376(44), 415 Bowen, W., 132, 179 Boyce, B. E., 3(23), I07 Boyko, I,. D., 259(109), 260(109), 261, 271 Brand, R. S., 302, 347 Brandon, C. A., 125, 127, 167, 178 Braun, W. I-I., 296, 346 Brdlik, P. M., 265(142), 272 Briggs, D. G., 311(71), 347 Briller, R., 123, 124, 125, 167, 178 Brodowicz, K., 300(46), 301, 347 Bromley, L. A., 248(74), 261, 270, 271 Brotz, W., 127(28), 178 Brown, A. H. O., 66(15), 106 Brown, A. R., 232(65), 270 Brown, T. W. F., 3(14), 65, 71(14), 91(14), 93(13, 14), 95, 96, 97(14), 106 Brunt, J. J., 244(73), 261(73), 270 Bundy, F. P., 376(44), 415

C Capatu, C., 128(34), 178 Captieu, M., 103, 106 Cardner, D. V., 304(63), 347 Caren, R.P., 351(7), 363(27), 390(7,27, 66), 393(54, 66), 406(100), 409(7), 41 1(107), 414, 415, 416, 417 Carlson, R., 117(5), 129(5), I77 Case, K. M., 363(26), 415 Cavallini, A,, 261(1 lo), 271 Cess, R. D., 294(18), 346,362(21), 363(21), 364(21), 365(21), 377(21), 379(21), 396 (21), 398(21), 415 Chan, C. K., 388(59), 391(59), 394(72), 398(72), 416 Chang, K. I., 265(147), 272 Chao, B. T., 134, 135, 144, 179, 227, 269 Chapman, A. J., 262(122), 271 Chato, J. C., 73(16, 17), 75, 79, 107, 253 (83), 270 Cheesewright, R., 291, 346 Chen, J. C., 378(48), 379(48), 415 Chen, M. M., 250(78), 251, 252, 257, 270 Chiu, S. C., 363(26), 415 Choi, H., 268 Christensen, R. M., 404(90), 417 Chu, N. C., 130, 132, 171, 172, I78

Chu, P. T., 25, 35, 107 Chung, P. M., 256(90), 270 Churchill, S. W., 304(65, 66), 347, 378(47, 48), 379(48), 415 Cimick, R. C., 144(67), 147(67), 156(67), 179 Citakoglu, E., 231(63), 264(63), 266, 269 Claassen, L., 296(35), 346 Clark, J. A,, 256, 270 Coates, N. H., 119, I77 Cohen, H., 3(20), 77, 78, 84, 87, 96, 97(20), 107 Colak-Antic, P., 322(84, 85), 348 Colclough, C. D., 3(21, 22), 66(21), 93 (21, 22), 95, 96(21, 22), 107 Collier, J. G., 3(23), 107 Cooper, M. G., 359(16), 414 Corruccini, R. J., 357, 358(12), 414 Coston, R. M., 384(55), 411(107), 413(108), 416, 417 Cramer, E. R., 132, I79 Crandall, S. H., 324(93), 348 Crane, L. J., 299(42), 347 Cravalho, E. G., 363(25, 27), 364(25), 390(27), 415 Crawford, J. E., 265(134), 272 Cresswell, D. J., 38, 70(88), I09 Crosser, 0. K., 259(106), 271 Cruddace, R. G., 299(43), 347 Cunnington, G. R., 351(7), 364(33), 367 (38), 375(43), 386(58), 387(33), 388(59), 390(7, 63), 391(59), 394(58), 399(33), 401(58), 402(58, 85), 409(7), 414, 415, 416, 417 Curievici, I., 128, 129, 178 Czekanski, J., 34, 71, 106

D Danziger, W. J., 121, 125, 178 Darling, R., 3(22), 66(22), 93(22), 95(22), 96(22), I07 Davey, A., 324,348 Davies, C. N., 126, I78 Davies, G. A., 227(58, 59), 229(58, 59), 231, 269 Davies, J. T., 184, 185, 198(4), 199(4), 202(4), 268 Davies, T. H., 2, 3, 4, 73, 74, 98, I07

42 1

AUTHORINDEX Davis, S. H., Jr., 265(134), 272 Deans, H. A., 259(106), 271 Dearing, D. L., 405(96), 406(96), 407(96), 41 7 deForge Deelman, A. S., 3(23), 107 Del Casal, E., 296(34), 304(59), 343(34), 346,347 Demetri, E. P., 199(14), 200(14), 201(14), 203(14), 268 Denny, V. E., 258(98), 265(133), 270, 272 den Ouden, C., 46,47, 48(104), 107, 110 Dent, J. C . , 262(125), 271 Depew, C. A., 122, 127(18, 20), 129, 130, 132, 161, 169, 172(19), 178, 179 Diaguila, A. J., 3(36), 9, 38 71(36), 91(36), 93, 94, 95, 101, 107 Dickson, P. F., 399(77), 416 Dieperink, G. W., 46, 47, 48(104), 107, I10 Dillard, D. S., 360(19), 414 Dinulescu, H., 128(35, 36), 129, 178 Dmitriev, B. A., 3(27), 107 Doig, I. D., 137, 145, 156, 179 Domoto, G. A., 364(31), 38.5, 386(31, 3 3 , 390(57, 6 9 , 391(68), 393(65, 68), 407 (57), 415, 416 Donaldson, I. G., 3(28), 63, 64, 107 Dring, R. P., 311(72), 317(81), 328(99, IOO), 330(99), 347, 348 Dukler, A. E., 262(128), 271 Dunkle, R. V., 391(67), 416 Dupre, A., 188(6), 268 Dussourd, J. L., 143, I79

E Eckert, E. R. G., 11, 12, 37, 38, 91(106), 97, 107, 110, 317(77), 322(83), 333, 347, 348 Ede, A. J., 288, 346 Edwards, D. K., 394(75), 395(75), 416 Edwards, J. P., 97, 107 Eichhorn, R., 54, 110, 140, 147, 179, 294(17), 311(68), 346, 347 Elgin, D. R., 413(108), 417 Ellerbrock, H., 3(33), 107 El-Saiedi, A. F. I., 134, 144, 179 El-Wakil, M. M., 258(97), 270 Emmons, H., 206, 269 Erik, S., 262(126), 263(126), 271

Eucken, A., 205, 268 Everaarts, D. H., 49, 102, 107 Eyring, H., 218(46), 269

F Farbar, L., 114, 116, 122, 125, 127(1, 18, 20), 129, 177, 178 Fatica, N., 204, 233(21), 236, 238, 243, 268 Feder, J., 202, 268 Fedorovich, E. D., 222(51), 269 Fick, J. L., 402(84), 417 Finston, M., 286(2), 346 Fishenden, J., 296(32), 346 Fisher, S. A., 3(114), I10 Fletcher, L. S., 390(61), 391, 416 Flood, H., 202, 268 Florschuetz, L. W., 227, 269 Fogelberg, C. V., 408(104), 417 Forsberg, C. H., 364(35), 386(35), 390(57), 409(57), 415, 416 Forslund, R. P., 259( 104), 271 Forstrom, R. J., 300(47), 301, 347 Foster, C. V., 5 , 11, 29, 30, 107 Fox, G. R., 405(97), 417 Fox, H. W., 190, 227(8), 229(8), 268 Foyle, R. T., 11, 30, 107 Fragstein, C., 362(23), 415 Frankel, N. A., 262(123), 271 Fransen, J. W. M., 102, 107 Freche, J. C., 3(36), 9, 38, 71(36), 91(36), 93, 94, 95, 101, I07 Frenkel, J., 190(9), 192, 199(9), 268 Fried, E., 359(17), 414 Friedrich, R., 3(37), 91(37), 93, 94, 107 Fries, P., 3(38), 107 Fujii, T., 299(44), 302, 347 Fulk, M. M., 378(50), 381(50), 407(50), 415

G Gabel, R. M., 3(39, 40, 41), 91(39, 40, 41), 95, 97, 107, 108 Gacesa, M., 261(117), 271 Galli, A. F., 119(9a), 177 Garrett, W. D., 183(2), 199(2), 268 Gasterstaedt, J., 135, 179

422

AUTHORINDEX

Gebhart, B., 283(1), 286(1), 289, 290(12), 292(12, 15), 297(36), 299(45), 300(45,48), 301, 304(55, 67), 305(45), 310(67), 311(70, 72, 73), 312(75, 76), 317(80, 81), 320(73), 324, 326(36, 9 3 , 327, 328, 330(98, 99), 332, 333, 334, 335(90), 339(102), 342(102), 343, 346, 347, 348 Gtnot, J., 3(42, 44), 71, 79, 80, 91(44), 96, 97(42, 44), 103, 108 Gerretsen, J. C. R., 102, 110 Gerstmann, J., 255, 270 Gifford, W. E., 3(43), 82, 108 Gilcrest, A. S., 402(84), 406(100), 417 Gill, A. E., 292, 324, 346, 348 Gill, W. N., 296(34), 304(59), 343, 346, 347 Ginwala, K., 258, 262(103), 271 Glaser, P. E., 351(6), 352(6), 390(6), 414 Glassford, A. P. M., 399(78) 404(78), 416 Goglia, G. L., 199(13), 200, 201, 268 Goldstein, R. J., 31 1(71), 317(77), 347 Golovanov, L. B., 402(86), 417 Gomelauri, V. I., 258(99), 271 Gorbis, Z . R., 119, 129(10), 167, 177, 180 Gose, E. E., 236(68), 237, 270 Gosman, A. D., 99, 100, 108 Greebler, P., 369(39), 415 Gregg, J. L., 248, 249, 251(76), 261(114), 270, 271, 286(3), 287(5), 303(54), 325(5), 346, 347 Griffith, P., 204(19), 206(19), 207, 210, 231(64), 232(64), 233(64), 241, 255, 268, 270 Grigull, U., 11(106), 91(106), 110, 207(30), 262(126), 263(126), 269, 271 Grizzle, T. A., 125(27), 127, 178 Grober, H., 262, 263, 271 Grosh, R. J., 296, 346 Gunness, R. C., Jr., 334, 348 Gyorog, D. A., 390(61), 391(61), 416

H Habetler, G., 73(53), 76(53), 108 Hahne, E. W. P., 91, 108 Hahnemann, H., 41, 108 Hale, D. V., 404(88), 417 Hallman, T. M., 169, 180 Hamilton, D. C., 73(47), 76(47), 108 Hammitt, F. G., 3(48), 25, 35, 107, 108

Hammond, M. B., Jr., 404(89), 417 Hampson, H., 265(138), 272 Hanke, C. C., 407(102), 417 Hanna, 0. T., 304(62), 347 Hanratty, T. J., 156, 180 Hansen, A. G., 296, 346 Hare, E. F., 231(62), 269 Hartnett, J. P., 20, 32, 33, 35, 40, 108, 109, 250(77), 251(77), 261, 270, 271 Hasegawa, S., 9, 10(52), 21(51, 52), 23, 36, 37, 99, 108 Hassan, K. E., 256(91), 270 Hauser, D. R., 407(103), 417 Hawes, R. I., 129, 178 Hawkins, G. A., 350(1), 414 Heighway, J. E., 296(28), 346 Hellman, S. K., 73(53), 76(53), 108 Hellums, J. D., 304(63), 347 Hering, R. G., 296, 346 Hewitt, H. C., 227(57), 269 Hiby, J. W., 127(28), 178 Hieber, C. A., 324, 328(91), 332(91), 334(91, 92), 348 Hil1,P. G., 199, 200(14), 201, 202(16), 203, 268 Hinkle, B. L., 137, 179 Hoffman, E. J., 258(101), 271 Hohnstreiter, G. F., 144(67), 147(67), 156(67), 179 Holben, C. D., 408(104), 417 Holland, E., 129(37), 178 Hollwegen, D. J., 365(36), 372(36), 415 Holmes, R. E., 262(122), 271 Holzwarth, H., 3(54), 11, 91(54), 108 Hottel, H. C., 362(22), 363(22), 364(22), 365(22), 415 Hughes, D., 265(146), 272 Humphreys, H. W., 298(40), 347 Humphreys, J. F., 73, 108 Humphreys, R. F., 261(118), 271 Hunter, B. J., 378(50), 381(50), 407(50), 415 Hurlbut, F. C., 358(15), 414 Husar, R. B., 295(23), 345, 346, 348

I Ibele, W. E., 253, 270 Isachenko, V. P., 237(69), 238(69), 270 Isakson, V. E., 408(104), 417

AUTHOR INDEX Ivanov, V. L., 95, 101, 103, 108 Ivanovskii, M. N., 241(71), 242, 255(71), 270 Ivey, R. K., 358(14), 408(14), 414

J Jablonic, R. M., 71(117), 104(117), 111 Jackson, T. W., 11, 12,37, 38,97,107,108 Jacobs, H. R., 258(96), 270 Jaeger, H. L., 202(16), 268 Jagannathan, P. S., 361(20), 382(20), 384(20), 385(56), 394(56, 69, 72, 73), 397(56), 398(70, 72, 73), 414, 416 Jakob, M., 206, 220(26), 256(91), 258, 268, 269, 270, 350(1), 414 Jallouk, P. A., 49, 50(64), 51(64), 53, 54(58, 64), 55(64), 56(64), 57(64), 60, 62(64), 67, 69(64), 70, 71, 103(64), 108, 109 Japikse, D., 3(65), 5, 10(63), 12(59), 26, 27, 28, 29, 30, 31, 35, 36, 46, 48, 49, 50, 51(59, 64), 52, 53, 54(64), 55(64), 56, 57(64), 59, 62(60, 64), 65, 66, 67, 69(59, 64), 70(59, 64), 71, 91(65), 92(65), 100, 103, 104, 108, 109 Jeffs, A. T., 405(93), 417 Jepson, G., 127(29), 178 Jew, H., 24, 109 Johnson, C . L., 365(36), 372(36), 401(81), 415, 416 Johnson, H. A , , 317(79), 347 Johnson, V. J., 415 Johnston, G. H., 3(66), 109 Johnstone, R. K. M., 220(47), 269 Jones, M. C . , 364(32), 386(32), 399(77), 415, 416 Jusionis, V. J., 265(133), 272

K Kada, H., 156, 180 Kaganer, M. G., 351(5), 352(5), 390(5), 399(5), 414 Kappler, G., 3(11), 82(11), 83, 106 Katsuta, K., 207(31), 210(31), 242, 269 Katz, D. L., 204, 233(21), 236, 238, 243, 268

Kaufman, A., 3(67), 97(67), 109 Keller, C. W., 386(58), 394(58), 401(58), 402(58), 416 Kemeny, G. A., 3(68), I09 Kerimov, R. V., 120, 129(12), 178 Kezios, S. P., 261, 271 Khanna, R., 3(23), 107 Kierkus, W. T., 300(46), 301, 347 Killian, E. S . , 243(72), 270 Kirby, G. I., 129(37), 178 Klein, J. D., 378(49), 379(49), 415 Klemens, P. G., 360(18), 361(18), 414 Klett, D. E., 358(14), 408(14), 414 Knoner, R., 3(10, 11, 69), 82(10, 11, 69), 83, 106, 109 Knowles, C . P., 324(95), 326(95), 327, 328, 330(98), 332, 333, 348 Koh, J. C. Y., 250, 251, 270 Kosky, P. G., 229(61), 269 Kozhinov, I. A., 265(141), 272 Kramer, T. J., 132, 137, 138, 142, 143, 144, 150, 155, 156, 163, 170, 172, 179 Kroger, D. G., 218, 222, 255(42), 264(131), 269, 272 Kropschot, R. H., 350(4), 352(4, 9), 378 (50), 381(50), 394(4), 399(9), 404(9), 407(50), 414, 415 Kruzhilin, G. N., 259(109), 260(109), 261, 271 Kuiken, H. K., 288, 346 Kulhavy, J. T., 3(70), 75(70), 109 Kunes, J. J., 3(71), 75, 109 Kurtz, E. F., 324(93), 348

L Lahey, F. J., 302, 347 Lancet, R. T., 259(104), 271 Lapin, Yu. D., 73, 95(55c), 97(72), 101 (55b, 55c), 108, 109 Larkin, B. K., 378(47), 415 Larkin, B. S., 3(73, 74), 76(73), 87, 88, 89, 109 Larsen, F. W., 32, 33, 35(50), 40, 108, I09 Laufer, J., 126, 148, 178 Lawrence, W. T., 73(17), 75, 107 Lebedev, P. D., 265(135), 272 Lee, J., 262(127), 263, 271 Lee, M. S., 231(64), 232(64), 233(64), 270 Lee, Y., 3(76), 80, 82, 83, 109

424

AUTHORINDEX

LeFevre, E. J., 210(35, 36), 232(36), 240, 244(36), 265(36), 269, 288, 346 IeGrivks, E., 3(42, 44), 71, 79, 80, 91(44), 96, 97(42, 44), 103, 108 Leppert, G., 256(87), 270 Leslie, F. M., 20, 21(78), 26, 27, 34, 39, 71(78), 109 Lewis, E. W., 256, 270 Lieblein, S., 118, 177 Lienhard, J. H., 357(13), 358(13), 361(13), 384(13), 414 Lightfoot, E. N., 304(64), 347 Lighthill, M. J., 9, 10, 11, 12, 15, 18, 19, 20, 22, 23, 27, 31, 36, 41, 42, 43, 46, 49(80), 99, 109 Lin, C. C . , 126(31), 178 Lin, S. H., 265(140), 272 Linehan, J. H., 258(97), 270 Linetskiy, V. N., 256(92), 270 List, R., 222(52), 225(52), 269 Liu, V. C., 24, 109 Livingood, J. N. B., 11, 97, 108 Lloyd, J. R., 345(107), 348 Lock, G. S . H., 3, 5 , 14, 29, 42, 44, 45, 54, 58, 59, 60, 102, 106, 109 Lockwood, F. C., 21(83), 24, 25, 26(90), 36, 52, 70, 99(43a), 100(43a), 108, 109 Long, E. L., 3(84), 84, 109 Lorenz, J. J., 205(23), 268 Lothe, J., 202, 268 Lowell, R. L., 304(60, 61), 347 Lucas, H. G., 119, 177 Luikov, A. V., 357(10), 373(41), 414, 415 Lurk, H., 317(79), 347 Lynch, F. E., 73(47), 76(47), 108

M Mabuchi, I., 294(19), 346 McAdams, W., 182(1), 255, 268 McCarthy,H. E., 137, 156, 161, I79 McCormick, J. L., 207(29), 208, 210(29), 211, 214, 232(29), 234, 235, 236, 239, 240(70), 243, 265(33), 269, 270 Macdougall, G., 190(7), 268 McGee, J. P., 118(8), 177 MacGregor, R. K., 390(62), 396(74), 399(74), 416 Mack, F. E., 406(98), 417

Macosko, R. P., 259(105), 271 Madden, A. J., 304(57), 347 Madejski, J., 3(85), 76(85), 109 Mahony, J. J., 302(49), 347 Malloy, J. F., 350(2), 352(2), 414 Manganaro, J. L., 304(62), 347 Mann, D. B., 350(4), 352(4), 394(4), 414 Manushin, E. A., 95(55c), 101(55c), 108 Marschall, E., 264(129), 265(142), 271, 2 72 Martin, B. W., 3(9, 92), 9, 10(87), 11, 17, 18(87), 21(83, 87), 22, 23, 24, 25, 26, 27, 29, 35, 36, 38, 39, 46, 52, 55, 70, 99, 100, 106, 109 Martin, J. H., 317(78), 347 Mathers, W. G., 304(57), 347 Matin, S. A., 253(82), 270 Maulbetsch, J. S., 259(108), 260(108), 271 Mehta, N. C., 156, 179 Meisenburg, S. J., 265(139), 272 Menold, E. R., 311(69), 347 Merte, H., Jr., 256(88), 270, 296, 346 Metiu, H., 215(38), 264(38), 269 Mihail, A., 3(97), 110 Mikhalchenko, R. S., 406(99), 417 Mikic, B. B., 205(23), 233(66), 242(66), 268, 270, 359(16), 414 Mikielewicz, J., 3(85), 76(85), 109 Miles, R. G., 253(80), 270 Miller, E. N., 121, 178 Miller, J. H., 262(121), 271 Mills, A. F., 221, 258(98), 265(133), 269, 270, 272 Millsaps, K., 303(51, 52), 347 Milne, P. A., 29(6), 34, 106 Milovanov, Y . V., 241(71), 242(71), 255 (71), 270 Min, K., 134, 144, 179 Minkowycz, W. J., 258(94, 9 3 , 265(94), 270 Misra, B., 251(79), 270 Mital, U., 80, 82, 83, 109 Mitchell, R., 3(93), 109 Mojtehedi, W., 227(59), 229(59), 269 Mollendorf, J. C., 289, 290(12), 292(12), 31 1(73), 320(73), 334(73), 339(103), 346, 347, 348 Moorhouse, W. E., 3(94), 90, 109 Morley, M. J., 114, 116, 125, 127(1), 177

425

AUTHORINDEX Morris, W. D., 2, 3, 4, 73, 74, 98, 107, 108, 109, I10 Morse, H. L., 141, 179 Mortenson, E. M., 218(46), 269 Moskowitz, S. L., 3(41), 91(41), 95(41), 97(41), 108, 110 Mostinskiy, I. L., 265(136), 272 Moussez, C., 3(97), 110 Mucciardi, A. N., 236(68), 237(68), 270 Muller, K. G., 127(28), 178 Murray, D. O., 409(105), 417 Murray, W., 261(118), 271 Murto, P. J., 410(106), 417 Musse, S., 103, 106 Myers, J. A., 258(100), 271 N Na, T. Y., 296, 346 Nachtsheim, P. R., 324(94), 348 Nandapurkar, S.S., 261(115), 271 Navon, U., 140(62), 179 Nicol, A. A,, 261(117), 271 Nimmo, B., 256(87), 270 Nishikawa, K., 9(52), 10(52), 21(51, 52), 23(52), 36(51, 52), 37(52), 99(52), I08 Norman, J. R., 127(30), 178 Nosov, V. S., 119(1I), 178 Nusselt, W., 218, 244, 256, 258, 269

0 O’Bara, J. T., 243(72), 270 Ockrent, C., 190(7), 268 Ogale, V. A., 3(93, 98), 36, 66, 71(98), 93(98), 95, 96(98), 97(98), 109, 110 O’Leary, J. P., 118(8, 9), 177 Olson, J. H., 137, 156, 161, 179 Omelyuk, V. A., 103, 104, 110 Orr, C., Jr., 171, 180 Ostrach, S., 20, 63, 110, 288, 296(28), 346 Otto, E. W., 186(5), 268 Ozisik, M. N., 265(146), 272

Parker, J. D., 227(57), 269 Parmley, R. T., 413(108), 417 Pepov, K. M., 103(9a), 106 Pera, L., 297(36), 299(45), 300(45), 304(55, 67), 305(45), 310(67), 324, 326(36), 335(90), 339(102), 342(102), 343, 346, 347, 348

Peretz, D., 128(34, 3 9 , 129(35), 178 Peshin, R. L., 135, 179 Peskin, R. L., 123, 124, 125(25), 150, 170, 178, 179

Peterson, A. C., 208(32), 238(32), 239, 240, 269

Petrick, M., 258(97), 270 Petrov, N. G., 265(141), 272 Pettyjohn, R. R., 370(40), 372(40), 415 Pfeffer, R., 118, 177 Piret, E. L., 304, 347 Plapp, J. E., 323, 324(87), 348 Pogson, J. H., 390(62), 396(74), 399(76), 416

Pohlhausen, K., 303(51, 52), 347 Poljak, M. P., 71(117), 104(117), 111 Poll, A., 127(29), 178 Polymeropoulos, C . E., 317(81), 327(96), 328(97), 332, 333, 347, 348 Ponter, A. B., 227(58, 59), 229(58, 59), 231, 269

Poots, G., 253(80), 270 Pope, D., 209(48), 210(34), 221(48), 233(34), 244(48), 264(130), 265(34, 48, 130), 269, 271 Potter, C. J., 209(48), 210(34), 221(48), 233(34), 244(48), 264(130), 265(34, 48, 130), 269, 271 Pound, G. M., 202, 268 Prater, P. G., 405(92), 417 Preckshot, G., W. 261(116), 271 Prins, J. A., 296, 346 Pucci, P. F., 102, 110 Puzyrewski, R., 224(53), 269

Q Quan, V., 122, 127(21), 129(21), 178

P Palanikuman, P., 80, 110 Palmer, D. C., 364(32), 386(32), 415 Palmer, L. D., 73(47), 76(47), 108

R Rajpaul, V. K., 130, 175, 178 Regalbuto, J. A., 144, 179

426

AUTHORINDEX

Reny, G. D., 404(88), 417 Rich, B. R., 296(31), 346 Rideal, E. K., 184, 185, 198(4), 199(4), 202(4), 268 Rohh, J., 405(94), 417 Robinson, A. F., 3(101), 93, 94, I10 Roblee, L. H. S., 243, 270 Rohsenow, W. M., 211(37), 218, 220, 222, 249(75), 255(42), 259(108), 260(108), 264(131), 268, 269, 270, 271, 272 Rolling, R. E., 362(24), 415 Romanov, A. G., 102, 103(9a), 106, I10 Roper, C . H., 137, 145, 156, 179 Rose, H. E., 135, 156, 179 Rose, J. W., 210(35, 36), 231(63), 232(36), 240, 244(36), 264(63), 265(36, 132), 266, 269, 272 Rosenecker, C. N., 119, 177 Rossetti, S., 118, 177 Rosson, H. F., 258(100), 259(107), 271 Rotem, A., 296(35), 346 Rouse, H., 298, 347 Roy, S., 292(16), 346 Ruckenstein, E., 21 5(38), 264(38), 269 Rufer, C. E., 261, 271 Runcorn, S. K., 3(102), 110 Russell, D. J., 390(62), 416 Russell, K. C., 202(16, 18), 268 Ryan, J. M., 401(82), 417 S

Saddy, M., 258(94), 265(94), 270 Sadek, S . E., 265(143), 272 Sarofim, A. F., 362(22), 363(22), 364(22), 365(22), 415 Saunders, 0. A., 16, 44, 110, 296(32), 322(82), 346, 347 Saville, D. A., 304(65, 66), 347 Schenk, J., 46(104), 47, 48, 110 Schetz, J. A., 311(68), 347 Schlichting, H., 28, I10 Schmidt, E., 3(107), 11, 77, 87, 91, 93, 94, 110

Schneider, W. F., 101, 110 Schoher, T. E., 3(41), 91(41), 95(41), 97(41), 108 Schorr, A. W., 299(45), 300(45, 48), 301, 305(45), 347 Schrage, R. W., 216(39), 217, 218, 269

Schrodt, J. E., 378(50), 381(50), 407(50), 415

Schuderherg, D., 117(5), 129(5), I77 Schuh, H., 288, 298(6), 302(6), 346 Schuster, J. R., 261(113), 271 Scott, R. B., 350(3), 352(3), 401(3), 414 Scriven, L. E., 234, 270 Sehan, R. A., 221, 254, 269, 270, 379(53), 386(53), 389(53), 394(53), 397(53), 416 Selcukogh, Y. A., 407(101), 408(101), 417 Selin, G., 255, 257, 272 Sergazin, 2. F., 265(135), 272 Sevruk, I. G., 299(41), 347 Shafrin, E. G., 229, 230, 269 Shanny, R., 140(62), 179 Shekriladze, I. G., 258(99), 271 Shelton, J. T., 244(73), 261(73), 270 Shen, S. F., 294(20), 346 Shetz, A,, 54, 110 Sheynkman, A. G., 256(92), 270 Siegel, R. S., 169, 180, 256(89), 270, 312 (74), 347 Silver, R. S., 258, 265(102), 271 Singer, R. M., 261, 271 Slegers, L., 254, 270 Sleicher, C. A., Jr., 174, 180 Slobodyanyuk, L. I., 103, 104, 110 Small, S., 147, 179 Smith, M. E., 406(98), 417 Smith, W., 127(29), 178, 220(47), 269 Smuda, P. A., 390(61), 391(61), 416 Soehngen, E., 322(83), 333, 348 Soliman, M., 261, 271 Somers, E. V., 3(68), 109, 304(56), 347 Soo, S. L., 121, 135, 144, 145, 147, 156, 178, 179 Spalding, D. B., 299(43), 347 Sparrow, E. M., 169,180, 248,249,250(77), 251(76, 77), 256(89), 258(94, 9 9 , 261, 264(129), 265(94, 140), 270, 271, 272, 286(3), 287(5), 294(18), 295(23), 300(47), 301, 303(54), 325(5), 345, 346, 347, 348, 361(21), 363(21), 364(21), 365(21), 377 (21), 379(21), 396(21), 398(21), 415 Spencer, D. L., 253, 265(147), 270, 272 Spencer, J. D., 119(9a), 177 Spizzichino, A., 363(28), 415 Springer, G. S., 357(11), 358(11), 414 Stachiewicz, J. W., 7, 106 Stappenbeck, A., I10

427

AUTHORINDEX Stegun, I. A., 374(42), 415 Steigelmann, W. H., 118(6), 177 Stern, F., 265(144), 272 Stewartson, K., 296(33), 343, 346 Stoddart, D. E., 29(6), 34, 106 Stone, A. A., 3(111), 75(111), 110 Strong, H. M., 376(44), 415 Stuart, J. T . , 323, 348 Subbotin, V. I., 241(71), 242(71), 255(71), 2 70 Sucio, S. N., 3(112), 91(112), 97, 110 Sugawara, S., 207(31), 210(31), 242, 269 Sukhatme, S. P., 218, 220, 269 Sukomel, A. S., 120, 129(12), 178 Syromyatnikov, N. I., 119(11), 178 Szewcyzk, A. A., 324(88), 348

Trezek, G. J., 144(67), 147(67), 156(67), 179 Tribus, M., 174(77), I80 Trotter, D. P., 144, 179 Tseitlin, L. M., 96, 103(112b), 110 Tsvetkov, F. F., 120, 129(12), 178

U Umur, A., 204(19), 206(19), 207, 210, 241, 268 Uskov, I. B., 71(117), 96, 103(112b), 104 (1 17), 110, 111

V T Tabbey, J., 3(40), 91(40), 95(40), 97(40), 107 Taitel, Y . , 265, 272 Takata, K., 224(54), 269 Takhar, H. S., 295(22), 346 Tamir, A., 265(145), 272 Tanner, D. W., 209, 210(34), 221(48), 233(34), 244(48), 264(130), 265, 269 271 Tantam, D. H., 405(94), 417 Tatchell, D. G., 99(43a), 100(43a), 108 Tchen, C . M., 135, 179 Thomas, D. G., 132, 134, 179 Thomas, M. A., 232(65), 270 Thomson, W., 192, 268 Thornton, P. R., 20, 63, 110 Tien, C. L., 121, 122, 127(21), 129(17, 21), 167, 172(17), 178, 180, 351(8), 357(13), 358(13), 361(13, 20), 362(24), 363(25, 27), 364(25, 31, 32, 33), 375(43), 377(46), 379(46, 52), 382(20), 384(13, 20), 385(31, 56), 386(31, 32), 387(33), 388(59), 390, 391(59, 60, 68), 393(64, 65, 68), 394(56, 69, 72), 395(64), 397(56), 398(70, 72), 399(33), 414, 415, 416 Timmerhaus, K. D., 360(19), 414 Tkachenko, G. M., 71(117), 103, 104, 110, Ill Tobin, R. D., 394(75), 395(75), 416 Tolman, R. C., 196(11), 268 Trefethan, L., 205(22), 233(22), 268

Vance, R. W., 405(95), 416 van de Hulst, H. C., 363(29), 365(29), 415 Van Dyke, M., 303(53), 347 Van Wylen, G. J., 199(13), 200, 201, 268 van Zoonen, D., 131, 179 Velkoff, H. R., 262(121), 271 Verschoor, J. D., 369(39), 415 Viskanta, R., 379(51), 415 Vizel’, Y. M., 265(136), 272 Vliet, G. C., 384(55), 416 Volmer, M., 202, 268 Votta, F., Jr., 265(144), 272

W Wachtell, G. P., I I8(6), 177 Waggener, J. P., 118(6), 177 Wahi, M. K., 130, 179 Wakeshima, H., 224(54), 269 Wald, A., 159, 180 Walker, L. F., 3(113), 66(113), 110 Waller, P. R., 129(37), 178 Wallis, G. B., 258, 265(102), 271 Walters, S., 104, 110 Wang, D. I., 376(45), 415 Wang, L. S., 377(46), 379(46, 52), 390, 415 Watson, R. G. H., 244(73), 261(73), 270 Welch, J. F., 204, 206(20), 231(20), 268 Welsh, W. E., 20, 33, 35(50), 40(50), 108 Wen, C. Y . , 121, 178 Wenzel, H., 221(50), 269

428

AUTHORINDEX

West, D., 209(48), 210(34), 221(48), 233(34), 244(48), 264(130), 265(34, 48, 130), 269, 271 Westwater, J. W., 204, 206(20), 207(29), 208, 210(29), 211, 214, 231(20), 232(29), 234, 235, 236, 238(32), 239, 240, 265(33), 268, 269 Whitelaw, R., 117(5), 129(5), 177 Wilcox, W. R., 304(58), 347 Wilhelm, D. J., 218, 219(45), 269 Wilkie, D., 3(114), I10 Wilkinson, G. T., 127(30), 178 Willing, H., 199(14), 200(14), 201(14), 203(14), 268 Willson, E. D., 202(16), 268 Wilson, A. H., 364, 415 Wilson, C. T. R., 199(12), 268 Winter, E. R. F., 5 , 10(63), 26, 28, 29, 30, 31, 35, 36, 46, 49, 50(60, 64), 51(64), 52, 53(64), 54(64), 55(64), 56(64), 57(64), 62(60, 64), 66, 67(64), 69(64), 70(64), 71(64), 103(64), 109 Worthington, W. H., 3(115), 75(115), 110

Y Yaffee, M. L., 98, 111 Yamagata, K., 9(52), 10(52), 21(51, 52), 23(52), 36(51, 52), 37(52), 99(52), 108 Yang, J. W., 262(124), 271 Yang, K. T., 253(81), 270, 286(4), 295 (21), 311(69), 312(4), 346, 347 Yih, C. S., 298, 302(38), 347 Yovanovich, M. M., 359(16), 414

Z Zecchin, R., 261(110), 271 Zeh, D. W., 296(34), 304(59), 343(34), 346, 347 Zeldovich, Y. B., 298(37), 347 Zierman, C. A., 402(85), 406(100), 417 Zisman, W. A., 190, 227(8), 229, 230, 231(62), 268, 269 Zuber, N., 227, 269 Zysina-Molodjen, L. M., 71(117), 104, 111

Subject Index A Absorption coefficient, 369 Adiabatic stratification, 278 Aluminized Mylar, 382 Aluminum oxide, 120 Amplification rates, 326 Anomalous-skin-effect, 364, 385 Augmentation of heat transfer, 114

B Body forces, 276 Boltzmann constant, 362 Bond number, 186 Borosilicate glass, 367 Boundary layer equation of, 282 thickness of, 287 Boundary layer flow in thermosyphons, 8, 39 Boussinesq approximation, 277, 279 Bubble formation, 295 Bulk cavitation, 191 Buoyancy, 275 mass-diffusion-induced, 286 normal component of, 296 coupling, 324, 327 combined mechanisms of, 303 Buoyancy force, tangential, 296 Buoyant jet, 282, 297 Burn-out, 79

C Calorimeters, 401 Calorimetry, 400 Capillarity, 186 Catalyst, alumina-silica, 114 Cavity sizes, 213

Charge uniformity, 134 Clausius-Clapeyron equation, 198 Cohesion, work of, 189 Combined buoyancy mechanisms, 303 Compatibility, chemical, 40 Composite structure, 352 Condensation, 81, 87 bulk, 223 correlations for, 238 dropwise, 182, 204 film, 182, 244 in forced convection, 257 in liquid bulk, 226 laminar film, 79, 244 nuclei for, 197 of mixtures, 264 rotating, 261 turbulent film, 262 Condensation coefficient, 219 Condenser, 79 Conduction, 35, 43, 44 Conduction, transient regime, 31 1 Cone flows, 296 Confluence analysis, 159 Contact angle, 229 Containment problems of, in thermosyphons, 40 Convection in stratified media, 290 term, 219 velocity, 277, 325 Convection models, 57, 58 Coolant primary reactor, 1 1 7 Cooling of electrical machine rotors, 3 of gas turbine blades, 3 of internal combustion engines, 3, 75 of nuclear reactors, 3 of transformers, 3

429

SUBJECT INDEX

430

Coriolis effects, 80 Coriolis force, 37, 84 influence on liquid film, 80 Critical point, 93 Critical radius, 196 Critical state, 90 Cryogenics application of, in thermosyphons, 3 Cryogen refrigerator, 410 Cryopumping, 182 Cubic packing, 373 Cylinder, 303

D Debye temperature, 364 Diameter effect in thermosyphon, 30 Diffusion equation, 304, 305 Diode behavior, 82 Disturbance amplification of, 328 amplification characteristics of, 326 amplitude distributions of, 326 symmetric, 335 frequency of, 325 symmetric, 335 temperature, 323 two-dimensional velocity, 323 Doppler shift velometer, 141 Double integral method, 312 Droplet formation, 80 Droplet removal, 234 Dryout, 87, 88

E Eddy diffusivity, 172, 174, 176 Eddy viscosity, 163 Effectiveness of heat exchanger, 120 Electron number density of, 364 relaxation time of, 364 Electrostatic probe, 144 Emissivity, total normal, 386 Energy equations derivatives of, 326 Orr-Sommerfeld, 326 Entrance orifice influence of, on thermosyphon, 24 Equilibrium across a curved surface, 192 Equilibrium film pressure, 227

Evaporation in thermosyphon, 87 Evaporation coefficient, 218 Exothermic regeneration stage, 114 Experimental technique, 136 Exponential growth rate, 339 Extinction coefficient, 377 Extinction index, 363

F Film-fracture mechanism, 206 Filtering mechanism, 332 Finite difference method, 99 Fins, 3, 76 Flat plate isothermal, 283 vertical, 283 Flow back, 243 base, 326 cone, 296 induction, 242 instability of laminar, 321 natural, 335 separation, 335, 342 transients, 310 transition, 321 Flow model development, 145 Flow visualization, 24, 343 light interruption technique, 139 use of fish flakes for, 51 with dye injection, 51 Flue gas, 114 Fluid mechanics of suspensions, 134 Fourier number, 313 Free boundary flows, 274 Free convection, 32, 274 Free energy of formation, 197 Frequency dominant, 334 Friction factors, 175

G Galerkin-Zhukhovitskii variational method, 76 Gas mass velocity, 121 Geothermal power, 63 Glass microspheres, 133

SUBJECT INDEX Glass particles, 1 19 Grashof number, 288, 313, 324 local, 324 temperature-gradient, 303

H Hagen-Rubens relation, 363 Heat capacity of suspensions, 114 Heat exchanger fins, 3, 76 Heat pipes, 3 Horizontal surfaces, 296

I Inclination effects on thermosyphon, 65 Inclined surfaces, 296 Instability general aspects of, 339 hydrodynamic, 322 laminar, 322 thermal, 322 Insulation absorption and scattering in, 369 cryogenic types of, 354 effect of conduction in, 356, 358 evacuated multilayer, 354 evacuated porous, 354 high vacuum, 354 multishield type of, 350 physical properties of, 366 spectral transmission of, 367 super-, 355 thermal diffusivities of, 355 Interfacial temperature, 216 Interferograms, 3 15 Interferometer (Fabry-Perot), 141 Interphase mass transfer, 218 Inviscid asymptotes, 335

J Jets buoyant, 339 nonbuoyant, 339

K Kinetic theory, 218 Knudsen number, 357 Kolmogoroff microscale, 115

43 1 L

Lagrangian integral time scale, 149 Laminar flows instability of, 321 transition of, 321 Laminar sublayer, 123 Laser Doppler velometer, 136 Latent heat, 190 Lead, 122 Leading edge effect, 280, 3 11 propagation of, 321 Leidenfrost boiling, 83, 87 Linear stability theory, 323 Line source, 287 Liquefied natural gas (LNG), 405 Liquid-drop model, 199 Liquid metals, 32, 33 Liquid-vapor interface phenomena, 21 5 Loading ratio, 116 Longitudinal rolls, 345 Low energy surfaces, 229

M Mean free path, 357 of phonons and electrons, 361 Microscale properties, 115 Minimum transport velocity, 132 Mixed convection, 75 Mixing, 43, 44, 57 Mobility, 146 Multichannel systems, 74

N Natural convection, 274 boundary-layer simplifications of, 280 external, 274 governing equations for, 275 internal, 274 Nusselt number for, 288 plumes, 322 similarity solutions, 282 transients, 3 10 Neutron cross section, 117 Noncircular cross section, 36 Noncondensables, 23 1, 264 Nonlinear effects, 336, 341 Non-Newtonian behavior, 296 Nuclear reactor cooling, 114

432

SUBJECT INDEX

Nuclear reactors, 76 Nucleate boiling, 84 Nucleation, 89, 183 in bulk phase, 199 of solid phase, 203 process of, 182 rate of, 202 Nusselt number for natural convection, 288

0 Optical cross-correlation, 138 Optical sensing technique, 145 Optical thickness, 377 Orr-Sommerfeld, 323 Ovens, 3 Overshoot, 313, 314 temperature, 320

axisymmetric flow, 301 boundary conditions of, 299 convected energy, 299 end effects of, 301 forced, 297 instability in, 335 line source, 299 temperatures of, 300 thermal, 300 unperturbed, 337 Porous plate, 294 Prandtl number, 288 limiting values of, 334 Precooling cryogenic equipment, 82 Pressure drop in condensation, 258 Pressure field, 296 Pressure term, 279, 293 Promoters, 210, 231

P Pametrada project, 65, 93 Particle drag law, 155 Particle dynamics, 123 Particle-fluid interaction, 132 Particle-particle interaction, 123 Particle Reynolds number, 124 Particles critical size of, 125 flux, 175 mass flux and density measurements of, 143 number density of, 123 solid spherical glass, 122 Particle slip, I50 Particle velocities, 137, 175 influence of particle size on, 122 measurement of, 137 terminal, 119 Penetration depth, 124 Permafrost, 3, 83 Perturbation methods, 39 Phonons, 361 Planck constant, 362 Planck’s law, 362 Plastic films reflectance of, 388 Pluglike flow, 122 Plume, 282, 297 axisymmetric, 339

R Rayleigh number, 281 Reflective shields, 382, 385 Refractive index, 363 Relaminarization, 321, 334 Relaxation time, 126 Reynolds number, 340 Reynolds stress, 152 Rotating test rig, 38 Roughness effects, 79

S Scattering coefficient, 369 Schmidt number influence of, 308 Separation, 343 flow, 342 Shear layer, 322 Sherwood number, 304 Similarity, 284 for mass diffusion, 285 for thermal transport, 285 variable, 283 Similarity flows, 9, 16, 17, 20, 21 Slip regimes, 358 Smoke filaments, 344 Source distributed, 294

SUBJECT INDEX Spacers, 382 Spectral normal reflectance, 388 Spiral-shaped strips, 117 Spreading coefficient, 189, 229 Stability, 52, 55, 91 Stability, 340 condition for, 291 equations of, 323 experimental values of, 328 limits of, 325 neutral, 324, 327 of laminar flows, 307 plane, 340 Stability limits predicted values of, 324 Stabilizing surface, 342 Stefan-Boltzmann constant, 363 Stefan-Boltzmann law, 362 Stokes stream function, 302 Stratification, 131, 132, 345 adiabatic, 290 effect of, 278 stable, 290 Streaking camera, 137 Stream function generalized, 283 Subcooled liquid drops, 225 Substrate material, 233 Suction, 294 Supersaturation pressure ratio, 194 Surface free energy, 185 Surface latent heat, 187 Surfaces flat, 343 horizontal, 339, 343 slightly inclined, 339 Surface tension, 184 critical value of, 229 Suspension density profiles of, 157 Suspension flow, 128, 171, 175 Swirling flow, 128

T Temperature boundary condition exponential variation in, 289 Temperature jump, 217 Thermal accommodation coefficient, 358 Thermal boundary conditions

433

nonuniform conditions in, 285 power-law variation of, 286, 296 Thermal capacity effects of, 310, 314 parameter, 3 13 surface, 327 Thermal conductivity apparent values of, 353 effect of pressure and void size on, 371 indirect methods for determining, 404 of low temperature solids, 360 Thermal constriction resistance, 359 Thermal coupling effect, 324 Thermal diffusivity measurement of, 404 Thermal entrance region, 115 Thermal triode damping, 83 Thermosyphons, 1, 3 analytical systems, 39 applications of, 3, 83 circular open liquid metal systems, 33 classification of, 3, 4 closed loop, 72, 93, 97, 104 closed systems, 49, 55, 57, 93 condensing, 97 constant wall heat flux in, 20, 35 constant wall temperature in, 102 effects of rotation of, 38, 65, 73 evaporating, 97 exchange mechanism in, 43 favorable pressure gradients in, 28 filling of, 86 governing equations for, 99 inclination effects in, 37, 39, 40 inclined, 67 inclined open systems, 38 influence of filling on, 78, 81 influence of leading edge on, 97 instability in, 34 liquid metal, 101 liquid metals in, 33 maximum heat flux in, 85 mixed convection systems in, 3, 75 noncircular, 36 noncircular closed systems, 46 noncircular cross sections in, 36 noncircular open, 36 nuclear boiling in, 84 open, 92 optimal Prandtl number in, 46

434

SUBJECT INDEX

perturbation methods for, 39 pressurization effects in, 40 rotating liquid metal, 71 rotating two-phase, NaK, 80 roughness effects in, 79 semiclosed, 66 stagnant region in, 22, 49 stagnation phenomena in, 24 two-phase, 77, 93 two-phase and critical states in, 3 two-phase flow in, 78, 84 Total extinction coefficient, 368 Transient flow response, 31 1 Transient response, 314 Transients, 320 delay of amplification in, 334 vigorous, 334 Transition, 10, 17, 19, 27, 28, 42, 321 natural, 332 Transverse curvature, 303 Turbine blade cooling, 97, 104 Turbine cooling, 91 Turbulator, 1 18 Turbulence, 16, 34 generation of, 22 onset of, 20 Turbulence promoters, 117 Turbulent bursts, 334

Two-flux model, 379 Two-phase flow, 78, 84

U Uniform heat flux surface condition, 325 Universal velocity law for suspension, 116

v Vapor blockage, 88 Velocity measurements of, 137, 142 measurements of, in gas-solid flows, 137, 142, 159, 175 Vertical needles, 303 Viscous dissipation, 279, 292 Void fraction, 114 Volumetric heat capacity, 353 Vortex layer, 342

W Wave number, 326 Wedges, 296 Wien’s displacement law, 362 Work of adhesion, 227 Work of separation, 191