Advances in Heat Transfer, Volume 10

T

ADVANCES IN

HEAT TRANSFER

Volume 10

Contributors to Volume 10 SESIM ABUAF RICHARD C. BIRKEBAK ROBERT COLE CLIFFORD J. CREMERS CHAIN GUTFINGER

J. 11. HELLMAN S. L. LEE

Advances in

HEAT TRANSFER Edited by

James P. Hartnett

Thomas F. Irvine, Jr.

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

State University of New Y m k at Stony Brook Stony Brook, Long Island New York

Volume 10

@ ACADEMIC PRESS

1974 0

New York

A Subsidiary of Harcourt Brace Jovanovicb, Publiehers

London

COPYRIGHT C 1974, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN % N Y FORM OR BY ANY MEANS. ELECTRONIC OR MECHANICAL. INCLL'DING PHOTOCOPY, RECORDING. OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WIlXOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 63-22329

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS List of Contributors . Preface .

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Contents of Previous Volumes .

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Thermophysical Properties of Lunar Materials : Part I

Thermal Radiation Properties of Lunar Materials from the Apollo Missions RICHARD C. BIRKEBAK I . Introduction . . . . . . . . I1. Apollo Samples: Location and Description 111. Remote Sensing Results . . . . . IV. Thermal Radiation Measurements . . V. Summary . . . . . . . . . Nomenclature . . . . . . . . References . . . . . . . . .

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Thermophysical Properties of Lunar Media :Part I1

Heat Transfer within the Lunar Surface Layer CLIFFORD J . CREMERS

I . Introduction . . . . . . . . . I1. Thermal Conductivity . . . . . . . I11. Specific Heat . . . . . . . . . IV. Thermal Diffusivity . . . . . . . V . Thermal Parameter . . . . . . . VI . Heat Transfer in the Lunar Surface Layer . VII . Reference Values of Thermophysical Properties Nomenclature . . . . . . . . . References . . . . . . . . . . V

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39 42 59 64 70 72 79 80 80

CONTENTS

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Boiling Nucleation ROBERTCOLE I . Introduction

86 87 92 111. Honiogcncous Sucleation . . . . . . . . . . . IV . Superheat Limits . . . . . . . . . . . . . 95 i'. Heterogeneous Sucleation . . . . . . . . . . 111 VI . Suclratinn from a Preexisting Gas or Vapor Phase . . . 117 V11. Size Ilangt of Active Cavities . . . . . . . . . 127 VIII . Stability of Sucleation Cavities . . . . . . . . . 134 IX . 3Iinirnum Boiling Superheat. . . . . . . . . . . 148 Somenclature . . . . . . . . . . . . . . 162 References . . . . . . . . . . . . . . . 163 .

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I 1 . lundanicntal Equations of Surface Science .

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Heat Transfer in Fluidized Beds

CHAIM GUTFINGER A N D KESIMABUAF 1. I1. 111. IV .

Introduction . . . . . . . . . . . . General Description of Fluidized Bed Behavior . . . Hrat Transfer betwxm Solid Particles and a Fluid . . Heat Transfer between a Fluidized Bed and a Surface Somenclat ure . . . . . . . . . . . . Kefcrenccs . . . . . . . . . . . . .

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167 169 171 180 213 214

Heat and Mass Transfer in Fire Research

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S. L . LEE A N D J. JI . HELLMAN I. I1. 111. IV. V. VI . VTI . VIII .

Introduction . . . . . . Pyrolysis . . . . . . . Ignition . . . . . . . . The Pliinict . . . . . . . Fire Spread . . . . . . . Instrunwntation in Fire Rrsearch Fire Research and the Fire Fighter Concluding Remarks . . . . References . . . . . . .

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. 230 226 . 23.5 . 245 . 260 . 272 . 275 . 280 . 281 .

Author Tndes

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Subject Indes

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LIST OF CONTRIBUTORS NESIM ABUAF, Laboratory for Coating Technology, Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel RICHARD C. BIRKEBAK, Department of MechanicaE Engineering, University of Kentucky, Lexington, Kentucky ROBERT COLE, Department of Chemical Engineering and Institute of Colloid and Surfuce Science, Clarkson College of Technology, Potsdam, New York CLIFFORD J. CRERIERS, Department of Mechanical Engineering, University of Kentucky, Lexington, Kentucky CHAIM GUTFINGER, Laboratory for Coating Technology, Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel J. M. HELLMAN, Power Systems, Westinghouse Electric Corporation, Pittsburgh, Pennsylvania

S . L. LEE, Department of Mechanics, State University of New York at Stony Brook, Stony Brook, New York

Vii

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PREFACE The serial publication Advances in Heat Transfer is designed to fill the information gap between the regularly scheduled journals and university level textbooks. The 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. The favorable response to the volumes published to date by the international scientific and engineering community is an indication of how successful our authors have been in fulfilling this purpose. The Editors are pleased to announce the publication of Volume 10 and wish to express their appreciation to the current authors who have so effectively maintained the spirit of the series.

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CONTENTS OF PREVIOUS VOLUMES Volume 1

The 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 Boundary Layers PAUL M. CHUNG Lorn 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

CONTENTS OF PREVIOUS VOLUMES

xii

Volume 3

The Effect of Frcc-Stream Turbulence on Heat Transfer Rates ,J, KESTIS 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 Trarisfcr K. VISKASTA -4Critical Survey of the Major Methods for Measuring and Calculating Dilute Gas Transport Properties A. -4. WESTEXBERG AUTHOR ISDEX-SUBJECT

INDEX

Volume 4

Advances in Free Convection A. ,J. EDE Heat Transfer in Biotcch~iology ALICEIf. STOLL Effects of Reduced Gravity on Heat Transfer

ROBERTSIEGEL Advances in Plasma Heat Transfer E. R. G. ECKERTand E. PFENDER Exact Similar Solution of the Laminar Boundary-Lager Equations C. FORBES DEWEY, JR. and JOSEPH F. GROSS AUTHOR INDEX-SUBJECT

INDEX

Volume 5

Applic*ationof Monte Carlo t o Heat Transfer Problems JOHNR. HOWELL Film and Transition Boiling DKANE P. JORDAE; Convcct ion Heat Transfer in Rotating Systems F R . 4 N K ICREITH Thermal Radiation Properties of Gases

C. L. TIES Cryogenic Heat Transfer JOHX A. CLARK AUTHOR INDEX-SUBJECT

INDEX

CONTENTS OF PREVIOUS VOLUMES

xiii

Volume 6

Supersonic Flows with Imbedded Separated Regions A. F. CHARWAT Optical hlethods in Heat Transfer W. HAUFand U. GRIGULL Unsteady Convective Heat Transfer and Hydrodynamics in Channels E. K. KALININ and 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 The Electrochemical Method in Transport Phenomena T. MIZUSHINA Heat Transfer in Rarefied Gases GEORGES. SPRINGER The 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. ~ K A U S K A S Natural Convection in Enclosures SIMONOSTRACH Infrared Radiative Energy Transfer in Gases R. D. CESSand S. N. TIWARI Wall TurbuIence Studies 2. ZARIE. AUTHOR INDEX-SUBJECT

INDEX

xiv

CONTENTS OF PREVIOUS VOLUMES Volume 9

Advaiiccs in Thcrmosyphon Technology 1).JAPIKYE Heat Transfer to Flowing Gas-Solid lfixtures CREICHTON A. DEPEW and TEDJ. KRAMER Condensation Heat Transfer HERMAN ~ I E R T EJR. , Xatural Convection Flows and Stability B. GEBHART Cryogenic Insulation Heat Transfer C. L. TIESand G. R. CUNNINGTON AUTHOR INDEX-SPBJECT

INDEX

ADVANCES IN HEAT TRANSFER

Volume 10

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Thermophysical Properties of Lunar Materials: Part I THERMAL RADIATION PROPERTIES OF LUNAR MATERIALS FROM T H E APOLLO MISSIONS

RICHARD C. BIRKEBAK Department of Mechanical Engineering, University of Kentucky, Lexingfon, Kentucky

I. Introduction . . . . . . . . . 11. Apollo Samples: Location and Description 111. Remotesensing Results . . . . . IV. Thermal Radiation Measurements . . A. Definitions . . . . . . . . B. Measurement Techniques . . . . C. Experimental Results . . . . . V. Summary. . . . . . . . . . Nomenclature . . . . . . . . References . . . . . . . . .

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I. Introduction

For eons men have looked upon the moon, speculated about its nature, and wondered about its origin. Men have dreamed of visiting the lunar surface and have written books on their speculations. The Apollo program has culminated these aspirations and made them a reality. The landing of men on the lunar surface and subsequent return of lunar samples has led and is leading to a better understanding of the origin of the solar system and moon-earth system. There have been a number of soft landings on the moon in the past several years including five unmanned Surveyor flights and five manned Apollo flights by the United States, and several unmanned Luna flights by 1

2

RICHARD C. BIRKEBAK

the Soviet Union.’ Several of these missions included provision for measurement of lunar surface temperatures and heat fluxes. Ironically, the first flight which included such experiments was the Apollo 13 flight which was aborted before a landing could take place. A successful heat flow experiment was installed by the Apollo 15 crew at Hadley Rille and it returned the first direct measurements of heat flux and temperature from the moon. An unfortunate accident by one of the Apollo 16 astronauts terminated the Apollo 16 heat flow experiment. The other actual measurements of lunar temperatures have been remote measurements from earth in which infrared or microxave radiation emitted by the dark moon, either at lunar nighttime or during an eclipse, was analyzed to j4eld an apparent radiation temperature. The successful landings on the moon of the Apollo flights and the return of samples of lunar surface material has permitted the measurement of the thermophysicai properties necessary for heat transfer calculations. These explorations have also proven what was previously hypothesized from remote thermal radiation measurements and laboratory studies [l], that the lunar surface is covered to a depth of several meters or more, a t least in the mare regions, with a layer of fine particulate soil. There is a hard substratum below these “fines” and there are many rocks and sometimes boulders scattered about as well. However, these occur randomly and may be considered perturbations in the porous, powdery surface layer [2, 31. Man has learned much about the moon’s optical properties, temperature, surface features, and thermophysical properties by using remote sensing techniques from astronomical, infrared, microwave, and radar measurements. The returned lunar samples are allowing scientists to compare physical properties obtained directly from these samples with those inferred from remote sensing results. Thc thermal radiation properties described herein have and are being used in heat transfer calculations [4-7) for the prediction of lunar surface temperatures, spacecraft temperatures, and temperature variation of scientific equipment placed on the lunar surface. The proper design and operation of such systems or structures which may be constructed for the lunar surface depend to a large degree on a complete knowledge of the thermal transport properties of the material in the lunar surface layer. Accurate data on the local thermal environment are also needed, particularly on the moon %-herethere are widc extremes of temperature. However, up to the Apollo 15 mission there has not been any direct measurements of 1 Subsequent to the writing of this paper the hpollo 17 mission was successfully completed. The returned lunar material is now being studied.

LUNARPROPERTIES I

3

the temperature field in the surface layer, nor had there been any reports until recently of the actual properties of the lunar material. We became involved in the Lunar Sample Analysis Program of the National Aeronautics and Space Administration in October 1966 when our group was selected as one of the initial 110 groups to study lunar material. Our group was selected to measure the thermophysical properties of lunar materials. This article reviews our results and those of other investigators on the thermal radiation properties of lunar material returned by the Apollo 11, 12, 14, and 15 missions and also, where possible, compares these results with remote sensing data.

FIG.1. Apollo mission landing sites (NASA).

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RICHARD C. BIRKEBAK 11. Apollo Samples :Location and Description

On July 20, 1969, astronaut Armstrong stepped onto the lunar surface in the Sea of Tranquility a t the location 0.67'N and 23.49"E. This site is shown on the photograph of the lunar surface, Fig. 1. During the extravehicular activity (EVA) phase the astronauts collected about 21.5 kg of lunar material. About one-half of the returned lunar material was finegrained soil (called fines) and half was selected rock fragments. All of the samples were Collected near the lunar module in an area approximately 7 x 22 meters. A complete description of the Apollo 11 samples is found in the Proceeditrgs of the Apollo 11 Luuar Science Covference [Z]. Each lunar specimen as cataloged by SASA's Lunar Heceiving Laboratory is specified by a number which identifies the Apollo mission and sample. A five-digit number is used, succeeded by two or three more digits; this refers to a particular split of the main sample. Documented sample histories include in many cases the sample location and orientation on the surface of the moon. The saniple series begin with 10000, 12000, and 14000 referring to samples from the Apollo 11, 12, and 14 missions, respectively.

FIG.2. Apollo 11 rock chip 10047.

LUNAR PROPERTIES I

5

FIG.3. Apollo 11 rock chip 10048.

Photographs of three rock chips from the Apollo 11 mission are shown in Figs. 2 4 , and illustrate the variation in textural nature of the rocks. These chips are from larger parent rocks. Rock number 10047 is a coarsegrained, vuggy, ophitic cristobalite basalt [2], 10048 a breccia [2], and 10057 a fine-grained, vesicular to vuggy, granular basalt [2]. The Apollo 11 lunar fines consisted of a distribution of small crystalline fragments and glassy fragments with a variety of shapes. Particle sizing of the fines [S] showed a range from 200 pm down to less than 1 pm, with most of the particles being a t the small diameter end of the range. Much larger rock fragments were in the initial soil sample but were removed prior t o this study [S]. The other Apollo fines have similar distributions when compared to that of Apollo 11 fines. The second manned lunar landing mission, Apollo 12, landed on the Ocean of Storms with coordinates 3.0"s latitude and 23.4'W longitude on November 19, 1969 [a]. During this mission two EVA'S were carried out which explored the lunar surface. Approximately 34 kg of material was collected with 82% being rock samples and the remainder of fines. The third lunar landing was made a t Fra Mauro a t latitude 3.66'5,

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RICHARD C. BIRKEBAK

FIG.4. Apollo 11 rock chip 10057.

longitude 17.48OW by the Apollo 14 lunar module on February 4, 1971. This niission provided an exceptionally rich harvest for lunar science. The Apollo 14 crew returned 43 kg of lunar material. The Apollo 14 samples may he contrasted with the samples returned from the Mare Tranquillitatis and the Oceanus Procellarum in that their chemical composition is quite different, and the Apollo 14 rocks exhibit characteristics that suggest they are ejects from the Imbrium Basin [9]. On July 30, 1971 the rip0110 1.5 mission, the fourth in the series, landed in the Hadlty-Apennine region of the moon at a latitude of 26.1°N and longitude of 3.65OE [lo]. Four EVA'S were performed on the lunar surface and for the first time a lunar Rover was used to transport the astronauts. Astronauts Scott and Irwin brought back 77 kg of samples from the Hadley region [lo]. To date, the oldest lunar material returned were the soils and breccias from Apollo 11 with ages of approximately 4.6 billion years [a]. The rocks

LUNARPROPERTIES I

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from Apollo 11 were younger by one billion years. In the oldest Apollo 12 material, the soils are one billion years younger than the Apollo 11 samples. 111. Remote Sensing Results

Prior to the Apollo, Surveyor, and Luna missions the only way that the surface of the moon was studied was by remote means, i.e., by optical and radar astronomical measurements. From those remote sensing results it has been possible to deduce many of the thermophysical properties of the lunar surface [l]. Hapke [l] reviews the optical remote sensing results prior to 1970 that appear in the literature. The general conclusions that he makes, and that are well known to people working in this area, are the five following characteristics. (1) The moon has a low albedo with an average value of near 7%. (2) The spectral albedo increases monotonically from 0.3 to 2.5 pm but has small and sometimes distinct variations in the spectrum. (3) The lunar surface backscatters strongly, i.e., the moon’s brightness peaks strongly a t full moon when the sun is almost directly behind the observer. (4) The degree of polarization of the lunar surface changes with phase angle (the angle between the observer and moon and sun) in a distinct manner. (5) The previous four statements hold for every type of surface feature on the moon. The most significant remote sensing results that have been reported to date are by McCord and Johnson [ll, 121, McCord et al. [13], and Adams and McCord [141. These data are the results of very careful experimental work. The ubiquity of the results over the lunar surface as stated in conclusion (5) is probably due to the fact that almost the entire lunar surface is covered with fines, but this does not mean that signification differences do not occur. The remote sensing data for normalized spectral reflectance is shown in Fig. 5. The results were measured for regions on the moon of approximately 18 km in diameter. Starting from the top curve and proceeding downward the results are for (a) the upland or highlands region, (b)-(d) the mare regions, ( e ) the upland bright craters, and (f) and (g) bright mare craters on the lunar surface. The curves have the form or trend as described in conclusion ( 1 ) , however, there are significant differences. All curves indicate an absorption band near or a t 0.95 pm. This band is due to clinopyroxene in the lunar fines and exposed rocks, and the depth of

RICHARD c. BIRKEB.4K

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FIG.5. Normalized bidirectional spectral reflectance (taken from McCord et al. [13]): (a) Sea of Moisture; (b) Sea of Tranquility; (c) Sea of Serenity; (d) Sea of Cold; (e) Tycho; (f) Sea of Moisture; (g) Aristarchus.

these bands is an indication of the amount of this mineral [14]. The slope of the curves is a function of the chemical composition of the lunar material and the amount of breccia and glass fragments in the fines [14]. McCord et al. El31 have normalized these results to a selected area in the Sea of Serenity. By doing this they have been able to emphasize spectral differences. The data in Fig. 5 have been normalized in this matter and the results are shown in Fig. 6. Clearly evident are different spectral types. From this type of procedure the composition and age of a given area on the moon can be inferred.

LUN.4R PROPERTIES

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9

Adams and JIcCord [14] have shown that the spectral reflectance for the Sea of Tranquility obtained by remote sensing agrees well with that obtained in the laboratory. The laboratory data also enabled hlcCord et al. [131 to correct their remote sensing results for atmospheric absorption, etc. The comparison of these results is shown in Fig. 7. According to Hapke [l] the most accurate presently available integral phase function distribution of reflected light from the moon was measured by Rougier [15]. A plot of his results is shown in Fig. 8. The normalized photometric distribution is presented as a function of phase angle. ( I n

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W J

LL W

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0.9

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HAVELENGTH ( p n )

FIG.6. Ratio of the reflectance of a lunar area to the Sea of Serenity (taken from MeCord et al. f13j): (a) Seaof Tranquility, (b) Sea of Moisture, (c) Luna 16 landing site, (d) Fra Mauro, (e) Le Monnier, (f) Sea of Moisture, (9) Plato, (h) Sea of Cold.

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c. BIRKEBllK

APOLLO II SITE

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071-

TELESCOPE MEASUREMN

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II

FIG. 7. Compnrison of telescopic and laboratory-obtained reflectances for the Sen of Tranquility.

engineering practice we would refer to these as normalized bidirectional reflpctances.) The lunar surface strongly backscatters the incident beam of sunlight. Hapke [lS] formulated a theory that predicted this effect. Later improvements were made by Alorozhenko and Yanovitsliii [l?]. In arriving a t his theory, Hapke assumed that the lunar material had a “fairycastle structure”; i.e., the grains of the soil stuck together in such a manner that the overall structure was very porous so that light could penetrate from any direction. From remote sensing results and laboratory studies on siniulated lunar material, Hapke as well as others came to the conclusion that thc surface of the moon was covered mainly ttith very small particles. This was evcntually borne out by the analyses of the returnad lunar samples and photographs returned of the lunar surface from the Apollo, Surveyor, arid Luna missions. As difficult as it is in the visible portion of the spectrum to obtain reliable remote sensing data, the problem in the infrared is even more so. The total

LUNARPROPERTIES I

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al U

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08-

[L

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120

180

Z

60

0 Phase Angle

180

120

60

FIG.8. Normalized bidirectional reflectance function.

emittance of the lunar surface has been usually assumed to be unity, a blackbody. However, the remote sensing results of Murcray [lS] and Murcray et al. [19] made from a mountain top and from balloons gave data that showed that the moon was not a blackbody. A result of theirs is shown in Fig. 9 for a spectral range from 7.0 to 13.5 pm. These results will be discussed later. If it were possible to obtain reliable emittance data, especially where the maximum value occurs [ZO], then one could determine something about the mineralogy of the lunar surface. The maximum emittance location has been shown by Salisbury et al. [20] and Cone1 [21] to depend on the chemical composition of the substances that make up a powered material.

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Mare lmbrium

t

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FIG.9. Remotely obtained spectral emittance of Mare Imbrium.

RICHARD C. BIRKEBAK

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IV. Thermal Radiation Measurements During the lunar day the dominating or controlling parameters that affect the temperature of the lunar surface or objects on the surface are the thermal radiation characteristics of the lunar material. These characteristics includc the bidirectional and directional reflectance and emittance of the lunar surface. It has been shown in Cremers et al. [S] that the directional solar albcdo has the greatest effect on lunar surface temperature than any other single thermophysical property. This variable reflectance effect is clearly showi in Fig. 10 where the temperatures differ substantially just after sunrise and before sunset. Table I compares the constant and variable property results for specific lunation times. The maximum differences occurs for a lunation fraction of 0.24 and 0.76 where the variable property calculated temperatures are 43.7"K and 64.1"K lower, respectively. In addition t o the necessity of knowing the thermal radiation characteristics for heat balance calculations, they also are useful in remote sensing work as shown in the previous section. The spectral signature of a substance can be used to identify, in part, its mineral constituents; that is, the location of absorption bands in reflectance curves can usually be associated with a given mineral. And the determination of the spectral

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10

Fraction of Local Lunation

FIG.10. Lunar surface temperature variation. p

and

E.

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Variable A@), t ( T ) ,- - - constant

LUNARPROPERTIES I

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reflectance gives us both thermal data as well as some spectral signature data as to the mineralogy of the material.

A. DEFINITIONS I n the literature on the photometry measurements on the moon one finds at least four different kinds of reflectances (albedos) used. Because the moon is a spherical body, it is natural that there arise albedos that take this into account. These definitions are presented here for the convenience of the reader. 1. Bond Albedo

The ratio of the amount of light reflected by a spherical body in all directions to that incident on the body is defined as the Bond albedo, B. 2. Geometric Albedo

As is often the case in photometry, albedo is defined in terms of a Lambertian surface. The geometric albedo p is the ratio of the brightness [radiance] of a body at zero phase angle to the brightness (radiance) of a Lambertian disk of the same angular diameter as the body and perpendicular to the sun's rays. The phase angle V , is the included angle between the source, sample, and observer. These two albedos are related to each other by the following equations: B = q-p where

TABLE I COMPARISON OF TEMPERATURE FOR CASEOF VARIABLE SURFACE PROPERTIES WITH THATOF CONSTANT SURFACE PROPERTIES~ Fraction of lunation, 7 0 0.24 0.25 0.50 0.75 0.76 0

(OK)

variable 389.3 161.2 134.4 94.7 86.1 125.4

(Noon) (Sunset) (Midnight) (Sunrise)

Cremers et al. [6].

Temp.

6

Cremen et al. [5].

p ( ~ ) , e(T)

Constantbp, 389.4 204.9 147.5 96.8 87.8 189.5

RICHARD C. BIRKEBAK

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and the quantity +( 1’) is called the integral phase function. It describes the manner in which the light from the lunar surface is distributed. The function Cp ( 1.) is taken to be 1 a t 1’ = 0.

3. A’omal Albedo ( A )

If the curvature effects are eliminated, that is, if we consider a flat surfwv, tht. analogy to the geometric albedo is the normal albedo ( A ) . It is thc brightness of an area a t an arbitrary angle of illumination, to the hrightncss of a Lambertiun disk under the same solid angle but at zero angle of illurnination. This albedo is the most frequent one reported because of the following reason. The photometric distribution function of the lunar surfacc behaves in such a manner that the normal albedo is independent of the direction of illumination and viewing. The fourth albedo found in the literature is the analogy to the spherical Bond albedo, it is th r hemispherical albedo. This albedo is the same as the dirrctional reflectance described next. 4. Directtom1 ReJlectance

Thc directional refiectance is used mainly in the calculation of the heat balance and temperatures of the lunar surface. It is defined in the following manncr: Let the incident radiation be contained in a solid angle An, oriented a t a specific anglc \k relative to the surface normal (Fig. l l a ) , and let reflected radiation be collected over the entire hemispherical space above the surface. We thcn define the directional hemispherical reflectance as ~ ( 9= )der,h/dei(Q)

(3)

nhcw de,,h is the reflected radiant energy that is collected over the entire hcmisphcrical spare and de, is the radiant energy contained in the incident h r w ~ In ~ . gcncral, the magnitude of de,,i, will depend upon the angle of illuniination of the incoming beam. Thc directional reflectance can also be understood to mean the following. Let t hr surface under study be illuminated hemispherically with diffuse radiation e , , h while the reflected radiance ir(0) is collected in a small solid anglc A9, (Fig. 1l b ). The hemispherical directional reflectance is defined as P(e> = ir(e)/ (ei,h/r) (4)

With the use of reciprocity, the reflectances in Eqs. (3) and (4) can be shown to bc. identical if tlie solid angles are the same, Anl = An,. p ( 9 ) = p(e)

for 6

= 9

(5)

LUNARPROPERTIES I

15

The measurement of p ( 0 ) with the integrating sphere is called the reciprocal mode.

5 . Bidirectional ReJtectance The angular distribution of reflected radiation from a surface can be described by the bidirectional rejectance. The bidirectional reflectance is defined as the ratio of the radiance of the reflected light in the direction of viewing e to the incident energy per unit time and surface area contained within a solid angle dQiin the direction Q, Pb(\k,

e)

=

?r

dirt*, 8)/dei(Q)

(6)

where dir is the reflected radiance and dei (9) is the incident flux of energy. 6. Directional Emittance

The directional spectral emittance of a sample is defined as the ratio of the radiance of radiation from the sample to radiance of radiation from a blackbody a t the same temperature, 4 0 , A,

T) = is(6, A, T)/iB(A,2')

(7)

Normal

Incident Radiation Reflected Radiation

Normal

T , [ $ z ; M Hemispherically Irradiated

(b)

FIG.11. Definition of coordinates: (a) directional hemispherical technique; (b) hemispherical directional technique.

16

RICHARD C. BIRKEBAK

where 9 is the angle of viewing, T the temperature of the sample, and is and iu are the radiance of the sample and blackbody, respectively. These thermal radiation property definitions are used in the following sections to describe the radiation characteristics of the lunar material. B. 3IEASUREMENT TECHHIQUES In this section we will briefly describe the experimental techniques used by us to determine the thermal radiation characteristics of lunar material. 1. Directional Reflectance Apparatus

The directional reflectance was obtained with a sample, center mounted in an integrating sphere reflectometer. The sphere coating was magnesium oxide. The sphere system was constructed with the sample held in a horizontal position, a necessity for powders, while by rotation of the sphere and external optics, angles of illumination or viewing up to approximately 75" are obtained. Since the theory of the integrating sphere is well known, it will be only briefly reviewd. Radiation directed on a test sample within the integrating spherc is reflected onto the sphere wall. If the wall is coated with a highly reflective and diffuse material, then any radiation hitting the wall is reflected diffusely throughout the sphere. The total radiation incident on a given area will be a summation of radiances from the multiple reflections. A detector mounted at the sphere wall measures radiance of the radiation striking a given area within the sphere. Our integrating sphere was operated in the reciprocal mode, that is, the samplc was illuminated by diffuse light from the sphere walls. The ratio of radiancr when the center-mounted sample was viewed to the radiance when the wall was viewed is the directional reflectance p ( 0 ) [Eq. (4)]. As represented in Eq. (5) this measurement is equivalent to illuminating the sample at an angle of incidence equal to the angle of viewing 0. A sketch of the apparatus used in this research is shown in Fig. 12. The integrating sphere was constructed of stainless steel hemispheres, 0.20 m in diameter, flanged and joined by a copper gasket seal. The interior of the sphere was smoked with NgO until a uniform coating of 2 mm thickness was obtained. Ports were provided on the sphere for the test sample, detector optics, light source, and vacuum pump. The actual sample transfer system [22] was Constructed so that the sample could be located along the diametral plane of the sphere and always held in a horizontal position under vacuum condition. The sample size was 25 mm in diameter by 6.0 mm in depth. The viewing optics were arranged so that the sample or sphere wall

LUNARPROPERTIES I

VIEWING PORT

17

* OPTICS AND SOURCE

AXIS OF ROTATION

' TO VACUUM MANIFOLD

FIG.12. Schematic of integrating sphere.

could be viewed by rotating the optical bench, Fig. 12. The spectral results were obtained with a Perkin-Elmer 112 U spectrometer having a tungsteniodine source and a lead-sulfide detector. The total or white light measurements were made with a 1000-W tungsten-iodine lamp (DXW) with a reflector and with a Kipp-Zonen CA-1 thermopile. Details of the design and construction are presented in a technical report, Birkebak and Cremers [23]. The measurements were obtained automatically by joining the spectrometer and its associated electronics to a minicomputer. 2. Bidirectional ReJEectance Apparatus The bidirectional reflectance measurements were made with a goniometric system designed so that the samples could be viewed and illuminated while under vacuum conditions. A schematic of the apparatus is shown in Fig. 13. The energy from the sample is reflected from three diagonal mirrors before it emerges from the system. It is then focused onto the entrance slit of the spectrometer or onto a detector. For this system the surface could be illuminated a t a fixed angle of illumination (\E) and the angle of viewing (0) varied or, by reversing the optical path, the angle of illumination was varied and the angle of viewing was held fixed.

RICHARD c. BIRKEBAK

18 Source ,OPtlCS

+;;r-, Fugn E

Rotating Arm

toble

I

To Vocuurn Moni f old

FIG. 13. Bidirectional reflectance system.

3. Spectral Emittance Apparatus

The nature of the sample dictated that we use a horizontal samplc in the measurements. The sample was mounted in a sample heating cup and positioned in a known radiation environment as shown in Fig. 14. Besides the sample holder a heated reference blackbody was placed in the environment chamber. Transfer optics were used to view either the sample or blackbody and to direct the encrgy into a Perkin-Elmer 112 U spectrometer.

Woler-Cooled Vacuum Chomber

Somple

8 Angle of Viewing

Holder or

Blackbody

FIG. 14. Schematic of emittance system.

LUNARPROPERTIES I

19

The spectrometer was interfaced with a Hewlett-Packard 2114B minicomputer. This syatem allowed us to automatically scan the wavelength range of interest and to perform the necessary operations required to obtain data. A complete discussion of the analysis of this technique is given elsewhere [24]. The working equation used by the system to obtain the spectral emittance is:

(8) where A(S), A(B), A(R) and Ts, TB, TR are the detector output and thermocouple readings for the sample, heated reference blackbody, and blackened surrounds, respectively.

C. EXPERIMENTAL RESULTS Before discussing the experimental results obtained on the returned lunar samples, the following facts should be kept in mind. A number of investigators have recognized that the density (packing, compactness) of the lunar fines will effect the measured properties [25-281. However, only Birkebak et al. [26] actually specify the bulk densities used in their experiments. Hapke et al. [25] discuss the change in albedo with packing and describe various ways to change the material density but do not give any quantitative results for the density. From the description given in the various papers reviewed it appears that many of the investigators sifted or gently poured the lunar fines into their sample holders. We have carried out several experiments using these methods and find that the bulk density can range from approximately 900 to 1100 kg/m3. However, these surfaces are very susceptible to vibrations and settle slowly. 1. Bidirectional ReJlectance Results

As important as the knowledge of the bidirectional reflectance distribution is to the heat balance calculations on structures on the lunar surface, only a limited amount of data is available: In their original papers, Birkebak et al. [29a, b] and Gold et aE. [8] and later, Gold et al. [27] and O’Leary and Briggs [30], present bidirectional reflectance results. As discussed in Section 111 on remote sensing, the geophysicists p r e sented their results for a fixed angle of observation and variable angle of illumination; whereas, in thermal property measurements we usually

20

RICHARD C. BIRKEBAK

FIG. 15. Normalized bidirectional reflectance-Apollo 11 sample No. 10084.68.

present the bidirectional reflectances for a fixed angle of illumination and variable angle of viewing. Therefore, direct comparison is not possible of the two results obtained in the laboratory on lunar fines. The results of Hirkebak et al. [29a, b] are shown in Fig. 15. The sample was illumination with white light (1000-W tungsten-iodine lamp) at angles of lo", 30", and 60". We have normalized the results with respect to the specular ray direction, = +O. A perfectly diffuse surface would give a horizont.al line. It is quite apparent that the lunar material backscatters very strongly in the direction of illumination.

FIG.16. Bidirectional reflectance as a function of phase angle V. Symbols: 0, Apollo 12 soil; 0, Apollo 11 soil; -, moon (normalized to Apollo 11); e, angle of viewing; A, 0.56 fim.

LUNARPROPERTIES I

21

The backscatter peak could not be measured with our present equipment. However, the peak is estimated to have a value as high as 2-3. This explains why the moon appears so bright during opposition (zero phase angle). In the laboratory if one illuminates the lunar fines at an angle of 30" and then moves one's head aver the hemispherical space above the surface, one can physically see this distribution of reflected light; a very interesting and rewarding experience. There is a slight wavelength effect that we have found and which has also been determined by O'Leary and Briggs [30]. As one approaches zero phase angle, and also for large phase angles, the color of the fines changes from a grayish-blue to a reddish hue. The results of O'Leary and Briggs [30] and Gold et al. [27] are shown in Fig. 16. Their results of reflectance are plotted against phase angle. The solid curve is the moon remote sensing data [31] normalized with respect to the normal albedo of Apollo 11. 2. Directional ReJectance Results We will first discuss directional reflectance made on lunar rock chips. Although these results are not useful in heat transfer calculations, they are useful in obtaining spectral signatures of the various minerals. Second, we will discuss the results obtained on the lunar fines, which, as we have stated previously, covers almost the entire moon to several meters or more in depth.

a. Lunar Rock Chips. The ApoIIo astronauts have brought back many rocks that weighed more than several kilograms. However, most investigators received only a small rock chip or thin section that weighed around 3.0 g. Because these chips were so small in size it was difficult to obtain representative reflectance results. Many of the returned rock had surfaces which could be identified from photographs as having been exposed to the lunar environment, and as a result one could sometimes readily see differences in surface properties. Adams and McCord [14], Birkebak et al. [29a, b], Adams and McCord [32], Conel and Nash [33], Gold et al. [S], Nash and Conel [34], and Perry et al. [35,36] have all presented reflectance spectra for various lunar rocks. The reflectance results for the three rock chips shown in Figs. 2, 3, and 4 are given in Fig. 17. Clearly evident are a number of distinct absorption bands. Adams and McCord [32] have presented diffuse reflectance for mineral separates for Apollo 12 basalt. They have demonstrated that the major absorption bands can be attributed to Fez+ in pyroxene. All observed bands are attributed to electronic transitions in iron and titanium. Adams and McCord [37] have determined the relationship between the wavelength of the two major bands and the pyroxene composition. One band occurs

22

RICHARD

-

c. BIRKEBAK

-1

9p 6

5'

06

I

08

I

10

I

12

1

14

I

16

I

20

18

Wavelength ( prn)

Fro. 17. Typical directional spectral reflectance of lunar rock chips-Apollo

11 mission.

between 0.9 and 1.0pm and the second near 2.0pm. The depth of the absorption bands is also a function of the average pyroxene composition and in the breccias, also of the glass content. The bands degrade from rocks to breccias to the fines material, and their depth or strength correlates with the percent increase of dark glass [14], a conclusion also reached by Cone1

0.5

I.o 1.5 2.0 WAVELENGTH ( p n )

2.5

FIG.18. Directional reflectance as a function of glass content-Apollo 12 mission. Symbols: a, 12063, 79 whole-rock powder; b, +20% glass; c, +55% glass; d, 12070, I l l surface fines; e, 1'2063, 79 whole-rock glass. (Reproduced from Adams and McCord I321.)

LUNARPROPERTIES I

23

and Nash [33]. The effects of the addition of artificially made glass from the crystalline lunar rock is shown in Fig. 18. Perry et al. [35,36] have made reflectance measurements on lunar samples from Apollo 11, 12, 14, and 15. The spectra were obtained from polished specimens and over a wavelength range of 5-500 pm. The resulting spectra are very complex and are of more interest to the geophysicists than to the thermal engineer. No meaningful thermal property data can be obtained from these results. b. Lunar Fines. Before we begin to discuss the directional reflectance properties of lunar fines it is desirable to reiterate the various parameters that affect the reflectance. It seems reasonable to assume that the bulk density or compaction of the material will affect the reflectance. The surface may become more compacted around equipment on the moon due to intentional packing or because of astronaut activities. Another reason for looking at the effect of density is that we still do not know what the exact density of the lunar surface is or how it varies over the lunar surface. Estimates [lo J of the soil density from in-place measurements range from 800 to 2150 kg/m3. However, it should be kept in mind that only the upper 10 mm or so actually affect the thermal radiation characteristics and the lower densities, therefore, could be reasonable estimates. When a fines sample is prepared by simply pouring it into a container and carefully leveling the surface, the bulk density achieved depends on the distribution of particle size in the fines. With this procedure followed for the Apollo 14 and 15 fines, bulk densities of nearly 1000 kg/m3 were obtained. Slightly higher values were obtained for the Apollo 11 and 12 samples. To achieve higher bulk densities [26] the fines were packed by use of a vibrating tool held on the sample holder edge. Initial smoothing and packing of the surface was achieved with a stainless steel spatula. Whether this procedure gives a surface texture close to that of the fines on the moon is open to question at this time. A second parameter that affects the directional reflectance is the angle of illumination. The studies of our group [26, 29b, 38-40] are the only ones, to our knowledge, that have been made on lunar fines. As we discussed earlier, the heat flux to or from the lunar surface is greatly influenced by the angle of illumination that the solar energy makes with the lunar surface [6].

(i) Spectral directional reflectance for fines. The smooth curves for our data which are shown in the figures in this section are fit through data points taken at 0.02 pm intervals to 1.0 pm and 0.05 pm interval to 2.2 pm. The spectral directional reflectance curves for fines from Apollo Missions 11, 12, 14, 15, and 16 are shown in Fig. 19. The results are for a bulk density of approximately 1600 kg/m3 for each sample and for an angle of illumina-

24

RICHARD

42rI

I

I

I

c. BIRKEBAK I

I

I

I

I

1

A

3 6 L-

-s -

-

34

-

32

-

30-

4

/ /

2826-

p -

24-

n"

22-

r

0 20-

-" 9

18-

160

I4

-

12I0

--

61 0 2

I 0 4

I 06

I 08

I 10

I 12

I

I

I

14

16

10

I 2 0

1I 2 2

Wavelength ( p m )

FIG.19. Spectral directional reflectance of lunar fines from Apollo 11, 12, 14, 15, and 16 missions. Angle of illumination S lo", bulk density Z 1600 kg/m3.

tion of approximately 10". It is readily apparent by comparing Figs. 17 and 19 that the spectral results for fines are very different than for rocks. It is difficult sometimes to identify absorption bands in the fines data. The pyroxene band near 1.0 pm is apparent in the Apollo 12, 14, and 15 samples but undetectable by us in the Apollo 11 and 16 samples. The results of Adams and Jones [41] and Conel and Kash [33] show a single, very shallow absorption band centered at 0.95pm for the Apollo 11 sample. Other investigators [27, 301, however, did not find any band features in this region of the spectrum. Both the results of Birkebak and Damon [40] and Adams and McCord [37] show an absorption band for the Apollo 14 sample centered at 0.93 pm and this band is due to pyroxene [37]. Also, a band is clearly evident at 1.8 pm. The variation in reflectance from one Apollo sample to another is associated with its chemical composition and glass content [25, 32-34, 37, 411. As the lunar fines become "lighter in color," an increase in reflectance, we find fewer opaque materials in the fines. Adams and McCord [37] have discussed the lighter appearances of Apollo 14 fines and have related it to the fines having lower overall iron and titanium content. The presence or

LUNARPROPERTIES I

25

absence of the absorption bands are a function of the dark glass content and the crystal/glass ratio. The disappearance of the pyroxene band near 1 pm may be caused instead by extensive impact melting and shock alteration of the soil [32]. Material taken from the core tube samples or trench below 70 mm or so from the surface has a higher reflectance than the surface fines. The darkening of the surface material takes place due to meteorite impactinduced vitrification and by regional contamination by iron- and titaniumrich mare material. The average composition of the Apollo 11, 12, 14, 15, and 16 fines are given in Table I1 [lo, 431. B. Glass [42] has reported that glass particles in the fines with low TiOzand FeO content ( "+$I

a -4na* k d T

-

75

(19)

where k and a are both functions of temperature. Consideration must now be given to the boundary conditions on Eq. (19). The condition at large depth to be imposed on Eq. (19) is that the heat flux goes to zero. The Apollo 15 heat-flux probe has shown that the average lunar heat flux is much less than that caused by insolation and cooling near the surface. In addition it is required that the problem be periodic with a period already mentioned. The condition at the surface is expressed by a heat balance between the incoming solar radiation and the energy which is emitted to space, plus that which is conducted into the surface layer. As the sun's angle with respect to the normal varies during the day, the fraction of incoming radiation which is absorbed also changes. These conditions are written as

The insolation term is z(7)

= s[1

- r ( T ) ] COS A COS@ + 2AT]

(21)

during the half-period of daytime, and I ( T ) = 0 during the half-period of nighttime. In Eq. (20), e is the total hemispherical emittance which is a function of temperature, and in Eq. (21)) T ( T ) is the directional reflectance which will vary with the sun's angle of incidence, and so it can be expressed as a function of the time variable r. A is the latitude and @ is longitude of the lunar site in question. S is the solar constant which was taken as 1395 W/m2 [68]. Recent measurements indicate that 1353 W/mz is perhaps a better value to use [69]. Equation (19) was solved on the IBM 360-65 digital computer using a modified Runge-Kutta scheme. Initially, the directional reflectance and temperature dependent emittance were not available for use in Eqs. (20) and (21). Results for the Apollo 11 and 12 sites for constant values of these properties are given by Cremers et al. [70] and Cremers et al. [71]. The noon, sunset, midnight, and sunrise temperatures are given in Table XVI. Later, when the directional reflectance and temperature dependent emittance became available, Cremers et al. [39] recalculated the Apollo 12 temperatures. The effect of a directionally dependent reflectance should be significant because if a constant reflectance is assumed, the surface immediately begins absorbing a given fraction of radiation a t sunrise and

CLIFFORD J. CREMERS

76

TABLE XVI CONPARISON OF bZE 9SURED AND CALCULATED LUNAR SURFACE

Source Apollo 11 Cremers [38] Apollo 1 1 Cremers [all Wesselink [4] Jaeger 151 Linsky 171 Pettit and Nicholson [2] Sinton [%I Saari 1721 Low 1621 Ingrao et a!. [73] Stimpson and Lucas 1671

Noon

Sunset

TEMPERATURES ("K)4

Midnight

395 389 370 368

152 134 144 178

101 95 98 97

374 389

181

120 122 104

Sunrise 92.9 86.1 90 89 89 109 90

393 386-390

14G200

100-112

From Cremers el a l . [70].

continues absorbing this fraction until sunset. In the real case, however, a t grazing angles the reflectance is near unity and so there is little absorption. Inclusion of the directional dependence of the reflectance has the important cffcct of moderating changes between night and day temperatures in periods near sunset and sunria.. Thv rrsults 0htainc.d for the complete variable property case for the Apollo 12 sitr are given in Fig. 14. A comparison between temperatures from the prescnt study and the results of the above analysis in which the reflrrtance and cmittance were held constant at mpan values is given in T a b h XVII and by the dashed line in Fig. 14. Notr that at noon there is practically no difference because the sun is directly overhead and the normal reflectance was that used in the constant surface property analysis. The only difference between the two calculations is in the emittance. The really significant differences in the two models are just before sunset and just after sunrise. The maximum differences occur a t 7 = 0.24 where the tenipcraturr is loner by 43.7"K in the variable property case and a t T = 0.76 when the temperature is lower by 64.1°1iin the variable property case. The temperatures at the Apollo 12 site were compared in Table XVI with thosc calculated for the ApolIo 11 site. It should be reemphasized that there were no total directional reflectances or temperature dependent emittances available when the latter calculations were made, and so constant radiative properties were assumed. The generally loit er temperatures in the Apollo 12 case are due primarily to the lesser amount of energy

LUNARPROPERTIES I1

77

400

300

5 200 W

a E al

+ I00

I

I

I

0.2

0

I

06 0.0 Fraction O f L o c a l L u n a t i o n

0.4

10

FIG.14. Temperature variation on the moon during a lunation.

calculated to be absorbed during the day. This difference is caused by taking the directional dependence of reflectance into account. The most significant differences occur again just before sunset and just after sunrise when the constant surface property model deviates the most from the actual situation. The differences in conductivity and diffusivity become TABLE XVII COMPARISON OF TEMPERATURES FOR THE CASEOF VARIABLE SURFACE PROPERTIES WITH THAT OF CONSTANT SURFACEPROPERTIESO Temperature (OK) .

Time 0 (Noon) 0.24 0.25 (Sunset) 0.50 (Midnight) 0.75 (Sunrise) 0.76 a

~

~~

Variable T , 389.3 161.2 134.4 94.7 86.1 125.4

From Cremers et al. [39].

~~

E

Constant T, 389.4 204.9 147.5 96.8 87.8 189.5

E

CLIFFORD J. CREMERS

78

most apparent beneath the surface. The amplitude of the diurnal variation at depths of 20 and 50 mm is on the order of 20°K greater in the Apollo 1I case. It is of historical interest to compare the present results with some prior calculations of lunar temperatures which were based on assumed properties. Thwv calculations wcre usually made in terms of the thermal parameter. For thc studirbs of Cremers et ul. [39, 70, 711, the value of the thermal parameter based on reference properties (discussed in Sect. VII) is about 2.4 X in SI units. For comparison, temperatures were calculated by Wessrlinli [4] ( y = 0.28 x 10-7, Jaeger [S] (y = 2.39 X and Linsky [7] ( y = 2.39 X 10-9, among others. Wesselink and Jaeger assumed constant properties throughout, and the numbers takcn from Linsky xwre calculated for an assumed medium with temperature depcndcnt specific heat and thermal conductivity. The results of these calculations are given in Table XVI along with those from calculations based on the Surveyor studies. Also shotvn in Table XVI are the results of remote measurements of lunar surfarc trmpcratures. These include temperatures inferred from mcasuremeiits of infrared or microwave radiation during lunar nighttimes or during eclipses by Pcttit and Sicholson [2], Sinton [ 5 5 ] , Saari [72], Low [62], Ingrao et al. [73], and Stimpson and Lucas [67]. In the latter paper several

0

0

I

I

0.2

04

Dimensionless D e o t h

I

I

08

06 X/(4

T Qf

PI”*

FIG.15. Temperature variation on the moon as a function of depth.

1

10

LUNARPROPERTIES I1

79

separate determinations are made of each temperature and the results are presented here as a range. Considering the assumptions required to deduce temperatures from the data and the wide area from which the measured radiation originates, the agreement is quite good. Figure 15 shows the variation of temperature with depth for several times during the lunar day as calculated from Eq. (19). It is seen that the temperature wave damps out rapidly with distance because of the excellent insulating characteristics of the lunar surface material. In both the Apollo 11 and 12 cases, the daily variation in temperature drops to about 1' at about E = 0.85 which corresponds to a depth of only 0.172 meters. The steady temperature on the moon below this depth is 225'K, considerably below that measured for the Apollo 15 site.

VII. Reference Values of Thermophysical Properties

It is frequently of interest to have relevant reference values of the thennophysical properties available for computational convenience because in many analyses the constant property assumption will suffice. There are two ways of calculating meaningful reference properties for the typical lunar problems. One way is to calculate the integrated average of the property in question over the temperature range of interest. That is, for a TABLE XVIII CONDUCTIVITY, THERMAL DIFFUSIVXTY, AND THERMAL AVERAGE VALUESOF THERMAL P ~ R A M E T E RFOR THE APOLLO FINES SAMPLES' Thermal conductivity (W/meters-"K) Density (kg/mJ)

I x lo"

k*

x lo"

Thermal diffusivity (m'/sec) ti X 109

(I* x 109

7 x 10'

2.04 2.11 1.64

2.73 2.12 2.06

2.81 2.17 2.08

1.58 1.21 1.08

2.91 2.77 2.48

3.11 2.87 2.47

Apollo 11 1300 1640 1950

1.77 2.32 2.09

1.61 2.12 1.95

2.34 2.42 1.85

Apollo 12 1300 1640 1970

1.52 1.37 1.45

1.28 1.21 1.33

1.95 1.44 1.29

4

From Cremers el al. [38,411.

Thermal parameter (m" "K-sec1n/J)

*

y*

x 10'

CLIFFORD J. CREMERS

80

propwty p ( 2') the integrated average jj is given by fj

=

(Tmax

- Tinin)-'/

Tmax

p(T) dT

(22)

Tm*o

In Eq. (22) T,,,,, and T,,, are the extreme temperatures of the problem. A second way is to simply evaluate the property at a mean value of temperature. That is, p* = p("). The obvious temperature to use here is the average temperature of the lunar surface layer. As the properties k, a, and c of the lunar fines are nonlinearly dependent on temperature, there will be a difference in the two sets of calculations Cremcra C38, 41) carried out the reference property calculation for the Apollo 11 and 12 fines samples. The rcsults are given in Table XVIII. The tempPraturrs used are taken from Cremers ef al. [39, 701, respectively.

ACKSOWLEDGMESTS The aiithur wishes to express his gratitude to the National Aeronautics and Space Administration for financial support for the investigations of lunar thermophysical properties. Many thanks are also due to Beverly Martin and Lynda Young who typed the manuscript and to Carla Cremers who helped organize much of the author's own data as well as those of others. xOXENCL.4TURE

A

B C

I J k P P

Q T

S

s t

T X

Coefficient of Eq. (10) Planck blackbody function; coefficient of Eq. (10) Specific heat Insolat ion Radiative intensity Thermal conductivity Period of cyclic heat flux Heat flux per unit length Heat flux per unit area Surface reflectance General direction Solar constant Time Temperature Distance from surface

GREEKSYMBOLS Thermal diffusivity Thermal diffusivity a t average lunar temperature p Longitude y Thermal parameter (lipc)-l/* c Surface emittance e Angle from surface normal K Absorption coefficient h Latitude Dimensionless distance z/(47ra*P)"* p Density T Dimensionless time t / P a a*

So DSCRIPTS

c r Y

0

Conduction Radiation Frequency Surface

1. P. Epstein, Phys. Rev. 33, 269 (1929). 2. E. Petitt and S. B. Nicholson, Astrophys. J . 71, 102 (1930). 3. E. Pettit, Asfrophys. J . 91, 408 (1940).

LUNARPROPERTIES I1

81

A. J. Wesselink, Bull. Aslron. Znst. Neth. 10, 351 (1948). J. C. Jaeger, Aust. J. Phys. 6, 10 (1953). V. D. Krotikov and V. S. Troitaky, Sou. Astron.-AJ. 6, 841 (1963). J. L. Linsky, Icurw 5,606 (1966). “Apollo 11 Preliminary Science Report” NASA Spec. Pdl. NASA SP-214 (1969). “Apollo 12 Preliminary Science Report’lNASA Spec. Publ. NASASP-235 (1970). “Apollo 14 Preliminary Science Report” NASA Spec. Publ. NASA SP-272 (1971). A. E. Wechsler and P. E. Glaser, Zcarus 4,335 (1965). D. R. Stephens, High Temperature Thermal Conductivity of Six Rocks. Univ. of California, Lawrence Radiat. Lab., Rep. UCRL 7605 (1963). 13. T. Murase and A. R. McBirey, Science 170, 165 (1970). 14. N. Warren, E. Schreiber, and C. Scholz, Proc. Lunar Sci. Conf., M ,Geochim. Cosmochim. Acta Suppl.. 2, 3, 2345 (1971).* 15. K. Horai, G. Simmons, H. Kanomori, and D. Wones, Proc. Apolb 11 Lunar Sci. Conf.; Geochim. Cosmochim. Actct Suppl. 1,3, 2243 (1970).f Geodrim. Cosmochim. 16. R. A. Robie and B. S. Hemingway, Proc. Lunar Sci. Conf., M , Acta Suppl. 2, 3, 2361 (1971). 17. R. A. Robie, B. 5.Hemingway, and W. H. Wilson, Proc. Apollo 11 Lunar Sci. Conf., Geochim. Cosmochim. Acta Suppl. 1, 3, 2361 (1970). 18. J. A. Bastin, P. E. Clegg, and G. Fielder, Proe. Apollo 11 Lunar Sci. Conf., Geochim. Cosmochim. Acta Suppl. 1, 3, 1987 (1970). 19. K. Watson, I. The Thermal Conductivity Measurements of Selected Silicate Powders in Vacuum from 150”-350”K;11. An interpretation of the Moon’s Eclipse and Lunation Cooling as Observed through the Earth’s Atmosphere from 8-14 Microns. Ph.D. Thesis, California Inst. of Technol., Pasadena, 1964. 20. P. E. Clegg, J. A. Bastin, and A. E. Gear, Men. Not. Roy. Astron. Soc. 133,66 (1966). 21. R. L. Wildey, J . Geophys. Res. 72,4765 (1967). 22. A. E. Wechsler and P. E. Glaser, Thermal Conductivity of Non-Metallic Materials. A. D. Little, Inc., Cambridge, Massacuhsetts, Rep. NASA Contract NAS8-1567 (1964). 23. A. E. Wechsler and M. A. Kritz, Proc. Themz. Conductivity Conf., 6th, Univ. Denver p. 11-D-1 (1965). 24. A. E. Wechsler and I. Simon, Thermal Conductivity and Dielectric Constant of Silicate Materials. A. D. Little, Inc., Cambridge, Massachusetts, Rep. NASA Contract No. NAS8-20076 (1966). 25. H. S. Canlaw and J. S. Jaeger, “Conduction of Heat in Solids,” p. 334. Oxford Univ. Press, London and New York, 1959. 26. J. H. Blackwell, Can. J. Phys. 34,412 (1956). 27. J. A. Fountain and E. A. West, J. Geophys. Res. 75, 4063 (1970). 28. J. C. Mulligan, J. Geophys. Res. 75, 4180 (1970). 29. C. J. Cremen, Rev. Sci. Instrum. 42, 1694 (1971). 30. E. C. Bernett, H. L. Wood, L. D. Jaffee, and H. E. Martens, AZAA J. 1,1402 (1963). 31. P. A. Ade, J. A. Bastin, A. C. Marston, S. J. Pandya, and E. Puplett, Proc. Lunar Sci. Conf., 2nd, Geochim. Cosmchim. Acta Suppl. 2,3,2203 (1971). 32. C. J. Cremers, R. C. Birkebak, and J. P. Dawson, Proc. Apollo 11 Lunar Sci. Conf., Gwchim. Cosmochim. Acta Suppl. 1, 3, (1970). 33. A. P. Vinogradov, Proc. Lunar Sci. Conf., grid, Geochim. Cosmochim. Acta Suppl. 2, 1, 10 (1971); also in J. Brit. Interplanet. Soc. 24,475-495 (1971). 4. 5. 6. 7. 8. 9. 10. 11. 12.

* Publiihed by MIT Press, Cambridge, Massachusetts. t Published by Pergamon, New York.

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CLIFFORD J. CREMERS

34. V. S. Avduevskii, N. A. Anfimov, M. Y. Marov, S. P. Shalaev, and S. P. Ekonomov, SOU.Phys.-Dokl., 16, 55 (1971). 35. bf. G . Langseth, Jr., S. P. Clark, Jr., J. Chute, Jr., and S. Kerhm, Rev. Abstr. Lunar Sci. Conj., 3rd Lunar Sci. Inst., Houston, Contrib. No. 88, p. 475 (1972). 36. M. G. Langseth, Jr., A. E. Wechsler, E. M. Drake, G. Simmons, S. P. Clark, Jr., and J. Chute, Jr., Science 168, 211 (1970). 37. C. J. Cremers and R. C. Birkebak, Proc. Lunar Sci. Conj., 8nd, Geochim. Cosmochim. Acla Suppl. 2, 3, 2311 (1971). 38. C. J. Cremers, AZAA J. 9, 2180 (1971). 39. C. J. Cremers, R. C. Birkebak, and J. E. White, Znt. J. Heat Mass Transjer 15, 1045 (1972). 40. C. J. Cremen, Moon 4, 88 (1972). 41. C. J. Cremen, Zcurus 18, 294 (1973). 42. C. J. Cremers, Proc. Lunar Sci. Conf., Srd, Geochim. Cosmochim. Acla Suppl. 3, 3,3611 (1973).* 43. R. Fryxell, D. Anderson, D. Carrier, W. Greenwood, and G. Heiken, Proc. Apollo 1 1 Lunar Sci. Conf., Geochim. Cosmochim. Acta Suppl. 1,3,2121 (1970). 44. W. D. Carrier, 111, S. W. Johnson, R. A. Werner, and R. Schmidt, Proc. Lunar Sci. Conf.. 2nd, Geochim. Cosmochim. Acla Suppl. 2, 3, 1959 (1971). 45. T. Gold, &I. J. Campbell, and B. T. O’Leary, Proc. Apdlo If Lunar Sci. Conf., Geochim. Cosmochim. Acla Suppl. 1, 3, 2149 (1970). 46. E. A. King, Jr., J. C. Butler, and M. F. Carman, Proc. Lunar Sci. Conf., 2nd, Geochim. Cosmochim. A& Suppl. 2, 1, (1971). 737 47. T. Gold, E. Bilson, and M. Yerbury, Rev. Abstr. Lunar Sci. Conf., Srd, Lunar Sci. Inst., Houston, Contrib. No. 88, p. 318 (1972). 48. I). F. Winter and J. M. Sasri, Astrophys. J . 156, 1135 (1969). 49. €3. S. Hemingway and R. A. Robie, Rev. Abstr. Lunar Sci. Conj., Srd, Lunar Sci. Inst., Houston, Contrib. No. 88, p. 369.(1972). 50. H. Kanamori, N. Fugii, and H. Mizutani, J. Geophys. R e . 73, 595 (1968). 51. H. Kananiori, H. Mizutani, and N. Fujii, J. Phys. Earth 17,43 (1969). 52. K. Horai, S. Baldridge, and G. Simmons, Proc. Lunar Sci. Conf., Houston, 1971. 53. H. Mizutani, N. Fujii, Y. Hamano, and M. Osako, Rev. Abstr. Lunar Sci. Conj., Srd, Lunar Sci. Inst., Houston, Contrib. NO. 88, p. 549 (1972). 54. J. C. Jaeger, Proc. Cambridge Phil. SOC.49, 355 (1953). 55. W. M, Sinton, in “Physics and Astronomy of the Moon” (Z.Kopal, ed.), p. 407. Academic Press, New York, 1962. 56. R. W, Muncey, Nature (London) 181, 1458 (1958). 57. J. Reichman, A I A A T h p h y s . Conj., Srd, Los Angeles AIAA Paper 68-746 (1968). 58. J. I).Halajian and J. Iteichman, Zcazus 10, 179 (1969). 59. J. Ulrichs and M. J. Campbell, Zcurus 11, 180 (1969). 60. Earl of Rose, PTOC.Roy. Soc., London 17, 436 (1869). 61. B. C. Murray and R. L. Wddey, Astrophys. J. 139, 734 (1964). 62. F. J. Low, Astrophys. J . 192, 806 (1965). 63. J. W. Lucas, J. E. Conel, and W. A. Hagemeyer, J. Geophys. Res. 72, 779 (1967). 64. G. Vitkus, J. W. L u w , and J. M. Saari, A I A A Thermophys. Conf.,Srd, Los Angeles, AIAA Paper 68-747 (1968). 65. G. Vitkus, R. R. Garipay, W. A. Hagemeyer, J. W. Lucas, and J. M. Saari, Lunar Surface Temperatures and Thermal Characteristics, Surveyor VI A Preliminary Report. N A S A Spec Publ. NASA SP-l66,97 (1968).

a,

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66. G.Vitkus, R. R. Garipay, W. A. Hagemeyer, J. W. Lucas, B. P. Jones, and J. M. Saari, Surveyor VII A Preliminary Report. NASA Spec. Publ. NASA SP-113,163 (1968). 67. L. D.Stimpson and J. W. Lucas, AIAA Themwphye. Conf., hth, San Francisco AIAA Paper 69-594 (1969). 68. F. S. Johnson, J. MetmoZ. 11, 431 (1959). 69. M.Baker, Opt. Spectra 6,32 (1972). 70. C. J. Cremers, R. C. Birkebak, and J. E. White, AIAA J. 9, 1899 (1971). 71. C. J. Cremers, R. C. Birkebak, and J. E. White, Moon 3,346 (1971). 72. J. M.Saari, Zmrua 3, 161 (1964). 73. H.C. Ingrao, A. T. Young, and J. L. Linsky, in “The Nature of the Lunar Surface” (W. N. Hess, D. H. Menzel, and J. A. O’Keefe, ed.), p. 185. John Hopkins hess, Baltimore, Maryland, 1966.

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Boiling Nucleationt

ROBERT COLE Department of Chemical Engineering and Institute of Colloid and Surface Science Clarkson College of Technology. Potsdum. New York

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I. Introduction I1. Fundamental Equations of Surface Science A. The Laplace Equation B . The Kelvin Equation 111. Homogeneous Nucleation A. Stabilityof an Activated Cluster B . Rate of Appearance of Nuclei IV. Superheat Limits A . Simplified Classical Treatment B . Recent ExperimentalObservations C . Engineering Significance V. Heterogeneous Nucleation A. Nucleation from Plane Surfaces B . Nucleation from Spherical Projections and Cavities C. Vapor Trapping VI . Nucleation from a Preexisting Gas or Vapor Phase A. Experimental Evidence B . Behavior of Gas-Filled Cavities C . Behavior of Vapor-Filled Cavities VII . Size Range of Active Cavities A. Effect of Nonuniform Superheat B . Nucleation Criteria C . Analysis and Experiment D. Characterization of the Boiling Surface VIII . Stability of Nucleation Cavities A. Theoretical Treatment for Cylindrical Geometry . B . Experimental Findings

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86 87 88 89 92 93 94 95 95 99 107 111 111 113 116 117 118 120 122 127 127 128 129 133 134 134 146

t The basis of this work was completed while the author was on sabbatical leave with the Heat Transfer Group of the Eindhoven University of Technology. The Netherlands. 85

86

ROBERTCOLE

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I S . Minimum Boiling Superheat A. Low Thermal Conductivity Liquids B. Liquid Metals Nomenclature References

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148 148 157 162 163

I. Introduction

One of the problems which has stimulated active engineering interest in the theory of riucleation from liquids is the prediction of the minimum boiling superheat. Consider the subcooled forced low boiling data of Fig. 1. The lines having a slope of unity are in the nonboiling region. When the wall temperature is increased sufficiently to cause boiling to begin, the curve turns sharply upwards. It is clear that the forced flow convective heat transfer equations grossly underestimate the heat flow density in the boiling region. In certain instances nucleation or boiling may be undesirable. The formation of vapor bubbles as a result of pressure reduction, is in fact identical to boiling. Damage resulting from cavitation can produce serious and sometimes even catastrophic results. Hydraulic machinery, valves, fittings, spillway crests, and conduit entrances and bends are all subject

BOILING NUCLEATION

87

to cavitation damage. Knapp et al. [2] present an outstanding discussion of the occurrence of cavitation and its effects upon flow properties and equipment performance. Under some conditions, explosive vapor formation can occur when molten metals are quenched. In the paper industry, sounds have been reported far below the surface of quench tank liquors which become deep, powerful rumbles or earthquake-like detonations. Such explosions have occasionally caused the unit or entire building to tremble. In rare instances tank tops have been blown off, welded side seams have split, and tanks have been displaced from foundations [3]. This review is concerned primarily with nucleation as applied to engineering heat transfer systems. Accordingly, in Section 11,the Laplace equation is developed using as a model, a wavy surface such as might exist in film flow. The Kelvin equation is derived from thermodynamic principles and from it the Laplace Kelvin equation is obtained; the latter equation being more often found in the engineering literature. A brief presentation of homogeneous nucleation appears in Section I11using the older but conceptually simpler kinetic theory approach. In Section IV, superheat limits as predicted by the homogeneous theory are developed and compared with experimental observations. Reference is made to recent evidence that homogeneous nucleation may be of great importance in some engineering situations. Heterogeneous nucleation is treated in Section V. The geometric conditions for vapor trapping, also presented in Section V lead to the conclusion in Section VI that nucleation in most boiling systems occurs from a preexisting gas or vapor phase. Sections VII-IX are concerned with the determination of nucleation criteria, the stability of nucleation cavities, and the prediction of minimum boiling superheat. 11. Fundamental Equations of Surface Science

Certain equations are fundamental to the science of surfaces and hence to a study of liquid-vapor interfaces. Two of these, the Laplace equation and the Kelvin equation will be developed here. The Laplace equation relates the pressures on opposing sides of a curved interface to its radii of curvature, and will be derived using a model surface typical of that found in wavy film-flow. The Kelvin equation relates the equilibrium vapor pressure of a liquid drop or vapor bubble to the external pressure and to the sphere radius. The equation will be derived under the condition that the “availability” of the system is stationary with respect to small variations of sphere radius; this approach being very similar to the kinetic method employed in nucleation theory.

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88

A. THELAPLACEEQUATION The model consists of a wavy surface, one of whose radii of curvature is infinite. Thus the surface might be thought of as a wavy plane as shown in Fig. 2 ( a ) . An increment of the curve which results from a normal intersection of the x-y plane and the wavy film is illustrated in Fig. 2(b), together with the external forces acting on the surface. Under steady-state conditions, the equation of motion yields

C F = F,

18

+ F,

/,+A#

+ FpV+ F p L

=

0

(1)

where F, is the surface tension force, FpVis the force exerted by the vapor, and FpLis the force exerted by the liquid. Further,

F, =

ut(2)

Fp = P Asn(I)

and

(2)

Here t and n arc the unit tangent vector and the unit normal vector to the curve, respectively, and 1 represents a unit length along the surface, normal to the x-y plane. Substituting into Eq. ( 1 ) and forming the scalar product with i, the unit vector along the x axis i . CF = u cos y

I,+A~

- u cos y

la

+ (PL- Pv)As(sin y ) = 0

(3)

Dividing by As, taking the limit as As + 0, and using the definition of the derivative,

For constant

u,ignoring the trivial solution,

But dsldr

FIG. 2. Wavy gas-liquid interface (symbols with overhead arrows correspond to boldface symbols in the text).

BOILINGNUCLEATION

89

Fro. 3. Assumed equilibrium configuration for liquid-vapor single component system.

where R is the radius of curvature. Therefore, p L

-pv

=

g/R (7) For the more general surface where the second radius of curvature is not necessarily infinite, the equivalent expression is pL - p v =

+ R1’)

~(R1-l

(8)

Eq. (8) is known as the Laplace equation, When the surface is concave toward the vapor phase, the left side becomes Pv - PL.

B. THE KELVINEQUATION The system is illustrated in Fig. 3 and consists of a vapor phase (assumed spherical) at uniform temperature To and pressure Pv, immersed in a liquid phase at uniform temperature Toand pressure Po. The surroundings are also at uniform temperature Toand pressure PO. The equilibrium condition shall be considered to be that for which the “availability” remains stationary with respect to small variations of the bubble radius T . (aA/ar) =0 (9)

IT~.P~

The availability A is a measure of the maximum amount of useful work which can be extracted from a system during a given change in given surroundings, A U - ToS PoV (10)

+

Note that it is defined in terms of the constant temperature and pressure of the surroundings, whereas the Gibbs function G is defined in terms of the temperature and pressure of the system,

G

U - T S + PV

(11)

It is convenient to express the availability of the system as the sum of the contributions due to the liquid phase, the vapor phase, and the interface. Since the availability is defined in terms of the pressure of the surroundings, these terms are additive even though the pressure of the vapor differs from that of the liquid (by 2a/r according to the Laplace equation).

ROBERT COLE

90

Thc availabilky of the liquid is A L = moL(uL- TosL

+ Pov") = moLgL(To,Po)

(12) where u , s, L', and g are the intensive analogs of U , S, V , and G, and subscript zero means that the quantity is constant and evaluated at To and Po. As indicated in Eq. (12),the avaiIability and the Gibbs function are equivalent for the liquid phase because the temperature and pressure of t.he liquid are maintained the same as those of the surroundings. The availability of the interface is

The availability of the vapor may be expressed in terms of the availability at Toand Poby a Taylor series expansion about TO,PO,

+

A'(To, P') = A"(To, Po) ( a A v / a P ) I ~ o , ~ o ( p '- PO) 4(d2Av/aP2) I T ~ . P ~-( Pol2 P~

+

+ -..

(14) Using standard thermodynamic equaIities and assuming the vapor phase to behave ideally, A'(T0, Pv)= Av(To, Po)

For the condition

mv + 2Pov -

(PV -

Eq. (15) becomes

Av(To, Pv) = Av(To, Po)

m' (P' + 2PO'

- Po)2

Po and substihting from the Laplace equation for Pv - Po, m v 2a2 Av(To, P ' ) = A'(T0, Po) -POV Po+

+

Thus upon addition of Eqs. (12) , (13), and (18), the total availability is A = ( m - %')goL

= mgoL

+

20 + 4rr20 + moVgOV+ m-v -

$dpoV(goV

POV POT2

- SOL)

+ 4w*a

(19)

BOILING NUCLEATION

91

Note that the enclosed expression in the third term of Eq. (19) must be taken as unity as a result of the approximation expressed by Eq. (16), i.e,

I Hence,

+ Q(u/rPo)= 1 + + [ ( P v - Po)/Po] = 1 A = mgoL + -$K@piJv(gov - goL) f 4R?%

(20) (21)

From the equilibrium condition

(aA/ar) I T ~ , P ~

=

and therefore,

+

4 ~ r ~ p o ~-( goL) g 0 ~ 8wu = 0

(22)

goL - gov = 2u/poVr

At a flat interface ( r = 03 ) , the pressure in both phases is identical and equal to the equilibrium vapor pressure. Thus,

V

dgv = (R,To/M)

/

PO

dP/P

(26)

Pm

Therefore, gov =

gm

- (tl,To/M) In(Pm/Po)

(27)

goL =

gm

-

(28)

In like fashion, [(pm

- P0)/pL]

Substituting Eqs. (27) and (28) into Eq. (23) yields R~ToPo' P m lnM Po

poV

2u

PL

r

- - ( P m- Po) = -

or since

(R,T~pov)/M= Po = PL Eq. (29) may be expressed as

Equation (31) relates the equilibrium vapor pressure of a bubble to its radius. When the second term on the left is neglected, it is known as the Kelvin equation. In the boiling literature, Eq. (31) is generally found in somewhat differ-

ROBERT COLE

92

ent form. Expanding the logarithmic term,

(

, p-LP L ){1 pl. P____

-i(1 P, pL- P L

)+;(p-p~~~j...I

-PV ( P , - P L )

= 2a -;

r

PL

I “LPLI

5

1

(32)

In analogy to Eq. (16), for the condition (P, - PL)/PL

0 ; ( c ) t2 > t , ; ( d ) t* > t z ; ( e ) f1 > P.

Employing a quasi-steady-statc approach, application of the macroscopic momentum balance to the liquid in the capillary yields

where inertial effects have been neglected (ie., time rate of change of

total momentum of liquid within the capillary and influx of momentum by virtue of bulk liquid entering thc cavity), gravitational effccts have been neglected, the flow has been assumed laminar, and constant physical properties assumed. P,” and PoLare the vapor and liquid pressures at the meniscus and top of the cavity, respectively. The temperature distribution in the liquid, resulting from a moving plane source of strength,

is

where X is defined in Fig. 30. Equation (111) takes convective transport into account, but incorporates the thin thermal boundary layer assumption, i.e., that the thermal ivave does not penetrate far into the liquid phase. At the interface, X = 0 and Eq. (111) becomes

wherc

T,L

is the temperature of the meniscus. Equation (109) may also

BOILINGNUCLEATION

137

be expressed in terms of the meniscus superheat by means of the ClausiusClapeyron equation

P z v - PoL = hfUPV -(T,L - Th t )

(113)

where it has been assumed that Pv = P,. Eliminating Pzv- PoLbetween Eqs. (109) and (113) and rearranging

By combining Eqs. (112) and (114), a nonlinear integrodifferential equation is obtained for x as a function of t. Rather than attempt to solve this equation, Bankoff instead approximates Eq. (112) by the expression

Before proceeding it is convenient to express Eqs. (114) and (115) in dimensionless form,

where AT, is given by:

Equations (115) and (114) become, respectively, f = Dgr-'/2

where

D= B=

b P V

pLCpLAT, 4w

Rcu cos 6

Equations (120) and (121) can be combined to eliminate f and yield a

ROBERT COLE

138

nonlinear first order differentiai equation dS

Bv ds

+ D ~ T - ~-" 1 = 0,

~ ( 0= )0

(124)

A particular solution which satisfies the differential equation and the initial condition is v =c p (125) where C = [ ( D ' + 2B)"' - D ) / B (126)

This solution is however unsatisfactory in that it does not predict a reversal in the motion of the meniscus and hence contradicts the original hypothesis. Marto and Rohsenow [SS] have modified the Bankoff analysis [57] by assuming that initial condensation is due to the cooling effect associated with microlayer evaporation rather than penetration of cold liquid from the bulk. Following bubble departure, the residual vapor that exists within the cavity is assumed to be at the static saturation temperature and the liquid at the meniscus a t all times in thermal equilibrium with this constant temperature vapor (except at t = 0 ) . The cooled liquid is assumed to have initial temperature To, which is slightly greater than the liquid saturation temperature but less than the static vapor saturation temperature. It is presumed that as a result, condensation begins to occur and the interface recedes into the cavity. (According to the Bankoff model [57], condensation should stop at this condition; compare Figs. 31 and 32.) Heat is received by the liquid from condensation at the interface and conduction from the cavity walls (which initially are presumed to have cooled to a minimum temperature To as a result of microlayer evaporation). As the liquid temperature increases, the interface motion slows and at some point

(C:

(b)

(C)

(d)

(el

FIG. 32. Diagram of assumed nucleation cycle for Marta-Rohsenow model (661: (a) t = 0;(b) ti > 0; (c) t 2 > t, (d) t* > g; (e) t3 > t*.

BOILING NUCLEATION

139

if the bulk liquid temperature equals the saturation temperature, condensation stops, the nucleus begins to grow again, and the cavity remains active. The nucleation cycle is depicted in Fig. 32. Application of the macroscopic momentum balance to the liquid in the cavity yields P,V - POL = (2u cos B)/R, (127) where in addition to neglecting inertial effects and assuming constant physical properties as in reference [57], viscous effects have also been neglected. Note however that these assumptions are all consistent with the physical model; inclusion of the viscous force for example would make the meniscus temperature a function of its velocity and hence time, whereas it has been assumed constant in the model. Application of the ClausiusClapeyron equation then yields

Tk, 2ucos8

(7)

TmL- TLt = PVbP

The temperature distribution in the liquid is obtained by solution of the energy equation. In order to maintain the problem one-dimensional and yet include conduction from the wall, this effect is assumed to be felt directly in the liquid as a time dependent heat source. The assumption is questionable for normal low thermal conductivity liquids, but may be reasonable for liquid metals. In any event, the wall heat flow density must be included if the meniscus velocity is to exhibit a minimum. Thus the energy equation, with initial and boundary conditions is

P ( 0 , t ) = TmL=

TL(a,1 ) = T,(t)

T:,

=

const

(133) (134)

where T, is the waII temperature, amis the surface area of the meniscus, and V is the capillary volume occupied by the liquid. Equation (134)presumes that far from the interface, the liquid temperature within the cavity

ROBERT COLE

140

o,+Aq

-

__ _ _ _ _ _ _

!i

8 ZL

w6a Z Z W

= $E d+z kj:

5;

-_____

I TMAX

ww

Yg

1L

=t 2i 2s 8Y -I

L , N

1

I

equals the variable cavity wall temperature. Thus, as h -+ and Eq. (134) is applied,

Q(l)/

(pLCpL) =

)TIME

00

in Eq. (129),

dTw(t)/dt

(135)

The solution of Eqs. (129)-(135) (for the interface temperature) is

where

The transient response of the wall during the period of interface travel is obtained by considering a semi-infinite body initially at uniform tempera-

BOILING NUCLEATION

141

ture, subjected to a step change in heat flow density for a finite time period. As illustrated in Fig. 33, the time period of the step change corresponds to . period of interface travel begins at bubble the bubble growth period T ~The departure when the wall temperature is a minimum. The solution is

Equation (139) which is the equation of interest is now approximated by a somewhat simpler expression and in order to relate T,,, to TmLand Tminto To,a series of approximations and empirical expressions are introduced. Substituting the resulting expression into Eq. (136),

Combining Eqs. (140) and (128), a linear integral equation is obtained for the interface position as a function of time,

the solution, obtained by use of the Laplace transform, is

When comparing this expression with that of Bankoff [57], it is of interest to note that the second term on the right side is that which is due to the condensation effect. Hence the two expressions are not similar, even though each includes a term proportional to t112. The stability requirement is that dx/dt = 0, and that the value of x for which this condition exists, be less than the cavity depth L.The usefulness of the expression resulting from Eq. (142) is severely restricted however by the empiricisms which have been introduced. The equation is also (as a result) dimensionally incorrect, thus defeating attempts to nondimensionalize it. For the sake of completeness (and to indicate the parameters) ,

where AT- is defined by Eq. (119), qo' is the average surface heat flow

ROBERTCOLE

142

T

FIG.34. Wall surface temperature variations during bubble formation and departure (Shai and Rohsenow [67]).

density, f is the frequency of bubble formation, K is the thermal diffusivity, and superscripts S and L refer to the solid and liquid, respectively. Shai and Rohsenow [67] apparently considered the wall temperature variation employed by Marto and Rohsenow [SS] to be inappropriate for Iiquid metal systems; the proper variation being of the form shown in Fig. 34 for stages 3 and 4. The latter represents the wall temperature variation during the recovery period following bubble departure and for liquid metal systems is perhaps 10-100 times longer than the bubble growth period represented by stages 1 and 2 (used by Jlarto and Rohsenow [SS]) . After the bubble has left, cold liquid from the bulk (at its saturation temperature) is presumed to contact the surface. The resulting initial temperature distribution is shown in Fig. 35. In the solid, the distribution is assumed parabolic up to a distance x = L’ where it becomes equal to the average gradient during a full cycle. The temperature difference at the wall surface (x = 0 ) at t = 0, 6,(0,0), is assumed to be the temperature required to initiate a bubble at the active cavity. The following set of equations describe the problem:

Initial conditions ( t

o ix I L’,

=

0),

e 8 ( ~0) ,

=

es(o, 0) + e,l(o, O ) X

+ e:(o,

0) ( x 2 / 2 ) (146)

BOILINGNUCLEATION

143

U

FIQ.35. Initial temperature distribution in the solid and in the liquid (Shai and Rohsenow [67]). 0

5 Y,

eL(Y,o) = 0

Boundary conditions (x = L', y 3 a),

eL(m,t)

=

o

Interface conditions (x = y = O ) ,

eye, t )

=

e.(o,

t)

The solution for the temperature distribution in the solid is

(147)

ROBERTCOLE

144

-

(t

+

Z) I)&( erfc

where This result is shown graphically in Fig. 36. In order to determine the stability of the cavity, it is assumed that the cavity radius-to-length ratio is such that the resistance to radial heat transport is an order of magnitude less than the resistance to axial transport L / R 10 for liquid metals. For this geometry then, the temperature distribution in the liquid metal (in the cavity) is equivalent to the temperature distribution in the solid. At the same time, the vapor temperature in the cavity is kept constant a t

>

where B1is a constant for a particular fluid. Condensation is presumed to proceed as long as the liquid temperature is less than the vapor temperature. When the two become equal, the meniscus penetration stops (at X , from the surface) and the cavity remains armed. The critical distance is found from the conditions 8, = ev, ae,/at

=

0, at x

=

X,,,

t

=

t,,

(155)

by means of Eqs. (152) and (154). The resulting expression is shown graphically in Fig. 37 as a function of B1. The critical distance is made

TABLE XI1 MAQNITUDE OF PARAMETERS

0.50 1.00 1.92 10.00

0.959 1.OOO

1.035 1.102

1.59 1.58 1.57 1.56

0.005 0.163.5 0.201 0.247

BOILING NUCLEATION

145

FIQ. 36. Temperature variations in the solid, Eq. (152),8, = 1.92 (Shai and Rohsenow [67]).

0.7L

0.60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (X/l3

Fro. 37. Maximum meniscus penetration from Eq. (156) (Shai and Rohsenow [67]).

ROBERTCOLE

146

dimensionless by the relaxation length L' (see Fig. 35). The cavity length > X , necessary for a cavity to remain armed is found to be

L

Numerical values of q, E, and F a r e given in Table XII.

B. EXPERIMENTAL FINDINGS Evidence that cavity penetration following bubble departure may indeed occur was first presented by Wei and Preckshot [SS]. High speed photography was used to obtain pictures of bubble nucleation, growth, and departure from heated glass capillaries having a depth of 1 mm and diameters of 1.5, 1.0, and 0.7 mm. A sequence of frames reproduced from the high speed pictures indicated a pocket of vapor in the cavity following bubble departure which produced the succeeding bubble. The vapor pocket grew and eventually emerged from the cavity as a vapor bubble. Interestingly, a liquid film was observed to exist on the capillary walls throughout the entire process. Kosky [69] employed high speed photography for the specific purpose of investigating nucleation site instabilities. Experiments were conducted using a heated glass capillary test cavity of 0.1-mm diam. and 20-mm

5pm,rb, ,

departure

I

/

W M ~ ~ departure

....

0

0 002

0.004 Time b e d

0.006

FIG.38. Motion within capillary nucleation site (Kosky [69]).

BOILINGNUCLEATION

147

FIQ.39. Minimum heat flow densities for stable boiling (Shai and Rohsenow [67]): circles, natural convection; crosses,8 , (0,0 ) ,stable boiling point; squares, Ow (0, 0 ) , stable boiling limiting point; cavity length, 1 = 0.150 in.

depth. Fifteen runs were reported with conditions from 3 X lo2W/m2 to 4.3 X lo4 W/m2 a t pressures ranging from 2.67 to 101 kN/m2 and subcoolhgs to 21°K. A plot of penetration vs. time is shown in Fig. 38 for a bulk water temperature of 295.2"K and saturation temperature of 3122°K. Little penetration occurred a t the lower pressures where the bubbles were hemispherical with an entrapped liquid microlayer under the bubble base. At higher pressures however, where the bubbles tended to be spherical and no microlayer could exist, significant penetration did occur. Thus the Kosky investigation tends to support the Bankoff and Shai models which assume initial condensation as a result of penetration of cool liquid from the bulk. Shai and Rohsenow [67] conducted experimental studies with liquid sodium boiling from a cylindrical cavity having a diameter of 0.27 mm and a depth of 3.8 mm. The limiting points were obtained by decreasing the power, after stable boiling had been achieved at high heat flow density, until boiling stopped. The limiting points shown in Fig. 39, designated by

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148

the square symbols were actually stable boiling points which deactivated with further slight decrease in power. Upon deactivation, stable natural convection began and the \Val1 temperature increased. Shown in Fig. 39 is the 3Iarto-Rohsenow equation [Eq. (143)], Eq. (156) solved for go’, and an empirical modification of the latter to take into account presumed convective effects in the liquid. The empirical modification is

With h’ = 1200 Btu/hr ft‘ experimental results.

O F ,

Eq. (157) is in good agreement with the

IX. Minimum Boiling Superheat A. Low THERMAL CONDUCTIVITY LIQUIDS The nucleation hypothesis proposed by Hsu and Graham [62] and desvribed in Sect. VI1,B was the catalyst which generated a scries of papers M hose objc1ctive was the prediction of the minimum superheat required to sustain nucleate boiling. From the analysis of Hsu [Sl] for the size range of active cavities, it is apparent that the discriminant in Eq. (106) must be greater than or equal to zero for the cavity to be active (othern-ise r* is imaginary). The discriminant is zero when

and although the discriminant is actually negative only for values of OW lying betwern the two roots, the lowcr one seems to have little significance. Hence 8,+ greater than the value given by Eq. (158) is predicted to be a necessary (but certainly not sufficient) condition for the existence of sustained bubbling from a surface which contains a spectrum of potentially active cavities. Similar expressions are obtained for the case of constant heat flow density. For that condition, Eqs. (106) and (1%) are still valid, but with 8, everywhere replaced by q’ 6/KL. The actual uses of Eqs. (106) and (1%) arc severcly restricted by a lack of knowledge regarding the proper value to use for 6 in a given system. Since 6 is not really defined, the only way to calculate it is through use of one of the final equations. Thus if incipient superheat data are available under one condition, Eq. (158) may be used to obtain the value of 6, which if assumed constant, may be used to predict the superheat or size range of cavities under other condi-

BOILING NUCLEATION

149

REFERENCE EXPERIMENTAL POINTS IN CALCULATING WC,

1

.

1

, , , 1 , 1

100

u1BCOOLING. 6&

I

l&-&

(OF?

,

I , , , ,

I000

FIQ.40. Effect of subcooling on incipient boiling temperature (Hsu [Sl]).

tions. Application of this procedure by Hsu is shown in Fig. 40 with quite favorable results except at very low values of subcooling. The analysis of Bergles and Rohsenow [70] is very similar to that presented by Hsu [Sl] except that the boundary layer thickness is more precisely defined and a graphical solution was, at least initially, considered to be more convenient. Eq. (94) is presumed to be a necessary condition

TV = T i t

+ 2uTsLst/pVhf4rn

(159)

where as indicated, the temperature is considered to be that of the vapor in the nucleus. The physical properties are to be evaluated at T;,. Thus given the system pressure, the uniform vapor temperature within a nucleus can be represented graphically as a function of nucleus radius as shown in Fig. 41. The size of the bubble nucleus is generally smaller than the thickness of the laminar sublayer in a forced convection flow, hence the temperature distribution near the wall can be approximated by the linear relation

TL = T , - (4’y/KL)

(160)

This distribution for three values of q’ is represented by the dashed lines on Fig. 41. It is postulated that the hemispherical nucleus of radius r, will grow if the liquid temperature at a distance y = r, from the wall is greater than the vapor temperature given by Eq. (159). This criterion requires that all of the liquid surrounding the vapor nucleus be able to transfer heat to the nucleus in order for it to grow. No doubt a nucleus will grow if the liquid temperature at some distance from the wall less than r,, exceeds the vapor temperature. The criterion thus seems to represent an upper limit for the wall superheat required to produce steady bubbling. On the other hand, the means of detection may not be actual observation of the bubble action but rather an observed increase in heat flow density (as on a plot of 4’ vs A T ) . Since the first few bubbles are unlikely to increase the

ROBERT COLE

150

’t

I

BUFFER ZONE

FIQ.41. Initiation of ljubble growth in forced convection (Bergles and Rohsenow 1701).

rate of heat transfer, the wall superheat associated with this indirectly determined appearance of initial bubbling will, most likely, also be higher. The graphical procedure proposed by Bergles and Rohscnow [70] for estimating the inception of boiling is illustrated in Fig. 41. Having plotted Eq. (159),the liquid temperature distribution is obtained from the relations

It is presumed that T W Lis known and that the heat transfer coefficient h’ can bo estimated from standard forced flow nonboiling correlations. Thus for a given q’, Eq. (161) relates the wall temperature and the temperature gradient. According to the criterion, steady bubbling will occur a t thc heat flow density which brings the liquid and vapor temperature curves tangent to each other T L = T’ at y = r, (162)

It is apparent t.hat once tangency occurs, a slight increase in heat flow

BOILING NUCLEATION

151

density (or wall temperature) results in the activation of a wide size range of additional cavities having radii both smaller and larger than that corresponding to the tangency condition. Although agreement with experimental data was quite good, there are two additional conditions which must be fulfilled in order for the method to work. First, a full spectrum of cavity sizes must exist and second, the cavities must contain entrapped gases or vapor. Fortunately, these conditions do exist for many commerically finished surfaces. A similar approach was independently proposed by Sat0 and Matsumura [71]. Rohsenow [65] has more recently indicated a problem which may occur when the heat transfer coefficient h' is low, e.g., low velocity forced flows or in natural circulation flows. The situation then arises where the slope (dTL/dy)y,o as calculated from Eq. (161) may be tangent to the vapor temperature curve, Eq. (159), at a point beyond the maximum existing cavity size. The first liquid temperature profile to intersect the vapor temperature curve at the maximum cavity size would then satisfy condition (162), but not condition (163), and consequently a higher than normal superheat would be required. Under normal conditions, the heat flow density at incipience is given by

obtained from Eqs. (159) and (161) using the conditions specified by Eqs. (162) and (163). For the condition where because of a low heat transfer coefficient, tangency does not result and Eq. (163) can not be satisfied, the heat flow density at incipience is determined by Eqs. (159), (161), and (162) as

Equation (165) is plotted in Fig. 42 for various values of rmax.Rohsenow [65] indicates that although much of the available experimental data is well represented by Eq. (164), another large body of data exists to the right at higher superhe%ts. None however falls to the left. The trend prediced by Eq. (165) is indicated to be difficult to observe. As noted by Davis and Anderson [72], a similar situation may also exist under conditions of normal velocity forced flows if the surface contains only a limited size range of cavities. Again, the tangency condition expressed by Eq. (163) would not be met and the heat flow density at incipience might be as expressed by Eq. (165). As indicated by Davis and Anderson [72] and as is apparent from Eq. (165), specific information on the maximum cavity size is necessary to apply the analysis.

152

ROBERT COLE

Frost and Dzakowic [73] have reported an empirical extension of the Bergles-Ilohsenow analysis [70] to fluids other than water and to natural convection systems. The approach is to modify Eq. (163) t o yield d T L / d y = dT\’/drn

at

1/ =

(Npr)nrn

(166)

The former is rationalized on the basis that y is the depth of superheated liquid adjacent to the heated surface which has temperature in excess of T’, and in single-phase heat transfer, the Prandtl number will dictate the shape of the temperature profile within this boundary layer. As shown in Fig. 13 howver, when compared with experimental data, and with the Berglcs-Rohsenow analysis, the modified approach does not appear to yield substantial improvement except perhaps a t very high and very low reduced pressures. In Fig. 43, X is defined as

Included are data from both forced and natural convection systems. A much different approach has been taken by JIadejski [74]. Having obviously been motivated by the paper of Hsu [Sl], he asks, why must the activation of a riucleus be governed by the superheat at the top of the bubble, rather than that existing at the bubble base (i.e., at the wall)? Further, JIadejski apparently disagrees with both the Hsu [Sl] and Griffiith and Wallis [59] interpretation of Eq. (94).Griffith had assumed that the temperature identified as T L in Eq. (94), which was derived assuming uniform superheat, might be interpreted as T , (the surface temperaturc) in a nonuniformly superheated system. Hsu had assumed that

Fro. 42. Incipient heat flow densities for 10s velocity flons, Eq. (165) (Rohsenon 1651).

BOILINQ NUCLEATION

153

5.0

Hydrogen Water PreonA Benzene X Nitrogen Oxygen 0 Neon Ethanol d-pentane 0 Carbon T e t r a c h l o r i d e 0 Acetone a Kerosene 0

+

0.04 0.002

l

c

n

0.005

t

n

l

0.01

.

1

I

I I I I I I I

0.05

*

0.1

1

I

L

'

'

0.5

1

1

1

1

1.0

REDUCED PRESSURE, Pr

FIG.43. Dimensionless incipient boiling conditions versus reduced pressure (Frost and Dzakowic [73]).

this same temperature should be interpreted as the vapor temperature for a nonuniformly superheated system. Madejski considers the original interpretation to be correct (i.e., TL).Starting with the Laplace-Kelvin equation [Eq. (34)], generalized to allow for a nonspherical surface, 1 p - P' L (Pa- PL) = .(-1 + -)

RI Rz P , - PLis then approximated by P,' (TL - T,J , where P,' is the slope PL

ROBERT COLE

154

of the equilibriuni vapor pressure curve, to give

where if R1 = Rz, the nucleus is spherical and TLis constant. On the other hand, if TL is a function of position as in the thermal layer adjacent to a heating surface, then the radii of the active nucleus is a function of position and uill depend not only on the wall superheat, but also upon the temperature gradient. The model employed by Nadejski is shown in Fig. 44 and assumes the nucleus to be symmetrical about the y axis, thus resembling a flattened spheroid. Defining a dimensionless temperature difference in terms of the wall superheat,

iL= BL/0w

=

(TL- Tsat)/(Tw.- Tsnt),

tL(O)

=

1

(170)

and noting that for the case of uniform superheat, Eq. (169) becomes AT, =

pL

2a

~pL - pv P,'R,

upon substitut.ing Eqs. (170) and (171) back into Eq. (169) and rearranging

where by definition,

I: FIG.44. Nucleation model (Madejski 1741).

BOILING NUCLEATION

155

To complete the definition of the problem, the temperature distribution is defined to be linear, so that

-

fL(y)= 1

( y / 6 ) for y

5 6,

f L ( y ) = 0 for y

>6

(174)

and the boundary conditions are specified to be

y(Rd

Ir=~c

=

0,

y(0) = b

tan 0 = tan ?r/2 = m, dy/dr =0 (175) In order to obtain an analytical solution to the problem, Eq. (172) is approximated by dyldr

=

AT AT,

and the mean value of R1/R2is obtained by assuming the nucleus to be elliptical in shape and = ?r/2rad,

+

+ (b/RJ2]

(Ri/Rz), = +[I

(177)

The solution to the problem as originally presented by Rfadejski [74] contained a minor error. The corrected solution as presented by Schmidt and Cole [75] is AT AT,

- ‘12 Rc (1 ki2

6

+ ;{I +

5

[l -

( 2 ~ ~ ~ f ) (178) ~ ~ } )

The dimensionless temperature gradient is

RJ6

=

[ 2 ( 2 - k12)]-1’2f ( 2 - ki2)[F(ki,

+T) -

- 2[E(ki, i n ) - E ( h , h ) ] )

F(ki, &a)] (179)

where F and E are elliptic integrals of the first and second kind, defined as

~ ( ke) ~=,

/a

8

(1

- k12 sin2 e)ll2 de

and kl is a parameter varying from 0 to 1. These results are shown graphically in Fig. 45. Unfortunately, there are as yet no experimental data available to compare with the Madejski analysis. The experimental difficulties consist primarily of determining the temperature profile in the liquid at or just prior to the appearance of a vapor bubble. Because of the thinness of the thermal layer (generally measured in micrometers) mechanical prob-

ROBERT COLE

156

i

51 I

0

1

2

I

1

3

1

Rc/b

FIG. 45. Uimensionless superheat versus dinlensionless temperature gradient from Madejski model. Eqs. (178) and (179) (Schmidt and Cole [75]).

ing techniques are questionable. Studies are currently being conducted [76] in which the resistance heating periods are of such short duration (5-50 msec) that the liquid temperature profile is given quite accurately by the conduction equation. Artificial nucleation sites of known size and geometry provide R,, while high speed close-up motion pictures at 16,000 frames/sec provide the time at which nucleation occurs. A technique not yet employed for nucleation inception studies, but one which holds great promise, is laser interferometry combined with high speed photography [77]. Not only can the temperature distribution in the liquid be accurately deterniirird at the moment of nucleation, but provided that the bubble grows symmetrically about a vertical axis, the liquid temperature profile surrounding the gro\ving bubble is revealed. The latter information is of value in determining the relative importance of SIarangoni flows (surface tension driven flows) [78, 79J at the liquid-vapor interface. Kenning and Cooper [SO] describe an experimental investigation of the flow patterns past an air bubble, which they use to formulate expressions for the initiation of boiling during forced convection. The accompanying analysis is based upon the criteria that the nucleus will grow when the

BOILING NUCLEATION

157

bubble surface temperature exceeds the temperature required to maintain the equilibrium value of excess pressure in the nucleus. The latter is expressed by

The bubble surface temperature is obtained for two limiting cases, NRJVpr>> 1 and N d p r > 1,

Upon comparison with experimental data, the best representation was obtained by changing the multiplying constant (0.93) to 1.33, and the exponent from 0.73 to 0.6, over the range

B. LIQUIDMETALS High thermal conductivity and chemical reduction power cause liquid metal systems to nucleate at higher wall superheats than predicted by the methods of the previous section. In terms of the Bergles-Rohsenow graphical procedure, for a given heat flow density, a high thermal conductivity results in a low liquid temperature gradient which in many cases will be tangent to the vapor temperature curve at cavity radii larger than any existing on the heating surface. Thus the first cavity to nucleate will tend

158

ROBERTCOLE

to be the largest existing cavity. The corresponding superheat is given by Eq. (165), but the maximum cavity radius must be known. The alkali metals, having high chemical reduction powers, tend to remove the oxide coating on a heating surface and consequently wet and penetrate the larger and wider cavities much more readily than do other coolants. The removal of additional existing large cavities as potential nucleation sites increases the incipient superheat even more. Other factors which tend to increase the superheat are the lower slope of the vapor pressure curve in the regions of practical interest and the increase, rather than decrease, in solubility of inert gases with increasing temperature. Holtz [Sl] suggested that the pressure-temperature history of the cavity affected the incipient superheat and proposed an equivalent cavity model to account for the effect. The depth of penetration of liquid into a cavity is presumed to be determined by the maximum liquid pressure PL’ and corresponding liquid temperature TL’ to which the system has been subjected. At this condition, i.e., the point of maximum penetration, the surface is concave toward the liquid (a nonwetting condition) as the oxide coating has not yet been reduced. Once nucleation occurs, the surface flips and becomes convex toward the liquid. This sequence of events is shown in Fig. 46. The equivalent cavity model assumes that nucleations will occur from unwetted cavities having mouth radii less than the cavity radius of the position of greatest penetration. Thus the equivalent cavity radius is given by req =

2a( TL’) PL’

- Pv(TL’)

The pressure difference then required for nucleation is

FIQ.46. Holtz model for cavity nucleation in alkali liquid metal systems: (a) a t initial filling; (b) a t deactivation; (c) at incipient vaporization (Deane and Rohsenow 1871).

BOILING NUCLEATION

159

!

P I (psi01

FIG. 47. Incipient boiling superheats obtained by Chen [85], compared with the Holtz model, Eq. (187) and the Chen model, Eq. (188) (Dwyer [86]). Experimental conditions for potassium: TL’= 1180”F, Pv’= 4.25 psia; Tsst = 1413”F, P L = 16.0 psia.

and the corresponding superheat is obtained by assuming P v = P , and using either the Clausius-Clapeyron equation or vapor pressure data. Comparison of this model with experimental sodium data by Holtz and Singer [82, 831 yielded only qualitative agreement. Heat flow density, in addition to temperature-pressure history was found to influence the incipient superheat. A subsequent paper by Singer and Holtz [84] ascribed the heat flow effect to inert gas diffusion out of the cavities. Chen [85] modified the Holtz model to account for the inert gas entrapped in a cavity. The limiting expression for narrow cavities was

where Go is a measure of the amount of inert gas entrapped. Because of the difficulty of obtaining advance information on this quantity, it is treated as an empirical constant. Equation (188) was compared with experimental potassium data, varying the deactivation pressure PL’, the deactivation temperature TL’,and the boiling pressure PL.Figure 47 is typical of the

ROBERT COLE

160

t

.

0

P l (psi01 Fic. 45. Correlation of Chen's data [85),by the model of Dwyer [SG]. Experimental conditions for potassium: TL' = 1180"F,Pv' = 4.25 psia; TL* = 1413"F, P L = 16.0 psia.

results so obtained, a value of Go equal to 8 X lo-'' in.-lbf/"R being used in all cases. The trend of the data is certainly well predicted and the agreement might even be considered good if it were not for the degree of scatter which is evident. A further modification of the Holtz model has been suggested by Dwyer [SS]. Shown in Fig. 48 are the same data points presented in Fig. 47. The solid line through the data, corresponding to the left ordinate is obtained by assuming a loss of inert gas from the active cavities between the time of maximum boiling suppression and the time of incipient boiling. The solid line corresponding to the right ordinate shows that the assumed loss over the time period of the experiment amounts to approximately one-third of the original gas content. An additional factor incorporated into the Dwyer analysis is the inequality between the bubble radius at maximum boiling suppression r' and that a t incipient boiling r, In Fig. 48, this ratio is taken as 0.7. At maximum boiling suppression the pressure difference is

where the third term on the left represents the partial pressure of inert gas

BOILING NUCLEATION

161

in the cavity and p' incorporates the assumed geometric factors, the mass of inert gas in the cavity, and the gas constant. At incipient boiling,

Equations (189) and (190) are used t o calculate the incipient boiling wall superheat when p', p/p', and r/r' are known or treated as parameters. As indicated earlier, in Fig. 48, @/p' = 1 and r/r' = 0.7, and 0 = p' varies in.-lb,/OR. from an initial value of 3.1 X 10-15 to 2.0 X Dean and Rohsenow [87] obtain essentially the same equations as Dwyer without having to rely upon oxidation and partial reduction of the surface as part of the mechanism. Assuming the existence of reentrant-type cavities on natural surfaces, as shown in Fig. 49, at deactivation the vaporliquid interface hangs at the inner mouth and is concave toward the liquid. For this condition, PL' - Pv' - Po'(t) = (2a' sin +)/Td (191) Cavities having mouth radii greater than rd will deactivate while those having radii less than rd will remain armed. At nucleation

Pv $- PQ(t) -. PL= 2U/rd

(192)

Equations (191) and (192) are essentially equivalent to Eqs. (189) and SODIUM

i 1

VAPOA LIQUID INTERFACE AT DEACTIVATION

FIG. 49. Incipient boiling model for reservoir cavity (Deane and Rohsenow [87]). Deactivation, PL' - Pv' - Po'(t) = (%'sin + ) / r d ; nucleation, PV Po(t) --PL =

+

2u/r4.

162

ROBERTCOLE

(190). The above model was first suggested by Bankoff [SS] in an invited lecture presented a t the International Symposium on Cocurrent GasLiquid Flow, University of Waterloo, Waterloo, Ontario, 1968.

ACKNOWLEDGMENTS Much of this review was presented as a series of lectures during the 1971 Fall Semester at the Technische Hogeschool Eindhoven, Nederland where the author was privileged to spend a full year on sabbatical leave. The author is indebted to Professor Dr. D. A. de Vries for his continued support and encouragement, to the engineers, faculty, and students who took the time to attend his lectures, and to his very good friend Dr. S. J. D. van Stralen without whose advice, encouragement, and kind words, this review might never have been written.

NOMENCLATURE a

A A' b b'

B 3 1

CL

CW

C c 1

CP, c3

Surface area Availability, defined in Eq. (10) Defined by Eq (103) Height of bubble above surface, Fig. 23 Dimensionless pressure, defined by Eq. (61) Constant, defined by Eq. (123) Constant Gas concentration in liquid phase Gas concentration in a saturated liquid across a flat interface Constant, defined by Eq. (126) Constant, defined by Eq, (137) Constants

c4, cs CP

d d'

D

ED E=

f fl

f? f3

14 F 9

G

Specific heat defined in Fig. 10 Defined in Fig. 14 Constant, defined by Eq. (122) -4ctivation energy for diffusion Activation energy for cluster of size x Frequency factor Defined by Eq (74) Defined by Eq. (86) Defined in terms of Eq. (86) Defined by Eq. (87) Force Gibbs function per unit mass or chemical potential Gibbs function, defined by Eq. (11)

Measure of quantity of inert gas entrapped in a cavity Planck constant Heat transfer coefficient Latent heat of vaporization Rate of formation of nuclei per unit volume Boltzmann constant Parameter varying from 0 to 1 Thermal conductivity Depth of cylindrical cavity Relaxation length, Fig. 35 Mass Molecular weight Number of molecules per unit volume Active cavity population density Number of cavities having radii in the range r to T AT Prandtl number Reynolds number Pressure Slope of equilibrium vapor pressure curve Average surface heat flow density Heat flow density Rate of internal hest generation per unit volume Bubble radius Cavity mouth radius Equilibrium cluster radius Nucleus radius

+

BOILING NUCLEATION Radius of spherical projection or spherical cavity Maximum existing cavity radius Radius of curvature Radius of cavity Gm constant Entropy per unit mass Entropy Time Dummy time variable Temperature Bubble surface temperature Equilibrium temperature corresponding to pressure within bubble nucleus Bulk liquid temperature Internal energy per unit mass Internal energy Volume per unit mass Velocity of liquid phase Friction velocity Volume Average thermal velocity of a vapor molecule Coordinate direction Defined by Eq. (167) Thermal layer thickness, defined by Eq. (107) Coordinate direction Compressibility factor Effective surface tension, defined by Eq. (77) Angle defined in Figs. 12-14 Angle defined in Figs. 12-14 Contact coefficient, defined by Eq. (153) Angle defined in Fig. 2 Thermal layer thickness Dimensionless temperature difference Dimensionless distance

e eL K

X A, p V V’

€ P U T 7.

4

+

163

Contact angle measured through liquid phase Temperature difference TL - T m Thermal diff usivity Coordinate direction, defined in Fig. 30 Heat of vaporization per rnolecule Absolute viscosity Kinematic viscosity Defined by Eq. (93) Dimensionless temperature difference Density Surface tension Dimensionless time Bubble growth period Angle defined in Fig. 15 Defined by Eq. (110)

SUBSCRIPTS AND SUPERSCRIFTS 0 Reference state 03 Refers to a flat interface, i.e., an infinite radius of curvature * Refers to an activated state b Refers to value a t top of bubble nucleus 0 GaS 1 Interface k Critical bubble condition L Liquid m Meniscus n Nucleus P Pressure S Solid sat Saturation value S Solid T Total V Vapor W Refers to value at wall U Surface tension

REFERENCES 1. W.M. Rohsenow and J. A. Clark, “Heat Transfer and Fluid Mechanics Institute.” Stanford Univ. Press, Stanford, California, 1951. 2. R. T. Knapp, J. W. Daily, and F. G. Hammitt, “Cavitation.” McGraw-Hill, New York, 1970. 3. W. Nelson and E. H. Kennedy, Paper Trade J . 140, No. 29,50 (1956). 4. R. Becker and W. Doring, Ann. Phys. (Leipzig) 24, 719 (1935).

164

ROBERT COLE

5. 31. Volnier, “Kinetic der Phasenbildung.” Steinkopff, Dresden-Leipzig, 1939. [Engl. transl., “Kinetics of Phase Formation,” Ref. AT1 No. 81935 (F-TS-7068-RE). ti. 7. 8. 9. 10. 11.

12. 13. 14. 1.5. 16. 17. 18. 19. 20.

Clearinghouse Fed. Sci. Tech. Inform., Springfield, Virginia. J. Frenkel, “Kinetic Theory of Liquids.” Dover, New York, 1955. H. Reiss, I d . Eng. Chem. 44, 1284 (1952). J. I a t h e and G. M. Pound, J . Chem. Phys. 36, 2080 (1962). 11. Reiss, J. L. Katz, and E. R. Cohen, J . Chem. Phys. 48, 5553 (1968). Sucleation Phenomena, I d . Eng. Chem. 44, 1269 (1952). .I.S. Michaels, chm., “Symposium on Nucleation Phenomena.” Amer. Chem. SOC., Washington, D.C., 1966. J. P. Hirth and G. M. Pound, Progr. dfater. Sci. 11, 1 (1963). A. C. Zettlemoyer, ed., “Nucleation.” Dekker, New York, 1969. 1:. B. Kenrick. C. Y. Gilbert, and K. L. Wisrner, J. Phys. Chem. 28, 1297 (1924). L. J. Briggs, J . Appl. Phys. 26, 1001 (1955). I La; this

case they called the “thermal

1 This correlation has been reported in the literature [2, 191 with the wrong coefficient 0.3 rather than 0.03.

HEATTRANSFER IN FLUIDIZED BEDS

175

FIG.2. Correlation of data for gas-solid heat transfer for Re < 100 (from Kothari [381).

olK1'

O . ~ ' O . l*

V Donnadlou. [32]0 HoortJrrond WKIkblnS. 1211) 0 Wolton st ol.125J A Kottonrlng at 01. c221 ' I0 ' ' loo ' " 1000

'"""I """'

"""'

Re

(2) L < La; here resistance t o heat transfer between gas and solid must be considered.

The height of the active section L , was given for fluidization in air as:

La/d, = 0.18Re/(l

- c)

(15)

As seen from Eq. (15), the height of the active section is of the order of 10-100 particle diameters for most fluidized beds met in practice. For the case of L > La,Gelperin et al. [40] suggested the following correlation: Nu, = BRePr

(16)

which for air simplifies to:

Nu, =

0.24 Re d, Lmt(1 - Emf)

For the case of L < La, Gelperin and Einstein [41] prefer to plot their data in terms of Nu, Pr+.33 vs Re/e. Figure 3 provides such a plot that these authors have prepared from data of many investigators [20, 22, 24, 25,27,32,34,35,37,42-551. In order to keep matters in the right perspective, the present authors have added to Fig. 3 the dotted line which represents Kothari's [38] correlation. This was done by arbitrarily assuming B = 0.5 and Pr = 0.72. The fact that the simple correlation passes through the data only points to the unlimited possibilities of a log-log plot.

176

CHAIM GUTFINGER AND NESIM ABUAF

D. THEORETICAL IfODELS Referring to Fig. 2 one notes that coefficients for fluid-to-particle heat transfer are much lower than one u-ould expect considering the exccllent heat transfer characteristics of fluidized beds. The high heat, transfer rates are mainly due to the large surface area of solids available for the contacting operation rather than high heat transfer coefficients. The minimum heat transfer coefficient for a single sphere in an infinite fluid, for the case where convective heat transfer to the fluid may be neglected, is: as Re+O (18) (Nup)min = 2 As seen in Fig. 2, there are heat transfer coefficients encountered in fluidized beds which are several orders of magnitude lower. 1. The Microbreak Model of Zabrodsky

In this model [56], it is postulated that the true heat transfer coefficient cannot fall below a value given by Rowe [57] as: ~u~~~ = 2 4 1

-

(1 -

4l/31

(19)

Thc solid may be viewed as arranged in horizontal rows. The apparent low Nusselt numbers stem from part of the fluid bypassing several rows of particles and then mixing with the fluid that went through the particles. The gas temperature for the nth mixing after the nth microbreak in the solid is a function of the fluidization velocity. No clue is given to the number of “microbreaks” present in a given bed. In a later paper Zabrodsky [58] developed the idea of low effectivevalues of Xu appearing simultaneously with high true values of Nutr due to the nonuniformity of gas distribution. An expression valid for low Peclet numbers was developed for the calculation of the effective Nu:

For low Pe, Eq. (20) will predict reduced effective Nusselt numbers. 2.

The Bubbling Bed Model

Iiunii and Levenspiel [l8, 19) have extended Davidson’s [13] single bubble model to a bed uith a rising crowd of bubbles. According to this model most of the gas rises through the bed as a swarm of bubbles, while a small fraction of it rises through the emulsion phase. Each bubble is surrounded by a thin layer of emulsion called the cloud. Heat and m s interchange occur from the bubble through the cloud and

HEATTRANSFER IN FLUIDIZED BEDS

177

FIG.3. Fluid-to-particle heat transfer as compiled by Celperin and Einstein [41] with Kothari's correlation [38] added. I. Data of Walton et al. [25], Richardson et al. [27], Donnadieu [42], and Juveland et al.

WI.

11. Heertjes et al. [21], Brun et aE. [44], and Lindin et al. [45]. 111. Anton [26]. Line a-a: single sphere, Rowe et al. [59]. 1. Heertjes [46] 9. Kazakova et al. [51] 2. Ravdel et al. [47] 10. Ciborowski et al. [52] 3. Donnadieu [32] 11. Beilin et al. [53] 4. Shimanski [48] 12. Sunkoori et al. [35] 5. Crishin 1491 13. Kettenring et al. [22] 6. Rosental [34] 14. Haruaki [54] 7. Bradshaw et al. [24] 15. Frantz [37] 8. Vasanova [50] 16. Peters et al. [55]

into the bed. When a bubble passes through a bed it is followed by a wake of particles. The solids mix inside the bed by rising to the bed surface in the wake behind the bubble and descending to the bottom through the emulsion phase. A small voIume fraction of solids (less than 2%) is also present within the bubbles. The emulsion phase is maintained a t minimum fluidization velocity, while the bubbles are assumed to rise with a velocity u b > 5umf. If uo and U b are the superficial gas velocity and the bubble velocity respectively, then the fraction of bed consisting of bubbles 6 is given by: 6=

-

(&l U m f ) / U b

=

1 - ( Lm f/ L )

(21)

Heat transfer coefficients may be defined for the bed depending on the temperature used. If the temperature of the leaving gas, consisting mainly of bubble gas, is measured, we have what may be called an overall driving force T f b - T. with a corresponding coefficient hoverall.If the gas temperature inside the bed is measured, say, by a suction thermocouple, an

CHAIM GUTFINGER AND NESIM ABUAF

178

average temperature Trbetween the bubble and emulsion gas will be read. This will result in an apparent driving force T, - T, and a corresponding apparent heat transfer coefficient h.PP. The two coefficients are related by: (22) The heat transfer coefficient is evaluated by equating the heat lost by the gas passing through a bed section to the heat transferred to solids: 6happ

urprcp, dTr

=

=

hoverall

happa(Tr - T,) dL

(23) Now it is assumed that gas enters the bed as hot bubbles. An accounting of heat loss by bubble gas in terms of the flow pattern of the bubbling bed model and the local, or true, heat t-ransfer coefficient, gives : by [ heatinlost bubble

gas

1=I

heat taken up by solids in bubble

1

$-

to [ heat transferred cloud 1 gas

or, in symbols: -PrUbCpf-

dTrb = ?’bhta’(Tfb- Ts) dL

+ H(Tfb - Ts)

(24)

whrrc h , is the local or true heat transfer coefficient between single particle and gas; Y b is the ratio of solids volume in bubble; a’ is the specific surface area of solids, and H is a transfer coefficient between bubble and cloud given by :

Combining Eqs. (22)-(25), one obtains an expression for the apparent heat transfer coefficient : 1 Nuapp= -(ybNut 1--E

+ 6kr

Here Xut is the local Nusselt number for a particle immersed in the bubble far from interfering particles, moving with a terminal particle velocity. One may use the equation of Rowe et al. [59] for its computation: Nu

=

2

+ 0.69 Re’/*Pr*”

(27)

For a given bed of solids Eq. (26) is of the form: Nuapp= A (Re)

+B

(28)

where A is a constant which increases with particle size and B is a very small constant depending on solid content in the bubble.

HEATTRANSFER IN FLUIDIZED BEDS

FIG.4. Comparison of the calculated curves A’ heat transfer in fluidized beds.

-

179

H’ with reported data on gas-solid

Using the bubbling bed model, Kunii and Levenspiel [19] were able to correlate the experimental data of Heertjes and McKibbins [21] and Kettenring et al. [22]. Figure 4 shows the calculated versus the experimental heat transfer coefficients. The excellent fit was possible by adjusting the bubble diameter db. As there are no fluidized beds with single-diameter bubbles, one should view db as an adjustable constant for each particle diameter rather than a measured physical quantity. Still, the seemingly correct power on Re and also the reasonable values required for db show that the bubbling bed model can explain experimental data, including the drastic reduction in A uidized bed heat transfer coefficients as compared to the one for single particles.

E. LIQUIDSOLID FLUIDIZED SYSTEMS Although not as popular as gas-fluidized beds, liquid-fluidized systems lately find important industrial applications such as the design of naturaluranium-water-fluidized nuclear reactors [SO], heat removal from water in desalination processes [Sl], and the use of fluidized particles inside vertical water heat exchangers in order to prevent scale deposition on the heat transfer surfaces [SZ]. Liquid-solid systems generally give rise to particulate fluidization over the whole range of liquid velocities employed. The works of Holman et al. [SO] and Sunkoori and Kaparthi [35] represent typical experimental heat transfer studies in liquid-fluidized beds. Zahavi [Sl] views the heat transfer process as a Wiener process. A more

CHAIM GUTFINGER AND NESIM ABUAF

180

applied aspect of the problem is presented by Trupp [63] who lists 203 references in a review of fluidization relevant to design of a liquid-fluidized bed nuclear reactor. Jfost of the heat transfer studies in liquid-fluidized beds look for similnrities to gas-fluidized bed systems in approach and in the resulting empirical corrclations. As seen from line 12 in Fig. 3, the data of Sunkoori and Kaparthi fall within the range of data for gas-fluidized systems. The present authors believe that due to the more predictable behavior of liquid-fluidized beds a more basic approach to the problem is possible. It seems like the work of Richardson [12] on particulate systems is a step in the right direction.

IV. Heat Transfer between a Fluidized Bed and a Surface The heat transfer rates betwecn a surface and a fluidized bed are much highcr than in single gas flow. In order to explain this phenomenon and predict heat transfer rates for design purposes, several investigators have studied this probleni experimentally and many models have been proposed. The literature in the field is tremendous. The authors have tried to examine, study, and classify this gigantic amount of data, but still it is obvious that they cannot pretend to have covered the whole field. A.

lrARIABLES AFFECTING THE

HEATTRANSFER RATE

In addition to the complexity of the phenomenon, one also realizes that the number of variables affecting the heat transfer rate is very high. A simple consideration of the effect will lead to the following list of the variables involved (one should mention a t this point that all kinds of radiation effects are going to be excluded from the analysis until very high temperature effects will be introduced later) : properties of the fluid, pr, p , cDf, kr ( 2) properties of the granular solid material, d,, p s , cfi, k,, & (3) conditions at minimum fluidization, u,r, t , r (4) fluidization conditions, uo, t ( 5 ) geometric variables, D, D H , LO, LH (1)

A simple dimensional analysis nil1 provide the following nondimensional parameters, relating all the effective variables [2]

With such a large number of nondimensional variables, it is quite easy to visualize the experimental difficulties to be encountered when the effect of

HEATTRANSFER IN FLUIDIZED BEDS

181

only one variable on the heat transfer rate is investigated. The same point also explains the difficulty and, in general, the impossibility of comparing experimental results of different investigators. Any comparison between graphs which represent the variation of the heat transfer coefficient with respect to one variable, without all the others being specified clearly, must be primarily qualitative in nature.

B. MECHANISMS PROPOSED FOR THE HEAT TRANSFER PHENOMENA BETWEEN A FLUIDIZED BED AND A SURFACE Many investigators have tried to model the wall-bed heat transfer phenomena in a fluidized bed and several mechanisms have been proposed. We will try to classify these models in a way similar to that of Kunii and Levenspiel [a], and also present and explain the development of the different ideas through time. After presenting the different models, we will try to relate them, determine the criteria that show which mechanism controls the heat transfer rate, and aid in the selection of a model which will represent a particular experimental or design situation. 1. Steady State Conduction across the Gas Film

In this model heat is conducted through the gas boundary layer near the heat transfer surface. In order to explain the high values of the heat transfer rates obtained experimentally, this gas layer was assumed to be scoured by solids moving along the heat exchange surface, decreasing the boundary layer thickness near the wall. Such a model has been developed by Leva et al. [64-66], Dow and Jakob [67], and Levenspiel and Walton [SS]. Dow and Jakob [67] studied the behavior of the fluidized gas-solid mixtures and classified it as the “dense phase” and the “lean phase” according to the gas velocity. In the “dense phase” fluidization, they observed three types of behavior: bubble formation, slugging, and mechanically smooth fluidization. Their model applied to the mechanically smooth, dense phase fluidization. The motion of the particles in the fluidized bed revealed a pattern as shown in Fig. 5. The motion is primarily upward a t the center of the tube and downward along the tube wall. Temperature measurements showed that the whole bed was at a constant temperature except for a small region a t the bottom (referred to later as the active section [41]) and a narrow layer near the tube wall. Heat is transferred radially through a thin air film to the particles and air moving downward outside this film. The particles carry the heat to the bottom of the bed where thermal equilibrium is attained instantaneously with the incoming cold air. The bombardment with small particles prevents the formation of a laminar boundary layer, and leaves a very thin

182

CHAIMGUTFINGER AND NESIM ABUAF

HEATEI-

COLD AIR IN (Ti) FIG.5. Heat transfer mechanism 1 and motion of the particles.

laminar sublayer and a thicker turbulent layer. Their experimental results are correlated by a simple dimensional analysis as an expression which will be presented in Tables 1-111. Levenspiel and Walton [SS] present a similar model where the resistance to the heat flow is due to a laminar gas layer which is destroyed by solid particles passing through it. Thus the average thickness of the laminar layer is much less than in an empty tube. They assume the particles are of uniform diameter d,, stationary, and arranged in equally spaced horizontal layers. The boundary layer formation at the tube wall is similar to that of a flat plate. At points of contact with the stationary solids, the boundary layer is destroyed and starts once again, Fig. 6. The distance between two successive layers is given as : D, = ~cE,/6(1- E ) 129) As 6f is the average boundary layer thickness between two points, then the

HEATTRANSFER IN FLUIDIZED BEDS

183

FIG.6. Laminar boundary layer thickness between two layers.

heat transfer coefficient is given by:

h, = kJ6+ and ti+ = (10/3) (@J%pr)”y

where = [(I

and

+

m 3 / 2

/3 = 0.041 (1

- 831

- E) (Dr%pr/p)

‘I2

(33)

These models do not take into account the influence o the s o L particles on the heat transfer phenomena. The picture presented and the mechanism proposed cannot be considered to be complete. 2. Unsteady Heat Conduction by Single Particles in Direct Contact with the

Heat Exchange Surface That high heat transfer coefficients are obtained in gas-fluidized beds, that thermal equilibrium is very quickly established between the gas and the particles in the bulk of the bed, that the entire thermal gradient is restricted in the immediate region adjacent to the wall, and that groups of randomly packed particles convey heat from the surface to the bulk of t,he

CHAIMGUTFINGER AND NESIM ABUAF

184

160 FLUID

DISTANCE 100

(P) TW

Tb

DISTANCE (/A

FIG.7. Unsteady heat conduction through a single particle (Botterill et al. [71]).

bed has already been mentioned. Photographic studies near the wall also showed that particles tend to associate in groups, move slowly along the wall, or remain in contact with it a certain time. These observations led Botterill et al. [69, 70, 721, to propose that heat is transferred from the vertical surface to the adjacent layer of particles arriving together at the wall. They confined their theoretical considerations to an isolated solid particle lying in contact with the wall. If the residence time of the particle near the wall is long, heat will penetrate further into the bulk and if the residence time is short, heat exchange calculations can be limited to only one particle, the nearest to the surface. By neglecting conduction through the point of contact, radiation effects, and thermal convection through the surrounding fluid, they solved numerically the unsteady heat conduction between a plane surface and an isolated spherical particle surrounded by static gas, initially a t the temperature of the bed (Fig. 7 ) . Conduciion equation:

where subscript i

=

1 (solid) and i

=

2 (fluid).

Heat exchange across the solid boundary and the fluid phase:

HEATTRANSFER IN FLUIDIZED BEDS

185

This system of equations was solved numerically for a 200-p particle with residence times up to 70 msec. The average rate of heat transfer to a single particle during a given residence time was calculated by integration of the instantaneous rate. The overall heat transfer rates for a given particle residence time was estimated from the average rates, taking into account the number of particles in contact per unit area of the heat exchange surface and scaling it for unit temperature difference. Botterill et a2. [72-751, extended this sample single particle model to different particle sizes and materials. The overall heat transfer coefficient is given by the following expression: h, = (170dp/~)[1 - eXp( -4765/dp2)](~b/~sb)''3

(Btu/hr-ft'-"F)

(37) where r is the residence time expressed in milliseconds. They conducted experiments, where the residence time of the particles at the surface was varied by either stirring the bed or moving the heater at various speeds. Their results showed the validity of their model in their system for short residence times. They also observed that the heat transfer coefficient is a decreasing function of the residence time, the shorter the residence time the higher the heat transfer rate. Botterill et al. [75] also extended their model to the case of heat transfer into two adjacent particles touching the surface in order to clarify experimental results obtained at larger residence times. Couderc et al. [76) studied experimentally the heat transfer rates obtained when the stirrer velocity was kept constant but the flow rate of the air was increased. They observed that the heat transfer coefficient increased with the flow rate at low stirring speeds, but decreased at the high stirring speeds. They reasoned that at constant stirring speed, the residence time is shorter for high flow rates and, therefore, according to Botterill, the heat transfer rate should increase, but in reality they observed that it decreased in some cases. They introduced a new parameter [77], the average distance between the particle and the wall. Although the change in the average distance is small, due to the low thermal conductivity of the gas, the thermal resistance could be considerable. They developed a mathematical model similar to that of Botterill with the addition of the new parameter. The unsteady heat conduction equations were solved numerically and their results were compared with their experimental data. Gabor [78,79) presented two models for the unsteady state heat transfer from a wall to a fluidized bed. One model was based on a string of spheres of indefinite length normal to the heater wall. It seemed geometrically close to reality, but required a large number of computations. The second approxi-

186

CHAIM CUTFINGER AND NESIM ABUAF

FIG.8. Transfer of packet to a wall.

mate model was based on heat transfer through a series of alternating gas and solid slabs. Gabor [80] also compared experimental results measured as a function of residence times, and found them to agree with his calculations based on his approximate alternate slab model. This model can be applied to mechanically smooth, dense phase (particulate) fluidization, but it is obvious that it is far from representing a lean phase fluidization or a fluidized bed with strong agitation and bubbles. 3. Unsteady Transfer of Heat to “Packets” of Particles Which Are Renewed by ?’iolent Disturbances i n the Core of the Fluidized Bed

In an attempt to present a mechanism for the heat transfer in bubbling dense beds, >lickley and Fairbanks [Sl] introduced the picture of a small group of particlm moving as individual units (packets). Their reasoning was based upon previous experimental observations where the void fractions of dense phase bubbling beds were recorded and found to be close to the void fractions of quiescent beds. In a dense fluidized bed, each particle may be expected t o be in contact with several neighbors most of the time. The packets are not permanent, they have a finite persistence in time, their void fraction, density, heat conductivity, and heat capacity are assumed to be the same as those of the quiescent bed. Their model can be seen in Fig. 8. A packet at bed temperature Tt, comes into contact with a flat surface at a temperature T,. Unsteady heat transfer starts upon contact. If A , is the contact area between the packet and the wall, and the packet is homogeneous, the heat transferred after a time t is given by:

HEATTRANSFER IN FLUIDIZED BEDS

187

and the local instantaneous heat transfer coefficient by: Experimentally recorded local heat transfer coefficients will be the time averages of all the local instantaneous coefficientsoccurring during a period of time at a particular location of the surface. If $ ( t ) is the frequency of occurrence in time of packets having an age t, or over a period of time, the fraction of the total time during which the surface is in contact with packets of ages between t and t dt, then the local average coefficient is given by:

+

or

and averaging over an entire isothermal area

h , = A-1

I,

hwLdA

and

h,

=

(kmpmcmS)112 where Slf2= A+

I, Si" d A

(43)

Mickley and Fairbanks [Sl] apply these derivations to the following idealized bed models. a. Slug Flow of Solids Over the Surface. In a fluidized bed with very low gas flow rates, there is no turbulent mixing, and solids are observed to move upward at the center and downward along the outside walls of the bed (Fig. 9). Assuming that the solids move downward with a constant velocity us,the age of all the packets a t a distance 1 from the top is always t = l/u, The calculation for the average heat transfer coefficient for a heater of length LHas presented by Mickley and Fairbanks [Sl] gives:

b. Side Mixing. With large heat transfer surfaces and highly turbulent beds, the sidewise transfer of solids will also have an important effect. To represent this case (Fig. lo), they presented the model that as the solid is

188

CHAIM

GUTFINGER .4ND NESIM ABUAF

FIG.9. Slug flow of solids over a surface,

moving don.nnard some of it is exchanged with some other solid at Tb, brought in sideways from the core of the bed. The average replacement of packets at the wall per unit time by means of side mixing, was defined as s, resulting in: h

w

= ~ (krnprncrns)

(45)

Mickley et al. [82] measured instantaneous and time-averaged heat transfer coefficients in a fluidized bed. The instantaneous values fluctuated sharply, low values being attributed to gas bubbles and high values to the

FIG.10. Downflow with side mixing.

HEATTRANSFER IN FLUIDIZED BEDS

189

sudden appearance of a fresh packet of emulsion. With their data they were able to find the time fraction that the surface was exposed to bubbles and the average bubble frequency near the wall. They also found that the fluctuations in the local heat transfer coefficient can be directly related to the movement of the solids in the vicinity of the surface. One essential deficiency of the model is that the instantaneous heat transfer coefficientis inversely proportional to the square root of the age of the element. If t = 0, the rate is 00 and at infinite age, the heat transfer rate is 0. Baskakov [SS] proposed a similar model with an added contact resistance to the heat transfer, located at the wall, facing the surfaces of the particles in contact with the wall. Thus, the heat transfer coefficient for very small values of age does not become infinite, but is determined by the finite contact resistance at the wall. The contact resistance to be added is given by: 1 h,

--‘v

dP dc,[In(k,/dkt) -

(46)

11

where d is the depth a t which a temperature gradient exists, and he recommends d = 0.1 to be used in Eq. (46). Baskakov’s model [83] still does not solve the problem that for infinitely large values of age, the heat transfer coefficient tends to zero. In all these penetration-type models the residence time of the particle is of fundamental importance. Mickley et al. [Sl, 821 found it by measuring the temperature fluctuations in the heater. Ruckenstein [84] used an instability theory to predict the renewal frequency. Pate1 [85,86], measured heat transfer coefficients and particle residence times at identical wall locations under the same conditions. He also presented two surface renewal models which are essentially extensions of the Toor and Marchello [87] model to describe the heat transfer between the wall and the fluidiaed bed. Model I (Fig. 11). A packet initially at the bed temperature Tb arrives at the wall T, at time t = 0. The packet is assumed to have the same properties as the bed at minimum fluidization. The packet receives heat from the wall through a contact resistance and returns to the bed after a length of time. During the residence time, heat penetrates a distance 2; beyond this distance the temperature is constant and equal to the bed temperature Tb. For an exponential age-density distribution function,

$ ( t ) = 7-l exp(-O/?),

e > 0;

$ ( t ) = 0,

otherwise

(47)

where 0 = a,t/z2 is a dimensionless time and 7 = mean dimensionless residence time of packets at the surface.

190

CHAIM

GUTFINGER AND

XESIM A B U A F

PACKET

T ( x,t) CONTACT

(INITIAL TEMPERATURE Tb

FIG.11. MODELI (Pate1 [%I).

The average heat t,ransfer coefficient is given by:

with asymptotic values for zero and infinite time respectively,

The value of the contact resistance l / h c was taken from Baskakov [SS], Eq. (46).

X o d e l I1 (Fig. 12). The fluidized particles are assumed to be spherical, each possessing the same diameter d,. A particle at the bed temperature TI, arrives at the heating surface T,. I t receives heat from the fluid adjacent to the wall, nhich is assumed to be stagnant and at wall temperature T,. While the particle is a t the surface it also loses heat by conduction to a packet of particles of thickness z situated between the wall particle and the bulk of the bed. The packet was assumed to have the same properties as the bed at minimum fluidization conditions. For an exponential age distribution [Eq. (47)], an average Nusselt number was calculated :

HEATTRANSFER IN FLUIDIZED BEDS

191

I WALL (Tw)

I

I W A U PARTICLE

----

Ttt)

1

INITIAL TEMPERATUR

Tb I

FIG.12. MODELI1 (Patel [85]).

where and

3Pm y=---

x

2 P a d,

and the asymptotic values for zero and infinite time, respectively, are:

kr x Nu(0) = 2 1 - - ~ k m dp

(53)

c = 1 for square packing and c = ($)1’2 for hexagonal packing. Experimental results were compared with the theoretical predictions. The penetration-type model of Toor and Marchello [S7] presented no contact resistance but had a finite characteristic length. The model of Mickley and Fairbanks [Sl] had no contact resistance and assumed the characteristic length to be infinite. Baskakov [ S S ] added the contact resistance, but still its characteristic length was infinite. Patel [SS] included a contact resistance and a finite characteristic length. Agrawal and Ziegler [88] used a similar surface renewal model with a generalized gamma distribution to represent the expected residence times of the particles. Chung et al. [S9] also presented an expression for estimating the heat transfer coefficient in a fluidized bed by using the same concepts.

192

CHAIMGUTFINGER AND NESIM ABUAF

I FLUCTUATION OF

I

I I

I

I Srmyllffi FORCE

I I I

I TIME

FIG.13. Experimental data of Drinkenburg et al. [go].

Employing the concept of a multiple capacitance contact time distribution, they found:

where b = d,/ ( 2a,r)'12and r is the mean residence t,ime. The prtdicted values of the heat transfer rate were compared to the model of Botterill el al. [69] and Dow and Jakob's [67] experimental results. Their model also yielded a maximum value for the Nusselt number for any gas-fluidized system Numax= 13.5

(56)

A long list of references was presented to show that no experimental value above their maximum was yet recorded. To finish this section, the authors a.ould like to present some experimental results obtained b y Drinkenburg el al. [go]. They measured instantaneous temperatures, static pressures, and shear stresses. The experimental results showed that all these properties varied as periodic functions of time. The period could be measured from the recorded fluctuations. From these facts they proposed that the heat transfer is an unsteady phenomenon, and is dependent on the overall bed circulation showing a stick-and-slip flow character, Fig. 13.

HEATTRANSFER IN FLUIDIZED BEDS

193

b;”1 I

I

-t-LATERAL PARTICLE EXCHMGE

FIG.14. Model of Wicke and Fetting [92j.

4. Steady Conduction through the Emulsion Layer Van Heerden et al. [91] observed that the heat capacity of the solid particles per unit volume is about a thousand times greater than that of the gas. The mean particle velocity is much lower than the gas velocity. Therefore, the largest portion of heat will be transferred by the moving particles. They assumed that the interstitial gas between the particles almost immediately follows the temperature of the particles, and the gas only provides the suspension of the particles. They observed that there was almost no radial motion of the particles as they moved downward along the wall. They concluded that the radial heat transfer would be determined by the thermal conductivity of the suspension. They indicated that the heat transfer coefficient would be large for short heat exchange surfaces and smaller for longer sections, and this was verified experimentally. Wicke and Fetting [92] presented a similar model, Fig. 14. Heat (pw) from an exchange surface was first transferred by conduction through a gas layer whose thickness was 6 ~ This . heat was then divided into two components:

194

CHAIM

~

WALL (Tld

\

GUTFINGER tlND

I

XESIM

0 ' 0 0;

ABUAF

FLUIDIZED CORE (Tb)

FIG.15. General description of Kunii and Leverispiel [2]. yz, heat taken by solids flowing parallel to the surfacc in a second zone of emulsion of thickness 6,; and yr, heat transferred into the core portion of the bed by interchange of solids.

Xeglecting y,, their model led to:

2 h w L ~ / K= 1 - CXP( -2LHkr/6&,)

(57)

a-here K = p s ( 1 - emf)c,u,6,. ( u sis the average particle velocity along the heat transfer wall.) In the heat transfer from surfaces to fluidized beds, Gorelik [93] considered the thermal resistance to occur in two layers, along the wall and at the boundary layer. The first layer closer to the wall had a high porosity and the second layer near to the core had the bulk porosity which was assumed to be constant. His numerical results were in agreement with his experimental data. 5 . Geiwral Description of Runii and Levenspiel

The complex picture of the flow field near a heat exchange surface in a fluidized bed can be visualized as prcsented by Iiunii and Levenspicl [ a ] , Fig. 15. In reality four mechanisms operate together: (1

A film of gas 6~ coats the surface. Its thickness 6~ is large or small

HEATTRANSFER IN FLUIDIZED BEDS

195

depending on whether a bubble of air is near the surface or the emulsion is uniform and close to the surface. (2) Some solid particles are in direct contact with the heat exchange surface, (3) There is a layer of emulsion with thickness 6, which flows along the wall. (4) Part of the emulsion layer is replaced occasionally by fresh emulsion coming from the core of the bed or by bubbles rising along the wall. Depending upon the fluidization conditions and the position of the heat exchange surface in the bed, emphasis may be placed upon one or another of these conditions.

C. COMPARISON BETWEEN

THE

PROPOSED MECHANISMS

The heat transfer coefficient between a fluidized bed and a surface is determined by the resistance of the gas-solid emulsion which is near the wall. The total resistance RT can be divided into a film resistance Rfilmand a resistance of either the emulsion or the bed which can involve packets or bubbles R b e d : R t o t a i = R t i l m Rbed. Patel [85, 861, followed a similar approach in his proposed mechanism as presented in the previous section. The film resistance Rfilmnear the heat transfer surface is determined by the steady conduction through a gas film whose thickness is altered by the presence of particles, mechanism ( l ) , and by the unsteady conduction through the solid particles in direct contact with the heat exchange surface, mechanism (2). The contribution of each one will depend on its value as compared to the other one. For dense phase fluidized beds with small particles, the heat transferred by conduction through the particles is quite important. For large particles and high gas velocities the heat transfer can be determined by the gas film near the wall. Now let us compare the heat transferred by thermal conduction through the film Rfilmand the heat picked up by the packets or the emulsion layer with an average residence time ? and sitting near the heat transfer surface

+

Rbed.

If the contact time of the emulsion elements with the surface is very short due to fast circulation, then the film resistance is the controlling mechanism. This short mean contact time is given by Kunii and Levenspiel [2] as:

CHAIMGUTFINGER AND NESIM ABUAF

196

The resistance of the bed R b e d or emulsion packet sitting between the contact film resistance and the bed is mainly determined by the hydrodynamics of the fluidized bed. We can observe two extreme bchaviors of the emulsion layer near the heat exchange surface. The packet of emulsion may contact thc surface for a very short time, all the heat transferred from the wall to the emulsion goes into the heating of the packet, mechanism (3), or the packet can remain a t the surface long enough so that steady state is achieved and the emulsion layer near the wall acts as a resistance to the heat conducted away from the wall. Yoshida et aE. C94) considered a thin layer of emulsion with thickness 6, which suddenly comes in contact with a heat exchange surface. After a time 1 , thr emulsion is replaced by a fresh element from the main body of the bed. The instantaneous heat transfer coefficient is derived as :

h,, =

(kmPmCpJrt)l/2

[

1

w

+ 2 C exp(-mz62/a!t) mil

]

(59)

for short times, and h, = 1

(km/ae)

{ + 2 2 cxp[m2r2(at/6ez) I} 1

(60)

m=l

for long times, where a! = lim/pmcm To calculate the average heat transfer coefficient, they used two surface renewal models.

1. Random Surface Renewal Model

The residence time distribution was given by : $ ( t ) = +p/i

(61)

where T is the mean age of the emulsion elements leaving the surface. This model is representative of the residence time distribution near a heat exchanger in the main core of the bed, which is continuously contacted by rising bubbles. For rapid replacement of the emulsion: 2 = (a?)”*/6e

1

= (a~)”2/6~

These results are shown in Fig. 16.

and

HEATTRANSFER IN FLUIDIZED BEDS

1

3.0[

197

RAPID RENEWAL

SLOW RENEWAL, OF EMULSION MECHANISMl3lAND ( 4 ) TOGETHER

MECHANISM (31 UNSTEADY CONDUCTION THROUGH EMULSW

STEPDl CONDUCTION T W W G H EMULSION

0‘

I

1.0

,

I

4.0

2.0 3.0 2 =(amT)”v6s

FIG.16. Steady and unsteady conduction through emulsion.

2. Uniform Surface Renewal Model The residence time distribution was given by: #(t) = P ) 0

< t < 7.;

# ( t ) = 0,

t

> .T

(62)

All elements stay the sa.me length of time on the surface. This distribution is representative of an emulsion flowing smoothly along a heat exchange surface. With this model Yoshida et al. [94] also derived similar expressions for the heat transfer coefficient: h,

=

( 2 / d ~(kmpmcm/t)1‘*, )

for short contact times

hw

=

km/&,

for long contact times

They also extended these results to a surface immersed in a bubbling bed. They presented expressions for the heat transfer coefficient for the case when the bubble frequency at the surface and the time fraction that the surface is exposed to bubbles are specified.

D. EXPERIMENTAL RESULTS

In this section we will present the influence of the various factors on the heat transfer coefficients as determined from experimental observations. Tables 1-111 present a list of nondimensional correlations reported by different investigators while trying to correlate their experimental findings.

198

cH.4IM

GUTFINGER AND NESIM ABUAF

1. Properties of the Fluid a. Density p r and SpecQic Heat cPf of the Fluid. The dependence of the heat transfer coefficient on the specific heat of the gas is not yet clearly determined. The existing data do not present a common trend. The product pIc, being three orders of magnitude smaller than pat%, one would expect the heat transfer coefficient t o be independent of prep,. The very commonly used ratio of psc,/p~cp,does not have a physical foundation. Gelperin and Einstein [41] and Baskakov [95] have predicted an increase in the heat transfer rate with an increase in the specific heat of the gas or the product of the two properties prcpf, a t high pressures and gas velocities.

b. Gas I'iscosity. Most of the authors agree on the result that the wall heat transfer coefficient decreases with an increase in the gas viscosity. c. Thermal Conductivity of the Gas (kf). The thermal conductivity of the gas is the property which has the greatest influence on the heat transfer coefficient. As reported by Leva [S] in a fixed bed the heat transfer coefficient increases with tho gas thermal conductivity h, a The relation presented by Wen and Leva [96] correlated h, a kf6'. Jacob and Osberg [9T] studied the effect of this parameter on the heat transfer coefficient and their experimental data was correlated b y : hw = hwo(l

- e)[1

- exp(-pkr)l

where h,o and p are empirical constants (values given in Jacob and Osberg ~971). TABLE I

NONDISIENSIONAL CORRELATIONS FOR EXTERNAL WALLP Correlation

Investigator

Remarks

kr h, = 0.64 -Gv

Leva and Grummer [65]

q, fluidization efficiency

c

Leva [a]

Wen and Leva [96]

*

Gelperin and Einstein [41].

kr

e$J =0.16(

i,

1.bp.b

)

0.'

R, bed expansion ratio

HEATTRANSFER IN FLUIDIZED BEDS

199

TABLE I-(continued) NONDIMENSIONAL CORRELATIONS FOR EXTERNAL WALLS“ Investigator Levenspiel and Walton C68 1

Correlation

kr

D, =

r

kr

-hwDc

Remarks

d,G

d,

-lm

- 0.0018(y)(ij)

XdP ~

6(1

= [(I

- c)

+ a*)a/* -

a = 0.041(1

G.

Toomey and Johnstone log El121

Van Heerden et al. [91]

hwD = 0.55 -

ki

(“tp) =

D

0.575 1 o g c F )

+ 0.130

hg =p 0.58 ( B kr

G/c

(): (z) 0.65

Dow and Jakob [67]

=

- c)

0.11

up, particle velocity (ft/sec)

yy.4s

B, shape factor

(continued)

CHAIMGUTFINGER AND NESIM ABUAF

200

TABLE I - - ( ~ ~ t i % ~ e d ) ON DIMENSIONAL

CORRELATIONS FOR

Investigator

Correlation 1.560

-h mc,G

Bartholomew and Katz ~1481

h,

MickIey and Trilling

0.0118-

pm, solid

concentration in fluidized mixture

(y) DIG1

_ h, _--0 . 7 2 CPtG’

.(1 -

hw

-.prt/a

= 2.0

-0.87

G’, mass velocity based on void area D’, effective diam. of free area across bed

$1

Re”.”(l -

,)a

8

CPfG

(-0.44

Huntsinger [152]

2 s

d,”

Brazelton [149]

Das and Sarkar [l51]

+ ln(Re Ga4j3 p,GO

=

Remarks

- 0.0120) -0.227 Prtls Ga0.U

-

Clool

Gamson [150]

EXTERNAL w.4LLW

2)

Ir

_ kt

Nu

=

1.4(Re)@=(Pr)-I

Nu

=

0.055 Re

Kao and Kaparthi Cll.51 Lemlich and Caldas [153] Richardson and hlitson C99 1

h,D - = 119 Pro 4 ( 5 3 ) ( 3 kr

.Rep

(~)o’a””

Ref based on free-fall velocity

N

=

0.020

- (3.45 + :-;)

HEAT TRANSFER IN FLUIDIZED BEDS

201

TABLE I1 NONDIMENSIONAL CORRELATIONS FOR IMMERSED BODIES~ Investigator

Correlation

Remarh

a, const. depends on

orientation of tube Horizontal tubes

Horizontal tubes

Vertical tube

Miller and Logwinuk c98 I

h,,.,

Baerg et al. [lo41

= 49 log

.-0.00037

P8b

dP

nonfluidized bed density

Psb,

h w d p = 0.033(1

Wender and Cooper C1181

-

$(x)

-

Reo.w*cR S t Pr2la= 2.52 Re*,*(l

Gamson [150] Gelperin et al. [154]

Nu

6(1

Gelperin and Einstein [41].

- e)

=

0.73 Reo.**

correction tor location

pfCpf

CR,

- e)-O'

Vertical tube

kt

Horizontal tube

202

CH.4IM

GUTFINGER AND

T\TESIM

ABUAF

TABLE 111 ~0XDIMEXSION.iLCORRELATIONS FOR

THE h h X I M U M

HEATTRANSFER COEFFICIENTSa

-~

Correlation

Investigator Varygin and Martyushin [I561 Baerg el al. [lo41

Zabrodsky

Nu,

hwmar

=

=

0 . 8 6 Gao

ReoDt= 0,118Gap 5.

239.5 log (7.05 X

lO-'p,b)

nonfluidized bed density

Psb,

d,

[a]

Gelperin et 01. [l55]

Remarks

Nu,

=

DH,heater tube diam-

0.64 GaO-"

z,

eter distance between axes of tubes

Sarkits [l57]

Numar= 0.0087 G a o 4 PI0 33 ($45

ReoDt= 0 . 2 Gao 6 (laminar region)

Sarkits [I571

Numnr= 0.019 Gao.8 PrO W

Reopt = 0.66 (turbulent region)

Traber el al. [158)

Numar= 0.021 Gao'PI0"

Reopt = 0.55 GaO

Jacob and Osberg [97]

Itwmax =

ho(l

-Re:

Chechetkin Cl.591

Numax=

Ruckenstein [1601

Numsr = Re$:

a

- t)[1

0.0017 dp0.a

- exp( --pkf)] ho, p, emp. constants

PrO 4

GaO.14 Pr1/3

Reopt = 0.209 Ga0.52 Re,,,

=

0.09 Ga03

Gelperin and Einstein [41].

Additional data of Van Heerden et al. [91], Wicke and Fetting [92], and Jacob and Osberg [97] were correlated by Leva [S J for the heat transfer rates, and h , was found to be proportional to k, to the power 0.60-0.64. Miller and Logwinuk [98) and Richardson and Nitson [99] also agree that h , increases with k r t o the power 0.5-0.66.

HEATTRANSFER IN FLUIDIZED BEDS

203

2. Properties of the Granular Solid Material a. Diameter (d,) and Shape of the Particles (+,-sphericity). Dow and Jakob [67], Wicke and Fetting [92], Jacob and Osberg [97], Mickley and Trilling [loo], Capes et al. [lol], Levenspiel and Walton [SS], and Leva and Grummer [SS], all agree that the heat transfer coefficient decreases when particles with larger diameters are used. The results of Sarkits et al. [102, 1031, as reported by Gelperin and Einstein [41] show that for the laminar flow region the heat transfer coefficient varies inversely with particle diameter, and for the turbulent flow region it has a direct variation. Gelperin and Einstein [41] also present Variggin’s data, where the heat transfer coefficient first decreases rapidly with an increase in the particle diameter, then levels off, and slightly increases once again for very large particle diameters (4.8mm) . The sphericity of the particles & also affects the heat transfer coefficient. Baerg, Klassen, and Gishler [lo41 and Mersmann [lo51 both report higher values for rounder and smoother particles. b. Density of the Solid ( p s ) . Levenspiel and Walton [SS] and Ziegler and Brazelton [lOS] both observed experimentally that the wall heat transfer coefficient increases with the increasing density of the solid ps. Sarkits et al. [102,103, 1071, record that the dependence of the heat transfer coefficient on the solid density becomes weaker as the flow becomes more turbulent. However, in both laminar and turbulent flows h, increases with the density of the solid. c. SpeciJic Heat of the Solid (cps). Dow and Jakob [67], Ziegler and Brazelton [lOS], Sarkits et al. [102, 103, 1071, and several other investigators all agree on the fact that the heat transfer coefficient increases with an increase in the specific heat of the solid, cw.

d. Thermal Conductivity of the Solid (lc,). Campbell and Rumford [lOS], Ziegler et al. [lOS, 1091, Miller and Logwinuk [98], Wicke and Fetting [92], Reed and Fenske [llo], and Bannister [ill], either measured experimentally the heat transfer rates over a wide range of solid thermal conductivities, or reviewed existing results. All found that the effect of the thermal conductivity of the solid k, on the heat transfer coefficient is modest, and thus concluded that h, is independent of k,. Chung et al. [89] developed an expression for the heat transfer coefficient and showed that the latter was independent of the solid thermal conductivity when d,(2a,i)-1’2 was less than 1 (a,is the thermal diffusivityof the solid particle, and 7 is the mean residence time), which is the case for all usual fluidized bed laboratory experiments.

204

CHAIMGUTFINGER AND NESIMABUAF

-

I

f

I

I

I

, I I

Gmf

I

I Gopt

FIG.17. Typical heat. transfer coefficient versus mass velocity.

3. Conditions at Minimum Fluidization (u,f, e m f ) Toomey and Johnstone [112] used a cylindrical fluidization chamber and a centrally located cylindrical heater. Their data was correlated by:

hwdp/k,

3.75[(dp~rn[pf/~)log(G/umt) 7.47

(63) whcrc the gas velocity at minimum fluidization was introduced. This exprmsion should only be used at high gas velocities, because a t minimum fluidization it predicts a zero value for the heat transfer coefficient, a result far from reality. =

4. Fluidization Conditions a. SuperJiciaZ Velocity or Gas M a s s Velocity ( G = p f u o ) . As swn in Fig. 17, where the heat transfer coefficient is plotted versus the gas mass velocity G, the dependencr is not a simple one. If one tries to correlate h , a Gn, the valuf>of the exponent n will depend on the range of the experiments. For this reason n-c have a large number of results with different exponents Dow and Jakob’s data [6T] show that h,,. a: GO.8, van Heerden et al. [91] obtain h , a Go 45, Micklcy and Trilling [loo] correlate as hw a and Bhat and Weingaertner [113] observe th a t h , a One point upon which all researchers agree is the shape of the rising and falling branches of the h,v vs G curve and the fact that there exists a maximum value of the heat transfer coefficient a t a specific superficial gas velocity uo or gas mass velocity, G = pruo.Tables listing the nondiinensional correlations for both the rising and falling sides of the curve, for the maximum heat transfer coefficients, and for the special gas flow rates when the h,,, is attained are presented by Einstein and Gelperin [114]. Gelperin and Einstein [41] show from Variggin’s results that the h, vs G curve is steeper for small particles and flatter for larger particles.

b. Bed Porosity

(E).

It is a commonly observed and accepted fact,

HEAT TRANSFER IN FLUIDIZED BEDS

205

Couderc et al. [76, 771, Rao and Kaparthi [115], etc., that the heat transfer coefficient varies inversely with some function of the bed porosity E. This effect was explained by the film theory: That, as the population of the particles near the wall is decreased, the heat transfer rate mill also decrease. But an increase in the porosity is obtained by an increase in the gas flow rate which under some conditions may have an opposite effect on the heat transfer coefficient.

5. Geometric Variables a. Bed Diameter ( D ) . The experimental results and the different nondimensional correlations do not show a clear and predictable variation of the heat transfer coefficient with the bed diameter. b. Diameter of the Immersed Heater ( D H ) . Jacob and Osberg [97] and several Russian investigators mentioned by Gelperin and Einstein [41], all agree that the wall heat transfer coefficient increases with a decrease in the immersed heater diameter DH, and that the heat transfer rate diminishes as the diameter of the heater is increased; it becomes independent when DH > 10 mm [41]. c. Fluidized Bed Height (L) or Static Bed Height (Lo). Van Heerden et al. [91], Leva [S], Gelperin et al. [ll6], and Xlickley et al. [82] showed that there is no dependence between the heat transfer coefficient and the fluidized bed height or the height at the fixed or static state. This independence is not true if one records its data at a section very close to the air entrance grid, where an active section exists and some entrance effects can be expected. Bibolaru [117] studied experimentally the dependence of the heat transfer coefficient with the height of the fluidized bed and found that at constant air velocity, the heat transfer rate decreases with an increase of the bed height. d. Length of the Heat Exchange Surface (LH). The investigations carried out by Toomey and Johnstone [112], Dow and Jakob [67], and Van Heerden et al. [91] showed that the heat transfer coefficient decreases with an increase in the length of the heat exchange surface LH. Wicke and Fetting [92] presented their results as:

hw

=

(Lmf/L)(K/2LH) { 1 - exp[-2kfL~/(kK)]}

(64)

where K = p , ( l - ~ m r ) ~ p r ~ Mickley s6e. et al. [SZ] used a vertical coaxial cylindrical heater and observed that the heated length had a relatively small effect on the heat transfer coefficient. In correlating different existing data Wender and Cooper [llS] concluded that for a fluidized bed with an external heat transfer surface, the

206

CHAIMGUTFINGER AND NESIMABUAF

condition LH/D > 7, was a good one in order to assume that the heat transfer coefficient is independent of the heater’s length. For internal heat exchange surfaces no effects were observed [llS]. Gelperin et al. [llS] review the work of Wicke and Fetting [92], Zabrodsky [119], Van Heerden et al. [91], Dow and Jakob [67], Baerg et al. [104], concerning the effect of the heat exchange surface height on the heat transfer coefficient. Their experimental investigations showed that if the heat transfer surface is located above the active section of the bed L,, then the heat transfer coefficient is independent of the heater’s height. Immediately near the gas distributing grid, the heat transfer coefficient is higher than the rest of the bed. If the heat exchange surface is located within the stabilization zone of the bed, then the local heat transfer coefficient will decrease from its value a t the bottom, to its constant value of the core of the bed, at a distance of 30-80 mm from the grid.

6. Other Factors a. Entrance Efects. Entrance effects as mentioned by Leva [S], Gelperin et al. [ll6], Nickley et al. [82], are common in almost all flow phenomena, and are caused by the sudden modification of the flow pattern. One cannot prevent entrance effects from affecting data. The active section in the fluidized bed is very important and one should take into account its existence when planning experiments. b. Temperature and Pressure of the Fluidized Bed. Levenspiel and Walton [SS] found that the heat transfer coefficient is independent of the temperature of the fluidized bed. Rabinovich and Sechenov [120] found that h , showed an increase with the bed temperature and pressure. A t high temperatures one should remember to take into account the effects of radiation on heat transfer. c. Properties of the Fluidized Bed at Minimum Fluidization (pm, km). Miller and Logwinuk [98] and Nickley et al. [Sl, 821 observed that the heat transfer coefficient increases with the square root of the density or the thermal conductivity of the fixed bed as predicted by the “packet” theory for heat transfer. E. IMMERSED BODIES In the experimental work reported, investigators used external and internal heat transfer surfaces. By external, we mean that the wall of the fluidized bed container itself was used as the heat exchange surface. By internal we mean that bodies with different shapes immersed in the bulk of the fluidized beds were used as the heat exchange surfaces. In order to be able to make a proper comparison between the two, one has to find experiments performed simultaneously a t identical experimental conditions.

HEATTRANSFER IN FLUIDIZED BEDS

207

Toomey and Johnstone El121 reported for the same equipment and similar conditions, heat transfer coefficients for an external wall and an immersed heater. At low gas flow rates, the coefficients for the internal heater were higher than the exterior wall coefficients. As the gas flow rate was increased this ratio approached unity. Raju et al. [121] also made experimental studies and reported that the heat transfer coefficient for an internal surface is higher than that for an external wall. These results cannot be generalized. The important point to keep in mind is that the fluidized bed can exhibit a nonhomogeneous character with regard to the hydrodynamics and heat transfer data. Sometimes with welldeveloped fluidization it may even happen that h,,, > h,,,,. While working with immersed bodies, one has to consider that the heat transfer coefficient may vary with location, first with respect to the air-distributing grid and second at a given plane, if the heater is at the center or not. The results from these variations are sometimes contradictory and one should keep in mind that they depend strongly on the local hydrodynamic conditions in each case. These facts were emphasized by Fritz [122], who investigated the heat transfer between internal surfaces and fluidized beds of various geometrical forms. He found the heat transfer coefficients are affected by the nature and position of the heating surface, the form of the bed, and the flow of the gas. 1. Spheres

Baskakov et al. [123] derived by solving the Laplace equation with the proper boundary conditions, an expression for calculating the local heat transfer coefficient at the outer surface of a hollow spherical heater, from the known temperature distribution on the surface. The latter was determined experimentally,

where 8 is the angle measured from the upstream pole, TI and R1 are the temperature and radius of internal surface, Tz(8) and R2 are the temperature and radius of the outer surface, and Tz,, is the mean integral temperature of the outer surface. Their results show that h, reaches a maximum at the equatorial zones of the sphere (60"-loo"), h, is higher at the pole facing the flow direction than at the pole situated downstream. As the gas flow rate is increased, the maximum h, shifts to smaller values of 8, and tends to equalize over the surface. Use of larger particles tends to equalize the h, over the surface. The distribution of h , is more pronounced for the immersed heater with the larger diameter.

208

CHAIM GUTFINGER AND NESIM

ABUAF

Galloway and Sage [124) and Ziegler and Brazelton [lo61 also used spherical probes and made heat transfer measurements. Ilchenko and Sfakhorin [125] also studied experimentally the heat transfer from a spherical probe. The wall heat transfer Coefficient increased linearly when the fluidized bed temperature was raised from 300" to 650°C, above that, the dependence was nonlinear. This effect was attributed by the authors to radiative effects. 2. Flat Plates

Iiorolev and Syromyatnikov [1261 studied experimentally the hydrodynamics of a fluidized bed in the vicinity of a plate submerged into it. The local porosity xvas measured by an x-ray apparatus, and the velocity of the gas phase was determined by a pitot tube. Their results showed that the iocal porosity a t the plate is higher than thc average value in the bed, and the gas velocity near the surface is higher than within the bed. They performed exprriments where a plate was cooled in a fluidized bed with and without a metal mesh surrounding it. Higher heat transfer rates were obtained when the plate was without the net. Raskakov and Fillipovskiy [127] built a flat plate calorimeter consisting of different heating elements. Thcy determined thc relationship between the average rate of heat trarisfcr and thv size and orientation of the plate and also thc variation of the local heat transfer coefficient along the plate for different parameters. They observed that the maximum heat transfer rates were obtained when the plate was in the vertical position, h , decreased when they used larger plates. When the plate thickness was increased, the rate of hcat transferrcd was affwtcd by the shape of the bottom. The local heat transfcr coefficients decreased from the bottom of the plate upward. Fillipovskiy and Baskaliov [1281 published a second experimental investigation where temperature distributions were measured near the heated plate. When the heated surface was turned upmud, a. layer was formed on the plate and the heat transfer coefficient decreased. 3. Small Cylinders

Kirk and Hudson [129] used two small immersed cylinders for both heat and mass transfer studies in fluidized beds. They observed that the Reynolds analogy between hcat and mass transfer does not apply for transfer processes in fluidized beds. The extra increase of the heat transfer coefficient as compared to the mass transfer was attributed to the role of the particles themselves in the transfer phenomena. This led them to conclude that a model like that of Botterill was more realistic.

HEATTRANSFER IN FLUIDIZED BEDS

209

4. Immersed Tubes

a. Vertical Tubes. Vreedenberg [130, 1311, placed a tube in various positions in a fluidized bed and determined the heat transfer coefficients. His experimental data were correlated by: h, dp/kr = a(Gv/Gmfvmt)0'35 (66) where v is the kinematic gas viscosity and a is a constant depending on location and orientation of the tube. Miller and Logwinuk [98] determined the heat transfer coefficient between two concentric vertical cylinders. Their data were correlated by:

Baerg et al. [lo41 also reported heat transfer data for an immersed vertical tube. Their results are 25% higher than the heat transfer coefficients for an external wall. Their data were correlated by: h,

=

- 55 exp[-O.O12(G - 0.71psb)l

h,

(68)

where hw,,,

= 49 10g(O.O0037p,~/d,)

and Psb is the nonfluidized bed density. Wender and Cooper [ll8] correlated experimental data of several investigators by using an internal vertical tube as the heat exchange surface and came up with the following expression :

where CR is a correction factor for nonaxial tube location [llS]. Berg, Baskakov, and Sereeterin [132, 1331, found that the heat transfer coefficient changed along the height of the vertical cylindrical heater. Genetti et al. [134] studied the heat transfer from internal bare and finned vertical tubes. b. Horizontal Tubes. Yreedenberg's [135] data for horizontal tubes covering a large range of particle diameters and solid density were correlated by:

(

hw -dp - 0.66 GDps(l kr PfLt-c and

">"""

Pr0.3

for

r?) k)

< 2050

(70)

210

CHAIM

GUTFINGER AND

NESIM

ABUAF

For the values in between, he suggested to use the arithmetic average of the tn-o values. Gelperin and Einstein [41] presented measurements of t>helocal heat transfer coefficients around the perimeter of horizontal tubes. The heat transfer rate is largest a t the equatorial lateral zones and smallest a t the upstream and downstream poles. This nonhomogeneity was related to the motion of the particles near the surface. Bartell el al. [136] once again compared heat transfer data between bare and finnrd horizontal tubes. c.

Tube Bundles.

i. Bundles of vertical tubes. Gelperin and Einstein [41] summarize the results obtained with bundles of vertical tubes. The value of the heat transfer coefficirnt increases for tubes located at the center line, slight deviations are obtained for the peripheral tubes. As the number of the tubes in thp bundle is increased the gas flow is more uniform and the heat transfer rate tends to be equal for each tube. The bed height and the distance to the gas inlet grid have little effect. When the ratio of the distance between tvio adjacrnt center lines and of the tube diameter is smaller than 2, the heat transfer coefficient is observed to decrease. Thry report the following correlation [41] for vertical bundles: h, d,/kr

=

0.75 Ga0.**[1 - (&/Z)]0'14,

Z / D T= 1.25-5.0

(72)

\there DT is the tube diameter and z is the horizontal distance between two adjacent center lines. ii. Bundles of horizonla1 tubes. Lese and Iiermode [137] studied the heat transfer from a horizontal tube in the presence of other unheated tubes. They also studied the effect of the number, size, and configuration of the other tubes on the heat transfer coefficients. The heat transfer from bundles of horizontal tubes placed in line and in staggered arrangement was investigated by Dahlhoff and Von Brachel [138] and also b y Gelperin et al. [139]. Correlations are presented by Gelperin and Einstein [41] as: In line tube arrangement,

Z / D T= 2-9,

h, d,/kr

=

0.79 GaO.**[l- (DT/Z)]0'25

(73)

in staggered tube arrangement, (74)

HEATTRANSFER IN FLUIDIZED BEDS

21 1

F. FLUIDIZED BED COATING As an application of heat transfer between a fluidized bed and an immersed surface we will describe here the problem of fluidized bed coating. When a hot object is dipped in a bed of fluidized plastic powder, a film of fused plastic coating will be formed on its surface. The coating thickness depends on the object temperature, the fusion temperature of the powder, the immersion time in the bed, the physical properties of the object and the powder, as well as on the heat content of the object. If the object possesses a very large heat content, it can be considered as an infinite heat source, and its temperature could be taken as constant during the coating process. The case of constant-wall-temperature fluidized bed coating was analyzed by Gutfinger and Chen [140,141]. Recently their analysis was extended to the case of variable object temperature [142,143]. Experimental data on coating are reported in the literature by Pettigrew [144] and Richart [145], while some industrial applications are described by Landrock [146]. Consider a flat plate with an area A , and halfwidth w , which is dipped vertically in a fluidized bed. The object is initially at a temperature, TlVo, which is higher than the softening temperature of the coating material, T,. The plastic coating material in contact with the object surface will melt and begin to form a layer on the plate. The process now involves the transfer of heat from the plate to the continuously growing film, and then into the fluidized bed. Usually the object is metallic, possessing a high thermal conductivity and a finite heat capacity, and therefore its temperature will remain uniform but decrease with time, i.e., the body is treated as a lumped parameter system. Heat from the plate is transferred through the coating film whose properties, pc, c,, k,, are assumed to remain constant during the process. The surface temperature of the coating film is assumed to be constant and equal to the melting or softening temperature of the coating polymer T,. Although the rate of heat convected from the coating surface into the bed is obviously dependent on the fluidization conditions and the temperature gradient, in this analysis the heat transfer coefficient between the plate and the fluidized bed is assumed to be independent of location and direction, and is taken to be constant during the coating. The temperature within the fluidized bed Tb is assumed to be uniform and constant. With these assumptions, the one-dimensional heat conduction problem with a moving boundary, describing the coating process, is given by the following set of equations:

CHAIMGUTFINGER AND

212

NESIM ABU.4F

---___ -----_

--- Af I

0

1

1.0

2.0

1

3.0

4.0

2

FIG.18. Plot of final dimensionless coating thickness A( and final dimensionless wall temperature 8, I versus dimensionless parameter 2 for various dimensionless melting temperatures em.

The boundary and initial conditions are:

T(0,O) =

Two

T ( 0 ,t )

=

Tw(t)

T(6, t j

=

T,

+

[~scfi(

Trn - Tbj

+ A] 6

(80)

and 6(0)

=

0

(811

Equation (79) is the heat balance at the surface of the coating film. It equstcs the heat conduction to the surface with the heat convected into the fluidized bed, plus the heat absorbed b y the coating material which sticks to the plate and forms the film. Equation (80) expresses the fact that the heat loss by the body during the time interval (0, t ) is equal to the heat transferred to the fluidized bed, plus the heat consumed in bringing the coating film temperature from its initial value Tb to its final value T. Equations (75)-(81) were solved [142] by a heat-balance integral technique similar to one used by Goodman [147], for the special case of negligible latent heat, of fusion A. This case corresponds t o noncrystalline

HEATTRANSFER IN FLUIDIZED BEDS

213

polymers usually used in coating. Good agreement was obtained between theoretical predictions and experimental results. Figure 18 provides a plot of dimensionless final coating thickness and object temperature as a function of a coating parameter 2:

The final coating thickness 6f is the one at which the coating terminates. In dimensionless form it is defined as :

Dimensionless final object temperature is: Bwf =

(Twf -

/( TWO- Tb)

(84) Figure 18 is the illustration the practicing coating technologist will be most interested in, as it shows the highest coating thickness possible for a given set of coating parameters as well as the drop in wall temperature at the point where the final thickness is achieved. Tb)

NOMENCLATURE Specific surface, surface of solid per unit volume of bed Specific surface of solid, 6/dp Heat exchange area Contact area of packet and surface Area of immersed surface Biot modulus, h,d,/k. Specific heat of packet, specific heat of quiescent bed Specific heat of coating Specific heat of fluid Specific heat of solid Bubble diameter Diameter of sphere of same specific area as that of the particle Diameter of the fluidized bed Diameter of the immersed heater Distance between two successive layers Time fraction that surface is exposed to bubbles Acceleration of gravity Fluid mass velocity, udf Galileo number (also called Archimedes number), dp3m (ps--Pt) g/P2

Heat transfer coefficient for the contact resistance at the wall Heat transfer coefficient between fluid and particle Local or true heat transfer coefficient between particle and fluid Heat transfer coefficient between fluidized bed and surface Local average heat transfer coefficient Instantaneous local heat transfer coefficient Bubble-to-cloud heat transfer coefficient Tube diameter Thermal conductivity of coating Fluid thermal conductivity Thermal conductivity of packet., thermal conductivity of quiescent bed Thermal conductivity of the solid Length Bed height Active bed section Length of heat exchange surface

CHAIM GUTFINGERAND SESIM ABUAF Bed height a t minimum fluidization conditions Static fluidized bed height Bubble frequency a t the surface Particle Nusselt number, hdP/lzi Xusselt number, h d P / k r Pressure Pressure drop Peclet number, d,u,/mr Prandtl number, c,,p/kr Heat flow rate from the wall into the packet Instantaneous heat transfer rate Radius of sphere Reynolds number, dpu@pr/p Reynolds number at minimum fluidization, dou,cpf/p Average replacement of packets a t wall by side mixing Time Temperature Fluidized bed temperature Temperature of fluid Temperature of gas in bubble Initial bed temperature Melting or softening temperature Solid temperature Wall temperature Initial wall temperature Bubble velocity Superficial velocity of minimum fluidization Superficial velocity at fluidization conditions Solid velocity Halfwidth of plate Coordinate system Penetration distance Coordinate system Vertical distance between two adjacent tube center lines

z

2

Horizontal distance between two adjacent tube center lines Coating parameter, Eq. 82

GREEKSYMBOLS Thermal diffusivity of fluid Thermal diffusivity of solid Solids volume fraction inside bubble Coating thickness, volume fraction of bed occupied by bubbles Final coating thickness Thickness of gas layer Thickness of emulsion layer Average boundary layer thickness near the wall Dimensionless final coating thickness Void fraction at fluidization conditions Void fraction at minimum fluidization Angle ( T m - Tb)/(Two - Tb) Dimensionless wall temperature Heat, of fusion Fluid viscosity Fluid kinematic viscosity Bulk density at fluidization conditions Density of coating Fluid density Density of packet Bed density at minimum fluidiaation conditions Density of the solid Settled bed density Residence time Mean residence time Sphericity of solid particles Residence time distribution function

REFERENCES 1. J. F. Davidson and D. Harrison, eds., “Fluidization.” Academic Press, New York,

1971. 2. D. Kunii and 0. Levenspiel, “Fluidization Engineering.” Wiley, New York, 1969.

3. S. S.Zabrodsky, “Hydrodynamics and Heat Transfer in Fluidized Beds.” Gosenergoizdat, Illoscow, 1963. [Engl. transl. by F. A. Zenz, M I T Press, Cambridge, Massachusetts, 1966.1

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Heat and Mass Transfer in Fire Research

. .

S L LEE Department of Mechanics. State University of New York at Stony Brook Stony Brook. New York

. .

J M HELLMAN Power Systems. Westinghouse Electric Cmporation Pittsburgh. Pennsylvania

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A . The Phenomena of Fire B. Fire Classification C. Thechemical Natureof Wood I1. Pyrolysis A. Experimental Methods of Studying Pyrolysis B. Interpretation of Experimental Results C . Mathematical Models of the Pyrolysis Process . D. Relation of Pyrolysis Testing to Fire Protection I11. Ignition A . Ignition Sources B. Fire-Safety Oriented Ignition Tests C . Ignition-Radiation Induced D . Ignition-Wildland Fires I V . The Plume A . Plume Phenomena B. Analytical Plume Studies C. Experimental Fire Plume Studies V . Firespread A. Firebrand Spotting B. Evaluationof FuelLoadof Fire C. Theories for Fire Spread D. Experimental Determination of Fire Spread Rates E. Large-Scale Fire Spread Studies 219

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VI. Instrumentation in Fire Research A. Exhaust Hoods B. Wind Tunnels C. Fuels D. FabricTesting Methods E. Flow Visualization F. Temperature Measurement G. Velocity Measurement. VII. Fire Research and the Fire Fighter A. Equipment Designed for the Fire Fighter B. Building Codes. C. Product Testing VIII. Concluding Remarks References

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272 272 272 272 273 273 274 274 275 275 277 279 280 281

I. Introduction Ever since primitive man discovered that controlled fire was useful and uncontrolled fire deadly, the combustion of liquid and solid fuels has been subjected to intensive investigation. Much time and effort has gone into the study of determining the most favorable burning conditions for commonly used fuels such as coal and oil, but relatively little technical effort, until recently, has been devoted to a study of the burning of materials not usually considered fuels, such as furnishings, homes, and forests. The need for research in this area is great. In the last decade in the United States alone, fire took an annual toll of more than 11,000 lives, and the direct fire losses from just fires in buildings exceeded one and onehalf billion dollars. These losses, coupled with a worldwide effort to encourage scientists and engineers to look a t human-oriented problems, nurtured the recognition of a relatively new discipline, $re research. This discipline is different from the more established ones in that even most of the elementary problems therein still elude the possibility of a fundamental treatment, no matter how crude. Another difference is the broad spectrum of technical and technological areas that are encompassed by the title of fire research. The reason for the inclusion of so many diverse areas is that for hundreds of years fires have occurred in many diverse places and from many diverse causes. Over this span of time, cells of expertise have grown with direct application intended in each of these areas; firemen have learned the “best” way to fight house fires, foresters have learned the “best” way to prevent and fight forest fires, and insurance company researchers have determined “acceptable” (at least from a claims point of view) limits of fire danger for the various buildings, installations, devices, and industrial practices.

HEATAND h h s TRANSFER I N FIRERESEARCH

221

To these and other similar groups have now been added fluid mechanics and heat transfer scientists and engineers among others interested in applying their knowledge and method of approach to the problem of unwanted fires. As a result of the varied interests of the people involved in fire research, there is often considerable interplay among the various groups; the inputs to one problem often come directly from the outputs of many other problems, and subsequently its outputs may very often find their way back to serve as inputs to the same other problems. The aim of this article is to present a review of recent technological advances in the field of fire research, along with some of the basics of the fire problem. It is hoped that this broad approach will enable researchers unfamiliar with the various aspects of fire research to obtain an overview of the subject, some of the problems and presently available methods of solution, and to permit those researchers working within one branch of fire research to visualize their work within an overall framework and possibly see new applications for their experimental and analytical expertise. Review papers on more specialized aspects of fire research have been written by Emmons [l], Lee [a], Thomas [ 3 ] , Lawson [4], and Berl [ 5 ] .

A. THE PHENOMENA OF FIRE The study of a single small burning candle illustrates many of the significant processes going on in a larger fire as well as showing, by comparison with the phenomena occurring in larger fires, how inadequate simple scaling procedure can be. For the candle flame shown in Fig. 1, heat from the combustion zone is transmitted to the solid wax and melts an exposed portion of it. The melted wax is then gradually drawn by capillary action into the wick where further absorption of heat transforms it into a volatile fuel vapor. This vapor is oxidized in the diffusion flame, producing combustion products and heat. Part of the generated heat is transmitted back to the candle to melt additional wax for the continued supply of fuel vapor for the sustained burning of the flame. The resulting hot gas mixture is driven upward by buoyancy, forming a natural convection plume above the flame. This upward motion constitutes part of a recirculation flow field around the flame both to provide fresh air needed for combustion and to help shape the geometry of the flame. Figure 2 illustrates the transfer mechanisms in a large fire. Note that a large-scale fire can contain a special spread mechanism, often dominant, which is not present in a candle flame. This process, known as spotting, occurs when burning pieces of fuel material (firebrands), such as leaves,

S. L. LEE A K D J. 3'1. HELLMAN

222

HOT

PLUME

A

"

. L

\ 4

DIFFUSION FLAME

'

MOLTEN WAX

SOLID WAX

FIG.1. Candle flame and associated phenomena.

-

AMBIENT WIND

--

4BURNED I

1 I

y'i {

I \

\',, k , '

RADIATIVELY HEATED AREA (PYRQYSIS L DEHYDRATION OCCUR)

HEAT CONDUCTION

FIG.2. Wildland fire and associated phenomena.

HEATAND MASSTRANSFER IN FIRERESEARCH

223

tree branches, or partially burning pieces of paper, are carried by the wind, either ambient or fire induced or a combination of the two, to a new location. If ground fuel conditions are right, these firebrands can ignite a new fire. From this reasoning, the study of a simple free burning fire should cover three main interrelated disciplinary areas : (a) The generation of combustible fuel vapors from the fuel bed material and the ignition process; (b) the diffusion controlled flame; and (c) the natural convection plume, including its interaction with the ambient wind and firebrands, if any.

B. FIRECLASSIFICATION In fire literature, various types of fires have been identified and given names. The National Fire Protection Association has established the following classificationsfor building and industrial fires [S] :

Class A : Fires involving ordinary combustible materials (such as wood, cloth, paper, rubber, and many plastics), requiring the heat absorbing (cooling) effectsof water, water solutions, or the coating effects of certain dry chemicals which retard combustion. Class B : Fires involving flammable or combustible liquids, flammable gases, greases, and similar materials where extinguishment is most readily secured by excluding air (oxygen), inhibiting the release of combustible vapors, or interrupting the combustion chain reaction. Class C: Fires involving energized electrical equipment where safety to the operator requires the use of electrically nonconductive extinguishing agents. Class D : Fires involving certain combustible metals, such as magnesium, titanium, zirconium, sodium, potassium, etc., requiring a heat absorbing extinguishing medium not reactive with the burning metals. These classifications have been primarily used for the specification of suitable extinguishment materials for the various types of fires. Another set of fire classifications delineating different types of very large-scale, or mass, fires has been recognized. A conflagration is a fire that develops moving fronts under the influence of wind or topography, and the hot, burning area is usually confined to a relatively narrow depth. A fire storm is defined as a fire in which many parts of the entire fire area are burning simultaneously. Such a fire is essentially stationary, with little outward spread. It is usually identified with a towering convection column, extending to heights of the order of up to a few miles. Feeding the convection column

arc. intense ground level winds drawn inward toward the fire. This airflow is believed to be a major reason for the lack of significant outward spread of the flaming region. In the Hamburg fire storm, witnesses recalled seeing automobiles being tossed about by the winds generated near the fire storm. A mouiny fire storm, so named by Countryman [7] for a wind-driven fire storm, is a fire storm that spreads under the influencc of wind or topography. Under certain conditions of fuel, wind, and topography, numerous firebrands can spot-ignite large areas ahead of a large fire, and the resulting developing fire can take on many of the characteristics of a fire storm, yet continue to move rapidly into unburned areas. This type of fire has been observed in u-ildland mass fires, and because of the very large areas that can be ignited in such natural and man-made disasters as carthquakes and nuclear attacks, it is possible that fire storms can also flare up in urban areas. Of the three types of mass fires, the conflagration is at least conceptually understandable, while the fire storm and moving fire storm are potentially the most dangerous and a t present least predictable. -1graphic description of the fire storm phenomena, for a relatively small-scale fire storm, has been given by Pirsko el al. [ S ] : About 1230 P.S.T. a high voltage transformer shorted a t an irrigation pump house, rreating a very hot ignition source. The resulting brush fire spread uphill against the gradient wind but under the influence of eddies on the lee side. Near the top of the hill, the fire began to turn east. Major forward progress was by spot fires that jumped as much as a quarter mile ahead of the front. rin experienced fire control officer on the scene estimated that because of the spotting, forward spread to the east aas about 10 times faster than normal for this area. The fire approached Arroyo Paredori one mile east from the pump house around 1400 to 1430 P.S.T. As the main fire and spot fires joined near the Arroyo, many small whirl-winds were seen to start. Wind sweeping southward down Arroyo Paredon whirled rlnckwise to the west where the slope flattened out. In the unstable air vertical development was unhindered. Many more spot fires started as entire Inirnirig bushes, 5 to 10 feet tall, were carried aloft and dropped into unburned vegitation. These fires built up great quantities of heat. Whirls continued to form, rise, and break off at the top. Firefighters could hear the roar of shock waves from the intensely burning brush. Then many of the small \vhirls seemed to coalesce, arid the whole fire area started to rotate clockwise. This massive fire-bearing whirlwind built up in height and started to roll downslope to the south. It plucked brush and small oak trees from the ground and carried them aloft. The intense whirl continued its vertical growth. The southward movement steered the fire into the area of homes, a chicken ranch, arid avocado orchards. That the whirl carried fire for about 200 feet after leaving the wildland fuel was shown by scorched and burned avocado trees and burned chickens a t the ranch. .4s it passed the ranch, the fire whirl ignited some walls. Eggs stacked for shipment were cooked in their crates. The Hhirlwind then sheared off the roof of a 7-rOOIn house, blew many windows inward, and lifted venetian blinds over the tops of the remaining walls. Streaking through the avocado orchard, the whirl tore some trees

HEATAND MASSTRANSFER IN FIRERESEARCH

225

out of the ground, broke tops of some, and smashed others. Limbs and trunks in its path pointed in the direction the lire whirl traveled. A 4-foot diameter oak tree was uprooted and twisted toward the center of the path. Four people were injured by flying material and falling debris. Two houses were destroyed-one of two stores and 10 rooms; three automobiles aere smashed; and a garage and part (section) of the chicken ranch were blown apart. A barn filled with hay was hoisted in the air, pulled apart, and never found again. Boards, plaster, corrugated sheet metal, and other building parts were scattered over an extensive area. The fringe of the fire whirl passed over a Carpenteria fire department tanker and sucked out the rear window of the truck cab. A fireman standing on the rear platform was pulled up vertically so that his feet were pointed to the sky while his hands clasped the safety bar. Continuing south through the avocado orchard to a two lane road, the whirlwind turned at a right angle to the east for about 250 feet and then made another right angle turn to the south down an orchard lane. I t rammed a 3 by 6 by f inch piece of plywood 3 inches deep into an oak tree, then lifted and moved overland to the south toward the Pacific Ocean. At the main fire, the ridgetop fuel break and the direction of the prevailing wind made control of the north fire perimeter comparatively easy. Two spot fires on the next ridge to the east were controlled by 2200 the same day.

C. THECHEMICAL NATUREOF WOOD

A description of the material wood, which is involved in many unwanted fires, is given by Murty Kanury and Blackshear [9]. Wood is the name given to the main tissue of the stems, roots, and branches of the so-called “woody plants.” It consists of “cells” or “fibers” which can be isolated by chemical means. Some of these cells (prosenchyma) are vessels, tracheids, and libriform fibers for transport of water and air and for giving mechanical strength. The diameter of these cells varies from 0.02 to 0.5 mm. Other cells (parenchyma) contain living protoplasm, cell nuclei, metabolites, and reserve food such as starch, fats, dyestuffs, and resins. These cell fibers range in length from 0.5 to 1.5 mm and in width from 0.07 to 0.1 mm. The porosity 4 (ratio of the volume of the pores to the volume by the cell walls) of real woods lies somewhere in the range 40-75%. The specific gravity of the actual woody substance is approximately 1.5. The influence of the porosity and the moisture content M (percent based on the oven dried weight) on the specific gravity of real wood is: = 1

+

1.5(1 - +/loo) 0.0135 M ( l - #1/100)

Chemically, wood is composed (by weight) of 50% C, 6% H, 0.10% N, 0.40% ash, and the rest oxygen. Wood is basically a combination of three

326

s. L. LEE .4ND J. Jf. HELLMAN

main components-about 20y0 lignin, 40% hemicellulose, and 40y0 cellulose and other carbohydrates. Cellulose is a polymeric residue, attacked neither by alkalides nor acids, its fiber diameter being about 60 A. The molecular weight of cellulose is found to be about lo6.The nondissolving residues of wood (which are later identified as cellulose) have a formula somewhat like CsHlOOswhich is isometric with starch. That this residue is enclosed within the cells has been shown. From the chemical formula, it can be seen that the degree of polymerization of cellulose is approximately 104. Hemicellulose consists of certain wood polyoses and carbohydrates which accompany many celluloses in nature. It is soluble in alkalies and easily saccharized by dilute mineral acids. While Gay-Lussac’s (1839) work on the elementary composition of wood gave the impression that wood is a homogeneous substance, Payen (1838) showed that membranes from wood fail to contain cellulose. By treating whole wood with nitric acid and alkali, he separated cellulose from some “incrusting” substances. Lignin, along with hemicellulose, is easily destroyed by acidic reagents. Lignin is that substance which cannot be converted into sugar. The “incrusting substance” Payen found was a mixture of polyoses and lignin. Lignified membranes were discovered to take place in color reactions with certain organic reagents. Colorimetric methods are extensively used to study lignin. Lignin is found to be of relatively lower molecular weight (about 1OOO). The elementary composition of lignin is found to be C47H52016 or C,,H3,0,(OH)j(CH30)j. It has a refractive index of about 1.6. Further information on the chemical nature of wood can be found in the books by Browning [lo].

11. Pyrolysis When a solid or liquid fuel burns, the molecular structure of the fuel is modified by the action of the heat reaching the fuel. The net result of this chemical change is to produce a combustible vapor. This process, called pyrolysis, has been under intensive investigation, since one possible method of control of fires is to eliminate or reduce the intensity of the sources of the flammable vapors. The pyrolysis process is often treated as a “wave” traveling through the material. I n a coordinate system affixed to the wave, density and temperature profiles would appear as in Fig. 3. If a more complete understanding of the heat and mass transfer within the pyrolyzing fuel can be achieved, it will be possible to better control the burning process. Theoretical work in this area is closely associated with available experi-

HEATAND MASSTRANSFER IN FIRERESEARCH -V*

227

VELOCITY OF PYROLYSIS WAVE

RAW

WOOD

WAVE FRONT I

I

"r"

DENSITY

"VVY

OLWIITY

FIG.3. A pyrolysis wave in a coordinate system moving with the wave.

mental findings, and there is much discussion in the literature as to the validity of various experimental procedures.

A. EXPERIMENTAL METHODS OF STUDYING PYROLYSIS 1. Digerential Thermal Analysis and Thermogravimetric Analysis

In differential thermal analysis (DTA) a sample of the material under test is instrumented with a thermocouple and placed in a controlled environment (i.e., nitrogen or air) along with a similarly instrumented and sized inert block. The assembly is then placed in an oven, and the temperature increased at a predetermined rate. The temperature difference between the sample and the inert material is recorded. Chemical reactions occurring within the sample as the raw material undergoes chemical changes a t the various ambient temperatures are either exothermic or endothermic, and show up as spikes on the differential temperature readout. The location and magnitude of these spikes can then be used to obtain information as to the identification of the reactants. Sample DTA plots are shown in Fig. 4. In thermogravimetric analysis (TGA), the sample is attached to a sensitive electronic balance, the assembly placed in a controlled environment, and the temperature raised at a predetermined rate. The rate of loss of weight is recorded, and the rate of generation of vapors as a function of

S. L. LEEAND J. M. HELLMAN

228

I-

0

YI

1

I,

-u 0

w

so

200

I00

300

Te WfWh(*c) %

TGA

-----------\

f x 0

w

f

U 0

IP'C

w

-

100

-

60

-

I

. I

I

1

80

100

I

I 200

I

I

300

60

40

20

1

4 0

Temptratur ('C)

FIG.4. Differential thermal analysis of cellulose treated with 5% zinc chloride by weight (top) and of pure cellulose (bottom) [ll].

temperature can be obtained. Figure 5 illustrates a TGA thermogram for cellulose both with and without a fire retardant. 2. Mass Spectrometry and Gas Chromatography When mass spectrometry and gas chromatography are used to study pyrolysis, the sample is placed in a controlled environment, heated at a known rate, and the vapors injected into the instrument. The pyrolysis gases can then be identified by reference to standard tables. Experimental

HEaT AND MASS TRANSFER I N FIRERESEARCH

229

work in this area has been done using both cellulosic materials [14] and a simulated heated char [lS]. 3. Densometric Analysis

In this type of analysis, the sample is instrumented with thermocouples to provide temperature profiles within the sample as a function of time as the sample is burned from the outside in. Right circular cylinders of cellulose have been used in this type of experiment with reasonable success. The cylinders are ignited around the periphery, and x-rays or gamma rays (from a 13'Cs source) used to take a picture along the axis of the cylinder at progressive times during the burning. From measurements of the optical density variations of the resulting negatives, it is possible to correlate the physical density of the cellulose inside the sample with the temperature profiles for various times, producing a time history of the pyrolysis occurring within the sample. Blackshear and Murty Kanury [lS] originated this method using x-rays, and Nolan et al. [17] and Brown [lS] have obtained results with the I3'Cs source. OF EXPERIMENTAL RESULTS B. INTERPRETATION

There is considerable variation in the reported results for the heat of reaction during pyrolysis, a key element in any theory of fire spread. 100

I0

t

\1\'\'\\\ vvll---

omo*

'I-

20

0

FIG.5. TGA thermograms of cellulose and (NH4)ZHPOI treated cellulose heated at 25"C/min in a flow of 100 cm3/min air [12].

230

S. L. LEE AND J. M. HELLMAN

Roberts has reviewed the literature on wood pyrolysis [19, 203, and points out that published experimental estimates of the heat of reaction range from an endothermic value of 370 J/gm (for cellulose) to an exothermic value of - 1700 J/gm (for wood). He concludesthat the following factors caused variations in the reported kinetics of the wood pyrolysis reactions. (a) The rate and course of the pyrolysis reactions in cellulose are very sensitive to catalytic and autocatalytic effects. Values for the apparent activation energy of the pyrolysis reactions have been obtained over the range E = 40-230 kJ/mole; a reaction with E = 125 kJ/mole appears to be the most important one during the pyrolysis of cellulose in bulk samples of wood. (b) Lignin pyrolyzes appreciably more slowly than does cellulose a t temperatures greater than 340°C. Its pyrolysis does not appear to be sensitive to catalytic or autocatalytic effects. (c) The physical structure of wood restricts the free movement of volatile products of pyrolysis until it suffers macroscopic changes, such as fissuring, which generally occur a t temperatures of 300-320°C. Before the local occurrence of such changes, conditions strongly favor autocatalysis. Subsequently the products can escape relatively easily, and the importance of this effect is reduced. (d) If the rate of temperature rise in a slab of wood is low or if it is heated only to temperatures of the order 300-320°C, it is to be expected that the kinetics of the pyrolysis reactions will be highly dependent on experimental conditions because the effects of the low pressure region will predominate. Conversely, if this low temperature region is passed through quickly, the kinetics will be more consistent. With regard to the overall pyrolysis of wood, Roberts reaches the following conclusions.

(a) Under some conditions, such as reduced pressure, the primary process of pyrolyzation can be endothermic. (b) Under normal conditions, a t atmospheric pressure in continuous slabs of wood, the primary pyrolysis of wood is exothermic. The difference may be caused by a change in the reaction mechanism of cellulose due to autocatalytic or catalytic effects. (c) If wood is heated to temperatures in excess of 320'33, the heat of reaction is approximately constant a t - 160 to - 240 J/gm of products evolved. It is estimated that 65% of this heat generation is due to the pyrolysis of lignin and 35% to the pyrolysis of cellulose. This heat of reaction originates in the primary pyrolysis of the wood materials and does not require the assumption of secondary reactions for explanation. (d) If wood is heated to temperatures not in excess of 320"C, the heat

HEATAND MASSTRANSFER IN FIRERESEARCH

231

of reaction is highly dependent on experimental conditions and may rise to -1600 J/gm. This change is due to secondary reactions; as the wood structure is progressively degraded with increasing temperature, the escape of volatile products is facilitated and the contribution of secondary reactions is reduced. Secondary reactions can contribute an extra - 1200 J/gm to the heat generated by the primary reactions. These conclusions are far from universal acceptance. At this time, much further research is needed into the chemistry of pyrolysis to permit a more basic understanding of the process and obtain more reliable values of the heat of reaction, either exothermic or endothermic, at the various locations in the structure of the pyrolysis wave.

C. MATHEMATICAL MODELSOF

THE

PYROLYSIS PROCESS

The typical model for analytical study of the pyrolysis process is a slab of wood exposed to heating on one side and insulated and impermeable to gas flow on the other. Assumptions are made as to the chemical reactions occurring within the porous, pyrolyzing wood, and for the various thermal properties of the material. Vapor generation rates, temperature profiles, and other pyrolysis related quantities can then be determined. Panton and Rittmann [Zl] have studied the case of a plane slab initially a t a uniform temperature suddenly heated on one side. Two cases are investigated, both assuming first order reactions with an Arrhenius rate equation. In one case, a single reaction, in which a virgin solid goes to a second solid plus a gas, is considered. The second case looks at a more complicated kinetic scheme: solidl + solidz solidl --f gasz solidz--f solidr

+ gasl + gass

They assume that the thermal conductivity is proportional to the density, and that the flow of vapors produced has no effect on the energy balance of the pyrolysis process. Figures 6(a)-6(c) illustrate some of the results of this study. Kung [22] uses a plane slab geometry and assumes transient conduction, internal heat convection of volatiles, Arrhenius decomposition of virgin material into volatiles and residual char, endothermicity of the decomposition process, and density, specific heat, and thermal conductivity of the char determined from a linear interpolation between values of the virgin wood and solid char. Murty Kanury [23] assunies linear temperature profiles in the interior of a wood-like solid, a critical temperature for pyrolysis, instant removal of the pyrolysis products upon their formation, and that

232

TIME(t1

(0)

FIG.6a. Total gas generation rate vs time for various values of the heat of reaction, R: E = 40; dimensionless frequency factor, F = 10-8; dimensionless radiation absorption boundary condition, CR = 1. Maxima occur a t approximately the same time, however, the duration of the reaction is markedly changed [21].

(b)

DISTANCEW

FIG.6b. Profiles of reaction rate a t various times for an exothermic reaction; E = 40, F = 10-8, R = -0.3, GR = 1. The front surface is X = 1. The maximum reaction rate a t the back wall is above 9, while at the surface it is about 4 1211.

HEATAND MASSTRANSFER IN FIRERESEARCH

233

W I4

a ; I

0 IV 4 W

K

(C)

DISTANCE I X )

FIG.6c. Profiles of reaction rate at various times for an endothermic reaction; E = 40, F = R = +0.3, GR = 1. The maximum reaction rate occurs at the surface [21].

the thermal properties of char and wood are dependent only upon the solid density and not temperature. A major difficulty in interpreting the results of this type of analysis is that there is not sufficient experimental data on the thermal properties of wood in the various stages of pyrolysis to provide a good check of the results or the validity of the assumptions made in obtaining the results. Many of the theoretical analyses performed when using the types of assumptions outlined in this section will provide good agreement with one set of experimental data but not with another. Assuming an exothermic heat of reaction where, in reality, it is endothermic, or vice versa, should be one of the obvious causes for noncorrelation of any theory with a certain set of experimental data.

D. RELATION OF PYROLYSIS TESTING TO FIREPROTECTION In an attempt to reduce damage from fires that do start, various chemical additives have been developed over the years to slow the spread of fire.

S. L. LEEAND J. 31. HELLMAN

234 2000

r

rControl

4

0

8

12

16

20

24

28

32

36

24

28

32

36

Elapsed time (min )

2000

1

r

/-Control

800

0

4

8

I2

16

20

Clapred time (min )

FIG.7. Effect of fire retardants AS (top) and DAP (bottom) on the weight loss rate of pine cribs (131.

One of the testing methods used to determine the effectiveness of a proposed fire retardant is by DTA and TGA. The rationale of the rating procedure is based on the observation that fires can burn either by glowing or flaming. Due to its slower rate of spread, the glowing burning fire is usually preferable from the traditional fire fighting point of view. Pyrolysis studies have indicated that wood can undergo two types of thermal degradation [13]. One, occurring at temperatures lower than 270°C, produces char, carbon dioxide, water vapor, and little combustible gas. The second type, occurring at temperatures above 340°C,produces little char and a large amount of combustible gases. This high energy path produces burning with flames. For a fire retardant to be effective,it must act to decrease the

HEAT AND hIASS

TRANSFER I N FIRERESEARCH

235

activation energy associated with the glowing burning process, causing this type of burning, rather than the flaming burning, to predominate in a fire. Typical chemicals used against forest fires for this purpose are diammonium phosphate (DAP), ammonium sulfate (AS) , and monoammonium phosphate (MAP). The phosphate-based chemicals have the added advantage of also acting as a fertilizer to the remaining vegetation, speeding the recovery time of the forest from the ravages of a fire. Figure 7 illustrates the weight loss rate of pine cribs as a function of the amount of DAP and AS used. Note the strong suppression of the rate of weight loss. This results in a smaller quantity of combustible gases produced, and hence a slower fire spread. 111. Ignition

A.

IGNITION SOURCES

I n almost every location where some type of fuel for a fire exists, ignition sources are quite often present. These sources can be in the form of lightning in a forest, a spark from an automotive ignition system, or a lit cigarette dropped from the hands of a smoker as he falls asleep. At times there is no fire as a result of an ignition source, while at other times a deadly fire may result. Another type of ignition is possible without an ignition source [24, 251. This process, known as spontaneous generation or self-ignition, can be explained by thermal explosion theory, Fig. 8. In usual exothermic reactions of interest to the fire problem, the rate of the reaction approximately doubles for every 10°C increase in temperature. If the heat generated by the reaction cannot be transmitted away fast enough, the rate of the reaction and the temperature will continue to increase until flaming burning occurs. This type of reaction can occur in many materials, as listed in Table I. Some of these reactions are thought to be initiated by the heat generated by biological decay of organic material. Since the rate of pyrolysis of a material is a function of the type of heating as well as the chemical nature of any impurities present, i t is not possible to precisely determine the ignition temperature of a type of material. Representative figures for various materials have been published and are listed in Table 11. In general, the ignition temperature of wood is of the order of 400"F, and 150°F is the highest temperature to which wood can be continuously exposed without risk of ignition. At temperatures intermediate to these values it is possible for pyrolysis to occur over an extended period of time, producing charcoal, a material which can undergo

TABLE I

MATERIALSSUWECT fro SPONTANEOUS HEATING'

Msterinl

Tendency to spontaneous henting

Usual shipping container or storage method

Precautions against spontaneous heating

Remarks

Charcoal

High

Bulk, bags

Keep dry. Supply ventilntion

Coal, bituminous

Moderate

Bulk

Store in small piles. Avoid high temperatures

Corn-meal feeds

High

Burlap bags, paper bags, bulk

Fertilizers: organic, inorganic, combination of both

Moderate

Bulk, bags

Fish meal

High

Bags, bulk

Linseed oil

High

Tank cars, drums, cans, glass

Manure

Moderate

Bulk

Mnterinl should be processed carefully to maintain safe moisture content and to cure before storage Avoid extremely low or high mois- Organic fertilizers containing nitrates must be carefully preture content pared to avoid combinations that might initiate heating Keep moisture 6 1 2 % . Avoid ex- Dangerous if overdried or packposure to heat aged over 100°F Avoid contact of leakage from con- Rags or fabrics impregnated with this oil are extremely dangerous. tainers with rags, cotton, or Avoid piles, etc. Store in closed other fibrous combustible materials containers, preferably metal Avoid extremes of low or high Avoid storing or loading uncooled moisture contents. Ventilate the manures piles

Hardwood charcoal must be carefully prepared and aged. Avoid wetting and subsequent drying Tendency to heat depends upon origin and nature of coals. High volatile coals are particularly Iiable to heat Usually contains an appreciable quantity of oil which has rather severe tendency to heat

Oiled fabrics

High

Rolls

Tung nut meals

High

Paper bags, Bulk

Varnished fabrics

High

Boxes

Waste paper

Moderate

Bales

Wool wastes

Moderate

Bulk, bales, etc.

0

National Fire Protection Association [26].

Keep ventilated. Dry thoroughly Improperly dried fabrics extremely before packing dangerous. Tight rolls are comparatively safe. Material must be very carefully These meals contain residual oil processed and cooled thoroughly which has high tendency t o heat. before storage Material also susceptible t o heating if over-dried Process carefully. Keep cool and Thoroughly dried varnished fabventilated rics are comparatively safe Keep dry and ventilated Wet paper occasionally heats in storage in warm locations Keep cool and ventilated or store Most wool wastes contain oil, etc., in closed containers. Avoid high from the weaving and spinning moisture and are liable t o heat in storage. Wet wool wastes are very liable to spontaneous heating and possible ignition

h-

z

tr

z

hm m

S. L. LEEAND J. 11. HELLMAN

238

GENERATION

X 3 J

u.

2 w

I

LOSSES


I 10.5, ., 106, 164 Block, J. -4., 268, 283 Bolsamo, S.R., 32, 37 Bondarchuk, T. P., 203(107), 217 Botterill, J. S. bl., 181(70, 72), 185(72, 73, 74, 733, 192, 216 Bouvier, J . E., 107(35), 164 Bowell, E. L. C., 2(7), 35 Bradshaw, K. D., 172, 177, 215 Brauer, F. E., 109, 164 Brazelton, W. T., 183, 200, 203(109), 208, 217, 218

Briggs, F., 19, 21, 24(30), 36 Briggs, L. J., 96, 97, 98(1>), 164 Brown, U. J., 229(17, 18),281 Brown, J. K., 262, 283 Browning, B. L., 226, 281 Brun, E. A., 177,216 Brundrett, G. W., 184(71), 216 Bryson, J. O., 263, 283 Butler, J. C., 57(46), 82 Butt, XI. H. D., 185(75), 216 Byram, G. &I., 267, 283

C Cain, G. L., 184(71), 185(75), 216 Caldas, J., 200, 218 Campbell, J. R., 203, 217 Campbell, hi. J., .5(8), 19, 20, 24(27), 35, 36, *57(45),73(39), 82 Capes, C. E., 203, 217 Carman, M. F., 37(46), 82 Carrier, W. D., 111, 53(43), 56, 82 Carslaw, H. S., 48, 81 Cassidy, W. A., 19(25), 24(25), 36 Charette, M. P., 7, 8, 9, 36 Chechetkin, A. V., 202, 218 Chen, J. C., 1.59, 160, 166 Chen, W. H., 211(140, 141), 218 Chung, B. T. F., 191, 203, 217 Chute, J., Jr., 53(35), 74(35, 36), 82 Ciborowski, J., 177, 216 Clark, H. B., 119, 131, 132, 165 Clark, J. A., 86, 163 Clark, K. K., 229(15), 281 Clark, S. P., Jr., 53(35), 54(36), 74(35,36), 82

Claxton, K. T., 177(59), 178(-59), 216 Clegg, P. E., 34, 37, 44(18), 45(18), 46, 48 (18), 33(18), 57, 62(18), 68(18), 81 Clements, H. B., 263, 264, 267(89), 283 Cohen, A. J., 19(25), 24(25), 36 Cohen, E. R., 92, 164 Cole, R., 153, 156, 165 Colner, D., 275(105), 284 Conel, J. E., 11, 21, 22, 24(32, 33, 34), 36, 71(63), 73(63), 82 Cooper, G. T., 200, 201, 205, 209, 217 Cooper, K. W., 118(50), 165 Cooper, hi. G., 136, 166 Copes, C. E., 171(17), 215 Corlett, R. C., 260, 282 Corty, C., 119, 165 Couderc, J. P., 185(77), 205, 216 Countryman, C. M., 224, 265, 270, 271, 281, 283

Cox, J. E., 107(35), 164 Crank, J., 243(33), 281 Cremers, C. J., 2, 4, 5, 6, 12, 13, 16(22), 17, 19, 20, 21, 23(6, 26), 31(46), 34, (4, 6), 35, 36, 48, 52, 53(37), 54(37 38, 39, 40, 41, 42), 57, 58(38, 41), 69 (32, 37, 38, 39, 40, 41), 70(37, 38, 39, 40, 41), 71, 75(39, 71), 76(41, 70), 77, 78, 79(41), 80(38, 39, 701, 81, 82, 83

D Dahlhoff, B., 210, 218 Daily, J. W., 87(2), 163 Das, C. N., 200,218 Davidson, J. F., 169(1, 5), 170(13), 171, 174, 176, 214, 215 Davis, E. J., 151, 165 Dawson, J. R., 19, 20, 21, 23(26, 29, 40), 24, 28, 29(40), 31, 36, 53(32), 54(32), 69(32), 81 Dean, C. W., 158, 161, 166 Deinken, H. P., 177(43), 216 Del Notario, P. P., 261(73), 272(73), 283 Denega, A. I., 177(51), 216 De Ris, J., 260, 267(87), 282,283 Dietrick, J. R., 109(36), 164 Doring, W., 92, 163 Dollfus, A., 19(28), 36 Donnadieu, G. A., 173, 175, 177(44), 215, 216

AUTHORINDEX Dougherti, J. E., 177(43), 216 Dow, W. M., 181, 192, 199, 203, 204, 205, 206, 216 Drake, E. M., 54(36), 74(36), 82 Drinkenburg, A. A. H., 192,217 Drbohlav, R., 169(4), 215 Dwyer, 0. E., 159, 160, 166 Dzakowic, G. S., 152, 153, 165

E Ebert, C. H. V., 246(41), 282 Einstein, V. G., 174(40, 41), 175(40), 177, 181(41), 198, 200, 201(154), 202(155), 203, 204(41), 205(116), 206, 210(139), 215, 217, 218 Ekonomov, S. P., 53(34), 58(34), 64(34), 70(34), 71(34), 82 Elliott, D. E., 189(71), 216 Elliott, E. R., 267(89), 283 Elliott, T. C., 278(111), 284 Elmas, M., 211, 218 Emelianov, D. S., 177(53), 216 Emmons, H. W., 221, 243, 245(1), 250, 255, 257, 269, 281, 282, 283 Endo, H., 269(94), 283 Enjalbert,, M., 185(76, 77), 205(76, 77), 216

Epstein, P., 40, 80 Ergun, S., 170, 215 Ermakov, G. V., 104, 164

F Fairbanks, D. F., 186, 187, 188(82), 189(81,82), 191,205(82), 206(82), 216 Fan, L. T., 196(89), 203(89), 217 Faust, A. S., 119, 165 Fenske, R. M., 203, d l 7 Ferron, J. R., 173, 216 Fetting, F., 193, 202, 203, 205, 206, 217 Fielder, G., 34, 37, 44(18), 45(18), 48(18), 53(18), 62(18), 68(18), 81 Fillipovskiy, N. F., 208, 218 Fisher, J. C., 111, 165 Flory, K., 109, 164 Fountain, J. A., 48, 51, 52, 56, 57, 81 Frandsen, W. H., 244(40), 266, 269(99), 282, 283 Frantz, J. F., 172, 174, 177, 215

287

Frenkel, J., 92, 164 Fritz, J. C., 173, 215 Fritz, W., 207, 217 Frost, W., 152, 153, 165 Fryxell, D., 55, 82 Fugii, N., 64(50, 51), 67(51, 53), 82

G Gabor, J. D., 185, 186, 116 Galloway, T. R., 208, 217 Gamson, B. W., 200, 201, 218 Garg, D. R., 243(36), 282 Garipay, R. R., 73(65, 66), 82, 83 Garlick, G. F. J., 19(28), 36 Garris, C. A., 253, 254(48), 255(48), 256 (48), 257, 282 Gear, A. E., 46(20), 57(20), 81 Geake, J. E., 19(28), 36 Gelperin, L. G., 207(123), 217 Gelperin, N. I., 174(40, 41), 175, 177, 181 (41), 198, 200, 201, 202, 203,204(41), 20$(116), 206, 210(139), 216,217, 218 Genetti, W. E., 209, 210(136), 218 George, C. W., 234(13), 281 George, P. M., 267(89), 28.3 Gilbert,, C. S., 96(14), 97(14), 98(14), 99 (14), 164 Gilsinn, D., 275(105), 284 Gishler, P. E., 201(104), 202(104), 203,206 (104), 209(104), 217 Glaser, P, E., 42, 43, 45, 48, 50, 58(22), 59 (22), 64, 68, 81 Glass, B., 25,36 Glassman, I., 267, 283 Gold, T., 5(8), 19, 21, 24(27), 27, 36, 37, 57, 82 Goodman, T. R., 212, 218 Gorelik, A. G., 194, 217 Graham, R. W., 128, 148(61), 149, 165 Gray, V. E., 280(118), 284 Green, N. W., 109(40), 164 Greenwood, W., 55(43), 82 Griffth, P., 122, 123, 125, 126, 127, 128, 130, 131, 132(59), 133, 152, 166 Grimmett, E. S., 209(134), 210(136), 218 Grishin, M. A., 177, 216 Gross, D., 235(25), 258, 265, 268, 280 (117, 118), 281, 282, 283, 284 Grummer, M., lSl(64, 65) 189, 203, 216

ACTHORINDEX

288

Guiglion, C., 185(77), 216 Gutfinger, C., 211(140, 141, 142), 212, 218

J

Jackson, R., 170(16), 215 Jacob, A., 187, 189, 199,202,203,205, 217 Jaeger, J. C.,40, 70, 71, 72, 78, 81,82 Hafer, C. A., 31, 281 Jaeger, J. S., 48, 76, 81 Hagemeyer, W-.A,, 71(63), 73(63, 64,65), Jaffee, L. D., 50(30). 51(30), 59(30), 6s 82, 83 (30~ 81 Haiajian, J. D., 73(58), 82 Jakob, hl., 181, 184, 192, 203, 204, 206, Hamano, Y., 67(33), 82 216 Rammitt, I?. G., 57(2), 163 Johanson, L. N..173, 21/i Han, C.-Y., 132, 265 Johnson, F. S., 29, 37, 75(68), 83 Hapke, B., 2(1), 7(1), 9, 10, 19(25),21(31), Johnson, G. M., 234( 13), 281 24(%3},35, 36 Johnson, S. W., 56(44), 82 Harrison, I)., 169(1, 21, 170, 171, 174, 814, Johnson, T. V., 7, 8, 9, 35, 36 215 Johnstone, H. F., 199. 204, 205, 207, 217 liaruaki, F., 177, 216 Jones, B. P., 73(66), 83 Harvey, E. K , llS(51, 32, 53), 121(.50, 51, Jones, R. L., 24(41), 36 62, 531, 165 Jueland, A. C., 177, 216 Hawthorn, 11. D., 188(82), 189(82), 205 ( 8 3 , 206(82), 216 Heertjes, P. XI., 172, 175, 177(46), 179, K

H

215,216

Heiken, G., 55(43), 82 Kagan, Y., 104, 105, 164 Hellman, J. AX., 261, 272, 283, 284 Kang, W. K., 171(17), 215 Hemingway, B. S.,44(17), 59(17), 60(16, Kanomori, H., 43(15), 44(15), 45(15), 60 17,49),61, 63(17), 64(17, 49), 69(17), (1.5), 62(15), 64(15,50,51), 65(15), 67, 70(17), 81, 82 81, 82 Hengstenberg, D., 105(24), 106(24), 164 Kaparthi, R., 173, 177(35), 179, 200, 205, Heselden, A. J., 259(65), 282 215, zir Hickerson, C. W., 224(8), 253(8), 281 Kasturirangan, S., 125, 127, 166 Ilirth, J . P., 92(92), 164 Katz, D. L., 109(43), 110, 164,200, 218 Holman, J. P., 179(60), 193(86), 216 Katz, J. L., 92, 105(24), 106(24), 164 Holtz, 11. E., 138, 159(83, 84), 166 Kazakova, E. A., 177(45, .51),116 Hooper, F. C.,121(38), 165 Kennedy, E. H., 87(3), 107(32), 164, 169 Hoori, Z., 179(62), 216 Kenning, I).B. R., 156, 166 Horai. K., 43, 44(13), 4.5, 60(15), 62, 64, Kenrick, F. B., 96, 97, 98(14), 99, lG4 65(lq5,X), 67, 81,81 Kerhm, S., 53(35), 74(35), 82 Hottel, II. C., 269, 283 Kermode, K.I., 210, 218 Hsu, Y. Y,,128(62), 129, 130, 132, 133, Kettenring, K. N., 172, 175, 177, 179, 115 148, 1.52, 165 Khitrin, L. N., 233(24), 281 Huang, W. C., 260,282 Kinbara, T., 269, 283 Hudson, F. L., 208, 218 King, I?. A., Jr., 57, 82 Huige, S. J. J., 192(90), 217 King, N. K., 265,283 Humphreys, H. W., 250(45), 282 Kirk, L. A,, 208, 218 Hunt, G. It., 11, 32, 36, 37 Klassen, J., 201(104), 202(104), 203, 206 Huntsinger, H C., 200, 218 (104), 209(104), zir I-fwang, C. L , 196(89), 203(89), 227 Knapp, R. T., 87, 118, 119, 121, 163, 165 Koppel, L. B.,193, 203(109), 217 I Korolev, V. N., 208, 218 ?. Korotyanskaya, L. A,, 210(139), 218 Ilchenko, A. I., 208, 217 Kosky, P. G., 146, 165 Ingrao, H. C., 76, 78, 83

AUTHORINDEX Kothari, A. K., 174, 175, 177(38), 216 Kritz, M. A., 48, 81 Krotikov, V. D., 41, 71, 81 Kruglikov, V. Y., 201(154), 218 Kung, H. C., 231, 281 Kunii, D., 169(2, 6), 170(2), 171(19), 174 (2, 19), 176(19),179, 180(2), 181, 194, 195, 196(94), 197(94), 214, 216, 217

L Lamb, W., 19(28), 36 Landrock, A. H., 211, 218 Langseth, M. G., Jr., 53, 54, 74(35), 82 Law, M., 243, 259(65), 279(114), 281, 282, $84 Lawson, D. I., 221, 278(112),281, 284 Lebedev, P. D., 174(40), 175(40),215 Lebofsky, L. A., 7, 8, 9, 36 Lee, S. L., 221, 247,248(43), 249,250,253, 254(48), 255(48, 50, 57), 256(48), 257, (62), 260, 261(75), 262, 269(75), 272 274(60), 281, 282, 283, 284 Lee, T. G., 280(118), 284 Lemlich, R., 200, 218 Lese, H. K., 210, 218 Leva, M., 169(8), 181(65, 66), 189, 202, 203, 205, 206, 215, 216, 217 Levenspiel, O., 169(2), 170(2), 171(19), 172(25), 174(2, 19, 25), 176(19), 175(25), 177(25),179,180(2), 181(68), 182, 194, 195, 196(94), 197(94), 199, 203, 206,2l4, 215, 217 Lewis, B., 247, 282 Lewis, J. B., 177(59), 178(59),216 Linan, A., 243(37), 282 Lindin, V. M., 177, 216 Ling, C. H., 255, 282 Linsky, J. L., 41, 47, 57, 71, 73(7), 76(73), 78(73), 81, 83 Lipsett, S. G., 107(33), 164 Lipska, A. E., 229(14), 281 Loftus, J. J., 280(117, 118), 284 Logan, L. M., 11, 32, 36, 37 Logwinuk, A. K., 201, 202, 203, 206, 209, 217 Lohneiss, W. H., 253(49), 282 Long, G., 107(34), 164 Lothe, J., 92, 164 Low, J. W., 71, 73(62), 76, 78, 82 Lowndes, R. P., 21, 23, 36

289

Lucas, J. W., 71,73(63,64,65,66,67), 76, 82, 83 Lyons, W. E., 16(22), 36

M McBirey, A. R., 43, 81 McCardell, R. K., 107(37), 164 McCord, T. B., 7,8(14), 9, 21, 22(14), 24, (32, 37), 25(32, 37), 36,36 McElroy, W. D., 118(50, 51, 52, 53), 121 (50, 51, 52, 53), 165 McGuire, J. H., 279(113), 284 McIlhinney, A. E., 203(101), 217 McKibbins, S. W., 172, 175, 177(21), 179, 215 Madejski, J., 152, 154, 155, 166 Makhorin, K. E., 208, 2f7 Malan, D. H., 243(33), 282 Manderfield, E. L., 172(22), 175(22), 177 (22), 179(22),216 Marchello, J. M., 189, 191, 217 Markvart, M., 169(4), 616 Marov, M. Y., 53(34), 58(34), 64(34), 70 (34), 71(34), 82 Marshall, W. R., Jr., 174(39),215 Marston, A. C., 34, 37, 53(31), 68(31), 81 Martens, H. E., 50(30), 51(30), 59(30), 68(30), 81 Marto, P. J., 138, 140, 142, 166 Martyushin, I. G., 202, 218 Matekunas, F. A., 156, 166 Matsumura, H., 151, 157(71), 165 Mersmann, A., 203, 217 Mesler, R. B., 109(40, 41), 164 Michaels, A. S., 92, 164 Mickley, H. S., 186, 187, 188, 189, 191, 200, 203, 204, 205, 206(8l), 216, 21 7 Miller, C. O., 201, 202, 203, 206, 209, 217 Miller, R. W., 109(37), 164 Mitson, A. E., 200, 202, 2f7 Miautani, H., 64(50, 51), 67(51), 86 Mlodinski, B., 177(52), 216 Moore, G. R., 102, 164 Moore, T. W., 179(60), 216 Moreno, G. F., 261(73), 272(73), 28.9 Morozhenko, A. V., 10,36 Morton, B. R., 255(51, 52, 53, 541, 282 Mukhlenov, I. P., 202(158), 203(102, 103, 107), 817, 218 Mulligan, J. C., 48, 81

AUTHORINDEX

290

Muncey, R. W., 71, 72(56), 82 Murase, T., 43, 81 Murcray, D. G., 11, 36 Murcray, F. H., 11, 36 Murgai, M. P., 255(56), 282 Murray, B. C., 71, 73(61), 82 Murray, J. D., 170, 215 Murty Kanury, A,, 225, 229, 231, 243(39), 281, 282 Muzichenko, L. B., 177(51), 216 Myers, J. E., 172, 177(24), 215

N Nakanishi, E., 109(39), 164 Napalkov, G. N., 174(40), 175(40), 215 Nash, I). B., 21, 23, 24(32, 33, 34), 36 Nelson, W., 87(3), 107(32), 111, 163, 164,

Perry, C. H., 21, 23, 36 Peters, K., 177, 216 Petitt, E., 40, 73(2), 76, 78, 80 Pettigrew, C. K., 211, 218 Philpot, C. W., 228(11), 234(13), 265(11), 281, 283 Pieters, C., 7, 8, 9, 36 Pirsko, A. R., 224, 253(8), 281 Plitt, K. F., 276(108), 284 Pomarentsev, V. M., 202(158), 218 Pound, G. M., 92(12), 164 Prasad, C., 266(83), 283 Preckshot, G. W., 146, 165 Puplett, E., 34, 3r, 53(31), 68(31), 81 Putman, A. A., 260, 283

16.5

Nicholson, S. B., 40, 73(2), 76, 78, 80 Nielsen, H. J., 255, 275(106), 282 Nilsson, E. K., 275(106), 284 Nobel, A. P. P., 193(91), 199(91), 202(91), 204(91), 205(91), 206(91), 217 Nolan, P. F., 229, 281

0 O’Leary, B. T., 3(8), 19, 21, 24(27,30), 35, 36, *57(45),82 Olson, R. L., 172(25), 174(25), 175(25), 177(25), 215 Orlichek, A., 177(55), 216 Orloff, L., 258(63), 260, 282 Osako, hl., 67(