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

D.

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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


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,


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.


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

78

D. JAPIKSE

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

ADVANCES I N THERMOSYPHON TECHNOLOGY

79

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.


"

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;
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


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,


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



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
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 , -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 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)


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)

-


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


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


-

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,

390

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


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




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.

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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

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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,

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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

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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-

CRYOGENIC INSULATION

HEATTRANSFER

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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

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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

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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

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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