330
Ind. Eng. Chem. Process Des. Dev. 1984,23, 330-334
Reactions of Particles with Nonuniform Distribution of Solid Reactant. The Shrinking Core Model M. P. Dudukovi6 Chemical Reaction Engineering Laboratory, Uepartrnent of Chemical Engineering, Washington Universe, St. Louis, Msouri 63130
Reaction of particles that have a nonuniform distribution of the reacting solid component (coke, active adsorbent) and follow the shrinking core model is examined. The relationships between solid reactant conversion and time differ from those for a particle with uniform solid reactant distribution. Tlmes required for complete conversion can be several times higher or lower, depending on the solid reactant distribution, than the one necessary for complete reaction of a uniform particle. Geometric instability of the reaction interface for all particle shapes may be prevented with proper profiles of the solid reactant.
Introduction Gas-solid noncatalytic reactions are often encountered in chemical and metallurgical industry. Reduction of various ores, adsorption of pollutants, and catalyst regeneration are some of the common examples. When the solid particle is nonporous, or at high temperature, the reaction rate can greatly exceed the rate of gas diffusion into the solid reactant and the reaction takes place in a narrow zone which moves inward in time. This situation is well approximated by the popular unreacted shrinking core model which, for a first-order reaction, has been described in standard reaction engineering textbooks (Levenspiel, 1972; Smith, 1981). Although numerous extensions of the model have been proposed, including treatment of various reaction orders (Szekely et al., 1976) and different non-nth order rate forms (Szekely et al., 1976; Erk and DudukoviE, 1982), all approaches consider the solid particle to have a uniform distribution of the solid reactant. This simply is not the case in many situations. A definitive profile of the active adsorbent is often found in a solid particle which consists of an inert solid support matrix due to the impregnation procedure used to deposit the active component. The coke profile in a deactivated catalyst to be regenerated is frequently far from uniform. This profile is strongly affected by the operating conditions under which the catalyst was used and can be very steep with the highest coke concentration at the surface of the pellet for a high-temperature catalyst. In contrast, in a low-temperature catalyst coke may be preferentially deposited in the center of the pellet when consecutive reactions are responsible for coke formation. While coke burnoff in the shrinking core regime received attention (Weisz and Goodwin, 1963) no attempt was made to account for the nonuniform coke distribution. Finally, ore concentration in rocks to be processed is generally nonuniform also. The objective of this paper is to examine what effect the nonuniformity of the solid reactant profile in the particle has on conversion-time relationship under the conditions when the shrinking core model is valid. Of particular interest is the development of expressions for reaction time required for complete conversion and their comparison to the equivalent expression for a particle of the same size and with the same amount of solid reactant but with uniform solid reactant distribution. Model Development A general gas-solid noncatalytic reaction of stoichiometry given by eq 1 is considered.
The following assumptions are made: (i) Diffusion into the solid reactant is much slower than reaction rate and the shrinking core model can be used. (ii) Particle size and shape do not change with reaction. (iii) Diffusion in the solid product layer is in the Knudsen regime. (iv) Pseudo-steady-state assumption can be used. (v) Reaction is first order in gaseous reactant but not dependent on local solid concentration. (vi) There are isothermal conditions. It is also assumed that the profile of the solid reactant concentration is known and is a function of one position variable only, e.g., radial position in a spherical or long cylindrical particle. Some examples of solid reactant distribution are given in Figure 1. The mean solid reactant concentration, C,, in a particle with nonuniform reactant distribution is given by
The same reactant concentration, CBo,would be present in a particle of the same shape and size with a uniform solid reactant distribution and with the same amount of solid reactant as contained in the particle with nonuniform distribution. We can then define the pointwise dimensionless solid reactant concentration by (3)
and for properly normalized profiles due to eq 2 it is now required that (4)
The governing equations for the shrinking core model can be written in dimensionless form as
l = 1 ; -1-dY - - 1-y Bi dE
0196-4305/84/1123-0330$01.50/00 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984 331
Dimensionless time 0 is based on characteristic kinetic reaction time and is defined by
bk C
&CAE
aCB$
acB,(vp/sex)
e=BApt=
(v
t
(10)
+ 1)
Integration of eq 8 yields an expression for dimensionless reaction time, 0, as a function of the position of the unreacted shrinking core, 5,
POSITION
where
,I
Solid reactant conversion can be related to the position of the unreacted core by
PARAMETER :p
X
= (v
+ Wu(Ec,l) = 1 - (v + l)Fu(O,tc)
Equation 4 gives the value Fu(O,l)= (v + quired for complete reaction, Be, is given by
0
0.2
0.4
0.6
0.8
..' N
Oe
PARAMETER:
A
4
0
a
0.2
04
e
0.6
0.8
1.0
Figure 1. Dimensionless solid reactant profiles of Table I11 in a spherical particle.
Parameter v takes the following values: v = 0 for a slab, v = 1 for a long cylinder, and v = 2 for a sphere. Equation 5 represents reactant gas diffusion through the solid product layer. Equation 6 equates the mass transfer rate across the boundary layer to the diffusion rate in the solid particle, and eq 7 asserts that the diffusion flux at the dimensionlessposition of the solid reactant core, Ec, equals the kinetic rate per unit surface of the core. In this manner the possibility of all three resistances in series: film mass transfer, solid product diffusion, and surface kinetics is accounted for. The mass balance on the unreacted solid results in
where yc = y ( t = E,, 0) is the dimensionless gas reactant concentration at the core and can be readily obtained from eq 5-7
Time re-
(14)
1.0
t
(13)
=
eek
+ Oef + Oed
(144
For the case of a cylinder, the L'Hospital rule has to be applied to the last term on the right of eq 11 and eq 14 to get a workable expression. Equation 14 illustrates the additivity of reaction times required for complete conversion. The first term on the right is the time required if surface kinetics controlled the rate, eek, the second term is the time required when film mass transfer controls the rate, e,, and the last one is the time necessary for complete conversion if diffusion through the solids product layer controls the rate, Ocd. Similarly, the three terms on the right-hand side of eq 11 give the contribution of kinetic resistance, film mass transfer resistance, and diffusion through solid product resistance, respectively, to the reaction time necessary to move the shrinking core from the outer surface at E = 1 to position tC.Using eq 13 one can relate reaction time to solid conversion, X.
Examples and Discussion In order to appreciate the formulas (ll),(13), and (14), it is instructive to consider the examples of possible solid reactant distribution in a spherical particle presented in Figure 1. The distribution of the solid reactant concentration and the integrals necessary for evaluation of reaction time by eq 11 are tabulated in Table I. The expression for the time required for complete conversion is given in Table 11. Examination of Table I1 and eq 14 reveals that the time required for complete conversion when only external mass transfer controls the rate is independent of the solid reactant concentration profile for any particle geometry. This is to be expected since for a reaction rate which is independent of solid reactant concentration the time required t~ completely react a given amount of solid reactant, when film mass transfer controls the rate, is independent of solid reactant distribution and only affected by the gas
332 Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
4
I+
n -t,
^w o1 -+
I" I i
I
si
l
21
3
U
h h
w u
I
I:,
-
v
B
c:
-.
h
ri
-.
++
h
-..5+a + + h
"+
4
3-
+
I +
+ a
4
4
+
h
3r i
9
I
+
+
P
a
v
4, w
ri
+
4
la I
' -. -. +
ri
+
I
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
333
Table 111. Effect of Solid Reactant Profile on Time Required for Complete Conversion o f a Spherical Particle (v = 2 ) solid reactant profile, z ( E ) kinetic control, (ee,/e,,)k diffusion control, (Oez/ee,)d
-
(T3)
(P + 3)@ + 2)(P + p ( p 2 + 6 p + 11)
@
p 2 i 6p + 5 p 2 + 6 p + 11
p+3
pt5
P+ 1
P+ 2 2 [ ( ~ (+1 e - " ) - 2(1- e - " ) ] "(a - 2 ) t 2 ( 1 - e - " )
+ 2)(P + 3) p 2 + 6 p + 11
- (1 - E ) p ]
(T4)y ( l - t p ) (T5)
(a3/3)e-"(I-&) - 2) + 2 ( 1 - e-")
( a 2 / 3 ) ( 1- e - " ) "(01 - 2 ) + 2(1 - e - " )
reactant concentration on the outside surface which is determined by the film resistance. Under such conditions Da a but Da/Bi remains finite, and proper dimensionless time is measured in units of mass transfer time
-
O/Da. Table I1 also demonstrates that when surface kinetics controls the rate, the time for complete conversion is independent of the solid reactant concentration profile only for a particle of a slab shape, i.e., v = 0. For geometries of the particle other than a slab even when surface kinetics controls the rate, time for complete reaction, Oek, depends on the solid reactant profile. This is a direct consequence of eq 4 and 8 where in the later one yc = 1for the case of kinetic control. Time for complete conversion, Oez( ), is given by J&(t)d t and is, by eq 4, independent of the profile only for a slab, i.e., v = 0. The ratio of times required for complete conversion for a particle with a nonuniform and uniform solid reactant distribution, (OeZ(€) / e,,), is presented in Figure 2 for solid reactant profiles of Table I1 and for spherical particle geometry (v = 2). Time for complete conversion when diffusion through solid product layer controls the rate, Bed, is always affected by the solid reactant profile for particles of all geometries. The ratio of times required for complete conversion for a particle with nonuniform and uniform solid reactant distribution, (Oe2(t)/Oel), is presented also in Figure 2. The summary of the times required for complete conversion under kinetic or diffusion control for a spherical particle is given in Table 111. Dimensionless solid reactant profiles represented by expressions (Tl) and (T2) in Table I11 approach a uniform profile of unity when p 0. Profile (T5) in Table I11 also approaches unity as parameter cy tends to zero. Hence, the ratio (O,,(,,/O,,) for either kinetic or diffusion control for these reactant distributions always is at one, for p or a equal to zero as seen in Figure 2. As p or a become very large, profiles (Tl) and (T5), respectively, concentrate more and more of the reactant in the region close to the outer particle surface (Figure 1). In the limit as p a (Be(o/Oel)k = 1/3 and (Oez(t)/Oel)d = 0. On the other hand, asp becomes very large, profile (T2) concentrates more and more reactant in the particle center (Figure 1) and hence the ratio (Be2(6)/0,1) for both kinetic and diffusion control tends to infinity as p m. Profiles (T3) and (T4) of Table I11 in the limit as p approaches zero represent a logarithmic solid reactant profile given by 6/11 In [ l / ( l -E ) ] for (T3) and 6 In (1/€) for (T4). As p becomes very large both profiles become uniform and tend to unity. This is reflected in Figure 2, where the ratio (Oez/Oel) for both kinetic and diffusion control approaches 1 as p becomes very large. The important message of Figure 2 is that the time required for complete conversion of a particle with nonuniform solid reactant distribution can be many times larger
-
-
-
DIFFUSION CONTROL
0
1
2
3
4
5
4
5
P or a
DIFFUSION -KINETIC
*JI
-a
CONTROL CONTROL
6
~
0
1
2
3
P
Figure 2. Comparison of times required for complete conversion of solid spherical particles with nonuniform and uniform reactant concentration for reactant distributions of Table 111.
or smaller than reaction time of a particle of the same size with the same total amount of reactant but with uniform distribution. The discussion so far has been based on the assumption that the shrinking core model is actually followed. No consideration was given to the stability of the reaction interface. It has been shown some time ago (Beveridge and Goldie, 1968; Ishida and Wen, 1968) that geometric instability can occur for a spherical particle and first-order reaction at 5, I0.5 or X 1 0.875 at infinite Biot numbers and at much lower conversions at smaller Biot numbers. The necessary condition for geometric instability can, for our purpose, be represented by
The criterion simply states that the shrinking core interface may become unstable if the rate of shrinking of the core accelerates as the thickness of the product layer grows. Under such conditions nonhomogeneities in the particle could cause an initial unevenness in the reaction front which in turn triggers even higher rates at points of deeper
334
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 2, 1984
penetration causing an ever increasing unevenness of the reaction interface. When a nonuniform solid reactant concentration profile is present the necessary condition for avoiding geometric instability can be written from eq 15 by using eq 8 and 9 as
(16) For a spherical particle, u = 2, of uniform solid reactant concentration, z = 1, eq 16 predicts that geometric instability cannot occur at 5, > 1 / [ 2 ( 1- l / B i ) ]when Bi > l , which is the result of Ishida and Wen (1968). Equation 16 also shows that the potential for geometric instability in a spherical particle can be avoided at all conversions when d In z Flt.'O -Da
This can readily be accomplished by a solid reactant concentration profile that distributes the reactant preferentially toward the center of the particle such as (T2) of Table III. This profile would avoid geometric instability in a sphere at all conversion levels whenever p > Da. In contrast, eq 16 predicts that geometric instability for a slab, u = 0, of uniform solid reactant distribution, z = 1, can never occur. A solid reactant profile such as (Tl) of Table I11 could cause geometric instability at E, Ip [ 1 + Da(1 + l / B i ) ] / D a(1+ p ) if p C D a / ( l + Da/Bi) or at all levels of conversion if p > D a / ( l + Da/Bi). This shows that a properly selected solid reactant profile can prevent geometric instability in all particle geometries. Extension of the above concepts to nonisothermal situations, multiple reactions, and particles of changing size is possible. Conclusions When accurate predictions of times required for complete conversion of solid particles which undergo reaction according to the shrinking core model are desired, solid reactant concentration profile in the particles must be known. If reaction times are predicted based on reaction of solid particles with uniform distribution of the solid reactant and of the same mean concentration, large errors can result. This may lead to unnecessarily high or low estimates of required solids retention times in catalyst or adsorbent regenerators, ore processors, and other reactor vessels. For example, for a parabolic dimensionless solid reactant profile (Tl) of (5/3)t2 which concentrates the reactant toward the outer surface in a spherical particle the reaction time required for complete conversion is 5/9, under kinetic control, and 2/6, under product diffusion control, of the time required for conversion of an equivalent particle of uniform solid reactant profile. The extension of the shrinking core model to particles with nonuniform solid reactant concentration is presented here and should be used in predictions of conversion or times for complete conversion. The times required for complete conversion vary from those predicted for particles with uniform solid reactant distribution even in case of kinetic control for any particle of varying cross sectional area. Solid reactant profiles can be determined by electron microprobes or can be predicted based on the deactivation
or impregnation process that was used to deposit the solid reactant. Geometric instability of the reaction interface can be prevented with proper solid reactant distributions in particles of spherical or cylindrical geometry. On the other hand, geometric instability, which cannot occur in particles of uniform solid reactant concentration of a slab geometry, can be induced by solid reactant concentration profiles which distribute the reactant preferentially a t the outer surface. Nomenclature A , B , G, S = chemical species of eq 1 a , b, g , s = stoichiometric coefficients of eq 1 Bi = k,R/D, = Biot number for mass transfer CA = gas reactant concentration C h = gas reactant concentration in the bulk CB = solid reactant concentration CBo= mean coverage solid reactant concentration in the unreacted (fresh) particle De = effective diffusivity in the solid product layer Da = k$l/De = Damkohler number for surface reaction F,,= integral defined by eq 12 k , = film mass transfer coefficient k , = surface reaction rate constant based on unit surface N b = initial total amount of solid reactant in the particle p = parameter used in Tables 1-111 R = half-thickness or radius of solid particle r = actual position in the particle S,, = external area of the particle t = time V = volume of the particle solid conversion y = CA/CA, = dimensionless gas reactant concentration yc = dimensionless gas reactant concentration at the shrinking core
x"=
z = CB/CBo = dimensionless solid reactant concentration
Greek Letters a = parameter used in the tables 6 = b k B C A ~ / a C B o=Rdimensionless time
Be = dimensionless time required for complete conversion = dimensionless time required for complete conversion when diffusion through solid product controls the rate Bef = dimensionless time required for complete conversion when film diffusion controls the rate Bek = dimensionless time required for complete conversion when kinetics controls the rate Beif) = dimensionless time required for complete conversion or a nonuniform particle Bel = dimensionless time required for complete conversion for a uniform particle v = particle shape factor (0 = slab; 1 = cylinder; 2 = sphere) = r / R = dimensionless position variable E, = dimensionless position of the shrinking core Subscripts Bed
Subscripts d = diffusion through solid product controls the rate k = surface kinetics controls the rate Literature Cited Beveridge. G. S. G.; Goldie, P. J. Chem. Eng. Sci. 1988, 23, 913. Erk, H. F.; DudukoviE, M. P. Ind. Eng. Chem. Fundam. 1982, 22, 5 5 . Ishida, M.; Wen, C. Y. Chem. Eng. Sci. 1868, 2 3 , 125. Levenspiei, 0. "Chemical Engineering Kinetics", 2nd ed; Wiiey: New York, 1972; Chapter 12. Smith, J. M. "Chemical Engineering Kinetics", 3rd ed.; McGraw-Hili: New York. 1981; Chapter 14. Szekely, J.; Evans, J. W.; Soin, H. Y. "Gas-Solid Reactions"; Academic Press: New York, 1976; Chapter 3.4. Weisz, P. B.; Goodwin, R. D. J . Catal. 1983, 2 , 397
Receiued for review November 22, 1982 Revised manuscript receiued June 3, 1983 Accepted June 13, 1983
