through palladium-silver tubes. The constants for the model are determined from pure-gas permeation measurements. Extensive comparison of the model with experimental data shows close agreement over a wide range of compositions. The experiments used to test the model were all nominally conducted at 4OO0C, 100 psia of driving pressure, and 20 psia of back pressure. The model should be valid over the same range of conditions for which eq 3 is valid when the pure-gas constants determined for the range of interest are provided. Within the limits imposed by the assumptions, the model is also adaptable to a variety of geometries. Although helium was used as the nondiffusing contaminant in our experiments, the results should be the same for other nonpermeating gases or gas mixtures providing that they do not poison the palladium surface or react with the hydrogen (Serfass and Silman, 1965; van Swaay and Birchenall, 1960). A particular process can readily be optimized by using the model to examine the influence of the process variables on the hydrogen-isotope permeation rate. Nomenclature
A = a = b = C = Cl,z =
effective diffusion area, cm2 constant, mole cm/(min cm2 - v‘/psia constant, mole cm/(min cmz - d z ] total solubility of a gas mixture concentrations of a gas in solution a t the opposite surfaces of a solid 3 = diffusivity G = molar flow rate into a tube, mole/min K = constant obtained from pure-gas solubility measurements L = molar flow rate out of a tube, mole/min 1 = length of a cylinder, cm P = total pressure of a gas mixture, psia P1 = driving pressure, psia Pz = back pressure, psia
pt,,
R
= =
rl,z =
T
=
X
=
2
= =
Y
partial pressures of hydrogen isotopes i and j , psia permeation rate of a gas, moles/min cylinder radii corresponding t o CI and CZ,cm temperature of a gas, O K mole fraction of a hydrogen isotope on the drivingpressure side length of a diffusion path, cm mole fraction of a hydrogen isotope on the backpressure side
literature Cited
Ackerman, F. J., Koskinas, G. J., J. Chem. Eng. Data 17, 51 (1972). Barrer, R. M., “Diffusion In and Through Solids,” pp 144-203, Cambridge University Press, New York, N. Y., 194;. Bird, R. B., Stewart, W. E., Lightfoot, E. N., Transport Phenomena,” p 502, Wiley, New York, N. Y., 1960. Darling, A. S., P l a p m Metals Rev. 2, 16 (1958). Darling, A. S., in Proceedings of the Symposium on the Less Common Means of Separation, Birmingham, England, April 1963,” p 103, Institute of Chemical Engineers, London, 1963. Davis, W. D., “Diffusion of Gases Through Metals. 11. Diffusion of Hydrogen Through Poisoned Palladium,” Report KAPL1375, Knolls Atomic Power Laboratory, Schenectady, N. Y., 1985. deRosset, A. J., Ind. Eng. Chem., 52, 525 (1960). Hickman, R. G., J. Less-Common Metals 19, 369 (1969). Hunter, J. B., U. S. Patent.2,773,561 (Dec 11, 1965). Lewis, F. A., The Palladium Hydrogen System,” p 130, Academic Press, New York, N. Y., 1967. Rubin, L. R., “Permeation of Deuterium and Hydrogen Through Palladium and 75 Palladium 25 Silver at Elevated Temperatures and Pressures,” p 55, technical bulletin, Engelhard Industries, Newark, N. J., 1966. Serfass, E. J., Silman, H., Chem. Eng. (London) No. 192, CE266 (1965). Sieverts, A,, Krumshaar, W., Ber. Deut. Chem. Ges. 43, 893 (1910). Birchenall, C. E., Trans. AIME 218,285 (1960). van Swaay, M., RECEIVED for review April 22, 1971 ACCEPTED March 30, 1972 Work performed under the auspices of the U. S. Atomic Energy Commission.
Countercurrent Backmixing Model for Fluidized Bed Catalytic Reactors. Applicability of Simplified Solutions Colin Fryer and Owen E. Potter* Department of Chemical Engineering, Monash University, Clayton, Victoria 3168, Australia
The countercurrent backmixing model for gas-fluidized catalytic reactors is studied using the assumptions of Kunii and Levenspiel concerning gas transfer between the bubble, cloud-wake, and dense phases. The various simplifications which have been made previously in the solution of this model are examined and their regions of applicability determined. The solutions obtained indicate several areas in which experimental study of reacting systems will be useful in assessing the validity of the model; in particular, concentration prdiles in the bed will be useful in distinguishing between the countercurrent backmixing model and others.
T h e existence of gas backmixing in beds fluidized with gas has long been recognized (Gilliland and Mason, 1949, 1952). Simple two-phase models such as those reviewed and developed by Davidson and Harrison (1963) and Partridge and Rowe (1966a,b) do not take account of such backmixing. 338
Ind. Eng. Chem. Fundam., Vol. 1 1 ,No. 3, 1972
However, May (1959) and van Deemter (1961) have incorporated axial diffusion in the dense phase. It is now clear that this axial diffusion is dependent on the bubbles. Stephens, et al. (1967), pointed out that the upward flow of solids with the bubbles would lead to a downflow of solids
in the remainder of the bed, and that if this downflow were sufficiently rapid, a very simple mechanism of gas backmixing arose. Recently Latham, et al. (1968), and Kunii and Levenspiel (1968a) have developed very similar models on the basis of this mechanism. Latham, et al. (1968), assumed that the rate of gas transfer between bubbles and dense phase was as suggested by Davidson and Harrison (1963) and that the gas in the bubble, cloud, and wake was well mixed. Potter (1971) has discussed further the reaction in the wake itself and recommends the assumption made by Kunii and Levenspiel (1968a) that the transfer process can be divided into two stages: (a) bubble to cloud and wake and (b) cloud and wake to dense or particulate phase. Latham and Potter (1970) and Hamilton, et al. (1970), have employed tracer gas backmixing studies to determine wake volumes and overall gas exchange rates. Kunii and Levenspiel (196813, 1969) have solved the countercurrent backmixing model for first-order reactions with several simplifying assumptions; in particular, they have neglected variation of reactant concentration with height in the cloud-wake and particulate phases, apparently assuming no bulk flow of fluid in these phases. It is the purpose of this paper to apply the model to a catalytic reactor without simplifying the solution in this way and thus to determine the extent to which the previously used simplifications are useful.
b
Exit gar concentration r d hcH
=
!
4
Phase : b b M e wake-cloud particulate k+dc,l c,+dG cp+dcp htdh Reactant I I concentrations : h S i % I CP Superficial gas velocity: tu, tu, UGp
I
Fractional vdume :
II EB 1I I1 t II tr,
l-&(ltt.)
Porosity:
Gas exchange coefficient : inkt gas concentration C.,
I
Fluidising gas velocity
ut
I
Figure 1 . Representation of fluid bed reactor
bubble and cloud-wake. Then for exchange between cloudwake and particulate phase they assumed a Higbie (1935) type penetration model. The coefficients for gas exchange which result are
Theory
Gas Flows. Following Latham, et al. (1968), and Kunii and Levenspiel (1968b, 1969), the cloud volume is neglected, an approximation reasonable for fine particles. Alternatively it might be said that the cloud volume is lumped in with the wake volume. The model is portrayed in Figure 1. The superficial fluidizing gas velocity is
I; =
~
G
+ UGP+ UQC
(1)
B
Since bubble and cloud-wake rise together a t the same absolute velocity [IGC =
(2)
fweOUGB
The superficial velocity of solids carried up in the bubble wakes is r G B f u ( 1 - eo). I n the particulate phase, solids must therefore be descending with this same superficial velocity. The backmixing model assumes that the relative velocity of gas to solids in the particulate phase is the same as a t incipient fluidization, Le. UGP [1 - eB(1 fio)leO
+
+
[1 -
UGBfw(1 - €0) EB(1 f f m ) l ( l -
€0)
-uo - €0 (3)
Rearrangement of eq 1, 2, and 3 gives the superficial gas velocities in the bubble and particulate phases as cGB = CGP =
CO[l
c - cO[l - E B ( l . f f w ) ]
-
€B(l
+
+ f ? ~ ) ] [ leOfwl
Chemical Reaction. The reaction is first order in gas concentration and occurs in the presence of particles in the cloud-wake and in the particulate phase. As pointed out by Kunii and Levenspiel (1969), for reactions normally encountered, it is reasonable to neglect the contact of reactant with the very small amount of solids which may be dispersed in the bubble phase. A correction would be required for the analysis given here if particles were extremely reactive. The material balances on reactant gas over the differential element of height in the bubble, cloud-wake, and particulate phases, respectively, are
Le.
- -dCc- dh
(4)
-
ufw€O
(5)
The gas phase velocity in the cloud-wake phase is readily obtained from eq 2 and 4. Note that U Q p will be negative (Le., backmixing of gas is predicted) if I; exceeds a critical value C,,, where
Gas Exchange Coefficients. Kunii and Levenspiel (1968a) adopted the equation of Davidson and Harrison (1963) for gas exchange, but applied it to exchange between
dCp dh
+
- KCP(CP - C C ) ~ B ~ C P D- CBU
+
fw)l
UGP
(10)
It is useful to write these three equations with the variables in dimensionless form, viz.
resulting in
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
339
and Dz. Inspection of expressions for all the parameters and of eq 2 , 4 , 5 , and 6 shows that it is necessary to specify values of the variables UO,U , uA, De, Do, eo, eg, j w JH , and k . Of these variables, bubble rise velocity may be related to other variables by the usual expression where A1,A z , etc., are as given in the Nomenclature list a t the end of the paper. Boundary Conditions. Under backmixing conditions (Le., when U is sufficiently large to cause downflow of particulate phase gas), the following boundary conditions apply. (a) At the distributor level (h = 0, z = 0) all bubble gas is considered to derive from the incoming gas c 1
= 1
(15)
UA
=
+ 0.7l(gDe)’”
U - UO
(20)
and bubble volume fraction is given by
It may also be more convenient to replace H with Ho, using
It follows that the remainder of the incoming gas must combine with downflowing particulate phase gas to constitute the cloud-wake gas
- UGpCp
+ (Lr - UGB)CO
=
UGCCC
i.e.
cz = Bl + BzC3
(16)
(see Nomenclature for B1 and Bz) (b) At the top of the bed (h = H , z = l),gas leaving the bed is considered to be made up of all the bubble gas and some cloudwake gas, with the remainder of the cloud-wake gas providing the downflowing gas in the particulate phase c 2
=
c3
(17)
The required expression for the exit gas concentration also results
U C H = UGBCB
+ (U - UGB)CC
Le.
(see Nomenclature for
D1 and Dz)
Theoretical Solution
The three simultaneous differential equations, (12), (13), and (14), have been solved on an analog computer; the twopoint boundary conditions were handled by incorporating automatic correction of an estimate of the value of C3 a t z = 0 until boundary condition (eq 17) was satisfied. This solution proved extremely useful for preliminary investigation of system behavior, but could not handle the full ranges of parameter values required, in that it became unstable with respect t o the value of CIS a t z = 0 for high values of A 7 ;in particular, solutions could not be obtained for high values of k and Ho. The differential equations may be solved analytically as in Appendix I, the solution taking the form
+ RzeXSZ+ RaeXa‘ + +
so that we now need to specify UO,I; (or C/Co), De, DG, BO, fm, Ho, and k . Equations 19 and 18 have been used to predict concentration profiles and conversions for some 40 sets of values of the relevant variables, covering a wide range of each variable. Figure 2 shows typical results and includes profiles for the “average” concentration, which would be expected if gas were sampled continuously a t a point in the reactor in proportion to the fractions of the phases present
Note the existence of minima in the C3 and C ,, profiles in Figure 2a, a phenomenon predicted also by Latham, et al. (1968), and to some extent consistent with the observations of Calderbank, et al. (1967). Such profiles are in marked contrast to the predictions resulting from any other fluid bed models. The profiles shown under the conditions of Figure 2b (for slower reaction and slower gas exchange) are even more unexpected ; the average concentration increases from the bottom of the bed to the top, the result of the dominance of the effect of reaction in the downflowing dense phase gas. Experimental work currently under way will measure concentration profiles to check these predictions of the backmixing model. Another factor to be investigated is the prediction of discontinuities in concentration (extremely large under some conditions) between Co and C,, a t z = 0 and between C,, and Cxatz = H . Comparison with Simplified Solutions. Kunii and Levenspiel (1968b, 1969) have presented two simplified solutions. (a) Omit the terms dCc/dh and dCp/dh in eq 9 and 10, and assume CH = CB a t z = 1 in place of eq 18. The solution, in terms of the nomenclature of this paper, is
where
C1 = RleXi’
CZ = alRleXir azR2ehSZ aaR3eXaz C3 = plRleX”
+ P2R2eXzz+
(19)
&R3eXsz
where R1, RZ, Rat X1, h2, X3, a,02, cy3, 81, 8 2 , and 8 3 are related (as shown in Appendix I) to the parameters A1 to A,, and B1 and Bz. Thus the profiles of the three gas concentrations may be obtained and the outlet gas concentration determined from eq 18 with z = 1,requiring values of the parameters D1 340 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
[
Yc
k
+
((1
+ Ye-l)-’]-‘}-’
- BOIKCP
s k = 0.5 5-1 Q=5 cm \= 50 cm &= 1 cm s-1
0
20
U/Q= 10
&= 1 &= 0.5
C
5
z W
q=0.2 c d s - 1
T
~
-0
DlMENSlONLESS HT,
ol 0
1D
z
ks0.1 5-1 De= 10 cm H g 5 0 cm U=, 1 cm s-I u/u.= 10 fw=1 k0.5 4=0.2 vn2s-1
z W
L 1.o
-0O"
DIMENSIONLESS HT, z
5
'
1
0
'
' 2
~
0
'
' 5
2
'
3
0
3
' 7 5
BUBBLE EQUIVALENT DIAM., D, , cm
U
J
k = 0.5 5-1 H0=50crn Uo=1 cm s-1
u/u,= IO = ,f
1
&=0.5
Q.0.2
cm s-I
Figure 4. Comparison of model predictions: variation of bubble size. Simplified solutions: (a) Kunii and Levenspiel; (b) Kunii and Levenspiel, slow reactions, small bubbles; (c) assuming C p = 0
Figure 2. Concentration profiles predicted b y the backmixing model
:$q k-0.5
u/y=10
v)
!& = e-Kf'
fw= 1
C30.5
co
n 0 z > w 0
where
Q=0.2 cfr9S-c
-
5( b)
Kunii and Levenspiel (1968b) also derived a result for very fast reactions, assuming that no reactant would enter the particulate phase. They considered such a case to be extremely rare. However, for fast reactions, a simplification of the analytical solution presented above may be considered. (c) For
8
3(a)
z
P VI K
> W
8
7
k = 0.5 s-1 De=5 cm Ho- 50 cm U o = l cm 5-1 &= 1 = 50& ,., &=0.2cds-l -
De=5 cm H,=50 crn Uo=1 cm 5-1 u/y=10 t=1 Co= 0.5
8-
5-1
4 - 5 cm b=50 cm
(b) For slow reactions and small bubbles, further simplification leads to
0
5
IO
15
20
u/uo
25
30
Figure 5. Comparison of model predictions: variation of incipient and fluidizing gas velocities. Simplified solutions: (a) Kunii and Levenspiel; (b) Kunii and Levenspiel, slow reactions, small bubbles; (c) assuming CP = 0
&=0.2 ur2 s-1
-00
RATE CONSTANT ,k, S-I
Q=10 cm Hy50 cm U G l cm 5-1
uiu,.
10
fw=1
&,=0.5 &=0.2 cm2s-1
0
0
0
"
"
'
I
1 2 3 4 RATE CONSTANT, k, S-I
5
Figure 3. Comparison of model predictions: variation of reaction rate constant. Simplified solutions: (a) Kunii and Levenspiel; (b) Kunii and Levenspiel, slow reactions, small bubbles; (c) assuming C p = 0
fast reactions, one might virtually expect all reactant gas transferring from the cloud-wake region to the particulate phase to react in that phase, Le., the concentration in the particulate phase would be very low. The approximation C p = 0 a t all positions in the reactor may be reasonable. Appendix I1 presents the resulting solution. Figures 3, 4, and 5 present comparisons of conversions predicted by the "rigorous" solution of this paper with those resulting from the three simplifications outlined above. Of the eight variables, the effects on the comparison of k , De, Uo, and U / F 0 are shown. An extensive survey has indicated that the effect of other variables on the comparison is very small within the ranges of values: Ho,25 to 200 cm; DG, 0.1 to 1.0cm2sec-1;co,0.4to0.6;f,,0.5tol.O. The predictions of the Kunii and Levenspiel solution (simplified case (a) above) are in agreement with those of the rigorous solution for small bubbles, but deviate considerably for Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
341
0
-
1
-1
-
0
4
I
RATE ; 2CONSTANT, ; 4 k,: s-l
y where
1
KBP
-
1
KEC
+-K1C P
The differences between predicted' conversions and between their relationships to variables (as indicated in Figure 7 and other similar comparisons) are large enough to suggest that experimental work conducted at selected values of operating variables may be able to distinguish successfully between the various models on the basis of overall conversion data alone. Summary and Conclusions
Simplified solutions to the backmixing model for fluid bed 342 Ind. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
9-10 cm y=50cm lb-1 cm 5-1 u / y 10
t=1
&=0.5
&0.2 c m w
E
solution of this paper
values of De larger than about 8 to 10 cm (Figures 3a, 4). The effect of all other variables on this comparison is small, although the influence of reaction rate becomes important a t larger values of D e . Figure 6 shows the combined effect of De and k on the comparison of predicted conversions. It is clear that the more rigorous solution should be used for De values greater than 10 cm. The further simplification for slow reactions and small bubbles (simplified case (b)) is obviously rarely applicable. I n general terms, k must be less than 0.1 sec-l and bubble equivalent diameter must be less than 1.0 cm for the result to approach that of the rigorous solution. The solution for C8= 0 (simplified case (c)) gives results close to those of the more rigorous solution for all but low reaction rates. For example, it is satisfactory for k > 3 sec-l a t D, = 5 cm, k > 1 sec-1 a t De = 10 cm, k > 0.5 sec-' a t D, = 20 cm, with other variables a t the values used in Figure 3. The agreement is improved at lower values of UOand U / U O but is considerably poorer a t higher values of these variables (see Figure 5). One further comparison of interest is that between the backmixing model and the simple two-phase models of Davidson and Harrison (1963). Figure 7 shows this comparison over limited ranges of values of variables; it includes predictions of the two-phase models with exchange rates as proposed by Davidson and Harrison (1963) and also with their exchange rate per unit bubble volume ( & / V B )replaced by REP,the overall coefficient resulting from the two-step transfer process of Kunii and Levenspiel (1968).
m
w >
= conversion by more rigorous
Figure 6. Comparison of model predictions: combined effect of reaction rate constant and bubble size
q backmixhg model
LI:
CQ) = conversion by Kunii & Leverspiel. simplified dution (a) C
-
0
0
1
2
3
4
RATE CONSTANT,k,
1 5-1
Figure 7. Comparison of rigorous solution of backmixing model with Davidson and Harrison two-phase models. Davidson and Harrison models: (d) well-mixed gas in particulate phase; (e) plug flow of gas in particulate phase; (f) well-mixed gas in particulate phase, Q replaced by K E p ; (9) plug flow of gas in particulate phase, Q replaced by KEP
catalytic reactors have been compared with the solution resulting when various simplifications are not made. The solution of Kunii and Levenspiel (simplified solution (a)) deviates significantly from the more rigorous solution for De > 10 cm. The further approximation proposed by the same authors for slow reactions and small bubbles (simplified solution (b)) is rarely applicable, giving a reasonable result only if k
