Heat Transfer in Packed Beds J. J. BARKER
The first half of an extensive review of the literature on heat transfer between fluids and particulate solids. This half is concerned with packed beds and emphasizes the fluid-to-particle heat transfer
coefficients
for
spheres,
cubes, and commercial packings. The results indicate that there is a surprising lack of data in many areas which could be corrected by simple experiments. The second half will deal with f Iuidized beds.
T review .
is an outgrowth of a report to Brookhaven Natlonal Laboratory on the status of the field of particle-to-fluid heat transfer in packed and fluidized beds. An attempt was made to include all the known data on fluid-to-particle heat transfer coefficients. This was impossible, not only because it is improbable that all sources were found, but also because some of those which were uncovered were unavailable for study. However, it is believed that a representative majority of data on the subject is included. Of the information found, nothing is withheld, even though it is quite different from information generally reported by the majority of investigators. Many references were found on subjects related to heat transfer in particulate systems, e.g., mass transfer coefficients, pressure drops, eddy diffusion, and mixing. It is hoped that enough references on these related topics are given to serve as a useful starting point for more thorough searches in the fringe areas. his
I
t__ 10
I co
1000
10 OCO
REYNOLDS NUMBER
Figure 7. General correlations of heat transfer coeficients between stationary particles in packed beds and moaing Juids. Superimposed numbers refer to corresponding literature reference
Experiments at Brookhaven National Laboratory with randomly packed spheres have shown that orderly clusters form within beds especially near flat surfaces, and that hydraulic or mechanical shocks are capable of causing a progressive growth of such ordered structures with reduction of the average voidage. Investigations are currently under way to find out if ordered clusters occur within the the interior of beds and, if so, to what extent. One lucky observation has already shown that a tightly packed layer of 1/8-inch spheres occurred in the middle of a supposedly random bed about 9 inches long in a tube about 11,’4 inches in diameter. Clusters of course, have important effects on the flow distribution and heat transfer behavior of the coolant. Therefore, these observations showing the 44
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
existence of clusters in “random” beds suggest that caution is in order when attempts are made to interpret the data from experiments in which the fine details of the method of forming the bed or of its ultimate configuration are omitted. General Conclusions
Particle-to-fluid heat transfer coefficients in packed beds have been measured mostly for air, although a few other gases have been used. One series of tests was made with water. Data for other liquids were not found. AUTHOR J . J . Barker is a consulting engineer in JeTzcho, Long Island, A7ew York.
L
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i
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CT 0
4.-
0.1
az a E
3 8
I
RANDOM SPHERES
Figure 2.
.-
I
100
10
g4
\
Correlation of heat transfer coeficients in beds of randomly Packed spheres
0.1
= z
z8
ORDERED SPHERES
0.01 10
100
1000
10,000
REYNOLDS NUMBER
Figure 3. Correlation of heat transfer coeficients in beds of spherespacked i n ordered arrays
Correlations exist for beds of randomly oriented spheres, for beds of spheres oriented in cubic or rhombohedral arrays, for cubes, cylinders, granular materials, and commercial packings. In addition, the data for the ordered arrays were obtained using models with large ratios of wall to wall areas. Heat Transfer Coefficients
Data found in the literature are summarized in Figures 1 through 5 and Table I. Figure 1 treats all the data while the remaining figures treat data for the packing materials indicated. Figure 1 indicates that, whatever the type of packing, there is general agreement among most of the investigators. The agreement is generally within a factor of about 2 over Reynolds numbers ranging from 10 to
100,000 and especially in the most common range from 200 to 4000. I t is interesting to note that j , = 0.1 is within a factor of 5 of the probable value for 15 < R e < 105. Figure 2 shows data for beds of randomly packed spheres of uniform diameter. Note that the pioneering work of Furnas (77A) produced data lying in the center of the point spread produced by later workers, indicating that his data are essentially correct in all important respects. T h e results of Norton (766A) are considerably lower than those of other investigators and are suspect on that account. However, it should be noted that the low results are consistent with the high gas temperatures. T h e results of Dabora ( 5 3 A ) , and Lancashire (129A), which were also obtained with high gas temperatures (1300' F. to 2200' F.), also lie below the general VOL. 5 7
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TABLE I.
S U M M A R Y OF E X P E R I M E N T A L WORK ON
Ref.
Material
OF.
Atm.
12A
H Z 0
240
2.7
13A
air
140-200
1
17A
air
89-300
2 3A
air
70
Material
P7
steel
spheresb
1
Celite, kaosorb
cylinders spheresb
0.16-0.223 0.185-0 344 0.12-0 2 0.07-0.26 0 125-0.188 0.07
800-2000
0.0818
70-90
2 . 9 8 5 diam. X 0.661 .16“
2000
12 5 diam. X 132” 1 , 7 5 diam. X 37” ? d i a m . X 1” ? diam. X 2”
50A
air
90
1
air
1900 1300
68
0.73
glass alumina steel Sic
air
55A
air
77
1
65A
air
70-200
1
71A
air
100-1300
1
75A
air
ca. 70
1
77A
air
80-1 60
1
Ilb/lP
241-244
117/5
0.173, 0,165
140-200
5 75 d i a m . X 5-25”
110-310
4 diam. X 4”
l/.i, 6/32
1/s,
0.689
alumina
pebblesb
0.448
> 1300
glass
spheres
0.129 0.248
100 i 20?
Johns-Slanville T y p e VI11 Celite
spheres
0.626
54
steel & various
spheres & 9 various b
0,342
70--200
iron
spheresb
0.73, 1.25, 1.91
100-1 300
Celitealundumkaolin
spherese
0.673
assumed shallow
0,2595
1 Cu sphere in wooden
57-1 1,000 57-11,000 40-1700
8000-60,000
6, 11, 22 6 22,44
3
S constant r a t e drying
S heat o n e sphere
spheres
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0.01 10
1
1
I
100
1000 REYNOLUS NUMBER
10,000
Figure 4. Correlation of heat transfer coeficients in beds uf cubes, cylinders, and commercial packings
values obtained by others. Is the proper inference that good data are difficult to obtain at the elevated temperatures, or is if that the correlation does not account for fundamental effects which come into play at high temperatures? Figure 3 shows results for beds packed with uniform spheres in ordered arrays. All the results were obtained with steady state techniques. Baldwin (72A), Lydersen (742A), and IYadsworth (227A) used electrical resistance heating of instrumented spheres. Galloway (75A), and Sen Gupta and Thodos (197A) used the constant drying rate technique. The fact that steady state techniques have been used exclusively suggests that experiments with transient techniques might be of value and interest. Although the thermal cycling of transient methods would not be expected to rearrange an ordered bed, it might affect the geometry of a randomly packed bed and exert some influence on the local heat transfer coefficients. Most of the data on cubes, cylinders and commercial packings have been obtained with steady state techniques, but Glaser (82A) used a transient technique in which step changes in fluid temperatures are coupled with flow reversals. The latter method is useful because it permits calculation of the heat transfer coefficient from temperature measurements made at the center and ends of the bed, and with the equation h = 2WCT,/ATgr
where h = heat transfer coefficient (B.t.u./hr.ft.2 " F.) W = weight of solid in the bed (lb.) C = specific heat of solid (B.t.u./lb. " F.) A = heat transfer area of solid (fte2) 7 = period (hr.) T , = temperature change at center of bed (" F.) T , = temperature change of inlet fluid (" F.) The characteristic dimension used by Glaser is the outside diameter of the Raschig rings. Taecker and 48
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Hougen (21I A ) , who investigated Berl saddles and partition rings, in addition to Raschig rings, correlated data with the diameter of a sphere which has the same surface area as that of a piece of the actual packing. Figure 5 indicates that surprisingly few investigators have made studies of heat transfer in packed beds of granular materials. Lof and Hawley (741A) worked with granite particles in the size range 0.314-1.312 inches in randomly packed beds with void fractions of about 0.43. Solntsev (204A) worked with granular particles of basalt, silica gel, and activated carbon in the size range 0.08-0.24 inch. There is an obvious need for further studies of granular, packed beds. Most of the data have been obtained with air, although a few other gases have been used and only one liquid (water) has been employed in heat transfer experiments in packed beds. This means that the assumed variation with the Prandtl number, as indicated by the choice of j , = St.Pr''8 as the correlating parameter, is nothing more than a guess. The range of materials, sizes, and temperatures is sufficiently broad to engender confidence that any important variation introduced by these factors has not been overlooked. Many different experimental techniques have been used and show consistent results. Part of the variation shown in Figure 1 stems from differences in voidage, although the differences are not great (0.26-0.78) and their effects are not strong. Another source of differences is believed to be the large wall effect which is present in the works of a number of investigators. Even when the heat transfer or heat capacity effects of a wall are accounted for, there is still a flow effect which remains and must exert some influence on heat transfer, particularly in a random bed where variations in voidage must induce lateral flow currents which would be prevented or distorted by a wall. There is more need for additional data than for more strain on the existing data through further correlation
I I I I I I
I
I
l 1 1 1 1 1 ~
I
I
I I l.lIl1
I
I
I
0.1
0.01)
I l l l l ’
100
I
I
I I I I l l /
1000
I I Ill1
10,ooo
REYNOLDS NUMBER
Figure 5. materials
Correlation of heat transfer coeficients in beds of granular
schemes. Attention must be paid to the manner in which the bed is formed and whether or not it changes geometry during the course of experiments. More data with fluids of widely varying Prandtl numbers are also needed. Lack of uniformity in the reporting of results suggests that it would be of great value to prescribe a standard method. Such a method would, in addition to making data more accessible, prevent some of the graphmanship that is occasionally resorted to in covering deficiencies in experimentation. R E F E RENCES (1A) Acrivos A. “Method of Characterirtics Technique-A plication to Heat and Mass T r d s f e r Problems (for Flow of Fluid through a &xed Bed),” Ind. Eng. Chem. 48, 703 (1956). (2A) Adivarahan, P. “Heat Transfer in Consolidated Porous Media with Flowing Fliiids,” Dissertation Abstr. 22, 312’7 (1962). (3A) Aerov, M . E. “ T h e Hydraulic Resistance of a Fixed Granular Bed,” Intern. Chem. Eng. 3, 272 (1963). (4A) Aerov M. E. Umnik, h-,N . ”Heat and Mass Transfer in a Granular Bed,” Soviet Phv;. Tech. Phvs. . 1,. 1203, 1212 (1957) (English translation). (SA) Amundson, N. R . “Solid-Fluid Interactions in Fixed and Moving BedsFixed Beds with Small Particles,” Ind. Eng. Chem. 48, 26 (1 956) ; “Fixed Beds with Large Particles,” Ihid., p. 35. (6A) Anzelius, A. “Ubcr Erwarmung Vkrmittels Durchstromender Medien” (On heating by means of percolating media) Z . Ang. M a t h . Mech. 6 , 291 (1926). (7A) Aris. R . “Shape Factors for Irregular Particles, 11. T h e Transient ProblemHeat Transfer to a Packed Bed,” Chem. Eng. Sci. 7, 8 (1957). I . “The Steady State Problem”-Diffusion and Reaction, Ibid., 6 , 262 (1957), (8A) Ark, R . , Arniindso:: K , R . , “Some Remarks on Lon itiidinal Mixing or Diffusion in Fixed Beds, Am. Inst. Chem. Eng. .I. 3, 280 (19575. (9A) Arthur, J. R., Linnett, J. W., “Interchange of Heat Between a Gas Stream and Solid Granules,” J . Chem. Soc. (London) 1947, 416. (10A) Ausman, J. M. “Heat and Mass Transfer in a Porous Catalyst Pellet During Regeneration,” Disiertation Abstr. 22, 3956 (1962). (1lA) Awberry, J. H., “Heat Flow Through Granulated Material,” Phil. Mag. 12, 1152 (1931). (12A) Baldwin D . E,, Jr., “Heat Transfer in Beds of Oriented Spheres,” D . Eng. Sci. ?Thesis, Carnegie Inst. Tech. 1961. (13A) Ball, W. E., “Heat Transfer Properties o f a Packed Bed: Determination by a Frequency Response Technique,” Dissertation Abstr. 19, 494 (1958). (14A) Bar-Ilan, M., Resnick W. “Gas Phase Mass Transfer in Fixed Beds a t Low Reynolds Numbers,” fnd, Ekg. Chem. 49, 313 (1957). (15A) Baron, T., “Generalized Graphical Method for the Design of Fixed Bed Catalytic Reactors,” Chem. Eng. Prog. 48, 118 (1952). (16A) Baumeister, E. B., “Fluid Particle Heat Transfer in Packed Beds,” Dissertation A b ~ 1 7 .17, 1281 (1957). (17A) Baumeister, E. B., Bennett, C . O., “Fluid-Particle Heat Transfer in Packed Beds,” A m . Ins/. Chem. Eng. J . 4, 69 (1958). (18A) Benennti R . F., Brosilow,, C. B., “Void Fraction Distribution in Beds of Spheres,” Ibii., 8, 359 (62). (19A) Boegli, J. S., Deissler, R. G., “Measured Effective Thermal Conductivity of Uranium Oxide Powder in Various Gases and Gas Mixtures,” NACA-RM ES4LlO. (20A) Bcrisevich, V. A , , “Investigation of Heat Exchange in the Movement of a Disperse Medium in Pipes,” AID Rep. T-63-63, Transl., 1963. (21A) Bowers, T. G., Reintjes, H . , “A Review of Fluid to Particle Heat Transfer in Packed and Moving Beds,” Chem. Eng. Prog. .Symp. Ser. 57, 69 (1961). (22A) Bradshaw. R . D . , Bennett, C. O., “Fluid-Particle Mass Transfer in a Packed Bed,” Am. Inst. Chem. Eng. J . 7, 48 (1961). (23A) Bradshaw, R. D . , Meyers, J, E. “Heat and Mass Transfer in Fixed and Fluidized Beds of Large Particles,” Ibrh., 9, 590 (1963). (24A) Brailsford, A. D., Major, K . G . , “The Thermal Conductivity of Aggregates of Several Phases, Including Porous Materials,” Brit. J . AppI. Phys. 15, 313 (1964). (25A) Bretsznajder, S., Lesniewicz, L., Jaszczak-Skorupska, M.,“Hydraulic
Resistances and Heat Transfer for a Vibrating Layer in the Solid-Gas Sysrem,” Bull. Acad. Polon. Scf., Ser. Sci.,Chim., Gcd. Gaogruph. 7, 573 (1959) (in English). (26A) Bretsznajder, S., Ziotkowski, D . , “Effective Thermal Conductivity of Granular Catalytic Beds.-I. Dependence of Specific rhermal Conductivity of Granular Beds on the Manner of Packing,” Ibid., 579 (1959). (27A) Brinn M. S . “Heat Transfer to Granular Material ” AEC-116 Univ. of Delaware,’and E.’I. du Pont & Co., Heat Transfer Lecturies; Vol. 1, December 1947. (28A) Brotz, W., “Untersuchungen uber Warmeleitung, Stofftransport und Druckabfall in durchstromten Schuttgutern (Investigation of Thermal Conductivity Mass Trans ort and Pressure Loss in Flow through Packed Beds),” Chem. Ins: Tech. 23,40k)(1951). (29A) Bunn J. M. “Two-Dimensional Flow Through Porous Media,” Dissertotion Abslr. 21, i 8 8 l (i961). (30A) Bunnell. D . G., Irvin, H . B., Olson, R. W., Smith, J. bl., “Effective Thermal Conductivities in Gas-Solid Systems,” Ind. Eng. Chem. 41, 1977 (1949). (31A) Businger, J. A , , “Warmtetransportproblemen bij de luchtbehandeling von ranulaire Materialen in estorte toestand (Heat Transfer in Aerated Packed 8eds of Granular Mateiial8,” Irigenicur 6 8 (29), 87, July 20, 1956. (32A) Campbell, J. M., Huntington, R . L. “Heat Transfer and Pressure Drop in Fixed Beds of Spherical and Cylindricai Solids-Part I . Pressure Drop and Packed Bed Characteristics,” Petrol. Refiner 30 (12), 127 (1951). “Part 11, Heat Transfer and Temperature Gradients,” I b d , 31 (2) 123 (1952). (33A) Carberry, J. J. “ A Boundary-Layer Model of Fluid-Particle Mass Transfer inFixed Beds,’’ Am,>nst. Chcm. Eng. J . 6 (3) 460 (1960). (34A) Carhex;t J. J..) Wendel, M., “A Computer Model of the Fixed Bed Catalytic e Adiabatic and Quasi-adiabatic Cases,” Ibid., 9 , 129 (1963). Reactor: (35A) Carman, D. C., “Flow of Gases Through Porous Media,” Butterworths, London, 1956. (36A) Chen J. C. Churchill S. W., “Radiant Heat Transfer in Packed Beds,” Am. h i . dhem. Ekg. J . 9, 35’(1963). (37A) Chen, J. C . , “Radiant Heat Tiansfer in Packed Media,” Ph.D. Thesis, Univ. of Michigan, 1962. (38A) Chennakesavan, B., “Heat Transfer to Liquid Streams in a Packed Tube Containing Large Packings,” Am. Inst. Chem. Eng. J . 6 , 246 (1960). (39A) Chervyakov, S. S., “Experimental Study of the Effect of Sphere Vibration on Heat and Mass rransfer in a Turbulent Air Flow,” J . Engr. Phys. 6 (6), 31 (1963). Publ. in Russian with English abstracts. (40A) Chu, J. C . , Kalil, J., Wetteroth, W. A,, “.Mass Transfer in a Fluidized Bed,” Chcm. Eng. Prog. 49, 141 (1953). (41A) Chu, P. L., Wang, H . S., “Heat Transfer Through Packed Beds-Temperature Distribution,” K’o Hsueh T’ung Pao, 178, 1957. (42A) Chu Y . C. Storrow J. A. “Heat Transfer to Air Flowing Through Packed Tubes,” khem. kn,?.Sci. 5: 230 (1952). (43A) Chuchanov, Z. F., “Heat and Mass Transfer Between Gas and Granular Material,” Intern. J . Heat Mass Trans. 6, 691 (1963). (44A) Ciborowski J. Roszak, J., “Heat Transfer in Fluidized Systems-111. Discussion of Resul;s ahd Conclusions Concerning the Mechanism of Heat Transfer,” Chcm. Stosowana 3, 15 (1959). English Summary. (45A) Clark, W., “Development of a Theoretical Procedure for Prediction of Static Forces in a Stationary Bed of Particulate Solids with Experimental Verification of the Resultant Equations,” Dissertation Abstr. 24, 1964. (46A) Coberly, C. A,, Marshall, Jr., W. R., “Temperature Gradients in Granular Beds,” Chem. Eng. Prog. 47, 141 (1951). (47A) Colburn, A. P., “Heat Transfer and Pressure Drop in Empty, Baffled, and Packed Tubes,” Ind. Eng. Chem. 23,910 (1931). (48A) C,olburn, A. P., Chilton, T H., King, W. J . , ”Heat Transfer and Pressure Drop in Empty, Baffled and Packed Tubes,” frans. A m . Ins/. Chem. Eng. 26, 166 (1931). (49A) Converse, A . O., “The Effect of Velocity Profile on Axial Dispersion in Packed Beds,” Am. Insl. Chem. Eng. J . 6 , 344 (1960). (504, Co page, J. E., London, A. L., “Heat ‘Transfer and Flow Friction Characteristics ofsorous Media,” Chem. Eng. Prog. 5 2 , 57 (1956). (51A) Creutz, E., “Laminar, Turbulent, and Transition Gas Flow in Porous Media,” N u t [ . Sci. Eng. 20, 28 (1964). ahora E. K. “Regenerative Heat Exchange with Heat Loss Consider(5%?nfl’ A F d S R TN’57-613, A S T I A No, 136 603, August, 1957. (53A) Dahora, E. K., Moyle, M. P., Phillips, R., Niciiolls, J. A , , Jackson P. L. “Description and Experimental Resdts of Two Re enerative Heat Exchingers,’: AFOSR Contract No. AF18(600)1199, U. Mich. T% 58-226, Feb. 1958. ( 5 4 A ) Dayton, R . \V., Fawcett, S. L., Grimble, R . E,$ Sealander C. E. “Improved Measurement: of Surfdce Heat Tranrfer by the Method of C’yclic +emper,itiire Variations,” Rep. BMI-747, Battelle Mernori.ii Institrite, Columbus, O., 1952. (55A) DeAcetis, J., Thodos, G., “Mass and Heat Trmsfer in Flow of Gases Through Spherical Pnckings,” Ind. E n f . Cham. 5 2 , 1003 (1960). (56A) Deisler, P. F., Jr., Wilhelm, R. H . , “Diifusion in Beds of Porous Solids,” Ind. Eng. Chem. 45, 1219 (1953). (57A) Denton W. H., “ T h e Heat Transfrr and Flow Resistmce for Fluid Flow Through Randomly Packed Spheres,” Proc. Inst. hlech. Engr. (London) 370 (1951). (58A) Denton, W. H., Robinson, C. H., Tibbs, R . S., “ T h e Heat Transfer and Pressure Loss in Fluid Flow through Randomly Packed Spheres. I,” AERE. HPC-35. (59A) Diepschlag, E., “Resirtances to Flow of Gases Through Beds of Granular Material,” Feuerungrtech. 23, 133 (1935). (GOA) Downing, D. G., Yeh, G . C., “Pressure Drop Characteristics in Beds of Compressible Solids,” Paper resented at the 54th N.itl. Meet., AIChE, Las Vegaa, Nev., Sept. 20-23, 1!64. (6lA) Dunskii, V. D., “ O n the Mec11;rnismof €lent Transfer Between a Surface and an A itated Bed of Dispersed hkiterinl under Vacuum,” Inlern. Chem. Engr. 4, 405 (1968. (62A) Dyankonov, G. K . , Semenov, G. A., Izuest. Aknd. #auk USSR Otdel. Tekh. Nauk No. 7, 109 (1955). (63A) Eckert, E. R . G., S arrow, E. M., Bele, MI. E. I., Goldsrein, R. J., “Heat Transfer Bibliography,” %tern. J . Heal .\.lass Trans. 6 , 761 (1 963). (64A) Elmer, N., “W;ierinediirclig.in~ durch r r o e s e Koerper bei gleichzeitigem Stoffdurchsatz” (Heat Transmission tiiroiigli oroiis M.iteri.ils with Simultaneous Passage of Substance Through Capillnry Nrtwork In Direct Flow and Counter Flow), W m . Z. 5 , 125 (1955). (65A) Elukhin, N. K . , Starosvitskii, S. I., “Heat Exchange and Hydraulic Resistance in Dumped Packings of Regenerators,” Inlern Chem. E n g . 4, 114 (1964). (6GA) Epstein, N., “Correction Factor for Axial Flow in Packed Beds,” Can. J . Chem. Engr. 36, 210 (1958). (67A) Ernst R . “Moving Bed Heat Transfer in Heat Exchangers,” Chem. In!. Tech. 32, i J (1’960).
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(68A) Fahien, R . W., Smith, J. Vi., “Mass Transfer-Packed Beds,” Am. Init. Chem. Ene. ~. J . 1, 28 (1955). (69A) Froment, G . , “Mass and Heat Transfer in Packed Beds,” Ind. Chtm. Beige. 24, 619 (1959). (70A) Fulk, M . M . , Devereux, R. J., Schrodt, J. E.; “Heat Transport Through Powders,” Proc. Cryog, Eng. Coni. 1956, 163 (1957). (71A), F u r y , C. C.: “Hear Transfer from a Gas Stream to a Bed of Broken Solids, I , Ind. Eng. Chem. 22, 26 (1930). (72A) Furnas, C . C., “Heat Transfer from a Gas Stream to a Bed of Broken Solids11,“Ibid.. 721 (1930). (73A) Furnas, C. C., “Heat Transfer from a Gas Stream to a Bed ofBroken Solids,” U.S. Bur. Mines Bull. S o . 361, 1932. (74A) P b o r J. D. Mecham, 1%‘. J. ”Radial Gas Mixing in Fluidized-Packed Beds, Ind.’Eng. C k . Fundamenlals 3: 60 (1964). (75A) Galloway L. R . , Lomamicky, W., Epstein, N., “Effect of Packing Configuration on ’Mass and Heat Transfer in Beds of Stacked Spheres,” Can. J . Chem. Enp. 35, 139 (1957). (76A) Gamson, B. W., “Heat and Mass Transfer-Fluid Solid Systems,” Chem. Eng. Prog. 47, 19 (1951). (77A) Gamson, B. W., Thodos, G., Hougen, 0. A,, “Heat, Mass and Momentum Transfer in the Flow of Gases Through Granular Solids,” Trans. A m . Inst. Chem. Eng. 39, 1 (1943). (78A) Gei’perin, K, I., Kvasha, V. B., “Optimum Temperature Conditions in Chemical Reactors,” Khim. Prom 1961, 51. See also “Mass Transfer in the Rectificaticn Cooling of Chemical Reactors,” Ibid. 1960, 406. (79A) Giona, A . R., “Heat Transfer in Catalytic Granular Beds,” Ingenere ( M z l a n ) 36. 111 (1962). . . (80A) Giona, A . R., Passino, R . , “Heat Transfer Coefficients and Temperature Gradients in Bedsof Granular Solids,” Colore 31, 315 (1960). (81A) Glaser, H . , “Heat Transfer and Pressure Drop in Heat Exchangers with Laminar Flow,” MAP-VG, 96-818T, March 1, 1947. (8 2A) Glaser, H., “Instationaere Messung des Warmeuebertragung von Raschigringschuettungen. (Nonstationary measurement of heat transfer in Razchig ring packing),” Chem.-Ing. Tech. 27, 637 (1955). (83.4) Glaser, H., “Warmeubergang an Kugelschiittungen (Heat Transfer in Packed Spheres),” Chem. Ing, Tech. 34, 468 (1962). (84A) Glaser, M. B., “Simultaneous Heat and Momentum Transfer in the Flow of Gases through Packed Beds,” Disserlation Abstr. 17, 2538 (1957). (85A) Glaser, M , B., Thodos, G. “Heat and Momentum Transfer in the Flow of Gases Through Packed Beds,” A m . Inst. Chem. Eng. J . 4, 63 (1958). (86A) Glueckauf, E., Watts, R . E., “Hear Transfer through Charcoal Beds,” AERE c/m 337, 3 p. (1958). (87A) Goldstein, S., “The Mathematics of Exchange Procesbes in Fixed Columns: I. Mathematical Solutions and Asymptotic Expansions,” A219, 151 (1953). (88A) Goldstein, S., Murray, J. D . , ”Mathematics of Exchange Processes in Fixed Columns: 111. Solution for General Entry Conditions, and a Method of Obtaining Asym totic Expressions ” Proc. Roy. Soc. (London) A252, 334 (1959). IV. L i m i t i n k d l u e s , and Correciion Terms for the Kinetic Theory ; Solution with General ntrk Conditions,” Ibid. p. 348. V. Equilibrium-Theory and Perturbation Sclutioks, and Their Connection with the Kinetic-Theory j Solutions for General Entry Conditions,” Ibid. p. 360. (89A) Golub? S. “Graphic Method for Determining the Condition? of Burning Limestone, Siroil: Materzaly 4, 12 (1958); Ibid., p. 11. (90A) Gopalarathnam C. D . Chennakesavan B. Laddha, G. S., “Heat Transfer in Packed Beds,” J . kct. Ind.’Res. (India) A21,’lSi (1962). (91A) Gopalarathnam, C. D., Hoelscher H . E. Laddha G . S. ”Effective Thermal Conductivity in Packed Beds,’’ Am. Z d t . Che;. Eng. J: 7, 24b (1961). (92A) Gottschlich, C. F., “Axial Dispersion in a Packed Bed,” Ibid.,9, 88 (1963). (93A) Green, L., Jr., “Gas Cooling o f a Porous Heat Source,” J . ofAjp1. M a c h . 19, 173 (1952). (94A) Greenkorn, R. A,, “Flow Models and Scaling Laws for Flow through Porous Media,” Znd. Ens. Chem. 56 (3), 32 (1964). 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50
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