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Radiative Transfer Through Plane-Parallel Clouds Of Small Particles

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Radiative Transfer Through Plane-Parallel Clouds Of Small Particles

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Content RADIATIVE TRANSFER THROUGH PLANE-PARALLEL CLOUDS OF SM ALL PARTICLES by Charles Franklin Sanders, Jr. A D isserta tio n Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial F u lfillm ent o f the Requirements fo r the Degree DOCTOR OF PHILOSOPHY (Chemical Engineering) August 1970 71-7738 SANDERS, Jr., Charles Franklin, 1931- RADIATIVE TRANSFER THROUGH PLANE-PARALLEL CLOUDS OF SMALL PARTICLES. University of Southern California, Ph.D., 1970 Engineering, chemical University Microfilms, A XEROX C om pany, Ann Arbor, M ichigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 9 0 0 0 7 This dissertation, written by Ch9.rl.e5.. .Frank].in.. Sanders.,.. Jr. ................. under the direction of //.Is Dissertation C o m mittee, and a p p ro v e d by all its members, has been presented to and accepted by The G r a d u ate School, m partial fulfillment of require ments of the detjree of D O C T O R O F P H I L O S O P H Y Dean Date... August.. 1970. Chairm an ACKNOW LEDGM ENTS The author would lik e to express his indebtedness to the following persons and organization for th eir aid in th is work. Dr. J. M . Lenoir - for his in t e r e s t, encouragement, and guidance during the course of th is research. The members of the D isserta tio n Committee: Dr. C. J. Rebert and Dr. Melvin Gerstein - for th e ir cooperation and contributions. Mrs. P h y llis Osborn - for preparation of the rough d raft of the manuscript. Mrs. Sergene Zimmerman and Mrs. Rita Hary - for preparation o f the fin a l draft of the manuscript. Mrs. C. F. Sanders, Jr. (the author's w ife) - for help in preparing the manuscript and her prayers and encouragement during the course of the research. The San Fernando Valley S tate College - for making a v a ila b le the ser v ic es of th e ir computing center. TABLE OF CONTENTS Page ACKNOW LEDGM ENTS i i LIST OF FIGURES ..................................................................................................... v LIST OF TABLES .........................................................................................................v i i i NOM ENCLATURE ............................................................................................................. xi I. INTRODUCTION ............................................................................................................. 1 II. THEORETICAL BACKGROUND ................................................................................... 8 A. P article-R adiation Interactions ..................................................... 8 B. Single Scattering ........................................................................................ 15 C. D iscrete Flux Approximations for M ultiple 21 Scattering ................................................................................................... D. Solution of the Equations for P lane-Parallel Clouds ........................................................................ 27 E. An Empirical Solution .............................................................................. 29 III. DESCRIPTION OF THE EQUIPMENT ..................................................................... 38 A. Infrared Spectrometer .............................................................................. 38 B. Sample Cell ...................................................................................................... 42 C. Microscope ................................................... 44 D. B a la n c e ................................................................................................................. 45 E. P a rtic les ............................................................................................................ 45 IV. EXPERIMENTAL PROCEDURE .................................................................................... 48 A. Sample Preparation ...................................................................................... 48 i i i Page IV. (Con't.) B. P a rtic le Size Measurement................................................................. 50 C. Infrared Transmission Measurements .......................................... 52 V. EXPERIMENTAL RESULTS ....................................................................................... 56 A. P a r tic le Size Measurements ............................................................... 56 B. Cell Blank Measurements ..................................................................... 61 C. Transmission of the Collimated Beam ........................................ 62 D. Transmisssion o f the D iffuse Radiation ................................ 63 VI. CORRELATION OF RESULTS .................................................................................. 75 A. The Collimated Transmission ............................................................. 75 1. Aluminum Oxide P a r tic le s .......................................................... 7^ 2. Graphite P a r tic le s ......................................................................... 90 B. The D iffuse Transmission .................................................................... 101 1. The D iffuse Extinction Cross Section < j > ........................... 104 2. The Multiple Scattering Parameter iJ jq ............................. H 7 VII. SUM M ARY ........................................................................................................................ ]26 VIII. REFERENCES ................................................................................................................. I 33 APPENDIX A: Tables of Data and Results ........................................ 136 APPENDIX B: Fortran Program for Determining < ( > and ip0 . . . 163 i v Page 10 36 39 43 51 64 65 66 67 68 69 70 71 LIST OF FIGURES S cattering Angles and Coordinates ...................................... The E ffect of the Parameter ip on the Transmission Through a Plane-Parallel Cloud ..................................... Beckman IR-2A Spectrometer ...................................................... Sample Cell and Assembly of One Side .............................. Feret Diameter I llu s tr a te d ...................................................... Transmission of Aluminum Oxide P a rtic les in Carbon T etrachloride with a Collimated Source, A = 2y Transmission of Aluminum Oxide P a rtic les in Carbon D isu lfid e with a Collimated Source, > = 8.5y . . Transmission of Graphite P a r tic le s in Carbon T etrachloride with a Collimated Source, A = 2y Transmission o f Graphite P a r tic le s in Carbon D isu lfid e with a Collimated Source, A = 5.5y .. Transmission of Aluminum Oxide P a rtic les in Carbon Tetrachloride with a D iffuse Source, A = 3.5y . Transmission of Aluminum Oxide P a rtic les in Carbon D isu lfid e with a D iffuse Source, A = 5.5y .......... Transmission of Graphite P a r tic le s in Carbon T etrachloride with a D iffuse Source, A = 4y . . . Transmission o f Graphite P a r tic le s in Carbon D isu lfid e with a D iffuse Source, A = 6y ............... v Figure Page 14 Extinction E ffic ie n c ie s of Aluminum Oxide P a rtic les in Carbon Tetrachloride ................................................ 79 15 Extinction E ffic ie n c ie s o f Aluminum Oxide P a r tic le s in Carbon D isu lfid e .......................................................... 80 16 Extinction E ffic ie n c ie s o f Aluminum Oxide P a rtic les ------- 81 17 Extinction E ffic ie n c ie s o f Non-absorbing Spheres with R efractive Indices near 1.0 .................................................. 82 18 Extinction E ffic ie n c ie s o f Absorbing Spheres, n' = 1.29 (1 - ik) ..................................................................................... 86 19 E xtinction E ffic ie n c ie s of Graphite P a r tic le s in Carbon Tetrachloride ......................................................................... 94 20 Extinction E ffic ie n c ie s of Graphite P a rtic les in Carbon D isu lfid e .................................................................................. 95 21 Extinction E ffic ie n c ie s o f Graphite P a r tic le s ......................... 97 22 Extinction E ffic ie n c ie s of Graphite P a r tic le s Compared to Calculated Values .......................................................... 99 23 Extinction C o efficien ts with Collimated and D iffuse Sources ........................................................................................... 107 24 "Diffuse" Extinction E ffic ie n c ie s o f Aluminum Oxide P a rtic les in Carbon T etrachloride ................................ 109 25 "Diffuse" Extinction E ffic ie n c ie s of Aluminum Oxide P a rticles in Carbon D isu lfid e ......................................... 110 vi Figure Page 26 "Diffuse Extinction E ffic ie n c ie s of Graphite P a rticles in Carbon Tetrachloride ............................................. I l l 27 "Diffuse Extinction E ffic ie n c ie s of Graphite P a rtic les in Carbon D isu lfid e ...................................................... 112 28 Variation o f the "Diffuse" Extinction E ffic ie n c ie s o f Aluminum Oxide P a rtic les with Phase Angle .................... 113 29 "Diffuse" Extinction E ffic ie n c ie s for P o s itiv e Phase Angles ................................................................................................ 116 30 The E ffect o f Forward Scattering on the Parameter ip0 . . . 119 31 The Parameter ip0 as a Function of the D ifference in the R efractive Indices of the P a rticles and the Liquid .. 121 32 Predicted D iffuse Transmission for Aluminum Oxide P a rtic les in Carbon T etrachloride, d3o = 2 0 .2p, A = 3.5 y , for D ifferen t Values of the Parameter tp0 . . 123 33 Predicted D iffuse Transmission for Graphite P a r tic le s in Carbon D isu lfid e , dp = 5.47P, ^ = 6P, for D ifferent Values of the Parameter ip0......................................... 124 vii LIST OF TABLES Table Page 1 Measured Size D istrib u tions for Aluminum Oxide ...................... 57 2 Measured Size D istrib u tions for Graphite P a r tic le s ........... 58 3 E xtinction Cross Section and E xtinction E fficiency of Aluminum Oxide P a rtic les in Carbon T etrachloride .............................................................................................. 77 4 Comparison o f E xtinction C o e fficien ts for Aluminum Oxide P a rtic les in Carbon T etrachloride Obtained by D ifferen t Experimental Procedures ..................................... 89 5 Extinction Cross Section and Extinction E fficien cy of Graphite P a r tic le s in Carbon Tetrachloride ............... 93 6 Values o f TanB for Graphite P a r tic le s ............................................ 100 7 Multiple Scattering Parameters for Aluminum Oxide in Carbon Tetrachloride ...................................................................... 105 8 Transmission of Cell Windows ................................................................... 137 9 Calculated Normal and Hemispherical R e f l e c t i v i t i e s by Liquid-Cell Window Interface .................................................. 138 10 Liquid E xtinction C o efficien ts .............................................................. 139 11 Transmission of Aluminum Oxide P a r tic le s in Carbon Tetrachloride with a Collimated Source ................................. 140 12 Transmission o f Aluminum Oxide P a rtic les in Carbon D isu lfid e with a Collimated Source ........................................... 141 vi i i Table Page 13 Transmission of Graphite P a rtic les in Carbon T etrachloride with a Collimated Source ................................. 142 14 Transmission o f Graphite P a rtic les in Carbon D isu lfid e with a Collimated Source ............................................ 143 15 Transmission of Aluminum Oxide P a rtic les in Carbon Tetrachloride with a Collimated Source ................................ 145 16 Transmission o f Aluminum Oxide P a rtic les in Carbon Tetrachloride with a D iffuse Source .......................................... 146 17 Transmission o f Aluminum Oxide P a rtic les in Carbon D isu lfid e with a D iffuse Source .................................................. 147 18 Transmission o f Graphite P a r tic le s in Carbon Tetrachloride with a D iffuse Source .......................................... 148 19 Transmission of Graphite P a r tic le s in Carbon D isu lfid e with a Diffuse Source .................................................... 150 20 Extinction Cross Section and E xtinction E fficien cy of Aluminum Oxide P a rtic les in Carbon T etrach loride.. 152 21 Extinction Cross Section and Extinction E fficien cy of Aluminum Oxide P a rtic les in Carbon T etra ch lo rid e.. 153 22 Extinction Cross Section and E xtinction E fficien cy of Aluminum Oxide P a rtic les in Carbon D isu lfid e ............ 154 23 Extinction Cross Section and Extinction E fficie n c y of Graphite P a rtic les in Carbon T etrachloride ....................... 156 ix Table Page 24 Extinction Cross Section and Extinction E fficiency of Graphite P a rtic les in Carbon D isu lfid e ............................ 157 25 M ultiple S cattering Parameters of Aluminum Oxide in Carbon D isu lfid e ................................................................ 160 26 M ultiple Scattering Parameters of Graphite in Carbon Tetrachloride ............................................................................. 161 27 Multiple Scattering Parameters of Graphite in Carbon D isu lfid e ....................................................................................... 162 x NOM ENCLATURE A — y C2- o 2 a -- phase angle =2x(np-nm) b — fr a ctio n of scattered radiation in hemisphere toward source B — s ca tterin g parameter for two-flux model C - 1-fwo C(Ca , Cs» Ct) - - p a r tic le cross sec tio n (subscripts refer to absorption, s c a t t e r , to ta l ex tin ctio n ) Cg — p a r t ic le geometric c r o ss-se c tio n cos 0 - - forwardness of sc a tte r d — p a r t ic le diameter D — b % E - d i e l e c t r i c constant f — fr a ctio n o f scattered radiation in the forward d ir ectio n i — v ^ r I — in t e n s it y , radiant energy flu x d en sity per unit s o lid angle o f divergence IB — in t e n s it y o f black-body radiation Io in t e n s it y on entry into system o f in t e r e s t *m ““ imaginary part o f K{Ka, Ks» Kt) - - ex tin ctio n c o e f f i c i e n t (subscripts absorption, s c a t te r , to ta l e x t in c t io n ) , cm"* re fer to I - - length; cm L — path length; distance; cm Le " e f f e c t i v e length, cm n - - real part of r e fr a c tiv e index xi n 1 - - complex r e fr a c tiv e index = n(l - i<) p( ) - - phase function with measure of d irectio n in parentheses Q - - energy flu x (energy/time) r -- radius Re -- real part of R |_ — re flecta n c e o f the cloud boundary at x = L R q - - r e flecta n c e of the cloud boundary at x = 0 t - - thickness o f p a r tic le v -- volume fraction so lid s x - - s i z e parameter: (ud/x) for spheres, ( 2t tt/X ) for flakes x, y , z - - Cartesian coordinates X(Xa , Xs , X t ) — Mie e x tin ctio n e f f ic ie n c y (subscripts refer to absorption, scatter,an d total ex tin ctio n ) X^ -- d if fu s e ex tin ctio n e ffic ie n c y ct - - a b so rp tiv ity , absorptance e — defined by tang = npic /|n p -n j e — em issiv ity 6 - - polar angle k - - absorption index e K aA/4nn X - - wavelength, microns y - - cos<f>, where c f > is polar angle v — frequency p — r e f l e c t i v i t y a — (oa, os, at) -- cross sectio n of p a r tic le s per unit volume (subscripts refer to absorption, s c a t t e r , and to ta l e x tin c tio n ) xi i o0 — e le c tr ic a l conductivity x — op tical thickness < p - - m ultiple sca tterin g parameter ip - - azimuth or bearing angle, sc a tte r in g parameter — m ultiple sca tterin g parameter w0 - - albedo for s c a t te r , the r a tio o f s c a t te r to to ta l ex tin ctio n c o e f f ic ie n t K S/(KS + K a ) Q , - - so lid angle, steradians Subscript Usage a, s , t - - absorption, s c a t t e r , and total ex tin ctio n h, n -- hemispherical, normal m - - measured in the medium p — measured in the p a rticle +, - - - used with in t e n s it y , in a unidimensional system, to ind icate flux in the d ir ectio n of increasing or decreasing value of the principal coordinate x, y , z - - used with in te n s it y to in d ica te component in d irectio n of the coordinates 1 I. INTRODUCTION In recent years there has been an increasing engineering in t e r e s t in infrared ra d ia tiv e tra n sfer through media which both absorb and s c a t te r the rad iation . At high temperatures most of the heat transfer from p a rticle s moving in pneumatic flow is by rad iation . Problems involving heat transfer from hot flu id iz e d s o lid s in chemical p rocesses, unburned propellant p a r tic le s in rocket engine exhausts, and e f f e c t s o f atmospheric p a rticu la te matter on infrared d etectin g systems are im mediate examples o f current engineering involvement. Earlier engineer ing in t e r e s t had been lim ited to some works with p a r tic le s o f fu e ls and soot in ind u strial furnaces. Previously the primary in t e r e s t had been in stu d ies o f s t e l l a r atmospheres by p h y s ic is ts and astronomers. Radiation is energy in transport in electrom agnetic form. The behavior of radiation in the infrared region is not unlike that o f other forms o f electrom agnetic ra d ia tio n , e s p e c ia lly v i s i b l e l i g h t and microwaves, those forms immediately below and above infrared in the wave length spectrum. In f a c t , the laws o f physical and geometrical o p t ic s , normally thought o f in reference to v i s i b l e l i g h t can also be applied to infrared or microwave rad iation . Furthermore, some of the previous works ap plicab le to infrared ra d ia tiv e tra n sfer through an absorbing-scattering medium were actu ally concerned with v i s i b l e l i g h t or microwaves. A substantial background in the basics o f physical and geometrical o p tic s - d if fr a c t io n , r e f l e c t io n , r e fr a c tio n , e x t in c t io n , etc. - would be a very d esira b le preparation for undertaking problems in infrared ra d ia tiv e tr a n sfe r, and a thorough in v e stig a tio n o f the problem cannot leave out con sideration o f sim ila r problems with other forms o f electrom agnetic waves. The basic physical phenomenon o f in t e r e s t in the an a ly sis of pro blems with media which absorb and s c a t t e r is the in tera c tio n between the electrom agnetic wave and the sc a t te r in g p a r t i c l e . Other than the o p tical properties o f the p a r t ic le and those o f the surrounding medium, the important parameter governing that in tera c tio n i s the s i z e of the p a r t ic le r e la t iv e to the wave length o f the ra d ia tio n . It is worth noting that the o p tical properties which are u sually expressed as a r e fr a c tiv e index, but may also be given by e le c t r ic a l con d uctivity and d i e l e c t r i c constant, are a lso dependent on the wave length. For spheres o f arbitrary s i z e r e la t iv e to the wavelength the b a sic theory o f l i g h t sca tte r in g by p a r tic le s was f i r s t formulated by Mie (1). Mie considered the problem o f a plane electrom agnetic wave s t r i k ing a s in g le sphere of arbitrary diameter. Even fo r such a simple geometry the so lu tio n to the problem took the form o f a very complex s e r ie s o f Bessel functions and Legendre polynomials. The complexity o f the so lu tio n has led to consideration o f sp ecial cases fo r which the so lu tio n may be s im p lif ie d . Additional e f f o r t s have been devoted to the problems a sso cia ted with other shapes, the presence of more than one p a r tic le , e t c . . Van de Hulst (2) gives a very complete t r e a t i s e o f these problems. A c o lle c t io n o f more recent works is given by Kerker (3). Much o f the owrk which has been done on sc a tte r in g by p a r tic le s has been r e s tr ic te d to s in g le independent sc a t te r in g . With s in g le in - 3 dependent sca tte r in g the in te n s ity scattered by a cloud of p a r t i c l e s , N in number, is simple N times the in te n s ity sca ttered by a s in g le p a r tic le . Likewise, the energy removed (e x tin c tio n ) is a lso N times that removed by a s in g le p a r tic le . This simple prop ortion ality r e quires that the incident radiation on each p a r tic le is e s s e n t i a l l y that of the original beam in cident on the cloud. In a cloud of p a r t i c l e s , each p a r tic le is also exposed to ra d i ation scattered by other p a r t ic le s . If the e f f e c t s o f that radiation are s ig n if ic a n t r e la t iv e to those o f the origin al beam then m ultiple sca tterin g e x i s t s . The simple p ro p o rtio n a litie s no longer e x i s t . How ever, as long as the p a r tic le s act independently the in tera ctio n between the individual p a r tic le and the in cident radiation is governed by the same laws which govern s in g le independent sc a t te r in g . However, the problem o f finding the in t e n s it i e s in sid e and o u tside the cloud is an extremely d i f f i c u l t mathematical problem. The problem o f m u ltiple independent sc a tte r in g is usually c a lled ra d ia tiv e transfer and has been studied in many ram ification s. Chandrasekhar (4) presents a very complete and comprehensive treatment. Hottel and Sarofim (5), Sparrow and Cess (6 ), and Love (7) have each included in th eir tex ts synoptic discussion s o f those topics o f most importance to engineering ca lcu la tio n s o f heat tra n sfer. Some o f the most current contributions to th is area include those of Marteney (8); McAlister, Keng, and Orr (9); Burkig (10); Novotny and Yang (11); Bobco (12); H ottel, Sarofim, Evens, and Vasalos (13); H ottel, Sarofim, Vasalos, and Dalzell (14); Edwards and Bobco (15); Glicksman (16); 4 Emanuel (17); and Muntes and Love (18). An extensive summary o f p e r t i nent published works was a lso given by Nagy (19). The problem o f ra d ia tiv e transfer through a p a rticu la te cloud can be solved by finding the p a rticu la r so lu tio n to the equation of trans fe r which meets the imposed boundary co n d itio n s. The equation o f trans fe r is dl w-K* r M - - K t I * (l-w0)Kt IB+ / l(n)p(0)ds (!) -Mfl The equation is derived by considering a volume element within a par t i c u l a t e cloud, and the changes in the in t e n s it y I o f a ray tra v ers ing a distance d£ through the cloud. The product of e x tin c tio n co e f f i c i e n t and the i n t e n s it y I accounts for radiation scattered from the o rigin al d ir ectio n as well as th a t absorbed within the volume element. The second term on the right hand sid e of the equation accounts for energy emitted from the volume element. The albedo u > 0 is the fra ctio n o f incident radiation sca ttered and (1 - ai0 ) is the fr a ctio n absorbed. The term IR is the in t e n s ity o f the black-body rad iation in equilibrium with the local temperature. The l a s t term of the equation represents ra d ia tio n with in te n s ity I(ft) in cident on the volume element from a ll d ir e c tio n s and sca ttered in the same d ir ectio n as th a t of the primary ray o f in t e r e s t. The term p(0) is a s c a t t e r ing function giving the fr a c tio n of sca ttered radiation per s o lid angle n as a function of the angle 0 measured between the d ir ectio n of the fix ed emergent beam and those o f the incident beams. The product o f I(ft) and p(G) must be integrated over a l l p o ssib le s o lid an g les, 4t t in t o t a l. It is the integral term o f the equation which leads to great d i f f i c u l t i e s in solving the equation. There are many systems for which no so lu tio n s have been obtained. There have been, however, some s o lu tio n s of sp ecial circumstances for which the integral is re a d ily r e d ucible. In addition, there have been some more general so lu tio n s employing d iffe r e n t techniques o f numerical approximation o f the in te g r a l. The reader is referred to Chandrasekhar ( 4 ) , Hottel and Sarofim (5 ) , Sparrow and Cess ( 6 ) , and Love (7) for d escrip tio n s of th ese. Most o f the approximation techniques require the use of large high-speed e le c tr o n ic computing equipment. Only the d is c r e t e -f lu x models, the "two-flux" and "six-flux" models, appear to o f f e r the engineer without ready a v a i l a b i l i t y to high speed computers a workable approach to the problem. I t was with th is in cen tiv e that Lenoir and Nagy (20) developed an experimental technique o f determining the parameters necessary to giv e a complete so lu tio n fo r ra d ia tiv e transfer through a p lan e-parallel cloud. Nagy measured the transmission through clouds of small spheres suspended in liqu ids by means o f a Beckman IR-2 spectrometer. He made measurements for both collim ated and d iffu s e in cid en t radiation for small glass spheres and small aluminum oxide spheres. He used as suspending liqu ids carbon te tra c h lo rid e and carbon d i s u l f i d e , both o f which have regions of transparency in the infrared spectrum. From his measurements Nagy was able to ca lcu la te the absorption and sc a tte r in g cross sectio n s o f the p a r tic le s and the fra ctio n s 6 scattered in the forward and reverse d ir e c tio n s , a ll of which are necessary for ap plication of the "two-flux" model. He was able to show that the clouds behaved q u a lit a t iv e ly as might be expected from know ledge o f physical properties and s in g le s c a tte r in g . He was unable, how ever, to arrive at any general co r rela tio n s for applying the 'two-flux" model to systems other than those studied. Nagy's work was lim ited to two m a teria ls, g la s s and aluminum oxide, both d i e l e c t r i c , and a lim ited range of p a r tic le s i z e s , 6.36 and 12.1 microns diameter for the alumi num oxide p a r tic le s and 8.65 microns diameter for the g la s s . For de termining general purpose c o r r e la tio n s , d iffe r e n t m a teria ls, including e le c tr ic a l conductors, non-spherical shapes, and a wider range of p a r tic le s iz e s should be included among those systems studied. The work described in th is d is s e r ta t io n was i n i t ia t e d to provide information for other systems so that general purpose co rrela tio n s could be obtained. The experimental techniques developed by Nagy were used to measure transm issions with aluminum oxide spheres o f 16.1 and 20.2 microns diameter and three d iffe r e n t s iz e s o f fla k e shaped graphite p a r tic le s with average fa c ia l diameters o f 2 .9 4 , 5 .4 7 , and 10.3 microns. The graphite p a r tic le s were both non-spherical and also e l e c t r i c a l l y conducting. Included in the d is s e r ta tio n is a sectio n g iving the th eoretical background which i s a synopsis of sin g le sc a tte r in g , and a d iscu ssion o f the “two-flux" and "six-flux" models for ra d ia tiv e transfer through a plane-parallel cloud. It is shown that the "two-flux" model is the same as the "six-flux" model when axial symmetry e x i s t s . As a further development the so lu tio n o f the "two-flux" model is rearranged so that the solution can be described by two m ultiple sc a tte r in g parameters. The parameters are such that when determined em pirically they include corrections which may be necessary to improve the accuracy o f the "two- flux" model. The d is s e r t a t io n also includes a descrip tion o f the experimental procedures and the re su lts obtained using both collim ated and d if fu s e radiation sources. The r e s u lts obtained when using a collim ated source are compared to the expected r e s u lts according to the theory o f s in g le s c a tte r in g , and a lso to Nagy's resu lts for p a r tic le s o f the same m aterial. The re su lts with the d iffu se source are interpreted in terms o f the empirical form o f the so lu tio n developed in the th eo retica l se c tio n . From the data two co rrela tio n s were obtained, one for each of the empirical parameters determining the empirical s o lu tio n . Sugges tio n s for future in v e s tig a tio n s are made. 8 II. THEORETICAL BACKGROUND The infrared radiation r e fle c te d and transmitted by a cloud o f p a r tic le s is dependent upon the ra d ia tiv e in tera ctio n o f each par t i c l e with the radiation incident upon i t . Most engineering problems involve d if fu s e radiation rather than collim ated beams in cident on the cloud, and c lo u d -p a r tic le d e n s it ie s such that a s in g le ray o f radiation may in t e r a c t with a number of p a r tic le s before re-emerging from the cloud as r e fle c te d or transmitted rad iation . Unfortunately, exact s o lu tio n s to those problems have been e lu s iv e except for the sim plest geom etries. Several approximate so lu tio n s have been proposed. In general these approximate so lu tio n s are related to or embody concepts rela ted to the e f f e c t o f a s in g le p a r t ic le on a collim ated beam. It is convenient to begin with a consideration o f the in tera ctio n s of in dividual p a r tic le s with in cident radiation. A. Parti cl e-Radiation Interactions A collim ated beam traversing a cloud o f absorbing-scattering par t i c l e s is attenuated partly by absorption and partly by d e fle c tio n s from the d ir ectio n o f the propagation o f the beam. The fraction al de crease in in t e n s it y o f a monochromatic beam is given by dl = K t d£ (2) I where I is the in t e n s it y o f the beam in the d irectio n o f propagation, I is the distance traversed, and is the total attenuation or ex tin c tio n c o e f f i c i e n t . In the absence of s c a t te r K t equals the absorption c o e f f i c i e n t K a ; in the absence of absorption i t is the s c a t te r c o e f f i c i e n t , K s ; and in the presence of both absorption and sc a tte r i t equals the sum of the two. If the c o e f f i c i e n t K, with the u nits reciprocal lengths or area per unit volume, is divided by the number of p a r tic le s per u n it volume the re su ltin g area term is known as the p a r tic le cross sec tio n C. The r a tio of C to the geometrical cross sec tio n Cg is known as the Mie e f f ic i e n c y fa cto r X. The terms K, C, X, may carry subscripts a, s , or t , to ind icate re la tio n to absorption, s c a t te r or to ta l e x tin ctio n . In order to completely describe the in tera ctio n o f a p a r tic le with the in cident beam i t is necessary to describe the d irectio n a l d i s trib u tio n of the scattered rad iation . This d is tr ib u tio n i s customarily given by the phase function, p(9,ip), which can be developed as fo llo w s. If a primary beam of in te n s ity of I0 and small divergent angle is incident on a p a r tic le o f sc a t te r cross sec tio n C s the to ta l s c a tte r throughout an angle S I of 4t t steradians is given by Q ( d , \ p ) dfi = I0CS (3) 4 T T the function Q(6,^) represents the sca ttered flux in the d irectio n 0 , ip, per unit angle o f divergence o f the in cident beam and per unit angle of divergence in s c a t te r . (See Figure 1 .) I f the s c a tte r were is o tr o p ic , that is of equal in ten sity in a ll d ir e c t io n s , Q would equal the to ta l sc a t te r divided by the to ta l s o lid angle, or 10 z X Figure 1. S cattering Angles and Coordinates 11 Now l e t the r a tio o f the in te n s ity scattered in the d irection 0, \p, to that scattered by an iso tro p ic s ca ttere r be the phase function or P (e.») ■ e -<’> (5) / Q(0,^)dfi The radiation scattered in the d ir ectio n 0, < p is given then by Q(e^) = p(0,^) — 1 (6) The functions Q(0,i|O or p(0,^) are often represented graphi c a lly . Such graphs are known as sc a t te r diagrams. If the p a r tic le is symmetrical or i f the functions represent the s c a t te r from many random ly oriented p a r tic le s the diagrams are independent o f < p and are normally presented as functions o f 0 in polar coordinates. For that case dft = 2 n sin0d0 (7) or df2 = 2?rd(-cos0) (8) or d Q = 27rd(l-cos0) (9) the l a s t integration parameter being used so that the integrating para meter runs from 0 to 1 as 0 changes from 0 to t t. Then, p(0) = j \ ----------- ^ ------------- (10) / Q (0 )d ((l-c o s 0 )/2 ) * C and a r e c t ilin e a r plot of Q(0) vs ( l - c o s 0 ) / 2 permits d irect v i s u a l iz a tio n by areas, o f the fra ctio n of sca ttered radiation lying within any range. The area under a p(0) vs ( l - c o s 0 ) / 2 curve is u nity. 12 In some radiation ca lcu la tio n s the forwardness of s c a t te r r e presented by cose is a convenient ca lcu la tio n parameter. The f o r wardness of s c a tte r is the integral of the component of the in te n s it y o f the scattered ra d ia tio n , in the d ir ectio n of propagation o f the primary beam, divided by the integral of the in te n s ity : The function cos0 is a measure o f the imbalance o f forward and back ward rad iation , p o s it iv e fo r p referential s c a tte r in the d ir ectio n of propagation, and negative for backward s c a t te r . For iso tr o p ic sc a tte r or sc a tte r with forward backward symmetry the term is 0 . Two other convenient terms occur frequently in radiation c a lc u la tio n s . The f i r s t o f these is the s i z e parameter x. For spheres x is equal to 7 rd/X where d is the diameter of the sphere and X is the wave length. The second term o f importance is the albedo for s in g le s c a t te r . The albedo for s in g le s c a t t e r w0 is defined as the r a tio o f the sc a tte r ex tin c tio n c o e f f i c i e n t to the total ex tin ctio n co e f f i c i e n t , th at is J^Q(Q)cosQdQ I Q(6)dft Xirr ( 11) p(9)cos8dfl 47r (13) 13 The term l-eo0 represents the ra tio o f the absorption c o e f f i c i e n t to the to ta l ex tin c tio n c o e f f ic ie n t . The in tera ctio n o f a p a r tic le with incident radiation on i t can be completely described by i t s geometrical cross s e c t io n , it s to ta l ex tin c tio n e f f ic ie n c y fa c to r , i t s albedo, and it s phase fu n ction . Some of these fa cto rs in turn depend upon the substance o f the p a r tic le and the substance of the surrounding media. When the radiation is incident on the p a r tic le surface some o f i t w ill be r e fle c te d back into the surrounding medium and some of i t w ill pass across the p a r tic le bound ary into the p a r tic le . Of the radiation passing into the p a r tic le some of i t may be absorbed. The unabsorbed portion f i n a l l y re-emerges in to the surrounding medium as scattered ra d ia tio n . This sca ttered radiation plus the i n i t i a l l y reflected radiation make up the to ta l sca tterin g o f the p a r t ic le . The parameter which determines the r e f l e c t i v i t y of the p a r tic le surface is the r e la t iv e r e fr a c tiv e index, i . e . the r e fr a c tiv e index o f the material o f the p a r tic le r e la t iv e to th a t of the material of the surrounding medium. In order to include substances which absorb radiation i t is necessary to use the complex r e f r a c t iv e index n1 o f each substance given by n 1 = n (1- i k) = n-inK (15) Here n is the real part of the r e fr a c tiv e index, and n< is the complex part. The absorption index k o f a substance is given by where A is the wave length in a vacuum. From the equations of pro pagation o f electrom agnetic radiation i t can be shown that n ( l - i x ) = (E-2icrQ/v)* (17) and (18) (19) where v is the frequency o f the rad iation , and E and a0 are the d ie l e c t r i c constant and conductivity o f the media at that frequency. The d if f i c u l t y in applying these equations is that the d ie l e c t r i c constant and conductivity must be measured a t the frequency in q uestion. However, these r e la tio n s do lead to some s ig n if ic a n t gen era liza tio n s about in su lators and conductors as c la sse s o f m aterials. For d ie l e c t r i c s the value o f < is e s s e n t i a l l y zero except in spectral regions corresponding to absorption bands. However, even with < equal to 0.001 the tr a n sm issiv ity over a specimen 1 m illim eter thick is only 0.001 so th a t th is does not correspond to transparency. Accordingly when k is equal to zero, n For d ie l e c t r i c s n ty p ic a lly varies from 1 to 4. For metal conductors the conductivity is very large and in the range where the r a tio o f conductivity to frequency is much la rg er than the d i e l e c t r i c constant. Thus and The value of ~1 for < corresponds to extreme opacity. The value of n may be 30 or 40 times that of a d i e l e c t r i c . B. Single Scattering The rigorous so lu tio n o f Maxwell's equations fo r the in tera ctio n of an electrom agnetic wave with a sphere o f arbitrary s i z e was obtained by Mie (1) in 1908. The so lu tio n appeared as a very complex s e r ie s of Bessel functions and Legendre polynomials. The necessary inputs to the Mie so lu tio n are the complex r e fr a c tiv e index, n ', and the s i z e para meter, x = nd/X. With the aid o f high speed computers, rigorous c a l cu la tio n s for the sc a tte r in g function, p (e ), and/or the e x tin c tio n co e f f i c i e n t s for sc a tte r in g and absorption, can be c a lcu la te d , and have been, by a number o f in v e s tig a to r s . Such numerical evaluations are q u ite tedious and the values o f r e fr a c tiv e indices for which so lu tio n s are a v a ila b le are too few to include many o f the p a r tic le s o f in te r e s t to engineering heat transfer ev alu ation s. However, under sp ecial c i r cumstances such as very small p a r tic le s or values o f the r e fr a c tiv e in dex near one, or near i n f i n i t y , the so lu tio n s take on s im p lifie d forms. For example, as the ra tio of the p a r tic le dimension r e la t iv e to the Oo V 16 wave length decreases, the so lu tio n equations approach those of Rayleigh sc a tte r as a lim itin g case. Rayleigh considered the r e s u lt o f electromagnetic wave generation by dipoles s e t in motion by the primary beam. The contribution to the scattered rad iation by the dipoles i s e a s i l y calculated when the e l e c t r i c in ten sity and the phase angle throughout the p a r tic le are equal to those o f the in cid en t rad iation . These conditions are usually s a t i s f i e d when the maximum p a r tic le dimension is le s s than 0.2 o f the wavelength measured jji the p a r tic le . In a general form to include p a r tic le s which absorb some o f the rad iation the ex tin c tio n e f f i c i e n c ie s are: where Im denotes the imaginary part of the r e su lta n t complex number. I f the expression for the complex r e fr a c tiv e index is inserted the equations become ( 20 ) Xa = - 4x Im 2-l 2+1 ( 2 1 ) X - 1 y" (rn2( l - < ) 2- l ] [ n 2(l-K )2+2]+4nV2} 2+36nV2 (22) S 3 * ( [n2 (1 -K2)+2]2+4nV2}2 and X 24n2K X (23) a = [n2( l- < 2)+2]2+4n‘*K 2 I t should be noted that the e f f i c i e n c y fa c to r calcu lated from the equations for values o f x much l e s s than one are much sm aller than u n ity , breaking down the concept o f physical blocking of the radiation by the p a r tic le . When the r e fr a c tiv e index o f the p a r tic le is near one, the r e s t r i c t i o n on the s iz e o f the p a r tic le s for which the lim itin g re la tio n s are va lid may be relaxed. There are p a r tic le s which s a t i s f y the con d it io n s |n - 11« 1 and x | n - l | « l , and the v e lo c it y of the radiation in s id e the p a r tic le is approximately equal to th at in the space su r rounding the p a r tic le . The d iffer en ces in phase o f the radiation leavin g the p a r tic le s can be e n t ir e ly attribu ted to the d iffer en ce in path length over which the ra d iation tr a v e ls . Under those circum sta n c e s, the contributions o f d if f e r e n t i a l volume elements can be added to determine the interference e f f e c t s , and the r e s u lts expressed as a co rrectio n to be applied to R ayleigh's equations. Van der Hulst (2) gives a compilation o f correction factors for a v a r i e t y o f shapes, in cluding spheres, e l l i p s o i d s , spherical s h e l l s , circ u la r c y lin d e r s, thin d i s c s , and randomly oriented rods and d is c s . I t is probably well to note that for spheres an increase in p a r tic le s i z e causes s c a t te r in the forward d ir ectio n to p rev a il. For small p a r tic le s of metals and other m aterials with a large re fr a c t iv e index, an a lte r n a tiv e expansion o f Mie's equations is a v a il- 18 and (25) An in te r e stin g feature about s c a t te r from small spheres with large n is th at, unlike Rayleigh sc a tte r the in t e n s it y d istr ib u tio n is strongly directed towards the source. When the r e fr a c tiv e index is near one, the e f fic ie n c y fa cto r for ex tin ctio n o f non-absorbing spheres o f arbitrary s iz e can be found by The f i r s t four maxima in (and the values o f a a t which they occur) are 3.17 (4 .0 9 ), 2.40 (1 0 .7 9 ), 2.25 (1 7 .1 6 ), 2.18 (23 .5 2 ); the f i r s t three minima (and th e ir lo ca tio n ) 1.54 ( 7 .6 3 ) , 1.73 (1 4 .0 0 ), and 1.81 (20.33). In the presence of a small absorption index, such that n < « l in addition to | n - l | « l , the e x tin c tio n and absorption e f f i c i e n cies are given by (26) where a = 2x |n -11 (27) Xt = 4 Re { F( i a )} (28) X, = 2 F (4xn<) a (29) where F(y) = i + S 2 + ( 30 ) 19 The e f f e c t o f increasing absorption is to decrease the value o f the f i r s t maximum and the amplitude o f the flu ctu a tio n s in Xt and to s h i f t i t s f i r s t maximum to smaller values o f a. The absorption cross s e c tio n increases monotonically with increase in k over the range covered. For large opaque p a r tic le s the ex tin ctio n e f f ic ie n c y c o e f f ic ie n t approaches a value o f 2 as the s iz e o f the p a r tic le increases re l a t i v e to the internal wave length. The fa cto r 2 is the lim itin g value predicted by Mie's rigorous so lu tio n . It is p h y sica lly in terpreted as being from d ir e c t blockage o f the incident radiation by the cross sec tio n of the p a r tic le and an equal e f f e c t due to Fraunhofer d if f r a c tio n . For opaque specular partial r e fle c to r s the p ertinent sc a tte r in g parameters are given by where is the hemispherical r e f l e c t i v i t y and p(0 ) the specular r e f le c t io n a t an incident angle 6 . xs = ph (31) xa = eh = 1 " ph (32) r _ (surface area) n Ls 4--------- M h (33) P(e) = P ( ^ ) / P h (34) o (35) For opaque d iffu se r e fle c to r s (which case a lso includes randomly oriented p a rticle s with no surface in d en ta tio n s), the pertinent s ca tterin g parameters are given by Xs = p (36) Xa = 1 - p = e (37) Cs = (surface area) (38) p(e) ■ (s in e - e cos e) (39) It is well to note th at these equations given do not include the e f f e c t s o f d if fr a c tio n . The in tera ctio n of the incident rad iation and the p a r tic le may be considered independent of the neighboring p a r t i c l e s . Then, the absorp tion c o e f f i c i e n t , the sca tter c o e f f i c i e n t , and the phase function in a cloud o f p a rticle s are simply the sums of the individual p a r tic le con trib u tio n s in a u n it volume. Let N(r) dr be the number o f p a r tic le s per u n it volume of cloud having a c h a r a c te r is tic dimension between r and r + dr. Then le t t in g Ka and K s denote the bulk c o e f f ic ie n t s (the to ta l cross sec tio n s for absorption and s ca tterin g per unit volume of cloud). 21 where Ca ( r ) , Cs ( r ) , and p (e ,r ) are the cross sectio n s for absorp tion and s c a t te r , and the phase function for individual p a r tic le s of s i z e r. The assumption of independent in tera ctio n s between the rad ia tion and the p a r tic le s is apparently good down to a center d ista n ce o f separation of about three r a d ii. For very large p a r tic le s and very small p a r tic le s r e la t i v e to the wave length the in teg ra ls may e a s i l y be evaluated to give e x p l i c i t expressions fo r the sc a tte r in g absorption parameters. When working with p a r tic le s o f irregular shape and s i z e i t is sometimes convenient to use the volume fr a ctio n v o f p a r tic le s in a u nit volume of cloud instead of the number of p a r t ic le s per unit volume o f cloud. In that case s p e c i f i c cross s e c tio n s , aa» os* and at> based on the volume o f the p a r tic le s present can be defined by C. D iscrete Flux Approximations for M ultiple S catterin g Of the approximate so lu tio n s proposed for problems in m u ltip le sca tterin g the d is c r e t e flu x representations o f f e r a r e la t iv e sim p l i c i t y which makes them adaptable to the so lu tio n o f engineering pro blems. The "two-flux" model proposed by Schuster (2 1 ), and extended by Hammaker (2 2 ), and by Churchill and Chu (23), assumes that a l l of the radiation s c a t te r can be represented by two components, one in the (43) (44) (45) 22 d ir ectio n of the propagation of the primary beam and one opposed to that d ir e c tio n . On th is basis the equation o f transfer can be r e w ritten as two equations, one for the forward component and one fo r the backward component. I f the p o s it iv e x d ir ectio n is taken as the d ir ectio n of the primary beam o f propagation the two equations become The in te n s ity o f the component in the forward d ir ectio n is represented by Ix+» the component in the reverse d ir ectio n by Ix_. The terms f and b are the fraction s o f the radiation that would normally be sca ttered into the hemispheres facing and opposed to the d ir ectio n of propagation. If the phase function for the p a r tic le s in question is known then f and b can be calculated from In the equations for the component tr a n sfe r, the product fw0 Ix+ ™ the f i r s t term on the rig h t hand s id e o f the equation accounts fo r fo r ward sca tte r in g of the component in question. The product bco0 Ix _ is the contribution to the forward flu x due to backward sca tterin g by the component in the opposite d ir e c tio n . The l a s t term on the rig h t repre sents the contribution due to emission by the p a r t ic le s . IB(x) = 0, (46) (47) O (48) (49) -i 23 I f the emission is considered to be n e g lig ib le , the tra n sfer equations become (50) K t dx + ^wo^x+ dIx- (51) These equations have been solved to c a lc u la te the fra ctio n s o f a plane p a ra llel beam that are r e fle c te d or transm itted by an absorbing- sca tterin g sla b . Churchill and Chu (24) have compared values of r e fle c ta n c e and transmittance calcu lated by the "two-flux" method with the exact so lu tio n s . They found that the "two-flux" method gives good agreement for integrated re fle c ta n c e and transmittance for d if fu s e in cident radiation but does not provide a r e li a b l e measure o f the e f f e c t of angle of incidence for a p la n e-p a ra llel incident beam. According to Churchill and Chu the "two-flux" method g iv es a much truer representa tion o f the scattered radiation when the in t e n s it y of the radiation in cid en t on a p a r tic le is symmetrically d istrib u te d than when i t is collim ated. Churchill and Chu (23) extended the "two-flux" method by assuming that the radiation was scattered by a p a r t ic le in s ix mutually per pendicular d irectio n s - forward, backward, and to the four s id e s . The fra ctio n scattered into the s o lid angle formed by de is 2irp(e) s in 6 de/47r, and i f the forward and radial fr a ctio n s of that s c a t te r are taken as c o s 20 and s i n 20 , then the fra ctio n s scattered in each d ir ectio n 24 are represented by r r / 2 p (e ) c o s 2e sinede = i p(y)y2dy (52) o where y = COS0 (53) p (e ) c o s 2e sinede = ±J p(y)y2dy (54) s 6 = (1- f 6~b6)/4 (55) Here the su bscript 6 is used to in d ica te that the fraction s scattered are those associated with the "six-flux" model. A s in g le value for the s id e sca tterin g assumes that the p a r tic le s are e ith e r symmetrical or randomly oriented and that the four sid e -s c a t te r e d components are a ll equal. Equations of tra n sfer may now be w ritten fo r s ix i n t e n s i t i e s , the forward and backward components along the three axes of any orthogo nal system of coordinates. In Cartesian coordinates, these equations are given below, with the term for radiation emission omitted from each equation. Kj.dx _ (^6wo“^^ x+ + ^6wo^x- + s 6U )o(*y+ + *y- + *z+ + *z-) (56) dl x+ 25 ■ K^dy = (f 6«*>o-,|) Iy_ + b6^oIy+ + S6^0 ( IX+ + Ix_ + I z+ + I z_) (59) dIz+ K^dz - (^6wo“^^z+ + b6wo^z- + S6wo(^x+ + Tx- + *y+ + ^y-^ (50) d Iz- K^.dz " (^6wo”^ ^ z - + b6wo^z+ + S6wo ^ x + + ^x- + *y+ + ^y-^ (51) When the i n t e n s it i e s at the boundaries are s p e c ifie d the above s e t of lin e a r d if f e r e n t ia l equations may be solved sim ultaneously. Symmetry may reduce the number o f equations to be solved. For example, in the case o f a collim ated beam incident normally on a plane p a ra llel d isp ersion , the sid e -s c a tte r e d components are independent of y and z, and are given by “qS6 (I x+ + Ix-) Iy+ " 1y - = ! z+ l z- = 1 - “ of 6 - “ob 6 - 2w0s6 (62) Hottel and Sarofim (5) s t a t e that the transmission and refra ctio n from p lan e-parallel dispersions are much more c lo s e ly approximated by the "six-flux" than the "two-flux" method. For a d iffu s e radiation at the boundaries the "six-flux" method gives adequate re su lts for a ll the conditions s tu d ie s , according to them. I t is probably well to note at th is point th a t, for cases in v o lv ing axial symmetry, the same pair o f d if fe r e n t ia l equations can be obtained whether one s ta r ts with the "two-flux" model or the "six-flux" model. These equations are where C and D are evaluated e ith e r by C = 1 - fw o (65) D = bu) o ( 6 6 ) or C = 1 - f 6u)0 (67) 4 s 2 w2 ( 68) D = b a). + 6 0 - w0 ( f 6 + b6 + 2s 6) An important observation or two should be made at th is point about the fa c t that both methods lead to the same pair o f d if fe r e n tia l equations. It seems obvious that the "two-flux" model is ju s t a special case (axi-symmetric) o f the "six-flux" model and i t is not surprising that the "two-flux" model is less accurate for non-axisymmetric problems. A second important consideration a r ise s in applying the d i f f e r e n tia l equations to experimental data obtained under axisymmetric con d it io n s . From the data the best values o f C and D can be evalu ated. With respect to the "two-flux" model values of C and D lead to three equations 27 1 - f»0 = C (69) bu)Q = D (70) f + b = 1 (71) which can be solved for f , b, and w0 . However, there are i n s u f f i c i e n t equations to determine f 6, bg , s 6, and % as required for the "six- flux" model. For the "six-flux" model experiments need to be carried out under non-axisymmetric conditions. D. Solution of the Equations for P lane-P arallel Clouds The two d if fe r e n tia l equations o f the d is c r e t e flu x model can be solved d ir e c t ly for transfer through an absorbing-scattering cloud en closed by two p a ra llel planes. Consider the net transmission o f radi a tio n through a cloud of thickness L. The radiation enters a t the su rface x = 0 and leaves the cloud somewhat attenuated at x = L. I f the values o f the forward and reverse fluxes are known at the boundaries th e ir values at any point in between are given by the equations: I I0+sinh[AKt (L-x)] + IL+sinh(AKt x) (72) x+ sinh(AKt L) I I0 _sinh[AK (L -x)] + I|__sinh(AKtx) (73) x- si nh(AK^L) where A =*y C2 - D2 Here the in t e n s it i e s o f the forward and backward flu x e s at the two 28 boundaries x = 0 and x = L are given by IQ+ and IQ_, I|_+ and IL_. Now i f Eq is that part o f the rad iation leaving the surface at x = 0 th at has been transmitted into the cloud through the bounding surface at x = 0 and RQ and are the r e f l e c t i v i t i e s o f the two bounding surfaces o f the radiation incident upon them then •o+ * Eo + V o - <7 4 > and IL_ = R lIl+ (75) These two equations in combination with the previous two can be com bined to give a so lu tio n for the forward in te n s ity a t x = L r e la t iv e to the in t e n s it y of the radiation transmitted into the cloud from the source. The equation is: *L+ A E^~ = [(l+R0RL)C-(R0+RL)D]sinh(AKt L)+A(l-R0RL)cosh(AKt L) (7 6 ^ When no p a r tic le s are present between the two planes, or in-other- words when only the suspending medium occupies the space between the two surfaces the forward in te n s ity at x = L r e la t iv e to the entering in te n s ity is given by the equation IL+\ 1 Eo I v=o eaL - R0 R|_e"aL (77) Where a represents the absorption c o e f f i c i e n t of the suspending medium between the planes. The l a t t e r equation gives the lim itin g value o f the forward flu x r e la t iv e to E0 as the volume fra ctio n 29 s o lid s approach zero. The transm ission of the cloud r e la t i v e to the transmission o f j u s t the suspending or surrounding medium, in the absence of any p a r t i c l e s , is given by the equation To take into account the absorption o f the medium surrounding the p a r t ic le s , the total e x tin ctio n c o e f f i c i e n t K t o f the cloud must be w ritten as I f the volume fr a ctio n o f s o lid s in the cloud is small r e la t iv e to u n ity , then the ex tin ctio n c o e f f i c i e n t becomes Nagy used the so lu tio n as given here, but in a s l i g h t l y rearranged form, to determine f , b, and w0 experim entally for small spheres of aluminum oxide suspended in carbon te tra ch lo rid e and also carbon d i s u lf id e , and for small spheres o f g la ss suspended in carbon t e t r a chloride. E. An Empirical Solution One o f the problems or questions associated with the a p p lica tio n of the "two-flux" model to the radiant tra n sfer between two p a ra llel planes separated by an absorbing-scattering medium is the value o f L to be used in the s o lu tio n . Hottel and Sarofim (5) suggest s u b s ti- AeaL(l-R 0RLe ' 2aL) [(l+R0RL)C-(R0+RL)D]sinh(AKt L)+A(l-R0RL)cosh(AKt L) (78) = a^.v + (1 - v)a (79) K t = o t v + a (80) 30 tuting L£ , the equivalent cloud th ick n e ss, for L, the actual th ick n ess, in order to account fo r the three-dimensional motion of the photons in traversing the d ista n ce L. The d if fe r e n t ia l equations for the "two-flux" model become, then, Furthermore, they said that the value Le = 1.76 L should be used. The fa cto r 1.76 comes from considering a ll o f the radiation transferred from one o f two i n f i n i t e p a ra llel planes to an incremental area o f the second plane as transmitted perpendicular to the plane rather than d istrib u te d over the e n tir e hemisphere. The equivalent length Lg i s defined by This equation y ie ld s the r e s u lt Le = 1.76 L. However, the integral expression is based on the assumption that the in t e n s it y o f radiation is uniform over the en tire hemisphere, i . e . , the phase fu n ction , p (0 ), is constant. The equivalent length Lg should be determined by K t (Le/L)dx = ' C *x+ + D Tx- d l x+ (81) and (82) e"Kt*-/cos0 COS0 d (s in 2e) o (83) KtL/cose c o s 0 p (e )d ( s in 2e) (84) 31 and the r a tio Le/L would depend on the phase function p(0 ). For example, Le/L has been computed for several d if fe r e n t phase functions as shown below. p(9) (Le/L) p(e) = 1 (I so tr o p ic ) 1.76 p(e) = 8/3 (sin e - 0 cos e) (Large d if f u s e spheres) 4.32 p(0) = 0.75 (1 + c o s 20) (Rayleigh s c a tte r in g ) 1.52 p(6) = 1 + 0.25 cos 0 1.57 p(0) = 1 + 0.75 cos 6 1.28 Thus, i t can be seen th at Le /L could take on a range o f values and vary as the s i z e o f the p a r tic le varies r e la t iv e to the wave length. The dependence o f Le /L on the d is t r ib u tio n o f the scattered r a d i ation is o f no p a rticu la r consequence when seeking an a n a ly tica l s o lu tio n for which the sc a tte r in g properties o f the cloud are known. One simply evaluates Le/L as above. However, when sta r tin g with ex p eri mental data and trying to determine the absorbing-scattering parameters of the p a r t i c l e s , wo> and b, as needed for the "two-flux" model, the v a ria tio n of Le/L is an o b sta cle. By measuring the transmission between two p a ra llel planes separated by an absorb in g-scattering medium, only three independent equations are a v a ila b le and Le/L c o n s titu te s a fourth unknown. An a lter n a te approach is needed. By defining B as the ra tio o f C to D the two d if f e r e n tia l equations for the "two-flux" model can be rew ritten as 32 DKt (Le/L)dx " " B !x+ + *x- (85) DKt (Le/L)dx = ' B ! x- + 1 dIx ( 86 ) x+ The so lu tio n f o r the transmission through a cloud o f p a r tic le s between two p a ra llel planes becomes then Thus, the s o lu tio n is given in terms o f the actual cloud thickness L and two parameters B and 4 > which characterize the ab so rb in g -sca tter ing properties o f the cloud. The two parameters could be ca lcu la ted from the o p tica l properties o f the p a rticle s and t h e ir suspending medium. Most important, they can be determined em p irica lly from ex perimental data without concern for what value to use for the ra tio I t ' s p o ssib le to put the so lu tio n in s t i l l another form by sub s t it u t in g the exponential forms for the hyperbolic s in e and co sin e, and L _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (07) E0 [(l+R0RL)B-(R0+R|_)]sinh(<f>vL)+ Y F T (1 -RoR|_)cosh(<f)VL) where (f is defined by < j > = D V B2-l a t (Le /L) = A a t (Le/L) (88) dividing both numerator and denominator by V bz - 1. 2 (89) [(l-RoRL )-^e“*vL + [(1-ML)+^ e (* ) V L 33 where = 0 + R0Rl ) B - R0 - RL v = 2-kl 2 L (90) B2 - 1 When e ith e r RQ or equals zero, the s o lu tio n becomes I, L+ Eo (1 - ^ )e -(f,vL + (1 + i|))e<|,vL and \p becomes (91) B - R , when R. = 0 (92) V F ^ i 0 or B - R o \p = . when R i = 0 (93) V b 2 - i Furthermore the nature o f the so lu tio n is such th a t B>1, so that when RQRL« 1 Eo (1 - ijj)e"^vL + (1 + ^)e^vL (94) and ip = ______ V b2 - l i B - * R i . . Ip = — f= - (95) I t is worthwhile to examine the behavior of the complete solution as given by equation (8 9 ). If the d erivative o f the ratio I|_+ /E0 is taken with respect to the quantity (vL) the fo llow in g equations are obtained. 34 d(IL+/E0 ) _ 2<t>{[(1 -R0RL) - ^ ] e ' <i)vL- [(1 -R0RL+^]e^vL} d(vL) {[ (1 ~RqRl ) -'JJ ] e ”< ^ )VL+[ (1 -R0RL )+^]e^vL} 2 and d£n(IL+/EQ) d){[(l-R0RL) ^ ] e ' (f>vL - [ ( l - R 0R L) + ^ e <t)vL} -cbvL cbvL d(vL) [ ( l- R 0RL) - ^ e + [l-R 0 R|_)+'P]e (96) (97) If (vL) -v » d £n(I,+/E0 ) / v d(bt) 0 - - * < 98> The l a s t equation leads to a p o s s ib le physical in terp reta tio n of the parameter < j> . The reader w ill re ca ll that fo r collim ated incident radiation of in te n s ity I0 the transm ission through a cloud of p a r t ic le s , with e x tin ctio n due only to the p a r t ic le s , and for s in g le s ca tterin g ^ ■ - «t <»> where I^+ and L are measured in the d ir e c tio n o f IQ. The para meter was the average ex tin c tio n cross sec tio n per unit volume o f p a r tic le s in the cloud. By analogy we might think of the parameter < J > as a "diffuse" e x tin ctio n cross sec tio n per unit volume o f p a rticle s in the cloud. The parameter B can be given some physical meaning by examining i t in terms o f its d e fin it io n through the "two-flux" model. According to t h a t formulation o f the problem 35 B = 1 : f “° bt»)„ (100) or (1 - o j0 ) + b u )0 ( 1 0 1 ) The f i r s t term in the numerator is the fraction o f in cident radiation which is absorbed by the p a r t i c l e s . The term bw0 represents the fr a ctio n of in cident radiation sca ttered in the backward d ir e c tio n . The numerator, th erefore, represents the total fr a ctio n not sca ttered in the forward d ir ectio n due to both absorption and b ack -sca tterin g . Keep ing in mind that the "two-flux" model is only an approximation, one could say th at the parameter B represents the r a tio of the total radiation not sca ttered forward to that scattered backward. By examining the l a t t e r two equations above we can see th at as the value o f the albedo o> 0 approaches one (complete s c a t t e r i n g ) , the value of the absorption s c a tte r in g parameter B approaches one. As the value o f the albedo becomes zero (complete absorption) the value o f B becomes i n f i n i t e . The e f f e c t that various values of the parameter B has on the so lu tio n for radiant tra n sfer through a cloud bounded by two plane surfaces can be seen by examining i t s rela tio n sh ip to the para meter !/> . The parameter ip has no singular physical s ig n ific a n c e , but is a convenient c o lle c t io n of terms, including some related to the boundary con d itions. The e f f e c t of various values of \p on the form o f the s o lu tio n can be seen by examining Figure 2 where the logarithm o f the 36 .0 0.6 0 .4 0.2 0.1 0.06 1.6 0 0.4 1.2 0.8 vL Figure 2. The E ffect of the Parameter ^ on the Transmission through a P lane-Parallel Cloud 37 ra tio I l +/Eq is p lotted versus the o p tical cloud depth as expressed by the product (vL). I t was shown e a r lie r that at larger values o f (vL) the slopes o f the curves approached a common value, the magnitude of which was given by the value o f the d iffu s e ex tin c tio n parameter < t> . It can a lso be shown that the i n i t i a l slopes o f the curves are related to the parameter ip. In f a c t, Thus, the i n i t i a l slopes o f the curves in Figure 2 d if f e r by a fa cto r tp, or the value o f ip determines the amount o f curvature in the curves. By r e la tin g the parameter ^ to B, i . e . one can see that 'P approaches (1 + RqRl) when B becomes very large (e x tin c tio n due p rin cip a lly to absorption) and \p becomes very large as B approaches a value of unity (no absorption). Furthermore, B as defined can never have a value le s s than unity and the value of \p w ill be greater than one in a ll cases where (R0 + R[_) is le s s than one. In other words, when the r e f l e c t i v i t i e s at the boundaries are low the value of ip w ill be greater than one. as (vL) 0 ,,, = 0 + RoRL> B - R o - RL ( 102 ) V b 2 - 1 38 III. DESCRIPTION OF THE EQUIPMENT A. Infrared Spectrometer The measurements o f infrared transmissions in th is research were made with a Beckman IR-2A spectrometer as modified by Nagy. The spectrometer i s shown schem atically in Figure 3. The spectrometer contained rock s a l t o p tical components and thus permitted measurements at wave lengths from 1 to 15 microns. The primary unit could a ctu a lly be p h y sica lly separated into three d if fe r e n t sec tio n s c o n s is tin g of: 1) a radiation source s e c t io n , 2) a sample s e c tio n , and 3) a monochrometer and d etectio n se c tio n . The f i r s t sec tio n was the source of a collim ated beam which was transmitted through the sample s e c tio n . The source sec tio n consisted o f a Nernst glower, a chopper, a mirror, and a co llim a tin g le n s e . The Nerst glower was a hollow tube 3 /8 inches long and 1/8 inch in diameter composed o f ceramic material and heated by a steady e l e c t r i c current. Part o f the radiation emitted from the glower f e l l on a photo tube. The photo tube was the sensing element in a feed back c i r c u i t which maintained the in t e n s it y o f radiation from the glower a t a constant l e v e l . The radiation from the glower was re fle c te d by mirror toward the sample sectio n and at the e x i t of the sample sectio n a rock s a l t lense collimated the radiation in to a p arallel beam. Between the Nernst glower and the mirror was a mechanical chopper. The chopper con sisted of a mechanically rotated d is c . The d isc was divided roughly in h a lf along one of i t s diameters so that one h a lf o f the disc was open. The other h alf o f the d isc was opaque to the 40 rad iation . The chopper was constructed so that e ith e r g la ss or metal shades could be used as the opaque s e c tio n . The chopper d isc was located and orientated so that as i t rotated the radiation from the glower was interrupted for part o f each cy c le . The ro ta tin g speed of the d isc was such that the radiant beam emitted from the source sectio n was chopped at the rate of 10 c y c le s per second. The a l t e r nating radiant source was needed so that the signal at the detector could be amplified s u f f i c i e n t l y . At the entrance o f the sample sec tio n was a metal s l i d e which could be used to control li g h t to the c e ll sec tio n with e ith e r a fu ll open or a fu ll closed p o sitio n . The sample sectio n could a c tu a lly be considered in two p arts. The f i r s t s e c t io n , a liq u id c e ll compartment, was located rig h t at the point at which the collim ated beam entered the sample s e c tio n . I t con sisted o f an opening about 3.3 centim eters wide and 10 centim eters high, large enough to admit a liq u id sample c e ll c o n s is tin g o f two windows pressed together with a sample cavity in the middle. Between the liquid and gas c e ll compartments was a converging len se. The gas c e ll compartment included space for two cy lin d rica l c e l l s , one about 10 centim eters long and the other about 2 centim eters. At the e x i t of th is compartment was a converging lense which focused the image of the glower upon the entrance s l i t o f the monochrometer. The purpose o f the monochrometer was to decompose the transmitted beam into i t s wave length elements and to allow only a narrow band width to f a l l upon the d ete cto r . The entrance s l i t o f the monochrometer was 20 m illim eters long and could be varied in width 41 from 3 m illim eters to 0.01 m illim ete rs. Nagy had modified the entrance s l i t by covering the top and bottom o f the s l i t to reduce the height to only 0.5 m illim eters. From the entrance s l i t the beam was collim ated by a curved mirror and then dispersed by a prism. The beam traversed the prism tw ic e , owing to a plane l i t r o mirror on one s id e o f the prism. The mirror was ro ta ta b le and coupled to a wave length dial which indicated the wave length o f the lig h t focused on the e x i t s l i t . From the e x i t s l i t of the monochrometer the beam was r e fle c te d by the plane mirror and focused upon a thermocouple which served as a d ete cto r . The signal generated by the thermocouple was amplified by an AC am plifier tuned to a 10 cycle per second frequency. The e le c tr o n ic signal was then r e c t i f i e d to a d ir e c t current by use of a synchronous r e c t i f i e r c o n s is tin g of breaker points and a cam located on the shaft o f the same motor which drove the chopper. The DC out put signal from the r e c t i f i e r was then am plified again by a D C am p lifier. The signal from the d etecto r as am plified could then be balanced by a potentiometer coupled to a calib rated dial graduated in per cent of transm ission. Besides the basic spectrometer u n it, power supply, and am p lifier already mentioned, a constant temperature bath was a lso used which provided a source of water c irc u la ted through the spectrometer to maintain the o p tica l components and sample box at 77 degrees plus or minus 0 .2 degrees, Fahrenheit. 42 B. Sample Cell Nagy had modified the shorter o f the gas c e l l s by reducing i t s length so that i t would f i t in sid e of the liquid c e ll compartment. With the procedures developed by Nagy and used in th is research, i t was necessary to use the liq u id c e ll s ec tio n where the radiant beam was collim ated. The body o f the c e l l th at Nagy used was 1.30 cen timeters long with an internal diameter o f 2.39 centim eters. In order to reduce the radiant lo s s e s due to absorption by the suspend ing medium a c e l l with a shorter beam path was constructed for th is research. The c e l l , shown in Figure 4, was made by d r i l li n g a 15/16 inch diameter hole through a block of aluminum 2 inches high, 1 1/2 inches wide, and 1/4 inches th ick . The in sid e surface of the cy lin d rica l c a v ity was p o lish ed . This provided a c e ll body with a length o f 0.635 centimeters compared to that of 1.30 centim eters used by Nagy. The completely assembled c e ll co n sisted of the c e ll body, a front and back window, Viton-A gaskets between the windows and the c e ll body, 2 p la tes to hold the windows in place, and thin te flo n gaskets between the p la tes and the windows to prevent scratching and chipping o f the windows. As assembled, the sample cavity len g th , that is the d istance between the in sid e surface o f the front window and the inside surface o f the back window was 0.762 centim eters. For the c e ll windows Nagy had chosen windows made o f Servofrax (arsenic t r is u l f i d e ) by the Servo Corporation of America. This material was s elec ted because i t was a g la s s , nonhygroscopic, and Viton-A gasket Teflon gasket c e ll window Figure 4. Sample Cell and Assembly of One Side CO 44 easy to handle. It had the disadvantage of having a high r e fra c tiv e index (2.40) with a r e su lta n t high r e f l e c t i v i t y . In th is research both Servofrax windows and windows made o f Irtran-3 manufactured by Kodak were used. The Irtran-3 windows are more expensive but had the advantage of a lower r e fr a c tiv e index (approximately 1.4) which r e su lted in s i g n i f i c a n t ly lower lo s s e s due to r e f l e c t io n s . The Irtran-3 windows, however, did have the disadvantage o f low transmission beyond 9 microns and therefore were not used whenever carbon d is u lfid e was used as the liq u id in the c e l l . For the collim ated measurements both front and rear windows were always o f the same m a teria l. However, for the d if fu s e measurements the front window was a Servofrax window with one surface roughened by rubbing on a number 80 g r i t paper. This was the same window th at was used by Nagy. The l i g h t transmitted through the window was sca ttered in a ll d ir ectio n s and provided a d iffu se source fo r the p a r tic le cloud contained in the c e l l . C. Microscope A microscope was chosen as the means for measuring the s iz e o f the p a r tic le s used in th is research. The instrument used was a Spencer monocular microscope with a sub-stage condenser and a graduated mechanical sta g e. The microscope was f i t t e d with a 10X and a 44X o b jec tiv e and with a 10X ey e -p ie ce. The ey e -p ie ce had a sca le o f one-hundred d iv is io n s superimposed on i t s f i e l d o f view. For all measurements the combination of the 1 O X ey e-p iece and the 44X o b jec tiv e were used. This combination gives a nominal overall m agnification o f 440 power. This p a rticu la r combination o f lenses 45 was calibrated using a stage micrometer, a glass s l i d e with ca lib ra ted markings that divided a two m illim eter length into 100 d iv is io n s . By th is ca lib ra tio n each d iv is io n on the f i e l d o f view o f the ey e-p iece was found to correspond to 1.61 microns. With th is combination i t was p ossib le to measure p a r tic le s having diameters up to 100 microns. D. Balance For a ll weighing necessary in th is research a Metier a n a ly tica l balance d istrib u ted by Metier Instrument Corporation, Heightstown, New J ersey, was used. The balance had a 100 gram capacity. The sm a llest sca le d iv is io n on the balance was 0.05 milligrams. With these d iv isio n s i t was p o ssib le to weigh a sample to 0.01 milligrams + 0.005 m illigrams. E. P a rticles In th is study the transmission o f collim ated and d iffu s e radia tion was measured through two d if f e r e n t types o f p a r tic le s suspended in two d if fe r e n t liq u id s . The f i r s t type o f p a r tic le was a spherical aluminum oxide provided by Thermal Dynamics Corporation made from 99.7% pure alpha phase. This material was melted in an arc furnace, formed into micron sized d ro p le ts, and quickly quenched. The product was then put through a fiv e hundred mesh s i e v e , thus d eliv erin g a fin e powder whose maximum s i z e was 25 microns. The powder was then subdivided into cuts o f narrow p a r tic le diameter range. The separa tion was carried out in an e lu t r ia t o r as described by Nagy. In th is study two d iffe r e n t p a r tic le cuts were used for most of the stu d ies 46 with alumina, one having an average volume diameter o f 16.1 microns and the other having an average volume diameter o f 20.2 microns. Nagy had included cuts with volume average diameters of 6.36 microns and 12.2 microns in his work. The two d if fe r e n t p a r tic le s iz e s used here were suspended in both carbon tetra ch lo rid e and carbon d is u lf id e . Transmission o f radiation from both collim ated and d iffu s e sources were made. Nagy had made sim ila r measurements for the two smaller diameters suspended in carbon tetra ch lo rid e and for the cut with a volume average diameter o f 12.2 microns in carbon d is u lf id e . A few measurements were made in th is study with the 6.36 micron diameter p a r tic le s and collim ated radiation in an e f f o r t to check Nagy's r e s u lts . As a r e s u lt o f th is work and Nagy's, measurements were obtained on four d if f e r e n t s iz e s of aluminum oxide p a r tic le s sus pended in two d if f e r e n t liq u id s which cover a range o f the r a t io of average p a r tic le diameter to wave length running from about 0 .5 up to 10. The second type o f p a r t ic le s used in th is study was a com m ercially prepared grap h ite, "Airspun Graphite" provided by the Joseph Dixon Crucible Company. Three samples were included. They were designated as type 200-10, type 200-09, and type 200-08. These three samples had average diameter of 2.94 microns, 5.47 microns, and 10.3 microns, r e s p e c tiv e ly . These p a rticle s were prepared by crushing and grinding natural graphite. The various p a r t ic le cuts were made by centrifugal s e t t l i n g through a i r . The p a r tic le s appear q uite irregular in shape, lik e very jagged flak es o f c o a l, and c r y s t a llin e 47 in nature. All three p a r tic le s iz e s were suspended in both carbon tetra ch lo rid e and carbon d is u lf id e and transmission measurements made with both a collim ated and a d if fu s e source. The range of the ratio o f the p a r tic le diameter to the wave length included in th is s e r ie s was from about 0.2 to 5 or 6. 48 IV. EXPERIMENTAL PROCEDURE Several tasks were involved in the experimental procedure. These were the preparation of a sample of p a r tic le s suspended in the liq u id , the determination of the amount o f so lid s in the suspension, the measurement o f the p a r tic le s i z e s , and the measurement of trans mission of infrared radiation through a cloud of suspended p a r t ic le s . It i s convenient to consider each of these separately. A. Sample Preparation The f i r s t step when working with a p articu lar p a r t ic le s iz e and a p articu lar suspending liq u id was to prepare a stock suspension. To prepare the stock suspension a portion o f the s o lid powder was added to 25 or 50 cubic centim eters o f the suspending liq u id in a glass b o tt le with a metal cap f i t t e d with a Viton-A gasket. The amount of powder added was ca lcu la te d to give the maximum concentration o f p a r tic le s desired fo r transm ission measurements. Each time, the stock suspension was shaken vigorously to obtain a good d isp ersion . For each s e t of transmission measurements (a set o f transmission measurements c o n s is tin g o f measuring the transmission a t a v ariety of wave lengths for a given sample in the c e l l ) enough of the stock suspension was added to the spectrometer c e ll to f i l l or p a r tia lly f i l l the c e l l c a v ity . In f i l l i n g the c e l l a small fu n nel, a ctu a lly c o n sistin g o f the hollow cylin d er portion o f a hypodermic syringe f i t t e d with a large bore needle, was used. The stock suspension was f i r s t added to the funnel and allowed to drain com pletely into the 49 sample c e l l . In those cases where d ilu tio n s o f the stock suspension were d esired , the c e ll ca v ity was then f i l l e d to capacity by adding more of the suspending liq u id . By varying the proportions o f stock suspension and additional liq u id the volume fr a c tio n of s o lid in the suspension could be varied. The determination o f the precise amount o f s o lid s within the suspension contained in the spectrometer c e ll was determined a fter the transmission measurements. After completion o f a ll transmission measurements on a particu lar sample, the sample and c e ll were shaken vigorously and then emptied quickly and com pletely into a small glass b o t t le . The b o tt le s used were of approximately 5 cubic centimeters capacity with a metal cap f i t t e d with a Viton-A gasket. The top was tightened to seal the sample within the b o ttle and to prevent any evaporative lo s s e s . Subsequently, the b o ttle s were weighed, shaken vigorou sly, and then dumped into a tared evaporating dish made of aluminum f o i l . The b o ttle and cap were then reweighed and the total sample weight determined. After a ll o f the liq u id had evaporated from the evaporating d is h , the dish and remaining s o lid s were dried in an oven at 300 degrees, Centigrade, for several hours. On co o lin g , the dish and s o lid s were reweighed on the a n a ly tic balance to de termine the weight of s o lid s o r ig in a lly in the suspension sample. From the weights determined and the known d e n s it ie s of the so lid s and suspending liq u id , the volume fra ctio n s o lid s in the suspension could be c a lc u la te d . 50 B. P a r t ic le S ize Measurement For measuring the s iz e o f the p a r tic le s in the powder, the techniques reported by Nagy were used. A very minute sample o f the powder was placed on a microscope s l i d e by means of a spatula and a drop of d isp ersin g agent (TW EEN 20, by the Atlas Powder Company) was mixed with the sample of powder. A cover s l i p was placed over the mixture and, using the eraser on the end of a p e n c il, the cover s l i p was c a r e fu lly moved back and forth to provide a shearing force that would separate the p a r t i c l e s . The s l i d e was then mounted on the microscope stage and the p a r tic le s brought in to focus. Using the previously described graduated d iv is io n s in the e y e -p ie c e , the sample of powder was s iz e d , p a r tic le by p a r t ic le . When working with the aluminum oxide p a r t i c l e s , the measurement o f each p a r tic le diameter was straightforw ard. However, the graphite p a r tic le s were not spherical as the aluminum oxide p a r tic le s were and the s iz e measurement was not as d ir e c t . The n on-circular cross s ec tio n s o f the graphite p a r tic le s meant that the measured diameters would depend on the o rien ta tio n of the axis of measurement r e la tiv e to the p a r t ic le . For such p a r tic le s F eret's diameter i s a measure which can be used, and was used in t h is study. The Feret diameter, shown in Figure 5, can be described by con sid erin g a large number o f randomly oriented p a r tic le s projected onto the x-y plane. Consider the locus o f points corresponding to the perimeter o f a p a r tic le . The Feret diameter i s the d iffer en ce in the values o f the maximum value and the minimum value of the x-coordinates dp K- dp >♦ ________ ^ Direction of i Scan Figure 5. Feret Diameter Illu str a ted 52 o f a l l the points on the p a r tic le perimeter. Or, in other words, the Feret diameter is simply the length o f the p a r tic le cross section projected onto the x -a x is. In p ractice the diameter is obtained by simply maintaining the microscope e y e -p ie c e , with s c a le , in a fix ed p o sitio n p a rallel to one axis o f movement o f the stage of the microscope. The p a r tic le s are scanned and measured by alig n in g the zero mark o f the sca le with one extrem ity o f each p a r tic le and noting the maximum extension of the p a r tic le in a d ir ectio n p arallel to the measuring s c a le . If the p a r tic le s are randomly o rien ted , the reported measurements represent a measure o f the p a r tic le s i z e , the sig n ific a n c e o f which is discussed in d e ta il in the sectio n on experimental r e s u l t s . C. Infrared Transmission Measurements Before any infrared measurements could be made with the spec trometer, i t was necessary to allow for a s t a b ili z a t i o n o f the equipment. This was done c o n s is te n t with the manufacturer's in stru c tio n s . At l e a s t two hours before the measurements were made the power supply, a m p lifier, cooling bath, and pump were turned on. At l e a s t th ir ty minutes before any measurements, the Nernst glower was turned on. After the s t a b iliz a t io n period, the potentiometer on the spec trometer was balanced by turning the s e le c t o r switch to zero, clo sin g the metal l i g h t g a te , and manipulating the zero knob on the am plifier to p o sitio n the galvanometer needle to in d ica te no current. Follow ing t h i s , the wave length and s l i t width were s e t , the l i g h t gage 53 opened, and the s e le c t o r switch turned to "check." The p o sitio n o f the gain control was then varied until the galvanometer again was balanced. The e f f e c t of th is l a t t e r procedure was to s e t the gain so that the transmission scale on the spectrometer would in d ica te 100% transmission with no sample c e ll in p o s it io n . The same e f f e c t could have been accomplished simply by s e t t in g the s e le c to r switch to "read" and the transmission scale to "100 per ce n t." When a sample was placed in the liq u id c e ll c a v ity , the movement of the transmission knob to p o sitio n the galvanometer so th a t the potentiometer was balanced, then gave a reading o f per cen t trans mission on the transmission s c a le . The sm a llest d iv isio n o f the s c a le was 1 per cen t. By in terp o la tio n a second s ig n if ic a n t figu re could be obtained. When th e.in d ica te d transmission was le s s than 10 per cent the gain was increased to give the maximum indicated reading on the transmission s c a le . The indicated reading was then corrected for the increase in gain. The gain control o f the spectrometer consisted of a switch and a graduated rh eo sta t. The switch had three fixed p o sitio n s repre senting incremental gains o f 0 .1 , 1 .0 , and 10.0. By varying the p o sitio n of the rheostat the actual gain could be varied from 0 to 100 per cent o f the gain indicated by the s e le c to r sw itch. By vary ing the p osition of the rheostat and noting the p osition o f the transmission sca le necessary to balance the potentiometer, the dial on the rh eostat was c a lib ra te d . Using t h i s ca lib ra tio n permitted changing the p o sitio n of the rh eostat a f t e r the balancing o f the 54 instrument for 100 per cent transmission and before the actual measurement of the transmission through a sample, and properly accounting for the change in am p lification . For a ll wave lengths l e s s than 8 microns, the metal chopper was used according to the manufacturer's in s tr u c tio n s . For longer wave len g th s, the g la ss chopper was used. The manufacturer recommended the switch in choppers at longer wave lengths to elim inate the p o s s ib i l it y o f sca ttered l i g h t generated by the metal p iece. The glass chopper could not be used for the shorter wave lengths because i t was not opaque in that part of the spectrum. Nagy had experienced some d i f f i c u l t i e s with the spectrometer in his work. He noted that at the maximum gain o f the am plifier i t was d i f f i c u l t to obtain a p recise and steady balancing of the p o ten ti ometer due to c i r c u i t n o ise. He had reported no great success in reducing th is n o ise . In th is research, sim ilar d i f f i c u l t i e s were observed. Because the s iz e of the d e flec tio n s o f the galvanometer needle seemed to be independent o f the p o sitio n o f the transmission s c a l e , the p o ssib le error due to these flu ctu a tio n s was more s ig n if ic a n t at the lower end o f the transmission s c a le . This fa cto r lim ited the p recision of very low transmissions to worse than might have otherwise been expected. The f i r s t s e t of measurements made with the spectrometer were to obtain the tr a n sm issiv ity of the Servofrax and Irtran-3 windows over the a v a ila b le wave length range. The next measurements were to determine the tr a n sm issiv ity o f the suspending media, carbon 55 tetra ch lo rid e and carbon d is u lfid e . For a ll measurements, both with the collim ated source, and the d iffu se source, the s l i t width was s e t a t 0.30 m illim eters. The s l i t housing had been modified by Nagy so that the e f f e c t i v e height o f the s l i t was only 0.5 m illim eters. This allowed for an opening 0.3 m illim eters wide and 0.5 m illim eters high, 18 centim eters from the sample c e l l . Nagy had shown that for a subtended angle th is sm all, the correction to the measured trans mission to account for scattered l i g h t w ithin the subtended angle was in s ig n ific a n t. For the actual measurement of the infrared transm ission, the suspension was prepared and introduced into the sample c e ll as described previously. With the spectrometer s ta b iliz e d and the proper adjustments made, the c e ll was shaken vigorously and then placed into p osition in the liqu id c e ll c a v ity . The potentiometer was then quickly balanced and the transmission read. It was noticed that the in d ica tin g needle would remain constant for about 8 to 10 seconds and then gradually decay, ind ica tin g th at the suspension was gradually s e t t l i n g . With the smaller s iz e p a r tic le s s e t t l i n g was slower and the needle would remain constant for a longer period o f time. After each reading the cell was removed, shaken vigo ro u sly , and replaced, to insure r e p e a ta b ility of the measurement. 56 V. EXPERIMENTAL RESULTS A. P a rtic le Size Measurements The s i z e d istrib u tio n s of the aluminum oxide p a r tic le s used in th is study are given in Table 1. From these data the volume average diameters, d 30, were calcu lated to be 16.1 and 20.2 microns re s p e c tiv e ly . The d istrib u tio n s of both samples appear to be somewhat normal. For the aluminum oxide with the sm aller average diameter, 91 per cent of the p a rticle s by number are included between the diameters o f 13 and 19 microns. For the larger diameter sample, 93 per cent of the p a r tic le s by number are included between the diameters o f 19 and 24 microns. For the three samples of graphite used in th is study the measured s i z e d istrib u tio n s are given in Table 2. The sample with the sm allest average p a r t ic le s i z e , Dixon #200-10, was found to have an average Feret's diameter of 2.94 microns. For th is sample, the measured Feret diameters were a ll between 0 and 6 .5 microns with 90 per cent f a llin g between 1.6 and 4.8 microns. For the sample, Dixon #200-09, the aver age Feret diameter was found to be 5.47 microns. For th is sample the range of Feret diameters was considerably larger with 93 per cent of the measured values f a l l i n g between 1.6 and 10 microns. For the l a s t sample, Dixon #200-08, the average Feret diameter was found to be 10.3 microns. For th is sample the range o f measured values for the Feret diameter was the most ex ten siv e with only 90 per cent of the measured values included between 3 and 19 microns. The sig n ific a n c e o f the average Feret diameter has been pointed TABLE 1 MEASURED SIZE DISTRIBUTIONS FOR ALUMINUM OXIDE Percent o f P a r tic le s Included Microscope Units* d30= 1 6 .ly d30= 2 0 .2y 3-4 0.101 4-5 0.101 5-6 0.303 6-7 0.807 7-8 3.23 0.135 8-9 19.7 0.675 9-10 36.9 2.43 10-11 23.2 8.37 11-12 11.2 31.3 12-13 2.42 28.6 13-14 1.31 15.1 14-15 0.404 9.85 15-16 0.101 2.83 16-17 0.101 0.270 17-18 0.101 0.135 18-19 - 0.135 19-20 - 0.135 *0ne microscope u n it = 1.61 microns TABLE 2 M EASURED SIZE DISTRIBUTIONS FOR GRAPHITE PARTICLES Percent o f P a rticles Included Diameter ** ** ** Microscope Units* dp= 2.94y d^. = 5.47y d^. = 1 0 . 3y 0-1 5.98 0.654 0.538 1-2 59.0 13.7 1.075 2-3 31 .6 26.8 8.06 3-4 3.41 35.3 11.83 4-5 - 9.80 10.75 5-6 - 7.84 16.13 6-7 - 3.92 15.05 7-8 - 1.31 14.52 8-9 - 0.654 5.38 9-10 - - 2.69 10-11 - - 6.45 11-12 - - 3.76 12-13 - - 1.613 13-14 - - 0.538 14-15 - - 1.075 15-16 - - 0.538 *One microscope unit = 1.61 microns **Feret diameter 59 out by Walton (25). He demonstrated th at for p a r tic le p r o file s which are non-reentrant, the average Feret diameter is equal to the diameter o f the c i r c l e o f equal perimeter. Gebelein (26) examined thoroughly the relation sh ip between various s t a t i s t i c a l diameters including the Feret diameter. Gebelein computed for randomly oriented p a r tic le s o f a given shape the r a tio o f the average Feret diameter to the diameter of a c i r c le having th e same area. Examples of his re su lts for e l l i p s e s and rectangles are given below. Ratio of Feret Diameter to Diameter o f C ircle o f Equal Area Dimension Ratio E llip se Rectangle 1.0 1.0 1.1284 1.4 1.0212 1.1444 1.8 1.0649 1.1775 2.2 1.1170 1.2172 For the e l l i p s e , the "Dimension Ratio" is the ra tio of the major to minor diameter, and fo r the rectangle the "Dimension Ratio" is the ra tio o f the longest to the sh o rte st s id e . From G ebelein's r e s u lts one can see that i f the Feret diameter were used to compute the cross section of the p a r tic le s the computed value would be larger than the actual cross sec tio n o f the p a r tic le . A ctually, the ra tio of the com puted cross sec tio n to the actual cross section would be the square of the ratio o f the Feret diameter to the diameter o f the c i r c l e o f equal area. This would mean th a t a t the maximum dimension r a tio of 2 .2 the computed area would be 1.25 times the actual area for an e l l i p s e and 60 1.48 times the actual area fo r a rectan gle. It is worth noting that the sm a llest p a r tic le s were more nearly c ir c u la r in shape in general than the large p a r t i c l e s . This p a r tia lly accounts for the apparent increase in the d isp ersion of the measured values o f the Feret diameter as the average diameter of the sample in creased. It is also worth noting that some d isp ersion is due to the random o rien ta tio n of the p a r t ic le s . For example, even i f a ll p a r tic le s were of the same s i z e , non-circular p a r tic le s would show a dispersion in values o f the measured Feret diameter. In that sense, the d isp er sion o f the measured values o f the Feret diameter would be expected to be greater than the dispersion of the p a r t ic le s i z e s based on an actual measure of th e ir area. The graphite p a r tic le s appeared under the microscope as irreg u la r ly shaped fla k e s . A d ir e c t measure o f the thickness of these flakes was not obtained. However, an estim ate o f the thickness can be made from data given by the su p p lier. Accordingly, the Dixon sample #200-10, which had an average Feret diameter o f 2.94 microns was reported to have a surface area o f 11.5 square meters per gram. If the Feret diameter is taken as a clo s e approximation o f the c i r c le o f equal area to that of the p a r tic le , then the thickness o f the p a r tic le can be com puted from the given surface area data. Based on these value the average thickness o f the p a r tic le s was 0.083 microns. Expressed in another manner, the average ratio of thickness to diameter was 0.0278 61 B. Cell Blank Measurements In order to evaluate the e f f e c t of the p a r tic le s on the in cident rad iation , both for the collim ated and d iffu se sources, i t was neces sary to determine the tr a n sm issiv ity and r e f l e c t i v i t y o f the windows used in the c e ll and the ex tin ctio n c o e f f ic ie n t s for the suspending liq u id s . The windows o f both op tical m aterials, Servofrax and Irtran-3, were inserted in the collim ated beam and the transmission measured. Using the reported ind ices o f refra ctio n for the window m aterials and F resnel's laws o f refra ctio n the tr a n sm issiv ity o f the windows were calcu lated. The measured transmissions and the calculated tr a n s m is s iv ity at various wave lengths is given in Table 8 fo r both the Servofrax windows and the Irtran-3 windows. For the Servofrax windows there was e s s e n t ia lly no lo ss to absorption by the o p tical material of the window over the range o f wave lengths from 2 to 8 microns. As the wave length increased above 9 microns the amount of absorption within the window increased somewhat. With the Irtran-3 window there appear ed to be a very s l i g h t amount o f absorption (3 to 6 per cent) over the wave length range from 2 to 8 microns and more absorption as the wave length was increased above 8 microns. It should be noted th a t the total transmission for the Irtran-3 windows was higher than that of the Servofrax windows. That was due to the lower r e fr a c tiv e index. The r e f l e c t i v i t y o f each type of window with the suspending liq u id on one sid e and a ir on the other s id e was calculated for each com 62 bination o f liq u id and windows used in th is study. F resnel's laws of r e fle c t io n were used to ca lcu la te the r e f l e c t i v i t y f i r s t of a normal ray s tr ik in g the window, e . g . the collim ated beam, and also fo r d iffu se radiation s tr ik in g th e liq u id glass surface with equal in t e n s it y from all angles. The re su lts o f these ca lcu la tio n s are given in Table 9. In order to determine the liqu id e x tin ctio n c o e f f i c i e n t s , tra n s mission measurements were made with the c e l l s f i l l e d only with the suspending liq u id . The r e s u lts o f these measurements are given in Table 10 showing the liq u id ex tin ctio n c o e f f i c i e n t , a, for carbon d i su lf id e and carbon tetra ch lo rid e at various wave lengths from 2 to 11 microns. C. Transmission o f the Collimated Beam The transm issions through the c e ll fo r various p a r tic le clouds suspended in one o f the two liquid mediums are given in Tables 11 through 15. These values were normalized by dividing the measured transm ission by the transmission through the c e ll containing the same liq u id but no p a r tic le s . The r e la t iv e transmission expressed on a per cen t basis i s what is reported in the Tables. The concentra tion o f s o l i d p a r tic le s is given as the volume fr a ctio n s o lid s within the c e l l . Table 15 shows a lim ited number o f r e s u lts for aluminum oxide p a r t ic le s , d 30 = 6.36y. These measurements were made to compare the experimental d iffe r e n c e s between this study and that o f Nagy. For r e la t iv e low concentrations of s o l i d s the p lo t o f log trans mission versus s o lid concentration should appear as a str a ig h t lin e as given by the equation The slope o f the lin e is equal to the volumetric ex tin ctio n cross s e c tio n , o^, times the thickness o f the s o lid liqu id sample within the c e l l . The value of a t giving the b est le a s t squares f i t from the equation E£n ( f I ' » I measured ( w ) t LZv This sim p lifie d form r e s u lts from the fa c t that the lin e i s forced through the 100 per cent point at zero so lid s concentration. The lin e corresponding to the calcu lated volume of oj- along with the data are given in Figures 6 through 9. These four fig u res are representative examples of the data. All o f the forty-on e p lots are not given here. D. Transmission of the D iffuse Radiation The data obtained with the d if f u s e source are presented in Tables 16 through 19. The re su lts reported in the tables are the per cent transmission through p a rticu la te clouds normalized to the per cent transmission from the d iffu s e source at zero concentration of s o lid s . The r e la t iv e transmissions as a function of so lid concentrations are shown in Figures 10 through 13, which are examples of the forty-one p lo ts , to t a l. The lines shown on the fig u res were obtained by determining in each case the values o f and ip which gave the best " least squares" f i t of the data by the equation Normalized Transmission, Percentage 64 100 60 40 20 10 6 0 10 20 Volume Fraction Solids xlO1 * Figure 6. Transmission o f Aluminum Oxide P a r t ic le s in Carbon Tetrachloride with a Collimated Source, X = 2 \i Normalized Transmission, Percentage 65 100 30= 3o~ 20.2m 60 40 20 10 6 10 5 0 Volume Fraction Solids xlO1 * Figure 7. Transmission of Aluminum Oxide P a r t ic le s in Carbon D isu lfid e with a Collimated Source, X = 8.5y Normalized Transmission, Percentage 66 100 = 2.94u = 5.47u = 10.3m 60 40 20 10 6 0 10 5 Volume Fraction S olid s xlO5 Figure 8. Transmission of Graphite P a r tic le s in Carbon T etrachloride with a Collimated Source, A = 2y Normalized Transmission, Percentage 67 100 60 40 CD 20 10 0 5 Volume Fraction Solid s xlO5 Figure 9. Transmission o f Graphite P a r tic le s in Carbon D isu lfid e with a Collimated Source, X = 5.5p 100 O d ,0» 16.1v A d ,.* 20.2u < D o> ea 4 -> c a> u s- 0) Q_ C O >r“ ■ a a t N i . o 0 20 40 Volume Fraction Solids x!0" Figure 10. Transmission o f Aluminum Oxide P a rtic les in Carbon T etrachloride with a D iffuse Source, X * 3.5y Normalized Transmission, Percentage 69 100 60 40 20 16.ly 2 0 . 2 y 10 10 20 0 Volume Fraction S o lid s xlO1 * Figure 11. Transmission of Aluminum Oxide P a r t ic le s in Carbon D isu lfid e with a D iffuse Source, X = 5.5y Normalized Transmission, Percentage 70 100 o dp - 2.94u A dp * 5.47u [ | dp = 10.3u 60 40 20 0 5 10 15 Volume Fraction S olid s xlO5 Figure 12. Transmission o f Graphite P a r tic le s in Carbon Tetrachloride with a D iffuse Source, A = 4y Normalized Transmission, Percentage 71 100 60 40 O □ 20 10 6 4 2 10 0 20 30 40 Volume Fraction Solid s xlO5 Figure 13. Transmission o f Graphite P a r tic le s in Carbon D isu lfid e with a Diffuse Source, X = 6p 72 IL+ ___________________ 2e_____________________ ' R ' f I u T v=o = ( l - * ) e - * <v'f“ / ° t )L + (1+tp)e<S> < v+ “ / a t >L 0 0 In arriving at the above equation the d if fu s e surface at x = 0 was taken to have a r e f l e c t i v i t y , R 0 , o f zero. The exponents in the de nominator are s l i g h t l y d iffe r e n t from those given in previous equations, in order to account for absorption due to the liq u id surrounding the p a r t ic le s . The exponent terms r e s u lt from the total e x tin c tio n co e f f i c i e n t K f- being given by K ^ . = Oj.v + a (106) a being the absorption c o e f f i c i e n t for the liq u id . The le a s t squares f i t o f the data was made on the basis of compar ing the logarithm of the observed r e la t iv e in t e n s it y , Ip, to the logarithm o f the calculated r e la t iv e in t e n s it y . With the calculated value o f the r e la t iv e in t e n s it y denoted by IR^ the sum of the squares o f the d ifferen ces SM D was defined by n SM D = I (£nIR - £nIRC)2 0 0 7 ) where the su bscript "i" refers to a s in g le data s e t and "n" is the total sets o f data a t a given wavelength for one s i z e and type of p a r tic le in one suspending liq u id . The independent variab le associated with each data s e t was the volume concentration of s o lid s in the su s pension. The optimum values of c p and ip were determined by s a t is f y in g d(SMD) = 0 d<f> and * s M = 0 The value o f the parameter B was calculated from the optimum The equations above become, on s u b s titu tio n for the sum o1 squares o f the d if fe r e n c e s , SM D n r i Hi RP -| , 1 , [ ( M r - = 0 and n 3-fnIpQ ^ 1 t(^nIR " ^nIRC^~5^ ^ = 0 The p artial d erivatives o f I n i R q are given by W K (v+a/0 t )L [ ( I - * ) . - W ° t)L - ( l +* )e * (Vt“/at>L ] ^ " ( l- r t e " ,'(W a/at)L ♦ ( l +*)e*(V +C ‘/ ° t)L and d fn IR C e -+ (v+«/at )'- _ e (*+o/"t )L d i p , v -<J)(v+a/at )L 4>(v+a/at )L (l-ip)e + (l+dj)e 1 7 3 (108) (109) value of the ( 110 ) ( 1 1 1 ) ( 112) (113) An it e r a t i v e procedure was used to so lv e the equations. Values o f < J > and ip were assumed and residues calculated fo r the two equations as shown. 74 n b i n l D r RESA = E i ( t n I R C - i n l R C ) (114) 1 = 1 dcj> 1 n b l n l D r RESB = [(-ChIrc - £«Ir c ) (116) The required changes in the values o f < p a n d ip , A<p and A<p, re s p e c tiv e ly , to reduce the residues to zero were determined from the pair of equations *(RESA) d(RESA) A c f> 5 $ + Alp = " resa ( 116) i(RESB) MRESA) A < t > d<P + A l(; b'P = " RESB (117) New values for the residues were then calculated a ft e r changing the values of the parameters 4> and ip according to 'f’k + i s *k + A*k <118) V i = *k + ^k 019) where the index k refers to the number o f the it e r a t io n . The ca lcu la tio n s were carried out on a G E 435 d ig ita l computer. The partial d eriv a tiv e s o f the residues were determined numerically. The iter a tio n s were continued u ntil the values o f < J > and ^ for one iter a tio n d iffer ed from those of the previous ite r a tio n by no more than one per cent. 75 VI. CORRELATION OF RESULTS A. The Collimated Transmission It should be noted that in observing the data for the per cent transmission fo r the collim ated source versus volume fr a ctio n so lid s in the suspension, reported in the previous s e c t io n , th at the slopes of the lin e s representing the actual data varied with both the diameter or s i z e of the p a r tic le s and the wave length o f the incident rad iation . It i s desirab le to determine whether these v a ria tio n s can be correlated in some manner. Van de Hulst (2) discussed the merits of re la tin g the e x tin ctio n e f f ic ie n c y c o e f f i c i e n t , Xt , to the absolute value o f the phase lag a which is defined as tw ice the s i z e parameter, x, times the absolute value of the differen ce between the r e fra c tiv e index of the p a r tic le s and the r e fr a c tiv e index o f the medium. Nagy found that the measurements he made on the ex tin c tio n due to aluminum oxide p a r tic le s suspended in carbon tetra ch lo rid e and carbon d is u lf id e could be correlated as suggested by Van de Hulst. His r e s u lts showed reasonably good q u a lita tiv e agreement with a sim ila r approach used in th is study. In order to use th is approach i t was necessary to know the re fr a c tiv e in d ices of the d if fe r e n t p a r tic le s used and o f the suspending mediums. The r e fr a c tiv e indices for both carbon tetra c h lo rid e and carbon d is u lf id e were the same as those used by Nagy and were obtained from the reported works o f Pfund (27) and Kagarise (28). Nagy discussed the r e la t iv e merit of r e fra c tiv e indices in the infrared region reported by Malitson (2 9 ), Harris (30), and 76 Neuberger (31) for aluminum oxide. He chose the values reported by Malitson fo r his ca lcu la tio n s and the same values were used in th is study. No great amount o f information has been reported on the re fr a c tiv e ind ices o f graphite in the infrared region. However, Dalzell and Sarofim (32) developed a dispersion equation for predicting the op tical properties o f soot which they measured for the wave length range from 0.4358 microns to 10 microns. Their measurements included acetylene soot with a low hydrogen to carbon ra tio and propane soot with a higher hydrogen to carbon r a t io , but they found l i t t l e e f f e c t o f the hydrogen content on the o p tical properties of the so o t. Re fr a c tiv e indices calcu lated by th e ir dispersion equation were used in the ca lcu la tio n s for th is study. The calcu lated phase la g s , along with the r e fr a c tiv e indices of the p a r tic le s and suspending liq u id s , are given in Tables 3, 21, and 22. Some d ifferen ces ex iste d in the methods o f ca lcu la tio n for the phase angle and e x tin ctio n e f f ic ie n c y c o e f f i c i e n t s for the d if fe r e n t types of p a r t i c l e s . These d ifferen ces are discussed separately in the s ec tio n s below for the d iffe r e n t types of p a r t ic le s . 1. Aluminum Oxide P a rtic les For the aluminum oxide p a r tic le s in both carbon tetrach lorid e and carbon d is u lfid e the ex tin ctio n e f f i c i e n c y X ^ . was calcu lated by dividing the measured value o f volummetric ex tin c tio n cross sectio n a t by the r a tio of the geometrical cross sectio n to the geometrical volume, Og. The values for the volumetric ex tin ctio n cross sections and the e x tin c tio n e f f i c i e n c i e s are given in Tables 3, 21, and 22. TABLE 3 EXTINCTION CROSS SECTION AND EXTINCTION EFFICIENCY OF ALUM INUM OXIDE PARTICLES IN CARBON TETRACHLORIDE d 30 = 1 6 .ly X nP n m lnp“nm ! X 2x|np-nm| atxlO- 2 ,cm-1 h 2.0 1.734 1.447 0.287 25.3 14.53 23.7 2.59 2.5 1.722 1.443 0.279 20.2 10.30 24.4 2.66 3.0 1.708 1.442 0.266 16.83 8.95 24.9 2.72 3.5 1.697 1.442 0.255 14.43 7.36 28.3 3.09 4.0 1.673 1.441 0.232 12.63 5.87 32.4 3.54 4.5 1.653 1.440 0.213 11.22 4.78 36.1 3.94 5.0 1.625 1.438 0.187 10.10 3.78 33.8 3.69 6.0 1.535 1.433 0.102 8.43 1.720 20.6 2.25 7.7 1.472 1.424 0.048 6.56 0.630 8.31 0.909 9.0 1.216 1.404 0.188 5.62 2.11 24.9 2.72 78 The values of the ex tin c tio n e f f i c i e n c i e s are shown as a function of wave length in Figures 14 and 15. The e f f i c i e n c i e s vary markedly with the value o f the wave length, and the v ariation at any wave length appears to depend on the s iz e o f the p a r t ic le s . The strong v a ria tio n of the e x tin c tio n e f f i c i e n c i e s with wave length appear to be well accounted for by co r rela tin g the e f f i c i e n c i e s with the varia tio n in phase angle a, as suggested by Van de Hulst (2 ), where Figure 16 shows the data for aluminum oxide p a r t ic le s suspended in both carbon te tra c h lo rid e and carbon d i s u l f i d e . The data shown include v a ria tio n s in the s iz e parameter x from 3.33 to 3 2 .8 , and variations in the r e la t iv e r e fr a c tiv e index o f the p a r tic le s from 0.62 to 1 .2 . Some s c a tte r in the data i s not unexpected since the re la tio n sh ip between the e x t in c t io n e f f ic ie n c y and the phase angle a is a function o f the r e la t iv e r e fr a c tiv e index. For comparison, the re la tio n sh ip between the e x tin c tio n e f f ic ie n c y and the phase angle as reported by Van de Hulst (2) for r e la t iv e r e fr a c tiv e indices o f 0 .8 , 1 .0 * e, and 1.33 are shown in Figure 17. Comparing the values obtained experim entally with the th eo retica l curves reported by Van de Hulst shows that the maximum values fo r the ex tin c tio n e f f i c i e n c y c o e f f i c i e n t are w ithin reasonable agreement of the values given by the th eoretical curves. Furthermore, there appears to be reasonably good agreement between the absolute value of ( 120 ) 4 .0 3.0 2.0 1.0 0 10 2 4 6 8 12 Wave Length, Microns Figure 14. Extinction E ff ic ie n c ie s of Aluminum Oxide P a r tic le s in Carbon T etrachloride 3.0 + -> X 2.0 1.0 0 Figure 15. 4 6 8 10 Wave Length, Microns E xtinction E ff ic ie n c ie s o f Aluminum Oxide P a r tic le s in Carbon D isu lfid e C X > o 4.0 □ d 3o = 6 .3 6 y , CC1 O d 3o= 1 6 .l y , CC1 30 = 1 6 .l y , CS2 A dso= 2 0 .2y, CC1 yX d3o= 2 0 .2y, CS2 +j x Nagy's Data /A . 7 / □ •/ I a| = 2x|n n, Figure 16. E xtinction E f fic ie n c ie s o f Aluminum Oxide P a r tic le s CO 4 .0 3.0 ■p X 1.0+e 2x a n n, Figure 17. E xtinction E f fic ie n c ie s o f Non-absorbing Spheres with R efractive Indices near 1.0 00 ro 83 the phase angle a at which the maximum occurs, comparing the data and the th eoretical curves. However, the th eo r etica l curves a ll decrease to a f i r s t minimum value a t values of the phase angle between 7.3 and 7 .6 , and then show a second maximum value a t a phase angle o f 10.8 or 10.9. The th eo retica l curves pass through a succession o f progres s iv e ly smaller maxima and minima and approach a lim itin g value o f 2 for the ex tin ctio n e f f i c i e n c y . The actual measured r e s u lts show only one maximum and then appear to approach the lim itin g value of 2 with increasing phase angle without o s c i l l a t i o n below and above the lim itin g value. Several reasonable explanations may account for the d ifferen ces between the measured values and the predicted values of the ex tin ctio n e f fic ie n c y c o e f f i c i e n t . For a completely rigorous comparison between measured and theo re tic a l values o f the e x tin ctio n c o e f f i c i e n t , i t would be necessary to a c tu a lly measure the r e fra c tiv e indices o f the p a r tic le s and of the suspending liq u id . At some wave lengths included in th is study, the re fra c tiv e index o f the p a r tic le and the r e fr a c tiv e index o f the suspending liq u id were very clo se in value. The sm allest d ifferen ce was only s l i g h t l y more than 3 per cent o f the r e fr a c tiv e index o f the suspending liq u id . For such cases the value o f the phase angle a would change s ig n i f i c a n t ly for a rather small change in the r e fra c tiv e index of e ith e r the p a r tic le or the liq u id at that p articular wave length. The aluminum oxide p a r tic le s were prepared from the same sample of aluminum oxide powder that Nagy used to prepare his samples. Nagy 84 reported that the powder was highly c r y s t a llin e and included two forms, alpha and gamma, with the l a t t e r being the major form as shown by an X-ray d iffr a c tio n pattern. He suggested that sin ce these two forms did vary s l i g h t l y in r e fr a c tiv e index, the heterogeneity o f the p a r tic le s would e f f e c t the rays passing through due to internal r e fr a c tio n s . The maxima and minima displayed by the th eo retica l curves are the r e s u lts o f interferen ce patterns between a ray o f lig h t passing through the p a r tic le in te r -r e a c tin g with a ray o f l i g h t by passing the p a r tic le . The h eterogeneity o f the p a r tic le would have the e f f e c t of decreasing the sharpness of the interference patterns and dampening the maxima and minima values. Another important consideration concerning the op tical p roperties of the p a r tic le s and the liq u id s is the fa c t that real r e fr a c tiv e indices were used for both. Since both s o lid s and liq u id s display some absorption of infrared radiation a t almost a ll wave lengths used in th is study, th e ir r e fr a c tiv e ind ices must, in f a c t , be complex. The complex portion of the r e fr a c tiv e index of a substance is d ir e c t ly related to the absorption index of the m aterial. The e f f e c t on a wave passing through the substance is an attenuation o f the wave amplitude. Attenuation o f the amplitude o f eith er the ray passing by the p a r tic le through the suspending liq u id or the amplitude o f the wave passing through the p a r t ic le would, in eith e r c a s e , have a dampening e f f e c t on the maximum and minimum in the ex tin ctio n e f f ic ie n c y c o e f f i c i e n t versus phase angle curve. Furthermore, the e f f e c t would be greater for the larger p a r t i c l e s , and the r e la t iv e 85 e f f e c t would be larger a t the larger values of the phase angle. Van de Hulst d isc u sse s the e f f e c t o f the complex part of the r e fr a c tiv e index on the e x tin c tio n e f f ic ie n c y curves. He shows the r e s u lts of a number o f c a lc u la tio n s by Johnson, Eldridge, and Terrell (33) for a r e fr a c tiv e index given by the equation n' = 1.29(1 - tk) (121) They made a number of c a lc u la tio n s varying the value o f k from zero to i n f i n i t y . The e f f e c t s o f even a small value of k, such as 0 .0 5 , are very s i g n i f i c a n t . For example, the value of the f i r s t maximum is decreased from nearly 4 for k = 0 to a value of s l i g h t l y more than 3 for k = 0 .0 5 . Furthermore, at even that small value of k the f i r s t minimum and second maximum have p r a c t ic a lly vanished. At a value of k = 0.25 the f i r s t maximum has p r a c tic a lly disappeared, the curve showing a very s l i g h t peak with a maximum value o f about 2.5 and then gradually decreasing towards the asymptotic value o f 2 .0 . These three curves are shown in Figure 18 with the e x tin ctio n e f f i c i e n c y p lotted versus x , the s iz e parameter, in th is case instead o f the phase angle. When the experimental data are considered and compared to the theo r e tic a l expected values fo r m aterials with even r e la t i v e l y small complex parts in the r e fr a c tiv e index, the agreement between ex p er i ment and theory appears reasonably s a tis fa c to r y . A comparison between the r e s u lts o f th is study and those o f Nagy is also shown in Figure 16. Nagy made measurements with two d if f e r e n t s iz e s of aluminum oxide p a r t i c l e s , 6.36 microns and 12.2 microns 4.0 3.0 X Figure 18. E xtinction E ffic ie n c ie s o f Absorbing Spheres, n' = 1.29(1 - i<) C D 87 volume average diameter. A few data for the smaller s iz e p a r tic le s suspended in carbon te tra c h lo rid e were a lso repeated as a part of th is study. These data are a lso shown in Figure 16. The data for the sm allest p a r tic le s were obtained when i t became obvious that Nagy's data for diameters of 6.36 and 12.2 microns were in c o n siste n t with the data for diameters o f 16.1 and 20.2 microns, made in th is study. Examination o f the data indicated that the e x tin c tio n e f f i c ie n c ie s for diameters o f 16.1 and 20.2 microns appeared more c o n s is te n t with the values expected from theory. The additional data obtained for a diameter o f 6.36 microns were a l l obtained in an e f f o r t to determine whether d iffer en ces in experimental techniques could have accounted for the d iffer en ces between the r e s u lts of the two s tu d ie s . Of particular concern was the d iffer en ce in the methods of charging the sample c e l l , and a ls o the d if fe r e n t c e l l s used. Nagy used a c e ll with s ta in le s s s te e l w a lls while an aluminum c e l l was used in th is study. Furthermore, the l a t t e r c e ll gave a sample thickness o f 0.762 centim eters compared to 1.43 centim eters for th a t o f Nagy. In charging the sample c e l l , Nagy added the dry, so lid p a r tic le s to the c e ll and then introduced the liq u id . He weighed the amount of p a r tic le s added to the c e l l . In th is study, a stock suspension of p a r tic le s was prepared by mixing so lid s and liq u id in a b o t t le . The stock suspension was prepared, in g en era l, several days before i t was a ctu a lly used. The sample was charged to the c e ll by p a r tia lly f i l l i n g the sample c e ll with stock suspension and f i l l i n g the r e s t of the c e ll with c le a r liq u id . The concentration o f s o lid s was 88 determined a f t e r emptying the c e l l . Table 4 shows a comparison o f values o f the e x tin c tio n co e f f i c i e n t a t obtained by d if f e r e n t experimental methods, including Nagy's method of charging the sample c e l l , and including the exact sample c e ll used by Nagy. E s s e n tia lly the same r e s u lts were obtained regardless o f the experimental v a r ia tio n s. The r e s u l t s , however, were not c o n sista n t with those reported by Nagy. When a ll the data of Table 4 were treated c o l l e c t i v e l y , the values of the ex tin c tio n co e f f i c i e n t , o t , were found to be 5970 cm" ^, 6300 cm"^, and 1280 cnf^, a t wave lengths of 2, 3 and 6 microns, r e s p e c tiv e ly . The values reported by Nagy were 3920 cnH , 4500 cm- ^, and 2620 cm“l a t the same wave len g th s. The values reported here are in b etter agreement with the r e s u lts reported for diameters of 16.1 and 20.2 microns as well as those expected by theory, shown in Figure 17. Thus i t would appear the Nagy r e s u lts cannot be explained by the use of a d iffe r e n t sample c e l l , or a d iffe r e n t method o f charging the c e l l . Although Nagy's samples, and the samples reported here, were both taken from the same gross sample o f aluminum oxide, they were sepa rated at d if fe r e n t times. The r e s u lt s of both stu d ies would indicate that a c t u a lly two d if f e r e n t systems were studied. One explanation might be the e f f e c t of contamination. A small amount of contaminant with an appreciable absorption index, even though i t were only on the surface of the p a r t ic le s , would have the e f f e c t o f causing the p a r tic le s to behave as i f they had a higher value o f the complex portion of the re fra c tiv e index than the uncontaminated p a rticle s TABLE 4 COMPARISON OF EXTINCTION COEFFICIENTS FOR ALUM INUM OXIDE PARTICLES IN CARBON TETRACHLORIDE OBTAINED BY DIFFERENT EXPERIMENTAL PROCEDURES d 30 = 6.36y ----------- a cm-1 ---------- Wave Length, Microns Cell Wall Cel 1 Length Sample Charge 2 3 6 A1umi num 0.762 cm. Stock Suspension 5920 6170 1210 A1umi num 0.762 cm. Dry Powder + Liquid 6020 6470 1330 A1uminum 1.429 cm. Stock Suspension 5990 6340 1320 S ta in le ss Steel 1.430 cm. Stock Suspension 5990 6180 1210 co t£ > 90 would d isp la y . Such a p o s s i b i l i t y could very well explain the lower values reported by Nagy as demonstrated by the marked e f f e c t o f changing k from 0.05 to 0 .2 5 , discussed p reviously. Such an ex planation cannot be substantiated in the absence of the actual p a r tic le s which Nagy used. Thus, at th is time, the d ifferen ces between Nagy's re su lts and those reported here are unaccounted for. 2. Graphite P articles An attempt was a lso made to tr e a t the r e s u lts obtained for the three d if fe r e n t s iz e s o f graphite p a r tic le s with a collim ated l i g h t source in the same manner as was used to co rrela te the data for the aluminum oxide p a r t ic le s . Van de Hulst suggested that randomly oriented irreg u la rly shaped p a r tic le s might be treated as spherical by choosing the proper eq u ivalent dimension. The graphite has a complex r e fra c tiv e index with a complex portion o f the same order o f magnitude as the real part. In a d d itio n , the graphite p a r tic le s were not spherical but d is k - lik e in shape. Both o f these features r e quired some additional consideration beyond the procedures applied to the aluminum oxide p a r t ic le s . F ir s t, in determining the e x t in c t io n e f f ic ie n c y o f the p a r t i c l e s , i t is necessary to determine the e f f e c t i v e geometrical cross s e c t io n , Cg, of the p a r tic le s . For th ese c a lc u la tio n s the area of the face o f the d is k - lik e p a rticle s was taken as the area o f a c i r c l e having the same diameter as the measured Feret diameter. Furthermore, the p a r tic le s were considered as randomly oriented so that the e f f e c t i v e 91 geometrical cross s e c tio n would be le s s than that expected by summing the fa cia l areas of a ll the p a r t i c l e s . To take into account the random o rien ta tio n o f the p a r t i c l e s , the area of a thin disk pro je cte d onto a plane perpendicular to the d ir ectio n o f the in cid en t radiation was averaged over a ll p o ssib le o rien ta tio n s w ithin one hemisphere. That i s , the e f f e c t i v e geometrical cross sec tio n is given by The a p plication of the above equation y ie ld s the f a c t th at the e f f e c t i v e geometrical cross sec tio n Cg is on e-h alf the actual geo metrical cross sec tio n Cg. In ca lc u la tin g the e x tin c tio n e f f i c i e n c i e s , the volumetric e x tin c tio n c o e f f i c i e n t s were divided by the e f f e c t i v e c r o ss-se c tio n a l area rather than the actual c r o ss-se c tio n a l area. In ca lcu la tin g the s i z e parameter x and the value of the phase angle a, the random o r ie n ta tio n o f the p a r tic le s and i t s e f f e c t was again considered. For a spherical p a r tic le the dimension chosen is the diameter of the p a r t i c l e , which represents the thickness o f the p a r tic le in the d ir e c tio n o f the in cid en t ra d ia tio n . For the graphite p a r tic le s the corresponding dimension would be the thickness o f the d is k -lik e p a r tic le s corrected for th e ir o r ie n ta tio n . For the e f f e c tiv e cross section i t was found th a t the p a r tic le s could be charac te rized by using o n e-h a lf the geometrical cross s e c tio n . This fa c to r corresponds to a p a r tic le oriented a t an angle 0 measured from the d ir ectio n o f in cident rad iation such that cose = 0 .5 . For such a . / Cgcosedft = T . 1 C 92 p a r tic le , the thickness in the same d ir ectio n as the in cid en t radiation is twice the th ick n e ss, t , o f the p a r t i c l e s , everywhere except near the edges. Since the p a r tic le diameters were large r e la t iv e to th eir thickness (d /t - 3 6 .) the edge e f f e c t s can probably be ignored. Therefore, for ca lcu la tin g the s i z e parameter x for the graphite p a r t ic le s , the length dimension included in x was twice the thickness o f the p a r t i c l e s , t , or 2TTt x = ---------------------------- (123) X The re su lts o f these ca lcu la tio n s are shown in Tables 5, 23 and 24. The e x tin c tio n e f f i c i e n c i e s are a lso shown in Figures 19 and 20. One very in te r e s tin g comparison to make is to look at the e f f e c t of changing p a r tic le s i z e and wave length on the e x tin c t io n e f f ic ie n c y for the graphite p a r tic le s r e la t iv e to the e f f e c t th at these changes had on the e x tin c tio n e f f ic ie n c y o f the aluminum oxide p a r t ic le s . For graphite the ex tin c tio n e f f i c i e n c i e s varied only s l i g h t l y with wave length, appearing to be almost constant when carbon tetra c h lo rid e was the suspending liq u id and showing a s l i g h t increase with wave length when carbon d is u lf id e was the suspending liq u id . This was a s i g n i f i cant d iffer en ce in behavior compared to the aluminum oxide p a r tic le s (Figures 14 and 15). In the l a t t e r case there did not appear to be great d iffer en ces in the ex tin ctio n e f f i c i e n c i e s o f the two d iffe r e n t sized aluminum oxide p a r tic le s at the same wave len g th , but very s i g n if ic a n t changes in the ex tin c tio n e f f i c i e n c i e s as the wave length was changed. These variation s were well accounted fo r by the Mie TABLE 5 EXTINCTION CROSS SECTION AND EXTINCTION EFFICIENCY OF GRAPHITE PARTICLES IN CARBON TETRACHLORIDE dF = 2.94y X " ' p nm i np-nm1 X 2x|np-nm| o ^ l O -1* ,c n n Xt 2.0 1.906-11.037 1.447 0.459 0.256 0.235 5.38 0.700 3.0 2 . 1 6 7 - i l .295 1.442 0.725 0.1710 0.249 5.43 0.707 4.0 2 . 3 9 9 - i l .486 1.441 0.958 0.1281 0.246 5.64 0.735 4.5 2 . 5 0 3 - i l .564 1.440 1.063 0.1138 0.242 5.64 0.735 5.0 2 . 5 9 8 - i l .635 1.438 1.160 0.1024 0.239 5.59 0.729 7.7 3 . 0 1 0 - i l .942 1.424 1.586 0.0666 0.211 5.70 0.743 V O oo 1.4 1.2 1.0 0.8 0.6 10 8 2 4 6 Wave Length, Microns Figure 19. E xtinction E ffic ie n c ie s of Graphite P a r tic le s in Carbon T etrachloride 1.2 A— A— A 0.6 0 -0 -0 0 — 0 — 0 , 0.4 4 6 8 10 12 14 2 Wave Length, Microns Figure 20. Extinction E ffic ie n c ie s o f Graphite P a r t ic le s in Carbon D isu lfid e V O cn 96 theory and the ex tin c tio n e f f i c i e n c i e s correlated well with the phase angle. Figure 21 shows the ex tin c tio n e f f i c i e n c i e s o f the three s iz e s o f graphite p a r tic le s suspended in both liq u id s for a ll the wave lengths measured presented as a function o f the phase angle. It would appear that with the graphite p a r t i c l e s , as with the aluminum p a r t ic le s , the e x tin ctio n e f f i c i e n c i e s c o r rela te well with the phase angle. Both the changes in p a r t ic le s i z e and the d ifferen ces in suspending liq u id s appear to be well accounted fo r by the apparent co rrela tio n even though there is some s c a t t e r o f the data. It is important to determine whether or not the re la tio n between e x tin ctio n e f f ic ie n c y and phase angle is accountable for from the Mie theory. Van de Hulst (2) suggests th at when the absolute value o f the d ifferen ce between the complex r e fr a c tiv e index and one is considerably l e s s than unity the ex tin c tio n e f fic ie n c y can be calcu lated from a sim p lifie d equation involving two parameters. The f i r s t parameter is the phase angle a which we have already presented. The second para meter is tang where n< tang = p y — (124) when the medium surrounding the p a r tic le s has a r e fr a c tiv e index equal to one. Otherwise, np< tang = ------------------- (125) lnp " nml Under the r e s tr ic t io n s then that )n ' - 1 | « 1 and n < « l , accord ing to Van de Hulst the e x tin ctio n e f f ic ie n c y X j- is given by 2.0 0 d F = 2.94y, o 0 ■ r 1 0 d F = 2.94y, CS2 A d F = 5.47y, ccr A dp = 5.57y, c s o ;...... □ dp = 10.3y, O O -r i JIf dp = 10.3y, c s > ; & 7 $ ° © p / p # la l = 2 x lnp ' nJ Figure 21. Extinction E f f ic ie n c ie s o f Graphite P a rtic les 98 v . -a (ta n g ),c o s g . . , Xt = 2 - e j— )s in (a -g ) 4e- a ( t a n 6 ) (cosB)2 c o s(a _2B) + 4 ( cosg)2(.os 2g ' (]26) a a Van de Hulst used the above equation to ca lc u la te values of Xt for tang = 1.0. The re su lts are shown in Figure 22 along with the curve representing the data for the graphite p a r tic le s as shown in Figure 21. In ad dition , Van de Hulst suggested that when the phase angle a is small an even simpler expression fo r the e x tin ctio n e f f ic ie n c y can be used, namely Xt = ^a(tan a) + ^a2( l - t a n 2B) (127) This equation was used to c a lc u la te the curves for tang = 2.0 and tang = 3.0 a lso shown in Figure 22. For the graphite p a r tic le s in carbon tetra ch lo rid e and carbon d i s u lf id e the value o f tang varies from 1.224 to 3.23 with the bulk o f the values f a l l i n g between 1.3 and 1 .5 . The values of tang are shown in Table 6. By the rough comparison shown in Figure 22 i t would appear that the ex tin c tio n e f f i c i e n c i e s measured might be estimated quite s a t i s f a c t o r i l y by the Mie theory applied to absorbing spheres. Un fo r tu n a te ly , a lack of a v a i l a b i l i t y o f rigorous ca lcu la tio n s o f the ex tin ctio n e f f ic ie n c y for spheres with complex r e fr a c tiv e ind ices make more p recise comparisons between experiment and theory d i f f i c u l t . Even though the mass o f data appear to f a l l within the bounds that 99 Graphite Data 4 -> X 0.8 0.4 0.8 2x|n a n, Figure 22. Extinction E ffic ie n c ie s o f Graphite P a r tic le s Compared to Calculated Curves 100 TABLE 6 VALUES OF TAN B FOR GRAPHITE PARTICLES In Carbon Tetrachloride In Carbon D isu lfid e X tan 3 X tan 6 2.0 2.26 2.0 3.23 3.0 1.786 3.0 2.19 4.0 1.551 4.0 1.782 4.5 1.471 4.5 1.655 5.0 1.409 5.5 1.458 7.7 1.224 6.0 1.340 8.0 1.422 8.5 1.382 9.0 1.356 10.0 1.320 10.5 1.306 11.0 1.294 101 might be calcu lated from theory, the data for one s iz e o f graphite p a r t ic le in a p articu lar liq u id does not behave exactly as ca lcu la ted . Based on the v a ria tio n in the values of tang with changes in wave length the values o f would be expected to vary more than was ob served. In f a c t , from theory i t would be expected that values o f Xt would decrease some with increasing wave length rather than remaining e s s e n t i a l l y constant or increasing s l i g h t l y as they a c tu a lly do. One explanation for th is behavior might be th a t the actual r e fr a c t iv e index o f the graphite does not vary with wave length as ca lcu la ted by the equation o f Dalzell and Sarofim. The experimental r e s u lts suggest that the r e fr a c tiv e index of the graphite may be more nearly constant over the wave length range studied. In summary, i t would appear that the ex tin ctio n e f f i c i e n c i e s of randomly oriented irre g u la r ly shaped p a r tic le s can be predicted from Mie Theory for spherical p a r tic le s i f the proper equivalent dimensions are employed. The a b i l i t y to p red ict the behavior o f a wide v a riety of p a r tic le s exposed to collimated radiation may become q u ite important in attempting to co rrela te r e s u lts observed when the same p a r t ic le s are exposed to radiation from a d iffu s e source, for which no theory e x i s t s . B. The D iffuse Transmission A fter having considered the co rrela tio n of the r e s u lts with collim ated incident radiation and before discussing the co r rela tio n of the re su lts with a d if fu s e radiation source, i t would be appropriate to consider some of the p o ssib le errors and problems a sso cia ted with the two s e t s o f data. In both cases the actual measurement was the in te n s ity o f the radiation normal to the plane o f the c e ll window in cluded w ithin a very small s o lid angle. With the sample cell in place and f i l l e d with liq u id , but containing no s o lid p a r t i c l e s , the measured in t e n s ity of the radiation with a collim ated source was, in general, about 10 times that o f the in t e n s it y measured with a d iffu s e source at the same wave length. There was a s ig n i f i c a n t v a ria tio n in the level of in t e n s it y with wave length due to the c h a r a c te r is tic s of the radi ation source, the suspending liq u id , and the c e ll window. These c h a r a c te r is t ic s were discussed previously in the s e c tio n s on ex p eri mental equipment and methods. For example, with the collim ated source and the sample cell containing only liq u id ,th e measured in t e n s ity of the transmitted radiation varied from about 50 per ce n t of the in te n s ity with no sample c e ll at a wave length o f 2 microns down to as low as 15 per cent at the longer wave lengths. However, with the d iffu sin g front window mounted on the c e l l , and with no p a r tic le s in the liq u id , the measured in t e n s it y o f the transmitted rad iation was only about 2 to 3 per cent at a wave length of 2 microns and even lower at higher wave len gth s. With p a r tic le s suspended in th e liq u id the transmitted in te n s ity was o f course s t i l l lower. Having to work with the lower i n t e n s it ie s o f transmission introduced some experimental d i f f i c u l t i e s with the d if fu s e stu d ies which were not s ig n if ic a n t in the collimated measurements. The lower level o f the transmitted energy placed a practical lim it on the upper lim it o f concentration o f so lid p a r t ic le s for which s i g n if ic a n t measurement could be made. This was e s p e c ia lly true when 103 working with the graphite p a r tic le s which exhibited e x tin ctio n c o e f f i c i e n t s o f an order o f magnitude greater than those o f the spherical alumina p a r tic le s when expressed on a volumetric b a sis. For the smaller graphite p a r tic le s (the worst ca se) with as l i t t l e as 0.006 volume per cent s o lid s present in the sample c e ll the transmitted ra d i a tio n was as low as 10 per cent o f the transmitted radiation with no s o l i d s present in the c e l l . For a c e ll volume of 4.45 cubic c e n t i meters th is volume fr a ctio n s o lid s represented only 0 .6 milligrams of graphite. Even though a very accurate a n a ly tica l balance was used (by means of a vernier sca le readable to 0.01 m illigram s), i t was necessary to exercise extreme care in both the drying and weighing procedures in order to obtain reproducible r e s u l t s . It should be evident th a t at higher l e v e ls of transmission the s o lid s concentration would be lower, and the errors introduced in measuring the small weights present would become more important. On the other hand, increasing the amount of s o l i d s present in the c e ll reduced the transmission through the c e ll to the point where the measurements o f the transmission i t s e l f became d i f f i c u l t because of the n oise lev el in the measuring c i r c u i t s . These d i f f i c u l t i e s were considered in attempting to c o r rela te the r e s u lts of the d if fu s e stu d ies. The optimum values of the m u ltiple sca tterin g parameters reported herein were determined by giving equal weight to a ll data. It was recognized that very high values o f transmission (small concentrations o f s o lid s ) and very low values o f transmission were the most su spect. However, determinations o f the s c a t te r in g parameters were also made 104 giving lower weights to the data points which represented e ith e r very low s o lid s concentration or very low transm ission. These r e s u lts were not s i g n i f i c a n t ly d if f e r e n t from those reported. E arlier i t was shown that the equation for radiation from a d i f fuse source through a plane cloud o f p a r tic le s could be expressed in terms of two parameters < |> and B, both properties o f the p a rticu la te cloud. For each s e t o f experimental data the optimum values o f the parameters were determined fo r the two s iz e s of aluminum oxide p a r t i c le s and three s iz e s o f graphite p a r tic le s used in th is study as well as the smaller aluminum oxide p a r tic le s and glass p a r tic le s included in Nagy's work. An a n alysis of these r e s u lts fo llo w s. For purposes of correlation the parameters were considered in turn. Furthermore, the parameter B behaved badly fo r purposes o f c o r rela tio n , so the para meter ip0 which was defined as = V b2 - i ^128^ was used instead. The parameter ij;0 is the value o f ip when R0 = Rl = 0. The values o f both < j > and \pQ are given in Tables 7, 25, 26, and 27. 1. The D iffuse E xtinction Cross Section < j > In the d efining o f the parameter < |> i t was noted that < t > could be related to the total ex tin c tio n cross sectio n at for collim ated incident radiation through the "two-flux" model according to 105 TABLE 7 MULTIPLE SCATTERING PARAMETERS FOR ALUM INUM OXIDE IN CARBON TETRACHLORIDE 3 0 = 1 6 .ly * a=2x ( y nm ) 2.5 1330 1.00 11.29 1.450 3.0 1380 1.25 8.96 1.512 3.5 1350 1.69 7.37 1.477 4.0 1330 1.72 5.86 1.451 4.5 1320 1.54 4.79 1.434 5.0 1200 1.65 3.78 1.314 6.0 945 2.59 1.719 1.033 ^30 ~ 2 0 .2U 2 .0 832 1.81 18.21 1.139 2.5 831 2.19 14.16 1.140 3.0 813 2.72 11.25 1.115 3.5 864 2.77 9.25 1.183 4 .0 782 3.42 7.36 1.070 4 .5 724 3.93 6.01 0.993 5.0 683 4.61 4.75 0.936 6.0 697 4.72 2.16 0.956 106 < f > = A a t (Le/L) (88) This re la tio n sh ip is demonstrated by the values of < } > and a t , as shown in Figure 23. In observing the data as shown, several thoughts occur. F irst, there is a reasonable co rrela tio n between < }> , the "diffuse" ex tin ctio n volumetric cross s e c tio n , and a t , the to ta l ex tin c tio n cross se c tio n . The data include a ll the systems studied here as well as by Nagy (aluminum oxide, graphite, and g la ss p a r tic le s ; carbon tetra ch lo rid e and carbon d is u lf id e liq u id s ) . The re la tio n sh ip then between < } > and as shown in Figure 23 could be a means o f estim ating values o f < p from values o f a t , i f the l a t t e r were known, and no b etter methods were a v a ila b le. An additional point worth noting at th is time is th a t, when compar ing values of c ) > to a t there is no apparent d iffer en ce between the re s u lts for the aluminum oxide p a r tic le s used in th is study and the re s u lts from the aluminum oxide p a r tic le s used by Nagy. In correlatin g the data for the collimated stu d ie s there was an apparent unaccountable d iffer en ce in the r e la t iv e values of a^- reported by the two s tu d ie s . A conclusion which might be drawn from looking at Figure 23 i s that the unaccountable d iffer en ce was also present in the measurements with d if fu s e incident radiation. This seems to be further evidence o f an inherent d iffer en ce in the aluminum oxide p a r tic le s used in the two s tu d ie s. As a r e s u lt o f th is conclusion, only the data from th is study were considered for further c o r rela tio n s. < J > xlO , cm" 107 1000 100 O A1 jO,, d , 0- 1 6 .1 u . CCI, P A1,0| t CS2 A AljO,, d JO*Z0.2u, CCI, A AljO,, d 10-20.2u, CS2 0 Graphite, dp *2.94y, CCI, {} Graphite, dp =2.94u, CS2 O Graphite, dp *5.47y, CCI, p Graphite, dp s5.47u, CS O Graphite, dp =10.3y, CCI,, : p Graphite, dp - 1 0 . 3u, CS 10 i I M M ■ + ; :+; •*+ 1 i 4-i- XXX — X I I NAGY'S DATA (Ref. 19) *+- Glass, d JQ»8.65u, CCI, X A120 i , d J0*6.36u, CCU V AljO), dio*12.2u, CCU A A 1 i0 i, d*o*12.2u, CS2 r LL 10 100 1000 a t xlO , cm Figure 23. E xtinction C o e ffic ie n ts with Collimated and D iffuse Sources The nature o f the parameter < J > ( e s s e n t i a ll y a volumetric cross sec tio n fo r d if fu s e incident rad iation ) suggests for purposes o f co rrela tio n the normalization o f the values of < f > by d ivid ing by the geometrical volumetric cross s e c tio n . That r a tio which we sh a ll refer to as the "diffuse" ex tin ctio n e f f i c i e n c y , and symbolize by is given in Tables 7, 25, 26, and 27. The values of at various wave lengths are also shown in Figures 24 through 27. Inspection o f the figu res shows th at the values of the "diffuse" e x tin ctio n e f f i c i e n c y , X^, vary with both the wave length and the p a r tic le s iz e . The v a r i ation with wave length for the aluminum oxide, however, is not as d r a s tic in appearance as that observed with the values o f X^, the Mie ex tin c tio n e f fic ie n c y (Figures 14 and 15). When the values of Xt were considered, i t was found that the absolute value o f the phase angle a was a useful co rrela tin g para meter. In a sim ila r manner, an attempt was made to represent the values o f X^ as a function o f the phase angle a. Figure 28 shows th is for the aluminum oxide p a r t ic le s . There are several in te r e s tin g features worth noting about Figure 28. F ir s t, employing the phase angle a appears to account for d ifferen ces between suspending liq u id s. In other words, the data for a given p a r tic le s i z e f a l l t o gether regardless o f the suspending liq u id used. Second, there is a varia tio n with p a r t ic le s i z e s t i l l unaccounted for. Third, the values appear to be zero when the p a r tic le s and surroundings have the same r e fr a c tiv e index, as might be expected. In addition, the re la tio n sh ip between X^ and "a" is not symmetrical about the "a = 0" lin e or, 2.0 o d 30 = 16 . ly A d30= 2 0 .2y -e- x Wave Length, Microns Figure 24. "Diffuse" E xtinction E ff ic ie n c ie s of Aluminum Oxide P a r tic le s in Carbon T etrachloride o lO 3.0 -e- x 2.0 Wave Length, Microns Figure 25. "Diffuse" E xtinction E f fic ie n c ie s of Aluminum Oxide P a r tic le s in Carbon D isu lfid e ^ o 5.47- □ dF = 1 0 .3y -e- x 0.8 o-o-o— o 0.4 Wave Length, Microns Figure 26. "Diffuse" E xtinction E ffic ie n c ie s o f Graphite P a r tic le s in Carbon Tetrachloride 2.94y 5.47y 10.3y ■ © - X 0.8 ' 0 - 0 — c , Wave Length, Microns Figure 27. "Diffuse" E xtinction E f fic ie n c ie s of Graphite P a r tic le s in Carbon D isu lfid e O d30= 1 6 .l y , CC1 0 d 3 0 = 1 6 .l y , CS2 A d 30= 2 0 .2y, CC1 A d 30= 2 0 .2y, CS2 3.0 -© • X o—9c-a A — +12 +16 3 = 2x(np - nm) Figure 28. Variation of the "Diffuse" E xtinction E f fic ie n c ie s of Aluminum Oxide P a r tic le s with Phase Angle co 114 in other words, i t is necessary to d istin g u ish between p o s itiv e and negative values of the phase angle a, which wasn't necessary when considering the co rrela tio n o f values o f the Mie ex tin c tio n e f f ic ie n c y , Xj-. In th at ca se (r e fe r to Figure 16) the absolute value of the phase an gle, | a | , worked q uite w e ll. It is not d i f f i c u l t to appreciate how such a d iffer en ce between the collim ated and d if fu s e cases might occur i f one remembers that the e x t in c t io n c h a r a c te r is t ic s of the p a r tic le s are very dependent on the r e f l e c t i v i t y o f the radiation a t the in te r fa c e o f the p a r t ic le and the surrounding liq u id . When a collimated beam perpendicularly s tr ik e s an in te r fa c e between two substances of d if f e r e n t r e fr a c tiv e in d ic e s, the normal r e f l e c t i v i t y depends on the r e la t iv e r e fr a c tiv e index. The value of the normal r e f l e c t i v i t y does not depend on the order o f the two mediums r e la t i v e to the in te r fa c e , s in c e the value o f the re f l e c t i v i t y is th e same no matter which medium is f i r s t . However, when considering th e "diffuse" e x tin c tio n c h a r a c te r is tic s o f the p a r tic le , i t might be expected th a t the hemispherical r e f l e c t i v i t y would be more pertin en t than the normal r e f l e c t i v i t y . For a given in te r fa c e , and a given angular d is tr ib u tio n o f incident ra d ia tio n , the hemispherical r e f l e c t i v i t y w i l l not be the same regardless o f which medium is f i r s t r e la t i v e to the in terfa ce. When a ray o f radiation s tr ik e s the in terfa ce from the sid e o f the medium with the lower r e fr a c tiv e index, p a rtia l r e flecta n c e occurs at a l l angles of incidence. However, when a ray approaches through the medium o f higher r e fr a c tiv e index there i s a c r it i c a l angle (dependent 115 on the r e la t iv e r e fr a c tiv e index) such th at there is to ta l reflecta n ce whenever the angle o f incidence is equal to or greater than the c r i t i cal angle. Furthermore, at any given angle of incidence, other than perpendicular, the r e f l e c t i v i t y w i l l be d if f e r e n t depending on the re f r a c tiv e index o f the medium through which the ray approaches the in te rfa c e. If the product o f the d if fu s e ex tin ctio n e f f ic i e n c y and the p a r t ic le diameter is considered as a function of the phase angle a a co n s is te n t correlation can be found which includes both the aluminum oxide p a r tic le s and the graphite p a r t ic le s . The correlation fo r p o si t i v e values of the phase angle is shown in Figure 29. The curve through the data can be expressed as X^d = 23.3 (1 - e"9 ) 1- 82, a>0 (129) The diameter used in the equation was the spherical diameter fo r the alumina p a r tic le s and the Feret diameter for the graphite p a r t ic le s . The correlation has a standard error which is eq u ivalent to about 28 per cent o f the predicted value o f the e x tin c tio n e ffic ie n c y . For negative values o f the phase angle only a lim ited amount of data were a v a ila b le for c o r r e la tio n , and only fo r values of the phase angle between -3.99 and -7 .1 5 . By assuming a s im ila r form of the equa tio n given for p o s it iv e phase a n g les, a co rrela tio n can be given for the case o f negative phase angles. That co rrela tio n would be X.d = 6 0.3 (1 - e 3 ) 1’ 82, a<0 (130) 10.0 - - o X 1.0 70 3 0.1 1.0 O Aluminum Oxide, d3o=16.1y, CC1*, 0 Aluminum Oxide, d3o=16.1y, CS2 A Aluminum Oxide, d3o=20.2y, CCU 0 Aluminum Oxide, d30=20.2y, CS2 t e , dp =2.94y, CCli, t e , dp =2.94y, CS2 t e , dp =5.47y, CCli, t e , dp =5.47y, CS2 t e , dp =10.3y, CCli, 0 Graphite, dp =10.3y, CS2 0 Graph 0 Graphi □ Graph! a Graph! : O Graph! 10.0 a = 2x(n_ - n_) v p m Figure 29. "Diffuse" Extinction E ffic ie n c ie s for P o s itiv e Phase Angles C T l The equation simply amounts to the same r e su lt as i f the absolute value o f the phase angle were used in the previous equation and the value o f thus obtained then m u ltip lied by a fa c to r o f 2.6. The l a t t e r cor r e la t io n is ce rta in ly tenuous and should be used with caution. Fortu n a te ly , p o s itiv e phase angles predominate in most engineering a p p li cation s . The sig n ific a n c e o f the p a r t ic le s i z e in the correla tio n is not understood at th is time, and any attempt at explanation would be mere sp ecu la tio n . It would also be well to note at th is point that the data for graphite do sc a tte r about the curve shown more than the data fo r aluminum oxide. This is probably due to e ith e r inaccuracies in estim ating the r e fr a c tiv e index for grap h ite, or the influence o f the complex part of the r e fr a c tiv e index. No attempt has been made to consider the l a t t e r at th is p o in t. There is no doubt that considerable additional data for d if fe r e n t p a r tic le s and suspending flu id s would be d e s ir a b le to improve the co rrela tio n . N evertheless, providing that a s u it a b le co rrela tio n can be formed for the parameter ip0 , an empirical method fo r predicting d iffu s e ra d ia tiv e tra n sfer through a plane- p a ra llel cloud w ill have been e sta b lish e d . 2. The M ultiple Scattering Parameter, ip0 It was shown e a r lie r that the parameter \p was given by 118 where "B" was a sc a tte r in g parameter which was conveniently a proper ty only o f the p a r tic u la te cloud. The parameter B, however, does not appear to be a d esira b le one fo r co r rela tio n o f the r e s u lt s . Many of the values o f "B" determined from the data had values very c lo s e to unity. For values of "B" near unity small errors in the value o f "B" lead to large errors in the parameter ip. This d i f f i c u l t y can be overcome by working with another parameter which may be appropriately c a lle d , > 0 ". The parameter ip can be w ritten in terms o f ip0 as Note that when RQ and RL are equal to zero, the value o f w ill be given by the value of ip0 . Earlier, the behavior o f the so lu tio n for d if f u s e transmission through a p lan e-p arallel p a rticu la te cloud was examined with resp ect to changing values of "B". This was done by considering the parameter B as defined by the "two-flux" model. That an alysis can and has been extended to consider the behavior o f the d iffu s e sca tte r in g parameter ip0 . Figure 30 shows values of \p0 calculated for d if fe r e n t values of the albedo, u > 0 , and d if fe r e n t values o f the fra ctio n o f sca ttered radiation sca ttered forward, f . Examination o f the behavior of shows that s u c c e s siv e ly decreasing values of the albedo oo0 , cor responding to increasing absorption, lead to values o f \p0 c lo s e r and c lo s e r to u nity. Furthermore, values o f ip0 approach unity as a lim it (131) 119 10.0 .0 .0 .0 0.8 .0 1.0 0 0.2 0.6 0.8 0.4 f Figure 30. The E ffect o f Forward S ca tterin g on the Parameter 120 as the value of "f" approaches a value o f u n ity , corresponding to a ll sca tterin g in the forward d ir e c tio n . In other words i t might be said that increasing the amount of radiation scattered backward by eith e r larger values o f caQ, or smaller values of f, r e s u lts in larger values of the parameter ^0 . I f i t is d esira b le to in terp ret < J j0 , one could consider i t as a measure of back-scattering. Large values o f ip 0 tend to ind icate that s i g n i fic a n t amounts of the in cident radiation are scattered backward. One should be aware, though, that large values o f \p0 do not n ece ssa rily mean that the fra ctio n o f radiation scattered forward is small. S ig n i fic a n t back-scattering can occur when the absorption is low, even though the majority o f radiation is scattered forward. These observa tion s may be helpful in in terp retin g the behavior o f the parameter ip0 as determined from the data. Before correlatin g the values of iJ j0 a new value of was calcu lated for each s e t o f data using the value of < ) > as predicted from the previously given co r r e la tio n s. Thus, the value o f is the optimum value for each s e t o f data in conjunction with the value of the parameter < } > which would be used for ca lcu la tin g purposes. The values o f ip0 are also given in Tables 7, 25, 26, and 27. A co rrela tio n between the values of < p0 and the absolute values of the d iffer en ces between the r e fr a c tiv e indices of the p a r tic le s and the surrounding liq u id s , |np - nm|, is shown in Figure 31. There is a considerable amount o f s c a tte r in the data. However, the correlation does include the cases with negative phase angles. It was not p o ssib le 10.0 1.0 — O Aluminum Oxide, d30= 16.1y, CCU 0 Aluminum Oxide, d30= 16.1y, CS2 A Aluminum Oxide, d30=20.2y, CC1u — / { Aluminum Oxide, d30= 20.2y, CS2 0 Graphite, dp =2.94y, CC1„ 0 Graphite, dp =2.94y, CS2 □ Graphite, dp =5.47y, CC1^ JLf Graphite, dp =5.47y, CS2 O Graphite, dp =10.3y, CC1„ — 0 Graphite, dp =10.3y, CS2 - r r - 4 - - - / 7 A . P ] A A % y X s a a - A A p o ' 1 ^ 0.1 o - O ^3 £ 0 .4 Figure 31. The Parameter < |j0 as a Function of the D ifference R efractive Indices o f the P a r t ic le s and the Liqu 122 to see any s ig n if ic a n t d ifferen ce between p o sitiv e and negative values o f the phase angle. The correlation is not good, but fortunately the predicted values o f transmission are not extremely s e n s i t i v e to errors in the value o f the parameter ip0 . The e f f e c t s of rather large p o ssib le errors in ip0 are shown in Figures 32 and 33. In the f i r s t of these fig u r e s , the transmission through aluminum oxide p a r t i c l e s , d30 = 20.2 microns, suspended in carbon te tr a c h lo r id e , with A = 3.5y, is shown in com parison with the predicted values using the co rrela tio n s given fo r < t > and The predicted values are shown by the s o lid lin e . The dotted lin e s show the predicted values for values of ip0 on e-h a lf (but not le s s than 1 .0 ) and twice the proper value. Figure 33 is the same so rt of example for graphite p a r t ic le s , dp = 5.47 microns, A = 6y, s u s pended in carbon d is u lfid e . A range o f one-half o f \p0 to twice ip0 is rep resen ta tiv e o f the maximum degree of variation between the curve shown in Figure 31, the co rrela tio n for ^Q, and the individual data on that fig u re. As can be seen in Figures 32 and 33, the d iffer en ces between the actual trans missions and the predicted values could certa in ly be improved with more p recise estim ates of ^0 . However, the correlation presented seems to ler a b le for many engineering c a lc u la tio n s . It could be noted also that a value o f 2.0 for ip0 would be crudely s u ita b le for most of the sets o f data. A physical interp retation of the behavior o f the parameter ip0 seems complex. The value o f \p0 appears to have a minimum when the Transmission, Percentage 123 100 40 20 10 4 2, Volume Fraction Solid s x l 0 “ Figure 32. Predicted D iffuse Transmission for Aluminum Oxide P a r t ic le s in Carbon T etra ch lo rid e, d30=2O.2u, A=3.5u, fo r D iffe r e n t Values of the Parameter ij> 0 Transmission, Percentage 124 100 0 10 20 Volume Fraction Solids xlO5 Figure 33. Predicted D iffu se Transmission for Graphite P a r t ic le s in Carbon D is u lf id e , dp=5.47u, ^=6y, fo r D iffere n t Values of the Parameter ip 0 d iffe r e n c e in the r e fr a c tiv e indices is about 0.7. The increasing values o f for larger d iffer en ces does not seem odd in that in creased r e f l e c t i v i t y o f the p a r tic le s and le s s absorption and greater back sca tte r in g might be expected under those circumstances. However, the apparently larger values o f \p0 for r e fr a c tiv e ind ice d iffer en ces less than 0.7 is not understood. One of the problems with interp reting the data is that the various properties were not varied independently. For example, the larger d iffe r e n c e s in r e fr a c tiv e indices in general correspond to longer wave lengths and smaller values o f the s i z e para meter x. With the amount of data a v a ila b le i t was not p o ssib le to assay the importance o f each of these d iffe r e n t parameters on the values o f The re la tio n sh ip between the value o f and the d iffer en ce in r e fr a c tiv e indices o f the p a r tic le s and liq u id was the best found. It may well be th at the quantity (H E L - 1) should have nm been used for the independent v a ria b le, but the d iffer en ces between the values o f np for the liq u id s were not great and the choice between n D (nn - nm) and - 1) seemed to f a l l to the former fo r the data v nm a v a ila b le. With the in clu sio n o f measurements on other system s, a more complete understanding of the behavior o f the s ca tterin g parameter should be p o ssib le. 126 VII. SUM M ARY Transmission o f infrared radiation through clouds o f spherical aluminum oxide p a r tic le s and clouds of irreg u la rly shaped graphite p a r tic le s were made with both collim ated and d if fu s e incident ra d ia tio n . Prim arily, two d if f e r e n t s iz e s of aluminum oxide p a r tic le s and three d if fe r e n t s iz e s o f graphite p a r tic le s were each suspended a t various concentrations in each o f two liq u id s , carbon te tra c h lo rid e and carbon d i s u l f i d e . The aluminum oxide p a r tic le s had volume average diam eters, d 30, of 16.1 and 20.2 microns. The graphite p a r tic le s were f l a k e - l i k e with average fa c ia l diameters o f 2.9 4 , 5 .4 7 , and 10.3 microns. The average diameter was the Feret diameter in the l a t t e r case. In his work, Nagy had included g la ss spheres, and two s iz e s o f aluminum oxide spheres. In a ll three cases the diameters were l e s s than those for aluminum oxide included in th is study. Nagy measured the transmitted radiation with both collim ated and d if fu s e incident rad iation . However, his work was lim ited to d ie l e c t r i c m aterials and spherical shapes. Furthermore, i t was d i f f i c u l t to draw any general conclusions from the lim ited r e s u l t s , e x p e c ia lly with regard to the transmission o f d iffu s e rad iation . This study attempted to overcome some o f those d e f ic ie n c ie s by including p a r tic le s which were not d i e l e c t r i c , and in addition n o n-sph erical. The experimental techniques developed by Nagy were u t i l i z e d , with some m od ification s. The same Beckman IR-2A 127 Spectrometer, as modified by Nagy, was used, but a new sample c e ll was constructed, both to f a c i l i t a t e sample handling and to provide a th in ner c e l l . The new c e ll provided a cloud thickness of 0.762 centimeters compared to the 1.43 centim eters used by Nagy. Using a thinner sample c e ll permitted in clusion o f greater p a r tic le concentrations. In ad d it io n , when working with carbon tetra ch lo rid e as the suspending liq u id , the plane Servofrax windows were replaced by plane Irtran-3 windows. The Irtran-3 windows had a r e fr a c tiv e index clo se r to that of the liq u id , which was experim entally d esira b le to reduce the amount o f r e f l e c t io n from the c e ll windows. From the data obtained with collim ated incident ra d ia tio n , the Mie e x tin ctio n e f f i c i e n c i e s were determined. The Mie ex tin c tio n e f f i c i e n c ie s obtained for the aluminum oxide p a r tic le s of 16.1 and 20.2 microns diameter included in th is study, correlated well with the absolute value of the phase angle |a| defined by The measured ex tin c tio n e f f i c i e n c i e s exhibited maximum values f a l l i n g in the range o f 3.3 to 4 .0 , and occuring between absolute values o f the phase angle o f 4 and 5. This is reasonable agreement with theory which predicts maximum values of 3.2 to 3.96 fo r r e la t iv e r e fr a c tiv e indices between 1.0 and 1.33. The maximums should occur at a phase angle with a value clo se to 4 .0 . The Mie e x tin c tio n e f f i c i e n c i e s for the f la k e - l i k e graphite p a r tic le s were a lso determined. In doing so , i t was necessary to con 128 sid er the random o rien ta tio n of the p a r t ic le s . It was shown that for f l a t irregular d is k s , randomly orien ted, the average projected geo metrical cross section was on e-h alf the surface area of one face o f the disk. This consideration was used in ca lcu la tin g the geometrical cross s e c tio n . For a "typical" o rien ta tio n the e f f e c t i v e thickness of the p a r tic le in the d ir ectio n o f the incident radiation was twice the actual thickness. For that reason, twice the p a r tic le th ick n ess, instead of the diameter, was used as the p a r tic le dimension in ca lcu la tin g the phase angle. With the geometrical cross sec tio n and phase angle de termined by the approaches ju s t described, the Mie ex tin ctio n e f f i c i e n c ie s fo r the graphite p a r tic le s correlated well with the phase angles. Furthermore, the co r rela tio n between ex tin ctio n e f f ic ie n c y and phase angle appeared to be a reasonable approximation to that fo r spherical p a r t i c l e s . The method of co r rela tin g the d if fu s e r e s u lts was based on an em p ir ica l approach derived from the "two-flux" model fo r d if fu s e ra d i a tiv e tra n sfer through a p lan e-p arallel cloud. The so lu tio n equation for transmission through a plan e-parallel cloud based on the "two-flux" model was rearranged in terms of two m ultiple sca tterin g parameters < f > and % which were both properties o f the p a rticu la te cloud. The im portant s ig n ific a n c e o f the parameters < J > and \p0 is th a t, when de termined em p irica lly , they include any necessary corrections to improve the accuracy of the "two-flux" model. The parameter < f > was shown to have the c h a r a c te r is t ic s o f a "diffuse" e x tin c tio n c o e f f i c i e n t . Furthermore, the experim entally de- 129 termined values o f < J > were found to be c lo s e ly related to the exp eri mentally determined values o f a^., the s in g le sca tte r in g e x tin ctio n cross s e c tio n , for a ll o f the p a rticu la te clouds included in th is study and Nagy's. These findings suggested a d if fu s e e x tin ctio n e f fic ie n c y defined as the ra tio of 4 > to the geometrical cross sec tio n o f the p a r tic le s . Of a ll the attempts to co rrela te the d if fu s e r e s u lts the most successful appeared to be one re la tin g a lumped parameter, con s i s t i n g of the product o f the "diffuse" ex tin c tio n e f f ic ie n c y and the p a r tic le diameter, to the phase angle a. It was necessary to d i s t i n guish between p o s it iv e and negative values o f the phase angle. A re la tio n sh ip between the lumped parameter X^d seemed equally s u ita b le for a ll of the p a r tic le -liq u id systems studied herein. Attempts to correla te the measured values of the parameter were not q u ite as su ccessfu l as for the parameter < (> . The parameter ip0 was shown to be a measure o f the back-scattering properties of the cloud and dependent on experimental measurements of transm issions for low concentrations o f p a r t ic le s . A lack o f a s u f f i c i e n t number of measurements at low concentrations o f s o lid s with some of the systems studied probably contributed to the apparent s c a t te r in the ex p eri mental values o f ip0 . N ever-the-less i t was p o ssib le to r e la te ip0 to the d iffer en ce between the re fra c tiv e indices o f the p a r tic le s and of the surrounding liq u id . Both the co rrela tio n for predicting < J > and the one for predicting \p0 could c e r ta in ly be improved. The importance o f the p a r tic le d i ameter is not completely defined. As presented, the co rrela tio n s may predict doubtful values of the parameters when applied to p a r tic le diameters ou tside the range included w ithin the data used to develop the co r rela tio n s. Data for more system s, e s p e c ia lly including larger and sm aller diameters, other r e la t iv e r e fr a c tiv e in d ic e s, and non- spherical shapes are needed. In s p it e o f the d esire for even b etter correlation s for and 4 > 0 those presented here do represent a s ig n if ic a n t step in the de velopment o f engineering ca lcu la tio n s for d if fu s e ra d ia tiv e transfer through p lan e-p arallel absorb in g-scattering clouds. It is necessary to know the p a r tic le dimensions and the r e fr a c tiv e indices at each wave length for both the cloud p a r tic le s and the continuous phase surround ing the p a r t ic le s . From the dimensions o f the p a r tic le s the geometric cross sec tio n per unit volume of the p a r tic le s can be ca lcu la te d . For non-spherical p a r tic le s the e f f e c t i v e cross s e c tio n , which is dependent on the p a r tic le o r ie n ta tio n , must be determined. Then for each wave length the phase angle a can be determined according to equation 120, and used with the equation a 1 .82 X.d = 23.3(1 - e ) f when the value o f the phase angle is not less than zero, to determine the value of X^, the "diffuse" e x tin c tio n e f f i c i e n c y . For negative values o f the phase angle the value o f X^ may be determined by the equation X^d = 60.3(1 - e 3 ) 1' 82 These values should be used with caution. By m ultiplying the geometric 131 cross sec tio n per unit volume o f the p a r tic le s by the "diffuse" ex tin c tio n e f f ic ie n c y the "diffuse" ex tin ctio n c o e f f i c i e n t < f > is obtained. From Figure 31 a value o f the sca tterin g parameter \p0 can be estimated. The m u ltiple sca tterin g parameters and along with the volume fra ctio n o f p a r tic le s within the cloud are s u f f i c i e n t to determine the transmission through a p lane-parallel cloud of th ick ness L. When the boundaries o f the cloud are n o n -reflectin g and absorption occurs only within the p a r tic le s the transmission is given by Eo (l-i/>0 ) e _< t> vL + (l+^0 )e^vL To include the e f f e c t s o f boundary surface r e fle c t io n s and absorption within the continuous phase equation 131 must be used in conjunction with (l-R 0RL-ij;)e“(f> (v+a ) L + (1 -R0RL+^)e< ()^v+^ L With e ith e r of the above equations the ca lcu la tio n s can be done quickly and e a s ily without an e le c tr o n ic computer. One could not devote so much e f f o r t to a project without uncover ing several questions whose answers might be best completed through additional experimental in v e s tig a tio n . How well the Mie e x tin ctio n e f f i c i e n c i e s can be estimated for a cloud o f p a r tic le s with a wide range of p a r tic le s iz e s is worth some consideration. A l o t more i n 132 formation on the behavior o f irreg u la rly shaped p a r tic le s is needed to determine more p recisely the proper dimensions to use with correlation s and equations based on spherical p a r t i c l e s . P a r tic u la r ly , for the problem o f d if fu s e transfer i t would be d esira b le to obtain additional data for a wider range of p a r tic le s i z e s and a wider range o f physical properties for both the p a r tic le s and the surrounding f lu id . Further more, i t would be highly d esira b le to a ctu a lly measure the r e fr a c tiv e index for the p a r tic le s and flu id used, rather than rely on values r e ported in the lit e r a t u r e . Then the individual importance of geom etri cal dimensions and pertin en t physical properties could be assessed with greater degree of confidence. 133 VIII. REFERENCES 1. Mie, G., Ann. Physik, 2 5 , 377 (1908). 2. Van de H ulst, H.C., Light S catterin g by Small P a r t i c l e s , John Wiley and Sons, I n c ., New York (1957). 3. Kerker, Milton, e d ., Electromagnetic S c a tte r in g , The MacMillan Company, New York (1963). 4. Chandrasekhar, S ., Radiative T ransfer, Dover P u b lica tio n s, In c ., New York, N.Y. (1960). 5. H o t te l, H.C., and Sarofim, A .F ., Radiative T ransfer, McGraw-Hill, I n c ., New York (1967). 6. Sparrow, E.M., and Cess, R.D., Radiation Heat T ransfer, Wadsworth Publishing Company, Belmont, C aliforn ia (1966). 7. Love, T .J ., Radiative Heat Transfer, Charles E. Merrill Publish ing Company, Columbus, Ohio (1968). 8. Marteney, P .J ., NASA Contractor Report NASA CR-211, A p ril, 1965. 9. McAlister, J .A ., Keng, E.Y.H., and Orr, C., J r . , NASA Contractor Report NASA CR-325, November, 1965. 10. Burkig, V.C., NASA Contractor Report NASA CR-811, June, 1967. 11. 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Paper 69-WA/HT-44 presented at the ASM E Winter Annual Meeting, Los Angeles, C a lifo rn ia , November 16-20, 1969. 15. Edwards, R.H., and Bobco, R.P., "Radiant Heat Transfer From Isothermal Dispersions With Isotrop ic Scattering." Paper 67-HT-8 presented at the ASME-AIChE Heat Transfer Conference and E xhibit, S e a t t le , Washington, August 6 -9 , 1967. 16. Glicksman, L.R., Journal o f Heat T ransfer, 9 1 , 502-510 (November, 1969). 17. Emanuel, G., International Journal o f Heat and Mass Transfer, 12, 1327-1331 (1969). 18. Munter, W.A., and Love, T .J ., "Measurement of Radiative Heat Transfer Through Clouds o f Dust." Paper 69-WA/HT-41 pre sented a t the ASM E Winter Annual Meeting, Los Angeles, C a lifo rn ia , November 16-20, 1969. 19. Nagy, A.R., J r . , "Absorption and S catterin g o f Thermal Radiation by a Cloud o f Small P a r tic le s ." Unpublished Ph.D. d is s e r t a t io n , U niversity o f Southern C a lifo rn ia , 1967. 135 20. 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Neuberger, M., "Optical Properties and Thermal Conductivity o f Aluminum Oxide," EPIC Report, No. S-6 (February 1965). 32. D a lz e ll, W.H., and Sarofim, A .F ., Journal o f Heat Transfer, 9 1 , 100-104 (February, 1969). 33. Johnson, J .C ., E ldridge, R.G., and T e r r e ll, J .R ., S c i . Rep. , 4 , M.I.T. Department o f Meterology. APPENDIX A TABLES OF DATA AND RESULTS 136 TABLE 8 TRANSMISSION OF CELL W INDOW S SERVOFRAX ------------------- IRTRAN-3 - - Wave Length, M i crons 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 Measured Transmission, Uncorrected 70% 70 70 71 71 71 70 62 54 57 Transmissivity 0.99 0.99 0.99 1.00 1.00 1.00 0.99 0.88 0.76 0.80 Measured Transmission, Uncorrected 88% 89 91 91 92 92 90 81 60 Transm issivity 0.94 0.94 0.96 0.96 0.97 0.97 0.94 0.84 0.62 CO TABLE 9 CALCULATED NO RM AL AND HEMISPHERICAL REFLECTIVITIES BY LIQUID - CELL W INDOW INTERFACE Carbon D isu lfid e- Carbon T etrachloride- Carbon T etrachloride- Servofrax Servofrax Irtran 3 Wave Length, li crons pn ph pn ph pn ph 2.0 0.203 0.241 0.216 0.256 0.0302 0.0663 2.5 0.202 0.240 0.215 0.255 0.0299 0.0660 3.0 0.201 0.240 0.215 0.254 0.0296 0.0656 3.5 0.201 0.240 0.214 0.253 0.0293 0.0648 4.0 0.201 0.240 0.214 0.253 0.0291 0.0639 4.5 0.202 0.240 0.213 0.253 0.0282 0.0620 5.0 -------- 0.213 0.253 0.0274 0.0600 5.5 0.206 0.247 -------- -------- ---------- --------- 6.0 0.210 0.250 0.213 0.252 0.0258 0.0569 7.7 -------- -------- 0.213 0.253 0.0240 0.0541 8.0 0.191 0.229 -------- ---------- 8.5 0.191 0.229 -------- -------- ---------- --------- 9.0 0.191 0.230 0.214 0.254 0.0208 0.0495 9.5 0.191 0.229 -------- --------- --------- 10.0 0.191 0.229 10.5 0.191 0.229 11.0 0.190 0.229 CO 00 139 Wave Length, Microns 2.0 2.5 3 .0 3.5 4.0 4 .5 5.0 5.5 6.0 7.7 8.0 8 .5 9.0 9 .5 10.0 10.5 11.0 TABLE 10 LIQUID EXTINCTION COEFFICIENTS ------------ a, Carbon D isu lfid e 0 0.00575 0.00575 0.817 0.01806 0.149 0.202 0.193 0.366 0.0207 0.0965 0.0977 0.1808 0.163 0.315 cm- 1 -------------------------------- Carbon Tetrachloride 0 0 0.01385 0.0239 0.0230 0.0469 0.160 0.309 0.245 4.53 TABLE 11 TRANSMISSION OF ALUM INUM OXIDE PARTICLES IN CARBON TETRACHLORIDE W ITH A COLLIMATED SOURCE dso = 16. ly Volume Wave Length, microns Fraction Solids 2 2.5 3 3.5 4 4.5 5 6 7.7 9 0.001551 6.23 5.73 7.09 4.21 3.93 2.46 3.14 9.51 34.6 6.56 0.001134 12.07 12.08 10.87 6.46 6.70 5.18 6.26 17.52 51.8 12.20 0.000531 43.9 38.0 39.3 39.3 26.4 20.5 23.4 42.0 68.5 35.9 0.000395 45.6 47.6 46.8 46.2 34.9 32.4 34.1 53.7 76.5 45.9 0.001163 24.6 24.6 16.04 14.52 13.75 10.35 8.70 14.78 58.5 16.30 0.000647 41.8 39.8 39.8 35.1 32.4 25.6 24.5 35.6 74.4 32.6 0.002400 6.16 6.12 4.98 4.18 3.64 2.96 2.11 4.13 44.4 4.04 0.000665 40.8 40.8 38.4 36.0 33.9 29.6 24.5 32.9 80.1 33.5 TABLE 12 TRANSMISSION OF ALUM INUM OXIDE PARTICLES IN CARBON DISULFIDE W ITH A COLLIMATED SOURCE d3fl = 16.1y Volume Fraction Sol ids 2.0 3 4 4.5 Wave Length 5.5 6 i, microns 8 8.5 9 9.5 10 10.5 11 0.0001095 77.7 77.7 81.0 83.4 89.3 86.5 75.2 74.9 76.4 72.9 73.5 74.8 73.6 0.000314 49.8 41.9 53.2 63.9 73.8 71.0 49.6 45.0 43.1 47.3 43.7 46.7 46.9 0.000504 33.3 28.2 41.7 48.0 64.4 61.1 23.8 26.2 30.1 29.2 26.9 24.0 25.4 0.001054 8.72 7.33 9.51 19.1 33.3 40.3 6.51 6.71 7.06 7.90 6.65 6.28 5.42 0.001201 30.3 10.22 20.2 17.89 42.7 ^30 = 40.4 2 0 .2y 9.16 10.9 11.4 13.9 9.13 12.08 13.68 0.000756 42.6 29.3 30.2 25.0 55.0 60.3 24.3 20.1 20.8 28.6 9.65 19.70 20.5 0.000316 66.6 50.5 59.3 70.0 79.0 86.7 57.8 53.0 50.1 57.0 53.3 49.5 55.6 0.0001945 71.7 65.2 64.2 74.0 80.00 93.3 62.1 65.0 66.6 68.2 65.0 68.3 64.4 TABLE 13 TRANSMISSION OF GRAPHITE PARTICLES IN CARBON TETRACHLORIDE W ITH A COLLIMATED SOURCE Volume Fraction dF = 2.94y Wave Length, microns Sol ids 2 3 4 4 .5 5 0.0000915 5.25 5.33 5.43 5.66 5.79 0.0000388 18.53 19.30 18.92 19.12 20.1 0.00001572 54.2 53.0 53.1 50.2 51.1 0.0000552 11.05 10.46 8.88 9.31 9.07 dF = 5.47y 0.0000935 8.02 8.02 8.62 8.27 8.02 0.0000414 30.6 25.4 29.4 31.3 26.9 0.0000255 51.1 50.5 50.2 50.6 49.7 0.0000545 25.0 23.6 24.0 23.4 22.8 dF = 1 0 .3y 0.0000414 31.5 31.6 31.7 32.4 30.8 0.0000664 20.1 19.21 19.80 17.04 17.10 0.0001119 7.85 8.40 8.91 9.08 6.05 0.0000363 39.5 38.9 40.0 42.3 42.3 7.7 5.28 19.70 52.1 8.31 8.10 27.1 48.8 20.5 28.4 18.20 9.10 39.5 TABLE 14 TRANSMISSION OF GRAPHITE PARTICLES IN CARBON DISULFIDE W ITH A COLLIMATED SOURCE dp = 2.94y Vol ume Fraction Solids 2.0 3 4 Wave length, 4.5 5.5 microns 6 8 8.5 9 10 10.5 11 0.0000376 31.2 31.2 30.1 31.8 26.1 33.0 26.0 30.6 27.6 27.8 28.3 27.3 0.0000391 31.2 31.2 29.3 32.1 26.1 34.0 26.0 32.1 28.7 29.3 29.3 27.3 0.00001925 62.0 62.7 65.1 68.3 59.4 69.0 64.5 65.7 64.5 62.1 61.5 66.4 0.0000218 52.9 53.0 52.7 58.7 50.3 57.1 50.4 53.3 51.7 51.3 51.0 53.7 0.0000766 19.78 19.72 20.1 21.8 dp = 5.47y 16.0 16.67 14.51 16.23 15.0 13.56 14.0 14.35 0.0000336 46.1 46.4 44.4 47.2 43.5 46.0 41.2 42.5 40.9 39.4 41.7 41.0 0.0000191 68.1 66.9 68.2 68.4 62.2 69.4 63.5 66.0 63.8 64.2 62.5 64.0 0.0000371 41.6 40.4 40.3 44.11 38.4 40.6 39.2 39.8 40.1 37.3 37.5 37.4 oo TABLE 14— Continued Volume Wave Length, microns Fraction Sol ids 2.0 3 4 4.5 5.5 dp = 10. 6 ,3y 0.0000306 60.2 58.0 58.6 60.7 55.5 57.5 0.0000342 59.6 59.7 56.0 60.7 56.6 57.5 0.0000684 33.0 33.0 30.0 29.2 28.9 27.6 0.0000962 21.5 20.3 18.22 17.47 16.95 14.6 8 8.5 9 10 10.5 11 55.0 57.4 57.5 57.8 57.0 58.6 55.0 55.8 54.9 57.8 53.7 54.1 28.6 26.8 27.4 26.6 27.0 27.1 16.54 16.0 16.15 16.6 16.43 15.7 TABLE 15 TRANSMISSION OF ALUM INUM OXIDE PARTICLES IN CARBO N TETRACHLORIDE W ITH A COLLIMATED SOURCE q3o “ o. joy Volume Wave Length, microns iction Solids 2 3 4 5 6 0.000365 21.2 16.08 24.8 48.5 70.4 0.000256 37.8 41.1 43.6 65.8 82.0 0.0001132 51.3 54.6 63.5 79.3 89.0 0.0001296 50.2 48.4 59.5 75.8 88.0 0.000425 14.35 13.93 43.7 43.9 64.0 0.000323 22.7 19.75 30.8 52.0 75.1 0.0000397 74.6 73.2 81.4 90.3 95.2 0.0001258 50.1 46.1 75.5 70.4 86.8 0.000736 4.35 3.25 7.34 22.8 47.8 0.000283 10.82 10.0 ____ ____ 59.2 0.0000833 40.6 41.0 ------ ------ 82.7 0.0001182 29.9 26.3 ------ ------ 79.3 0.0001700 29.3 24.3 ------ ------ 75.0 0.000265 9.80 8.16 ____ ____ 60.3 0.0001287 29.2 28.5 ------ ------ 78.8 0.000238 15.6 16.15 ------ ------ 70.3 Remarks Nagy's sample preparation technique used Aluminum c e l l , L=1.429 cm. S ta in le ss s te e l c e l l , L=1.43 cm. (c e ll used by Nagy) TABLE 16 TRANSMISSION OF ALUM INUM OXIDE PARTICLES IN CARBO N TETRACHLORIDE W ITH A DIFFUSE SOURCE d 30 = 16. ly Volume Wave Length, microns Fraction Sol ids 2 2.5 3 3.5 4 4.5 5 6 7.7 0.000871 48.2 39.0 35.8 30.0 30.2 33.2 35.2 35.9 0.000384 84.4 77.8 77.6 72.8 71.8 73.1 74.2 70.8 70.4 0.00251 9.55 8.31 7.31 6.29 6.52 7.04 8.21 10.8 ------ 0.001698 16.5 16.0 14.5 12.9 12.3 14.0 14.3 21.8 23.76 0.000767 52.4 44.9 42.7 35.7 35.9 38.2 37.3 39.8 53.1 ^30 _ 2 0 .2y 0.00412 5.45 5.04 4.58 3.84 4.32 4.76 5.10 5.09 2.30 0.00203 18.4 16.6 15.5 14.6 13.7 13.2 13.2 13.5 14.23 0.001126 37.8 35.5 31.3 28.7 27.3 28.1 28.7 30.0 34.2 0.000563 64.1 61.3 56.1 53.8 54.0 52.6 51.2 48.8 55.2 TABLE 17 TRANSMISSION OF ALUM INUM OXIDE PARTICLES IN CARBO N DISULFIDE W ITH A DIFFUSE SOURCE Volume Fraction Solids 2 3 4 0.001188 20.8 21.8 26.5 0.001024 26.2 24.8 30.7 0.000223 63.3 63.8 72.0 0.000401 51.1 52.9 61.9 0.000159 82.5 84.0 87.4 0.0000975 90.7 93.4 93.0 0.00221 16.3 14.1 18.2 0.000574 48.8 44.6 49.8 0.000295 74.1 74.1 78.3 0.0001445 86.8 82.3 82.5 0.0001253 92.2 88.5 91.5 0.001397 23.2 22.7 27.3 d3o ~ 16. ly Wave Length, microns 4 .5 5.5 6 8 8.5 34.0 44.6 45.0 2.70 5.1 39.5 46.6 48.2 3.60 6.58 75.8 80.2 65.0 25.4 37.0 68.7 73.1 57.0 14.8 29.1 92.9 89.1 78.8 32.4 55.5 98.1 92.8 84.5 41.8 71.2 Q. CO o I I 2 0 .2y 26.7 32.9 26.9 1.17 2.61 58.3 70.7 57.3 13.4 22.8 80.7 85.0 72.9 30.3 53.9 85.0 93.8 89.0 54.3 70.7 91.3 96.5 94.3 68.7 84.8 33.8 45.2 44.0 5.57 9.75 9 9.5 10 10.5 11 3.3 3.8 4 .2 3.6 4.1 3.94 6.84 9.40 6.27 6.83 27.4 31.3 38.2 31.9 38.2 16.6 22.6 28.9 23.4 26.8 35.2 44.0 59.4 51.0 52.5 47.0 57.8 75.3 67.3 71.8 1.74 2.02 1.74 1.70 1.72 14.0 18.6 21.9 18.5 19.9 37.5 43.2 53.5 49.5 53.3 56.7 66.0 69.8 68.1 70.7 77.0 80.4 82.5 77.2 84.3 7.46 8.54 5.79 4.83 5.04 TABLE 18 TRANSMISSION OF GRAPHITE PARTICLES IN CARBON TETRACHLORIDE W ITH A DIFFUSE SOURCE Volume Fraction Solids 2 3 0.0001070 4.31 4.18 0.0000455 21.8 21.2 0.0000224 49.5 47.8 0.0000185 57.6 52.8 0.0000371 31.8 29.2 0.0000467 21.0 19.8 0.0001186 8.67 7.06 0.0000958 12.95 11.3 0.0000587 24.3 21.7 0.0000545 32.5 29.8 0.0000261 55.7 54.8 0.00000740 81.9 79.8 dF = 2.94 \i Wave Length, microns 4 4.5 5 3.71 3.83 3.78 20.0 19.3 19.4 44.2 46.0 46.2 52.4 52.0 52.2 27.0 27.8 27.9 18.5 18.0 17.5 dF = 5.47y 6.84 6.96 6.58 11.05 10.05 10.0 20.2 19.35 18.9 26.7 27.5 27.0 52.7 50.8 52.1 79.0 79.1 74.2 6 7.7 9 3.60 3.77 3.5 18.8 15.8 15.2 45.3 42.3 40.9 51.0 50.9 49.2 26.8 23.3 23.4 17.0 14.3 13.7 5.80 4.75 3.18 9.53 8.25 7.0 18.2 14.95 19.1 25.6 23.0 25.7 50.2 48.8 48.2 78.5 79.2 70.8 - p» 00 TABLE 18--Continued dp = 1 0 .3y Volume Wave Length, microns Fraction Solids 2 3 4 4.5 5 6 7.7 9 0.0001416 8.28 7.21 6.36 6.15 6.25 5.53 4.75 4.35 0.0001052 15.05 15.15 14.8 14.25 13.95 12.55 11.25 9.55 0.0000735 25.8 24.4 21.9 22.1 21 .6 20.5 17.7 19.8 0.0000521 37.0 24.5 33.5 31.6 29.9 30.5 29.2 28.8 0.0000333 53.3 50.0 46.5 48.4 49.1 48.2 47.0 43.0 0.00001235 81.5 80.7 75.9 78.7 76.3 77.0 76.0 71.5 TABLE 19 TRANSMISSION OF GRAPHITE PARTICLES IN CARBON DISULFIDE W ITH A DIFFUSE SOURCE dF = 2.94y Volume Fraction Sol ids 2 3 4 0.0000942 10.8 7.55 4.57 0.0001624 2.84 2.32 2.22 0.0000523 15.0 14.92 14 30 0.0000387 23.2 22.6 2'. .0 0.0000322 27.5 26.9 25.8 0.0000389 23.8 23.4 23.4 0.0000666 18.35 18.13 16.88 0.0000671 22.4 21.4 18.61 0.0001952 5.46 4.81 3.41 0.0001763 5.12 4.73 3.63 0.0001077 13.22 12.05 10.42 0.0000446 23.6 21.0 19.7 0.000226 3.21 2.83 2.19 0.0001426 6.80 5.75 4.97 4.5 5.5 6 8 2.51 2.67 3.76 1.445 1.655 1.054 1.663 13.78 13.35 12.0 14.97 21.4 22.0 20.7 23.4 25.7 27.0 25.5 28.8 22.6 23.6 21.1 25.6 dF = 5.47y 15.88 14.94 13.60 11.78 17.50 15.92 15.08 13.08 2.72 2.25 1.222 1.375 2.81 1.87 1.345 1.715 9.23 8.07 5.93 5.90 18.4 17.55 15.5 14.4 1.97 1.52 1.11 1.088 4.01 3.33 4.44 2.58 8.5 9 ] £ 11 4.60 6.18 11.3 11.7 2.111 2.29 3.03 3.66 14.68 15.76 16.81 15.55 23.7 25.4 26.55 26.0 28.8 30.1 31.9 32.6 25.2 25.4 26.4 26.7 11 .38 10.64 11.13 13.37 12.78 11.35 1.182 2.04 - - 1.506 1.99 - - 4.25 3.67 2.79 14.86 15.0 11. 36 2.12 1.69 _ — cn o TABLE 19--C ontinued dp = 10.3y Volume Fraction Solids 2 3 4 4.5 5.5 6 8 8.5 9 10 11 0.000634 0.641 0.481 0.295 0.231 „ 0.00420 1.975 1.295 0.912 0.820 0.732 ------ 0.941 — - - - - 0.000243 4.60 4.70 3.65 3.79 3.73 1 .94 2.28 — 2.61 — 2.96 0.00Q1242 11.9 10.83 10.97 11.55 10.91 8.87 7.74 — 8.49 8.15 8.88 0.0000638 30.2 29.6 28.9 36.8 27.0 26.1 24.4 — 23.6 24.8 20.4 0.0000268 58.9 58.2 58.1 57.3 52.4 53.7 54.0 — 52.2 50.7 52.2 0.000271 2.78 2.66 2.03 1.583 1.73 1.435 1.795 - - 1.597 1.168 - - TABLE 20 EXTINCTION CROSS SECTION AND EXTINCTION EFFICIENCY O F ALUM INUM OXIDE PARTICLES IN CARBON TETRACHLORIDE d 3 o = 6.36)j X np nm p m 2T0 1.734 1.447 0.287 3.0 1.708 1.442 0.266 4.0 1.673 1.441 0.232 5.0 1.625 1.438 0.187 6.0 1.535 1.433 0.102 X 2x |n - n . | P m , „ 2 1 a^xlO ,cm h . 10.0 5.74 59.7 2.67 6.66 3.55 63.0 2.82 5.00 2.32 44.8 2.00 4.00 1.496 26.3 1.173 3.33 0.580 12.8 0.572 tn no TABLE 21 EXTINCTION CROSS SECTION AND EXTINCTION EFFICIENCY OF _ : OXIDE PARTICLES IN CARBON TETRACHLORIDE d 30 = 20.2m ii i i 2 i A n n | n - n | x 2x|n -n | a xlO“ ,cm" X P m p m p m t t 2.0 1.734 1.447 0.287 32.8 18.8 16.8 2.30 2.5 1.722 1.443 0.279 25.4 14.2 17.1 2.34 3.0 1.708 1.442 0.266 21.8 11.6 19.7 2.70 3.5 1.697 1.442 0.255 18.10 9.25 21.2 2.90 4.0 1.673 1.441 0.232 16.35 7.60 22.2 3.04 4.5 1.653 1.440 0.213 14.54 6.20 25.7 3.52 5.0 1.625 1.438 0.187 11.90 4.45 27.9 3.82 6.0 1.535 1.433 0.102 10.92 2.23 21.5 2.95 7.7 1.472 1.424 0.048 8.25 0.792 5.59 0.765 9.0 1.216 1.404 0.188 7.28 2.74 21.4 2.93 cn co ^ TABLE 22 EXTINCTION CROSS SECTION AND EXTINCTION EFFICIENCY OF ALUM INUM OXIDE PARTICLES IN CARBO N DISULFIDE d3Q = 1 6 .ly X nP nn lnp-nml X 2x|np- n J a^xlO ^cnr1 Xt 2.0 1.734 1.585 0.149 25.3 7.53 29.7 3.24 3.0 1.708 1.577 0.131 16.83 4.41 33.1 3.62 4.0 1.673 1.565 0.108 12.63 2.73 27.0 2.95 4.5 1.653 1.558 0.095 11.22 2.13 20.0 2.19 5.5 1.600 1.522 0.078 9.20 1.435 13.0 1.42 6.0 1.535 1.457 0.078 8.43 1.315 12.5 1.37 8.0 1.344 1.660 0.316 6.32 3.99 34.1 3.73 8.5 1.280 1.645 0.365 5.85 4.27 34.0 3.72 9.0 1.210 1.637 0.427 5.62 4.80 32.9 3.60 9.5 1.150 1.633 0.483 5.32 5.15 32.0 3.50 10.0 1.084 1.628 0.544 5.05 5.50 34.2 3.74 10.5 1.000 1.624 0.624 4.81 6.00 34.7 3.79 11.0 1.000 1.620 0.620 4.60 5.70 35.4 3.87 cn 4^ TABLE 22— Continued d 3 o= 2 0. 2p X A m h „ _rU 1 p m' X 2x|n - n l 1 p nr at x l0 “ ,cnf Xt 2.0 1.734 1.585 0.149 32.8 9.78 14.82 2.03 3.0 1.708 1.577 0.131 21.8 5.71 24.6 3.37 4.0 1.673 1.565 0.108 16.35 3.53 20.0 2.74 4.5 1.653 1.558 0.095 14.54 2.76 18.23 2.50 5.5 1.600 1 .522 0.078 11.90 1 .855 10.15 1.39 6.0 1.535 1.457 0.078 10.92 1.702 8.64 1.18 8.0 1.344 1.660 0.316 8.20 5.18 25.7 3.52 8.5 1.280 1.645 0.365 7.71 5.63 26.0 3.56 9.0 1.210 1.637 0.427 7.28 6.21 25.7 3.52 9.5 1.150 1.633 0.483 7.10 6.85 22.2 3.04 10.0 1.084 1.628 0.544 6.55 7.12 27.0 3.70 10.5 1.000 1.624 0.624 6.24 7.78 25.6 3.50 11.0 1.000 1.620 0.620 5.95 7.38 24.5 3.36 (XI cn TABLE 23 EXTINCTION CROSS SECTION AND EXTINCTION EFFICIENCY OF GRAPHITE PARTICLES IN CARBON TETRACHLORIDE dF 5.47y A nm m In -n l p m X 2x|n -n | p m i* a t xlO , cm-1 X t 2.0 1 . 9 0 6 - i l .037 1.447 0.459 0.477 0.438 3.52 0.853 3.0 2.167— i 1.295 1.442 0.725 0.318 0.462 3.68 0.893 4.0 2 . 3 9 9 - i I .486 1.441 0.958 0.248 0.456 3.54 0.859 4.5 2 . 5 0 3 - i I .564 1.440 1.063 0.212 0.452 3.53 0.856 5.0 2 .5 9 8 -i1.635 1.438 1.160 0.1910 0.443 3.67 0.890 7.7 3 . 0 1 0 - i I .942 1.424 1.586 0.1240 0.393 3.74 0.883 dF 10.3m 2.0 1 . 9 0 6 - i l .037 1.447 0.459 0.900 0.827 3.20 1.422 3.0 2 . 1 6 7 - i l .295 1.442 0.725 0.600 0.873 3.19 1.418 4.0 2 . 3 9 9 - i l .486 1.441 0.958 0.450 0.864 3.13 1.390 4.5 2 . 5 0 3 - i l .564 1.440 1.063 0.400 0.851 3.16 1.404 5.0 2 . 5 9 8 - i I . 635 1.438 1.160 0.360 0.834 3.39 1.506 7.7 3 .0 1 0 - i1.942 1.424 1.586 0.234 0.742 3.22 1.430 c n TABLE 24 EXTINCTION CROSS SECTION AND EXTINCTION EFFICIENCY OF GRAPHITE PARTICLES IN CARBON DISULFIDE dp = 2.94y A n' n |n -n 1 X 2x|n -n | k ot xlO“ , cm-1 X P m p m p m t 2.0 1 .9 0 6 -i1.037 1.585 0.321 0.256 0.1646 3.84 0.500 3.0 2.167-11.295 1.577 0.590 0.1710 0.202 3.82 0.498 4.0 2 . 3 9 9 - i l .486 1.565 0.834 0.1281 0.215 3.90 0.508 4.5 2 . 5 0 3 - i l .564 1.558 0.945 0.1138 0.215 3.56 0.464 5.5 2 .6 8 7 - i1.699 1.522 1.165 0.0933 0.217 4.34 0.565 6.0 2 .7 7 0 - i1.760 1.457 1 .313 0.0855 0.224 3.48 0.454 8.0 3 . 0 4 7 - i l .972 1.660 1.387 0.0641 0.1785 4.26 0.555 8.5 3 .1 07-i 2.021 1.645 1.462 0.0603 0.1768 3.76 0.490 9.0 3.163-1*2.070 1.637 1.526 0.0570 0.1740 4.05 0.528 10.0 3 . 268-i 2.164 1.628 1.640 0.0513 0.1685 4.07 0.530 10.5 3.317-1*2.211 1.624 1.693 0.0489 0.1659 4.07 0.530 11.0 3.364-1*2.257 1.620 1.744 0.0466 0.1629 4.04 0.527 < J 1 TABLE 24— Continued dp = 5.47y A " p nm 1 np-nm1 X 2x|np-nmj 0-j-xlO“^cm-1 h 2.0 1 .9 0 6 - i1.037 1.585 0.321 0.477 0.306 2.88 0.700 3.0 2 . 1 6 7 - i l .295 1.577 0.590 0.318 0.375 2.92 0.708 4 .0 2 . 3 9 9 - i l .486 1.565 0.834 0.248 0.398 2.92 0.708 4.5 2.503-11.564 1.558 0.945 0.212 0.402 2.74 0.665 5.5 2.687-11.699 1.522 1.165 0.1733 0.404 3.23 0.784 6.0 2 .7 7 0 -i 1.760 1.457 1.313 0.1590 0.417 3.02 0.733 8.0 3 . 0 4 7 - i l .972 1.660 1.387 0.1192 0.331 3.32 0.805 8.5 3.107-12.021 1.645 1.462 0.1122 0.324 3.16 0.767 9.0 3.163-12.070 1.637 1 .526 0.1060 0.354 3.28 0.795 10.0 3.268-12.164 1.628 1.640 0.0955 0.313 3.44 0.835 10.5 3.317-12.211 1.624 1.693 0.0909 0.308 3.38 0.820 11.0 3 .364-i 2.257 1.620 1.744 0.0866 0.303 3.36 0.815 cn oo TABLE 24— Continued dp = 1 0 .3y X M 1 P nm lnp“nm 2.0 1 .9 0 6 -il.0 3 7 1.585 0.321 3.0 2 . 1 6 7 - i l .295 1.577 0.590 4.0 2 . 3 9 9 - i l .486 1.565 0.834 4.5 2 . 5 0 3 - i l .564 1.558 0.945 5.5 2.687-11.699 1.522 1.165 6.0 2.770-11.760 1.457 1.313 8.0 3 . 0 4 7 - i l .972 1.660 1.387 8.5 3 .1 07-i 2.021 1.645 1.462 9.0 3.163 -i2 .0 7 0 1.637 1.526 10.0 3 .2 6 8 -i 2.164 1.628 1.640 10.5 3.317-1*2.211 1.624 1.693 11.0 3.364-1*2.257 1.620 1.744 X 2x|np-nm| C T t xlO"^cm-1 xt 0.900 0.578 2.10 0.935 0.600 0.707 2.15 0.955 0.450 0.752 2.30 1.022 0.400 0.757 2.27 1.009 0.327 0.762 2.39 1.063 0.300 0.789 2.47 1.098 0.225 0.624 2.43 1.080 0.212 0.620 2.45 1 .089 0.200 0.612 2.44 1 .085 0.1800 0.609 2.41 1 .072 0.1713 0.580 2.46 1.092 0.1638 0.573 2.46 1.092 CJ1 K £ > 160 TABLE 25 MULTIPLE SCATTERING PARAMETERS OF ALUM INUM OXIDE IN CARBON DISULFIDE d 3 o = 16. ly X < f > Yo a=2x(np-nm) x* 2.0 1340 2.41 7.53 1.464 3.0 1420 2.46 4.42 1.546 4.0 1400 2.05 2.73 1.528 4 .5 1110 1.73 2.13 1.218 5.5 602 1.85 1.434 0.660 8.0 2920 4.40 -3.99 3.19 8.5 2740 1.61 -4 .3 4 3.00 9.0 2620 3.69 -4.80 2.87 9.5 2560 2.25 -5 .1 4 2.80 10.0 2880 1.30 -5.50 3.15 10.5 2940 2.14 -6.01 3.21 11.0 3030 1.71 -5.70 3.31 d 30 = 2 0 .2U 2.0 783 2.93 9.46 1 .173 3.0 854 3.42 5.54 1.281 4 .0 699 2.77 3.43 1.047 5.5 563 1.78 1.800 0.844 8.0 2210 4.08 -5.01 3.32 8 .5 1880 1.59 -5.45 2.82 9.0 1884 3.09 -6 .0 2 2.82 9.5 1940 2.27 -6 .4 5 2.64 10.0 2280 2.44 -6 .9 0 3.42 10.5 2230 2.86 -7.54 3.33 11.0 2350 2.48 -7 .1 5 3.52 161 TABLE 26 MULTIPLE SCATTERING PARAMETERS OF GRAPHITE IN CARBON TETRACHLORIDE d = 2.94y F' <0x10-2 ipg a=2x(np-nm) dF = 5.47y 2.0 371 1.00+ 0.236 0.482 3.0 363 1.00+ 0.248 0.473 4 .0 372 1.04 0.246 0.484 4.5 370 1.02 0.242 0.481 5.0 372 1.19 0.238 0.486 6.0 378 2.65 0.228 0.494 7.7 351 2.63 0.212 0.458 9.0 385 0.200 0.503 2.0 254 1.00+ 0.439 0.615 3.0 279 1.00+ 0.462 0.677 4.0 269 1.00+ 0.458 0.653 4.5 269 1.00+ 0.450 0.651 5.0 275 1.05 0.443 0.666 6.0 295 2.42 0.424 0.714 7.7 315 2.56 0.394 0.745 9.0 369 0.372 0.894 dF = 1 0 .3y 2.0 223 1.58 0.827 0.991 3.0 228 1.53 0.871 1.014 4.0 231 1.81 0.864 1.027 4 .5 239 1.96 0.848 1.063 5.0 233 2.19 0.836 1.036 6.0 253 3.74 0.801 1.123 7.7 268 3.70 0.743 1.190 9.0 282 0.702 1.252 TABLE 27 162 MULTIPLE SCATTERING PARAMETERS OF GRAPHITE IN CARBON DISULFIDE 2.0 3.0 4.0 4.5 5.5 6.0 8.0 8.5 9.0 10.0 11.0 <J>xl0‘ 192 221 230 295 281 308 288 250 243 207 183 dp = 2.94y 4.63 2.33 2.09 2.75 2.54 2.13 5.52 4.68 4.40 3.89 4.43 a= 2x(V nm) 0.1648 0.202 0.214 0.216 0.218 0.225 0.1781 0.1767 0.1741 0.1684 0.1628 A< j ) 0.250 0.287 0.300 0.385 0.366 0.402 0.375 0.325 0.316 0.270 0.239 dF = 5 . 47y 2.0 3.0 4.0 4.5 5.5 6.0 8.0 9.0 10.0 11.0 126 132 153 167 188 216 200 231 181 321 3.16 1.46 2.87 1.13 1.71 1.87 7.97 9.58 10.4 13.8 0.307 0.376 0.398 0.401 0.405 0.418 0.331 0.324 0.313 0.303 0.306 0.319 0.371 0.405 0.456 0.524 0.484 0.560 0.439 0.778 dF = 1 0 .3P 2.C 3.0 4.0 4.5 5.5 6.0 8.0 9.0 10.0 11.0 74.0 83.9 97.5 108 132 182 119 161 191 146 2.19 1. 00+ 1.04 1 . 00+ 1.16 2.64 5.06 6.24 7.78 7.26 0.568 0.709 0.752 0.757 0.764 0.789 0.625 0.611 0.591 0.572 0.329 0.373 0.433 0.481 0.586 0.811 0.531 0.716 0.848 0.647 APPENDIX B FORTRAN PROGRAM FOR DETERMINING < f > AND ^ 163 164 EVALUATION OF SCATTERING PARAM ETERS DIMENSION V (20), TR(20), RS(4), DR(4,4), RSN(4), DX(4), AX(10,11), 1 RSM(4), WTF(20), TITL(15) C O M M O N X,T,ALP 1 READ 27, (TITL(I),1=1,12) CALL EOFTST ( 5 0 ,JT) G O TO (2 ,2 6 ), JT 2 CONTINUE PRINT 28, (TITL( I ),1=1,12) READ 29, W,RZG,RLG,ALP,T,X PRINT 30 CC=1.01 CCC=1.01 CG=0.9*X DG=1.1 DD=0.7 NV=2 M V=NV+1 LV=NV+2 NTRL=0 NM=0 NZ=0 READ 31, N,(V(K),TR(K),K=1 ,N) DO 3 K=1 ,N 3 WTF(K)=1.0 C=CG D=DG RZ=RZG RL=RLG 4 DO 6 1=1,NV RS( I)=0.0 D O 5 K=1 ,N VF=V(K) L=0 CALL FLX (C,D,RZ,RL,VF,L,P) C0M=P/TR(K)*100. L=I CALL FLX (C,D,RZ,RL,VF,L,P) 5 RS(I)=RS(I)+P*ALOG( COM)*WTF( K) 6 RSM(I)=0.0-RS( I ) DO 13 J=1,NV CN=C DN=D RZN=RZ RLN=RL IF ( J - l) 8 ,7 ,8 7 CN=CCC*C DX(J)=CN-C 165 G O TO 11 8 IF (J - 2 ) 1 0 ,9 ,10 9 DN=CCC*D DX(J)=DN-D G O TO 11 10 CONTINUE 11 DO 13 1=1,NV RSN(I)=0.0 DO 12 K=1,N VF=V(K) L=0 CALL FLX (CN,DN,RZN,RLN,VF,L,P) C0M=P/TR(K)*100. L=I CALL FLX (CN,DN,RZN,RLN,VF,L,P) 12 RSN(I) =RSN(I)+P*ALOG(COM)*WTF( K) DR( I , J) = (RSN(I)-RS(I))/DX(J) 13 AX( I,J)=DR(I,J) DN=D RLN=RL RZN=RZ DO 14 1=1,NV 14 AX(I,MV)=RSM(I) DO 15 I =M V ,10 D O 15 J=LV,11 15 AX(I,J)=0.0 M =NV CALL SOLVGJ (AX,M) ZF=1.0 NTRL=NTRL+1 IF (NTRL-50) 16,16,25 16 C=CN+DD*AX(1,MV)*ZF IF (C) 24,24,17 17 CONTINUE D=DN+DD*AX(2 ,MV)*ZF IF (D-0.99) 24,24,18 18 CONTINUE IF (ZF-0.999) 4,19,19 19 CONTINUE TST=ABS((C-CN)/CN) IF (TST-0.01) 20,20,4 20 TST=ABS((D-DN)/DN) IF (TST-0.01) 21,21,4 21 CONTINUE PRINT 32, W,T,C,D,ALP,RL,NTRL 22 PRINT 33 DO 23 K=1,N VF=V(K) L=0 166 CALL FLX (C,D,RZ,RL,VF,L,P) PP=P*100. ER=TR(K)— PP ERR=TR(K)/PP 23 PRINT 34, TR(K),PP,ERR,WTF(K),ER CFA=D*D-1.0 CFC=D*D+RL*RL BETA=-RL/CFA+SQRT(RL*RL+CFA*CFC)/CFA BSQ=BETA*BETA BSQM=BSQ-1.0 SQBSQ=SQRT(BSQM) C0X=C/X PRINT 35, COX,BETA,BSQ,BSQM,SQBSQ G O TO 1 24 ZF=ZF*DD NZ=NZ+1 IF (NZ-60) 16,16,25 D=(C+D)/2. C=1.1*D NM =NM +1 IF ( NM-25) 4 ,4 ,2 5 25 PRINT 32, W,T,C,D,ALP,RL,NTRL G O TO 22 26 CALL EXIT 27 FORM AT (12A6) 28 FORM AT (1 HI ,12A 6///) 29 FORM AT (6E12.6) 30 FORM AT (1H0,3X,11HWAVE LENGTH,5X.9HCELL THK.,8X,3HPHI,13X,3HPSI,14 1X,2HAL,15X,2HRL,12X,6HTRIALS,//) 31 FORM AT (1 2 ,2X,6E12.6/(4X,6El2 .6 )) 32 FORM AT (1 HO, 6 ( 2X, FI3 . 4 ) ,5X, 1 6 / / / / / ) 33 FORM AT (1H0,20X,7HTRANSM.,1 2X,9HPREDICTED,13X,5HRATI0,13X,1OHW T. F 1 ACTOR,//) 34 FORM AT (1H0,10X,5(10X,F10.6)) 35 FORM AT (1 HO,///lOX,5(5X, FI 5 . 1 0 ) / / ) END SUBROUTINE FLX (PHI,XI,RZ,RL,VF,K,P) SUBROUTINE FLX (PHI,XI,RZ,RL,VF,K,P) C O M M O N X,T,ALP XK=( VF+ALP/X)*T G=PHI*XK H=ALP*T EXH=EXP(H) EXG=EXP(G) A=1.+XI B=1.-XI DEN=A*EXG+B/EXG IF (K) 2 ,1 ,2 1 P=2.*EXH/DEN RETURN 2 IF ( K -l) 4 ,3 ,4 3 P=XK*(B/EXG-A*EXG)/DEN RETURN 4 IF (K-2) 6 ,5 ,6 5 P=(l./EXG-EXG)/DEN RETURN 6 PRINT 7 RETURN 7 FORM AT (9X.31H ERROR UN FLX CALLING STATEMENT) END SUBROUTINE SOLVGJ (A,N) SUBROUTINE SOLVGJ (A,N) DIMENSION A (1 0 ,l1 ), KS(IO), B(10), C (ll) D O 1 1=1.10 1 KS ( I ) = I K =1 M =N+1 2 J=K+1 T=A(K,K) IF (T) 3,11,3 3 A(K,J)=A(K,J)/A(K,K) J=J+1 IF (J-M) 3 ,3 ,4 4 1=1 5 IF ( I-K) 6 ,8 ,6 6 J=K+1 7 A(I,J)=A (I,J)-A(I,K )*A(K ,J) J=J+1 IF (J-M) 7 ,7 ,8 8 1= 1+1 IF ( I-N) 5 ,5 ,9 9 K=K+1 IF (K-N) 2 ,2 ,1 0 10 RETURN 11 L=K+1 DO 12 IA=L,N T=A(IA,K) IF (T) 13,12,13 12 CONTINUE 13 DO 14 JA=1 ,M C(JA)=A(K,JA) 168 A(K,JA)=A(IA,JA) 14 A(IA,JA)=C(JA) G O TO 3 END

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Creator Sanders, Charles Franklin, Jr. (author)

Core Title Radiative Transfer Through Plane-Parallel Clouds Of Small Particles

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Lenoir, John M. (committee chair), Gerstein, Melvin (committee member), Rebert, Charles J. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-450544

Unique identifier UC11362390

Identifier 7107738.pdf (filename),usctheses-c18-450544 (legacy record id)

Legacy Identifier 7107738

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Document Type Dissertation

Rights Sanders, Charles Franklin, Jr.

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

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