The inconsistencies in published data and methods of estimating the actual coefficients for particle-to-gas heat transfer in fluid beds are discussed. Calculations for two fluidized bed reactions show that the particle-to-gas temperature difference will nearly always be negligible, but significant temperature differences may exist between gas bubbles and the dense bed
heat
transfer PETER H A R R I O T T A N D L. A. BARNSTONE recent review by Barker (7) lists 242 references
A dealing with heat transfer to particles in a fluidized
bed. The data are summarized in a plot of Colburn j-factor us. Reynolds number which shows great differences in the published correlations. The j-factors for N R= ~ 1 cover a 10,000-fold range, and the slopes of the I n an earlier review, graphs range from - l / 2 to ‘/2. Frantz ( 3 ) rejected some of the inconsistent data and presented separate correlations for “true” and “apparent” coefficients. However, his recommended correIEtions give such low values of the Nusselt number (0.1 to 10) that they are not realistic for particle-to-fluid heat transfer. Lack of a suitable correlation will probably inspire still more studies, but before starting, it would be well to consider the inherent difficulties of the problem and the possible value of the data from theoretical and practical viewpoints.
Peter Harriott is Professor of Chemical Engineering at Cornell University. A t the present time he is on sabbatical leave at the Centre Nationale de la Recherche ScientiJique, Institut de Recherche5 sur la Catalyse, France. Leonard A . Barnstone is an engineer for Esso Research and Engineering Co. AUTHORS
VOL. 5 9
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4 APRIL 1 9 6 7
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One reason for the scatter in published data is that the coefficients were calculated assuming plug flow of gas through a bed of solids, which is an incorrect model. Backmixing of gas occurs above the distributor plate, and a sizable fraction of the gas may rise in bubbles which bypass most of the solid. Thus a decrease in particle size, which may have little effect on the bubbling and backmixing phenomena, will have little effect on the gas temperature profile, and the apparent coefficient based on particle surface area will decrease. A second source of inconsistencies is that heat transfer in a fluid bed is rapid enough in spite of the nonideai flow to make the driving force almost zero over most of the bed. Hot gas entering a bed of fine solids that are 400" F. cooler comes to within 0.1 F. of the particle temperature in a fraction of an inch (3). Even if it were possible to allow for the backmixing effect, which has not yet been attempted, it is still difficult to get separate measurements of gas and solid temperature within the bed. Finally, any coefficients measured for the first half inch of bed are of doubtful theoretical value, since the bed density, solid velocity, and bubble pattern would not be the same as in the main part of the bed. From a practical standpoint, a fluid bed that is used to heat or cool a gas stream can generally be designed by heat balance, assuming temperature equilibrium of the gas and solids at the gas exit. This is certainly true for particle sizes from 300-mesh to 20-mesh. which cover most fluidization studies. For shallow beds of large particles, say to inch. the temperature change might be only 90 to 99YG of the equilibrium value, but even for these cases, a correlation of heat transfer efficiency us. flow might be simpler to use and less subject to error than a correlation of heat transfer coefficients. The data showing a 10,000-fold range ofj-factors would reduce to a very narrow band of efficiencies, with most of the points at 1 0 0 ~ o ! The nearly perfect performance of fluid beds as heat exchangers doesn't rule out the possibility of significant temperature differences in fluidized reactors or adsorbers, where heat is generated within the bed. Calculations are shown later for a typical solid catalyzed reaction and for a homogeneous reaction, where the solid serves merely as heat ballast. The model used assumes that all gas in excess of the minimum needed for fluidization passes through the bed as mushroom shaped bubbles or void spaces (2, 6, 8,9 ) . The bubbles move more rapidly than gas in the dense bed and a cloitd of gas, somewhat larger than the bubble, circulates through the void space and the layer of solids around the bubble.
H E A T TRANSFER I N T H E DENSE PHASE Because the particles are moving relative to each other, heat transfer from solid to gas in the dense phase should be at least as rapid as in a fixed bed at the same gas velocity and void fraction. However, the particle Reynolds numbers are often less than 1, and fixed bed correlations are not reliable at such low flows. Eone of the fluid bed correlations in Reference 1 or 3 can be used since they correspond to Nusselt numbers of 56
INDUSTRIAL A N D ENGINEERING CHEMISTRY
2 X IO-' to 0.2 at *VRe = 1, compared to the minimum h-usselt number of 2.0 for an isolated sphere. For particles in the dense phase of a fluid bed, the minimum Kusselt number is even greater than 2, according to the following reasoning. If the average distance betu een particles is d / 2 , the film thickness can be taken as d / 4 , the distance to the center of the gas stream. However, the heat transfer coefficient is based on the average gas temperature and not the temperature in the middle of the flow channel, so the effective conduction distance is about d / 8 . If we make a slight allowance for particle motion, the figure is rounded off to d, 10. corresponding to a minimum Nusselt number of 10. Some recent frequency response tests under Brodkey at Ohio State indicated Niisselt numbers of 10 to 100 for the dense phase, but the technique is not accurate enough to give other than order-of-magnitude results.
H E A T TRANSFER F R O M GAS BUBBLES Different rates of heat generation could lead to temperature differences betu een the bubbles and the dense phase of a fluidized reactor. Heat transfer from bubbles to dense bed would also be important in tests where hot gas was passed into a bed of cool solids. Although bubbles in the bottom inch of bed would not be typical of the bed as a whole, this transfer mechanism can partially account for the low apparent coefficients that have been reported. I n many studies coefficients were calculated from the temperature gradient near the bottom of the bed using the plug flow equation:
- 1, In 1, ___tf
-
ts
=
'aL - h__
(1)
UPC,
If heat transfer occurs solely by exchange of bubble gas with dense bed gas at a rate qbf/cu. ft/sec. per cu. ft. of bed, the corresponding equation is:
Thus the effective interchange rate can be calculated from the apparent heat transfer coefficient :
(3)
To check the magnitude of q b f , suppose 100 p of catalyst particles are fluidized with air at 0.5 ft./sec. The Reynolds number based on superficial velocity is about IO, and according to the correlation of Frantz ( 3 ) , the apparent Nusselt number is 0.012. The void fraction in the dense bed is about 0.6 and the minimum fluidization velocity about 0.05 ft./sec. : h' =
0.012 X 0.016 3.27 x 10-4rt. ~~
=
0.59 B.t.u./hr./sq. ft./" F
q0f = 0.59 X 7300 X 0.45 -_ X 1 = 64 sec 0.07 X 0.24 0.50 3600
--1
The value of qb for mass transfer is about 0.5 sec.-I, according to the kinetic study of Lewis, Gilliland, and Glass (6). T h e 100-fold higher value for heat transfer is not surprising, since heat is transferred from bubbles to the dense phase by the solid as well as by the gas. Another estimate of qbt can be obtained from the mass transfer data and a model of the exchange process. Gas circulates u p through the bubble, through a shell of particles moving down around the bubble, and between particles in the wake region beneath the bubble. Some transfer occurs by molecular diffusion from the outside of the bubble cloud to the dense bed. Davidson and Harrison (2) predict that 80 to 90% of the transfer is by molecular diffusion, but this seems high, since others (4) found no difference in residence time curves using helium and carbon dioxide as tracers. The major exchange may come from gas carried down with the solids that flow around the bubble and are shed at the wake region. Bulk transfer also occurs when bubbles break up, coalesce, or change shape because of turbulence in the bed. These bulk flow mechanisms are assumed to account for half of the mass transfer, and the heat transfer rate is estimated from the bulk flow portion of the mass transfer. For each cubic foot of gas exchanged by bulk flow, a corresponding volume of solid particles, or (1 - e d ) / e d cu. ft., is assumed to carry heat from the bubbles to the dense phase. The total heat transferred by bulk flow is the sum of that carried by the gas and the solid, which is expressed as a n effective gas interchange rate as follows:
For the example chosen
qbt
= 0.25[1
80 X 0.2 X 0.4 + 0.07 X 0.24 X 0.6
1
= 160 sec.-l
T h e analysis indicates that almost all the heat transfer is by exchange of solids. The contribution of conduction through the gas was neglected because it would be the same magnitude as the diffusion term for mass transfer.
C A T A L Y Z E D E X O T H E R M I C REACTION T h e hydrogenation of ethylene is a rapid exothermic reaction that has been studied in a fluid bed (6). Suppose that 90yo conversion of a n equimolar mixture is obtained in a 1-ft. deep bed of 100 p of catalyst particles a t 1 atmosphere, 100' C., and 0.5 ft./sec., then, 0.5 273 F = - X - = 1.02 X 359 373 Qav.
= 0.90(0.5)(1.02
mole/sec./sq. ft.
B.t.u. X 10-8) 58,800 -mol. 27 B.t.u./cu. ft./sec.
The minimum fluidization velocity is about 0.05 ft./sec. and the Reynolds number based on this velocity is 0.1. Since fixed bed heat transfer correlations are not available for such low Reynolds numbers, the Nusselt number is assumed to be 10.
k h G
= 0.018 for CzHe
10 X 0.018 = 550 B.t.u./hr./sq. ft./' F. 3.27 x 10-4
ha = 550 X 7300 = 1.12 X l o 3 B.t.u./cu. ft./sec./'F. 3600 AT1 = 27/1120
=
0.024' F.
If the heat is released uniformly in the bed, the difference between solid temperature and gas temperature would be only 0.02' F. If we allow for greater heat release near the bottom of the bed and in the layer of solid around the gas bubbles, local differences up to 0.05' F. might be expected. For particles 400 p in size (40-mesh), ha is about 12 to 1 6 fold smaller ( e d is lower), and temperature differences up to 1" F. would be expected for the same rate of heat release. This difference might be detected by using a suction thermocouple to get the gas temperature, but it is still too small to have a significant effect on the reaction rate. T h e bubble temperature will be greater than the dense bed temperature even though all the heat is generated inside the catalyst particles. Catalyst in the bubble cloud and in the wake sees gas richer in reactants and generates more heat than catalyst in the dense phase. T h e heat capacity of gas is small, and the bubble gas and surrounding solid will be at almost the same temperature. T h e fraction of the solid in contact with bubble gas varied from 5 to 150j, in the study by Glass (6). T h e theory of Pyle and Rose (7) predicts that for u = 10 U C , ed = 0.6, and a wake filling of the bubble, the volume of the wake and surrounding cloud is equal to the void volume. For a void volume of 2OyOcorre, sponding to a bed expansion of 1.25, this means that 20% of the catalyst is in contact with bubble gas. For our example, 15y0 of the catalyst is assumed to see bubble gas, and '/3 of the reaction is assumed to occur here. Qbubble
=
'/3
x
27 = 9 B.t.u./cu. ft. bed/sec.
The heat exchange coefficient is estimated from the gas exchange rate and the solid capacity:
0.25 (80 X 0.2 X 0.4/0.6) = 2.7 B.t.u./sec./cu. ft./' F. If all the heat generated in bubbles is transferred to the dense phase (and from there to the wall) there would be a significant temperature drop :
AT2 = 9/2.7 = 3.3" F. Although the calculations are speculative, they show that temperature differences are more likely to occur VOL.
59
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57
between bubbles arid dense phase than between particles and the surrounding gas. Both differences will be negligible for inost fluid bed reactors, where several feet of catalyst bed are used and the heat generation rates are an order of magnitude lower than in the example chosen.
UNCATALYZED E X O T H E R M I C REACTION Homogerieous reactions may be carried out in a fluidized bed to take advantage of the rapid preheating and almost uniform bed temperatures achieved by fluidization. Consider the chlorination of methane in a 3-foot deep bed of 100-p sand particles at 350" C. aiid 2 atmospheres. Assume 509; Clp and 50% C H 4 in the feed and a velocity of 0.5 ft./sec.: AH
=
kp
=
-43,200 B.t.u./lb. mole 1.0 sec.-'/atni.-'
(5)
If the gas is heated to 350" C. without appreciable conversion, the initial rate of heat generation is
Q
= 1.0 (1.0 atrn.)(0.00122 mole Clz/cu. ft.)(43,200) == 53 B.t.u. sec. cu. ft. of gas bed
;=:
Q E ~
53 X 0.6
=
32 B.t.u./sec./cu. ft. dense bed Based on ha = 1120, as in the previous example, the solid in the dense phase could receive this much heat with a very small driving force. There would be a slight drop in solid temperature near the gas distributor, as solid originally at 350' C. heated the gas in a fraction of an inch to say 349" C. and then was reheated by reacting gases. There might also be a drop of a few dcgrees from the bottom of the bed to the top of the bed similar to that reported for the ethylene hy-drogeiiation (6),because of the finite rate of axial mixing of solid. The bubbles will generate heat faster than the dense bed, arid will tend to reach a maximum temperature near the bo.ttom of the reactor, where heat generation equals heat transfer to the dense phase. For a bed expansion of 25yG, the volume of the bubble voids would be 207, of the expanded bed. The additional gas in the bubble cloud and wake is about 10% of the bed ( 7 ) , making 307, in all. The heat generation rate in the bubbles is therefore Qbubble =
53 ( 0 . 3 ) = 16 B.t.u./sec./cu. ft. of bed
'The heat excharige rate should be about twice as great for sand as for porous catalyst, because of the greater heat capacity of the solid. hzaz g 5 B.t.u./sec./cu. ft./ " F. AT2 = 16/5 = 3.2" F.
A difference cif 3' F. could be measured with a probe aiid ~7ouldaffect the kinetics of the reaction. If the reactor were operated at 400" C., where k z is 6.0 sec.-l atm.?, the calculated AT2 would be 19" F., which is about half way to the critical temperature difference, where the rate of change of heat generation equals the 58
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
rate of change of heat removal. Operation above the critical temperature difference leads to runaway reactions. Flames and minor explosions were observed in a fluidized bed chlorinator (40 to 70 p particles) when using high chlorine concentrations and high reactor temperatures (5). Excess HC1 in the product also indicated local hot spots where pyrolysis occurred. The critical teniperature for pyrolysis was given as 350 O C. for 50% Cln, suggesting somewhat poorer heat transfer than in the calculated example, but still confirniing the general magnitude of the heat transfer predictions.
R E C O M M EN D A T l ONS T h e heat transfer rates influence the selection of a particle size for a fluidized reactor. Increasing the particle size increases A TI, but this difference is probably negligible for particlcs u p to at least 400 i . ~ . Increasing the particle siie increases the fraction of gas flowing through the dense phase and should also increase the exchange rate between bubbles of a given size and dense phase. Some data do show an increase in qb with particle size (6),but more studies are needed covering a wide range of particle size in a given apparatus. A s far as measuring the heat transfer rates directlj~, it seems best to carry out tests where the differences might be appreciable, that is to study a rapid catalyzed or uncatal) zed chemical reaction. The chlorination of methane is a good reaction for further stud). NOMENCLATURE particle area, sq. ft./cu. ft. of dense bed heat capacity of gas heat capacity of solid particle size gas flow rate, moles/sec./sq. ft. true gas-particle heat transfer coefficient apparent gas-particle heat transfer coefficient thermal conductivity second-order rate constant volumetric transfer coefficient for bubbles to dense phase bed depth Reynolds number, d u p / p or duop/p Nussel: number = h d / k gas interchange rate, cu. ft./sec./cu. ft. of dense bed effective interchange rate for heat transfer, set.-' heat generation rate bubble temperature feed temperature dense phase temperature gas temperature solid temperature temperature difference, gas to solid temperature difference, bubble to dense phase superficiaI gas velocity minimum fluidization veIocity void fraction of dense bed gas density solid density
REFER ENC ES (1) Barker, J. J., I W DEsa. . CIILM.5 7 (j),33 ( M a y 1 9 6 5 ) . (2) Davidson, J. F,, Hariison, D., "Fluidized Particles," p. 106, Cnmhridgr University Press, 1963. (3) Frantz, J. F., Chem. Eng. 69, 89, Oct. 1, 1962. (4) Gilliland, E. R., Mason, E. A , , Oliver, R.C., IND.E N G .C F ~ E 45, M . 1 1 7 7 (1953). ( 5 ) Johnson, P. R.,Parsons, J. L., Roberts, J. B., Zbid., 51, 499 (1959). (6) Lewis,','i. K., Gilliland, E.R., Glass, W., il.1.Ch.E. J . 5 , 419 (1959). (7) P>le, D. L., Row, P. I,.,Chem. Eng.Sci. 20, 2 5 c1965). (8) Rowe, P. N., Chern. Eng. Progr. 60 ( 3 ) ;75 (hIarch 1964). (9) Rowe, P. N., Cliem. E n g . Progr. S'ntp. Sei. 58 (381, 42 (1962).
