Application of a Porous Pellet Model to Fixed, Moving, and Fluidized Bed Gas-Solid Reactors James W. Evans* and Shinghai Song Department of Materials Science and Engineering, University of California, Berkeley, California 94 720
A previously developed model for the reaction between a porous solid and a gas is generalized to the case of time-dependent gaseous reactant concentration external to the pellet. The application of this generalized model to fixed bed, moving bed, and fluidized bed gas-solid reactors is then described. Results of calculations are presented in dimensionless form and are of sufficient generality to embrace a large number of cases which might be encountered industrially. Elementary graphical or numerical *techniques by means of which the results may be extended to other combinations of parameters are described. The work is intended to provide guidelines for the preliminary design of such reactors.
Introduction Recently several models have been developed for the reaction of a porous solid pellet with a gas by Ishida and Wen (1971), Calvelo and Smith (1970), Szekely and Evans (1971), and Sohn and Szekely (1972). These and other authors are unanimous in their rejection of the classical “shrinking core model” (or “shell progressive model” as it is sometimes known) as being unable to represent accurately the reaction of porous pellets. While the shrinking core model may be fitted to the experimental data on a particular pellet, its use in extrapolation to pellets of different structure (different porosity, etc.) is fraught with danger in that the model contains no allowance for structural properties. The model put forward by Szekely and Evans (1971) considers the porous pellet to be made up of a large number of spherical “grains” which are reacting topochemically. Figure 1 is a diagram of a spherical pellet reacting according to the grain model. The model allows for limitation of the progress of reaction by intergranular diffusion of gaseous reactant as well as by the chemical step at the shrinking reaction surfaces within the grains. Sohn and Szekely (1972) have presented a dimensionless version of this “grain model” and generalized it to include pellets which are spherical, cylindrical, or plate-like and which, in turn, are made up of spherical, cylindrical, or plate-like grains. These investigators were able to present their results in the form of a set of general curves, giving the extent of reaction as a function of time with a “general reaction modulus” as parameter. In the case of the plate-like grains, the local rate of reaction within the pellet is independent of the local extent of reaction, and the grain model becomes equivalent to an earlier one presented by Ishida and Wen (1968) and to a model recently put forward by Tien and Turkdogan (1972) for the reduction of iron oxides. While the results of such calculations are most useful for study of pellet reactions in the laboratory, they have been derived with the assumption of a constant gaseous reactant concentration external to the pellet. Consequently, they cannot be applied, a priori, to industrial reactors (fluidized beds, shaft furnaces, etc.) in which considerable variation in the gaseous reactant concentration encountered by the pellet may be expected. It is the purpose of this paper to show how the results for constant ambient concentration can be generalized to the case of time-dependent gaseous reactant concentration external to the pellet and, subsequently, to illustrate the application of the grain model to fluidized bed, moving bed, and packed bed gas-solid reactors. The grain model is also applicable to the reaction of a 146
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
solid with a solute contained within a solvent which completely fills the pores of the solid, provided the molar solute concentration in the solvent is low compared with the molar concentration of reactive solid within the pellet (molar density in the case of a pure solid) (Luss, 1968). Such conditions are frequently met in the leaching of ores.
Mathematical Formulation The assumptions made in the formulation of the grain model are discussed by Sohn and Szekely (1972) and will be merely listed here. The assumptions are as follows: (1) The pseudo-steady-state approximation is appropriate for describing the concentration of A within the pellet. (2) The resistance due to external mass transfer is negligible. (3) Diffusion within the pellet is either equimolar counterdiffusion or is at low concentration of the ’diffusing species. (4) The system is isothermal. ( 5 ) The solid structure is .macroscopically uniform and unchanged by reaction. (6) Diffusion of the gaseous reactants through the product layer of the individual grains is not rate limiting. (7) The “viscous flow” contribution to mass transfer within the pellet is negligible (Evans, 1972). The only significant change in the formulation for the generalized model is to replace the constant gas concentration external to the pellet, C A , ~ by , a fluctuating concentration, €A,&, where C A , O is the gas concentration at some arbitrarily selected point (e.g., the gas inlet point to the reactor) and 0 is a function of time or position in the reactor or both. The dimensionless equation describing the diffusion of the gaseous reactant within the pellet is, for first-order surface reaction V*Z* - .2+(Ft-’ =0 (1) where the Laplacian operator is in dimensionless form and the concentrations have been rendered dimensionless by division by ~ A , o . u is a dimensionless reaction “modulus” defined by
and is perhaps best considered to be a dimensionless pellet size. (See Nomenclature for meanings of symbols.) The equation for the reaction of a grain is a(/at* = - +e (2) where the dimensionless time t* is given by
(3) The extent of reaction of the pellet is given by
n
0 m
~ w w w w w w w w w w m ~ ~ ~ ~ w e - ~ ~ Z N N N N N N N N N N 1 , m+ N m wN d H 3.0.0 wNwm c9 ~ rl u3. m. m. m. m. m. m. m. m. m. m. m. d . . . N . . ri . 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 REACTINQ PARTICLE
n!
4
3
II
Ib
4
m. 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
'? IO
I
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
wwwwwwwwwwwwwwwwwwwww wwwwwwwwwwwwwwwwwwwww
? ? Q, Q, 0.
Q), Q,, Q), Q)
Q!
? Qi ? ? a? ? ? ? Qi
Q)
w
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
E N L A R G E D VIEW AT LOCATION x
Figure 1. The grain model for gas-solid reactions. . _
000000000000000000000
/s
,
W d we-d0mQ,00w0Q,d0Q,dm0wc1N m OWwd~wwm~,wrle-NmOu,HWNwm w.e -. c1. e-. e-.w .w .w .m .y '? 9,-1c? N, *,rl. 0. 0.
(4)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
which may be differentiated to give i i i i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
lqFp-lFg(Fg-lII, dv
0 I
O
L1v
dv
~ p - 1
Let us write this
9
m
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
c9
dX/dt* = 0p (6 1 where p is a function of X,a, F,, and F,. Due to the linearity of the equations, fl depends on fl only through the dependence of X on 8. Consequently, we may produce a table of values of fl as a function of X, C, F,, and F, by calculations with the concentration external to the pellet held constant (at eA.o)from
rl
i00000000000000000000
n!
?a II
rl
Ib
b
4
and subsequently use this table of fl values in the righthand side of (6) for the more general case, where the concentration external to the pellet fluctuates. Values of /3 are presented in Table I for F, = 3 (spherical pellet), Fg = 2 and 3, and a range of values of X and 6 = u / d Z F p F g .For values of 6 outside the range given, the value of p may be approximated by (Sohn and Szekely, 1972)
c? IO
forF, = 3. Calculation of the fl values was performed using a slightly modified version of the computer program pre0 m 0 u, 0 m 0 m 0 m 0 m 0 In 0 m 0 u, 0 19 0, by Evans (1970). The usual tests were performed 0 0 r i r l N h l m m d d m m w w e - ~ w ~ Q , ~ sented ? for insensitivity of the results to change in the step size 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 used in the finite difference approximations. The values have two significant figures of accuracy. Table I also inInd. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
147
cludes /3 values for F, = 1; these have been calculated using an analytical solution to the governing equations (Ishida and W,en, 1968)
P
-
coth u
3F,(:
1‘
U2
Table 11.
a F.
=
2
Fg = 3
for Fp = 3 and
0.3
0.5
0.8
1.2
1.8
3.0
1.87 2.72
1.69 2.37
1.40 1.88
1.10 1.42
0.808 1.02
0.522 0.652
or
P =
- 3 ~ 2+ ( ~ / u ~ ) [ u m Qth
+
-
11 (3/a2)[a coth ( U T ) - a2q cosech2 (q)J -(a2/3)(1 ~ ) ( l 27) (a2/3)(1 1)’ [UT coth (q) - 11 (1 $[a m t h (UT) aZq cosech2 (UT)] (UQ)
+
+
-
+ -
(11)
for F , = 3, F , = 1, and X L 3(l/u) coth u - l / u z where 7 is given by X =1
- v3 +
(37/u2)[aq m t h ( U T )
-
13
Equation 9 also applies for Fg = 2 and 3 a t X = 0 and yields the values for /3 listed in Table JI. Comparison of these analytical values with the numerical values in Table I lends additional credence to the figures in Table I. Application to Fixed Bed Reactor The model used for a packed bed is a familiar one of plug flow of gas, isothermal behavior, uniform spherical pellets, and negligible diffusive or dispersive mass transfer in the axial direction. Furthermore, it will be assumed that mass transfer to the pellet surface from the gas stream contributes a negligible “resistance” to the progress of reaction. A differential material balance on the gaseous reactant yields
-G
-a8=
(1
- Q(1 b
aY
G)P,
dt
+
(fp
+
ae
-&(12)
where G, is the superficial mole velocity of the gaseous reactant (superficial linear velocity of the gas stream multiplied by the molar density of the inlet gas or multiplied by molar concentration of gaseous reactant in the inlet gas should the gas not be pure). An appropriate choice of ~ A , O in this case is the inlet concentration, whereupon the boundary condition on the above equation becomes
8
=
1at y = 0
(13)
Except at elevated pressures, the third term in eq 12 is small compared with the second and may be neglected. Defining a dimensionless coordinate, y*, given by
12 becomes
with obvious boundary condition and initial condition (assuming the gas residence time is negligibly small compared with the time for pellet reaction) X=Oatt* = O , y L O (16) Equation 15, together with the boundary condition, can be manipuiated into the form 148
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
where X is a “dummy variable.” The numerical solution of eq 17 with initial condition (16) and /3 values from Table I is a matter of applying elementary techniques, such as Simpson’s rule and the Runge-Kutta procedure. Some numerical results are presented in Figures 2-4 for the case of spherical pellets made up of spherical grains. Curves for beds having values of i less than 0.3 lie very close to those of Figure 2, as might be expected from the insensitivity of /3 to ;a t low values of ;(see eq 8). As might have been anticipated from the form of equation 15, Figure 2 shows that after an initial period (roughly after t* = l ) , the plot of the extent of reaction within the bed takes on a sigmoid shape, which then travels down the bed with unvarying profile and a constant dimensionless velocity of unity. Reaction is confined to a fairly narrow zone in the bed. Figure 3 illustrates similar behavior for the case i = 1, except that now the width of the reaction zone is considerably larger. For i = 3 (Figure 4), the reaction zone extends throughout the whole of the bed (although, of course, for a longer bed, the reaction zone might well occupy only part of the bed). As already mentioned, the reaction modulus ;can be interpreted as a dimensionless pellet size. As a consequence, pellets with small values of i might be expected to react rapidly compared with pellets of large i due to less hindrance of reaction from intrapellet diffusion in the smaller pellets. This behavior is seen in the entrance regions of the bed on comparing Figures 2-4. The extent of reaction at y* = 0 and (say) t* = 1 is highest for i = 0.3 and lowest for ;= 3. Deep within the bed, however, the situation is reversed. For example, at t* = 1 and y* = 3, the extent of reaction is highest for 6 = 3 and lowest for 6 = 0.3. This may be rationalized on the grounds that for a bed made up of the smaller, more reactive pellets, the gaseous reactant is depleted close to gas inlet. Consequently, pellets remote from the gas inlet have no opportunity to react until the pellets closer to the inlet are almost completely reacted. At values of i higher than about 3, p becomes approximately proportional to l / i 2 (see eq 8). As a consequence, a plot of X against y * / 2 with t * / 2 as a parameter yields the same curves for different values of 6 I 3. Figure 4 can, therefore, be used to predict the initial stages of reaction of beds with ;> 3 provided the abscissa and parameter are redefined as indicated in parentheses. Application to Moving Bed Reactors Here, assumptions identical with those described under application to fixed beds are made, together with the additional assumption that the solids move through the reactor (of uniform cross section) in plug flow. Attention will be restricted to steady-state behavior. The differential equation describing the concentration of gaseous reactant is (18)
Y*
Figure 2. Fixed bed reactor. Extent of reaction us. position in bed for reaction modulus = 0.3.
X
Figure 4. Fixed bed reactor. Extent of reaction us. position in bed for reaction modulus = 3.0.
0
-
OR
'04-
Ia
0
20
3.0
4.0
( Y V W
e-----
'-
50 0 01
Y*
Figure 3. Fixed bed reactor. Extent of reaction us. position in bed for reaction modulus = 1.0.
A similar differential material balance on the solid reactant yields
O'
10
IO
Y
Figure 5. Cocurrent moving bed. Extent of reaction and gaseous reactant concentration us. position in bed for gas flow equal to the stoichiometricrequirement.
(20) where G, is the mole velocity of the solid reactant and has a positive value for solids flow in the same direction as the gas. Let
bG,JG,
R The boundary conditions on (22) are cocurrent flow: X = 0 a t y* = 0 countercurrent flow: X = 0 at y* = L* where
L*
E
4Go(1 -
=
6,)(1
-
6")
GP,V,
(21) -
(22)
Figure 6. Cocurrent moving bed. Extent of reaction and gaseous reactant concentration us. position in bed for gas flow equal t o twice the stoichiometricrequirement.
L
from stoichiometry
x - xo= - R(e -
1)
(23)
where Xo is X a t y* = 0. Substituting (23) in (20) and integrating give for cocurrent flow
where flow
x
I
is a "dummy variable," and for countercurrent
These integrals can be readily integrated, given Table I, using Simpson's rule or Gauss quadrature.
Figures 5-7 are the results of calculations for the cocurrent case with spherical pellets made up of spherical grains. In Figure 5 , the gas feed rate to the bed is set equal to that required stoichiometrically for complete reaction of the solid feed ( R = 1).The sigmoid shapes of the curves arise from the semilogarithmic plot; reaction rates are highest a t the entrance to the bed, as would be expected. As might have been expected, a shorter bed length is required to bring about a given extent of reaction of pellets or gas with pellets of smaller i values. In Figure 6, twice the stoichiometrically required feed rate of gas is used. In agreement with expectation, the extent of reaction of the solid a t a given position in the bed is increased above that for R = 1. Figure 7 presents results for an undersupply of gas to the reactor, a situation which might be encountered where Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
149
I
001
0.1
YX
10
0 01
10
Figure 7. Cocurrent moving bed. Extent of reaction and gaseous reactant concentration us. position in bed for gas flow equal to half the stoichiometric requirement.
1
10
01
IO
L*
Figure 9. Countercurrent moving bed. Extent of reaction and gaseous reactant concentration us. position in bed for gas flow equal to twice the stoichiometric requirement. I n.
‘7
I
01 0.01
I
I
O.1
L
.-. .*...-
I
1.0
,
L
Figure 8. Countercurrent moving bed. Extent of reaction and gaseous reactant concentration us. position in bed for gas flow equal to the stoichiometric requirement.
Figure 10. Countercurrent moving bed. Extent of reaction and gaseous reactant concentration us. position in bed for gas flow equal to half the stoichiometric requirement.
high gas conversions rather than high solids conversions are sought. Figures 8-10 are concerned with reactors through which the solids pass in countercurrent flow to the gas. Comparison with the corresponding Figures 5-7 reveals a significantly greater conversion for solids leaving a countercurrent reactor than a comparable cocurrent reactor, especially at high conversions. Because of the asymptotic behavior of /3 at extreme values of & (see eq 8), the curves for & = 0.3 in Figures 5-10 nearly coincide with curves for 6 < 0.3. Also, a plot of 0 and X against y * / 2 (for the cocurrent case) or OF and X F against L*/? (for the countercurrent case) for any G > 3 yields a curve nearly coincidental with a similar plot for & = 3. Figures 5-8 are, therefore, useful for any value of 6 .
ticle size and low gas conversion has been considered by the authors in an earlier publication (Evans and Song, 1973)) and that the bed is isothermal. Finally, it is assumed that mass transfer external to the particle contributes negligible “resistance” to the progress of reaction and that particle residence time in the reactor is independent of particle size. As a consequence of the assumption of perfect mixing of the solids, the mean gas concentration encountered by the particle during reaction can be equated to the spatial mean gas concentration within the dense phase external to the particles. In the case of fluidized beds, it is convenient to choose this as the concentration C A , ~which , choice leads to a value for 0 of unity. In pragmatic terms, this ideal of perfect mixing will be approached when the particle makes many “cycles” of the bed during the course of reaction. The choice of CA,Oas a mean gas concentration within the emulsion, then, leads to the grain model eq 1 and 2 (and associated boundary and initial conditions) becoming identical with those of Sohn and Szekely for the case of constant concentration external to the particle. Consequently, Sohn and Szekely’s results can be applied to the fluidized bed. In particular, their approximate equation
Application to Fluidized Bed Reactor
To a first approximation, a gently bubbling fluidized bed containing only large bubbles can be considered to be made up of well-stirred solid particles between which the fluidizing gas is passing in plug flow a t minimum fluidizing conditions. Gas passing through the bed in the form of bubbles makes an insignificant contribution to reaction. This behavior is assumed for the model discussed below. More sophisticated models, for example, the bubbling bed model (Kunii and Levenspiel, 1969), would, no doubt, yield more accurate results, but a t the expense of introducing further parameters (e.g., bubble size), and the purpose of this paper is to illustrate the application of the grain model to gas-solid reactors rather than to be all embracing. It is further assumed that all the solid particles are spherical and of uniform size (the case of nonuniform par150
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
t*
h
g(X)
+
&(X)
(26)
(1
-
(27)
where
g(X)
= 1
-
and p ( X ) = 1 - 3(1 - X ) 2 ’ 3
-
-
+
2(1
X)’$Fg
-
X ) (for F , = 3) (28)
is most useful. This equation is asymptotically correct for i m or & 0,is exact for X = 1, and has a maximum
08-
E0
2
o
4
F
/ R = 2 0
01
I
I
10
01
10
I
100
1;
LR
Figure 11. Fluidized bed. Extent of reaction us. dimensionless solid residence time t ~ * . 10
Figure 13. Fluidized bed. Extent of reaction us. dimensionless solid residence T R * . Gas flow equals twice the stoichiometric requirement.
08
-
06 06
X
04
02
01 01
I
I
10
rR
10
J
IO0
10
Figure 12. Fluidized bed. Extent of reaction us. dimensionless solid residence time T R * . Gas flow equals the stoichiometric requirement.
error (at i between 1 and 2) of about 10% for extents of reaction greater than 5070, compared with a numerical solution of (l),(Z), and (4). Rewriting eq 26 formally, the extent of reaction of a single particle after spending a time t* in the reactor is given bY
x
= f(t*,(T)
(29) (In practice, the evaluation of X , given t*, requires solution of eq 26 by interpolation.) The weight fraction of solids having residence time between t* and t* + dt* is given by
1
E(t*) dt* = t R * exp( - t*/tR*) dt*
Consequently, the mean extent of reaction of the solids leaving the reactor is given by i m E ( t * ) f( ~ , t *dt* )
I
IO0
Figure 14. Fluidized bed. Extent of reaction us. dimensionless solid residence time TR*. Gas flow equals half the stoichiometric requirement.
mensionless nominal residence time (33) where CA,OIis the gaseous reactant concentration a t the inlet to the reactor. T R is a more meaningful measure of reactor size than t ~ * . From assumption of plug flow of gas through the emulsion (34)
(30)
where tR* is the dimensionless nominal residence time of the reactor
X(i,tR*)
IO r R
R = molar feed r a t e of gas to emulsion phase of bed x b molar feed rate of solid to bed molar feed r a t e of gas to bed a t minimum fluidizing conditions X b N molar feed r a t e of solid to bed Consequently
(32)
This integral may be readily evaluated by Laguerre integration. Results are presented in Figure 11, X being plotted us. the dimensionless nominal residence time, tR*, with i as a parameter, for the case of spherical pellets made up of spherical grains. While Figure 11 has the advantage of being a general relationship between extent of reaction and dimensionless nominal residence time, it suffers from the disadvantage that its use requires a knowledge of CA,O,a quantity which is not known a priori. Consider, therefore, an alternative definition of a di-
(35) The curves appearing in Figures 12-14 have-been calculated from Figure 11 using eq 35. Curves for u < 0.3 almost coincide with the curves for i = 0.3 in Figures 11-14. If the curves of X against t R * (Figure 11) or against T R (Figures 12-14) are- replotted with t R * / i 2 or T R / ? (respectively) as abscissa, then curves for ;> 3 nearly coincide with the curve for ;= 3. Figures 12-14 show a higher conversion of solids for a given dimensionless reactor residence time for pellets of Ind. Eng. Chem.. Process Des. Develop., Vol. 13,No. 2, 1974
151
lower reaction modulus, i. Recalling that the modulus can be interpreted as a dimensionless pellet size, this is to be expected. Furthermore, increasing the gas flow above that required for complete reaction increases the extent of reaction of the solids leaving the reactor, again in line with expectations. While the above discussion has been confined to a single fluidized bed reactor, there is no difficulty in extending the approach to multiple beds. Conclusion The grain model for the reaction of a porous solid pellet with a gas has been generalized to the case of time-dependent ambient gaseous reactant concentration. The rate of reaction of the pellet under these circumstances is given by a product of dimensionless concentration and a second quantity, p, which is akin to the effectiveness factor of heterogeneous catalysis. A table of p values has been presented for a range of values of the reaction modulus, i, and an analytical approximation, valid outside this range, is given to enable calculation of (3 at other values of i. The application of this generalized grain model to fixed bed, cocurrent and countercurrent moving bed, and fluidized bed gas-solid reactors was then demonstrated. The results of calculations for such reactors are presented for spherical pellets made up of spherical grains and reaction modulus values between i = 0.3 and i = 3.0. Simple graphical procedures are described for extending the results to values of i outside this range. Mention is made of the elementary numerical techniques which may be used to calculate extents of reaction for other grain geometries (or in the case of moving beds, for other gaseous reactant feed rates) given the table of @ values. Rather simple models have been used for the flow of gaseous reactant (and solid in the moving and fluidized bed cases) through the reactor, but of course, there is no conceptual difficulty in applying the grain model to more sophisticated flow models. The purpose here has been to illustrate the power and generality of the grain model as modified to allow for a time-dependent gaseous reactant concentration external to the pellet. It is hoped that the results will be of value in the preliminary design and optimization of industrial shaft furnaces, fluidized bed reactors, and the like. Nomenclature A, = grain surface area A, = external surface area of pellet b = stoichiometric coefficient (moles of solid reacting per mole of gas reacting) C,A,o = gaseous reactant concentration external to pellet at some arbitrarily chosen point in reactor C A , O I = gaseous reactant concentration of reactor inlet
152
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2,1974
De = effective diffusivity of gaseous reactant within pellet Fg = shape factor for grain (3 for sphere, 2 for long cylinder, 1for plate) F , = shape factor for pellet G , = superficial mole velocity of gaseous reactant through reactor G , = superficial mole velocity of solid reactant through reactor g(X) = function defined by eq 27 k = chemical rate constant L, L* = reactor length, dimensionless reactor length p(X) = function defined by eq 28 R = factor defined by eq 21 t , t* = time, dimensionless time t ~ t ,~ *= nominal residence time of reactor (with respect to solids), dimensionless residence time V , = volumeofgrain V , = volume of pellet (including pores) X = extent of reaction of solid X = mean extent of reaction of solid leaving reactor y , y* = distance down reactor measured from gas inlet, dimensionless distance down reactor p = “effectiveness factor” given by eq 7 t and t, = porosity of pellet t V = void fraction of bed = dimensionless radial coordinate within pellet (except for eq 11) 0 = dimensionless gaseous reactant concentration external to pellet E = dimensionless position of reaction front within grain p s = true molar density of solid o and i = “reaction moduli” (defined following eq 1 and 7) 7 R = dimensionless reactor residence time (defined by eq 35 1
$ = dimensionless gaseous reactant concentration within
pellet Literature Cited Calvelo. Symp Evans, J Evans, J
A , . Smith. J. M . . R o c . CHEMECA. Svdnev , . (Inst. Chem. Eng. Ser , No 331, 1 (1970) S , Song, S , M e t Trans , 4, 1701 (1973) W , Ph D Thesis State University of New York, Buffalo, N Y ,
1970
E v a k - J . W., Can. J. Chem. Eng., 50, 811 (1972). Ishida, M., Wen, C. Y . , AIChEJ., 14, 311 (1968). Ishida, M., Wen, C. Y., Chem. Eng. Sci., 26, 1031 (1971). Kunii. D.. Levenspiel, O.,“Fluidization Engineering,” Wiley, New York, N . Y . , 1959. Luss. D.,Can. J . Chem. Eng., 46, 154 (1968) Sohn, H. Y . , Szekely, J., Chem. Eng. Sci., 27, 763 (1972). Szekely, J., Evans, J . W., Chem. Eng. Sci., 26, 1901 (1971). Tien, R. H.. Turkdogan, E. T., Met. Trans., 3, 2039 (1972).
Received f o r reuieu! July 17, 1973 Accepted November 23,1973
The calculations described in this paper were made possible by a gift of computer time from the Computer Center, University of California, Berkeley, Calif. Support of this research by the National Science Foundation under Grant GK-37461 is gratefully acknowledged.
