Ind. Eng. Chem. Res. 1993,32, 1839-1852
1839
Diffusion, Convection, and Reaction in Catalyst Particles: Analogy between Slab and Sphere Geometries Z.P.Lu, M. M. Dias, J. C. B. Lopes, G. Carta,f and A. E.Rodrigues’ Laboratory of Separation and Reaction Engineering, School University of Porto, 4099 Porto Codex, Portugal
of
Engineering,
Expressions for the calculation of the “augmented” intraparticle effective diffusivity have been obtained for diffusion, convection, and reaction problems in catalyst particles with slab and sphere geometry at steady state and in the transient regime. Analytical solutions for the time domain model equations for both geometries have been derived, and a numerical scheme for solving the spherical particle model equations is also presented. An equivalence ratio, 1/R, between the slab and the sphere characteristic dimensions has been obtained from these results for steady-state and transient operation. Finally, a comparison of the actual use of the sphere and the slab model equations is presented.
1. Introduction The coupling of reaction kinetics and mass transport processes in porous catalytic particles has been studied for a long time. A thorough account of earlier work can be found in Aris (1975). An enhancement of the observed reaction rate, Le., of the effectiveness factor, by intraparticle convection when reaction and mass transport rates are comparable has been reported by several authors (Nir and Pismen, 1977; Rodrigues e t al., 1982; Rodrigues and Ferreira, 1989). More recently, the importance of intraparticle convection when large-pore materials are used has been considered in bioreaction engineering (Young and Dean, 1987; Stephanopoulos and Tsiveriotis, 1989; Carrondo, 1990; Prince et al., 1991). Rodrigues e t al. (1982)have shown that the enhancement of the effectiveness factor resulting from intraparticle convection can be expressed in terms of an “augmented” intraparticle effective diffusivity. This augmented diffusivity is defined in such a way that the classical expressions for the effectiveness factor derived for the case of purely diffusive transport can be used for the case of simultaneous diffusion and convection simply by replacing the effective diffusivity with the augmented value. For simple geometries and intraparticle flows, analytical expressions for the augmented diffusivity as a function of intraparticle flow can be derived. The equivalence between various geometries has been recognized for a long time (Glueckauf, 1955;Kucera, 1965; Villermaux, 1974; Greco et al., 1975), and the sphere geometry has been widely used in writing particle model equations in diffusion and diffusion/reaction problems due to ita symmetry characteristics (Crank, 1955; Ruthven, 1984). When intraparticle convection is important, some analytical solutions for linear systems for the sphere geometry have been obtained (Nir and Pismen, 1977; Stephanopoulos and Tsiveriotis, 1989; Do, 1990; Carta et al., 1992) but, in most cases, they are not easy to use. An approximated solution for transient diffusion and convection problems in spheres was recently obtained by Frey e t al. (1992). The methodology of establishing the equivalence between two geometries allows the replacement of twodimensional models by one-dimensional models, which are easier to solve, but give similar results. For example,
* To whom correspondence should be addressed. E-mail address: (EAN) arodriga fe.up.pt + Present address: Department of Chemical Engineering, University of Virginia, Charlottesville, VA 22903-2442.
in the diffusion problem, the two-dimensional cylindrical geometry problem can be replaced by a one-dimensional spherical geometry problem (Greco e t al., 1975). In diffusion/convection/reactionproblems, the slab geometry is preferred because it either gives a relatively simple analytical solution (Nir and Pismen, 1977; Rodrigues et al.,1982,1984,1991,1992b; Rodrigues and Ferreira, 1989) or the equations can be easily solved by numerical methods for more complex problems (Lu et al., 1992). In a previous paper (Rodrigues et al., 1992a), an equivalence ratio, 1/R, between the characteristic dimensions of a semi-infinite slab and a sphere was defined for the description of the intraparticle diffusion and convection of a linearly adsorbed, unreactive tracer in a porous medium. This equivalence ratio was obtained by equating the transient response to a pulse injection of the tracer to a chromatographic column computed from two models: one describing intraparticle diffusion and convection with the semi-infinite slab geometry, and the other with the sphere geometry. When the flow field within the particles, in both the slab and sphere geometries, is assumed to be uniform, by equating the second moments of the pulse response in the two descriptions, a simple relationship between I and R was obtained. Their ratio varied between 1 / 4 5 , when intraparticle transport was dominated by diffusion, and 315, when intraparticle transport was dominated by convection. This relationship was also shown to exist even in the presence of adsorption kinetics (Rodrigues e t al., 1992b). In this work, the focus is on intraparticle mass transfer, i.e., diffusion and convection, coupled with first-order isothermal reaction at steady state and in the transient regime. The objectives of this paper are the following: (i) Obtain the augmented intraparticle effective diffusivity for the slab and the sphere geometries and the characteristic dimension equivalence ratio between slab and sphere at steady state. (ii) Solve the model equation for the slab and the sphere geometries in the transient region, and study the evolution of the augmented intraparticle effective diffusivity and of the particle effectiveness factor. (iii) Obtain the augmented intraparticle effective diffusivity for the slab and the sphere geometries and the characteristic dimension equivalence ratio between slab and sphere in the transient regime. (iv) Compare the results calculated from the sphere geometry model and those predicted from the slab geometry model using the characteristic dimension equivalence ratio.
0888-588519312632-1839$04.00/0 0 1993 American Chemical Society
1840 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 Slab
2. Diffusion, Convection, and Reaction at Steady
State
For fixed-bed reactors under steady-state operation,the reaction conversion depends on the effectivenessfactor of thecatalyst particles. The analogy betweencharaderistic dimensions for different particle geometries can be based on the equivalence of the particle effectiveness factors (Rodrigues and Ferreira, 1989; Rodrigues et al., 1992a).In this work, the case of uniform intraparticle flow perpendicular to a semi-infinite slab and in a sphere is considered. As shown by Carta et al. (1992), stream lines are nearly vertical, except in a thin boundary layer near the particle surface when particles have permeabilities of the order 1W2cm2. Therefore, the assumption of uniform flow in spherical pellets is quite satisfactory. Figure 1shows the slab and suhere geometries with the corresuondinn- coordinate syitems. The mass balance eauations for the slab and the sohere geometries were deri4ed for these conditions and iolved analytically by Nir and Pismen (1977) in the case of constant concentration at the particle surface. The effectiveness factors are given by for the slab geometry
nb
1: tz Figure 1. Schematic representation of slab and sphere geometry particle models.
lo~q---
_1 - _1 ‘1
Sphere
100
r2
= coth r, - coth r2
1
6eIDe
for the sphere geometry
--.
IO0
1 .1
100
a
steady state
1
1
1 0
10
b
steady state
where 1,,,+1/2 and I m + 3 p are the modified spherical Bessel functions of the first kind and
0 = (a2+ 44,2)’/2
(2a)
10
The model parameters are the intraparticle Peclet number (for slab and sphere, respectively)
50
(3)
IO0
1 1
the Thiele modulus (for slab and sphere, respectively)
-
If intraparticle transport is assumed to be purely diffusive (Ab, Xp 0, in the above equations), the relatio-mhips between the “apparent” effectivenep factcr, rn or qp, and an ‘apparent” Thiele modulus, @b or 4p, become tanh &,
fb =
and
for the slab 6b
10
100
hb Figure 2. Enhancement of intraparticle effective diffusivity by convection,ddD*,88 a function of intraparticle Peclet number, Ab, and Tbiele modulus,&, for the slab geometry at steady state. (a) dJDe v8 Q b for different Ab; (b) ddD. vs A b for different Qb. jjp =
”(
4p h h 4
-
-)1
for the sphere
(6)
4p
where
(5)
and & is the augmented intraparticle diffusivity which
'"1
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1841
T
0.7
a
d y state
ie/De
a
0.6
I
0.5
0.4
10
1
100
V."
.I
10
1
100
(% 10
steady state
"..
b
I
,
c
6e/De
steady state
3 5 7
0.6
b
10
e /R
15
0.5 20 30
0.4
1 1
10
100
0.3
50
'
100
10
1
h, Figure 3. Enhancement of intraparticle effective diffuaivity by convection, DJD,,as a function of intraparticle Peclet number, A,, and Thiele modulus, h,for sphere geometry at steady state. (a) b J D , YB 6 for different A,; (b) b J D , va A, for different b.
lumps both diffusion and convection (Rodrigues et al., 1982, 1991). 2.1. Augmented Diffusivity by Convection. The ratio between the "augmented" intraparticle effective diffusivity and the "real" intraparticle effective diffusivity, as a function of the intraparticle Peclet number and the Thiele modulus for both geometries at steady state, is obtained by imposing that )lbP?b
and V p = f,
(8)
This results in relationships of the type
which can be calculated numerically. Equation 9 applied to the slab geometry is plotted in Figure 2. Figure 2b shows that, in the reaction kinetics controlled limit, C$b 0, the relationship
-
which is the result obtained by Rodrigues et al. (19821, is verified. For large values of 4 b and small A, values, i.e., in the mass transfer controlled limit, bdDe = 1. Intraparticle convection plays an important role over a wide range of 4 b values, and its effect is negligible ody_forvery large Values Of 4b. For VdUeS Of 4 b >> 1 and 4 b >> 1, following the development of Rodrigues et al. (19821, the augmented diffusivity is given by
100
A, Figure 4. Equivalence ratio of characteristic dimensions of slab and sphere as a function of intraparticle Peclet number, X,, and Thiele modulus, QP,at steady state. (a) 1/R VB &, for different A,; (b) 1/R vs A, for different &p
De which shows that b$De becomes only a function of the ratio hb/24b. Figure 3 shows the b J D , relationship in the case of the sphere geometry. The calculation of the sphere effectiveness factor involves the numerical evaluation of a series. The number of terms needed to reach convergence is several hundreds, and it increases with the intraparticle Peclet number, which implies the estimation of high-order Bessel functions for high values of the argument. When an 8-byte word precision was used in the calculations, meaningful results were obtained only up to A, = 20. In order to obtain results for values up to A, = 60, a 16-byte word precision was required. Inspection of the results in Figure 3b shows that the general behavior of the &IDe for the sphere geometry is similar to that obtained for the 0, at high A, values, b J D , is seen slab. In the limit, 4, toapproach the asymptote Ap/lO. Cartaet al. (1992) have treated the transient problem of diffusion and convection of a linearly adsorbed, unreactive tracer in a spherical particle. This treatment yielded a relationship between the augmented diffusivity and the intraparticle Peclet number, A,, essentially identical to the curve in Figure 3b for 4, 0. The curve was approximatedby the expression
-
-
-
This expression is plotted in Figure 3b. The agreement 0 is good over an ample with the exact solution for 4,
1842 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 1.2
d
l$Ih=5, h,=10
1.o
0.8
C'
C'
0.6 0.4
0.2 I."
u.0
0.5
1 .o
1.5
0.d
u.0
2.0
0.5
z 1.2
l$Ib=l;hF10
1.5
2.0
z eb
1.o
C'
1 .o
0.8
1.o
0.6
^ ^
u.u
C'
0.4
0.2
1
el
ob=l0,h$O
L
II\
,001
0.6 0.4
0.0
u.0
0.5
1 .o
1.5
2.0
z
0.2
0.0
u.0
0.5
1 .o
1.5
2.0
z
C'
Figure6. Dimensionless intraparticle concentration profiles for slab geometry at various times for different set of intraparticle Peclet number, Ab, and Thiele modulus, &,. (a) &, = 1 and hb 0; (b) 4b = 1 and hb 10; (C) &, = 1 and A b = 40; (d) & = 5 and hb 10; (e) &, = 10 and hb = 10,
range of A, values, although the predicted asymptote is Ad9 instead of the value A,/lO found from the exact solution. This asymptotic value, on the other hand, coincides with that obtained by Rodrigues et al. (1992a) using a simplified analysis of the sphere problem for high A, values. 2.2. Equivalence between Slab and Sphere. Now, imposing that ?b = ?p (13) the relationships for the ratio between characteristic dimensions for slab and sphere as a function of the intraparticle Peclet number and the Thiele modulus are obtained as
IIR = F
~ ~ ( d p , b=)FRb(&,Ab)
(14)
Here, some limits for the ratio 1/R can be anticipated. When the Thiele modulus is small, 4,+0, i.e., slow reaction kinetics, IIR = l / d b at A, = 0, and this ratio increases with the intraparticle Peclet number up to 1/R = 213. This is the result obtained by Rodrigues et al. (1992~).For large values of the Thiele modulus, 4, a,i.e., at the
-
diffusion controlled limit, 1/R = 113, and this value does not change with the intraparticle Peclet number. Parts a and b of Figure 4 show 1/R as a function of the Thiele modulus for different intraparticle Peclet numbers, and as a function of the intraparticle Peclet number for different values of the Thiele modulus, respectively. This figure shows that 1/R is practically independent of A, and of 6, when they are both less than 1. Figure 4 can also be easily converted to RII as a function of Ab and 4 b as was done by Rodrigues et al. (1992a). 3. Diffusion, Convection, and Reaction in the Transient Regime
With the increasing interest of reaction coupled with separation (Vaporciyan and Kadlec, 1989; Sardin et al., 1992),understanding the mechanisms involved in the case of comparable reaction kinetics and mass transport rates in the transient regime is becoming more and more important. Do (1990) solved analytically a diffusion, convection, and reaction problem for the sphere geometry for an exponential kinetic rate and for Thiele modulus
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1843
r \ b
0.8
-
0.6
-
0.4
-
0.0' .uo1
.
...
'
.01
'
.1
.
1
.
oor
-....-.I
:uo1
10
.
.
'
.
'.'.I
.01
'
.'.
" . ' I
.1
'
. . . ""'
1
'
'
'
.
10
e b
1.0-
.
0.6
3Lb=10
0.8 i 0.4
qb 0.2
Y."
.uo1
.01
.1 o
1
10
0.0 .uo1
.01
.1
1
0,
b
Figure 6. Evolution of particle effectiveness factor for slab geometry for various seta of intraparticle Peclet number, Abt and Thiele modulus, &. (a) Tb vs & for different &, at Ab = 0; (b) Vb vs & for different &, at Ab = 10; (c) r)b vs ob for different Ab at f$b = 1;(d) r)b va eb for different A b at & = 5.
equal to 1. Here, analytical solutions for the slab and the sphere geometries in the transient regime are derived, assuming a first-orderreaction. As before, the intraparticle flow field is assumed uniform and perpendicular to the surface of the semi-infinite slab. 3.1. Time Domain Model Equations and Solutions for Step Response. The transient behavior of a catalyst pellet initially free of reactant and subjected to a step change in the reactant concentration at the particle surface is now considered. Analytical, time domain expressions for the effectivenessfactor are obtained for both slab and sphere geometries. 3.1.1. Analytical Solution for the Slab Geometry. An infinite slab of thickness 21is considered. The following equations may be written.
A;
4:
44; + A: + m2r2 + m2r2 + A; + m2r2exp( -
w ]) (16)
and the transient effectiveness factor is
species mass balance A;
4: boundary conditions c ' = 1 for z = O a n d z = 2
(15a)
4: + :A + m2r2 + m2r2 + A: + m2r2exp( -
w ]) (17)
-
If intraparticle transport is assumed to be purely diffusive (Ab 01,eq 17 reduces to
initial condition
c'= 0 at k$, = 0
(15b)
Here, c' = C/CO is a dimensionless concentration, = is the dimensionless time (Tdb = ebZ2/De is the intraparticle diffusion time constant), and co is the particle surface concentration which is assumed to be constant. The solution of eqs 15-15b is (for details see Appendix A) t/Tdb
The intraparticle dimensionless concentration profiles at different times for various sets of the Thiele modulus and the intraparticle Peclet number, calculated from eq
1844 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 loo
a
.01
.l
1
10
b '
loo
loo
b /
&/De
&?/De 10
10
.bo1
$b=5
.01
.1
1
'1
b '
10
100
'h
Figure 7. Evolution of enhancement of intraparticle effective diffueivity by convection, &/De, for varioue seta of the intraparticle Peclet number, Ab, and Thiele modulus, &,, for slab geometry. (a) ddD vs eb for different A b at &, = 1; (b) BdD, vs ea for different Ab at & = 5; (c) ~ J Das , a function of A b at different times &, for &, = 1; (d) b d ~as. a function of A b at different times 4 for &, = 5.
16, are shown in Figure 5. Again, high-precision computations (16-byte) are required and results could only be obtained up to hb = 40. This figure shows that, as the intraparticle Peclet number increases, the concentration profilesdevelop in a steeper and more nonsymmetric mode. When the Thiele modulus is high, regions of very low concentration exist inside the particle even at long times. The evolution of the effectiveness factor for the slab geometry, calculated from eq 17, for various intraparticle Peclet numbers and Thiele modulus values is shown in Figure 6. Since the particle is initially free of reactant, 9 b takes the same value for different values of 4 b and hb for short times. For higher intraparticle Peclet numbers, it takes shorter times to reach steady state and 9b has always a higher value. For higher values of the Thiele modulus, it takes a shorter time to reach steady state and 9b always has a lower value. Setting Vb(8b) = &(e,) a t a given time, the evolution of the ratio between the augmented intraparticle diffusivity and the real intraparticle diffusivity can be calculated, as shown in Figure 7a,b. For small values of the Thiele modulus, this ratio starts from unity, then goes through a maximum value, and finally reaches the steady-state value. This maximum value almost disappears for higher values of &,. The evolution of the enhancement of the intraparticle diffusivityas a function of intraparticle Peclet number a t different times is shown in Figure 7c,d for different values of the Thiele modulus. The enhancement of intraparticle diffusion by convection as a function of the intraparticle Peclet number starts from unity at time zero and then develops toward the steady-state value. A t intermediate times and for small values of the Thiele modulus, the enhancement relationship crosses with that a t steady state. For higher values of the Thiele modulus, the function gradually approaches the steady-state values.
3.1.2. AnalyticalSolutionfor the SphereGeometry. Assuming a uniform intraparticle flow field with a uniform surface concentration, the following equations and boundary conditions may be written.
species mass balance
boundary conditions c'= 1 for r = 1; c'finite at r = 0 (19b) initial condition c'= 0 a t 8, = 0 (19c) Here, rand p = cos 8 are the dimensionless sphere spatial coordinates and OP = t/Tdp (Tdp = E&~/D,)is the dimensionless time. The time domain solution of eqs 19-19c is (see Appendix B for details)
where
Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 1845 h =lo,$ = 3 P
I."
P
0.8 0.6 0.4 0.5
0.2
0.4
0.3 0.2
0.0 ."01
0.1
.01
.1
1
.1
1
0,
0.0 1.o
0.8 0.6 V
P
0.4
0.2 0.0 ."01
.01
e,
Figure 8. Dimensionless concentration contours at different times for sphere geometry: 6, = 3 and h, = 10.
C,
= ~01Jm+l~z(C,,,,,r)~3~z dr
(20a)
and Dmm
4
45,;
+ 44; + A;
S' O
Here, Jm+l/zandJm+3/2arethe spherical Bessel functions of the first kind, P , is the Legendre polynomial, and fmn are the roots of J,+l&,,,) = 0; m = 0, 1.2, ... and n = 1,2,3, ...
(20~)
The transient effectiveness factor is now
exp
Figure 9. Evolution of particle effectiveness factor for sphere geometry for varioua sets of intraparticle Peclet number, A,, and Thielemodulua,6.: (-)calculated fromeq22bynumericalmethod; ( 0 )calculated from eq 21; (- - -) predicted from eq 17 by using the equivalenceratio in the transient regime; (- -) predicted fmm eq 17 byusingtheequivalenceratioatsteadystate. (a)qPvs8,fordifferent A, at $p = 3; (b) qp vs 8, for different b a t A, = 20.
(-
4cm.,,2+44;+ 4
h2)4
calculation of a great number of terms, and each term involves the calculation of Bessel functions, Legendre polynomials, and numerical quadrature. For example, the calculations for Figure 8 took about 45 min of CPU time to perform on an IBM RISC/6000-530 computer. 3.1.3. Numerical Solution for the SphereGeometry. As an alternative to the time-consuming computations of the analytical solution, a numerical scheme to solve the spherical geometry problem is proposed in this section. Thismethod can be also applied when the model equations are nonlinear, i.e., when nonlinear adsorption isotherm is present, or the reaction order is different from one or zero, and the problem cannot be solved analytically. Particle model equations for the slab geometry have been solved by Lu et al. (1992). The model equations for the sphere geometry can be rewritten as
(21)
L, = ~ I m + 1 / zdr; ( ~E,,) ~ = 3 ~ z
c' = 1 for z = -1 and z = 1 c' = 1 for
Contour plots of the dimensionless concentration for A, = 5 and 4, = 3, at different times, 0, = 0.01, 0.1, 1, and m, are shown in Figure 8. Computation of eq 20 is timeconsuming. This is particularly true for high intraparticle Peclet numbers, since series convergence requires the
p = (1 - z2)'/';
act = 0 at ap
c'= 0 at 8, = 0
p =0
(22a) (22b) (22c)
where z is the dimensionless central axial coordinate, parallel to the intraparticle velocity, and p is the radial
1846 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
dimensionless coordinate from the z axes toward the surface of the sphere. The method of collocation on finite elements is widely used for solving boundary value problems (Finlayson,1980; Lu et al., 1992). Here, eq 22 was first reduced to a onedimensional problem in the z coordinate, by using collocation on finite elements in the p coordinate. The z coordinate was then transformed by y = ( z + 1)/2, and the resulting partial differential equations were solved by the PDECOL package (Madsen and Sincovec,1977). Various numbers of elements in the p and z coordinates were tested. Using 20 elements in both directions, the CPU time to reach the steady state (0, = 1) was around 10 min. Figure 9 shows the evolution of the sphere particle effectiveness factor calculated numerically (solid lines). The results from the analytical solution (eq 21) have also been checked as presented in Figure 9a in black circles, Since the calculations of eq 21 are very time-consuming, particularly for high intraparticle Peclet number, results = 3, A, = 0 and 4p = 3, are shown for two cases only: 10. The evolution of the effectiveness factor is very A? similar to the one obtained for the slab geometry (Figure 6),but it takes a shorter dimensionless time to reach steady state. This is because l2 < R2,so Tdp > Tdb. Also, the more pronounced effect of intraparticle convection was found to be at 4p== 3, while for the slab it had been found a t I#Jb
where
The zeroth moment of ( c ) is
and the first moment is
I#J~
= 1.
3.2. Laplace Domain Solution for Pulse Response. In this section,an alternative way of analyzingthe behavior of the transient response of a catalyst particle corresponding to a pulse of a reactive tracer at the particle surfaceis examined. The objectiveis to find an appropriate relationship between the augmented diffusivity and the intraparticle Peclet number for these conditions, as well as an expression for the IIR ratio of the characteristic dimensions of the slab and the sphere geometries. The same equations and initial conditions developed for the step response are used, but the boundary condition at the surface of the pellet is changed to
=o, 2
(23)
c’ = MP8(Op) at r = 1
(24)
c’=
M b 6(8b)
at
for the slab geometry, and
where rl and r2 are given by eq la. If intraparticle convection is not considered in the model equation, (30) and I
I
3.2.2. Transfer Function and Moments for the Sphere Geometry. As before, taking the Laplace transform of eq 19,and solvingthe partial differential boundary conditions problem, the transfer function for the sphere geometry is obtained as
for the sphere. The response of the particle to the tracer signal is described by the average particle concentration ( c ( t ) > ; the moment of the order n of the impulse response can be obtained by m, = (-1)”-d”g(s) (25) ds” where g(s) is the transfer function between the average particle concentration and the particle surface concentration, in the Laplace domain. The first absolute moment is defined by (26) In the next two sections, the transfer function and the moments for the particle model equations for the transient diffusion, convection, and reaction problem for both geometries are calculated. 3.2.1. TransferFunction and Moments for the Slab Geometry. Taking the Laplace transform of eq 15 and solving the resulting ordinary differential, boundary conditions problem, the transfer function for the slab geometry becomes P1
= m,/mo
where
8’ = (h2 + 4(42 + Tdps))1/2 The zeroth and first moments are now equal to
(32a)
Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 1847 100
1I
be/De 10
10 I
a
transient regime 100
I
transient regime a
GelDe
\
50
[ /-I?. 7 > , , \ 20
.1
1
.1
100
10
10
1
100
@P
10
b
:transient regime
transient regime IZI
De'De
'1
10
b
I
100
Figure 10. Enhancement of intraparticle effective diffusivity by convection, &/De, as a function of intraparticle Peclet number, Ab, and Thiele modulus, &, for slab geometry in the transient regime (full lines). For cornparison dashed lines represent the diffusivity enhancementfrom the equality of steady-state effectivenessfactors. (a) &/De vs @b for different Ab; (b) b J D , vs A b for different &.
Figure 11. Enhancement of intraparticle effective diffusivity by convection, bJD,, as a function of intraparticle Peclet number, A,, and Thiele modulus, @ , , for sphere geometry in the transient regime (full lines). For comparison dashed lines represent the diffusivity enhancement from the equality of steady-stateeffectiveness factors. (a) &/De vs 6, for different A,; (b)DdD,vs A, for different @ p
equating the first absolute moments (Rodrigues and Ferreira, 1989). However, when a reactive tracer is used, the meaning of the first absolute moment is modified to the average remaining time of the reactive trace and now the equivalence should be based on the productivity of the particle. This productivity is proportional to the reacted amount, i.e., proportional to the zeroth moment, mo, of the average concentration of the reactive tracer, and it is inversely proportional to the residence time of a unit reactive tracer, i.e., pdmo. The equivalence parameter is then mO3/m1,and the augmented intraparticle effective diffusivities for both geometries is given by mM3 e m 3 --where B is defined in eq 2a. If intraparticle transport is assumed to be purely diffusive, one obtains (35)
3.2.3. Augmented Diffusion by Convection. The moments of the average concentration for both particle geometries having been determined, the question now is which combination of these moments gives the right measure for particle performance. If reaction is not present, the tracer is conservative and the equivalence is based on the intraparticle diffusion/convectiontimes, i.e.,
mlb
mN3 fim3
-=mlp
for the slab
(37a)
for the sphere
(37b)
%b
%P
The numerical values of the &,/De ratios that satisfy these equalities are shown in Figures 10 and 11 in solid lines. For comparison, the numerical values of these ratios that lead to an equality of the effectivenessfactors at steady state,with a constant surface concentration, are also shown in dotted lines. The enhancement of the intraparticle diffusivity by the intraparticle Peclet number for the slab and the sphere geometries in the transient regime is very similar to that obtained at steady state. In the limiting cases, i.e., for very low and very high values of the Thiele modulus, they both tend to the same values. For intermediate values of
1848 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 0.7 I
1
e /R
transient regime .I
10
1
100
of the intraparticle Peclet numbers. However, for intermediate values of the Thiele modulus, the ratio 1IR obtained in the transient regime is always lower than that obtained at steady state. The reason is 2-fold. First, as the Thiele modulus increases, i.e., the reaction becomes faster, it is harder for the reactive species to reach the center of the particle in the transient regime. Second, while for the slab the area for mass transfer remains constant from one side to the other, in the sphere, this area decreases toward the center. When the Thiele modulus is very high, reaction takes place only near the particle surface, and both transient and steady-state values coincide. 4. Discussion and Conclusions
n7 V.,
I transient
regime
o.l,
3
b
I
0.6
e /R 0.5
0.4
0.3
'
I
10
1
100
h, Figure 12. Equivalence ratio of characteristic dimensions of slab and sphere as a function of intraparticle Peclet number, A,, and Thiele modulus, &, in the transient regime (full lines). For comparisondashed lines showthe steady-state equivalenceratio. (a) l / R vs &p for different A,; (b) l / R vs A, for different bP
the Thiele modulus, the augmented intraparticle diffusivity by convection in the transient regime is always lower than that at steady state. This is an averaged effect resulting from using eq 37a or 37b. Actually, the effect of intraparticle convection is significantly dependent on the intraparticle concentration. Figure 7, for example, shows that the value of b,$De during time evolution can pass through higher values than the steady-state value. The maximum observed in b$De versus 8b for t$b = 1 (Figure 7a) can be explained by the fact that intraparticle convection strongly affects concentration profiles inside the pellet; hence the penetration distance is higher than when mass transfer is by diffusion alone (see profiles at 8b = 0.05 in Figure 5a,c). Effectiveness factors calculated from those profiles will lead to higher values than those obtained from steady state (e.g., profiles at db = 10 in Figure 5a,c). When 4p 0, the same relationship is obtained at steady state (Figure 3b) and in the transient regime (Figure llb). 3.2.4. Analogy between Slab and Sphere in the TransientRegime. The ratio between the characteristic dimensions of the slab and the sphere in the transient regime is estimated by equating the ratios
-
-
-
Figure 1 2 shows that when intraparticle convection is not important, the limiting cases for 4p 0 and $p are IIR = lid5 and 1lR = 113,respectively. These resulta are those obtained from the steady-state analysis. When $ J ~< 1, the characteristic dimension ratio 1/R in the transient regime and at steady state coincide for each value 0)
Exact analytical solutions have been obtained for simultaneous diffusion and convection with first-order reaction in catalyst pelletswith slab and sphere geometries, at steady state and in the transient regime for initially clean pellets. A numerical method for the integration of the model equations has also been introduced, since, in this case, the direct numerical integration is, for some conditions, simpler than the numerical evaluation of the analytical solution. The solutionsdeveloped have been used to define general relationships between a convection-augmented intraparticle effective diffusivity and the intraparticle Peclet number. These relationships extended previous work, pertaining to transient diffusionlconvection of a nonreactive tracer in a slab and a sphere by Rodrigues et al. (1982) and Carta et al. (19921, respectively, to the more general case of transient intraparticle diffusion and convection of a reactive tracer in the two geopetries. The augmented diffusivity by convection, DJD,, as a function of the Thiele modulus and the intraparticle Peclet shown in Figures 2 and 3 for the slab and the sphere geometries, respectively, allows simple calculations of fixed-bedreactors using the classic knowledge of diffusion and reaction problems. In fact, for first-order reactions in isothermal plug-flow catalytic reactors filled with largepore slab particles, the steady-state outlet conversion is given by 1 - exp(-q&a), where Da is the Damkh6ler number and 9 b is the catalyst effectiveness factor when diffusion, convection, and reaction take place inside the catalyst. However, if the relationship of b J D e versus &, and h b is known, then the results from diffusion and reaction can be used and the _conversionicsimply 1- exp(-rlbDa) 1- exp(-(tanh 9bldb)Da)with 4 b = &/(&De). An analogy between the slab and the sphere treatments has been defined for these cases, based upon the ratio 1/R of the characteristic dimensions for the two geometries. This ratio has been defined in such a way that the simpler slab model equations may be used to predict intraparticle convection effects in a spherical pellet. The efficacy of this approach is shown for example, in Figure 9a,b in the transient regime and at steady state. This figure shows that exact calculations of the time evolution of the effectiveness factor in a spherical pellet are very well approximated by the slab model solution, eq 17, in which the replacements hb = Xpl/R, 4 b = and 8b = 8p/(l/R)2 have been made. As is shown by this figure, one can expect that the agreement between the slab solution and the sphere solutioncalculationsto be exact for long times when IIR is obtained from the steady-state analogy (Figure 4). On the other hand, both analogies come quite close to the exact solution over the entire range of times, allowing one to use the simpler slab model for practical calculations with problems coupling mass transport and reaction in
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1849 the transient regime. This analogy is, of course, valid only within the context of the validity of the assumption of a uniform intraparticle flow field. As shown by Carta et al. (1992), on the other hand, this assumption is not too restrictive, and the approach developed should be valid for many practical situations. The methodology presented here can be extended to generalized kinetics and nonisothermal particles allowing fmed-bed reactor calculations,but would require numerical solutions. Similar results, however, especially in regard to the analogy between slab and sphere geometry, could be expected.
cl=O for z = O a n d z = 2
(Ala)
at
(Alb)
c1
Replacing c1 = c2
ob=
0
+ c3 leads to two sets of equations
Nomenclature c = species concentration, mol/cm3 c' = dimensionless species concentration Da = Damkhiiler number De = intraparticle effective diffusivity, cm2/s k l = first-order kinetic constant, s-1
c 3 = 0 for z = O a n d z = 2
(A3a)
= 0; 8, 0 (A3b) The solution of eqs A2-A2b is easily obtained: c3
1 = half-thickness of slab, cm
m= zeroth moment of reduced averageparticle concentration r = dimensionless spatial coordinate in sphere rl, r2 = parameters defined by eq l a rl', r i = parameters defined by eq 27a R = sphere radius, cm t = time uo = intraparticle superficial velocity, cm/s z = dimensionless spatial coordinate in slab and in sphere
Greek Symbols 8, 8' = parameters defined by eqs 2a and 33a, respectively 7 = z-coordinate transformation = (z + 1)/2 ep = particle porosity t = root of the Bessel function 9 = effectiveness factor 8 = angular coordinate in sphere ob, 8, = dimensionless time for slab and sphere, respectively X = intraparticle Peclet number p = spatial coordinate in sphere ( p = cos 8) pl = first absolute moment of the average particle concentration, s p = dimensionless spatial coordinate in sphere f d = intraparticle diffusion time, s 4 = Thiele modulus
)
(
ml = first moment of reduced average particle concentration
sin(m;z) - eAp12exp b - - ~ ; + ~ m ~ (A4) r ~
M = strength of the impulse, mol.s/cm3
Equations A3-A3b are solved by introducing the transformation of variables C3(z,eb)
= J:c4(z,eb,E)
dt
(A51
which leads to the set of equations
whose solution is
Integrating as in eq A5 m
Superscripts = "apparent" values obtained if intraparticle convection is not considered as an independent transport mechanism
-
Subscripts b = slab p = sphere
Appendix A. Derivation of the Time Domain Analytical Solution for the Slab Introducing the transformation of variables c' = c1e+b6
+ 1, eq 15-15b reduce to
(
eXP
4
~ , +2 m2r2 b
'
e
b
-
T
)
(A8)
and substituting backwards, c' = 1- e4bN(c~+ cs), one obtains the solution of this problem as shown by eq 16. To check the validity of this solution, the following limiting conditions in the corresponding expression for the effectiveness factor (eq 17) were verified: (i) If no intraparticle convection is present, i.e., Ab = 0, the effectiveness factor reduces to eq 18, which is the solution (eq 8.47) obtained by Crank (1955). (ii) If no intraparticle convection and no reaction are present, i.e., Ab = 0 and 4 b = 0, eq 17 reduces to
1860 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
which is also the solution (eq 4.24) obtained by Crank (1955).
A, = 0 for r = 1; A, finite at r = O
Appendix B. Derivation of the Time Domain Analytical Solution for the Sphere
at 0, = 0 (B9b)
Introducing in eqs 19-19c the transformation of variables c' = cleWI2 gives
c, = e-'+/'
for r = 1; c, finite at r = 0 (Bla) c, = 0 ate, = 0 (Bib)
Now, substituting c1=
(B9a)
+
c2 e~p[-[(44,~ Ap2)/418,1results
where
Finally, replacing A,,, = AI, into two seta of equations
+ AZm, the problem is divided
in
Ate )
44; + + 4 , for r = 1; c2 finite at r = 0 (B2a) c2 = 0 at 8, = 0 03%)
c2 = exp(
The characteristic equation of eq B2 is
A,, = 0 for r = 1; A,, finite at r = 0
(Blla)
at 6, = 0 (Bllb) and
where P m ( p ) is the Legendre polynomial of order m. Assuming that the solution is of the form
4,
m(m + 1)(B12) r2 A,, = 0 for r = 1; A,, finite at r = O A,, = 0 at 0, = 0
and replacing in eqs B2-B2b dB, -de,
d2B, --+--dr2
2dB, r dr
B, = f1(m,8,) for r = 1; B,finite B,=O
+ 1)-Bmr2
(B5)
at r = 0
(B5a)
m(m
at 8,=0
(B12a) (B12b)
which can be easily solved. The solution of the eqs B1Blb is then
(B5b)
where
Using the relationship (Abramowitzand Stegun, 1965)
and substituting backward, one obtains the solution expressed by eq 20. As before, a validity check is done by verifying the limiting conditions in the corresponding expression for the effectiveness factor (eq 21). (i) If no intraparticle convection is present, i.e., A, 0,
-
and applying the orthogonal properties of the Legendre polynomials, f,(m,B,) can be written as
I1/2(~)4/2(
f o r a l l m l l and
'r1I2
y)
1 = - (B14) a
then the summation in m in the series of eq 21 is reduced to the first term (m = 0). Furthermore Now introduce the transformation B,(m,r,B,) A,(m,r,ep) + fl(m,e,).
=
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1851 " f l
Then eq 21 reduces to
-,E
4,2
+ n2p2e-(6p2+n2*2)8p
a,(e,) = 1
(I3181 n2(4,2+ n 2 2 ) which is the solution (eq 8.41) obtained by Crank (1955). (ii) If no intraparticle convection and no reaction are present, eq 21 becomes
where Jn+l/dPrnn) = 0. Transforming eq B24 and rearranging
7r n i l
which, again, is the solution (eq 6.20) obtained by Crank (1955). Alternative Solution for the Sphere. The solution of eqs 19-19c can also be obtained using the integral transform method. First, introduce the transformation of variables
dGmn + (Pmn2 + 4;)G,, de,
=
,-A
le p p'
4
F,,
(B27)
where
dp dr = F,, (B27b) Solving this first-order ordinary differential equation and applying the inverse transforms, the solution is obtained:
c'(r,p,O,) = 1- G(r,p,Bp)eX~r~2e-h~ze~~4 (B20) to obtain a nonhomogeneous PDE with homogeneous boundary conditions V?G - 4 : ~ + 4 ; e - ~ / 2 e ~ p ~=4aG de, G = 0 for r = 1; Gfinite at r = 0 G = e-'W/' at e, = 0
(€321) (B21a) (B21b)
Now choose the associated eigenvalue in 1.1:
-1
= -n(n + 1)F, n = 0, 1,2, ...
&[ d (1- p Z ) d F dp
(B22)
whose solution is F ( p ) = P n ( p ) , and choose the integral transform
G, = $-;G(r,p,O,) PnW dp
(B23)
Jn+1/2(Pmnr) dr (B28c) and the effectiveness factor of the particle:
Applying this transformation to eq B23 gives
where
where
Fn(r)= = fie-'+"/'P,(p)
dp
(B24a)
Now, repeat the procedure by choosing the associated eigenvalue in r
pd
r2
- I n ( n + l)F+ P2F = 0
(B25)
dr) r2 = fn(Pr),and whose solution is F(r) = (~/2@r)'/~J,+1/2(@r) choosing the integral transform
Literature Cited Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions; Dover Publications: New York, 1965. Aris, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts; Clarendon Press: Oxford, 1975. Carrondo, M. Lactic Acid Production with Immobilized-Cellsin Large Pore Supports. Personal communication, 1990. Carta, G.; Massaldi, H.; Gregory, M.; K h a n , D. J. Chromatography with Permeable Supports: Theory and Comparison with Experiments. Sep. Technol. 1992,2 (2), 62-72. Crank, J. The Mathematics of Diffusion; Oxford University Prese: Ely House, London, 1955.
1852 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 Do, H. D. Effect of Intraparticle Convection in the Diffusion of Vapors on a Large Pore Particle. Presented at the 18th Australian Chemical Engineering Conference, Chemeca’90, Auckland, New Zealand, August 1990. Finlayson, B. A. Nonlinear Analysis in Chemical Engineering; McGraw Hill: New York, 1980. Frey, D.; Schweinheim, E.; Horvath, C. The Effect of Intraparticle Convection on the Chromatography of Biomacromolecules. Biotech. Prog. 1993,9,273-284. Glueckauf, E. Theory of Chromatography Part 10-Formulae for Diffusion into Spheres and Their Application to Chromatography. Trans. Faraday SOC. 1955,51,1540-1551. Greco,Jr., G.; Iorio, G.; Waldram, S. P. Unsteady-State Diffusion in Porous Solids: Conditions for Equivalence in Various Geometries, Tram. Zmt. Chem. Eng. 1975,53,55-58. Kucera, E. Contribution to the Theory of Chromatography Linear Non-Equilibrium Elution Chromatography. J.Chromatogr. 1965, 19,231-248. Lu, Z. P.; Loureiro, J. M.; LeVan, M. D.; Rodrigues, A. E. Effect of Intraparticle Forced Convection on Gas Desorption from FixedBed Containing ‘Large-Pore” Adsorbents. Znd. Eng. Chem.Res. 1992,31,1530-1540. Madsen, N. R.; Sincovec, R. F. General Collocation Software for Partial Differential Equations; Lawrence Livermore Laboratory, 1917. Nir, A.; Pismen, L. M. Simultaneous Intraparticle Forced Convection Diffusion and Reaction in a Porous Catalyst. Chem. Eng. Sci. 1977,32,35-41. Prince, C.; Bringi, U.; Schuler, M. ConvectiveMass Transfer in Large Pores Biocatalysts: Plant Organise Cultures. BiotechnoL Prog. 1991,7, 195-199. Rodrigues, A. E.;Ferreira, R. M. Q. Convection, Diffusion and Reaction in a Large Pore Catalyst Particle. AZChE Symp. Ser. 1989,84,84-81. Rodrigues, A. E.; Ahn, B.; Zoulalian, A. Intraparticle Forced Convection Effect in Catalyst Diffusivity Measurements and Reactor Design. AZChE J. 1982,28,541-546. Rodrigues, A. E.; Orfao, J. M.; Zoulalian, A. Intraparticle Convection Diffusion and Zero Order Reaction in Porous Catalyst. Chem. Eng. Commun. 1984,27,237-337.
Rodrigues, A. E.; Lu, Z. P.; Loureiro, J. M. Residence Time Distribution of Inert and Linearly Adsorbed Species in a FixedBed Containing Large Pore Supports: Application in Separation Engineering. Chem. Eng. Sci. 1991,46, 2765. Rodrigues, A. E.; Lopes, J. C. B.; Dias, M. M.; Carta, G. Diffusion and Convectionin Permeable Particle: The Analogybetween Slab and Sphere Geometries. Sep. Technol. 1992,2,1-4. Rodrigues, A. E.; Ramos, A. M.; Loureiro, J. M.; Dim, M.; Lu, Z. P. Influence of Adsorption/Desorption Kinetics on the Performance of Chromatographic Process using Large Pore Supports. Chem. Eng. Sci. 1992b,47,4405-4413. Rodrigues, A. E.;Lopes, J. C. B.; Lu, Z. P.; Loureiro, J. M.; Dias, M. M. Importance of Intraparticle Convection in the Performance of Chromatographic Processes. J. Chromatogr. 1992c,590,93-100. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984. Sardin, M.; Schweich, D.; Villermaux, J. Preparative Fixed-Bed Chromatographic Reactor. In Preparation and Production Scale Chromatography; Granetsos, G., Barker, P. E., eds.; Chromatography Science Series 61;Marcel Dekker: New York, 1993. Stephanopoulos, G.; Tsiveriotis, K. The Effect of Intraparticle Convection on Nutrient Transport in Porous Biological Pellets. Chem. Eng. Sci. 1989,44,2031-2039. Vaporciyan, G. G.; Kadlec, R. H. Periodic Separating Reactors: Experiments and Theory. AZChE J. 1989,35,831-844. Villermaux, J. Deformation of Chromatographic Peaks under the Influence of MassTransfer Phenomena. J . Chromatogr.Sci. 1974, 12,822-831. Young, M.; Dean, R. Optimization of Mamallian-Cell Reactors. Bio/ Technology 1987,5,835-831. Received for review December 22, 1992 Revised manuscript received May 25, 1993 Accepted June 11, 1993’
* Abstract published in Advance ACS Abstracts, August 15, 1993.
