Since the parameters used in the corresponding-states theory are concentration independent, the correspondingstates theory represents a significant improvement over the Flory-Huggins theory in data modelling for many binary, polymer-solvent systems.
Acknowledgment The authors thank Amoco Chemicals Corporation for permission to publish these results, and R. W. McCoy for his assistance in designing the GC apparatus. The authors also thank the Texas Tech University Computer Center for the use of their facilities. Literature Cited Bondi. A , , "Physical Properties of Molecular Crystals, Liquids, and Glasses," pp 214-260, Wiley, New York, N. y . , 1968. Bonner, D. C.. Maloney, D. P., Prausnitz, J. M . , lnd. Eng. Chem.. Process Des. Develop.. 13, 91 (1974).
Bonner, D. C . , Prausnitz. J. M., Amer. lnst. Chem. Eng. J., 19, 943 (1973). Brockrneier, N. F . . Carlson. R. E., McCoy, R . W., AlChE J.. 19, 1133 (1973). Brockrneier. N. F., McCoy, R. W.. Meyer. J. A . . Macromolecules. 5, 464 (1972). Conder. J. R., Purnell. J. H., Trans. Faraday Soc.. 65,824 (1969). Flory. P. J . , J. Chem. Phys., 9, 660 (1941). Flory, P. J . . J. Amer. Chem. SOC.,87, 1833 (1965). Flory, P. J.. Discuss. faraday Soc., 49, 7 (1970) Heil, J. F.. Prausnitz, J. M., Amer. lnst. Chem. Eng. J.. 678 (1966) Hijrnans, J., Hollernan, T., Advan. Chem. Phys.. 16, 223 (1969) Huggins, M. L.. J. Chem. Phys., 9, 440 (1941). Newman, R. D., Prausnitz, J. M., J. Phys. Chem., 76, 1492 (1972). Newman, R. D., Prausnitz, J. M., AlChEJ., 19, 704 (1973). Patterson, D.. Macromolecules, 2,672 (1969). Patterson, D., Tewari, Y . B., Schreiber, H. P., Guillet, J. E.. Macromolecules, 4, 356 (1971). Prigogine, I . , "The Molecular Theory of Solutions " North-Holland, Amsterdam, 1957. Simha. R., Hadden, S. T.. J. Chem. Phys.. 25, 702 (1956). Smidsrfid. O., Guillet, J. E.. Macromolecules. 2,272 (1969)
Received for reoiew M a r c h 13, 1974 A c c e p t e d June 5, 1974
Heat Transfer in Fixed Beds Arcot R . Balakrishnan and David C. T. Pei* Department of Chemical Engineering. University of Waterloo. Waterloo. Ontario. Canada
The use of microwave power in the heating of solids in a packed bed results in a uniform temperature throughout the bed, almost instantaneously. Thus, heat transfer through the particle-particle mode between the pellets was eliminated and fluid-particle heat transfer coefficients alone were determined experimentally. Eight commercial catalysts with different physical, thermal, and transport properties were used as bed materials. Ar/Re,,', defined as the ratio of gravity force to the inertia force and the shape factor were found to be the important parameters for heat transfer in fixed bed systems. A model for the total heat transfer in a fixed bed is presented and comparison of results with the data reported in the literature showed good agreement.
Introduction Considering the many applications of fixed bed gassolid systems, it is not surprising to find a considerable amount of work reported in the literature and summarized in a number of textbooks (Davidson and Harrison, 1971; Zabrodsky, 1966). Therefore, only pertinent references and reviews are given here. Barker (1965) presented a n extensive review of such literature and a more recent review has been presented by Bhattacharyya (1973). In the reviews, it was pointed out that the application of many of these empirical and semiempirical correlations are severely limited as they cannot be safely extrapolated beyond the range of existing data and, secondly, discrepancies exist between the results of many investigations. The cause of the discrepancies among the empirical relations was generally attributed to the definitions of parameters and the experimental techniques used by different researchers. Since heat is transferred by several distinct mechanisms, many models have been presented in the literature which consider each mode or mechanism separately. The assumption here is that the contribution of each mode is independent of each other and they are additive. Yagi and Kunii (1957) proposed a model which predicts the effective thermal conductivity of the bed when there
is no flow of fluid through it. Wakao, et al. (1969), studied radiation heat transfer in packed beds and Wakao and Kat0 (1969) studied the effective conductivity of packed beds. They found that the area of contact has no effect on the effective thermal conductivity of the bed. Yovanovich (1973) studied conductivity of glass microspheres from atmospheric pressure to vacuum conditions. He used the Hertz theory for elastic contact to obtain the actual contact area in the model. Kunii and Smith (1960) used a model in which they assumed that only point contact exists between the spheres. Chan and Tien (1973) also used elastic contacts to obtain the area of thermal transfer. It is now generally accepted that a t very low pressures, the use of a model that takes into account the contact area is mandatory. On the other hand, Galloway and Sage (1970) used existing data and their own data to propose a generalized model for the mechanism of transport in packed, distended and fluidized beds. Bhattacharyya and Pei (1974), working with 0.126-in. diameter Fez03 spheres and 0.2-in. diameter Fen03 cylinders with L = D, i e . , a shape factor of 1, showed experimentally that the total heat transfer coefficient in a fixed bed consists of three parts, namely the effective conductivity of the bed, the effect of the presence of fluid on the conductivity, and finally the conInd. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
441
tribution of the fluid-particle mode Nutotal =
kf
+
Nufc
+ Nufp
(1)
where the effective conductivity of the bed is given by the classical Yagi and Kunii (1957) model
TUNING SCREWS
Bhattacharyya and Pei (1974) evaluated the effect of the presence of fluid on the conductivity and proposed the correlation
Finally, the contribution of the fluid-particle mode is represented by
Figure 1. Experimental setup: 1, dummy load; 2, waveguide; 3, rubber connector; 4, 2-in. copper tube; 5, bidirectional coupler; 6, to vacuum; 7, circulator; 8, water load; 9, air filter and dryer; .O, rotameter; 11, vacuum jacket; 12, test section; 13, compressed air supply; 14, safety valve.
0.25
RepPrfl/3 - ( j k ) f p = 0.018
(4 )
THERMOCOUPLE MICROMETER
The use of microwave power to heat a bed of metallic oxides has been shown to result in the bed attaining a nearly uniform temperature almost instantaneously by Ford and Pei (1968) and Bhattacharyya and Pei (1973). This eliminates thermal gradients within the bed and hence particle-particle heat transfer. The purpose of this investigation is to examine the effect of thermal diffusivity, shape, size, etc. on N u f p using the microwave heating technique and hence the need, if any, to extend the correlation given by eq 4, which applies to particles with a shape factor of unity only, to commercially available catalysts. Moreover, to validate the model, the total heat transfer coefficients were calculated for the materials used by Baumeister and Bennett (1958), Dayton, et al. (1952), Gamson, et al. (1943), and Lindauer (1967), and compared with the experimental data which were reported in the literature. Unfortunately, it was not possible to include the results of many other investigations owing to lack of data on the bed conditions a s also the physical and thermal properties of the materials used.
Experimental Apparatus The equipment used is shown in Figure 1. It consists of a microwave generator, a system of waveguides with the test section placed vertically through a section of the waveguide. A circulator was connected next to the generator to separate the forward and reflected beams and divert the reflected beam to a water load for total absorption. Bidirectional couplers were used to measure the forward and reflected powers. A set of tuning screws of the waveguide network just before the load was used for matching the standing waves for minimum power reflection-from the test load and the dummy load which was fixed at the end of the waveguide network to absorb the power not being absorbed by the load. Compressed air supply a t 45 psig was used as the source of the pneumatic system. This was passed through a stabilizing tank and an air filter-dryer. Air entered the 2-in. diameter test section through a calming section of 15 f t of 2-in. diameter copper tube. The test section as shown in Figure 2, consisted of a vacuum-jacketed Pyrex glass tubing, as Pyrex is nearly transparent to microwave power. The test section was partially filled with catalyst pellets to form the bed. Some of the pellets were tagged to thermocouples and posi442
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4, 1974
-
WIRES
ASSEMBLY I B R 4 S S I
2" I D COPPER T U B E R O O ISSI
-EXTENSION
- G L 4 S S ROD I l / B
014 I
SUPPORTING FLANGE lBR4SS1 -COOLING
2
COIL 1 B R 4 S S l
I D P Y R E X TUBE
-THERMOCOUPLE
TIP
WAVE GUIDE V4CUUM J A C K E T I 21; I O !1 ' 6 THICK S C R E E N TO SUPPORT T H E BED
IS T A I N L E S S S T E E L ) -CONNECTION
TO VACUUM PUMP
SUPPORT FOR SCREEN 1 5 S -SEALING RING
I
-TIGHTENING CAP 14LUMlNUM) -RUBBER RING SUPPORTING R O D I B R b S S 1 -CONNECTING FLANGE
f
C O L D 4117
Figure 2. Test section details.
tioned in different parts of the bed to monitor the bed temperature and also to ensure that the temperature was uniform throughout the bed. Thermocouples (copper-constantan) were also used to measure the inlet and outlet temperatures of the fluid stream. Fluid-particle heat transfer coefficients were determined by a steady-state (energy balance) method. The heat transfer coefficient is defined a s
Steady-state measurements of inlet gas, outlet gas, and bed temperatures were carried out for different flow rates. The maximum flow rates used were well below the point of incipient fluidization. Eight commercial catalysts were used as bed materials and these are listed in Table I, along with their physical and thermal properties. The data obtained are listed in Table I1 and shown in Figure 3 as a plot of (ih)fnagainst Re,,m. Discussion The present data represent the fluid-particle mode alone and, as shown in Figure 4, they can be correlated in
Table I. Thermophysical Properties of the Catalysts Catalyst
Shape
Vanadium pentoxide (Type 1) Vanadium pel toxide (Type 2) Nicke 1- molybdenum oxide Cobalt-molybdenum oxide I r o n oxide Nickel oxide Vanadium pentoxide Nickel oxide
Cylinders L/D = 1 Cylinders L / D = 1.5 Cylinders L/D = 2 Cylinders L/D = 2 Sphere Sphere Sphere Sphere
a
Size, in.
Btu/(lb)("F)
ft2/hr
7/32
0.415
0.0075
0.2614
84
1.8
7/32
0.263
0.0206
0.3027
54.31
1.9
0.325
0.0065
0.1077
51.8
1.75
'/8
0.297
0.0083
0.128
51.8
1.85
74
0.208 0.39 0.42 0.184
0.014 0.0058 0.0034 0.0123
0.3525 0.1569 0.12 0.02865
121.06 52.76 82.98 129.76
1.9 1.8 1.8 1.9
c,
'/4 1 " 6
%
ff,
ks
p,.
1
Btu/(hr)(ft)("F)
lb/ft3
Bed height, in. a
Bed diameter. 2 in. 01 IRON W I D E
5 05
SPHERES
[
SPHERES
NICKEL OXIDE SPHERES NICKEL OXIDE SPHERES
y2 o5
CYLINOERS~YD= I
5 O5
CYLINDERS IL/o 1 5 I
Co-Mo
CYLINDERSII/D'
MI-MO
C Y L I N D E R S I ~=~2
2
I I I
0.01
1 L
505
CYLINDERS
i y ~ =G5= l 7 i
V, 05
CYLINDERS /'/D= 15 &=08% NI-Mo CYLINDERS l L / 0 = 2 0 ~ ~ 0 8 3 1 Co-Mo CYLINDERSfL/D'2 0s.083i V,O5 SPHERES NICKEL OXIDE SPHERES NICKEL OXIDE SPHERES IRON WIDE SPHERES
Figure 3. Fluid-particle heat transfer data.
Figure 4. Fluid-particle heat transfer correlation.
the form ( j h ) f p = 0. 018[Ar/Re,,2]0~25@,3~76
(6 )
Equation 6, which is an extention of eq 4, includes the shape factor &. The Archimedes number, Ar, is defined as Dp3gpf(pp p f ) / p f 2 and Re,,,,, is D l , u f p f / p f ( l e ) . Therefore
where Fr is the Froude number and Frl,m is a modified Froude number for packed beds. Therefore eq 6 becomes
Based on the above correlation, i t is noted that for fluid-particle systems, the viscous, inertial as well as the gravity force plays an important role in heat transfer. The gravity force represents the compactness of the pellets; Le., the contact is better if the particles are heavier, and
is proportional to the density difference between the solid and fluid. The inertial force is responsible for the motion of the fluid. Therefore, the ratio of gravity-to-inertia force 1/Fr,m2 is an important parameter in correlating with the jh factor which includes the viscous force term. Having established the correlation for h f,,, the total heat transfer coefficient for a packed bed of the eight commercially available catalysts is estimated for our eq 1, 2, 3, and 6 and is shown in Figure 5 . It can be seen from Figure 5 that a general correlation relating Nutotal with Re, for different materials with a wide range of physical and transport properties is not possible. This has been generally observed in the literature where the correlation for the total heat transfer coefficients of one author working with one material is often a t variance with that of another author working with some other material. The total Nusselt number for a particular flow rate (Reiim = 300) for the materials used by Baumeister and Bennett (19581, Dayton, et al. (1952), Gamson, et al. (1943), and Lindauer (1967) was also calculated and tabulated in Table I11 and is shown in Figure 6. The computed values compared favorably with those obtained experimentally and reported by the original authors. This conInd. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
443
Table 11. Experimental Data
Material Vanadium pent oxide i/32 -in. cylinders L/D= 1
Vanadium pent oxide 7/32 -in. cy licde rs L / D = 1.5
Ni cke 1molybdenum oxide V8-in. cylinders L/D = 2
Cobaltmolybdenum oxide -in . cylinders L/D = 2
Iron oxide spheres -in. diameter
Vanadium pent oxide spheres 3/16 -in. diameter
Nickel oxide spheres y4-in. d i a m e t e r
444
Gas velocity, ft/sec
( jlz 1f p
Rem
7.8 6.8 6.5 6.3 6.0 5.4 5.2 4.6 10.8 10.2 9.4 8.4 8.0 6.9 6.3 6.0 6.8
0.0315 0.033 0.03 1 0.0325 0.036 0.0355 0.039 0.038 0.0245 0.024 0.023 0.026 0.025 0.028 0.030 0.0325 0.017
124 1 1077 1019 990 946 847 8 14 711 2324 2188 2024 1805 1722 1485 1347 1289 719
18,301,320 18,161,130 18,076,120 18,056,231 18,022,520 17,885,560 17,817,640 17,841,700 59,328,430 59,033,040 59,399,920 59,202,860 59,368,040 59,328,430 58,739,980 59,210,210 9,474,329
12.0 14.5 14.7 17.5 19.2 24.7 28.6 35.5 11.5 13.5 15.5 17.7 21.0 26.2 31.0 34.0 17.0
6.6 6.1
0.0171 0.0192
7 03 644
9,587,310 9,437,125
20.0 22.5
4.9 4.6 4.3 4.2 3.9 7.2
0.0195 0.022 0.02 0.0215 0.027 0.017
510 475 450 427 396 749
9,240,104 9,112,267 9.008,732 8.884, 932 8,914,872 9,027,942
39.0 40.5 45.0 51.0 57.0 16.0
6.1 5.2
0.0175 0.019
6.3 5.7
8,781,472 8,575,935
23.5 31.5
4.8 4.4 3.8 3.6 3.5 11.6 11.1 10.1 9.8 8.4 7.8 7.0 6.3 5.4 4.8 4.6 4.11 4.1 4.0 3.5 3.2 5.6 5.4 5.0 4.8 4.2 6.5
0.0195 0.0121 0.0122 0.0123 0.0122 0.027 0.028 0.029 0.0305 0.0295 0.0305 0.034 0.0373 0.028 0.0346 0.0374 0.0393 0.04 1 0.042 0.0413 0.044 0.042 0.044 0.04 1 0.05 0.047 0.038
470 44 1 375 357 349 1620 1549 1409 1363 1167 1083 968 87 1 586 517 493 474 435 429 371 341 750 720 650 620 535 855
8,480,784 8,704,598 8,665,448 8,626,543 8,592,32 1 15,336,970 15,314,850 i5,314,850 15,241,490 15,217,190 15,188,070 15,096,470 15,200,000 4 , 356,502 4,312,211 4,278,620 4 , 3 12,211 4,380,641 4,278,620 4,196,387 4,245,430 6,511,656 6,502,341 6,498,721 6,484,218 6,463,224 6,521,819
39.0 44.0 58.0 65.0 67.10 58.0 6.4 7.6 8.4 11.5 13.0 16.2 19.8 12.6 16.0 17.5 19.1 23.0 23.2 30.4 36.3 30.0 33.0 37.0 40.5 54.0 22.0
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
Ar
Ar/Re,'
or l/Frp
Table 11. (Continued)
Material Nickel oxide s p h e r e s -in. d i a m e t e r
'4
Nickel oxide spheres '/2 -in. d i a m e t e r
Gas velocity, ft/sec
(jlz)f,,
Rem
Ar
Ar/Rem2 or l/Fr,
6.2 5.8 14.4 13.0 12.9 11.8 11.0 9.6 8.2 7.2
0.039 0.04 0.027 0.028 0.029 0.03 1 0.029 0.034 0.041 0.039
815 776 4416 3987 3956 3619 3366 2937 2433 2137
6,520,03 1 6,519,218 120,236,900 120,163,607 120,103,924 120,091,621 119,807,600 119,800,724 114,109,200 114,001,002
24.2 27.8 6.1 7.5 7.6 9.2 10.5 13.5 19.2 24.9
Table 111. Comparison of Nutotal (Present Model) and Nutotal (Literature) Values
Reference
Material
Diameter, in.
e
E
0.125
0.52
0.35
24.8
0.156
0.52
0.35
0.25
0.52
0.129
Baumeister and Steel Bennett (1958) spheres Steel spheres Steel spheres Dayton, et a l . Glass (1952) spheres Gamson, et a l . Celite (1943) spheres Lindauer Tungsten (1967) spheres
Nu,, k e o / k , (Re, = 300)
Nufc
By present model
From literature
11.86
20.02
56.68
52.1
24.9
14.01
20.02
58.93
52.1
0.35
25.3
19.95
20.02
65.27
52.1
0.89
0.4
10.9
8.69
10.2
29.8
34.1
0.09
0.95
0.4
5.4
5.19
10.6
21.2
24.8
0.019
0.39
0.354
24.8
3.58
8.4
36.7
39.8
100
70
-
.
50
-
1 "205 111 2 IRONOXIDE 3 4
5 6 7
8
iJ/32 IN IN NICKEL OXIDE Y2 IN b o 5 (21 7/32 IN NICKEL OXIDE y4 IN 716 IN "2O5 Co-Mo '/B IN NI-MO '/B IN
a
CYLINDERS SPHERES SPHERES CYLINDERS SPHERES SPHERES CYLINDERS CYLINDERS
c BAUMEISTER AM5 BENNETT
0 DAYTON ET AL
.
0 GAMSON ET AL A LINDAUER
e 9
4
d
E
P
z'
P
30 -
2
20 -
0
4 10
30
50
NuToTAL ( F R O M LITERATURE)
Figure 6. Comparison of Nutotal values in the literature with values calculated by present model.
100
Figure 5. Plot of Nutotal us. Re, of catalyst samples used in present invistigation.
firms that i t is indeed possible to predict the total heat transfer rates in a packed bed if the contributions of the three distinct modes of heat transfer are evaluated independently.
Conclusions Based on the analytical and experimental results on fixed bed heat transfer, the following conclusions can be drawn., 1. The fluid-to-particle mode of heat transfer in a fixed bed was truly determined using microwave heating. 2. The data agreed with those reported in the literature Ind. Eng. Chern., Process Des. Develop., Vol. 13,No. 4,1974
445
by Bhattacharyya and Pei (1974) if the shape factor is taken into the correlation. 3. Analysis of the physical parameters of a fluid-solid system indicates that Ar/Re,,2 or the modified Froude number is the important factor for fluid-particle heat transfer in a fixed bed. 4. It is also seen that a general overall correlation relating Nutotal with Re, for different bed materials with a wide range of physical and transport properties is not possible. However, total heat transfer rates in a packed bed can indeed be predicted if the contributions of the three distinct modes of heat transfer are evaluated separately. Nomenclature Ap,e = total surface area of all the particles in the bed, '
ft2
At = cross-sectional area of bed, ft2 Ar = Archimedes number = D D 3 p p f ( ~-o pf)/ccf2, dimensionless cpf = specific heat of fluid, Btu/(lb) ( O F ) cps = specific heat of solids, Btu/(lb) (OF) D, = diameter of pellets, ft e = emissivity, dimensionless Fr = Froude number, uf/dD,g, dimensionless Frpm = Froude number for packed beds, uf(pf/D,g(p, p f ) ( l - t ) l - z , dimensionless g = acceleration due to gravity, ft/(hr2) Gf = fluid mass velocity, lb/(ftZ) (hr) hf, = conductive heat transfer due to fluid, Btu/ (ft2)(hr)("F) hf, = fluid-particle heat transfer coefficient, Btu/ (ftZ)(hr)("F) h,f = radiative heat transfer between fluid spaces {0.1952/[1 + [t/2(1 - t)][1 - e/e]]]itf + 273/100}3, Btu/ (hr)(ft2)( O F ) hr, = radiative heat transfer between packings, 0.1952 [e/2 - e] [tu 273/10013, Btu/(ft2)("F) (jh)f,= Colburn heat transfer factor = (hf,/ct,fGf)Prf23, dimensionless Gh)f, = Colburn heat transfer factor = (hf,/ ~ ~ , f G f ) Pdimensionless rf~~~, keO = effective thermal conductivity of a packed bed when there is no fluid flow, Btu/(ft)(hr)("F) kf = fluid thermal conductivity, Btu/(ft)(hr)("F) k, = thermal conductivity of solids, Btu/(ft)(hr)("F) 1, = effective length between pellet centers, f t
+
446
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4, 1974
1, = effective length of solid relating to conduction, f t 1, = effective thickness of fluid film adhering to solid, f t Nufc = Nusselt number = hfcD,/kf, dimensionless Nuf, = Nusselt number = hfpDD/kf,dimensionless Nutotal = Nusselt number = htotalDp/kf, dimensionless Prf = Prandtl number = c,,fuf/kf,dimensionless Re, = Reynolds number = D,ufpf/Ff, dimensionless ReDm = modified Reynolds number = Re,,/(l - e ) , dimensionless (tf)i" = inlet fluid temperature, OF (tf)out = outlet fluid temperature, "F tp = packing temperature, O F u f = fluid velocity, ft/hr Greek Letters = parameter in Yagi-Kunii model, l,,/Dp y = parameter in Yagi-Kunii model, L/DD t = void fraction of bed, dimensionless pf = fluid density, lb/(ft3) p p = particle density, lb/(ft3) $I = parameter in Yagi-Kunii model, L/D, &, = shape factor, dimensionless pf = fluid viscosity, lb/(ft(hr)
6
Literature Cited Barker, J. J., Ind. Eng. Chem.. 57 (4), 43 (1965). Baumeister, E. B., Bennett, C. O., AlChEJ., 4, 69 (1958). Bhattacharyya, D., Ph.D. Thesis, University of Waterloo, Waterloo, Ont., 1973. Bhattacharyya, D., Pei, D. C. T., J. Microwave Power. 314, 287 (1973). Bhattacharyya, D., Pei, D. C. T., Chem. Eng. Sci.. in press, 1974. Chan, C. K., Tien, C. L., Trans. ASME-J. Heat Transfer. 95, 302 (1973) Davidson, J. F., Harrison, D., "Fluidization." Academic Press, London, 1971. Dayton, R . W., Fawcett, S. L., Grimble, R. E., Sealander, C. E., Rep. BMl-747, Battelle Memorial Institute. Columbus, Ohio 1952. Ford, J. D., Pei, D. C. T., J. Microwave Power. 2, 61 (1968). Galloway, T. R . , Sage, 8. H., Chem. Eng. Sci.. 25, 495 (1970). Gamson, B. W., Thodos, G., Hougen, O.a., Trans. AIChe. 39,1 (1943). Kunii, D., Smith, J. M., AIChEJ.. 6, 71 (1960). Lindauer, G. C.,AlChEJ.. 13, 1181 (1967). Wakao, N . . Kato. K., J. Chern. Eng. Jap.. 2, 24 (1969) Wakao. N.. Kato. K., Furuva. N., Int. J. Heat Mass Transfer. 12. 118 (1969). Yagi, S., Kunii, D.,AlChEJ. 3, 373 (1957). Yovanovich, M. M., ASME Paper 73-HT-43, ASME-AIChE Heat Transfer Conference, Atlanta, Ga., 1973. Zabrodskv. S. S.."Hvdrodvnamics and Heat Transfer in Fluidized Beds." M.I.T. h e s s , Cambridge M a s s , 1966
Received for review March 25,1974 Accepted June 11,1974
