1856
Hayworth, C. B., and Treybal, R. E., Ibid., 42, 1174 (1950). Johnson, H. F., and Bliss, H., Trans. Am. Inst. Chem. Engrs., 42,
Subscripts 1,2 = ends of tower or section of tower C = continuousphase D = dispersedphase
T
=
Vol. 43, No. 8
INDUSTRIAL AND ENGINEERING CHEMISTRY
toluenephase
W = waterphase
LITERATURE CITED
(1) Blanding, F. H., and Elgin, J. C., Trans. Am. Inst. Chem. Engrs., 38,305 (1942). ( 2 ) Elgin, J. C., and Browning, F. BI., Ibid., 31, 639 (1935). (3) Geankoplis, C. J., and Hixson, A. N., IND. ENG.CHEM., 42, 1141 (1950).
EngF-iYring
331 (1946). Licht, W., Jr., and Conway, J. B., IND.EKQ. CHEW,42, 1151 (1950). Morello. V. S..and Poffenberaer. N.. Ibid.. 42. 1021 (1950). Nandi, S. K., and Viswanathln, T. R., Current Sci. (Indid), 15, 162 (June 1946). ENQ. (91 Sherwood, T. K., Evans, J. E., and Longcor, J. V. A., IND. CHEM., 31, 1144 (1939). (10) Woodman, R. M., J . Phys. Chem., 30, 1283 (1926). (11) Yost, J. R., M.S. thesis, University of Pennsylvania, 1949. RECEIVED December 16, 1950.
Heat Transfer and Heat Release
in Fluidized Beds
p*cess
development' I NORMAN
L. CARRI
A N D NEAL R. AMUNDSON
UNIVERSITY OF MINNESOTA, MINNEAPOLIS 14,
MI".
T h i s paper is a portion of an investigation into the interaction of fluids and solids i n fixed, moving, and fluidized beds in which the main concern is the operation carried on in the bed. This particular investigation was begun with the idea of determining in a theoretical way the temperature relations in a fluidized bed, depending upon the Bow and mixing assumptions. Formulas have been derived relating operating variables and physical parameters for heat transfer and heat release for two cases. The first assumed t h a t the conductivity in the solid might be a limiting factor, while in the second case conduction in the solid was neglected. I t is shown
for the case of heat transfer alone that for particles of the size ordinarily encountered in fluidization practice the two cases give almost identical results. With the assumptions made of random motion of particles in the bedand complete mixingof the fluid, the simple case discussed above is sufficient for design purposes. This offers a simplification which materially diminishes the calculations and should further aid the experimenter in the determination of heat transfer coefficients between the fluid and the particle. This work is being continued i n order to determine the effect of other fluid and solid mixing assumptions.
M
ticles from the bed presents a problem Those solution can bwt be obtained from considerations of probability. This problem has been discussed in the previously mentioned paper oE KaTten and Amundson (6). If solid is admitted to a bed containing S pounds of solid dt a rate of w s pounds per hour, the fractional part of an element, dm = wad+, introduced a t time Q = 0 still residing in the bed a t time 4 is given by - %?s@ __ e 8
EAT transfer and the release of heat in fluidized catalytic beds is a major factor in the design of these reactors as well as in their operation. In spite of the importance of this probleiii little is available in the literature either from an experimental or theoretical point of view. The problem of mass and heat transfer between particles and the fluid has been considered by Wilhelin and hIcCune ( I $ ) , Resnick and White (II), and Kettenring, Manderfield, and Smith (6). Kalbach ( 4 ) considered chemical reactions in fluidized beds from an over-all standpoint, while Carr and Amundson ( 1 ) and Kasten and Amundson ( 5 ) discussed reversible chemical reaction and adsorption, respectively, in fluidized beds. Experimental work on chemical reaction systems has been presented by Lewis, Gilliland, and McBride ( 7 ) and Len-is, Gilliland, and Reed (8). A great deal of other work is being done but reports have not been made. I n this paper the problem of heat release and heat transfer is considered from a theoretical point of view. It is supposed that a fluidized bed is simultaneously fed with a fluid stream and a solid stream. Effluent streams of fluid and solid are withdrawn at the same respective rates a t which they were admitted. The solid is assumed to be in the form of small uniform spheres of constant diameter which enter the reactor in a continuous fashion much in the manner of a fluid. Because their size is small and because the rate of solids circulation is usually large, this assumption is probably valid. If it is assumed that the particles in a fluidized bed are in completely random motion the withdrawal of these par1
Present address, Illinois Institute of Technology, Chicago, Ill.
and the amount of this element still in the bed a t time Q is given by -W S 9 __ (1) w,e S d~ Integration of this expression over all past time
should give the total mass of solid S in the bed. This it does. If one wishes to obtain the average residence times of particles in the bed the weighted mean of the time must be considered as
;Jmw,ee-
-Wad
S
de
This gives on integration by parts, S/wB,which is usually called the nominal holding time for such a reactor.
August 1951
1857
INDUSTRIAL AND ENGINEERING CHEMISTRY
If fluid is admitted to the bed at temperature TOand leaves at temperature Tl, and if the particles are admitted at temperature to,there will be a transfer of heat from the particles to the fluid or vice versa. Because different particles in the bed have resided there for different times each particle in the bed is in a different transient state in so far as its temperature is concerned. Hence a temperature gradient is set up in each sphere and the rate a t which heat enters or leaves a particular sphere depends upon the thermal conductivity, heat capacity, density, and the heat transfer coefficient a t the sphere surface. Because the temperature gradient at the sphere surface is continually changing, the equation for heat conduction in the sphere must be solved simultaneously with the over-all heat balance over the whole exchanger. If the particles are in random motion in the bed it seems reasonable to assume that there iF; complete mixing of the fluid in the reactor. Hence each particle, regardless of its particular transient state, is in contact with a fluid of unchanging temperature and this must be the temperature of the fluid as it leaves the reactor. With these facts in mind the mathematical system may be set up. DIFFERENTIAL EQUATION AND AUXILIARY CONDITIONS
If one considers a single sphere whose radius is R, the equation for heat conduction in that sphere, assuming spherical symmetry, is
where t i s the temperature a t a distance r from the center at time 6, and p is the amount of heat generated in the sphere per unit of time per unit of volume. I n general, q arises from a catalytic reaction and, therefore, is a rather complicated function of the temperature as well as the catalyst activity, activation energy, heat of reaction, concentration of reactants, and time of contact. As an approximation one might write q=a+Pt
If an element of mass of spheres is considered which entered at time + = 0, the rate at which heat is leaving the part still remaining in the bed at, time + is
However, the whole sphere mass in the bed consists of fractions of elements which have entered over all past times so that the rate a t which heat is leaving the solids in the bed and entering the fluid is
where To is the temperature of the entering fluid and wjand cf are the flow rate and specific heat of the fluid, respectively. This, of course, assumes the reactor is adiabatic. If heat losses occur at the reactor walls, terms allowing for these losses must be added to the right-hand side of Equation 7. The average temperature of spheres leaving the reactor can be obtained as follows. The average temperature of a single sphere is
This will be the average temperature of any mass of spheres which have resided in the bed for time Thus, the average temperature of an element wad+ which was admitted at time + = 0 is f. Of this element only
+.
-2084 wse
‘ d+
remains, and so by calculating the weigkted average temperature it is clear that the average temperature 3 of particles in the bed is
where a: and are constants depending upon the system. The differential Equation 2 then reduces to
If there is no heat generation in the spheres the average temperature of the spheres can be calculated from
If attention is concentrated on a particular sphere, then it enters a t time + = 0 with a temperature to so that the initial condition may be written t = to, when
+=0
(4)
Each sphere is in contact with a fluid of temperature T1and if the heat transfer coefficient a t the sphere surface characterizing the film resistance is h at
- ks(%>r=R
= h(t - T l ) ,when r = R
(5)
Equations 3, 4,and 5 make up a complete mathematical description of the sphere if the further assumption is made that the temperature in the sphere cannot become infinite, or t finite for 0 6 r I R
(6)
The solution of this system will be obtained later. Another relation involving the sphere temperature results if a heat balance over the whole reactor is made. The rate a t which heat leaves a particular sphere is
GENERAL SOLUTION OF THE PROBLEM
The method of solution which lends itself most easily to this problem is the Laplace transformation, details of which can be found in Churchill (3) or Marshall and Pigford (9). The transform of 1 is defined by
L[t(r,+)l=
f me-pmt(r,+)d+
JO
= H(r,p) =
H
The general procedure is to take the transform of the system, thus reducing it to one of simpler type which may be solved more easily. The solution to the original system is obtained by performing the inverse operation on the function H. The transforms of Equations 3 and 5 are 2 dH
+7 B
+ (6’ - $ ) H 01
where j3’ = ke ’ a‘ = -,and k,
=
-
($ + $)
(8)
k,
K = -,and caps
and the rate at which heat leaves a unit mass of spheres is respectively. r =Ois
The solution of Equation 8 which remains finite at
INDUSTRIAL AND ENGINEERING CHEMISTRY
1858
R Ip' t K and A is an arbitrary constant whose value must be determined by the auxiliary conditions. The constant A can be obtained by substituting Equation 10 into Equation 9 and solving for A , When this value of A is substituted back into Equation 10, there r e dt s as shown by Munro and Amundson (IO) vhere
w =
Vol. 43, No. 8
In order to evaluate the limit corresponding to the pole at p = P'K it is necessary to expand the trigonometric terms and cancel common factors. It develops that the residue of the first term is the negative of the residue of the second so that the sum of these two residues at p = p'K is zero. To evaluate the residues a t the poles corresponding t o the zeros of D = 0, denote these zeros by wn where
or where
D =
w COS w
+ - 1) sin w (e
(12)
_andwhere e = -
hR 12,
-It ip seen that E is related to the Kusselt number by the relation h ( p ) = w cos w
+ (e
- 1) sin w
and The inverse transform of Equation 11 can be found by utilizing the extension of the Heaviside expansion bheorem (S,pp. 168-170). In brief, one must determine the residues of the function
h'(p,) = (w, sin wn
R2 sin wn
= -[w:
2K4
H(r,p)eP+
a t the poles of H ( r , p ) . The inverse transform is the sum of these residues. In Equation 11 it is clear that the poles occur at the zero8 of the denominator. Hence, considering only the first term on the right-hand side of Equation 11, these poles occur for those values of p for which p(p
- P'K)D
=
0
where use has been made of D the pole p = p , 2R2h Iw?t(h rk.*
(w2
-
TI)^
=
-
E
+
R2 cos an)2Kw, €(E
- 1))
0, there results for the residue a t
- (a + Bto)R21sin
fWn -
R
- p ' R * ) [ w ? + E(€ - l ) ]sin wn
On summing over all t,he poles one obtains
Obviously, these are at p = 0 , p = P'K, and the zeros of D = 0. At a simple zero of the denominator p,, the residue can be obtained from the formula ( 3 )
If the function whose inverse transform is desired has the fractional form g ( p ) / h ( p ) , where g ( p ) and h ( p ) are analytic at p = p,,, g ( p , ) = 0 then the residue at the simple pole, p n , is (3)
where h'(p,) is the derivative of h ( p )a t p = pn. If the residue at p = p , is denoted by p n the complete inverse transform of H ( r , p ) is given by L-1 [ H ( r , p ) l
= z
p n n=l
where the summation is taken over all the poles. It can be proved quite easily that in general the poles of H ( r , p ) w e all simple poles, although the proof Kill not be given here. Also the zeros of D = 0 are all real as shown in Carslax and Jaeger ( 8 ) for physically reasonable values of E. Hence Equations 13 and 14 can be applied, and the residues Kill now be evaluated. Since there are poles at p = 0 and p = p'K in both the terms on the right-hand side of Equation 11, consider these first. Equation 13 can be applied directly a t p = 0 to obtain.
This formula gives the temperature a t a point in a sphere as a function of the radius variable and the length of time the particle has been in the reactor. In order to solve the problem originally posed it is necessary to use Equation 7 . Hence the derivative of Equation 15 must be obtained and this must be followed by an integration with respect to the time as indicated in Equation 7 . The manipulative details will not be presented, but after a good deal of juggling, one obtains WjCj(T1
- TO)^==
--
w
m-here a
TI RP~ 3hs(
+ i ) [ R dRF e+-
--
t a n R e - 1) tan R-
(E
1
+
w,R2
= --
KS
The convergence of the infinite series portion of this formula is very rapid since the nth term behaves essentially as ~ ~ - 4 The . roots of D = 0 for large values of n approach multiples of T so
August
.
1951
INDUSTRIAL AND ENGINEERING CHEMISTRY m
that the series converges about like Zn-4 which is rapidly con1
vergent. Equation 16 is the solution of the general problem which was the purpose of this paper. It would be desirable to construct charts or tables which could be used instead of this equation but the number of parameters is so large that this numerical work is prohibitive at this time. The calculations using particular parametric values are not difficult, and it is suggested that each problem be treated as a special case of the above. On using the formula for the average temperature of the particles in the bed one obtains after some manipulation ;=
a 3 - -6+ - R2p' -E
2
6e2A
(T 4- %)( R d / Pt s' n+R d /-P '1)- Rtand PR' d P ' )
1859
is available in the boek by Carslaw and Jaeger ( 2 ) . The calculations using this formula are straightforward because of the rapid convergence at least for small values of A. For large values of A the convergence is not so rapid. It is of interest to try to improve upon Equation 17 by summing the series-i.e., obtain an expression for the series in closed form. This can be done by using the following theorem whose proof is obvious and will not be presented here. If q ( w ) is an analytic function of a complex variable which has simple zeros and if f(w) is a rational algebraic function such f(z) = O( 121-2) and if f ( w ) has simple poles a t ai, az, . ., a, with residues bl, bz, ., b,, respectively, then E,
..
...
+
(E
W?(W:
n=l
w:(to - Ti) - (a' + @'to)R2 - P'RZ)[w;4+ E ( E - l)](w," + A - p'R2)
where the first summation is over the zeros of g(w) inside C , the second summation is over the poles of f(u)inside C, and C is a contour taken as a square with vertexes a t
SPECIAL CASES OF THE GENERAL PROBLEM
Various special cases of the general problem can be obtained quite easily. CASEA. Constant heat generation; p = 0, 01 # 0. I n this case the limit of Equation 16 must be taken as 6 + 0. There results
:>
(n + -
(flhi).
This theorem follows directly from well-known theorems in complex variables. The contour is chosen in this manner so that it passes through none of the poles of the integrand. Under the conditions of the theorem as n m the integral on the left can be shown to vanish so that
-
- 2bjg(ai)
Zj(wj) =
where the summations have the significance given above. In order to sum the series in Equation 17 let g ( w ) = w cos w
CASE B. Constant heat generation with no solid addition or removal; fl = 0, CY # 0, wa = 0.
* PI
= w / c / ( T ~-
-
1) sin
w
The values of wk are the roots of g ( w ) = &Le., the same as those in the summation in Equation 17. The poles off(w) occur a t
The residues at these simple poles can be calculated from the usual formula. Then
m
ff
(E
TO)
CASE C. Heat generation proportional to the temperature; CY=O,P#O.
n= 1
+
( 4 - P'R'
+ A ) ( 4 - fl'R2)[4 f
€(E
- l)]
CASE D. Same as case C with no solid added or removed; = 0,p # 0,we = 0.
Be2
=
=
az = -a1
a3 =
id2
a4 = -aa CASEE. Linear heat generation with no solids added or removed; w s = 0.
CASEF. No heat generation; CY = 0, p = 0. This is the case which corresponds to straight heat exchange in a fluidized bed between the particles and the fluid. Equation 16 reduces to
2 i d m ) [A
- bl ba = 2ifi[A
-
E(E
- I)]
bz =
bq
=
6~2
- €(E
- l)]
-68
Further details will not be presented but substitution into the formula of the theorem gives
Thus it is seen that for the case of straight heat transfer in a fluidized bed the formula relating temperatures with physical parameters is
n= 1
and where the summation is taken over the roots where y = Wac8 of the equation w
cos
w
+ (e, - 1) sin w
= 0
As mentioned previously the roots of these equations for physically realizable situations are real and simple, and, in fact, a table of the roots of this equation for various values of the parameter,
It is indeed somewhat surprising that such a complex physical problem leads to such a simple mathematical result. APPROXIMATE SOLUTION OF T H E PROBLEM
AB an approximation t o the rigorous solution one might consider the same problem from the standpoint that the total resistance to the transfer of heat is the bounding film between fluid and
Vol. 43, No. 8
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
1860
solid. I n this case one assumes that the sphere has a uniform temperature. The rate a t which the amount of heat in a single sphere is increasing is The special cases obtained in the previous section can be deduced here also but this is a trivial exercise and will not be repeated. Of some interest, however, is the case of straight heat exchange in which there is no heat generation. Equation 22 then reduces to
Heat is being generated in the sphere a t the rate 4 3
-7rR3(a
+ Pt)
hssuming that the exterior fluid is hotter than the solid, heat is entering the sphere a t the rate 47rR2h(T1 - t )
This formula is somewhat simpler than the rigorous one.
(20) DISCUSSION OF T H E TWO SOLUTIONS OF CASE F (NO HEAT GENER4TION)
Hence collecting terms and simplifying
T o this differential equation must be appended the initial condition t = to,when + = 0. The solution to the differential Equation 21 can be found quite easily to be
where
Certainly Equation 19 is a simple formula and little fault can be found with it from this point of view. However, it can be shown, as will be done, that the nonrigorous solution leads, in general, to results of about the same accuracy. I n a recent paper Kettenring, Manderfield, and Smith (6) have obtained a correlation of heat transfer coefficients with the Reynolds number in a fluidized bed. These authors have found that
This equation may be written in the form E
=
0.00675
(2)
(de)
From the initial condition it follows that C=to-
+
3hT1 CYR3h - OR
and, therefore, t =
%(to
+
- T I ) - R(CY @to) 3h - /3R
+
e-a+
+
3hTi CYR 3h - PR
This gives the temperature of a single sphere as a function of its residence time. From Equation 20 the rate at which heat is leaving the fluid and entering the solid per sphere is
For a unit mass of spheres this equation must be divided by 4/3 nRaps. Therefore, equating the rate a t which heat is entering the solid to the rate a t which the fluid is losing heat there results
In Figures 1 and 2 this equation has been plotted on the left-hand side with 6 as ordinate and Reynolds number as abscissa for various values of k,/k,. On the right-hand sides of Figures 1 and 2 , Bas defined in Equation 19 is plotted with E as ordinate and R as abscissa for various values of A. The values of B in Figure 2 must be small otherwise the range of data of Kettenring, Manderfield, and Smith is exceeded. If nom the same procedure is repeated using Equation 22 instead of Equation 19 the curves obtained are superimposable with those calculated above. The numerical error involved is less than the error involved in reading the curves and certainly within the error in experimental data. Hence Figures 1 and 2 are graphical representations of both the rigorous and approuimate solutions. An examination of the two formulas 13-ill show that a little foresight might have shown that the coincidence ib not unwarranted. If the value of A is small tan h& can be expanded in its Xaclaurin series
If one uses only the first two terms of the series then
or
This can be reduced to
" ( di+ < A
A
(E
- tan h z / x - 1)tan h z / a > =
Hence for small values of A the two formulas are about equivalent. For large values of A the fraction
dx - tan h d z
4% + ( e - 1)Gtan h d / a It is obvious that the average temperature of the spheres in the bed and hence of the spheres leaving the bed is
is approximately equal to one since e is very small in all cases considered here. Hence both Equations 19 and 22 reduce to
INDUSTRIAL AND ENGINEERING CHEMIST R Y
August 1951
1861
3E
- since in Equation 22, 3~ is A
n e g l i g i b l e compared with A. I n Figure 2 on the right-hand side then the function whose graph is shown should be
This is a straight line on a loglog plot and this clearly fits the graph as a little examination will show. Thus i t appears for the range of E values considered here one can safely neglect conduction in the particles. For the case of heat generation inside the particles this analysis may not be valid. Further examination should be given to the range of A values. T o recall, A is defined by
r
Figure 1.
-
d& us. B = " Plot of Reynolds Number __ for Various Values of k , / k s P t o - Ti wsR2 and A = ICs ~
~
Since for small A it is obvious that the approximate solution should be valid, consider only very large values Suppose that particles of I/,inch could be fluidized and
Low A range
REYNOLDS
Figure 2.
NO.
Plot of Reynolds Number
B
d G P
1)s.
B = y ___ - Tofor Various Values of kJk. and A = w,R2 t o - Ti KS High A range
INDUSTRIAL AND ENGINEERING CHEMISTRY
1862
Vol. 43, No. 8
suppose a circulation rate per hour of twenty times the solids holdup were used, then
doubtedly lies someplace between the two. Efforts to solve problems in the intermediate range have not proved fruitful and a great deal of difficulty has been encountered even in the stat'ing of a problem.
Thus one sees that to obtain a value of A equal to one the value of the thermal diffusivity must be small, and values of A as high as or higher than those shown in Figure 2 are improbable if not impossible. Values of K of the order of magnitude of 0.01 are certainly possible for some catalytic materials. In order t o illustrate the use of the graphs the following example will be considered.
ACKNOWLEDGMENT
I t is desired to transfer heat from a mass of hot particles to air in a fluidized bed, the particles entering a t 500" F. with a flow rate of 600 pounds per hour. The bed is 1 square foot in cross section and 246 pounds per hour of air enter. The air is to leave a t 400 O F. The particles are 20-28 mesh and the bed contains 60 pounds of solid. It is desired to determine what the entering air temperature must be for these conditions. dp = 0.00234foot S = 60pounds p a = 100 pounds/cubic foot tor = 600 poundsjhour cg = 0.30 B.t.u./hour/" F. k , = 0.228 B.t.u./hour/square footj" F./foot T I = 400°F. to = 500°F. WJ = 246pounds/hour G = 246poundsjhour A , a t 400 O F. = 0.025 centipoise k, at 400' F. = 0.0226 B.t.u. jhour/square foot/' F./foot c f at 400' F. = 0.25 B.t.u. /hour/ O F. From these data
K = " k= caps
o'226 = 0.00753 square footfiour (0.30) (100)
The authors viish to acknowledge the help of ,James Lee, John Gorman, and particularly Thomas E. Guenter for aid in the calculations and preparation of draa-ings. NOMENCLATURE
specific heat of fluid, B.t.u./hour/" F. specific heat of solid, B.t.u./hour/" F. diameter of particles = 2R, feet mass velocity of fluid, pounds/hour/square foot heat transfer coefficient, B.t.u. jhour/square foot/' F. h H = transform of t C/
=
= = = =
i
=.\/Fi
k , = thedmal conductivity of fluid, B.t.u./hour/square foot/ F. /foot k , = thermal conductivity of solid, B.t.u./hour/square foot/ O F./foot K = - k* CSPS
P = transform parameter corresponding to + P r
= rate of heat generation in solid, B.t.u./hour/cubic foot = radius variable in sphere, feet
R = radius of sphere s = mass of solid in exchanger, pounds t = temperature a t a point in a sphere, O F . t o = initial temperature of sphere, O F. 2 = average temperature of a sphere, F. t = average temperature of %articlesin reactor, F. To inlet fluid temperature, F. TI = outlet fluid temperature, F. W J = flow rate of solid, pounds/hour w s = flow rate of fluid, pounds/hour CY, 6 =: constants for linear heat generation
-
O
8'
=
fflh Plk
Y
=
WjCJ/Wscs
cy'
A = - -waRZ -=
KS
(0.00234)2(600) = o.oo183 (0.00753) (4) (60)
IC,/?€,
A = w,R2/KS
= 0.1
B
=
0,955 = Wac8
(s)( =
To
=
(246) (0.25) (600) (0.30)
400 500
- To - 400
=
Ps = P =
+
120" F.
I n order t o determine the temperature of the particles leaving the exchanger, one uses
i = to - ws
hR/h density of solid, pounds/cubic foot viscosity of fluid, pounds/hour/foot c b = time, hours w = R d p - Pf/R W n = root of w cos w (t - 1)sin w = 0 E
On Figure 1 starting a t a Reynolds number of 9.5 the heavy arrowed line is followed to k,jk, = 0.1 then to A = 0.001%. Reading vertically B is 955. Hence
(TI - To) = 459" F. cs
Hence the reduction in temperature of the solids is small but this is to be expected since the driving force for heat flow is limited by the outlet temperatyre of the fluid. This, of course, is the outstJandingdisadvantage in the use of such an exchanger. I n a countercurrent exchanger as considered by Munro and Amundson (IO)the particles leaving the exchanger at least approach in temperature the fluid entering the exchanger. CONCLUSIONS
The main result in this paper has been to shot\- that for the case of heat exchange without heat generation in the particles, intraparticle conduction can be neglected. This can almost certainly always be done for particles n-hich are in common use in fluidization processes. It has been assumed in this paper that the solids in the bed are in a completely random motion and that the fluid is completely mixed. These assumptions are probably untrue. I n a previous paper ( I O ) the problem of countercurrent flow of particles and fluid was considered and the true Rituation for a fluidized bed un-
LITERATURE CITED
Carr, N. L., and dmundson, N. R., J . Phys. R. Colloid Chem.. in press. Carslaw, H. S., and Jaeger, J. C., "Conduction of Heat in Solids," London, Oxford University Press, 1947. Churchill, R. V., "Modern Operational Mathematics in Engineering," pp. 168-70, Xew York, ,MoGraw-Hill Book Co., 1944. Kalbach, J. C., Chem. Eng., 54,No. 1, 105 (1947); 54,No. 2, 136 (1947). Kasten, P. R., and Amundson, N. R., IND.ENG.CHEM.,42, 1341 (1950). Kettenring. K . N., Manderfield, E. E., and Smith, J. hI., Chem. Eng. Progress, 46, 139 (1950). Lewis, TV. K., Gilliland, E. K.,and McBride, G. T., IND. ESG. CHEM.,41, 1213 (1949). Lewis, W.K., Gilliland. E. R., and Reed, W.A . , Ibid., 41, 1227 (1949).
Marshall, W ,R., and Pigford, R. L.. "Applications of Differential Equations to Chemical Engineering Problems," Newark, University of Delaware, 1947. hlunro, IF'. D., and Amundson, 9.B., I s n . EYG.Cfrmf.. 42, 1481 (1950). Resnick, W.,and White, R . R., Chem. Eng. Progress. 45, 377 (1948). Wilhelm, R. H., and McCune, L. K., IND. EXG.CHEM.,41,1124 (1949). RECEIVED Kovember 21 l Q 51
