ec-' J kah C, I 3( - THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING RADIANT HEAT TRANSFER IN PACKED MEDIA John Chun-Chien Chen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1960 February, 1961 IP-497

57 I I - vYn IVA u a

To my parents ii

ACKNOWLEDGEMENTS The author wishes to express his appreciation to the members of his committee for their guidance during the course of this work. Special thanks are due to Professor Stuart W. Churchill, chairman of the committee, for his initial suggestion of the topic and for his continued encouragement and aid during all phases of the work. The author is indebted to the Research Center of the OwensCorning Corporation for generous loan of experimental equipment and to the Monsanto Chemical Company for financial aid through the granting of a fellowship. Thanks are also due to the Industry Program of the College of Engineering for reproduction of the final thesis. The author wishes to thank his laboratory mates, Doctors J. A. Leacock, Go D. Towell, J. D. Hellums, and Mr. J. Hiestand for their many helpful suggestions. Finally, the author wishes to thank his wife for her constant encouragement throughout the period of the work and for her aid in preparation of the final manuscript. iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS....................*.......... iii LIST OF TABLES.................................... vii LIST OF FIGURES......................................viii ABSTRACT......4......4.......................................... 4xi PART I. INTRODUCTION...*.................... 1 PURPOSE AND SCOPE OF STUDY..*............................. 2 Purpose.....**.*......*.4**.....*.....4..... 2 Scope.............................................. 4 LITERATURE REVIEW....*.................. 7 Analytical Studies *..................................... 7 Experimental Studies........................... 14 PART II. ANALYTICAL DEVELOPMENT............................*... 16 GENERAL ONE-DIMENSIONAL CASE..................... 17 Radiant Energy Transfer.................. 17 Simultaneous Conductive and Radiative Transfer.............. 25 Dimensional Analysis...................................... 27 Approximate Solutions.................................. 30 Radiant Conductivity for the Media......................... 36 EXPERIMENTAL CASE..4......................... 38 Moduilated Transmission Equations *..,.................. 38 PART III. EXPERIMENTAL INVESTIGATION.*.......................... 41 DESCRIPTION............................................. 42 Background........42 Method............................................... 42 Apparatus..................46 Test Particles..................................... 55 PROCEDURES........................... 58 Measurement of Bed Height.................................58 Measurement of Transmission Flux.................... 59 Measurement of Incident Flux........................... 60 Procedure of Test Runs................................. 61 iv

TABLE OF CONTENTS (cont'd) Page EQUIPMENT CHECKS AND CALIBRATIONS....................... 63 Black Body Behavior of Source......................... 63 Linearity of Measurement System............................ 64 Temperature Response of Measurement System 6............... 66 Effect of Support Screen............................. 68 Wall Absorption........................................... 70 Lateral Variations.............................. 72 Calibration of Standard Screen........................ 75 RESULTS AND DISCUSSION.................. 81 Transmission Curves................................. 81 Attenuation Cross Sections.............................. 95 Radiant Conductivities.....................................98 Relative Importance of Radiation and Conduction.....99 Comparison with Analytic Estimates....................... 104 Effect of Temperature..................................... Effect of Packing Properties......................... 125 Correlation of Attenuation Parameters...................... 134 Experimental Precision................................. 138 PART IV. SUMMARY AND CONCLUSIONS.............................. 141 APPENDIX............................................... 146 DATA REGRESSION EQUATIONS.................................. 147 SAMPLE CALCULATIONS............................ 154 Regression for Transmission Curves...................... 154 Radiation to Conduction Ratios..............157 Analytic Estimates......................................... 158 Estimation of Conductivity from Correlation................159 EXPERIMENTAL DATA............................................. 161 Test Run Data.............................................. 161 Temperature Response Data........................... 181 Linearity Check Data...................................... 182 Screen Ratio Calibration Data.............................. 183 Screen Temperature Calibration Data....................... 86 TRANSMISSION VALUES FROM REGRESSION..................87 PROPERTIES OF TEST PACKINGS................................ 196 COMPUTER PROGRAMS..................................... 198 v

TABLE OF CONTENTS (cont'd) Page NOMENCLATURE........................... *** 205 REFERENCES.................. *. 209 vi

LIST OF TABLES Table Page I Test Particles......... *........,........o 55 II Attenuation Cross Sections...............,.....9*. 6. 96 III Radiant Conductivities.,..,.,.................. 98 IV Radiation to Conduction Ratios...........,....... 101 V Temperature Dependence of kir............,,...... 124 VI Test Run Data.................................,. 161 VII Temperature Response Data..,....,....o,,...so.o, oao... 181 VIII Linearity Check Data..1o.,..,.................,.... 182 IX Screen Ratio Calibration Data....,......,.,,.,.,.o. 183 X Screen Temperature Calibration Data......,.,...o...... 186 XI Transmission Values from Regression 7................. l 187 XII Properties of Test Packings...,,.o..,,,.......o., 197 vii

LIST OF FIGURES Figure 1 2 5 4 5 6 Geometry of Analysis Case.............................. Analysis Models......................................... Illustration of Radiant Intensity...................... Traverse of Flux through Elemental Volume.............. Cartesian Coordinate System............................ Flux Impinging on Detector........................... 7 Diagram of Apparatus.................................... 8 Photograph of Equipment.................. 9 Details of Source...................... 10 Details of Bed Assembly................. 11 Relative Positions of Apparatus.... 12 Photograph of Test Particles............. 13 System Linearity Check.................. 14 Temperature Response of System........... 15 Effect of Support Screen................. 16 Effect of Tube Wall Length.......... 17 Placement of Detector in Lateral Variation eleeeaeeeeeeloe * 0 * a 0 0 0 a a 0 0 a eeeeeeeeeeleeee eeeeeeeeeeeeeee ellleleeeeesee~ eeeeeaeeeeeeeee eeeeeeeeeeeeeee eeeeeeeeeeeeeee eeeeleeeeeeeeee Test......... Page 5 10 20 20 20 44 47 48 49 52 54 57 65 67 69 71 73 74 77 78 79 82 83 84 85 18 19 20 21 22 25 24 25 Effect of Lateral Variation...................... Position and Source-Area Dependence of Fs............... Temperature Dependence of Fs.......................... Temperature Calibration for St.................. Transmission Curves for GS-3......................... Transmission Curves for GS-4......................... Transmission Curves for GS-5........................ Transmission Curves for AS-3/16........................ viii

LIST OF FIGURES (cont'd) Figure 26 27 28 29 30 31 32 33 34 35 Transmission Transmission Transmission Transmission Transmission Transmission Transmission Transmission Radiation to Curves for Curves for Curves for Curves for Curves for Curves for Curves for Curves for Conduction AP-1/8...................... AP-5/32......................... AP-3/16........................ ss-1/8......................... SS-3/16...................... AG-4............................ AG-16........................... CG-16.......................... Ratio........................... Ratio........................... Page 86 87 88 89 90 91 92 93 102 103 106 107 108 Radiation to Conduction 36 Comparison with Analytic Estimates for AP-1/8........... 37 Comparison with Analytic Estimates for SS-3/16........ 38 Comparison with Analytic Estimates for GS-3..5....... 39 Temperature Effect for GS-35........................ 112 40 Temperature Effect 41 Temperature Effect 42 Temperature Effect 43 Temperature Effect 44 Temperature Effect 45 Temperature Effect 46 Temperature Effect 47 Temperature Effect 48 Temperature Effect 49 Temperature Effect for for for for for for for for for for GS-4 e e e e e e e....... * *........... GS-5................... AS-3/16.................. AP-1/8........................ AP-5/32......................... AP-3/16......................... SS-1/8........................... SS-3/16.......................... AG-4............................. AG-16 i...................*....... ix 113 114 115 116 117 118 119 120 121 122

LIST OF FIGURES (cont'd) Figure Page 50 Temperature Effect for CG-16...2....................... 123 51 Effect of Void Fraction on Scattering Cross Section.... 126 52 Effect of Void Fraction on Absorption Cross Section.. 127 53 Effect of Emissivity............................. 129 54 Effect of Particle Size on Scattering Cross Section.... 130 55 Effect of Particle Size on Absorption Cross Section... 131 56 Effect of Particle Size on kir, Parameters of Particle Material............................................ 132 57 Effect of Particle Size on kir, Parameters of Temperature...3.................................... 1. 58 Correlation of Scattering Parameter................ 136 59 Correlation of Absorption Parameter............... 137 x

ABSTRACT This thesis presents the results of an experimental investigation of radiant heat transfer in packed media. Steady state, one dimensional, simultaneous, radiative and conductive heat transfer in quiescent packed beds was analyzed theoretically. If a two flux model is used to represent absorption and scattering, radiation and conduction can be represented by a set of nonlinear differential equations. In this model the radiant properties of the media are specified in terms of an absorption cross section and a back scattering cross section per unit volume. With this representation, a minimum of three dimensionless dependent variables, one dimensionless independent variable, and six dimensionless parameters are necessary to define the system. The experimental program was designed to obtain measurements of the two attenuation cross sections for a number of typical packings. This was accomplished by making transmission measurements of modulated blackbody radiation and correlating with the two-flux model by least square regression. Twelve packings were tested, covering four types of materials (glass, aluminum oxide, steel, and silicon carbide), three types of shapes (spheres, cylinders, and irregular grains), and a range of sizes (3 mm., 4mm., 5mm., 1/8 inch, 5/32 inch, 3/16 inch, mesh 4, and mesh 16). Measurements were made at several source temperatures in the range of 800~F to 2000~F. Results showed that back scattering is the major mechanism of attenuation for packings of glass, aluminum oxide, and steel. The measured absorption cross sections for these packings are all less than seven percent xi

of the value of the corresponding back scattering cross sections. Absorption was found to be important only for packings of silicon carbide, for which the absorption cross sections may be as high as 60 percent of the value of the back scattering cross sections. In general, it was found that back scattering cross sections decrease with increasing particle size and solid emmissivity. The absorption cross sections were found to decrease with decreasing solid emissivity and with increasing particle size. Aluminum oxide cylinders are exceptions in that the absorption cross sections were found to increase with increasing particle size. Two graphical correlations of absorption and back scattering parameters are presented to enable interpolation and extrapolation of the experimental data for first-order estimates of the parameters for other packings and other operating conditions. Equivalent radiant conductivities calculated from the measured cross sections were found to be substantially different from a priori estimates calculated from various theoretical models which have been proposed previously. A possible explanation is that these models do not account correctly for scattering and solid-transmission effects. The radiant conductivities were found to increase with temperature of radiation, solid transmissivity and emissivity, and particle size. The temperature dependence was found to be quite different from the third power dependence usually assumed. In the temperature range of 1600~F to 2000~F, the exponent of temperature ranged from a low of 1.6 to a high of 5.6 in the various packings tested. Radiation was found to become important, relative to conduction, at temperatures of approximately 1000~F and to become predominant at temperatures of about 2000~F. For example, at 2000~F, radiation was found to xii

account for approximately 35 percent of the total heat transfer for packings of mesh 4 silicon carbide grains, 45 percent for 3/16 inch steel spheres, 60 percent for 3/16 inch aluminum oxide cylinders, and 85 percent for 5mm. glass spheres. xiii * * Xlll

PART I INTRODUCTION -1 -

PURPOSE AND SCOPE OF STUDY Purpose Particulate aggregates in the form of either packed beds or powders are important in a wide variety of applications such as catalytic reactors, pebble bed heaters, adsorbers, powder castings, solid propellents, and thermal insulations. In addition, many processes of technical interest, such as underground combustion for petroleum production or underground-contained nuclear explosion, deal with aggregate materials. In almost all these cases, the heat transfer characteristic of the packed aggregate is an important design and operational factor. Some investigations on heat transfer in packed media, both theoretical and experimental, have been reported in the literature. Recent comprehensive reviews are presented by Gorring and Churchill(ll)and by Hill and Wilhelm(l5). These previous investigations were mainly concerned with overall thermal conductivities and were generally limited to temperatures of less than 500~F. At such temperatures, conduction was found to be the dominant transfer mechanism(3 8 so that the concept of an "effective overall conductivity" could be practically utilized. However, this approach of grouping all effects under one overall conductivity is proving to be inadequate for recent technological demands of high temperature, close tolerance operations. It is recognized that radiation becomes increasingly important at higher temperatures and could easily become the same order of magnitude as conduction at temperatures above 1000~F(5). The heat transfer characteristics of the bed then become quite complex due to the coupling of conduction, a linear temperature function, with radiation, a fourth order temperature function. Moreover, at points -2 -

-3 - near walls, the radiant heat transfer becomes affected by the emission properties of the wall itself(20), so that any "effective conductivity" would be a function of the properties of the wall as well as of particle characteristics, The temperature distribution would no longer be linear as it is for the case of conduction controlling heat transfer. In order to understand and adequately treat this more complex case of simultaneous conduction and radiation, it was felt that the mechanism of radiant transfer needed to be isolated and studied in detail by itself. A number of authors (1, 3,6,31,3) have proposed equations to describe the radiation contribution to total heat transfer. However, direct experimental measurement of radiant transfer in packed beds has apparently never been carried out although several investigators(l5') have estimated radiation rates from measured total heat transfer rates. Due to this lack of actual data, even the relative importance of radiation and conduction is not clearly known at the present time. The objectives of this study were, therefore, to: a) find an experimental means of isolating radiant transfer for individual examination, b) obtain direct measurements of radiant heat transfer in a number of typical packings, c) clarify the relative importance of radiative heat transfer as compared to conductive heat transfer, d) check the validity of various methods of a priori estimations of radiant conductivity which have been proposed, by comparisol to experimental measurements, e) determine a practical means of characterizing radiant transfer properties of packed media, and n

f) provide a start toward correlating the bulk transfer characteristics with physical properties of the packing particles. Scope The analysis of this study dealt with the case of steady state heat transfer from an infinite flat wall at temperature T1 to a second infinite wall at temperature T2, across a quiescent gas-solid bed of packed particles, as shown in Figure 1. This one dimensional case was chosen for study because it lends itself to analysis and because many applications dealing with packed media have sensibly uniform cross-section temperatures and can be treated as one dimensional cases. The experimentally measured parameters, however, are applicable to general problems of radiant transfer in any geometry. The work of Rogers and Morrison(30) indicates that natural convection is negligible for the types of packings of interest and it was therefore not considered in this study. Only radiative and conductive heat transfer was taken into account. A dimensional analysis, based on the two-flux transport model of Hamaker ), was made to define the minimum number of variables and parametric groups necessary to specify the problem. Approximate solutions were obtained for the resulting dimensionless system of nonlinear differential equations. These solutions provide the local values of temperature, radiant energy fLux, and conductive energy flux at every point of the packed bed. An experimental method which was recently used by Larkin and (20) Churchill to study radiant heat transfer in fibrous insulations was found to be applicable also to close packed media. This method, utilizing transmission measurements of black body radiation, has the great advantage of isolating radiation effects for individual study, free from conduction and

-5 - t, Q t2 L Figure 1. Geometry of Analysis Case

-6 - convection. Twelve types of packings were tested at temperatures ranging from 800~F to 2000~F. The particles were of four different materials, glass, aluminum oxide, steel, and silicon carbide. Three different shapes, spheres, cylinders, and irregular grains, were covered by the samples. Nominal particle diameters ranged from 1.7 mm to 5 mm. An equivalent radiant conductivity, which is applicable at interior points of packed beds, was defined and calculated from the measured radiation parameters. This conductivity proved to be a useful means of characterizing radiant heat transfer properties of packed media, checking analytic estimates of radiant conductivities, and comparing the relative importance of radiative and conductive heat transfer. An ultimate objective would be to define the effects of such variables as temperature of radiation, size of particle, shape of particle, and bulk void fraction on the radiation parameters determined from experimental measurements, so that predictions could be made for the behavior of any packing under any operating condition. A start toward such a general correlation is provided by this work.

LITERATURE REVIEW Analytical Studies It is generally recognized that a quiescent bed of packed particles in gas provides a highly complex system for heat transfer. There are five transfer processes that could occur: (a) conduction in solid, (b) conduction in gas, (c) convection in gas, (d) radiation between solid particles, and (e) radiation between solid particles and gaseous voids. Conceivably all five processes could contribute to the overall heat transfer and since each process is dependent on a number parameters (i.e., particle size and shape, pressure, volume of voids, emissivity of surfaces, temperature, conductivity of solid, conductivity of gas, particle arrangement, etc.), their interaction would be difficult to treat rigorously. All existing analyses have been based on simplifying assumptions of one type or another. One assumption common to almost all treatments is that convective heat transfer is negligible (7,38,40) A number of studies have been made on the conditions necessary to initiate natural convection, by such investigators (17) (19) (30) as Horton and Rogers,) Lapwood, and Rogers and Morrison. However, no empirical or theoreticalequations to calculate the convective transfer in packed media has been proposed at the present time. A second type of simplification common to a large number of analytical treatments is to approximate the heterogeneous random mixture of solid particles and gaseous voids by some regular geometrical arrangement of arbitrary solid and gaseous bodies. The first work along this line was by (29) Rayleigh ) near the beginning of the century, when he solved the general potential problem of conduction in a medium of conductivity kl through which are distributed spheres or cylinders of a different conductivity k2. Maxwell(23) -7 -

-8 - also treated this same problem, but under the assumption that the spheres (or cylinders) were small in diameter relative to the distances between them so that there were no interactions. This assumption invalidates the results for packed beds where the particles are actually in contact. Eucken(9) was the first to suggest the possible application of this potential solution to thermal conduction in porous media. The next major contribution along this line was by Schumann and Voss(34) in 1934. They represented the heterogeneous bed by a square, divided into two parts by a hyperbolic curve running between two diagonally opposite corners, as shown in Figure 2-a. One part was taken to represent the solid phase of the actual bed and the other part to represent the gas phase. Conduction was thus calculated as occurring in series through the two phases. Wilhelm et al., have proposed slight modifications to Schumann and Voss' equations to account for conduction at contact points. Russell(32) in 1935, presented a model where the solid particles were in the form of cubes located at the centers of cubic lattices, as shown in Figure 2-b. The conductivity for this assembly was determined and taken (7) to approximate the conductivity of the real media. Deissler(7) and Woodside (9) in 1958, also worked with cubically packed models, though they both considered the particles to be spheres rather than cubes. The above studies all were concerned mainly with conductive transfer. Nusselt(27), in 1913, was the first to present a geometric model for estimating radiant transfer of heterogeneous media. He considered alternating layers of solid and gas where the relative thicknesses were dependent on the porosity of the medium (see Figure 2-c). The equation he obtained

-9 - for an equivalent radiant conductivity was, K, = 4LTJ3 (1) where L = thickness of gas layer. All later geometric analyses for radiant transfer were based on this or some modified form of this model. Damkohler(6), in 1937, introduced several correction factors into Nusselt's equation to obtain, Rr = +T ' T 3 (2a) where a = absorptivity of solid surfaces D = particle diameter T' = "fraction free area for radiation". Argo and Smith() gives the following as Damkohler's simplified expression, Kr 4(-2 )DpT T (2b) where 6 = particle emissivity. More recently, Laubitz(21) usedRussell's cubical model to estimate radiant transfer. This may be considered as another modification of Nusselt laminar model, with the solid layers being cut into cubes. The expression obtained by Laubitz was, Kr 4we a 1- p ) r) (3) where a = length of solid cubes p = fractional volume of solids 6 = emissivity of solids.

-10 - o-SCHUMANN AND VOSS b -RUSSEL'S. y/ / // /e //////// / ///. I/ /OF GAS SOLID/7W /////S/SEL' / //A// SOUI C- NUSSELT'S d-YAG! AND KUNII'S GAS GAS Figure 2. Analysis Models

-11 - Yagi and Kunii(0) used a composite laminar representation as shown in Figure 2-d to account for five transfer processes. Their result for quiescent beds was presented as a ratio of equivalent bed conductivity to gas conductivity, A ( I-,) Xs ^ 0 + ] +,C, plr B " (4) where ke kg hrs = equivalent bed conductivity = gas conductivity = heat transfer coefficient for radiation solid to solid t + 273 3 Kcal. - 0. 2-195 2p 100 m2. hr.0 hrv = heat transfer coefficient for radiation void to void '.= 0 195 L - (- P Kc l?..*0 k r.~ C p A, 1,p = properties of the bed. Schotte() also treated coupled conduction and radiation from the geometric point of view. He considered the energy transfered from a plane located on one side of a particle to a second plane on the far side of the particle. Conduction in the particle, parallel radiation across the voids, and conduction-radiation in series were all taken into account. The following expression was obtained by Schotte for equivalent radiant conductivity, }r = IL" _ + I rI- + R ++ pr 1( r (5) where ks r = conductivity of solid = radiant conductivity between particles - 6t U )T o kh ft,OF = void fraction

= emissivity Dp = particle diameter As an alternate to the geometric approach of the above works, some authors have proposed a kinetic or diffusion type of treatment. Rosseland(31) considered radiant transfer as a diffusion of photons through the media and obtain the following expression for equivalent conductivity, -t +-1 T3 (6) where 1= mean free path for the photons. Bosworth(3) has proposed that for a packed bed, 1 may be set equal to the particle diameter, Dp. A third approach is to represent the heterogeneous media by an equivalent homogeneous media, making it possible to treat the particulate problem as a continuous one and to define the heat transfer processes by differential or integro-differential equations with suitable boundary conditions. This approach has a distinct advantage in that it is not necessary to assume any specific geometric model or to arbitrary specify the paths of the various transfer processes. While continuous representation of what is actually a discrete process will not give exact solutions at any one point, it does give correct solutions for quantities averaged over a large volume. In fact, if the parameters defining the equivalent homogeneous media are obtained by actual experiment, it is theorectically possible to write down a system of differential equations that will rigorously define the true average heat transfer rate per unit area through the packing. (35) There are two major works in this line. Van der Held treated the general three dimensional problem by adding a radiation term to the

-13 - Fourier heat conduction equation, T T ^ (p,-JO)d _ g at (7a) where the second term represents the net accumulation of radiant energy of all wavelengths. At steady state, Van der Held obtained the following expression for an equivalent radiant conductivity at interior points of the packing, Rr = - 4- r3 (7b) where n = refractive index of equivalent homogeneous media a = absorption coefficient s = scattering coefficient. The other major work was by Hamaker(3) who, in 1947, suggested the two-flux treatment of the one dimensional problem. He proposed the following i, and a backward radiant flux, j. The third equation is an overall consystem of equations to treat simultaneous conduction and radiation, d? "' - 0-F = The first two equations describe the attenuation of a forward radiant flux, i, and a backward radiant flux, j. The third equation is an overall continuity equation for transfer of both conductive and radiative energies. Hamaker presented general solutions to the system of equations in terms of four arbitrary boundary constants, but the complete solution for the case of a finite bed was not given in detail. Since this present study is based on Hamaker's concepts, the system of equations, (8) were rederived, dimensionally analyzed, and solved in detail for the sake of completeness, (see Part II).

-14 - Experimental Studies Majority of data reported in the literature for heat transfer in pack beds are limited to temperatures below 500~F. At such temperatures, radiation is relatively un-noticeable, and since these data have been amply summarized by Waddams(36) in 1944, Wilhelm, Johnson, Wynkoop, and Collier(38) in 1948, and Yagi and Kunii(40) in 1957. they will not be discussed here. Only some data for relatively higher temperatures are reviewed below, In 1947, Lucks, Linebrink, and Johnson(22) made thermal conductivity measurements for packed beds of sands over a temperature range of 750~F to 2250~F. They reported that the overall conductivities of three tested samples Btu in. 3.2 - 8.8 hft. 2F at 2250~F. No effort was made to separate out radiation contribution. Bell(2) in 1948 also reported data for sands, in a temperature range of 1000~F to 2000~F, Values obtained for overall conductivities were of the same order of magnitude as Lucks et al's. Again, no analysis was made to study radiant contributions. Bunnell, Irvin, Olson, and Smith(4) investigated heat transfer in a two inch bed of 0,125 inch alumina cylinders, with air flow through the bed. Maximum test temperature was 400~C. Extrapolating their data to zero air velocity, an effective conductivity of 5 x ka was obtained (ka = true conductivity of air). Campbell and Huntington(5) also investigated packed bed with gas flow. Particles tested were silica-alumina cylinders, hydrated alumina cylinders, tabular alumina spheres, aluminum cylinders, and glass spheres, ranging in size from 0.194 to 1.0 inch. Test temperatures were between 200~F and 700~F. A single linear extrapolation to zero gas velocity Btu was used to obtain an overall conductivity of 0.31 Btufor all test hr.ft.~F

-15 - particles. Maximum deviation of data points from this correlation seemed to be t 51 percent. Campbell and Huntington also reported data for packed beds in vacuum. From this set of data they were able to qualitatively conclude that radiant contributions were already an "appreciable factor." Yagi and Kunii(40) present data for a number of packings in still air, with particle sizes ranging from 0.18 to 11.0 mm, and temperatures up to 840~C. Maximum experimental scatter, for values of overall conductivity + of the same packing, seemed to be approximately - 13 percent, a very respectable precision for experimental work in this field, Schotte(33) has analyzed Yagi and Kunii's data according to his proposed equations and reported that at the highest temperature, radiation contribution accounts for about 80 percent of the total thermal conductivity. Only recently has an experimental investigation designed to study radiant heat transfer in packed media been reported. Hill and Wilhelm5) made local total-conductivity measurements on a quiescent bed of 358 mm. alumina spheres at temperatures up to 1000~C. The conduction-conductivity was arbitrarily set equal to total-conductivity at 0~C, then extended to higher temperatures by the correlation of Wilhelm et al. 8) These conduction-conductivities were then subtracted from total-conductivities to obtain values for radiant-conductivities. By this means, it was estimated that radiant transfer contributes about ten percent to total heat transfer at room temperatures and about 55 percent at 1000~C. Accuracy of the measured conductivities was estimated to range from five to 25 percent.

PART II ANALYTICAL DEVELOPMENT -16 -

GENERAL ONE-DIMENSIONAL CASE As mentioned previously, the analysis of this study was based on concepts presented by Hamaker.(13) A more detailed derivation with some new interpretations are presented below in order to underline the basic assumptions, advantages, and disadvantages of this mode of treatment. Radiant Energy Transfer Heat transfer by radiation is inherently different from conductive or convective heat transfer. While the latter two are dependent on the conveyance of kinetic energy of molecular vibration, radiation is dependent on the transport of electromagnetic energy. As such, there are three basic phenomena that come into play in the transport of radiant flux through a media of packed particles —emission, absorption, and scattering. At a position r in the media, the specific intensity of radiant flux of wavelength N which is traveling in the direction Q may be denoted by, (, a) khr. ft sterardtnand the power impinging on an elemental area dA due to radiant flux traveling in the elemental solid angle d%, about the direction a, is then equal to i (rLA) dUn dA cos0M hr. where,. = angle between f and normal to dA. The next step, basic in this treatment, is to assume that properties of the discrete particulate media may be averaged over a bulk volume- to obtain pseudo-homogeneous properties. This representation is analogous to such concepts as the "average density" for a mixture of solids, and should be quite -17 -

-18 - good if a large enough sample is considered. Thus, while local point conductivities vary greatly from solid particle to void, the average conductivity over a large cross section is sensibly constant. On the basis of this assumption, the following bulk properties may be defined. SN = monochromatic scattering cross section per unit volume, the effective cross sectional area presented by a unit volume of packing that scatters the incident radiant flux, units of ft2/ft. AN = monochromatic absorption cross section per unit volume, the effective cross sectional area presented by a unit volume of packing that absorbs incident radiant flux, units of ft2/ft3. E^ = monochromatic emission cross section per unit volume, the effective cross sectional area presented by a volume of packing for the emission of radiation, units of ft2/ft3 Now consider a beam of radiant flux, L (ri,l) dLQ dLA as it passes through an elemental volume of packing, as illustrated in Figure 4. The changes in the intensity of LA (rIL) dl.A as it traverses the differential length dl are as follows: a) loss due to scattering and absorption = ' ^ L(, L)n d.A 1 di (6,5+Ax) b) gain due to emission = PA- I d lAdl A (Ex) c) gain due to scattering of other fluxes into da = J~L HAd.l J ix(i5 ) fV(..fL') ^5; where Pt. = Planck's radiation spectra for emission of wavelength 7 at temperature t. f^(,J' ) = angular distribution of scattering, the fraction of scattered flux which came from direction W' and is scattered into S.

-19 - -- -di' = integration over all directions of Q'. The net change in ixC,A )s dA is then, dl[c (,)ndAJ = - (S +A) (r (,5L) LQLA 1 + (CS, ) - dA di JQ c(i. Q c') a) d + CE) Pt4. dI Q A d-1 which may be rewritten as, L [^r.n)d -(SA + AA) i (r S) +(SA)^ J; rdQ) fS dQ' (9) (EA) 'Pth 4r The integro-differential Equation (9) is commonly called the transport equation. In the Cartesian coordinate system of Figure 5,, ( r,51) = L0 CX,,yz2, 4)0) and ~[c~r,~) =, a, ax, + a ay + a_ ~2 1and,+ ax a1 ay ai a+ al ax^a 4? al ai ae a + ae al aB aFor the one dimensional case, a; _ Li _ aL _~ ay az a- o and for any specific direction of Q, al a~ Thus,, () = LA (,,e) r)_ - d- '. (o) ax (10) dI dX al cos e i&i(Xe) dx

-20 - Figure 3. Illustration of Radiant Intensity Figure 4. Traverse of Flux through Elemental Volume 0 z Figure 5. Cartesian Coordinate System

-21 - Substituting into Equation (9), cos e Cxe) = - (5 + Ax) LC(X, 8) dX d..x: + C5) J Cx,e ) f Ce, ') 4d + (EA)PtA 4,r Evaluating the integral over the solid angle ', ^e @,<A: = - (5A + A)L(X,~) + (S6A) T JL. ^ (xe)fx Ce, e ) ste'c e' (11) + CE)?P ^ 4 this energy balance is often written in terms of total.intensity for black body radiation as, CoS (Xe) =- (5 +A), (X,e) (5)lr Jo, l- ICx, ')fj ) s^~' de m (S + (ET) T iO(x,@) = total specific intensity of all wavelengths S,A,E, = total cross sections per unit volume of all wavelengths for scattering, absorption, and emission, respectively cr = Stephen Boltzman constant T = OR f(G,Q' ) = aver. angular distribution function for all wavelengths. The relationships between these black body parameters and the monochromatic parameters may be derived by integrating Equation (11) over all

-22 - wavelengths, r 00 5 = 7J SA^ A (I 95^ A ) J); h / (X^ CD A T T JJ 'Otd Comparing Equations (12) and (13), it is seen that, c;~1, a JS (:c, e) ~X - fS x(x.,e) d Oe d).x, coA ~' f T, ) -(X,6')) J -Pr~~e, ~ L'k 55LA (X,(e') c= S~^ ^rX~e~ a S^i ^ ^^^ (13) (14) (15) (16) (17) 9')d, (18) Hamaker(l3)started his analysis with the two-flux equations, Zx~ =-(a+ Sb) + Sj 4 '-*T (8) ^d = (_+ )j - - a.g where i represents the flux of all radiant energy traveling in positive x direction and j represents the flux of all radiant energy traveling in negative x direction. The first terms in Equpation (8) represent the net loss due to absorption and backward scattering. The second terms represent gain due to backscattering of the opposite flux, and the last terms represent gain due to thermal emission. The assumptions inherent in this system of equations, Equation (8), may be pointed out by integrating Equation (12) over forward and backward hemispheres.

-23 - Over forward hemisphere l & ( ia c) cos SLeS = - 5tA)J OC(x.A0 Ke (19) + (5)(Czv.Jo Jo (cx,e'>fe,( e')s:'se d.'de0 + (E)i-T fj d5 - and over backward hemisphere, z. j- ~ (x,)cose;ede = - (+A) ^ (x, e)5 (20)e 4- (5)1 D) e r ( e:, ). e '5e' + (E)rT4/ s;^ e d Let, (x) = (X, ) C0 so; d (21) = radiant energy passing through a unit cross sectional area per unit time in +x direction j(C) = (X e) cose S9n dCe (22) = radiant energy passing through a unit cross sectional area per unit time in -x direction A.t = -bA (x ) stne d (23) AV = -.: J { % (:c6":s.'e6 =, j),: X, 6+j)5x ~ (24) S= +JL;d(X)I d' (25) =S =- i4- f> & (X,e) su;"e -6 = %n YJCd, e+i+) n ~e (26) SF= (^^)SJ ' (^e')f(9ee')5^~ e;6>'cLd ' (27) SIF = ( S. |, (r:}@) f(SW9) 5.R 3 sAna' d ' (27) S,5 = (,,)2f f-5 I- -,()fC' e d616' (28) F= _-(^i) Y-S- If f (x, e)O') f ) s, e', d e = 2f)-V 5 ^ (x,e+7) 'f(e+i e'-r) z kedBd (29) j -'e~o -'e'^o e el 5-B -(2,)^ + f, L (xa) -f@e,') es1 5s'e' {e Le' 1~~Je= (Z o' S- ( |, '+-n)f( +^e'+,~):os,,e.si,, '' (30)

-24 - Equations (19) and (20) may then be written as, -:-x) _- -S F - A, + E, T 5, + (1) and, =t SX j +Aj - 5Fj -5 - E rr (32) By definition of angular scattering function, J oJo f(eo )stde = I (33) and so, -- x e= s 6 a' a' $,L — 1 Z. J (X)SL0, o ad tTS fo e (X,)sLnr Je f fc C)e, - e d.' + Jc f (e @ ) 5r ] d+eI E6,L (,SF + $,3)L> (34a) similarly, 5' = Cs, sZF)j (34b) Therefore, Equations (31) and (32) reduce to, - =^-C(SZB+ A,I) SIB + ET (5) — a^- =-(S,+AB)J + s52B E T1 (36) Comparing Equations (35) and (36) with Equation (8), it is now readily seen that the conditions or approximations necessary for Hamaker's two-flux equations are as follows: (a) a=A1= A e,} J O(X,e>) s;nede J. (X, e+r)S; teL (37a) JO OOCX, 9O;1&OSB 0 l eeosesL. I+)S;K~S0e eed 40 0 0 e (b) Sb = S2B = SlB - ~ J;OJ ~* (x e') f Ce e') s. e sLe' de ee' Jo0;. (x, e+ r) s;. e cose de (37b) r~ r( J f. oc, e'i)f(+') f'e+e '+tr) s;..e 5;te' dLe de' f;s (X, ) s:. e cose d e J oo (C S S

-25 - (c) a =E (37t) Condition (37c) is the equivalent of Kirchhoff's law and may be considered as satisfied. Conditions (37a) and (37b) are equivalent to the requirement that the fraction of radiant flux traveling in +x direction which gets absorbed or backscattered be the same as the fraction that would be absorbed or backscattered if the radiant flux has been traveling in -x direction. These conditions are sensibly satisfied for particles that are either symmetric or are randomly packed. Under these conditions, the two flux equations of Equation (8) may be considered to be valid. Simultaneous Conductive and Radiative Transfer For all packings of the particle-size and temperature ranges considered, conduction and radiation are the two important heat transfer mechanisms and occur simultaneously. The coupling between them is by the process of absorption and emission of radiant energy. The governing relationship may be obtained by a thermal energy balance about a differential element, dA*dx, of the packed bed. Net accumulation of energy due to conduction is given by the familiar Laplacian term, k BtT C dxz where kc = bulk conduction-conductivity for packed media. The input of radiant energy is given by, a(i + j) and the output of radiant energy is La T CT4) Thus the continuity equation of heat flow is C Z X i Kc IT 4 OLC )-Z~T =78 d-e (38) where p = bulk density, Ca= bulk heat capacity, = time, T = absolute temperature,

-26 - For steady state, yk ~T- +. (c +j) - -D T r o (39) Four boundary conditions are required in order to completely define the one-dimensional system of Figure 1. Two conditions usually known are the wall temperatures, T CX= ) =T, ~R T(=L) =T, R If the emissivities of the two walls are also known, then the other two boundary conditions may be written as a radiant energy balance at each wall, (JX=o) = 6I,<T + I 1-,)Ij (X~O) j(x =L) = e6Tt~ + Ci-,).( Lc L) where C, = emissivity of wall at x = 0 At - emissivity of wall at x = L. To summarize, one dimensional, steady state, coupled radiation and conduction heat transfer through packed media is described by the following three simultaneous differential equations and four boundary conditions: 4- =- C(a sb). + s - (40) ij = - (C+ 56)j + es a T4 (41) KcI dt +.C- +j) - Za'T =0 (42) and, T( =o) = T, (43a) T (X-L) T (45b) (X —o) = +, T,4 + -e,)j CX-o) (43c) JCX L) =- Era,+ - (i- )L (X-=L)

-27 - The conductive heat transfer rate is, BtT tu- (44) Q = - cAx hr. fZ ( and the radiative heat transfer rate is, Btu Qr - ) (45) Qr = i-j, r. ft 5 from which an equivalent radiant conductivity may be defined as, Qr L-J kr - _ - -- _ dr_ (46) &X dC Dimensional Analysis From Equations (40) to (43), it is seen that the four variables, i, j, T, and x are functions of eight parameters a, sb, kc, L, T1, T2, 61, and 62. It may be expected that some of the eight parameters could be combined to obtain a minimum number of parametric groups, affecting a very useful reduction in the amount of numerical work necessary to specify the general solution. This minimum number and the grouping of parameters may be obtained (14) by the dimensional analysis technique of Hellums and Churchill, as follows. For the four variables in the differential Equations (40) to (42), substitute the dimensionless quantities, T T T X Xx (47) Jo r, where to, io, jo xo are arbitrary constants to be specified later. Substituting into Equations (40) to (42) and boundary conditions (43), + TC.^;(48) d&X =-(CLXo +s*xo)I + )J + o t (48) dX -(=-axsbx.)JT + x ) (X~")I ) (49)

-28 - er aKo Lo i o )' a. X~ CT to l + (0i: ~ _ )J - +2(.X OtO)^ = ~ (50) tXR. K, to lleto ( +o) T (X=o) = t (51a) (X ) = X to (51b) I(X= O) = ( ) () (X ) (51c) JCX=+-) = (-) ( O. ) I(X=o) (51d) It is seen that there-are ten independent groups of parameters which appear in Equations (48) to (51), (ta) (5o-), t ' ). (-L) T ) -I), JO, ( Lo,, ) and z Since four quantities, xo, io, Jo, to, are arbitrary, it is evident that ultimately the physical problem can be defined in terms of just six groups of parameters. If the four arbitrary reference quantities are chosen to be, x =L ft. L = T ~ 4 -tT. (52) ~. = -Ti' I-. f:t. j. = T, r. ft-t then the dimensionless variables are, X xL T- OT (53) TT I= <r (T JJ 3' ~_ri

-29 - and the six independent parameter groups may be chosen to be, 6= &L = dimensionless absorption coefficient = Sb L = dimensionless backscattering coefficient Y = qT L = dimensionless ratio of radiation to conduction from kc wall at X 0 (54) 7 \ T = boundary temperature ratio a, = boundary emissivity at X - 0 C =-= boundary emissivity at X - 1. The system of Equ.ations (40) to (42) may now be written in dimensionless form as, -- -C) + I J + J T+ dLX -, + Ii (I + J- ZT) = and the corresponding boundary conditions would be, T.(X=I) = i(X=-) =,+ c,-,)T (X=~) JaX ) = E2+ (I-^)I(XI) In terms of these dimensionless variables, Equation written as, QCX, - - - d-T &T(),~ Y (55) (56) (57) s (44) to (.6) (58a) (58b) (58c) (58d) may be re rT "LI (X)- JcX) (59) (60) Qr(X) and Rr(X) ~ - LT [I(X)- J(X) dX (61)

-30 - Approximate Solutions The system of Equations (55) to (57) are nonlinear, and no exact solution has been found. However, a number of approximate solutions have been suggested by various authors for the equivalent system of Equations (40) to (42), These approximate solutions are rewritten below for the dimensionless system, Equations (55) to (57), and are reviewed with regard to regions of applicability, Larkin and Churchill(20) obtained a solution by assuming that the temperature T(X) may be approximated as a linear function of position. This allowed the nonlinear terms, d T, to be replaced by a function in X so that Equations (55) to (57) becomes an ordinary system of nonhomogeneous linear differential equations. The solutions can then be written as, I( X)= c, slrh (X ^,-^ ) + K (X -') + t 4 4(,-z)r3 (I, - r2T2)3 q(YI+) Cy(r1-n p) (62) zCr,-^r)" r, -r (-r, )+ + ze-T)() +,t- ^24 ) X), = (C, A1- + C t) Co"k(X4 '- ) + '-c 5,_ cX) ~ank X'-~gL) + (6) 4(-"t)O3 2 C, -% T(,+P) + (z'^5) 24(-',-Tr )^r,'- +) (k^f ) (of_ fL) ( Be _ f-I

-31 - where = -c + = net attenuation coefficient, and cl, c2 are boundary constants to be determined from Equations (58c) and (58d). The region of applicability for this solution may be found by examining Equation (57). The approximation, -(X) - cX + C4, linear X function (64) c3, c4 = constants is equivalent to the approximation that &7 = c constant dX C) and Jz =0 (65) From Equation (57), it is seen that there are four cases which would satisfy this condition: a) L o 0 i.e., the packed media is sensibly nonabsorbent. b) i.e., the ratio of radiation to conduction is sensibly zero. c) O O S o, trio i.eo, absorption and radiation both exist but are sufficiently small so that the product, o(, approaches zeroO d) I + J - Z i.e., the total radiant flux at any point is sensibly equal to the local thermal emission. In practical cases, only conditions (a), (c), and (d) are of interest since the case of zero radiation degenerates to the usual Fourier conduction problem. For condition (a), another approach may be taken which leads to a simpler solution, as follows. Assuming that the packed media is quite nonabsorbent, c, - o I+P a-~

the nonlinear terms in Equations (55) to (57) drop out to give, &7r = -IJ + qJ - = -^ + jI dzY = o dX1 Note that while radiation and conduction can still occur simultE are no longer coupled in this case. The temperature function i. Equation (68), T(X) = cX + C4 and evaluating the constants, c3 and c4, by boundary conditii (58b) (66) (67) (68) aneously, they s obtained from (64) ons (58a) and r(X) - ($ -)X + I (69) Equations (66) and (67) may be solved rigorously to obtain the radiant fluxes, I(X) = cX + c (70) J(X) = cX + q (71) Boundary constants cl and c2 are evaluated from Equations (58c) and (58d) to obtain, - ie ( - (72) t, - &, + 6^ +6,6iCn-') (72) CL Substituting found to be.,(I+ qex)- C., -C-E,) (73) -6 i + Eu (60 ) ad (q 1) into Equations (60) and (61), the radiant heat transfer rate is q- T, rE, C -,) Qr _ -r sl ^e,+(^ *-+ (6eL^-<)] and the equivalent radiant conductivity is, = 'T L (I- s") r (q- + - eI-l-~ + (;1)(I - -'4) (74) (75)

Hamaker (3) proposed a solution based on approximating the fourth order temperature term by, Y r, ~4 +r 4r T3 (- T) (76) where T = some base temperature near the mean bed temperature. Hamaker's solution, transposed into dimensionless form is, I(X) A(,-, ) X B 5(pi'e)e x + C -'X —p' + D (77) J7X), A(+, )e"X + B (-+) + C^~r~'X ~+#')+ ~ D (78) +, c~-'X - ')4(X)A -CA /X - e- X + CT'X + D (79) where, L' _ j (8rx3 + - - +2) d = o(+ ZB (80) o( + 2 The four constants A, B, C, D were to be evaluated by boundary conditions. Since this evaluation was not given by Hamaker, it was necessary to do so in order to obtain a useful solution. Substituting functions (77) to (80) into the boundary Equation (58) and solving, the constants were determined to be, Aa b -, (81) - __ a, cl - ac, - a,,-ab, (82) C = _l['C.-,)A + (eT'- )B - (0- -') (83) D = 'A +B - 1 (84)

-.34 - where, a, = 6, (+dO') - ' (z-e,)[) ](e] (85) bh = 6, (+')-/ 'C(z-,) - r(e-)-0-iJ (86), C - -)(e) (87) a, =- eL c, +^' ) - ' <2-,.) (e'-) + e (88), 6 X z-e~Z E.) b, E= se (l+e') -t' (e-1)-l0 (e-,) - ep~j A(89) a (- ' )(z- e+2) (90) For large values of q', which is usually the case, it was found that many terms could be neglected and the above relations simplify to, 0. - _ _'_(z-_ _-' fQ_ a' d- (2-fi e (91),, =,+a' ) + (' (2-e,)( — +1) (92) + occzB (93) = _ L(i-')(-e) (94) += ( (95) C~= Cl-1-)(2-62) (96) 1 d + 2 # (96) By this solution, the temperature gradient is no longer limited to a constant, _r(X)_ _ d(4) d X 4TO3 dX and by differentiating Equation (79), -r(^X_ 4[-A,'X, -T"X ] /i-dXX 4 ( A g e X + IBe ia The conduction heat transfer rate would then be, Q (X)= -k, - ^ cTr' (_ Ao'eX +Bd'e-p 4L r3 The radiation heat transfer rate would be, Q,(X) = T+ ( I X) - JX)) (97) (59) (98) (60)

and by Equations (77) and (78), QC(X = DT; T (C-Ae A + B,-x - C) (99) The corresponding equivalent radiant conductivity is found by substitution into Equation (61), Or (X),- L iX (61) - 33_ TA X- Be + C r/ L o' (- A e'X + Be-"X) - CJ. (100) Equations (98) to (100) together with boundary functions (81.) to (84) and (91) to (96) represent the complete solution in accordance with Hamaker s approximation. It should be noted that the approximation Equation (76) is the equivalent of a truncated Taylor series,.. % T.~,(-r.)[^ "- Z-r,,. t- tr (r-r t)L0 -] + 4 X ( -& ["fdL<I neglected so that an estimate of the maximum error may be obtained from Taylor's remainder formula, = _ r_ 1 (T- r)L am > LSz JiLz z - vr — t)2'(101) = (m (X- Te) where Am is the maximum possible error in r and T, = for T > To 'T = Io for T< To In general, this approximate solution is quite good for small values of, i.e., for small temperature differences. It is to be recommended for most practical applications, and especially for application where local details are of importance.

-56 - Radiant Conductivity for the Media With the exception of the zero-absorptivity case, it is seen from the above solutions that I, J, T, and Qr are all functions of position and boundary conditions as well as of the packing properties. It is evident that any equivalent radiant conductivity, kr, would be a property of the system rather than a physical property of the packed media itself, such as kc is. Larkin and Chlrchill(20) have demonstrated that kr for an optically thin system is highly dependent on the values of boundary temperatures and emissivities. This fact may also be seen from Equation (100). Therefore, a value of kr is generally insufficient to specify the radiant transfer properties of a packed media. Rigorously speaking, only the two attenuation cross sections, a and sb, are true radiant "properties" of the packed media and can be used as a measure of its transfer characteristic. However, values of a and sb have little physical meaning and does not provide a sense of the ease of transfer, as kc does for conduction. Moreover, the relative importance of radiation and conduction is not evident from comparisons of a and sb to kc. What is desired is some sort of a conductance coefficient which woqld have physical meaning in terms of ease of radiant transfer and which would be a true "property" of the packed media itself —independent of boundary conditions. Such a coefficient was found by examining the expression of radiant conductivity for interiors of thick packings, where boundary effects may be expected to be at a minimum. Equation (100) gives the local conductivity at any point as, AX -O'X K( ~X) - - /.3 r3 -, B C $(A- 'I- bAe.-B x) ~ (100)

-57 - Equations (81), (82), and (91) to (96) show that the two constants, A and B, are of the form, A = die B = d2 where d1 and d2 are nonexponential constants. The constant a is directly proportional to the thickness of packing, L, as seen by Equation (80). Thus, for interior regions of thick packings where, L is large X 0 x~l X 1, it is seen that, i x,.~ (I-X ) AeTI = d > A e Be-'X e-X o Therefore, the exponential terms of Equation (100) may be dropped, - Q T;3ro 3L'[/ (e TOX-Be-"'X)+ C d_% - a T t '(-A eX + B e-'-) + C L dX T-r; 3,,3L ~' _.~~~3 (lo3)(103) 8 gTo a+ 2sb Thus a quantity kir may be defined, kir = interior radiant conductivity 8 - T3 Bt,.+ zSb r. ft. ~k (104) where T = operating temperature, ORo This function, kir, is seen to be independent of boundary conditions and has the physical meaning of conductance. It provides a simple expression for characterizing radiant transfer properties for packed media and may be used as a guide to the relative importance of radiation and conduction by direct comparison with kco The expression, Equation (104), was also derived by Hamaker(l) by a different procedure, but its significance was not actually utilized.

EXPERIMENTAL CASE In the experimental program of this investigation it was possible to modulate the transmission signal in such a way that only the direct and scattered radiant fluxes would be measured. Radiant energy which was absorbed and then re-emitted, along with any conducted energy, was not counted in the strength of the transmission signal. This modulated transmission led to simplified forms of the two-flux equations which could be solved rigorously. These simplified equations and their solution are described below. Modulated Transmission Equations In Equations (40) and (41), the (aC-T ) terms denote gain due to emitted radiant energy. If ie and Je are defined as radiant flux intensities for the experimental case, i.e., without emission flux, then the (acT ) terms may be dropped from the flux equations to give, -Le = - (- +- S) Le + e (105) = - an, Le 5j _ Q = - (+ je t )it~ j e- ~ (106) = -- KJe + SbLe. Boundary conditions for the experimental case are: a) at x = 0, ie(0) = So, a known input signal (107) b) at x = h ft., je(h) = 0, no reflection. (108) These conditions represent black body boundaries with, = = I Solving Equations (105) and (106) simultaneously, J. =- t dO +D z] (109) _d - | dJe 6L ae + a (110) -38 -

-39 - and substituting into Equation (106), - (aK - s,) = or, 2 2 2 where, m = an - f =-~ + 20-5a, always real. The general solution of Equation (111) is, LOec) = C, cos5 CmX) + C,^nh (mx) (112) and tIec) = crn Lm sn (m) + C m cosh (mx) (115) dx Substituting into Equation (109), \pI == ( CCry- C,. a Lh(mx ) b (114) - b (Cm++cl,) Coshh^CX) The constants cl and c2 may be evaluated from boundary conditions of Equations (107) and (108), ieo) = = c 0 osk co) + c;nh o) ci =S, and J )= = (rS0 + C4)s;nh Crrh) + (C^^ + 5,) col(nk) C = - 5 rM onh(nmh) + (incsh(h')] A ~ m eOS,(hl) + a, sh(m k)J Thus the flux transmitted through a bed of depth h is, i ^ f i L\ nkCm^) +- aKCos~<MI~k - <(mk)IJ e (k) = 5 [ nkmk) -5~ [ rv\ cosk(mrh) -4 -n SkQcnk)

Since SO is the flux incident on the bed at x =0, a normalized transmitted flux may be defined as, 5 kSC) ec~) So (115) rricosk (rnk)+ scdn s1 mA) Equation (115) is the analytical model used to fit experimental measurements of Sn and h.

PART III EXPERIMENTAL INVESTIGATION

DESCRIPTION Background Equations (99), (100), and (80) of the previous section show that the radiant heat transfer characteristics of any packed system may be described in terms of two attenuation cross sections for the packed media, a and sb. If values of these two parameters are known, together with boundary temperatures and emissivities, then the local radiant heat transfer rate may be obtained by Equation (99) and the corresponding radiant "conductivity" would be given by Equation (100). The interior radiant conductivity, kir, solely a property of the packed media, may also be described in terms of a and sb according to Equation (104). The obvious next step would be to obtain measured values of a and Sb for all types of particles. Ultimately, it would be desirable to be able to predict values of a and sb from basic particle properties. At the present time, no measurements of a and sb have been reported for packed beds. Some measurements for fibrous insulation at temperatures below 1000~F are reported by Larkin and Churchill(20) from a study of radiant heat transfer in porous insulations. The experimental investigation of this present study was designed to obtain sample measurements of a and sb for a range of packings, at temperatures up to 2000~F. Method The experimental method was based on the same black body transmission technique as that used by Larkirl and Churchill(20). A variable temperature source provided black body radiation which was passed through the test bed- of packed particles. The intensities of flux transmitted through various depths of the bed were measured by a thermopile detector and normalized with respect to the flux intensity at zero bed depth. -42 -

By mechanical chopping, the input signal from the source was modulated to approximate a ten cycle per second square wave. The total flux that impinged on the detector could be considered to be of four types: a) the flux that traveled through the bed by direct transmission through voids, b) the flux that traveled through the bed by multiple scattering off the particles, c) the flux due to absorption of radiant energy which was converted to thermal energy and then re-emitted as radiant energy at the temperature of the particles, and d) the random background flux from external sources. The background flux (d) was of random intensity and may be considered to have a constant root-mean-square value, VB, as shown in Figure 6. The absorbed radiant flux had a ten cps time function, but the thermal inertia of the particles damped out this function so that the re-emitted flux (c) had sensibly a constant intensity, VE. The transmitted and scattered fluxes, (a) and (b), traveled through the bed at the speed of light and therefore would retain the square wave function, causing a ten cps response, VT, at the detector. All these fluxe.s are additive so that the total intensity at the detector was of the type shown in Figure 6. The detector was designed to provide a voltage output directly proportional to the intensity of radiant flux impinging on it, so that quantities VT, VE, and VB may be treated directly as the voltage signal outputs from the detector. The detector signal was first passed through a primary transformer which served to eliminate all D.C. portions of the voltage. This sensibly shiftedthehorizontal axis of Figure 6 so that VE and VB were eliminated from the signal and only the VT signal passed on through to the amplifier. A narrow band amplifier was used which could be finely turned to amplify only

-44 - VT = transmitted and scattered flux signal VE = emitted flux signal VB = background flux signal t* X IZn VT Iz TIME -- Figure 6. Flux Impinging on Detector.

-45 - ten cps signals. This served to filter out any additional random frequencies which might have passed through the primary transformer. Thus the final signal output of the amplifier was proportional to solely the VT portion of the radiant fluxes impinging upon the detector. Normalizing VT(h) with respect to VT(h=O), a direct measure of the quantity Sn(h) was obtained, VT( ) S- c= (k) (116) V7Ah=O) 50ch- ) By measuring VT(h) at several packed depths, h, a set of data points of Sn(h) versus h was obtained. Using two of these points, a first-guess approximation for m and an could then be calculated by Equations (178) and (179). These values were then used to initiate an iterative regression. Equations (146), (165), and (166) were used to obtain progressively closer estimates of the true regression values of m and an corresponding to a minimum ~ error, based on the model of Equation (115). The attenuation cross sections, a and sb, were then calculated from these converged values of m and an by Equations (174) and (175). Derivations of these regression equations are given in the Appendix. It should be noted that while the source temperature and the black body radiation spectrum were variable, the test beds' temperatures remained sensibly constant at room level. In most practical applications this would not be the case, as packed beds are commonly operated at temperatures close to wall temperatures. This dissimilarity is not as serious as it may seem to be. Temperature affects absorption and scattering in two ways: (a) by changing wave length spectrum of the radiation, and (b) by changing physical properties of the packing. Of the two, the spectrum change is the more important. Physical properties of the solids are not expected to change greatly over the temperature range considered. In the experimental work,

-46 - the spectrum effect was reproduced by the variable-temperature source, and it was only the relatively minor effect of physical property change that was not accounted for. Apparatus The apparatus assembly used in the experimental studies is schematically represented in Figure 7 and a photograph is presented in Figure 8. Details are given below. The black body source was basically an open-end tubular furnace, as shown in Figure 9. The core was a silicon carbide (Norton's Crystolon RC-138 mix) tube closed at one end, of three inches internal diameter, 27 inches length, and 1/2 inch thick wall. Silicon carbide was chosen as the material because of its high emissivity, mechanical strength at elevated temperatures, good thermal conductivity, and low electrical conductivity. The first and third qualities were necessary in order to obtain uniform black body radiation and the other two properties were desirable for constructional ease. Heating coils of 18 gauge Chromel-A (nickel-chromium) were externally wound on the core and set in refractory cement (Norton's RA 1055). The coils were in three 20 foot length sections, each with its own Variac power control and ammeter. This sectional power control allowed local temperature adjustments to obtain uniform wall temperatures. Since heat loss was greatest at the top open end, heating coils were wound proportionally closer at that end. Thus the top 20 feet section of coils covered only 5.4 inches of the core, the middle section covered 8.1 inches, and the bottom section covered 13.5 inches. Each section was designed to provide a maximum of two kilowatts power input at 135 volts A.C. With this set-up, it was found possible to maintain wall temperatures uniform within 5~F from top to bottom, at temperatures up to 2000~F.

-47 - OPTICAL BENCH TOP SHIELD I — TEST PARTICLES BED HOLDER / BOTOM SHIELD -r~i BLOWER = a BLACK BODY SOURCE I I I I I I I I I II g I I w I{ I - THERMOCOUPL-ES POTENTIOMETER POWER CONTROLS I r ~ --- —----— iI _1 ci ~oo ---— A C " Figure 7. Diagram of Apparatus

4 I 4r OD Al[ 4 9 ia I:~ Figure 8. Photograph of Equipment

-49 - 0.5" SiC core - 0.5" refroct. cement 2" bubble Alundum - 3" Superex steel case screws Figure 9. Details of Source

-50 - A 2-1/2 inch diameter reflector-cone of aluminum oxide (Norton's Alundum RA 321) was set inside the core, raised a height of two inches from the bottom. This cone served to prevent a direct view of core bottom and thus increased the overall emissivity of the enclosure. It also presented a surface temperature closer to wall temperature than could be attained by the core bottom. Two types of insulations were set around the outside of the core. The inner high-temperature insulation was of bubble alundum (Norton's E165). It was packed to a thickness of two inches around and under the core. Preformed diatomaceous silica blocks (Johns-Manvill's Superex) were used for the external low-temperature insulation. Side blocks were of three inches thickness and bottom blocks were of 5-1/2 inches thickness. The entire assembly of core and insulations was set in a 1/16 inch steel casing. Core wall temperatures were measured by two chromel-alumel thermocouples, set three inches and 24 inches from the top of the core. The thermocouples were led in through ceramic insulators, and the hot junctions were kept in contact with the core wall by spring tension. Cold junction was kept at 32~F by an ice bath. Thermocouple EMF's were measured by a standard Leeds and Northrup potentiometer. The thermocouple assembly was not calibrated as only 5~F precision was required. The test bed was designed to approximate a one-dimensional system. Rigorously, this would require a bed of infinite lateral dimensions. Physically, this could be approached by the use of mirror walls which would provide perfect reflection of any side radiant fluxes so that net transfer would be zero in lateral directions. The actual test bed was constructed of a two inch diameter aluminum tube, highly polished on the inside. This tube was set on a screen which was in turn clamped between

-51 - two annular aluminum rings, as shown in Figure 10, The test particles were then packed in the tube, on top of the supporting screen. Tubes of various lengths, from one inch to five inches, and two different screens, one of NBS mesh 30 and one of NBS mesh 18 were used during the course of the experiments. The entire bed assembly was clamped to runners on an optical bench track, which was mounted vertically on the wall. The detector used in this study was the same as that used by Larkin and Churchill(20, custom built by Charles M. Reeder and Company, Its sensing element consisted of bismuth-antimony-tellurium thermocouples with a goldblack filmed junction. Element response to ten cps signal was specified as approximately 64 percent of D.C. The total sensing junction had cross sectional dimensions of approximately 0.1 x 0.1.5 inches. The element was mounted in an evacuated brass case with a potassium bromide window. All electrical leads were shielded to minimize noise signals. The junction was so fixed in its case that the sensing element could be positioned a few millimeters below the rim of the bed tube, as shown in Figure 11. The entire detector case was also clamped to runners on the optical bench. The ruled scale of the optical bench served to measure relative positions of detector, bed, and source. When not in use, the detector was kept in a dessicator in order to prevent fogging of the KBr window, Voltage signals from the detector were passed through a primary transformer. Thermador Corporation's type HG-3 with 90 db, magnetic shielding was used. This primary transformer served the dual purpose of eliminating D.C. signals and of providing a 1:102 step-up in ten cps signal strength. Output from the transformer was then fed to a ten cps narrow band amplifier, Electro-Mechanical Research's model 33A. This amplifier featured

BOLTS RING I VO I ROD Figure 10. Details of Bed Assembly

-53 - a maximum sensitivity of one microvolt for full scale deflection with an overall voltage gain of 6 x 106. Band width was specified as one cycle to either side of ten cps at half power points. Thus, sensibly only the ten cps portion of the signal from the detector would be amplified and measured. A calibrated attenuator having a total ratio of 1000 to one was incorporated in the amplifier, together with an uncalibrated gain control. Output could be measured by either a build-in milliammeter or by externally connected recorders. The built-in meter was of variable range and had a specified accuracy of better than ~ two percent of full scale. By judicious use of the calibrated attenuator and the dual range of the meter, it was found possible to maintain precision at the + two percent value for almost all readings. Several external recorders of both millivolt and milliampere types were tried, but precisions were found to be less than that of the built-in meter. All readings of the final experiments were taken from the meter. Modulation of original source signal was accomplished by mechanical chopping. A double bladed chopper was constructed out of asbestos sheeting attached to a wire frame. The bottom face of the blade was covered with aluminum foil to reduce absorption of energy from the hot source, The blade was shaped to have equal open and closed spaces in order to maximize signal response of the amplifier. Rotated by a 300 rpm synchronous motor, the chopper provided a close approximation to the desired square wave, ten cps, pulse signal. A blower was mounted to circulate air across the space between the source and the test bed. This served the dual purpose of (a) preventing convective heating of the bed and (b) of cooling the modulator and shields.

-54 - TOP SHIELD DETECTOR BED HOLDER I "2 0 CY BOTTOM SHIELD MODULATOR BLADE SOURCE / Figure 11. Relative Positions of Apparatus

-55 - Walls and control panels around the equipment were painted a dull black to approximate the condition of black body boundaries, as assumed in the derivation of Equation (115). However, it was discovered that still enough radiation from the source reflected off the surroundings to noticeably affect readings. To prevent this, two shields had to be installed. The bottom shield, with a 2-1/2 inch hole centered over the source, served to block edge radiant fluxes from the source. It was constructed of 1/4 inch beaver board with aluminum sheeting on the bottom and painted dull-black on top. The top shield was constructed out of dull-black felt cardboard, having a surface of high absorptivity. It was conically shaped to fit over the top of the bed and served to prevent any secondary reflections off walls and ceilings from reaching the detector. Relative positions of the apparatus and the placing of the shields for actual experimental conditions are indicated in Figure 11. Test Particles Twelve different packing particles were tested. They are listed and described under code names in the following table. TABLE I Test Particles Code Name Description GS-3 solid glass spheres, 3mm. in nominal diameter, supplied by Kimble Glass Co. GS-4 solid glass shperes, 4mm. in nominal diameter, supplied by Kimble Glass Co. GS-5 solid glass spheres, 5mm. in nominal diameter, supplied by Kimble Glass Co. AS-3/16 solid aluminum oxide spheres, 3/16 inch in nominal diameter, Norton's RA 5291.

-56 - TABLE I (cont"d) Code Name Description AP-1/8 aluminum oxide pellets, cylindrically shaped, 1/8 inch diameter x 1/8 inch length nominal size, Norton's SA 103. AP-5/32 aluminum oxide pellets, cylindrically shaped, 5/32 inch diameter x 5/32 inch length nominal size, Norton's SA 103. AP-3/16 aluminum oxide pellets, cylindrically shaped, 3/16 inch diameter x 3/16 inch length nominal size, Norton's SA 103. SS-1/8 carbon-steel spheres, 1/8 inch diameter, polished surface, Hoover's grade D balls. SS-3/16 carbon-steel spheres, 3/16 inch diameter, polished surface, Hoover's grade D balls. AG-4 aluminum oxide grain, mesh 4, irregularly shaped, nominal size 6.6 mm,, Norton's number 38 Alundum. AG-16 aluminum oxide grain, mesh 16, irregularly shaped, nominal size 1.7 mm., Norton's number 38 Alundum. CG-16 silicon carbide grain, mesh 16, irregularly shaped, nominal size 1.7 mm., Norton Crystolon (regular grade). All further references to the test particles will be by these code names. Properties of the packings, such as particle emissivity, particle conductivity, bulk density, and void fraction, are given in Table XII, in the Appendix. Figure 12 presents a photograph of samples of the test particle.

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PROCEDURES The experimental data required were values of normalized transmissions, Sn(h), at a number of packed heights, h. Three quantities had to be measured for this data: (a) bed height, h, (b) incident flux intensity, So(h = 0), and (c) transmitted flux intensity, S(h). Assuming a linear detector and a constant amplifier gain, the measured voltage signals VT(h) and VT(O) would be directly proportional to S(h) and So, respectively. In this case it would be valid to treat the amplifier outputs, VT, directly as the quantity S(h), since the normalized ratio is the same, S h (S(h) VT(h) Sn(h = ()- (116) So(h=O) VT(h=O) This simplification will be utilized in all further discussions. Details of the procedures used to measure h, So, and S were as follows. Measurement of Bed Height Bed height, h, was defined as the bulk packed height through which the radiant flux was transmitted. It was necessary to use bulk height rather than actual packing height since the latter could not be specified to better than + D. To obtain measurements of h, it was first necessary to determine bulk density, PB' for each type of packing. This was accomplished by packing a bed tube to the brim and weighing the entire assembly. Then, subtracting the tare and knowing the dimensions of the bed tube, it yas possible to obtain a value of pB as, -58 -

-59 - (weight packing & tube)-(weight tube) PB -1 2 (117) where D = tube I.D. H = tube height, ft. For each packing, the above determination was made with several different tubes of varying heights. In all cases, the variation of PB between determinations was only of the order of 1%. The arithmatic mean was then used as the true PB for the packing. These values are given in Table XII of the Appendix. In an experimental run, incremental samples of the packing would be weighed out previous to the test. These incremental samples were then added to the bed during the course of the test. The bulk height for the j-th data joint would thus be, K 2 - A A (118) where AWi = i-th incremental sample. Measurement of Transmission Flux The modulated transmission, S(h), was read from the output meter of the amplifier. At a packing height, h, the detector would be centered over the bed as shown in Figure 11 and the shields placed in position. With the source set at the desired temperature and the chopper turned on, the amplifier attenuator and meter range would be adjusted to bring reading on scale. At steady state, the meter reading, R, and attenuator ratio, At, would be recorded. All meter readings, for all data points were then adjusted to a common basis of unit attenuation by, R =Rm (119) At

-60 - During the course of each run, background signal would be measured by covering the bottom of the bed with an opaque piece of transcite and taking a reading on the meter. This reading, Rb, normalized to unit attenuation as with R, was considered to be a valid measure of all stray background signals. The true transmission signal was then given by, S(h) = R - Rb (120) In the actual experiments, the bed would be repacked and the value of S(hj) measured two to four times at each packed height, hj. The arithmatic mean value of S(hj) was then used as the true signal value for that data point. Measurement of Incident Flux For a one dimensional bed, the flux incident at the bottom of the packing would be the same as that reaching the detector if the bed had no packing. In the experiment, S(h=O), the zero packing signal, was measured and used as the normalizing incident flux, SO. A difficulty in measuring S(h=O) directly was that at temperatures above 800~F, the flux intensities without packing were so high that all readings were off scale even at maximum amplifier attenuation. While this problem could have been avoided by increasing the distance between source and bed, this was not desirable since the signal with packing would then be too low for accurate measurement. In the experiments, two aids were used to obtain values of SO. First, a standard screen was placed over the source opening of the bottom shield, serving to reduce the flux intensity by a constant factor. This screen was calibrated at temperatures up to 2000~F and the factor was determined to be 5.80, independent of temperature, bed location, and source opening.

-61 - A second aid was the signal-temperature relationship. It was found by actual measurements at temperatures up to 1700~F, that a plot of SO 4 versus source temperature raised to the fourth power, Ts, could be quite accurately correlated by a straight line. This correlation was then extrapolated to 2000~F in order to obtain values of So at the higher temperatures. Thus, the actual values of SO used in the experiments were determined as, S = 5.80 x S(121) where SO = signal value from temperature correlation, with standard screen. Procedure of Test Runs To summarize, the procedure of a typical test run was as follows: 1. Determine bulk density of test packing, B. 2. Weigh out incremental packing samples, Ali. 3. Heat up source to desired uniform temperature. 4. Align source, modulator, bed, detector, and shields (by plumb line). 5. Start modulation and wait for source temperature to attain steady state. 6. Obtain value of So, by meter reading if below 1700~F, by correlation if above 1700~F. 7. Measure background signal, Rb. 8. Dump and pack into bed the first incremental sample, AW. 9. Measure transmission flux reading, R. 10. Repack bed and repeat step 9 for two or three times. 11. Take average of all values from step 10 as the true value of R. 12. Add next incremental sample and repeat steps 9 to 11 for next data point. 13. Continue till all samples have been added or till transmission falls too low for accurate measurement. 14. Re-measure background signal, Rb.

-62 -15. Calculate S(h) for each data point by Equation (120), using arithmatic average of Rb. 16. Obtain normalized transmission, Sn(h), by Equation (116). 17. Plot Sn(h) versus h for desired transmission curve.

EQUIPMENT CHECKS AND CALIBRATIONS The analytical development and experimental method described in the previous sections were based on several assumptions regarding equipment characteristics. These assumptions were checked by either experimental tests or by calculations wherever possible. These checks, along with equipment calibration, are described below. Black Body Behavior of Source The attenuation cross sections, a and sb, were defined as total cross sections for radiation of all wavelengths, i.e., for black body radiation. Therefore, it was necessary that the experimental source approximate black body behavior as closely as possible. In theory, an opening in an enclosure approaches black body behavior as the opening decreases in size, since any radiation falling on the opening would tend to be trapped within the enclosure. In practical applications where an extended source is required, the opening has a finite area so that a fraction of incident radiation would be reflected back out. The absorptivity (or emissity) of slch an enclosure would evidently be equal to a 1 - - (fraction of incident radiation reflected back o.t). For any real enclosure, es would be a function of enclosure size and shape, opening size, shape, and location, and emissivity of wall inside enclosure. (12) Gouffe has derived equations for calculating effective enclosure emissivities from these parameters. His result for cylindrical enclosures may be written as, w ('- - A A) A AO Vw- A) (122) w Aw Aw where a = effective emissivity of enclosure -635 -

-64 - Ao A' Applying AW Ao Aw A' Ao Aw Ao A' = emissivity of wall inside enclosure = area of opening = area of wall of enclosure = area of equivalent sphere with diameter equal to length of the enclosure. Equation (122) to the source enclosure of this study; = emissivity of silicon carbide = 0.9 = x 3 = 7.06 sq. in. = (l x 52) + ( r x 3 x 27) = 262 sq. in. = x 272 = 2,290 sq. in. = 0.0271 = 0.0051 0.9 s 0.9 (1-.0271) +.0271 [1 + (1-.9)(.0271 -.0031)] = 0.994 * Thus it is seen that the effective emissivity of the test source was better than 0.99. This was judged to be sufficiently close to black body behavior. Linearity of Measurement System A linear detector and amplification system was required in order to obtain meaningful transmission curves. Basically, it was required that the final meter reading be directly proportional to the flux intensity impinging on the detector. The experimental system was checked for this characteristic by means of the reciprocal-distance-squared law. It is well known that flux intensity decreases as the reciprocal of the square of the distance from a point source. Therefore, if the detector should be placed at various distances from a point source and an output-meter

-65 - I0 0 500 Cf) SLOPE =-1.9 -.....o __ 5 10 50 100 500 DISTANCE SOURCE TO DETECTOR,Cm. Figure 13. System Linearity Check

-66 - reading taken at each distance, then a log-log plot of meter reading versus distance would be a straight line of slope -2 for a truly linear system. In actual test, only an approximate point source would be practical, so that the straight line relationship might only be expected at large distances. To check the experimental system, the black body source was heated to 354~F and a half inch circular opening in the bottom shield centered over it to serve as the approximate point source. The detector was then placed at various distances from the opening, along the optical bench. The packed bed frame was removed for this test. Detector signal at each distance was then read from the amplifier-output-meter and plotted against distance in Figure 13. Numerical values are listed in Table VIII of the Appendix. It is seen that the plot of Figure 11 does approach a straight line at large distances. Asymptotic slope of the straight line portion was measured to be -1.97, well within experimental accuracy to the expected value of -2. It was therefore concluded that the detector and amplification circuit was sensibly a linear system. Temperature Response of Measurement System The total radiant intensity absorbed by a grey body detector from a grey body source is given by, i = G- (sT4 - adTd ) (123) where G = geometric factor Ts = source temperature, ~R Td = detector temperature, ~R 6s = source emissivity ad = detector absorptivity

-67 - 800 500 300:t 3 6SLOPE = 4.01 O 50 30 I 2 3 4 5 6 7 8 9 10 SOURCE TEMPERATURE, 1000 OR Figure 14. Temperature Response of System

-68 - Since the experimental source was a black body and the detector system was linear, Equation (123) takes the form, Rm - (Ts4 - adT) (124) where Rm = output meter reading. For TS>Td, Equation (124) would be approximated by Rm T 4 (125) or, log Rm = 4 log Ts + const. Test values of Rm and Ts were obtained on the experimental equipment, with bed in place but with no packing. Source temperature was varied from 900~F to 1700~F and the results are tabulated in Table VII. Plotted on loglog coordinates in Figure 14, it is seen that a straight line is obtained as required by.Equation (125). Slope of the line was determined to be 4,01, very close to the expected value of 4.0. Effect of Support Screen In the experimental test beds, a bottom screen was used to support the packings, as shown in Figure 10. The assumption was that by normalizing all transmission measurements of packing and screen to that of screen alone, the attenuation effect of the screen itself would cancel out. This assumption would be valid if the screen attenuation remained constant, independent of the presence or absence of packings. To check this assumption and to obtain an order-of-magnitude estimate of the screen effect, two test runs were performed with identical packings but with two widely different screens. Runs 10 and 14 were both made with GS-3 packing, at a source temperature of 1200~F, and with a four inch length bed tube,

-69 - 1.0 10 -I C c 2 u, 10 I I I I I P II - I I I \ \ \ \ _ Izj~ ~ ~ ~ ~ -4 -_ VIII, _ 7 _ I I _ 'T \____N 10; MESH 18 SCREEN ------- --- 1-+ RUN-14; MESH 30 SCREE-N — ^ --- —---—. \ ==E ==%= ----....\ ------- - s -- ~ ---- ---- ---- ---- ---- ____ ^ ___ ____ ____ ____ ___~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' -- -- _______________ _Q __ _____,.__, \ -— ______________ \________ --------- ------ -— _____________- 0 RUN 10; MESH 18 SCREEN *i ~.U 4 ES 0 SRE -4 10 0 1 2 3 4 5 6 7 8 9 10 h,Cm. Figure 15. Effect of Support Screen

-70 - The two screens used were: Material NBS Mesh Opening, mm. Wire diam., mm, Run 10 copper 18 1.00 0.43 Run 14 brass 30 0.59 0.29 The data points for these two runs are plotted together on Figure 15 and it is seen that the two runs represent quite similar transmission curves, The regression values of a and sb and the corresponding kir were determined to be: 2 2 ft2 ft2 Btu a -- sIb — kir u diff. in kir f f3t3 hr.ft.~F Run 10 0.96 343 0.091 7~6% Run 14 0.84 372 0.084 The 7.6 percent variation in conductivity is of a noticeable magnitude and may place a limit on the-validity of the experimental data. Indications are that the screen effect was probably one of the relatively important drawbacks of the experimental set up. However, this order of magnitude variation is still quite acceptable when compared with general experimental precision commonly reported for packed bed characterizations. Wall Absorption It was desired to have the bed walls approximate mirrow-walls as closely as possible, i.e., to have zero absorptivity. Any real wall has some finite absorptivity which would tend to decrease flux intensity. However, effect of this flux decrease on transmission curves is only secondary since normalization tends to cancel out the absorption factor. In fact, if wall absorption with packing is the same as without packing, then the normalized

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-72 - transmissions, Sn(h), are not affected at all. In real cases, packing materials change the angular distribution of flux and so the total flux absorbed by the bed wall will not be the same for packed as for empty beds. To obtain an estimate of the effect of wall absorption, two runs were made under identical conditions but with bed tubes of different lengths. Runs 10 and 13 were both made with GS-3 packing at 1200~F. A four inch length bed tube was used for Run 13 and a three inch length tube was used for Run 14. If wall absorption were important, then the 30 percent change in wall length might be expected to show a noticeable effect on the transmission curves. Data points of Runs 10 and 13 are plotted together on Figure 16. It is seen that the points coincide quite closely on sensibly one transmission curve. Values of parameters obtained by regression were: ft2 ft2 Btu aft- sbSu kir - Diff. in kir ft? ftb hrftoF ~ ir Run 10 0o96 343 0.091 1.1o Run 13 o.88 341 0.092 The agreement is seen to be remarkably good, indicating that wall absorption effects were probably neglegible. Lateral Variations The correlation model, Equation (115), was based on the one dimensional case, where it was assumed that there were no variations in either packing properties or flux intensities in the lateral directions. In the actual case, there were three types of variations that could have been expected. First, the bottom face of the packed bed might not have been uni*formly illuminated by the incident flux. This could be due to either nonuniformity of the extended source or due to differences in distance between

- SENSING I 606WIO ELEMENT c RUN 17 RUN 40 Figure 17. Placement of Detector in Lateral Variation Test

-74 - \e 10 + RUN 40; DETECTOR SHIFTED -3 0 1 2 3 4 5 6 7 8 9 10 h,Cm. Figure 18. Effect of Lateral Variation

-75 - source and various points of the bed face. Second, any absorption loss at wall would cause a decrease in flux intensity in the vicinity of the wall. Third, it is expected that local void fraction would be greater near the wall than in the bulk of the bed. A measure of the overall lateral effect was obtained by making a run (Run 40) with the detector placed off center over the bed. The results were then compared to a similar run (Run 17) with detector placed in the normal center positions as illustrated in Figure 17, In Run 40, the detector was shifted off center as much as constructional features allowed, a distance of approximately one quarter of tube diameter. Both runs were made with GS-5 packing and at a source temperature of 1600~F. The data are plotted together on Figure 18. Values of regression parameters corresponding to the two runs were found to be: 2 t ft2 Btu aft s 2 kB -tu Diff. in ki ftRu3 1ft3 hr.ft.OF Run 17 0.81 151 0.396 1.2% Run 40 0.80 149 0.401 Once again, the agreement in transmission curves and regression parameters are seen to be very good. It was concluded therefore that effect of lateral variations on experimental results was negligible. Calibration of Standard Screen The standard screen used for measurement of incident flux had to be calibrated for two functions: So a) for values of the ratio Fs = -- So where, S = true incident flux intensity 0 SO = incident flux intensity measured through standard screen.

-76 - b) for values of So at various source temperatures. Values of the ratio Fs were obtained by direct measurements of S(h=0) with and without the standard screen. It was expected that Fs would be a constant independent of bed position, area of source opening, and source temperature. To be certain, measurements were made at a number of bed positions, with several different source areas, and over a range of temperatures. In Figure 19, Fs is plotted versus bed position for different source areas, at two parameters of temperature. It is seen that the best correlation is a straight horizontal line for each temperature, indicating that Fs was truly independent of bed position and source size. In Figure 20, the average value of Fs at each source temperature is plotted against temperature. It is seen that at temperatures below 800~F, Fs was not independent of temperature but decreased with decreasing temperature. Above 800~F, Fs was constant at a mean value of 5.80, independent of source temperature. Since all experimental runs were made at temperatures of 8000F or higher, this constant value of 5.80 was taken as the value for Fs. As explained in a previous section, it was necessary to obtain a calibrated correlation of So versus temperature for extrapolation to higher temperatures where So and So could not be measured directly. To use such a correlation, it was necessary to maintain constant gain at the amplifier and to have consistent placement of bed and detector. Under such conditions, S* would be a function only of source temperature, bed tube length, and type of bed support screen. Experimental measurements for So were obtained over a temperature range of 600~F to 1700~F, for two different bed tube lengths and with the mesh 30 support screen. Numerical values are tabulated in the Appendix, Table X. In Figure 21, So (in terms of meter readings) is plotted against source temperature (in ~R raised to the fourth power). It is seen

6.0 5.8 5.6 5.4 5.2 U.O 5i 0 z L.I bJ t0 cn 5.6 5.4 5.2 — f' -' --- —-. --- -- _ <__ —. —._ f — ---- — _ AVG. z 5.77 SOURCE DIA. I" o 22 2 3 4 2 l______1 __ It 570 *F ______ _ A-E._]__ _ AVG 5.46 A IJ -,1 -,1 40 50 60 70 80 BED POSITION, (FRAME Figure 19. Position and Source Area 90 100 SETTING, cm.) Dependence of Fs 110 120 130

6.6 6.4 6.2 6.0 5.8 F s 5.6 5.4 5.2 5.0 4.8 4.6 i tI I I0 -[~ - -1 1 1 Im ^.. - ---— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ __^__ _ ___ _ _______~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I I 400 600 800 1000 1200 1400 1600 1800 2000 2200 ts (oF) Figure 20. Temperature Dependence of Fs

-79 - 10( J 500.. 4 z (/3 400 /200 l a n 2+ 4" BED TUBEJ SUPPORT SCREEN 200 -------- 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 T, 10 ~R Figure 21. Temperature Calibration for So

-80 - that linear correlations fitted the data points very well. For the four inch bed tube, the dependence of So on source temperature was correlated by So(Ts) = 44.8 x (Ts~R) x 102 (126) and for the three inch bed tube, by, S*(TS) = 47.4 x (Ts~R)4 x 10-12 (127) where S is given in terms of meter readings normalized to unit attenuation at the amplifier. Relations (126) and (127) were used to obtain Valties of S* 0 at temperatures above 1700~F. It should be noted that the data points of Figures 20 and 21 were obtained over a period of two months. This showed that the standard screen was sensibly not affected by the atmosphere or by handling and that the correlations could be used over a period of several months without recalibration.

RESULTS AND DISCUSSION Transmission Curves A total of fifty-two runs were made with twelve different packings. For the glass particles (GS-3, GS-4, GS-5) it was possible to obtain measurements down to 8000F. For the other opaque particles (AS-3/16, AP-1/8, AP-5/32, AP-3/i6, SS-1/8, SS-3/16, AG-4, AG-16, CG-16) it was possible to obtain accurate measurements only at temperatures above 1500~F. The data for valid runs are presented in Table VI of the Appendix. As explained in the Description section, the final transmission value for each packing height was obtained from the arithmatic average of several measurements, with the bed being repacked between each measurement. These average transmission values are plotted as data points in Figures 22-33. Each figure represents one type of packing, with parameters of source temperature. Runs were made for the glass particles at temperatures of 800 ~F, 1200 ~F 1600 ~F 1800 ~F, and 2000 F. For all other particles, runs were made at 1600~F, 1800~F, and 2000~F. It is seen that for every run, the natural log of Sn(h) did approach a linear dependence on h at large values of h. Therefore, Equations (178) and (179) were valid for estimating first-guess values of m and an. By regression formulas developed in the Appendix, transmission curves were fitted to the data points according to Equation (115). These curves are plotted on Figures 22-33 to show the degree of correlation with data points. It is evident that the two-flux model of Equation (115) was very successful in fitting the experimental data. In every run, the regression curve seemed to be coincident with the "best curve" that could

-82 - 10 F...-^,^ ^ -—. \- \ -\ )... \ a 10 0 1 2 3 4 5 6 7 8 9 10 h, Cm. Figure 22. Transmission Curves for GS-3

-83 - 1.0 -I 10 I & n I I I I C u() I - I I I -2 10 - ~\ |==., _ _ RUN F 0 8 800 tyW~ -^A II 1200 + 16 1600 \\ 29 1800 \O \0 42 2000 \ \l\..___ \ __ _________.._ -3 I0 -4. 10 0 I 2 3 4 5 6 7 8 9 10 11 h,Cm Figure 23. Transmission Curves for GS-4

Sn(h) oI o,. o0 0 A) (D IR) ro.p (A 3 r0 Vl H0 a) F0 C I \J1 rC 3 U, m~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 io(:, =. m m / / j/~ / /^ c / / ^w ^ Q~~~~~~~~~~~~~~~~~~~~~~~~~~~~E + D> 0 W r) CU (D - z PO?O o o + Io 0 0 0 0 0 0 0 O0000 (0

-85 - 1.0 -1 I0 C ( (n -2 10 + 18 1600 -i 31 1800 O 44 2000 l____- "^ -__-~~ii i i ~i ^^^^^^~~~~~~~~~~~ -------— ^ —~~~ ____________I__1_______I_____I__I -3 I0 -4 I0 0 I 2 3 4 5 h,Cm. Transmission Curves for AS-3/16 Figure 25.

-86 - LO 10 -C.c c C) lid RUN OF + 20 1600 \o 32 1800 \\ x — 45 45 2000 \\ \\ _\.%.... l L 1 1 n I I '\\\-=^^E:^===EE.....I I -................ \X I3 -4 10 0 I 2 3 4 5 h,Cm. Figure 26. Transmission Curves for AP-1/8

-87 - I.0 -I 10 -,-, c nf RUN OF + 21 1600 o 33 1800 0 46 2000 )\\-....... --- —-..\<\ ---------— I ----- x \I^ --- —------ ----------- ----------- ----- -------------- \~~~~~~~ ~~..... --- —-------------......= ===,,-..-= ^^_^ ^zr~~~~~~~~~~~~~~z:~~....^^-= ---— ~~~~~ ^\"'~-.... -------- - \ --- — -------------------- V ~ s ------ 210.10 -3 I0 0 10 2 3 4 5 h,Cm. Figure 27. Transmission Curves for AP-5/52

-88 - 1.0 -I 10 -c c C) -2 10 I ^\\~ ~+ 22 +i6 13 34 1800 aN x --- —-- 0 47 2000 \xx M\ - __\\\..... - I~as XNNX ~ ~ \N ~\..... -----,~~~, \\-\. x I.... \ \ \ I - --—, --- —-----—..... --- — ~~~~~~'^ -3 10 -4 I0 0 I 2 3 4 5 h,Cm. Figure 28. Transmission Curves for AP-5/16

-89 - 1.0 i RUN OF + 23 1600 - \ _ 38 1800 10 0 51 2000 I-o0 2 3 4 Figure 29. Transmission Curves for SS-1/8 *V~~~~~~~~.. O 1 2 3 4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_ h, Cm.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Fiue2.......... uve fr S-/ 5

Sn(h) 0r 01 b.m 0 F-* (D 0 N UUP) U) H(1) O U) 0 2 I' -yJ (D Eo CO cr\ -) 3 I I to (0

-91 - 1.0 10' -d2 10 lo" I R~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I ~~~I I. Run ~F + 25 1600 \\ i ______________. El 35 1800 xi \\\ - 48 2000 A\I \\...,,, -- ~ -- = -. N... ~~~~~~~~~~~~~~.. ~~~~~~~~~~~~~.. _ - o Vx. x~~~~~~~~~~~~~~~ v, X\~ .~ ~~ \\N, \%\.. ~.. ===^=~~~~. w c () -3 10 -4 10 m 0 I 2 3 4 5 h, cm. Figure 31. Transmission Curves for AG-4

-92 - 1.0 120 -2 IO a I la Run OF \ + 26 1600 \ 0 36 1800.\... -z Q 49 2000. -. a~~~~ \ t. i iiiii i i i i_ C c n) -d I0 d4 I0 10s C ) 0.5 I 1.5 2 2.5 3 h, cm. Figure 32. Transmission Curves for AG-16

91-O0 0OJ saAJnD tOTBsTmsUSUJL *C a~jThLj 'U(3 6 4 911 O' I s'o 0 ii iai ----l ii -- 0002 0 1X 0081 L 1 E 0091 LZ + ----— o un —......!s ~ 0081..... 0091 ~~ +~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0I 9. 01 01 C, en.. z01 01 0*1 A -~6 -

-94 - have been drawn through the data points by eye. This fact is also evident from Table XI which compares the transmissions calculated from the regression curves to the experimental transmissions. The differences between the calculated and experimental values are seen to be generally small, of the order of 2-8%, and to be fairly well divided between positive and negative differences. A statistical measure of the correlation fit may be obtained from the standard deviation, defined as., -.nr / i(128) where i is given by Equation (143) n - number of data points used in correlation. Standard deviation, 5, has the physical significance that in 100 measurements of Sn, 95 of the measurements may be expected to fall within t 2 z6 7 a % of the value predicted by the correlation. Standard deviation values for the transmission curves are given in Table XI. It is seen that the deviations were lowest for glass particles and highest for silicon carbide particles. This verified an experimental characteristic which was noticeable during operation —that precision and reproducibility were best for packings with high transmissions. Several trends should be noted about the transmission curves of Figures 22-33. First, it is seen that the curves for glass particles show the greatest separation between temperatures. This is consistent with the fact that glass transmissivity is highly dependent on wavelength of the radiation. At 8000F, the wavelength of maximum intensity is 4.9 microns,

-95 - to which glass is mostly opaque. At 2000~F, the wavelength of maximum intensity is reduced to 2.2 microns which is almost in the optical range, and to which glass is much more transparent. The transmission curves for all the other test particles, which remain opaque to radiation of all wavelengths, do not show as wide a separation between temperatures. It is also seen that transmission curves for packings of high absorptivity tend to have steep slopes and to approximate straight lines. This is as expected since a packing of infinite absorptivity would have zero transmission, and the transmission curve would approach the limit of a vertical straight line. Among the test particles, CG-16 approached the closest to this limiting case. A side effect of steep transmission curves is the tendency to increase standard deviations since a small variation in h would correspond to a large variation in Sn. This may be seen from the curves for CG-16, Figure 3355 which approach all data points quite closely yet have relatively high standard deviations of 18-27%. Attenuation Cross Sections The main objective of this experimental investigation was to obtain values of the attenuation cross sections, a and Sb, for a number of typical packings. The values of these parameters determined by regression, for the twelve packings tested, are listed in Table II below. These values correspond to the transmission curves of Figures 22-33. The relative importance of absorption and back scattering is immediately evident from examination of Table II. For all test packings except CG-16, it is seen that back scattering is definitely the major mechanism of attenuation. This is especially true for the alundum and glass

-96 - TABLE II ATTENUATION CROSS SECTIONS Packing Particle ts( ~F) Run a (ft ft3 ft2 sb ( ft3 GS-3 800 1200 1200 1200 1600 1800 2000 800 1200 i6oo 1800 2000 GS-5 AS-3/16 AP-1/8 AP-5/32 AP-3/16 AG-16 800 1200 1600 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 7 14 10 13 15 28 41 8 11 16 29 42 9 12 17 40 30 43 18 31 44 20 32 45 21 33 46 22 34 47 0.390 0.843 0.969 0.884 1.54 1.88 2.21 0.209 0.521 0.988 1.33 1.59 0.178 0.437 0.812 o.804 1.09 1.24 2.74 2.70 2.84 1.06 1.49 1.47 1.00 1.32 1.54 1.96 2.18 2.77 977 372 343 342 190 153 129 878 298 149 114 95.0 952 296 151 149 114 96.8 496 483 449 1190 860 847 906 682 576 682 614 496 779 811 837 26 12.7 36 10.6 49 9.37

-97 - TABLE II (CONT'D) Packing ft2 ft2 Particle t (~F) Run a (ftt) Sb (ft7) AG-4 1600 25 2.05 789 1800 35 1.16 1090 2000 48 1.20 1010 CG-16 1600 27 272 420 1800 37 107 1050 2000 50 192 627 SS-1/8 1600 23 13.1 450 1800 38 17.0 387 2000 51 20.5 343 SS-3/16 16oo 24 6.66 370 1800 39 8.70 322 2000 52 9.26 314 particles (spheres, pellets and grains), all of which have absorption cross sections of less than 2% the value of the back scattering cross sections. For these packings it would be valid to assume that the medium is sensibly non-absorbent and to use Equations (74) and (75) to describe radiant heat transfer. The absorption cross sections are significant for steel spheres —having values of approximately 2-7% of that of back scattering cross sections. Silicon carbide packings are seen to have the greatest absorption cross sections, as might be expected since SiC particles with its black surface have a very high emissivity. The magnitudes of the absorption cross sections for CG-16 were determined to be approximately 5-60% of the magnitudes of scattering cross sections. Equations (75) and (74) can not be applied to packings with such high absorptivities, so it is recommended that the approximate Equations (99)

-98 - and (100) be used in describing radiant heat transfer for steel or silicon carbide packed media. Radiant Conductivities The internal radiant conductivity, kir, was calculated for each of the test runs by Equation (104). These values are useful in obtaining an estimate of the relative ease of radiant heat transfer in the different packings. In Table III, representative values of kir for each packing are tabulated, in approximate decreasing order. TABLE III RADIANT CONDUCTIVITIES (kir ft ) hr.ft. OF) Temperature ~F Packing Particle 800 1200 1600 1800 2000 GS-5 GS-4 GS-3 SS-3/16 SS-1/8 AS-3/16 AP-3/16 AP-5/32 CG-16 AG-16 AP-1/8 AG-4 0.0144 0.0156 0.0140 0.106 0.105 o.o840 0.401 0.401 0.314 0.161 0.131 0.120 0.0877 0.0661 0.108 0.0762 0.0504 0.0758 0.693 0.689 0.515 0.243 0.200 0.163 0.129 0.116 0.0714 0.0968 0.0918 0.0729 1.05 1.06 0.785 0.320 0.289 0.227 0.205 0.177 O.141 0.121 0.120 0.101 Several general trends are evident from Table (a) kir increases in the order -- alundum, steel, and glass. (b) kir increases for increasing temperature. (c) kir increases for increasing particle size. III: silicon carbide,

-99 - Packings of glass particles are seen to be the best radiant "conductors", especially at higher temperatures. Physically, this can be explained in terms of the transmission of low wavelength radiation, which is effectively counted as part of the unattenuated flux in Equations (55) and (56). For opaque particles, the conductivity results from a balance between back scattering and absorption. For alumina, which has a high surface reflectivity, attenuation is largely controlled by the large back scattering cross section, accounting for the relatively low conductivities. For silicon carbide and steel particles, absorption becomes important and the attenuation must be attributed to both back scattering and absorption. Increasing emissivity of solid increases absorption cross sections but decreases scattering cross sections. The overall effect generally is to increase radiant conductivities. The second two trends of kir, increasing with increasing temperature and particle size, are as expected. Almost all analytic works predict an increase in radiant conductivity for higher temperatures and larger particles, as shown by Equation (5) from Schotte's(33) study, Equation (6) from Rosseland's(31) study, and Equation (3) from Laubitz's(21) study. Of the two, the temperature effect was definitely more pronounced. These two trends, showing the effects of temperature and particle size, are discussed in greater detail in later sections. Relative Importance of Radiation and Conduction The ratio of heat transfer by radiation to that by conduction is of practical interest in defining the conditions where radiant transfer becomes important. Previously, this ratio had been experimentally determined

-100 - for only one packing -- Hill and Wilhelm's(l5) alumina spheres. From the experimental measurements of this present study, it was possible to determine the radiation-conduction ratios for all twelve of the packings tested. Conductive heat transfer is commonly specified by a bulk conductivity, kc. Wilhelm et al.(38) have proposed a semi-empirical method for calculating kc, based on Schuman and Voss' model.(34) By means of this method, values of kc were obtained for the test beds. The ratios kir/kc were then determined and used as a measure of the relative importance of radiation and conduction. Details of the calculation are given in the Appendix and the results are shown in Table IV. Some representative radiation-conduction ratios have been plotted against temperature in Figures 34 and 35. Figure 34 shows the effects of different particle materials and shapes. Glass spheres are seen to have the largest ratios and to have the greatest temperature sensitivity. For packings of 5 mm. glass spheres, radiation accounts for 50% of total heat transfer at a temperature of 1300~F. At higher temperature, the ratio increases rapidly so that at 20000F, radiation accounts for 85% of all heat transfer. The packings of opaque particles are seen to have much lower radiationconduction ratios, which are also less sensitive to temperature. At 1600~F, the radiant heat transfers are all less than 50% of total heat transfer, for 3/16 steel spheres, 3/16 alumina cylinders, 3/16 alumina spheres, and mesh 4 silicon carbide grains. At 2000~F, the radiant transfers increase to 61% and 58% for alumina cylinders and spheres respectively, while for the steel spheres and silicon carbide grains, the radiant transfers are still below 50%, at 45% and 35% respectively.

TABLE IV TO CONDUCTION RATIO RADIATION 800 OF kc kir kir kc 1200 ~F kir kc kir kE 16oo ~F 800 OF 2000 ~F k k kir kc kir kc k ir k kir kir c i rE7 kir kc kir kc~ GS-3 GS-4 GS-5 AS-3/16 AP-1/8 AP-5/32 AP-3/16 AG-16 AG-4 CG-16 ss-1/8 SS-3/16.136.136.136.014 o.io.015 0.12.014 o.io.151.151.151.091.105.106 0.60 0.70 0.70.167.167 j.167.154.098.098.098.132.115.246.354.314.401.396.120.050.066.088.076.076.108.131 1.88 2.40 2.37 0.78 0.51 0.67 0.89 0.58 0.66 0.44 0.37.173.173.173.161.103.103.103.138.121.254.372.515.689.693.163.092.116.129.097.073.071.200 2.98 3.98 4.01 1.01 0.89 1.26 1.25 0.70 o.60 0.29 0.54.178.178.178.168.131.131.131.144.126.259.389.785 1.o6 1.05.227.120.177.205.121.101.141.289 4.41 5.96 5.90 1.35 0.92 1.35 1.56 0.84 0.80 0.54 0.74 H 0 I O.354.161 0.45.372.243 0.65.389.320 0.82 B.T.U. Units of k = hr.t. ~F

-102 - 5.0 4.5 4.0 3.5 3.0 2.5 C, 2.0 jt. 1.5 1.0 0.5 0 PACKING 0 GS - 5 a AS - 16 El SS - 16 x AP - + CG - 4 ----- HILL & WILHELM S DATA 800 1000 1200 1400 1600 1800 2000 TEMPERATURE,OF Figure 34. Radiation to Conduction Ratio

-105 - 5.0 4.5 4.0 3.5 3.0 2.5 ___ PACKING _ 0 GS - 5 A GS - 3 i_ + AP - 3/16 X AP - 1/8 - - U 1%. Ia. 2.0 1.5 1.0 0.5 800 1000 1200 1400 1600 1800 2000 TEMPERATURE, OF Figure 55. Radiation to Conduction Ratio

Hill and Wilhelm's data(l5) have been replotted on Figure 34 for comparison. It is seen that their values, which were obtained for 3.8 mm. alumina spheres, agree quite well with the data for 3/16 inch (4.8 mm) alumina spheres of this investigation. Figure 35 shows the effect of particle size. Two types of particles are presented -- glass spheres and alumina cylinders. In both cases, it is seen that radiant transfer is more important for larger particles. Thus at a temperature of 16000F, radiation accounts for 65% of all heat transfer for 3 mm. glass spheres,'but accounts for 73% for 5 mm. glass spheres. Similarly, the fraction increases from 34% to 47% in changing from 1/8 inch alumina cylinders to 3/16 inch alumina cylinders. This dependence on particle size is more pronounced at high temperatures than at low temperatures, as shown by the increased divergence of curves in Figure 33 at higher temperatures. In general, the results indicate that radiation starts to become important at temperatures of about 10000F. The radiant heat transfer becomes increasingly important relative to conductive transfer as temperature increases, and becomes dominant at temperatures of about 1400~F to 22000F. Comparison With Analytic Estimates As mentioned in Part I, a number of authors have proposed equations for estimating radiant conductivities of particulate media. To compare these analytical estimates with the experimental data, radiant conductivities were calculated by three of these proposed equations for packings of AP-1/8, SS-3/16, and GS-3 particles. The equations used were: (a) Damkohler(6) - Argo and Smith(l), K - 4r( - D T3 (2b)

-105 - (b) Schotte (33), -= ' P. _ __ 4_ _ _ (5) 'Ks XK0 where Xr = 4 DpT3 (c) Rosseland(31) - Bosworth(3) 7k - r 1 T3 (6) where 1 = mean free path = Dp Details of the calculations are given in the Appendix. Results have been plotted together with experimental conductivities on Figures 36, 37, 38. It is seen that the agreement is generally not very good, either among analytic estimates themselves or between estimates and data. While the order of magnitudes are the same, variations range through factors of O - 8. The estimated values are seen to be all too high for the two opaque packings. For the glass packing, the estimates are seen to be too high at low temperatures and too low at high temperatures. One possible explanation for the poor agreement is that all the analytic estimates were based on extremely simple models, as described in Part I. The models assumed no interaction between particles, which actually may not be valid for packed media where the particles are in physical contact. It is conceivable that two particles together would have a larger effective scattering cross section than the two particles separately, due to the greatly reduced free path length in vicinities of contact points. Such a situation would of course reduce the equivalent conductivity of the

-106 - 1.0.8.6.4.2. I..08. - c.06..04.02.01 k3 k -=< R=~~~~~~~~I......~ I kir _ —^r --- —— _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ kir = Exp. Value kI Est. by DomkohlerArgo & Smith eq'n k2 ~ Est. by Schottl eq'n ks; Est. by RosselandBosworth eq'n.008 ------.006 - U. Ila.JA. — L.- -- 0 0 0 0 0 0 0 000 In48 _ _ _o _ _.004 1960 2060 2160 2260 2360 2460 2560 TEMPERATURE, TOR Figure 36. Comparison with Analytic Estimates for AP-1/8

-107 - 1.0.8.6.4.2 L 0 U. I Ag. I.08.06.04 k2 kir Exp. Value k, Est. by Domk8hlerArgo a Smith eq' n k2 Est. by Schotte eq'n lk Est. by Rosselond Bosworth eq' n.......... _~~ _ _ _ _ _. _. L _ _ _ _ I. _..02.01 I IUL LL IL IL.006- 0 - - I 0 0 0 0 0 0 I0 0 0 0 0 o 0 t1 0. c0 co 1960 2060 2160 2260 2360 2460 2560 TEMPERATURE, OR Figure 37. Comparison with Analytic Estimates for SS-5/16

-108 - LL 01 x Q: 3 bL 1.0 0.8 0.6 0.4 -- 0.2 0.1 0.08 0.06 0.04 0.02 0.01 l 0.008 0.006 U 0 0 0.004 D 1260 1460 1660 1860 2060 2260 2460 TEMPERATURE,OR Figure 38. Comparison with Analytic Estimates for GS-3

-109 - packed media, accounting for the low values obtained experimentally. One possible way to modify the analytic expressions in order to account for this increased attenuation is to use an effective particle diameter, D_, which is smaller than the actual particle diameter, Dp. This would serve to reduce the conductivities calculated by all three of the analytic estimates A second factor, that may account for part of the failure of the analytic estimates, is that none of the three estimates allowed for particle transmission. Thus the estimated conductivities fell too low for glass packings at high temperatures, where a fair portion of the flux transport may occur by direct transmission through the glass particles. To correct for this, some equivalent conductivity due to transmission would have to be added to the present expressions (2b), (5), and (6). This additive term would be expected to be directly proportional to material transmissivity which would in turn be a function of radiation temperature. The combined effect of the two suggested corrections, i.e., using D' and an additive kr p of transmission, would be to lower estimated conductivities for opaque particles.and to raise estimated conductivities for transparent particles. Such a modification would bring much better agreement between predictions and data, as evident from Figures 36-38. It is interesting to note that most of the analytic expressions for radiant conductivity are of the form, Xk 4 K Dp T3 X K (129) where K is some arbitrary function given by the specific analysis. This similarity between the various expressions have been noted previously by

-110 - Gorring and Churchill(ll) and by Hill and Wilhelm.(i5) The entire problem of predicting radiant conductivities may be reduced to a problem of finding the correct K function. In the three analytic models tested, the function K is seen to be: (a) Damkohler - Argo and Smith, K - (130) (b) Schotte, K -= ^(131) (c) Rosseland - Bosworth (132) K - The K function may also be separated out from the expression for kir, Equation (104), as follows. 4 a Dp T3 8rT3 (a + t b)(4 DpTr) _ Z p2 a - S sb,Dp~"~ (133) Comparing Equation (133) to Equations (130) - (132), it is seen that the K functions given by the three analytic models are basically too simple to handle the true situation. Rosseland and Bosworth's function is a set constant, while Schotte's and Damkohler - Argo - Smith's functions are dependent only on emissivity. In the real case, it is expected that K

-111 - would be a function of temperature of radiation, void fraction of packing, particle shape, size, and transmissivity as well as of emissivity. Thus, any close agreement between the analytic estimates and true conductivities would actually be fortuitous. The only exact method of obtaining K functions and conductivities at this time is to experimentally measure the attenuation cross sections, a and sb, and applying Equations (133) and (104), as was done in this study. Effect of Temperature All analytic expressions for radiant heat transfer predict that radiant conductivity would have a third-power temperature dependence. Equations (2b), (5), and (6) are typical examples. By Hamaker's formulation, kir would also be proportional to T3 if sb and a were temperature independent, as seen by Equation (104). However, the experimental data show that for most of the test packings, sb and a were affected by the temperature of the radiation. This is evident from Figures 39-50, where the experimental values of sb, a, and kir have been plotted against temperature for each test packing. As mentioned previously, the effect of temperature was most pronounced for glass particles. As an example, the value of sb changed from 977 to 129 and the value of a changed from 0.4 to 2.2 for GS-3 packing over a temperature range of 800~F to 2000~F. The variation of sb and a were much less for the other opaque packings. Some typical values are: Sb (ft-1) a (ft-1) 1600 F 2000 F 1600 F 2000 F AS-3/16 496 449 2.7 2.8 AP-1/8 1190 847 1.1 1.5 AG-16 779 837 12.7 9.4 SS-1/8 450 343 13.1 20.5

-112 - 1.0 4. F. 39 T r 10 i i " -s.16 U.^iiL,0 -. 00 0 0 0 0 O 0 1260 1460 1660 1860 2060 2260 2460 TEMPERATURE, R Figure 39. Temperature Effect for GS-3

-113 - l3e 101: I0 18 48 o C TEMPERATUR,R Figure 40. Temperature Effect for GS-4 1.0 -------------------------— ^ ^ - ^. 1260 1460 1660 1860 2060 TEMPERATURE,R Figure 40. Temperature Effect for GS-4 2260 2460

-114 - 0 10 1 IC I-. lC U0 01 I.C 1.0 Sb I0 " I_ I_ I\ I I "a.~~~~~^\ 1, 1 1-1 - 1 IL O0 IL Ii., m, IL S I - - I -I 10 1260 146 1660 1860 2060 2260 2460 TEMPERATURE, ~R Figure 41. Temperature Effect for GS-5

-115 - 2000 1000 100 I. 0 <) o 10 1.0 Sb.___2 _ IL o 0o L O 0 I =_ ~__ t8 t_ t) t........ 0~~~~~~~~~~~~~~~~~~~~~~~~~~,. 10 1.0 0 0: I 3 SD ic -I -2 I0 1960 2060 2160 - 2260 2360 2460 2560 TEMPERATURE, OR Figure 42. Temperature Effect for AS-3/16

-116 - 2000 1000 - --— 1___ --- —-_ _____ - - 10 Sb 100 ' 1.0 UI o IL U)~~~~~~~~~~~~~~~~~~~~~~~~~~~Ikir I0 0~~~~~~~~~ 0 1.. 0 00 0 0 0 0 I960 2060 2160 2260 2360 2460 2560 TEMPERATURE, OR Figure 43. Temperature Effect for AP-1/8

-117 - 2000 1000 100 I-: 1U. (N) I0 0 10 1.0 ~LL. LL..U. _., in:: IL CID 1 1-7 1 l~l l l l2l 11 1X 1.1 1 1.1.1. IrI 1 0~~~~~~~~~.. '.,. —~ m 10 1.0 U. 0 II 113 to au 10' -2 I0 1960 2060 2160 2260 2360 2460 2560 TEMPERATURE, OR Figure 44. Temperature Effect for AP-5/32

2000 1000 ii I ii i I i I i 100 -: L. m U).0 ir LU. IU. U. L. I.. ILL a 0 0 0 0 0 0 0 0 0 0 0 O 0 0 0 0 ~U 0. 0 0 - ii ii 10 1.0 U. o I r i,_ ao L 10'I -2 I0 10 1.0 1960 2060 2160 2260 2360 2460 2560 TEMPERATURE, OR Figure 45. Temperature Effect for AP-3/16

-119 - 2000 1000 100 IL.o o o 0 o0 0 0 __ ----- ~ _ — o -- -1 — --- — _ _i. 0 o o 0 0 0 0 0 0 0 0 _ _ _ 0 0 _ ~ 0 I O C c 0 -- ii ~ ~ ~ i,, i U. UL x 3: I 10 1.0 10 -I 10 1.0 10-2 1960 2060 2160 2260 2360 2460 2560 TEMPERATURE, OR Figure 46. Temperature Effect for SS-1/8

-120 - 2000 1000 100 IL.0 V) ( Sb _. kir L __ (~ ___ IL U L L 1 0 0 o 0 ~ ~ 0 ~0____ ~ _0 00 1.0 LL 0 II.or k~ 10 10 10 1.0 i-2 I0 1960 2060 2160 2260 2360 2460 2560 TEMPERATURE, OR Figure 47. Temperature Effect for SS-3/16

-121 - 2000 1000 100 1 -U. en o I0 1.0 1.0... Sb f~~~~ k. 0 M U. u. U,,- U. U. 6_ U. _ it) 0. - i i i i 10 1.0 L6 0 IU. x m cic J~ 10' lo-2 1o 1960 2060 2160 2260 2360 2460 2560 TEMPERATURE, OR Figure 48. Temperature Effect for AG-4

- 122 - 2000 1000 100 IL6 d. Vo a I - Sb _. kir LA..... i. IL. A. _ I. l 0 0 0 o o0 o 0 0 0 0 0 0 0 ~ _.___ -- _ ea i~~r i i ii i l o i0 Ic Lw 10 1.0 10 1.0 10 ' 1 2 1960 2060 2160 2260 2360 2460 2560 TEMRERATURE, OR Figure 49. Temperature Effect for AG-16

-123 - 2000 1000 100 lo V) u. ao 00. 1.0 It I&I IX L If I I IL -t —.. iro < 1 o 0 o 0 0 0 0 0 0 0 0 0 0 U_ ~_ Q_ ~ o ------- N s ^ -------- - 3-~~ i i i i l IlI I 10 1.0 LL 0 I IL I i L. m 10 I0 1960 2060 2160 2260 2360 2450 2560 TEMPERATURE, OR Figure 50. Temperature Effect for CG-16

-124 - It is evident that the attenuation cross section, sb and a, may not be assumed to be temperature independent. Therefore the temperature dependence of kir may not be an exact third power function. This effect was also found in Larkin and Churchill's(20) work. Assuming a general power function, ~K; ~ TP (134) the value of p would be different from 3.0 for packings where sb and a are temperature dependent. Some sample values have been determined, for the test packings, over a temperature range of 1600~F to 2000~F and are listed in Table V. TABLE V TEMPERATURE DEPENDENCE OF kir ki (_Btu ir hr.ft. ~F Packing Particle 1600 F 2000 F p GS-3 0.314 0.785 5.2 GS-4 0.401 1.o6 5.5 GS-5 0.401 1.05 5.5 AS-3/16 0.120 0.227 3.6 AP-1/8 0.050 0.120 4.9 AP-5/32 0.066 0.177 5.6 AP-3/16 0.088 0.205 4.8 AG-4 0.076 0.101 1.6 AG-16 0.076 0.121 2.6 CG-16 0.108 0.141 1.6 SS-1/8 0.131 0.289 4.5 SS-3/16 0.161 0.320 3.9 It is seen that the powers, p, are quite different from a constant value of 3. For the grain packings AG-16, AG-4, and CG-16, the values of p are less than 3. This is due to sb increasing with temperature, opposing

-125 - the effect of the T3 term in Equation (104). For all other packings, attenuation cross sections decrease with temperature, serving to magnify the temperature dependence of kir, as indicated by p values greater than 3. Effect of Packing Properties The data showed a number of qualitative trends in the effects of packing properties on radiant heat transfer. The most important single effect was that of particle transmissivity, as demonstrated by the large change in sb and a for glass particles between opaque conditions (low temperatures) and transparent conditions (high temperatures). This effect has been described in previous sections and will not be discussed further here. Probably the second most important effect was that due to particle shape. This effect is hard to define since there are no quantitative means of characterizing particle shape. It is necessary to use some secondary property which is related to particle shape and which can be measured. One possibility is void fraction of the packed media, a property that is dependent on particle shape. To test this possibility, the data for AS-3/16, AP-3/16, and AG-4 were compared. These three test packings are of the same material and particle size, differing only in particle shape. Experimental values of sb and a for these three packings are plotted against void fractions in Figures 51 and 52. It is seen that the results are irregular, with no evident trend. This is taken to mean that there is some other factor connected with particle shape which overshadows the effect of void fraction. It would be of interest to isolate and define this factor by more detailed investigations in the future.

-126 - 2000 1000 800 600 ILL., 400 20 200 _ -k - -| \^t s= 1600 OF __ _.___ | I- < tss 1800___I B. t,= 1800 OF 1 — _- S ---_- et t= 2000~F lII 4 l I oi - -4- -__ Inn 0.40 0.50 0.60 0.70 0.80 VOID FRACTION, p Figure 51. Effect of Void Fraction Cross Section on Scattering

-127 - 5.0 4.0 3.0 IU. Vb M L I I ___t|________ 1' ________ O 4-;c3 * t a 2000 OF i ______ __t_________ - 1800 *F t M - I 2.0 1.0 0.40 0.50 0.60 0.70 0.80 0.90 VOID FRACTION Figure 52. Effect of Void Fraction on Absorption Cross Section

-128 - Particle emissivity was found to have a noticeable effect on attenuation cross sections. Among the packings tested, there were two pairs which could be used to demonstrate this emissivity effect: (a) AS-3/16 and SS-3/1l69 (b) AG-16 and CG-1.6o The two packings in each pair are of the same particle shape and size, differing only in emissivities. Values of the cross sections for these four packings, at three temperatures, have been plotted against emissivity on Figure 5- The lines connect particles of each pairo General trends are seen to be (a) sb decreases with increasing emissivity, (b) a increases with increasing emissivity. This is consistent with physical expectations since a black, high emissivityv surface has lower reflectivity and higher absorptivi.tyo One last noticeable effect was that of partic'le size-, Figures =4 and 55 show the variation of sb and a, respectively, with equivalent particle diametero The lines connect data points for particles of the same material and shape, differing only in sizeo It should be noted that particle diameters were known only nominally so that these plots can only be interpreted qualitativel.y It is seen that the trend for sb is to decrease with increasing particle size,.as may be expected since the number of particles per unit volume decreases for larger particleso The trend. for absorption cross section, a, is not as clear -- decreasing for GS and SS packings but increasing for AP packingso The total. effect on radiant conductivity, kir3 is shown by Figures 56 and 57~ A quite definite trend is evidento Conductivities are seen to increase with increasing particle size for a.ll three types of packings and for all the temperatures testedo The effect also seems to be more pronounced at larger particle sizes and at higher temperatureso

-129 - l0 ao' - --- /A 1800 0/ (I,-..0 IO., ~~~~20 00 *F _ lot |tl^ _1800 'F -- -- -^ 1600.*F_. - | - -- - - - -- - - - | --- - Sb1 - a 0 AS - 3/16 1.0 A SS-3/16 + AG - 16 w X CG- 16 X Z z ~10Z' L — 1-1 --- —- 10 w U 0.2.4.6.8 1.0 PACKING EMISSIVITIES, E Figure 53. Effect of Emissivity

-130 - 9000 800 600 400 200 AP- i/8' AP-5/32 Ei AP-3/16 SS-I/8 SS -3/16 0 GS-3 GS-5 GS-4 r~ ~~~___ ____ I___ ____ ___ ___ U. u) 100 80 60 40 20.11 I i At 20nnOF I I. r I ' - -I I.12.13.14.16.17.18.19.20.21 NOMINAL PARTICLE DIAMETER, IN Figure 54. Effect of Particle Size on Scattering Cross Section

-151 - 40 IO I I I I 6 AP - 3/16 GS -3 2 0~A P - 5/32 APAP/8 AP —/8 GS —5 II.12.13.14.15.16.17.18.19.20.2 NOMINAL PARTICLE DIAMETER, IN. Figure 55. Effect of Particle Size on.Absorption Cross Section

-132 - 1.7 -4 GS-5 1.0 0.9 09/~ [|AT 2000 F 0.8 eGS-3 0.7 -L 0.6 O O. 0. 0.5 Fl | *X~'^. —' AP-3/16 0IAP-5/32 0.I AP - 1/8 0.1 0.12 0.14 0.16 0.18 Q20 0.22 NOMINAL PARTICAL DIAMETER, IN. Figure 56. Effect of Particle Size on kir Parameters of Material

.34.32.30.28.26 cm.24....22 -- UL.20 _ _o. 184.10.08 F.04 -- --------- ---------.02 0 ___ _-...12.13.14.15.16.17.18 NOMINAL PARTICLE DIAMETER, IN. Figure 57. Effect of Particle Size on kir,Parameters of Temperature.19

Correlation of Attenuati.on Parameters As mentionedi previously, it ultimately -would be desirable to have quantitative correlations for attenuation cross sections, sb and. a, in terms of the physical. properties of the materials and the geometrical. properties of the systemso With such correlations; it would be possible to predict sb and a3 and so to estimate radiant conductivity for any packing of any materiai. The pertinent physical properties are not known in sufficient Fetail. to obtain quant'itiative correlat;ion from t:he data of this study. Measurement of these properties woul.d be a task of the same magnitude as the thesiso A number of correlations were attempted between the measured cross sections and the available l.iterature values of the microscopic properties, io.e.o emissivity and, transmissivityo However, d.ue to the high degree of uncertainty and incompleteness in the literature values and also to the compl.exity of interacting effects of geometry (roughness) shape, size9 etco) and, tese microscopic properties; no simple correlations were evidenrto Two graphical correlations, Figures 58 and 599 are presented to enable interpolation and extrapo7lation of the experimental data for pred.ic - tion of attenuatfion cross sections for other packings, Such predictions are obv'iousl.y subject to consi..d.erab'le uncertain.ty but at least ind.icate first orlier effects The attenlua-tion cross sections are correlated. in the form of dJi. mensionless parameters9 C - D p (135) C s Sb Dp (136) 5 _ l - SP

-155 - since it was expected that sb and a would be both directly proportional to the total projected area of particles per unit volume. Thus, to a first order estimate, 4 'rrD Ca X X D x 6(- ) a independent of Dp andp (137) C X Lx 6 T X independent of D and 6p (138) IT 6 pp) In general, the attenuation cross sections are functions of particle size, shape, emissivity, transmissivity, and of radiation wave length. The effects of the last three factors are in turn functions of temperature of radiation, particle size, shape, and material. These secondary functional dependences are quite complex in their interactions so that the dependence on Dp and p can not be entirely eliminated by the use of the parameters Ca and C.s. Thus, Ca = fl(shape, Dp, 6p, E transmissivity,.) for all materials (139) or Ca = f2(shape, Dp, 6, T) for a specific material, (140) and Cs - gl(shape, Dp, p, ~, transmissivity, X) for all materials (141) or Cs 8 ga(shape, D-, (p, T) for a specific material, (142) Correlations with literature values of emissivity and transmissivity were attempted for the type of fl and gl functions, for all materials. A general trend of third power dependence on emissivity was evident. However, the degree of complexity necessary for the fl and gl correlations were

-136 - _ag _2 ----- 3/16 ALUMINA SPHERES~ 10.ss L \LMESH 16 ALUMINA GRAIN~ "10 w //>Q^ \3/16" STEEL.8 SPHERES i. ___ / mm GLASS SPHE l i" STEI 6 TPHER Fr 6 C/Ze-4mm. GLASS SPHERES, z -— 3mm. GLASS SPHERES 4 - Lu 1.0 Z.8 -. o0 Y.2 — T 1260 1460 1660 1860 TEMPERATURE, R Figure 58. Correlation of Scattering Parameter

4.0 I x \ - 2 P% I. a I of 0 0 0 In w z 0 a. 0 (I) U) U) w -J z 01 z w 1: 0. 0..u ---------------------— ^ - --- - J MESH 16 SiC GRAIN \ / l l O~~\ _/ 0. 4 - -----------— /8'STEEL SPHERES 1/16" STEEL SPHERES.2 — ^_ — -- -— I MESH 16 ALUMINA GRAIN. ___..MESH 4 ALUMINA GRAIN ____ 3/16" ALUMINA PELLETS --- - 3/16'ALUMINA SPHERES5/32 "ALUMINA PELLETS 4 I/8" ALUMINA PELLETS 3mm GLASS SPHERES g ~2 _ _ 5mm GLASS SPHERES _ 4mm GLASS SPHERES 4 0 0 * 00 00 o 0 0 0 0 0 0 2 L T T T T T I,~ ~. = 0.0 0.0 0.0 0.004 0.00; m 1260 1460 1660 1860 2060 2260 2460 TEMPERATURE, ~R Figure 59. Correlation of Absorption Parameter

higher than coul,. be ju>stified, in view of the uncertainies introduced by using literature values of physical properties The final. graphical. correlatiors of the f2 and, g2 type were found to be more suitable The correlation plots of Cs and, Ca versus T, Figures 58 and 59, give parametric curves for packings of different shape, material., size;, anr void fraction To use the correlation plots it is necessary to interpo:late between each of the four factors,, Values of these factors for the curves are given in Table XII of the Appen1.ix: It shoul.d be noted, that the data for silicon carbide grain (CG-1.6) have a relatively high degree of uncertainityo The maximum and minimum exhi.ited by' Cs an. Ca~ respectively, may be due to data scatter an,. may no' be actu.al phenomenao As an example of the use of these correlations, the radiant conductivity) kv r y for pack.ngs of 5/32 inch steel spheres at 1i7000F was estdimated to be Ool 84 Btu/hrofto ~Fo Deta:ils of this example are given in the Appenldi.x Experimental. Prec s:i. on The absolu'te accuracy of the experimental, data could, not be determined.ue to a lack of other data for comparison. However, it is known that the values obtained, were of the correct or.er of magnitudes, as indicated by the comparisons with, analytic estimates and by the good agreement of kir/k c ratio with Hill and, Wilhelms(sl5) valueso Internal experimental precision was judged to be quite good., relative to the usual precisions reported, for experimental work in this field._ As exarmples of usual precisions, the maximum, scatter of Campbell.

-139 - and Huntington's(5) data was approximately + 50%, Yagi and Kunii's(40) was + 13%, and Hill and Wilhelm's(l5) was + 20%. A common cause of this high scatter is that packed bed properties are often quite sensitive to the particle arrangement, so that repacking a bed would result in a substantial change in the data obtained. The experimental method of this study has the advantage of allowing easy repacking. It was thus possible to take measurements over a number of packings at each bulk height, resulting in an average measurement of much greater precision. There are several things which may be taken as indications of the degree of precision obtained. First, the standard deviations in Table XI provide a measure of the variations in transmission measurements within each run. It is evident that the degree of precision obtained was dependent on the type of packing. Best precision was obtained with glass packings, whose standard deviations were of the order 0.5% to 5%. Standard deviations for most of the test particles were around 2% to 10%. An indication of run-to-run precision may be obtained by examining Figures 39-50. In these figures, each point on the kir curve represents a separate run. Therefore, the scatter of these points from the smoothed curve would be a measure of the overall precision in the experimental values of kir. The worst case was that of CG-16, which shows a scatter of approximately + 20%. The second worst case was that of AG-4 which shows a scatter of + 10%. All the other cases show surprisingly good precision, with scattering in the order of only 1 or 2 per cent. A final indication of experimental precision may be obtained by comparing Runs 10 to 13 and Runs 17 to 40. Each pair of runs- ere

made for one type of packing at one source temperature, but each run was a separate measurement, with the equipment being shut down and reassembled between runs. Therefore, Runs 13 and 40 may be regarded as reproducibility checks of Runs 10 and 17, respectively. The variations in the measured kir are seen to be only 1.1% between runs 10 and 13, and 1.2% between runs 40 and 17. This degree of reproducibility was considered to be quite sufficient.

PART IV SUMMARY AND CONCLUSIONS

SUMMARY AND CONCLUSIONS This study was undertaken to investigate radiant heat transfer in beds of packed particles. The analysis and experimental method were formulated for the one-dimensional steady-state case, but the experimentally measured parameters are applicable to general problems of radiant transfer in packed media of any geometry. Hamaker's two-flux model() was taken as the basis for a defining set of nonlinear differential equations, Equations (40) to (42). Transport properties of the packed media were specified in terms of an absorption cross section, a, and back scattering cross section, sb. The assumptions inherent in this method of treatment were derived by integration of the general vectorial transport equation of monochromatic radiation. A dimensional analysis was made on the differential equations and boundary conditions to determine the minimum number of variables and parametric groups necessary to specify the one dimensional system. Various approximate solutions to the system of nonlinear differential equations were also examined, completed where necessary, and catagorized in terms of the following regions of applicability: a) cases of linear temperature distribution, b) cases of negligible absorption, c) cases of small temperature differences, and d) cases of no re-radiation. The attenuation cross sections, sb and a, were experimentally measured for the twelve types of packings listed in Table I. The experimental method was based on transmission measurements of modulated black-body radiation. Values of sb and a were obtained from the transmission measurements by regression on Equation (115), using an IBM 704 digital computer for the necessary iterative calculation. -142 -

-143 - An equivalent radiant conductivity, kir, was defined in terms of the attenuation cross sections, sb and a, and was calculated for each of the test packings at every experimental temperature. This radiant conductivity was then compared to conduction conductivity, kc, for each test packing in order to determine the relative importance of radiation and conduction as heat transfer mechanisms. Values of kc were estimated by means of Schumann and Voss' correlation(34) with Wilhelm et al.'s modification(38). The experimentally obtained radiant conductivities, kir, were also used to check a number of analytic estimates. Radiant conductivities estimated by equations of Damkdhler-Argo & Smith(l), Schotte(33), and Rosseland - Bosworth(3) were graphically compared to kir for packings of glass spheres, steel spheres, and aluminum oxide pellets. The effects of (a) temperature of radiation, (b) void fraction of packing, (c) size of particles, and (d) emissivity of particle surface, on sb, a, and kir, were graphically indicated. Though a number of qualitative trends were evident, it was not possible to obtain quantitative correlations because the pertinent physical properties of the packings were not known in sufficient detail. However, two graphical correlations, Figures 58 and 59, are presented to enable interpolation and extrapolation of the experimental data for first-order estimation of attenuation cross sections for other packings. The main conclusions resulting from this study may be summarized as follows: a) the problem of steady state, one dimensional, simultaneous, radiative and conductive heat transfer in packed media may be specified by a minimum of four dimensionless variables, defined by the system of Equation (53j, and six dimensionless parameters, defined by the system of Equation (54).

-144 - b) Equation (115), the solution of the two-flux model for the case of no re-radiation, correlates transmission measurements of modulated flux very well. Average standard deviation of the regressed transmission curves from experimental data points was only 5.6 percent for the 46 valid runs of this study. c) Values of the absorption cross section (a), the back scattering cross section (Sb)) and the equivalent radiant conductivity (kir), range widely between various types of packings. For example, at 2000~F, the lowest and highest values of a were 1.20 and 192 (l/ft.) respectively, among the twelve packings tested. Similarly, sb ranged from a low of 95.0 to a high of 1,010 (l/ft.), and kir ranged from a low of 0.101 to a high of 1.06 (Btu/hr. ft. ~F). d) Back scattering is the major mechanism of attenuation for radiant transport in packings of glass, aluminum oxide, and steel particles. For packings of silicon carbide particles, both absorption and back scattering are important as attenuating mechanisms. e) Radiation becomes important, relative to conduction, at temperatures of approximately 1000~F and becomes predominant at temperatures of about 2000~F. Thus, at 2000~F, radiation was found to account for approximately 35 percent of the total heat transfer for packings of mesh 4 silicon carbide grains, 45 percent for 3/16 inch steel spheres, 60 percent for 3/16 inch aluminum oxide cylinders, and 85 percent for 5mm. glass spheres. f) The temperature dependence of radiant conductivities may be substantially different from the simple the ird power dependence commonly assumed. In the range of 1600OF to 20000F, values of the exponent of temperature was found to be less than three for packings of grains

and to be greater than three for packings of spheres and cylinders. g) Radiant conductivity increases with increasing particle transmissivity and size. h) Increasing particle emissivity increases absorption cross sections and decreases scattering cross sections. The overall effect is generally to increase radiant conductivities. i) Equations available in the literature for predicting radiant conductivities a priori are generally too simplified to give better than order of magnitude estimates. The three equations tested were found to differ from experimental values by as much as factors of eight. j) This experimental-method, based on transmission measurements of modulated radiant flux, is shown to be applicable to studies of radiant transport properties of close packed media. It is the only method presently known that can isolate radiant transfer from conductive and convective transfer for individual examination. The major contribution of this study is in providing the first direct measurements of radiant heat transfer properties of packed media. The data obtained for the twelve test packings and the various trends which they showed are the first experimental information available for calculation of radiant heat transfer in packed media. In future investigations, it may well be fruitful to isolate and individually study the effects of various particle properties and system geometrics, with the ultimate goal of obtaining complete quantitative correlations for predicting radiant transfer properties of any packed system.

APPENDIX -146 -

DATA REGRESSION EQUATIONS Experimental measurements supplied values of Sn as a function of h. From this, it was desired to obtain values of a and sb by regression on Equation (115). As the fractional deviation (rather than absolute deviation) was approximately equal for all data points in a run, the least square error requirement specified was that L i S 5(kj -, Sh) (143) be minimized. In Equation (143), n denotes the number of data points, Sn(hj) is the experimental value of Sn for jth data point, and S'(h.) is the correlation value of Sn at hj. Since Equation (115) is a nonlinear model, analytic regression was not possible, and it was necessary to resort to iterative methods. A number of numerical regression methods were attempted, including straight-log iteration, steepest-descent iteration, and truncated Taylor series iteration. The Taylor series method was found to be most satisfactory in giving fast convergence and was the method finally used. The necessary relationships may be developed as follows. Consider a run with n data points. There exists some value of m and an such that ) is at a minimum, where ( is defined by Equation (143) and Sn is to be calculated by Equation (144), S j) = j("m, an) n cok (ik)+ an O5h k(mk) (144) 3 j~~ At the ith iteration, the ith approximate value of m and an differs

-148 - from this true value by, mi = m - mi Lan an - ani n n. (145) where m and an are the true converged values corresponding to minimal ), i i and mi and an are the ith approximations. Evidently for the i+l iteration, the best Values would be, mi+l mi + mi i+l i+l + i n n n (146) However, the quantities ani and ban are unknown and only an estimate of their values can be obtained, as follows. For the ith approximation, ' =. [S.S j) ) (147) where, i: fj. (e, A-) For any specific set of data points, sional function of the variables mi OL -- a Oam - aV mt' osh (n' ) + as sC (m'kj) d A.* 5<1k (148) ) may be considered as a two dimenand a1. At the desired minimum point, n f. C)

and also, a mv a m 2)rru~ -0 -0 (149) a 1 a CL VI (150) By Equation (145), a4? 3A a~n a L ad M(i = 0 - O (s g-i) (151) (152 ) since n and an are fixed constants for any specific data set. Then frorm Ecqation (1.7), ad m C.. c L vt j - J=' bn i. _ J_ 1 L - i j 2 O =- - S, ) (5r(A-) f)(-,,) =0 J = Ij (1 55) a5$ 3Ac~rL -- >z IS ) _~ L >i (Sr(J) - )J )A( )Arn (154) to continue, f j (m, an) is expanded in a Tatylor series about m and a, n Y;cmA)Qr, = f. + 6 O ( a i )L + La ( - ), am a,. + I ^( )( A)k. Z M ( ) Tt(,a)... 155)

-150 - where, ft.(mAO = 5~ (hj^, d) M(156) A -fj (m;,,)^ = 5,^C^^~'L~). (157) i i aam) =L' 1 ak j (158) (L =. )\ -- *0 j,h (159) aarL; aano 'kM If the approximations, mi and an, are close to the true values, m and a, then the higher order terms of Equation (155) may be neglected to give the truncated Taylor series, c(,,) j rn + (airr). +,A )' ' (160) From which, - ( afi,; (161) __f_ - (:~'i) (162) Substituting Equations (160) to (162) into Equation (153), - 5 a;=O = -)zK tS L(i) L I) (163)

-151 - and substituting into Equation (154), = zZ j-: L LI - (L. -) -f. I4 yI 'J A (am); - A m -Aa,(yj). I a aK Equations (163) and (164) can be solved simultaneously to obtain, i Dm 6 I= Da i Da Ca = (164) (165) (166) where, S (j) i= t f Z (?m fL,j1t I —f^ Jy\ j>, am\ar W14 VL I 13 fA a, a /\,(ij)(Sf ine s,,t,~>j) law'^ar\j; j - r1 t^e (kj) as; JI vi. 25v1::- (ifL).ll (167) (168) (169)

-152 - In these determinants, Sn(hj) = experimental value of Sn at hj (170) L.L m i -,oLo. (m'pj) + ^Sn(j) (l71) (f; \ - ( m"a";i)os~ "(rj) * (\j - 4i)5 (') (172) "rv' ~ — [a' Cosk (e'lj) + ' sk(Mk)3 and, 4.\ va -slnk (Y ^- &Ij) (175) Thus at the end of the ith iteration, the values of Lni and Aan may be estimated by Equations (165) and (166), with the relationships of Equations (170) to (173). The (i+l) approximation for m and an is then given by Equation (146). This iterative procedure may be continued until m and an converge to as many significant figures as desired. The values of radiation parameters a and sb can then be calculated from these converged values by, Sb = a2 - m2 (174) a a - sb (175) To initiate the iterative regression, it was necessary to obtain a fairly accurate first-guess for values of an and m. This may be done by

-153 - writing Equation (115) in exponential form, 6~^) nmtoskh(n) + a=nsh (Mk) (115) (mOa,)e + (m-an)eM ' (176) At large values of h, -Ynk e o and Equation (176) may be approximated as, 2 Yn SnCh) = (hwa ) emk or t(Sn) = - (1^ ) - "k (177) By using two experimental data points at relatively large h, a first approximation can then be obtained from Equation (177) as, -, rLMS (178) = 2 - ~ L (SI) ) - (179) In the actual experimental study, this first approximation calculation and the subsequent iterative regressions were programmed for computation on the IBM 704 digital computer. These programs are described in the last section of the Appendix.

SAMPLE CALCULATIONS The data of Run 41, for GS-3 at 2000~F, are worked out below as an illustrative sample calculation. The results for all other runs were determined in a similar manner. Regression For Transmission Curve The original test run data are tabulated in Table VI. There are eight data points for Run 41, with packing height, h, ranging from 1.72 cm. to 7.49 cm. Using the sixth and eighth data points, first-guess values for an and m were calculated by Equations (178) and (179) as,.oo Zq m~ = _ (o.00g7q9) 7.49- s^7 o.756 f[ -(o.7sG x 7.49) 1(,07) 3 - - 0. 523 c'. Calculated transmissions, S'(h), corresponding to these values of an and m were obtained from Equation (144) for each packing height. The < error was then calculated by Equation (147) and from that the standard deviation was determined by Equation (128) to be, i = 1.0 (-0 ) 7 777 Io -154 -

-155 - Corresponding to Equations (167) to (169), the values of the following three determinants were calculated, D = 4.43 Dm = 0.143 D = -4.682 a and the adjustments were determined to be, by Equations (165) and (166), Dm.143 m~ =- - 0.0322 D 4.43 0 Da 4.682 en D-= 4.43 -i.o6 Therefore, for the next iteration, m m = m~ + Am0 = 0.756 + 0.0322 = 0.788 an = m~ + Aa0 = 5.32 - 1.06 = 4.17 and the corresponding standard deviation was found to be, (146) (146) S' = 2.56% S Continuing successive this iterative regression, the following values were iterations obtained at m1 mVF Iteration 0 1 2 4 4 0.756 0.788 0.787 0.786 0.786 an cm 5.23 4.17 4.29 4.30 4.30 s (%) 7.77 2.56 1.68 1.67 1.67 Am cm.0322 -.0014 -.0005.0000.0000 1 Aan cmM -1.057 0.120 0.003 0.000 0.000 It is seen the end of an as, that convergence to three significant figures was obtained at the fourth iteration, giving final regressed values of m and -1 an = 4.30 cm or 24.1 ft.or 1- ft.1 or 131 ft..

-156 - The calculated transmissions, SA(h), corresponding to these regressed parameters were then calculated by Equation (144) and may be compared to the experimental transmissions, Sn(h), as follows: h, (cm) Sn(h) Sn(h) Sn - Si 1.72.0856.0839.0017 2.62.0394.0397 -.0004 3.43.0212.0209.0003 4.25.0108.0109 -.0001 5.07.00569.00574 -.00005 5.87.00299.00oo306 -.00007 6.58.00175.00175.00000 7.49.00088.00086.00002 The experimental values, Sn(h), are plotted as data points and the calculated values, S'(h), are represented by the transmission curve on Figure 22. The fit is seen to be quite satisfactory. The standard deviation of this converged regression was found to be 1.67 percent. The attenuation cross sections corresponding to these regressed values of m and an were calculated by Equations (174) and (175) to obtain, 2 2 Sb an- m =/112 - 24.12 = 129 ft.-1 a = an - Sb = 130.97 - 128.76 = 2.21 ft.1 The transmission values obtained by such regressions for all runs are listed in Table XI.

-157 - Radiation to Conduction Ratios By Schumann and Voss' correlation(4) with Wilhelm et al.'s modification(38), the conduction-conductivity of a packed medium may be estimated from the equation kc = kc + A (180) where kc = conductivity given by Schumann and Voss' graphical cdrrelation A = Wilhelm et al's correction. The graphical correlation for kc presents it in the ratio, k'/kg, as a function of ks/kg for parameters of Sp, where ks = conductivity of solid k = conductivity of gas ip = void fraction of packing. In Table XII, the values of ks and p for GS-3 at 2000~F are given as 0.63 Btu and 0.350, respectively. The conductivity of air at high temphr.ft. OF eratures is given by Glassman and Bonilla' The values used were: 800OF 1200 F 1600 F 1800OF 20000F Btu kg hr-ftF - 0.0303.o0569 00o427 00.454 o.o480 9 hr.ft. F Thus for GS-3 at 2000~F, 0,63 ks/kg 0. =480 13.1 and from Schumann and Voss' correlation, kc/kg = 37 kc = 0.178

-158 - Wilhelm's correction was calculated by, A = Mantlog (o.8. +. o0,0.Zq K 2)j ) 10 =.tnlo i (O-4 o.o+ 0 x l)o 1XI =.00ooo7 Therefore, the bulk conductivity, kc, for GS-3 at 20d0~F was determined to be, kc = 0.178 + 0.00076 Btu = 0.178 hr.ftF The radiant conductivity was calculated as kir = 8T3 a + 2sb 8 x 1.713 x 10-9 x 2460 (104) 2.21 + 2(129) = 0.785 Btu hr.ft.~F The ratio of radiation to conduction was thus, kir 0.785 kc 0.178 = 4.41 Analytic Estimates The three analytical estimates of radiant conductivity were calculated as follows. Estimate of Damkohler(6) and Argo & Smith(l ), krl = 4 (2- ) DpT3 (2b) where 6 = particle emissivity. Estimate of Schotte(33) kr2 = + (K kCS = 4 (5) kr0 = 4we D T3 r ~~p

-159 - Estimate of Rosseland(3 and Bosworth(3), Kr3 = 3DT3 (6) For GS-3 at 2000~F: a) krl = 4 x 1.713 x 10 20 65 x 0.00983 x 24603 Btu = o.483 hr.ft.F b) k~ = 4 x 1.713 x 10-9 x 0.65 x 0.00983 x 2460 r = 0.652 1 - 0.35 kr2 = 1 ~ - + 0.35 (0.652) 1 + 1 o.063 0.652 Btu = 0.436 Btu hr.ft. OF c) kr3 = 3 x 1.713 x 10o9 x 0.00983 x 24603 Btu = 0.334 hrft. ~F Estimation of Conductivity from Correlation To demonstrate how the graphical correlations, Figures 58 and 59, may be used to estimate radiant transfer properties of some packing not tested in this study, an example is worked out below for 5/32 inch diameter steel spheres operating at a temperature of 1700~F. Figures 58 and 59 present smoothed curves of Cs and Ca for steel spheres of 1/8 inch and 3/16 inch diameters. Since 5/32 inch steel spheres have approximately the same emissivity, shape, material, and bulk void fraction as these two tested packings, it is necessary only to interpolate for size and temperature. Interpolating between curves at 1700~F, a value of 8.0 is obtained for Cs from Figure 58 and a value of 0.23 for Ca from

-160 - Figure 59. Attenuation cross sections may then be calculated as, i-Sp I — l= Ca = Dp The radiant conductivity at interior of a packing of 5/32 inch steel spheres at 1700~F may be thus estimated as, gb - -r3 |<. +.(310) +o1t. o1V2 hr F

EXPERIMENTAL DATA Test Run Data Experimental data for the valid test runs are given below in Table VI. Values of background reading, RB, are the average values for each run. The column of S gives the transmission reading minus background; S = R - RB. Sn denotes flux signal normalized with respect to input flux; Sn - S/So. Source temperatures are nominal values, with possible variations of ~ 5~Fo TABLE VI Test Run Data Run Description AW, gm. h.cm. S Avg. S Avg. Sn 7 GS-3 ts=800~F so=836 RB=0.13 0 26.25 24.40 22.80 25.25 21350 0 1. o01 1.95 2.83 5.72 4.54 0 1.03 1.97 2.85 3.70 836 24.4 22o8 23.7 7.52 7.62 7.55 5349 3.43 3540 1 57 1.59 0,77 0.79 830 31o9 35.1 3553 10.7.1,2 11,4 6.07 5.95 6.02 3569 3552 3.57 836 23.6 7.56 3.44 1.,58 0,78 850 34.1 11.1 1 0.0282 o. oogo4 0, 00904 0.00415 0.00189 0.00093 1 0,0411* 0,0134 8 GS-4 ts=800~F So=830 RB=013 0 26,75 24.50 22.80 22.25 6o01. 0.00724 3.59 0.00432 -161 -

-162 - TABLE VI (cont'd) Test Run Data Run Description AW gm. h cm. S Avg. S Avg. Sn 22.50 24.55 24.55 4.57 5.51 6.46 2.07 2.02 1.07 1.07 0.60 0.63 2.04 1.07 0.62 0.00246 0.00129 0.00075 9 GS- 5 ts=800 "F so=855 RB=0.13 0 29.15 26.35 23515 23555 23.80 26.30 0 1.20 2.29 3.24 4.21 5.19 6.27 1.01 10 GS-3 ts=1200~F So=2,560 RB=0.06 26.25 855 30.6 25.4 26.8 8.82 9.37 9.12 4.37 4.47 2.57 2.60 1.41 1.41 0.76 0.75 182 175 173 79.0 74.5 78,.6 42.1 42.5 27.7 27.6 18.0 18.0 11.4 1.1.7 6.54 6.64 3.50 3.39 3.44 1.82 1.81 855 27.6 9.10 0.0106 4.42 0.00517 2.60 0.00304 1.41 0.00165 0.75 0.00087 1 0,0323* 177 0.0690 21.35 18.30 153.80 13565 13.75 13.50 21.00 20.90 2.54 3.07 3.59 4.12 4.64 5.45 6.26 77.0 0.0301 42.3 o.0165 27.7 0.0108 18.0 0.00703 11.5 0.00449 6.59 0.00257 3.44 0.00134 1.82 0.000711

TABLE VI (cont'd) Test Run Data Run Description zLW gm. h cm, S AVg. S Avg. Sn 11 GS-4 ts=1200~F So=2,610 RB=0.06 26.75 22.80 22.25 15.55 15.70 15.95 17.80 17.85 22.55 29.15 25.55 18.80 18.80 18.25 12 GS-5 ts=1200~F So=2,590 RB=0.06 1.05 1.91 2.76 5.97 4.58 5.27 5.95 6.82 1.20 2.17 2.95 5.72 4.47 5.24 6.02 6.78 7.74 1.01 1.90 258 217 228 109 103 59.5 60.2 41.2 41.9 30.4 29.1 21.0 21.5 15.9 13.7 9.55 9.19 5.66 5.72 203 203 87.8 88.0 59.5 59.0 58.8 38,5 58.5 25.4 24.3 26.0 16.8 16.4 10.9 10.7 7.39 7.64 4.54 4.66 203 212 198 80.8 85.0 84.0 lo6 59.8 41.5 29.7 21.53 15.8 9.27 5.69 203 87.9 59.2 58.6 25.2 16.6 10.8 7.51 4.60 228 0.0874 0.04066 0.0229 0.0159 0.0114 0.00816 0.00529 0.00355 0.00218 0.0784 0.0339 0.0229 0.0149 0.00975 0.00641 0.00417 0.00290 0.00178 18.75 18.80 18,50 23.15 13 GS-3 ts=1200~F So=2,780 RB=0.06 26.25 23.15 204 0.0734 82.6 0.0297

-164 - TABLE VI (cont'd) Test Run Data Run Description AW gm, h cm, S Avg. S Avg. Sn 18.50 15380 15375 21355 22.80 14 GS-3 ts=1200~F So=l, 980 RB=O.06 GS-3 ts=1600~F So=4,680 RB=0.09 15 24,40 23570 25325 18.30 13.65 20.90 21,00 26.25 24.40 23570 21,35 18.30 18.50 20.90 21.30 23.25 2.61 3.15 5.67 4,50 5.37 0,94 1.85 2.75 3545 3598 4.78 5.59 1.01 1.95 2.86 3.68 4.39 5.10 5.81 6.73 7.62 45.2 45'. 0 29.2 29.6 19.3 19.4 9.79 9.96 4.89 4.89 148 142 146 57.4 56.1 25.6 25.6 14.5 14o5 9.34 9.55 4,90 4.94 2,50 2.56 535 600 565 227 233 226 110 107 55.9 56.1 32.0 32.0 18.0 18.1 9.52 9.52 5.01 4.91 2.41 2.49 45.1 29.4 19.3 9.88 4.89 145 0.0732 56.7 25.6 14.5 9.44 4.92 2.53 0.0386 0.0129 0,00732 0.00477 0O00248 0.00128 0.01.62 oo0106 o 00694 0.00360 0o00176 567 0.121 229 0.0490 108 56.0 32.0 18.1 9.52 4.95 2.45 0.0231 0.0120.00oo684 0,00387 0.00204.oo00106 0.000524

-165 - TABLE VI (cont'd) Test Run Data Run Description GS-4 ts=1600~F So=4,680 RB=0.09 17 GS-5 ts=16oO~F So=4,680 RB=O.07 AS-3/16 ts=1600~F So=4,680 RB=O.13 AW gm. 51.40 24.65 22.50 17.80 17.85 22.80 22.25 24.55 24.55 55.50 23.80 18.80 18.80 18.75 18.80 23.55 23.15 27.70 17.70 17.20 14.65 1.98 2.93 3.79 4.48 5.17 6.04 6.90 7.83 8.79 2.29 3.27 4.04 4.82 5.59 6.36 7.33 8.29 0.77 1.27 1.75 2.16 340 341 187 185 185 114 120 78.3 8o.!i 80.1 54.0 53.5 32.1 30.8 32.2 19.4 19.4 11.4 11.53 6.89 6.77 292 288 165 169 112 112 76.5 76.1 50.0 49.5 33.2 32.8 20.0 20.0 12.4 12.6 311 300 105 109 42.0 44.4 44.4 21.7 21.6 h cm. D i Avg. S Avg. Sn 340 0.0727 185 0.0396 117 0.0250 79.2 0.0169 5358 0.0115 31.7 0.00678 19.4 0.00415 11.4 0.00244 6.83 0.00146 290 0.0620 167 0.0357 112 0.0240 76.3 0.0165 49.7 0.0106 33.0 0.00706 20.0 0.00428 12.5 0.00267 305 0.0652* 107 0.0229 43.6 0.00933 21.7 0.00464 18

-166 - TABLE VI (cont'd) Test Run Data Run Description AW gm. h cm. S Avg. S Avg. Sn 14.65 14.35 14.20 14,90 20 AP-1/8 ts-1600~F So=4,680 RB=0.12 19.30 14.25 2.56 2,97 3,36 3*78 0.81 1.41 1.76 2.13 2.50 2.86 10.6 10.6 5.57 5.48 2.72 2.90 1.42 160 163 163 40.7 41,7 42.6 20.5 20.7 10.7 10.9 5.88 5.88 3,35 3.35 1.34 1.39 0.54 0.60 0.63 10.6 0.00227 5.53 0. 00118 2.81 o.000601 1.42 0.000304 162 41,7 20.6 10.8 00o347* 0.00892 0.00441 0.00231 8.55 8.80 8.80 8,6o 5.88 0,00126 3,35 0.000717 1.38 0.00295 0.59 0,000126 13.05 13.20 3,5.96 21 AP- 5/32 ts=1600~F So=4,680 RB=0.14 27.50 14.25 8.95 8.45 8.75 1.28 1.94 2.35 2.75 3*15 96.2 104 103 27,5 29.4 31.8 30.9 15.0 16.0 15.8 8.36 9.18 9.41 4,71 5.o6 4.60 4.96 101 29.9 15.6 0,0216* o0oo640 0.00334 8.98 0.00192 4,83 0.00103

-167 - TABLE VI (cont'd) Test Run Data Run Description AW gm. h cm, Avg. S Avg, Sn 8.20 14.30 14.45 22 AP-3/16 ts =1600 F So=4,680 RB=O.10 30.70 9.30 3.53 4.19 4.87 1.33 1.73 2.13 2.53 2.94 3.71 2.90 2.o6 2.76 1.20 1.16 1.18 0.53 0.46 0.49 74.9 72.9 72.4 36.7 36.9 16.o 17.6 16.2 7.85 8.50 8.40 4.55 4,20 4.89 1.23 1.52 1.29 2.91 0.000622 1.18 0.000252 0.493 0.000105 73.4 36.8 16.6 0.0157 0.00787 0.00355 9.25 9.25 9.35 17.75 8.25 0.00176 4.55 0.000973 1.28 0.000274 23 ss-1/8 ts=1600 F So=4,950 RB=0.15 0.53 22.60 22.15 13.20 14.00 0.79 1.05 1.21 1.37 405 470 440 192 128 160 170 46.4 48.7 42.9 44.6 26.7 26.6 28.2 13.6 12.5 12.6 15.8 438 165 45.7 27.2 13.1. 0885* 0.0329* 0.00923 0.00549 0.00265

-168 - TABLE VI (cont'd) Test Run Data Run Description tW gm, h cm. S Avg. S Avg. Sn 0 13.85 22.45 13.70 22,35 SS- /16 ts1600 ~F So-4,950 RB=-013 141.60 22.70 1.53 1.79 1.96 2.22 1.69 1.96 2.22 2.49 2.76 3 12 7.97 7.62 6.86 8.57 3.07 2.92 2.67 1.57 1.45 1.67 o.86 0.72 0.78 33.4 17.3 17o3 1,6.2 15.5 9.82 8.88 9.32 4.17 4.60 4.54 2.53 2.47 2.79 1.29 1.20 1.67 7.75 0.00157 2.89 0o000584 1.56 0.000315 0.787 0,000159 33.4 16.6 0.00675 0.00335 22,15 22.70 22.55 30.40 9.34 0.00189 4.44 0o000897 2.60 0,000525 1.25 0.000253 25 AG-4 t =1600 ~F So=4,950 RB=O.10 27.20 16,70 0.98 1.58 1,98 2.36 169 186 166 38.0 37.4 37.9 38.9 16.6 16.3 18.0 7.85 7.33 8.00 174 38.1 17.0 0.0351,* 0.00770 0.00343 10.95 10,70 7.73 0.00156

-169 - TABLE VI (cont'd) Test Run Data Run Description AW gm. h cm. S Avg, S Avg. Sn 10.65 2.74 3.70 3.60 0.000727 3.54 3.57 10.60 3.12 1.95 1.86 0.000376 1.88 1.75 16.35 3.72 0.73 0.74 0.000149 0.75 26 AG-16 13.85 0.42 232 234 0.0473 ts=1600~F 233 So=4 950 236 RB=0.09 6.45 0.62 80,4 80.2 0.0162 80.1 6.35 0.81 29.7 29.7 0.00600* 29.6 3.90 0.93 20.2 20.1.o00406 20.1 3585 1.05 11.6 11.6 0.00234 11.5 6.45 1.25 4.41 4.39 0.000887 4.36 3.95 13.7 2.55 2.61 0.000527 2.67 3.85 1.49 1.60 1.61 0.000325 1.62 27 CG-16 4.85 O,161 403 317 0.o640 ts=1600oF 230 o=4,950 4.35 0.305 14.0 14.6 0.00295 RB=0.08 14.8 15.0 4.60 0.458 0.92 0.89 0.000180 0.86 4.20 0.600 0.14 0.14 0.000028 0.14 28 GS-3 44.75 1.72 500 496 0.0729 ts=1800~F 492 so=6,780 23.25 2.62 228 235 0.0345 RB=0.09 238 239 21.00 3.43 123 123 0.0181 123 21.35 4.25 64.1 64.3 0.00945 64.4

-170 - TABLE VI (cont'd) Test Run Data Run Description 29 GS-4 ts=1800~F So=6,780 RB=0.09 AW gm. 21.30 20.90 18.30 23.70 51.10 24.55 22.50 22.80 22.25 17.80 24.85 17.85 h cm. 5.07 5.87 6.58 7.49 1.97 2.91 3.78 4.65 5.51 6.20 7.16 7.84 1.95 2.93 3.91 4.86 5.63 6.41 7.18 S 33.9 33.5 17.8 17.6 10.0 10.1 5.01 5.09 628 630 339 350 207 203 123 126 77.4 75.5 77.1 51.5 50.8 29.9 29.6 19.9 19.8 652 679 672 381 357 391 227 214 223 130 137 134 89.7 87.1 59.9 61.6 41.4 39.0 38.9 Avg. S 33.7 17.7 10.1 5.05 629 345 205 125 76.7 51.2 29.8 19.8 668 376 221 134 88.4 60.8 39.8 Avg. Sn 0.00495 0.00260 0.00148 0.000742 0. 0925 0.0507 0.0301 oo0184 0.0113 0.00753 0.00438 0.00291 0.0982 0.0553 0.0325 0.0197 0.0130 0.00894 0.00585 GS-5 ts=1800~F So=6,780 RB=O.10 23.80 23.55 23.15 18.80 18.80 18.75

-171 - TABLE VI (cont'd) Test Run Data Run Description ZW gm. h cm. S Avg. S Avg. S 18.50 31 AS-3/16 ts =1800~F So=6,780 RB=o.15 27.70 17.70 17.20 14.65 14.65 14.35 14.20 14.90 7.94 0.77 1.27 1.75 2.16 2.56 2.97 3.36 3.78 4.18 27.4 27.4 499 514 175 190 166 66.5 72.5 68.9 33.9 33.8 34.8 17.0 15.7 15.2 8.22 7.87 8.38 4.70 4.58 2.37 2.37 2.35 1.30 1.25 27.4 0.00403 507 177 0.0745 * 0.0260 69.3 0.0102 34.2 0.00503 16.o 0.00235 8.16 0.00120 4.64 0.000680 2.36 0.000347 1.28 0.000188 32 AP-1/8 ts=1800~F So=7,192 RBg0.37 19.30 14.25 8.55 8.8o 8.80 8.60 0.81 1.41 1.76 2.13 2.50 2.86 299 250 295 86.9 83.5 78.7 39.6 41.8 47.7 43.8 22.1 23.6 22.6 11.9 12.0 11.6 6.85 6.68 6.58 281 0.0391 83.0 0.0115 43.2 0.oo600 22.8 0.00317 11.8 0.00164 6.70 0.00Q931 *

-172 - TABLE VI (cont'dd) Test Run Data Run Description LW gm. h cm. S 13.50 13.05 3.43 3.98 2.58 2.63 2.73 1.06 1.16 1.20 Avg. S Avg. Sn 2.65 0.000368 1.14 0.000158 33 AP-5/32 t,=18oo0F So=7,192 RB =. 33 27.50 1.28 1.94 8.95 8.45 8.75 8.20 14.30 2.35 2.75 3.15 3.53 4.19 4.87 1.33 171 188 168 57.2 66.5 59.7 30.6 31.7 31.8 17.9 18.0 17.9 10.1 10.1 10.0 5.90 5.72 5.89 2.40 2.33 2.47 1.00 1.07 0.95 113 125 122 54.o 57.7 52.9 24.1 25.8 28.1 13.3 14.1 12.6 6.91 7.41 7.24 176 61.1 0.oo849 31.4 o.oo436 17.9 0.00249 10.1 o.oo00140 5.84 0.000812 2.40 0.000334 1.01 o.ooo000140 122 0.0179 54.9 0.00807 26.0 0.00382 13.3 -0.00196 7.19 o.ooio6 0.0245 * 34 AP-3/16 ts=1800~F SO=6,780 Rg=0.18 30.70 9.50 9.30 9.30 9.20 2.14 2.55 2.94

-175 - TABLE VI (cont'd) Test Run Data ZW gm. h cm, Run Description s Avg. S Avg. Sn 9.40 17.70 3.35 4.12 3.27 3.58 3.37 1.04 1.05 1.00 3-39 0.000498 1.03 0.000151 35 AG-4 t s=1800F so=6,780 RB=0.14 27.20 16.70 10.95 10.70 10.65 10.60 16.95 16.55 o.98 1.58 1.98 2.36 2.74 3.12 3.74 4 33 260 259 270 54.9 55.0 50.7 22.9 25.0 24.1 11.5 12.1 11.9 5.55 5.47 6.08 3.16 3.46 3.36 1.40 1.37 1.26 0.54 0.54 263 0.0387 53.6 0.00788 24.0 0.00353 11.8 0.00173 5.70 0.000838 3.33 0.000490 1.34 0.000197 0.54 0.000079 * 36 AG-16 t =18o00F So=7,190 RB=. 18 13.85 6.45 6.35 6.45 6.50 3.90 3.85 3.95 0.42 0.62 o.81 1.01 1.21 1.33 1.45 1.56 432 470 451 148 149 57.9 59.9 23.5 24.0 10.0 10.2 6.07 6.52 6.44 3.92 3.83 2.44 2.46 451 0.0627 * 148 0.0206 58.9 o.00819 23.8 0.00331 10.1 0.00140 6.34 o.ooo881 3.88 0.000539 2.45 0.000341

-174 - TABLE VI (cont'd) Test Run Data Run Description AW gm. h cm. S Avg. S Avg. Sn 37 CG-16 ts=1800~F So=7,190 RB=0.08 38 ss-1/8 ts=1800~F So=7, 192 RB=0.23 39 SS-3/16 ts=1800~F So=6,780 RB=0.24 4.85 4.35 4.60 4.20 45.10 22.60 22.15 22.45 13.20 14.00 13.85 13.70 0.305 0.458 0.600 0.30 0.79 1.05 1.31 1.47 1.63 1.79 1.96 1.05 1.32 1.59 1.85 495 395 29.0 19.0 2.37 3.42 0.44 0.19 530 590 690 620 263 208 191 52.3 60.8 60.3 24.5 20.8 24.7 14.3 11.0 11.9 6.29 6.47 6.17 3.23 3.37 3.40 1.85 1.95 2.12 218 228 267 103 97.4 110 51.8 44.8 61.9 24.2 25.1 25.0 495 24.0 2.90 0.32 608 221 57.8 23.3 12.4 6.31 3.33 1.97 238 103 52.8 24.8 0.0619 0.00334.000403 0.000044 0.0845* 0.0307* 0.00803 0.00324 0.00172 0.000877 0.000463 0.000274 0.0350* 0.0151 0.00776 0.00365 88.oo 22.65 22.70 22.15

-175 - TABLE VI (cont'd) Test Run Data aZ gm, h cmo Run Description S Avg, S 22.70 22,55 30.90 40 GS-5 ts=1600~F So=4,362 RB=I.O 41 GS-3 ts=2000~F So=9, 50 RB=O. 10 42 GS-4 ts=2000~F So=9, 520 55,50 23.80 18.8o 18.80 18.75 1.880 23,55 23,15 44,75 23.25 21,00 21.35 21.30 20.90 18.30 23,70 51.10 2.12 2.39 2.76 2.29 3.27 4.04 4.82 5.59 6.36 7.33 8.29 1.7'2 2,62 3.43 4.25 5.07 5.87 6.58 7.49 1.97 11.9 11.4 13,1 6,76 6,66 6,71 2.86 3.26 3.21 263 273 310 163 166 103 105 72,0 69,5 46.8 46.4 31.7 31.3 20.2 20.2 12,1 12,5 820 810 372 379 200 203 103 103 54.0 54,3 28,7 28.4 16.7 16.6 8.40 8,35 976 1050 1010 12,1 6.71 3.11 282 164 104 70.8 46.6 31.5 20.2 12,3 815 375 202 103 54.2 28.5 16.7 8.37 1012 Avg. Sn 0.00178.ooo000986 0,000457 0.0647 0,0376 0.0238 0,01.62 0.0107 0000722 0.00461 0.00282 0.9 0,0394 0.0212 001.08 oo00569 0.00299 0.00175 0.000879 0.106

TABLE Test VI (cont'd) Run Data Run Description aW gm. h cm, S Avg. S Avg. Sn RB=O. 11 43 GS-5 ts=2000~F So=9, 520 RB=O.ll 44 AS- 3/16 ts=2000~F So=9, 520 RB=0.27 24.55 22.50 22.80 22.25 17.80 24.85 17.85 71.20 23.55 23.15 18.80 18.80 18.75 18.50 45.40 17.20 14.65 14.65 14.35 14.20 2.91 3.78 4.65 5.51 6.20 7.16 7.84 2.93 3.91 4.86 5.63 6.41 7.18 7.94 1.26 1.75 2.16 2.56 2.97 3.36 562 565 345 343 205 207 123 126 82.7 83.9 48.o 47.9 32.9 33.2 591 611 616 368 362 217 222 148 149 101 99.6 66.7 67.4 46.6 45.1 262 264 261 101 102 48.3 47.3 27.5 27.7 15.0 15,0 7.50 7.20 7.75 563 344 206 124 83.3 48,0 33.0 606 365 219 149 100 67.1 45.9 262 102 47.8 27.6 15.0 7.48 0.0591 0.0361 0.0216 0.0130 0.00875 0.00504 0.00347 0.0636 0.0383 0.0230 0.0156 0.0105 0.00705 0.00482 0.0275 0.0107 0.00502 0.00290 0.00158 0.000785

-177 - TABLE VI (cont'd) Test Run Data AW gmo h cmo Run Description S 14.90 3.78 4.18 45 AP-1/8 ts=2000~F So=10, 100 RB=Oo38 33-55 8.55 8.80 8,80 8.60 13,50 1.3.05 13.20 1.41 1.76 2.13 2,50 2.86 3.43 3.98 4.53 3470 3.32 3.63 1.87 1.91 1.o90 120 121 135 64.0 64,2 34.7 33.2 17.3 17.3 9.30 9470 3.70 4.10 1.50 1.65 1.70 0,67 0.69 0.85 Avg. S 3.55 1.89 125 64.1 34.0 1703 9.50 3.90 1.62 0.o74 0.000373 0.000198 0.0124 0.00637 0,00338 0,00172 0.000944 0.000388 0,o000161 0.000074 Avg. Sn 46 AP- 5/32 ts=2000~F So=10, 100 RB=O0 20 27.50 14.25 1.28 1.94 2035 8.95 2.75 282 272 292 97,0 9009 49.2 51.7 50,8 29.5 30.3 30.4 16.4 16.3 8,95 10.1 4.20 4 20 2.05 1.83 1.77 282 93.0 50.6 30.1 16o4 9.53 4 20 1.88 0.0280 0 00924 0.00503 0.00299 0.00163 0.ooo000947 0.000417 0.000187 8,75 8.20 14.30 14.45 3.15 3453 4.18 4.87

-178 - TABLE VI (cont'd) Test Run Data Run Description aW gm. h cm. S Avg. S Avg. Sn 47 AP- 3/16 ts=2000~F So=9, 520 RB=0.22 30.70 1.33 9.50 9.30 9.30 9.20 9.40 17.70 48 AG-4 ts=2000~F So=9, 520 RB=O. 15 43.90 10.95 2.14 2.55 2.94 3.35 4.12 1.58 1.98 2.36 2.74 3.12 3.74 4.33 188 199 208 87.1 93.0 94.0 41.8 46.5 50.0 43.0 20.8 22.0 21.0 10.1 10.1 11.6 5.80 5.70 5.78 1.63 1.73 1.48 86.3 91.0 81.6 35.0 37.0 39.9 18.2 19.2 17.8 9.40 9.oo 9.o8 4.80 5.20 5.10 2.08 2.22 2.10 0.95 0.97 0.93 198 91.4 45.3 21.3 10.6 5.76 1.61 86.3 37.3 18.4 9.16 5.03 2.13 0.95 0.0208 oo00960 0.00476 0.00224 0.00111 o.ooo605 0.000169 o.ooo00906 0.00392 0.00193 0.000962 0.000528 0.000224 0.000100 10.70 10.65 10.60 16.95 16.55

-179 - TABLE VI (cont'd) Test Run Data Run Description AW gm. h cm. S Avg. S Avg. Sn 49 AG-16 ts=2000~F So=10, 100 RB=0.17 50 CG-16 ts=2000~F So=10, 100 RB=O.10 51 ss-1/8 ts=2000~F So=10, 100 RB=0.24 20.30 6.35 0.62 0.81 1.01 6.50 3.90 3.85 3.95 4.85 4.35 4.60 4.20 1.21 1.33 1.45 1.56 o.16i 0.305 0.458 0.600 0.79 1.05 1.31 215 220 226 93.0 91.7 90.8 38.2 38.2 17.2 16.9 10.3 10.8 7.00 6.65 4.53 4.58 531 534 52,1 34.1 34.1 49.0 1.70 2.90 2.75 0.27 o0.63 0.33 0.47 242 236 213 73.0 67.0 90.0 83.0 40.2 32.0 30.2 24.4 12.9 14.5 14.4 8.20 7.05 9.80 7.16 220 91.8 38.2 17.0 10.5 6.80 4.55 532 42.3 2.12 0.0219 0.00912 0.00380 0.00169 0.00104 0.000676 0.000452 0.0529 0.00420 0.000211 0.000043 67.70 22,15 22.45 230 78.3 31.7 13.9 8.05 0.0229* 0.00778 0.00315 0.00138 0.000800 13.20 14.00 1.63

TABLE VI (cont'd) Test Run Data Run Description LW gm. h cm. S Avg. S Avg. Sn 13.85 13.70 52 SS-3/16 ts=2000~F So=9,520 RB=0.24 88.0 22.65 1.79 1.96 1.05 1.32 1.59 1.85 2.12 2.39 2.76 22.70 22.15 22.70 22.55 30.90 3.65 4.06 4.16 2.53 2.36 2.14 289 390 366 294 168 123 120 67.0 67.8 65.5 31.9 36.1 38.5 31.0 14.8 18.0 16.4 8.90 8.6o 8.4o 3.96 3.78 3.66 3.51 3.96 2.34 335 137 66.8 34.4 16.4 8.63 3.73 0.000394 0.000233 0.0352* 0.0144 0.00701 0.00361 0.00172 0.000906 0.000392 NOTE: Some data points were thought to be redundant or were evidently out of line with respect to other points of the same run. These points were considered to be invalid and have been marked by an asterisk, *, in the above table. They were not included in regression calculations.

Temperature Response Data Data for the response of the detector and amplification system are presented below. R denotes the measured signal corresponding to a source temperature of Tso Signals are expressed as output-meter readings, normalized to unit attenuation of the amplifier. The data was obtained with a 2-1/2" source opening (in the bottom shield), 3" length bed tube, mesh 18 support screen, and with the bed frame at a distance of 95 cm above the bottom shield. These data are plotted on Figure 14. TABLE VII TEMPERATURE RESPONSE DATA Source Temperature (Ts) Measured Signal 0F ~R R 915 1375 52.4 1187 1647 114 1402 1862 189 1464 1924 208 1704 2164 335

-182 - Linearity Check Data Data for the linearity check on the measurement system are presented below. All measurements were made at a source temperature of 354~Fo Source opening (in the bottom shield) was 1/2 inch in diameter. The distance between this opening and the detector is tabulated under "Distance", in units of centimeterso The corresponding signal measured is given in terms of output-meter readings, Rm. These data are plotted on Figure 13. TABLE VIII LINEARITY CHECK DATA Distance, cm 8 12 12 18 26 34 46 60 80 100 116 Measured Signal, Rm 605 290 287 142 74.0 44 5 25.0 14.7 8.4 5.0 3.9

-183 - Screen Ratio Calibration Data Calibration data for the standard ratio, Fs, are presented below. The notations of the headings are: ts = temperature of source, ~F Opening = diameter of source opening in bottom shielf, inches Setting = setting of bed holder on mounting track So = incident flux intensity measured without standard screen So* = incident flux intensity measured through standard screen Fs = So. So Avg. Fs = the average value of Fs for one source temperature TABLE IX SCREEN RATIO CALIBRATION DATA Avg. ts, ~F Opening Setting So So* Fs Fs 570 0.50 35 99.8 18.8 5.31 5.46 0.75 300 55.3 5.42 2.50 995 182 5.47 800 145 5.52 0.50 40 1000 181 5.52 0.75 1005 182 5.52 2.50 1000 180 5.52 845 152 5.56 50 1000 182 5.52 500 91.0 5.49 90 299 55.5 5.39 645 117 5.51 200 36.8 5.43 110 100 18.6 5.38 149 27.7 5.38 0.75 29.6 5.40 5.48 677 2.50 50 552 97.3 5.67 5.67 992 174 5.70 541 95.1 5.69 985 174 5.66 429 76.0 5.64 1000 175 5.71 410 72.9 5.63

TABLE IX (CONTtD) Avg. ts, ~F Opening Setting So So* F Fs 800 2.50 50 674 115 5.86 5.82 1000 173 5.78 648 111 5.84 856 147 5.82 884 152 5.82 1020 2.50 65 990 172 5.76 5.77 0.75 100 17.9 5.59 0.50 100 17.7 5.65 2.50 70 845 146 5.79 0.75 299 52.5 5.70 0.50 95.5 16.5 5.79 2.50 80 802 138 5.81 95 504 87.7 5.75 0.75 222 37.5 5.92 0.50 97.5 16.5 5.91 2.50 115 995 175 5.69 475 83.4 5.68 0.75 94.0 16.0 5.88 0.50 46.5 7.80 5.96 2.50 900 157 5.73 1200 0.75 50 982 171 5.74 5.78 2.50 90 802 138 5.81 810 139 5.83 0.75 50 252 43.9 5.74 1400 2.50 110 900 160 5.63 5.72 0.75 75.5 13.2 5.72 80 284 50.0 5.68 0.50 50 206 36.0 5.72 0.75 1003 172 5.83 1465 0.75 70 990 168 5.89 5.86 0.50 574 96.5 5.95 2.50 110 1023 178 5.75 0.75 270 46.0 5.87 0.50 52.5 8.95 5.85 1700 0.75 110 296 51.0 5.80 5.83 0.50 68.1 11.5 5.92 0.75 80 800 137 5.84 0.50 138 23.8 5.80 0.75 60 970 167 5.81 0.50 1000 172 5.81

-185 - TABLE IX (CONT'D) Avg. ts, ~F Opening Setting S SoF Fs F 1710 0.75 110 272.46.5 5.85 5.83 0.50 282 48.3 5.90 0.75 80 493 84.8 5.81 0.50 710 120 5.92 0.75 60 1000 173 5.78 0.50 196 34.2 5.73 1800 0.75 50 980 171 5.73 5.80 710 122 5.82 1000 173 5.78 599 102 5.87 2000 0.75 50 908 157 5.78 5.78

-186 - Screen Temperature Calibration Data Data for calibration of input signal measured through standard screen, So, as a function of source temperature are presented below. All measurements were with mesh 30 support screen. Parameters of two different bed tubes are presented. These data are plotted on Figure 21. TABLE X Screen Temperature Calibration Data Bed Tube Length, inches 3 4 Source Temperature ts,~F Ts4, OR4 678 1.68 x 1012 798 2.50 1200 7.59 1599 18.0 1696 21.6 677 1.67 x 1012 800 2.52 1200 7.59 1599 18.0 1696 21.6 Signal S* 76 115 355 850 1020 73 111 339 810 971

TRANSMISSION VALUES FROM REGRESSION At any packing height, h, the transmission of modulated flux may be calculated by Equation (115), using the converged values of sb and a obtained by regression. These calculated transmissions, Sn'(h), are tabulated below and compared to the experimentally measured transmissions, Sn(h). The difference, Sn - Sn, is then a measure of the fit obtained by the regressed transmission curves. The values of S are plotted as data points and the values of Sn' are represented by solid curves on Figures 22-33. Standard deviations, 6, are also tabulated below. Table XI Transmission Values from Regression Run 7 Sb, ft1 a, ft 977 0.390 8 878 0.209 6s' % h, cm. 5.07 1.o 1.95 2.83 3.72 4.54 2.14 1.97 2.85 3.70 4.57 5.51 6.46 3.33 2.29 3.24 4.21 5.19 6.27 3.81 1.01 1.83 2.54 3.07 3.59 4.12 4.64 5.45 6.26 s s n Sn.0282.0259.00904.00966.00415.00426.oo0189.oo00189.00093.00090.0134.0135.00724.00731.00432.00421.00246.00242.00129.00134.000750.000737.0106.0101.00517.00547.00304.00300.00165.00166.000870.000861.0690.0703.0301.o308.o165 0164.0108.0104.00703.00669.00449.00427.00257.00275.00134.00138.000711.000697 Sn-Sn '.0023 -.00062 -.00011.00000.00003 -.0001 -.00007.00011. oooo4.oooos.00004 -.00001.000009 -.0013 -.0007.0001.ooo0004.00034.00022 -. 0001.8 -.00004.oooo14 9 952 10 343 0.969 -187 -

-188 - Table XI (Cont'd) Transmission Values from Regression Run sb, ft a, ft h, cm. Sn S ' S-S' A n- n 11 298 0.521 1.99 12 296 3.46 1.03 1.91 2.76 3.36 3.97 4.58 5.27 5.95 6.82 1.20 2.17 2.95 3.72 4.47 5.24 6.02 6.78 7.74 1.01 1.90 2.61 3.15 3.67 4.50 5.37 o.940 1.85 2.75 3.45 3.98 4.78 5.59.0874.o4o6.0229.0159.0114.00816.00529.00355.00218.0784.0339.0229.0149.00973.00641.00417.00290.00178.0734.0297.0162.0106.00694.00360.00176.0732.0286.0129.00732.00477.00248.00128.c843.0410.0235.0163.0113.00793.00531.00358.00216.0732.0360.0226.0147.00981. oo65.00430.00288.00173.0714.0301.0165.oio6.0106.00695.00355.00176.0714.0287.0132.00739.00477.00247.00127.0031 -.0004 -.ooo6 -.0004.0001.00023 -.00002 -.00003.00002.0052 -.0021.0003.0002 -.00008 -. oooo8 -.ooo0009 -.00013.00002.00005 -.0004 -. ooo4 -.0003.0000 -.00001.00005.00000.0018 -.0003 -. 00007.00000.00001. 00001 13 342 373 0.884 0.843 1.47 1.46 14 15 190 1.54 0.647 1. o 1.95 2.86 3.68 4.39 5.10 5.91 6.73 7.62.121.0490.0231.0120.00684.00387.00204.00106.000524.119.0493.0233.0120.00683.00388.00204.00106.000522.002 -.0003 -.0002.0000.00001 -.00001.00000.00000.000002

-189 - Table XI (Cont'd.) Transmission Sb - ft-1, f Values from Regression <sy, % h, cm. Sn Run Sn Sn-Sn 16 149 17 151 o.988 0.812 2.74 1.o6 1.85 1.98 2.93 3.79 4,48 5.17 6.04 6,90 7.83 8.79 1.33 2.29 3027 4,04 4,82 5059 6.36 7.33 8.29 2.93 1.27 1.75 2.16 2.56 2.97 3.36 3.78 5.16 1.41 1.76 2.13 2.50 2.86 3.41 3096.0727.0396.0250 o0169 o01 5 00678 oo0045.00244.oo146 o0620.0357.0240 o0163,oio6.00706.00428.00267.0229 o00933.oo464.00227.oo018.ooo6oi.000304.00892.00441.00231 o00126.000717.000295.000126.0736.o4o6.0245. oi65.0112.00683.00420.00249.oo145.o626.0359.0238.0158.oO06.00714.00434.00265.0218.00950 o00470.00237 o 00118.000603.ooo0294.000294.00802.00448.00243.00132.000730.000295.000119 -o0009 -o0010.0005.0004 00003 -o00005 -,00005 -o00005.00001 -.ooo6 -.0002.0002.0005 o0000 -oooo0008 -.oooo6.00002.0011 -.00017 -,oooo6 - 0001.0.00000 -,000002 -.000010.00090 -o00007 -.00012 -ooooo6 -000oooo01.000000.000007 18 496 20 1188 21 22 906 682 1.00 1.96 4.38 1.94 2.35 2.75 3.15 3.53 4.19 4.87 5.96 1.33 1.73 2.13 2.53 2.94 3.71.oo640.00334.00192.00103.000622.000252.000105.0157.00787.00355.00176.000973.000274 o00598.00042.00336 -.00002.00192.00000.00109 -.00006.000643 -.000021.000255 -,000003.0000986.ooooo64.0149.00748.00378.00192.000956.000259.00ooo8.00039 -.00023 -.00016.000017.000015

-190 - Table XI (Cont'd) Transmission Values from Regression Run b, ft-1 a, ft-1 S'bPr a, r s' % h, cm. Sn Sn' s -S n n n 496 2.74 1.06 2.93 1.27 1.75 2.16 2.56 2.97 3.36 3.78 5.16 1.41 1.76 2.13 2.50 2.86 3.41 3.96.0229.00933.oo00464.00227.00118.000601.ooo3o4.000304.00892.00441.00231.00126.000717.000295.000126.0218.00950.00470.00237.00118.000603.000294.00802.00448.00243.00132.000730.00295.000119.0011 -.00017 -.oooo6 -.00010.00000 -.000002 -.000010.00090 -.00007 -.00012 -.oooo6 -. 000013.000000.000007 20 1188 21 22 906 682 1.00 1.96 13.1 6.66 4.38 1.94 2.35 2.75 3.15 3.53 4.19 4.87 5.96 1.33 1.73 2.13 2.53 2.94 3.71 8.43 1.05 1.21 1.37 1.53 1.79 1.96 2.22 6.26 1.69 1.96 2.22 2.49 2.76 3.12.oo640.00334.00192.00103.000622.000252.000105.0157.00787.00355.00176.000973.000274.00923.00549.00265.00157.000584.000315.000159.00675.00335.00189.000897.000525.000253.00598.00336.00192.00109.000643.000255.0000986.0149.00748.00378.00192.000956.000259.oo888.00500.00282.00159.000627.000341.000134.00633.00339.00186.000996.000533.000232.00042 -.00002.00000 -.oooo6 -.000021 -.000003.000ooooo64.0008.0oo0039 -.00023 -.0001i6.000017.000015.00035.oo00049 -.00017 -.00002 -.oooo000043 -.000026.000025.00042 -.oooo4.00003 -.000099 -.ooooo8.000021 23 450 24 370

-191 - Table XI (Cont'd) Transmission Values from Regression Run Sb ft-1 a, ft-1 % h, cm. S S' S-S 25 789 2.05 8.41 1.58.00770.00704.ooo66 1.98.00343.00333.00010 2.36.00156.00164 -.oooo8 2.74.000727.ooo804 -.000077 3.12.000376.000396 -.000020 3.72.00149.000129.000020 26 779 12.7 4.58 0.420.0473.0439.0034 0.620.0162.0172 -.0010 0.930.oo4o6.oo408 -.00002 1.05.00234.00234.00000 1.25.000887.000927 -.000040 1.37.000527.000532 -.000005 1.49.000325.000305.000020 27 420 272. 25.2. 160.0640.0494.0146 0.310.00295.00330 -.00035 0.460.00080.000221 -.000041 0.600.0000280.0000176.oooo0000104 28 153 1.88 0.784 1.72.0729.0729.0000 2.62.0345.0345.0000 3.43.181.01 81.0000 4.25.00945.00943.00002 5.07.00495.00494.00001 5.87.00260.00262 -.00002 6.58 o00148.00150 -.00002 7.49.000742.000731.000011 29 114 1.33 0.734 1.97.0925.0921.0004 2.91.0507.0509 -.0002 3.78.0301.0304 -.0003 4.65.0184.0184.0000 5.51.0113.0112.0001 6.20.00753.00753.00000 7.16.00438.00434 00004 7.84.00291.00294 -.00003 30 114 1.08 0.909 1.95.0982.981.0001 2.93.0553.0552.0001 3.91.0324.0325.000oooo 4.86.0197.0197.0000 5.63.0130.0132 -.0002 6.41.00894.00881.00013 7.18.00585.oo592 -.00007 7.94.00403.oo400.00003

-192 - Table XI (Cont'd) Transmission Values from Regression Run Sb,ft-l a, ft-l,9 % h, cm. S' n Sn.' S 31 483 2.70 7.27 1.27.0260.0229.0031 1.75.0102.0101.0001 2.16.00503.00509 -.oooo6 2.56.00235.00260 -.00025 2.97.00120.00131 -.00011 3.36.000682 000679.000003 3.78.000347.000336.000011 4.18.000188.000172.oooo000016 32 860 1.49 4.26 1.41.0115.0107.0008 1.76.00600.00597.00003 2.13.00317.00322 -.00005 2.50.00164.00174 -.00010 2.86.000931.000957 -.000026 3.43.000368.000371 -.000003 3.98.000158.000149.000009 33 682 1.32 4.40 1.94.00849.00785.000oo64 2.35.00436.00442 -.oooo6 2.75.00249.00253 -.oooo00004 3.15.00140.o0045 -.00005 3.53.000812.000851 -.000039 4.19.000334.000339 -.000005 4.87.000140.000131.000009 34 614 2.18 5.45 1.33.0179.0163.0016 1.74.00807.00807.00000 2.14.00382.oo4o8 -.00026 2.55.00196.00203 -.00007 2.94.oo00106.00105.00001 3.35.000498.000521 -.000023 4.12.000151.000141.000010 35 logo1090 1.16 10.2 1.58.00788.00659.00129 1,98.00353.00340.00013 2.36.00173.00182 -.00009 2.74.000838.000973 -.000135 3.12.000490.000521 -.000031 3.74.000197.000188.000009 4.33.0000790.0000711.0000079 36 811 10.6 4.77 0.620.0206.0190.0016 0.810.00819.00834 -.00015 1.01.00331.00351 -.00020 1.21.00140.00148 -.00008 1.33.000880.000881 -.0000ooooo 1.45.000539.000524.000015 1.56.000341.000326.000015

-193 - Table XI (Cont'd) Transmission Va-ues from Regression Run b, ft1 a, ft-1, h, cm. S Sn' S-Sn' 37 1054 107 18.6 o.160 0619.o460.0159 0.310.00334.00418 -.00084 o.460.000403.000380.000023 o.600 oooo440.oooo0000407.0000033 38 387 17.0 4.71 1.05.00803.00824 -.00021 1.31.00324.00306.00018 1.47.00172.00167.00005 1.63.000877.ooog000908 -.000031 1.79.000463.000494 -.000031 1.96.000274.000259.000015 39 322 8.70 7.57 1.32.0151.0143.0008 1.59.00776.00731.00045 1.85.00365.00385 -.00020 2.12.00178.00197 -.00019 2.39.000986.oo00101 -.000024 2.76.000457.ooo46.000051 40 149 0.804 2.09 2.29.0647.0636.0011 3.26.0376.0366.0010 4.04.0238.0244 -.ooo0006 4.82.0162.0163 -.0001 5.59.0107.0110 -.0003 6.36.00722.00740 -.00018 7.33.00461.00451.00010 8.29.00282.00277.00005 41 129 2.21 1.67 1.72.o856.0839.0017 2.62.0394.0399 -.0005 3.43.0212.0209.0003 4.25.0108.0109 -.0001 5.07.00569.00574 -.00005 5.87.00299.00306 -.00007 6.58.00175.00175.00000 7.49.ooo879.ooo856.000023 42 95.0 1.59 o.886 1.97.16.107 -.001 2.91.0591.0593 -.0002 3.78.0361.0355.0006 4.65.0216.0214.0002 5.51.0130.0131 -.ooo001 6.20.00875.00881 -.oooo6 7.16.00504.00508 -.oooo00004 7.84.00347.00344.00003

-194 - Table XI (Cont'd) Transmission Values from Regression Run S b, ft-1 a, ft-l s, s% h, cm. Sn Sn Sn n Sn-Sn 43 96.8 1.24 0.556 2.93 3.91 4.86 5.63 6.41 7.18 7.94.0636.0383.0230.0156.0105.00705.00482.0639.0379.0232.0156.0105.00708.00481 -.0003.0004 -.0002.0000.0000 -. 00003.00001 44 449 2.84 1.47 6.25 1.27 1.75 2.16 2.56 2.97 3.36 3.T8 4.18 6.72 1.41 1.76 2.13 2.50 2.86 3.43 3.98 4.53.0275.0107.00502.00290.00158.000785.000373.000198.0124.00637.00338.00172.000944.000388.000161.0000740.0248.0111.00561.00289.oo146.000766.000382.000197.0111.00622.00338.o0184.00102.000401.000163.0000oooo660.0027 -.0004 -.00059.00001.00012.000019 -.000009.000001.0013.00015.00000 -.00012 -.000076 -.000013 -.000002.ooooo8o 45 576 47 1.54 2.77 1.20 8.45 1.28 1.94 2.35 2.75 3.15 3.53 4.19 4.87 5.22 1.33 1.74 2.14 2.55 2.94 3.35 4.12 496.0280.00924.00503.00299.00163.000947.000417.000187.0208.00960.00476.00224.00111.000605.000169.oo9o6.00392.00193.0238.00934.00528.00304.00175.00103.ooo044.000162.0194.00oo953.00478.00236.00120.000594.oools8.000158.00723.00377.00203.0042 -.00010 -.00025 -.00005 -.00012 -.000083.000003.000025.0014.00007 -. 00002 -.00012 -.00009.000011.000011.00183. ooi83.00015 -.00010 48 1010 12.4 1.58 1.98 2.36

-195 - Table XI (Cont'd) Transmission Values from Regression Run Sb, t-1 a, ft-l is, % h, cm. Sn Sn' Sn-Sn 49 837 9.37 192 50 627 2.74 3.12 3.74 4.33 5.31 0.620 0.810 1.01 1.21 1.33 1.45 1.56 27.3 o.16o 0.310 0.460 o.60o 6.97 1.05 1.31 1.47 1.63 1.79 1.96 4.37 1.32 1.59 1.85 2.12 2.39 2.76.000962.000528.000224.000100.0219.00912.00380.00169.ooio4.000676.000452.0529.00420.000211.0000oooo430.00110.000594.000218.0000837.0202.00920.00404.00177.00108.000659.000419.0494.00370.000277.0000247 -.000138 -.000oooo66.00ooooo6.0000163.0017 -.000oooo8 -.00024 -.0000oooo8 -.00oooo4.000017.000033.0035.00050 -.000066.0000183 51 343 20.5 9.26.00778.00315.00138.ooo800.000394.000233.0144.00701.00361.00172.000906.000392.00785.00281.00149.000794.000422.000215.0138.00699.00363.00184.000930.000366 -.00007.00034 -.00011.ooooo6 -.000028.000018.ooo6.0002 -.00002 -.00012 -.000024.000026 52 314

PROPERTIES OF TEST PACKINGS,,.. ~ _ - _ _ Some properties of the test particles are listed below. These are the values used in the correlations presented previously. Many of the properties had to be estimated from limited data available in the literature. Reference numbers are given for such items. Some bulk properties were determined experimentally and these are marked by a superscript "e". Other properties were obtained from manufacturer's specifications. These are marked by a superscript "m". Dp = nominal equivalent diameter, ft. Material = basic material of solid particle. ks = conductivity of solid particle, Btu/hr.ft.~F. _6 = emissivity of solid particle. gs = density of solid particle, gm./cc. oB& = bulk density of randomly packed particles, gm/cc. Hi,= void fraction of packing. -196 -

TABLE XII PROPERTIES OF TEST PACKINGS 8000F 1200OF 1600oF 1800oF 2000~F Particle Material k Dp 6 k 6 k E k s f 8 B Sp GS-3 boro-0licate 0 o.0098(m) 0,63(24) 0.9(24) 0.63(24) 0.8(24 0 (4) 0 (24) 0.6(24) 1.45(e) 0.35(e) GS-3 boro-ilicate I0.0098m) 0.63 0.63 0.8 0.63 0.63 0.63 2.23 1.45 0. GS-4 Or 0.0131(m) (.45 )0 35 GS-5 0.0164(m) 1. 6(e) 059(e) AS-5/16 A1203(m) 0.0157(m) 2.22(25) 0.322(25) 2.29(25) 0.28(25) 2.56(25) 0.28(25) 3.95(25) 2.01(e) Q.49(e) AP-1/8 I1 0.0104(m) j 1.54(e) 066(e) AP-5/32 O.0150(m) 21(e) (e) AP-3/16 0.0157(m)12(e).67( SS-1/8 carbon steel(m) 0.0104(m) 17(24) 0.5(24) 16(24) 0.6(24) 15(24)1 0.6(24) 7.8(16) 4,76(e) 09, (e) SS-3/16 0.0157(m) 4.72(e) 0.40(e) AG-4 A1203(m) 0.0216(m) 1 2.22(2 (25) 02(25) 2.29(25) 0.28(25) 2.36(25) 028(5) 95(26) 155(e) 0.61(e) AG-16 | 0.0056(m) I 1.83(e) 0.54(e) CG-16 SiC(m) 0.0056(m) 8.75(25) 0.88(25) 8.2525) (25). 0.(25) 5.20(26) 1.68(e) 0 (e) sCc?. 7'~( )lo.g y~:. 6(). H!

COMPUER PROGRAMS Data regression for values of the attenuation cross sections was performed on an IBM 704 digital computer. The necessary equations have been developed in a previous section. The actual programs are presented here. Computation was carried out in two parts: a. regression on Equation (115) for values of an and m, b. calculation of Sb and a from the converged values of an and m by Equations (174) and (175). Schematic flow diagrams and Fortran statements of the programs are presented below. -198 -

-199 - FLOW DIAGRAM FOR ITERATIVE REGRESSION: FLOW DIAGRAM FOR CALCULATION OF CROSS SECTIONS: READ INPUT: RUN NO. NO. OF DATA POINTS BACKGROUND REGRESS VALUES OF an and m STANDARD DEVIATION,,8 CALCULATE sb and an BY EQUATIONS (174) and (175) I 1 WRITE AND PUNCH OUTPUT: RUN NO. m, an, s', sb, a

-200 - FORTRAN STATEMENTS FOR REGRESSION PROGRAM DIMENSION H(20),S(20),F(20),CH(20),SH(20),FN(20),DIF(20),R(20), X A(100),B(100),PHE(100) 55 READ INPUT TAPE 7,3,NURNHBG READ INPUT TAPE 7,4.(H(J),F(J),J=1 NH) READ INPUT TAPE 7,10,KEYITFITN WRITE OUTPUT TAPE 6,5,NUR WRITE OUTPUT TAPE 6,6,NHBGITN FNH = NH DO 99 J-1,NH 99 R(J) = 1.0 /(F(J)*F(J)) I = 1 IF (KEY) 57,56t57 56 K = NH J = NH - 2 B(I) = ELOG(F(J)/F(K)) / (H(K)-H(J)) EP = EXP(B(I)*H(K)) A(I) = 20*8(I) / (EP*F(K)) - B(I) GO TO 58 57 READ INPUT TAPE 78,IA(I)gB(I) 58 WRITE OUTPUT TAPE 6,12,KEYITF9A(I)*B(I) 59 PHE(I) = 0.0 DO 98 J=SINH EP = EXP(B(I)*H(J)) EN = 1.O/EP CH(J) = 0.5*(EP+EN) SH(J) = 0O5*(EP-EN) FN(J) = B(I)/(B( I )*CH(J)+A( I SH(J))

-201 - DIF(J) ' F(J) - FN(J) 98 PHE(I) = PHE(I) + (DIF(J)*DIF(J)*R(J) ARG = PHE(I)/FNH SD = 100.0SQRT(ARG) M ITF + I - 1 WRITE OUTPUT TAPE 6,11,M,A(I),B(I),SD WRITE OUTPUT TAPE 6~13,(H(J),F(J)~FN(J)tDIF(J),J=1~NH) IF (M-ITN) 60,65,65 60 SDB = 00 SDA = 0.0 SAB = 00 SAA 5 0*0 SBB = 0.0 DO 97 J=1~NH DEN = (B(I)*CH(J)+A(I)*SH(J)) * (B(I)*CH(J)+A(I)*SH(J)) PA = B(I)*SH(J)/DEN PB = (B(I)*A(I)*H(J)CH(J)+(B()*BI)*H(J)-A(I))*SH(J))/DEN DA = -DIF(J)*PA*R(J) DB = -DIF(J)*PB*R(J) AB PA*PB*R(J) AA = PA*PA*R(J) BB PBPBRPB(J) SDA = SDA + DA SDB a SDB + DB SAB = SAB + AB SAA a SAA + AA 97 SBB = SBB + BB CA s SBB*SDA - SAB*SDB CB = SDB*SAA - SDA*SAB

-202 - C = SBB*SAA - SAB*SAB DELA = CA/C DELB = CB/C WRITE OUTPUT TAPE 6916,SDASDBSABSAAsSBBCA~CBCDELADELB K = I+1 A(K) = A(I) + DELA B(K) = B(I) + DELB I K GO TO 59 65 X = FNH / (FNH-10O) V = SD * SQRT(X) WRITE OUTPUT TAPE 6,5,NUR WRITE OUTPUT TAPE 6~6,NHBGITN WRITE OUTPUT TAPE 6,11,M,A(I)gB(I) WRITE OUTPUT TAPE 6,11,MSDV WRITE OUTPUT TAPE 613~ (H(J) 9F(J),FN(J),DIF(J),J1,NH) HT = 0.0 WRITE OUTPUT TAPE 6.5,NUR WRITE OUTPUT TAPE 6,11~,MA(I )B(I) DO 95 L=1,40 EP = EXP(B(I)*HT) EN = 1O/EP CSH = O.5*(EP+EN) SNH = O.5*(EP-EN) T = B(I) / (B(I)*CSH+A(I)*SNH) IF (T-0.OO0001) 70,71,71 71 WRITE OUTPUT TAPE 6,9,HTT 95 HT = HT + 0.25 70 PUNCH 3~NUR~NH~BG

-203 - PUNCH PUNCH PUNCH GO TO 3 FORMAT 4 FORMAT 5 FORMAT 6 FORMAT X 7 FORMAT 8 FORMAT 9 FORMAT 10 FORMAT 11 FORMAT 12 FORMAT 13 FORMAT 14 FORMAT 15 FORMAT 16 FORMAT 8,A(I) B(1) 8,SDV 14,(H(J) F(J) FN(J) J=1,NH) 55 (2I7,F15.6) (F9*2,F18.6) (1H13AH (1HO//16H DATA POINTS I5/15H 14H ITERATION I5) (1H 7E17,8) (3E20.8) (1H F20*2,F20.6) (317) (1HO // I7,4E20.8) (1HO 217~2E20,8) (1HO 4E17.8/(1H 4E17.8)) (3E17.8) (1H F27.2F47.6) (1H /5E17.8/5E17*8) RUN 17 ) BACKGROUND F9.4/

-204 - FORTRAN STATEMENTS FOR CROSS SECTION PROGRAM 55 READ INPUT TAPE 793,NUR~NH~BG READ INPUT TAPE 7,8.AB READ INPUT TAPE 7,8,SD~V SB = SQRT (A*A-B*B) AA = A-SB WRITE OUTPUT TAPE 6t5,NUR WRITE OUTPUT TAPE 6~6~B~A~SB~AA~NH~SD~BG PUNCH 3~NURtNHtBG PUNCH 8~A~B PUNCH 8.SBAA GO TO 55 3 FORMAT (217,F15.6) 5 FORMAT (1H134H 8 FORMAT (3E20.8) 6 FORMAT ( 1HO//20H K CONSTANT F3Z X 35H ATTENUATION CROSS SECTION F1 X 39H BACK SCATTERING CROSS SECTIC X 34H ABSORPTION CROSS SECTION F2( X 21H DATA POINTS I27// X 28H STANDARD DEVIATION F23.2, 1] X 20H BACKGROUND F35.6 ) RUN 4.5// 19.5// )N F15.5// 5//H PER C 1H PER C 17 ) ENT //

NOMENCLATURE A Absorption cross section per unit volume for black body radiation; ft2/ft3. A Boundary constant, Equation (81). At Attenuation ratio on amplifier. AN Absorption cross section per unit volume for monochromatic radiation; ft2/ft3. a Bulk absorption cross section per unit volume; ft2/ft3. ad Absorptivity of detector. an Net attenuation cross section per unit volume; (a + sb); ft2/ft3. B Boundary constant, Equation (82). C Boundary constant, Equation (83). aDp Ca Dimensionless absorption parameter, 1 -. Btu CB Bulk heat capacity; bI, -u F ' C5 Dimensionless scattering parameter, bD 1-5p D Boundary constant, Equation (84). D Determinant in regression solution, Equation (167). Da Determinant in regression solution, Equation (169). Dm Determinant in regression solution, Equation (168). Dp Nominal particle diameter; ft. E Emission cross section per unit volume for black body radiation; ft2/ft3. EA Emission cross section per unit volume for monochromatic radiation; ft /ft3. FF Standard screen factor, ( s. f(QQ') Angular distribution function for black body radiation. i fj Calculated Sn for jth data point at ith iteration of data regression, Equation (i48). fk(,QO') Angular distribution function for monochromatic radiation. -205 -

-206 - NIOMENCIATURE (cont'd) h Packed bed height; ft. or cm. I Dimensionless forward flux intensity; Equation (53). i Forward flux intensity, Equation (21); Btuhr- hr.ft"B Btu ie Modulated forward flux intensity for experimental case; hr.ft~ ~i Specific intensity for monochromatic radiation; hBt --- u hr.ft steradian is Specific intensity for black body radiation; Btu/hr.ft.stetadian. J Dimensionless backward flux intensity, Equation (53). j Backward flux intensity, Equation (22); Btu/hr.ft.2. je Modulated backward flux intensity for experimental case; Btu/hr.ft 2. K Function in Equation (129). kc Bulk conduction-conductivity; Btu/hr.ft. F. kir Bulk radiation-conductivity at internal points; Btu/hr.ft. ~F. kr Bulk radiation-conductivity; Btu/hr.ft. F. ks Conductivity of solid particles; Btu/hr.ft.~F. L Length of bed across packirg; ft. ~m Attenuation parameter, (Nan + ); ft.2/ft.3. n Number of data points of a run. p Exponential power on temperature, Equation (134). Qc Conduction heat transfer rate; Btu/hr.ft.2. Qr Radiation heat transfer rate; Btu/hr.ft.2. R Output meter reading, normalized to unit attenuation. Rb Output meter reading for background signal, normalized to unit attenuation. Rm Output meter reading, unnormalized.?r ~ Position vector, Figure 4.

-207 - NOMENCLATURE (cont'd) S Scattering cross section per unit volume for black body radiation; ft.2/ft.3. S(h) Modulated flux intensity transmitted through bed of height h; Btu/hr.ft 2. Sn(h) Normalized transmitted flux intensity, (S(h)/So), experimental value; Btu/hr.ft.2. Sn(h) Correlation value of Sn(h); Btu/hr.ft.2. So Incident modulated flux intensity; Btu/hr.ft.2. St Value of SO through standard screen; Btu/hr.ft.2. Sx Scattering cro s section per unit volume for monochromatic radiation; ft. /ft.. sb Bulk back scattering cross section per unit volume; ft.2/ft.3. ts, Ts Source temperature; ~F or ~R. tl, T1 Hot wall temperature; ~F or ~R. t2, T2 Cold wall temperature; ~F or ~R. VT Voltage signal due to modulated flux. X Dimensionless position, Equation (53). x Position coordinate, ft. OL Dimensionless absorption coefficient, Equation (54). ~P Dimensionless back scattering coefficient, Equation (54). Dimensionless radiation-conduction ratio, Equation (54). ~6 Dimensionless temperature ratio, Equation (54). Sp Void fraction of packed media. Ss Standard deviation for transmission curves, Equation (128). Am Maximum T error in Hamaker's approximation. Ami Correction to m in regression, Equation (145). Znai Correction to an in regression, Equation (145). n

-208 - AW Incremental packing sample weight; lbs. or gms. ~G Emissivity of packing. 55 Emissivity of source. (N ^Emissivity of source walls. } wEmissivity of hot wall (boundary). ~Emissivity of cold wall (boundary). q ADimensionless net attenuation parameter, (a + I). ~ Directional angle, Figure 5. N Wave length of radiation; microns. e ^Bulk density of packed media; lb./ft.3.. TStephan-Boltzman constant; 1.713 x 10-9 Btu/hr.ft.20R4. 1 Dimensionless temperature, Equation (53). 10 Base temperature for Hamaker's approximation, Equation (76); OF or OR. Sum of errors squared, Equation (143). Azimuthal angle, Figure 5. Solid angle; steradian. Vectorial direction of flux propagation, Figure 5.

REFERENCES 1. Argo, W. B. and Smith, J. M. Chem. Eng. Prog., 49, 443 (1953). 2. Bell, J. C. Trans. Am. Foundrymen's Assoc., 56, 365, (1948). 3. Bosworth, R. C. L. Heat Transfer Phenomena. New York: John Wiley and Sons, Inc., (1952). 4. Bunnell, D. G., Irvin, H. B., Olson, R. W., and Smith, J. M. Ind. Eng. Chem., 41, 1977, (1949). 5. Campbell, J. M. and Huntington, R. L. Petroleum Refiner, 31, 123, (Feb., 1952), 6. Damk'ohler, G. "Der Chemic-Ingenieur." Eucken-Jakob, 3, Part I, 445 Akademische Verlagsgesellschaft M.B.H., Leipzig, (1937T. 7. Deissler, R. Trans. ASME, 80, 1417, (1958). 8, Dickson, J. H. Glass. NBwv2York: Chemical PUblishing Co., Inc., 1951. 9. Eucken, A. Beilage Zu Forschung auf dem Gebiete des Ingenieurwesens, Forshungsheft 353, Ausgabe B, (1932) 10. Glassman, I. and Bonilla, I. CEP Symposium Series No. 5, 49, 153, (1953). 11. Gorring, R. L. and Churchill, S. W. Paper presented at A.I.Ch.E. meeting, Washington, D.C., Dec. 1960. 12. Gouffe, A. Revue D'optique, 24,, (Jan.-Mar., 1945). 13. Hamaker, H. C. Philips Research Reports, 2, 55, 103, 112, 420, (1947). 14. Hellums, J. D. and Churchill, S. W. Paper presented at A.I.Ch.E.-A.S.M.E. Heat Transfer Conference, Buffalo, New York, 1960. 15.. Hill, F. B. and Wilhelm, R. H. A.I.Ch.E. Journal, 5, 486, (1959). 16. Hodgman, C.D. ed., Handbook of Chem. and Phys., 33rd ed., Cleveland: Chemical-Rubber Publishing Co., 1952. 17. Horton, C. W. and Rogers, F. T. J. Applied Physics, 16, 367, (1945). 18. LaFara, R. L., Miller, E. L., Pearson, W. E., and Peoples, J. F. Tables of Blackbody Radiation and the Transmission Factor for Radiation through Water Vapor. Navord Report 3171, U.S. Naval Ordnance Plant, Indianapolis, 1955. -209 -

-210 - 19. Lapwood, E. R. Proc. Camb. Phil. Soc., 44, 508, (1948). 20. Larkin, B. K. and Churchill, S. W. A.I.Ch.E. Journal, 5, 467, (1959). 21. Laubitz, M. J. Canadian J. of Phys., 37, No. 7, 798, (1959). 22. Lucks, C. F., Linebrink, O. L., and Johnson, K. L. Trans. Am. Foundrymen's Assoc., 55, 62, (1947). 23. Maxwell, J. Electricity and Magnetism, I, 314, Oxford: Oxford University Press, (1873). 24. McAdams, W. H. Heat Transmission. 3rd. ed., New York: McGraw-Hill Book Co., Inc., 1954. 25. Norton, F. H. Refractories. 3rd. ed., New York: McGraw-Hill Book Co., Inc., 1949. 26. "Refractory Grain - Properties and Applications." Norton Co., Worcester, 1959. 27. Nusselt, W. Zeitscher d. bayer Revisionsuer, Nos. 13 and 14, (1913). 28. Perry, J. H., ed., Chemical Engineer's Handbook. 3rd ed., New York: McGraw-Hill Book Co., Inc., 1950. 29. Rayleigh, L. Phil. Mag. V, 34, 481, (1892). 30. Rogers, F. T. and Morrison, H. L. Heat Transfer and Fluid Mechanics Inst., Stanford University Press, 1951. 31. Rosseland, S. Theoretical Astrophysics. Oxford: The Clarendon Press, 1936. 32. Russell, H. W. J. Am. Ceramic Soc., 18, 1, (1935). cc N) 33. Schotte, W. Thermal Conductivity of Insulating Powders. Eng. Res. Lab., E.I.duPont de Nemours and Co., Wilmington, Delaware, 1958. 34. Schumann, T. E. W. and Voss, V. Fuel, 13, 249, (1934). 35. Van der Held, E. F. M. Appl. Sc. Res., A3, 237, (1952), and A4, 77, (1953). 36. Waddams, A. L. Chem. and Ind., 206, (1944). 37. Waddams, A. L. J. Soc. Chem. Ind., 63, 337, (1944). 38. Wilhelm, R. H., Johnson, W. L., Wynkoop, R., and Collier, D. W. Chem. Eng. Prog., 44, 105, (1948).

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UNIVERSITY OF MICHIGAN 3 9015 02829 516311111111 I 3 9015 02829 5163