Znd. Eng. Chem. Res. 1994,33, 273-276
273
Effect of Intraparticle Convection on the Effectiveness of a Biporous Pellet+ V. K. J a y a r a m a n Chemical Engineering Division, National Chemical Laboratory, Pune 41 1 008, India
The effect of convective flow induced by an external pressure gradient on the performance of a biporous pellet has been studied. The effectiveness factor of such a pellet depends on the convective flow parameter, A, the Thiele modulus, and a,which characterizes the ratio of macropore t o micropore diffusion times. Considerable enhancement in effectiveness due to convection occurs, especially when macropore diffusion is the controlling step. Introduction
micropores
Many of the industrially used catalyst pellets are biporous. When porous particles are pelletized, bidispersed distributions emerge. These pellets have large pores called macropores, from which the micropores branch out. Due to the practical importance of these micro-macro pellets, a large number of analyses have been carried out for the prediction of their effectiveness and characterization of the design parameters (Carberry, 1962; Hashimotto and Smith, 1974; Ors and Dogu, 1979; Jayaraman et al., 1983; Namjoshi et al., 1984). The mean radius of macropores can vary between 100 and 1500 8. The alumina pellet used by Hashimotto and Smith (1974) had a mean macropore radius of 1200 8.Due to the presence of such large macropores there can be considerableconvectiveflow in the pellets. This convective flow can alter the concentration profiles in the pellet and hence the effectiveness to a great extent. It must be mentioned here that convective flow can arise within a pellet due to a change in number of moles during the course of a reaction (Weekman and Gorring, 1965) or due to an external pressure gradient. An analysis of simultaneousconvection (caused by an external pressure drop), diffusion, and reaction has earlier been considered in detail by the seminal papers by Komiyamaand Inoue (1974)and Nir and Pismen (1977a,b). Since then a number of analyses dealing with intraparticle convection have been made (Cogan et al., 1982; Rodrigues et al., 1982; Cresswell, 1985; Dogu et al., 1989;Rodrigues, 1991; Rodrigues et al., 1991a,b; Lu et al., 1992a,b; Quinta Ferriera et al., 1992). Manufacture of large pore catalysts and adsorbenta has made rapid progress to take advantage of the improved efficiency due to convection (Rodrigues et al., 1991a,b). In the present analysis an attempt has been made to solve the bidisperse catalyst pellet problem incorporating convective flow caused by an external pressure gradient.
cmi= cma at x = Lmi macropores
with the boundary condition cma= co at y = fLma
t
NCL Communication No. 5869. 0888-5885/94/2633-0273$04.50/0
(5)
The above set of boundary conditions for a biporous pellet are identical to the boundary conditions used by Nir and Pismen (1977a,b). These conditions can be used in packed bed reactors with high pressure drop and in reactors with small extents of conversion. By defining x=-, X Lmi
cmi='mi
Y = L ,
Cma cma=-
CO
CO
'ma
Theoretical Development The following are assumed to hold good in the subsequent development of the balance equations: 1. The pellet is a biporous slab with sealed edges. Both sides of it are maintained at a uniform concentration, and they bear a pressure difference. 2. Convective flow is restricted to the large pores. 3. An irreversible first order reaction occurs in the micropores. With these assumptions the balance equations for the micropores and the macropores can be written as follows:
(3)
a = (1- €)-
Lm82 Dma L:, m 'i
the balance equations can be made follows: micropores d2Cmi -dX2 -amiPCmi with the boundary conditions 0 1994 American Chemical Society
274
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994
-dCmi - 0 dX
at X = O
Cmi= Cma at X = 1
(12)
macropores
with the boundary conditions
C m , = l at Y = f l (14) By solving the micropore equations subject to the relevant boundary conditions, we get 0 Figure 1. Effect of convection parameter X on the effectiveness of the pellet (a = 1).
cosh(@miX) Cmi = Cma cosh(ami) Thus
3
a = 50
X=l
= Cmaami tanh(ami)
(16)
I x=5
2 1-10 3 1=20 4. 1 = 50
By letting
am: = ciamitanh(ami)
(17)
The macropore equation can be written as
!= 0.4-
Solving eq 18 subject to the boundary condition represented by eq 14, we get Cma =
sinh P2 exp(PIY) - sinh P1 exp(P2Y) sinh(P, - P1)
(19)
0.2-
0.1
I
10
0
Figure 2. Effect of convection parameter X on the effectiveness of the pellet (a = 50).
where Pl,2
=
$(
1 f (1 +
T) ) 4@.,,2
lJ2
(20)
Defining the effectiveness factor as observed rate in all micropores
'= intrinsic rate at pellet surface conditions An expression for it can be obtained as II=-
am:
(UP21 - (1/PJ 2 coth P1- coth P2 a @mi
(21)
An expression for the effectiveness factor in the absence of convection can be obtained (either by solving the design equations without the convective flow term or by putting X = 0 in eqs 20 and 21) as
Results and Discussion The effectivenessof a biporous pellet depends essentially on three dimensionlessparameters, viz., a,@mi, A (the other dimensionless parameter, @ma, depends on @mi and a).?mi is the well-known Thiele modulus whose characteristics have been well studied and well interpreted. The parameter a characterizes the ratio between the diffusion times of the macropores and micropores. For fixed values
of the Thiele modulus an increase in a increases the diffusion time of the macropore. Thus with an increase in the value of a the effectiveness of the pellet decreases and macropore diffusion becomesthe controlling step (Ors and Dogu, 1979). The parameter h represents the ratio of the macropore diffusion time to that of the convection time. This dimensionless parameter (characterizing the internal flow) can be related to the external flow parameters as follows (Nir and Pismen, 1977a,b):
( R e ) ( S c ) Kf Lma2 1 C i For Schmidt number typical for liquids and for Re N 100, this convective flow parameter was found to be N 10 for pellets with an average pore size of 100 A. From their experimental data on the oxidation of butene to maleic anhydride on catalyst pellets with mean pore diameter of lo4A, Rodrigues et al. (1982) estimated this parameter to be varying between 1 and 15. The influence of this parameter on the effectiveness of the biporous pellet has been plotted in Figures 1-3. It can be seen from the figures that there is considerable enhancement in the biporous pellet effectiveness for high values of a,i.e., when macropore diffusion is controlling. The enhancement in effectiveness above its value in the absence of convection has been plotted in Figures 4-6. It can be seen from the figures that at low values of a the maximum in enhanceA=
Ind. Eng. Chem. Res., Vol. 33, No. 2,1994 275 a = 50 I. x = 5
2 A-10
3 x=20 x = 50
4. 2.5
0
Figure 3. Effect of convection parameter A on the effectiveness of the pellet (a = 10). a = io
increasing amounts of convective flow. The actual enhancement in effectiveness and the maximum value of the enhancement depend on the ratio of the diffusion times of the micropores and the macropores. The result of the present work can be useful for deciding on the system parameters to obtain optimum reactor performance.
2 x-10 3.
Q Figure 6. Enhancement in catalyst effectiveness due to convection (a = 50).
x = 20
Nomenclature = concentration in the macropores C,, = dimensionless concentration in the macropores c,; = concentration in the micropores C,i = dimensionless concentration in the micropores D,, = diffusion coefficient in the macropores D,; = diffusion coefficient in the micropores K = permeability coefficient L,, = half-length of the pellet L,; = length of the micropore Re = Reynolds number Sc = Schmidt number u = intrapellet velocity 3c = micropore length variable X = dimensionless micropore length variable y = macropore length variable Y = dimensionless macropore length variable c,
9 Figure 4. Enhancement in catalyst effectiveness due to convection (a = 10). 3.5
1 a =i
;6’
1. A - 5 2 X-IO
3 x=20 4 x=50
2.5
Greek Letters a = parameter defined by eq 9
= parameter defined by eq 20 = pellet porosity A = parameter defined by eq 8a
c
4,; = Thiele modulus defined by eq 8 $ = ratio of effective diffusion coefficient to diffusion
coefficient
+ Figure 6. Enhancement in catalyst effectiveness due to convection (a = 1).
ment occurs a t higher values of the Thiele modulus. With an increase in the value of a the maximum shifts t o lower values of the Thiele modulus.
Conclusions The effectiveness of a biporous pellet (like in the case of a monoporous pellet) increases considerably with
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276 Ind. Eng.Chem. Res., Vol. 33, No. 2, 1994 Jayaraman, V. K.; Kulkarni, B. D.; Doraiaswamy, L. K. A Simple Method for the Solution of a Class of Reaction Diffusion Problems. AZChE J . 1983,29,521-523. Komiyama, H.; Inoue, H. Effect of Intraparticle Convection on Catalytic Reactions. J. Chem. Eng. Jpn. 1974, 7,281-286. Lu, Z. P.; Loureriro,J. M.; LeVan, M. D.; Rodrigues,A. E. Intraparticle Convection Effect on Pressurization and Blow Down of Adsorbers. AZChE J. 1992a,38,857-867. Lu, Z. P.;Loureiro, J. M.; Rodrigues, A. E. Single Pellet Cell Measurement of Intraparticle Diffusion and Convection. AZChE J. 1992b,38,416-424. Namjoshi, A. N.; Kulkarni, B. D.; Doraiaswamy, L. K. Initial Value Approach to a Class of Reaction-Diffusion Systems. AIChE J . 1984,30,915-924. Nir, A,; Pismen, L. Simultaneous Intraparticle Forced Convection, Diffusion and Reaction in a Porous Catalyst-I. Chem. Eng. Sci. 1977a,32,35-41. Nir, A. Simultaneous Intraparticle Forced Convection,Diffusion and Reaction in a Porous Catalyst-11. Chem. Eng. Sci. 1977b,32, 925-930. Ora, N.; Dogu, T. Effectiveness of Bidisperse Catalysts. AZChE J . 1979,25,723-725. Quinta Ferreira, R. M.; Costa, A. C.; Rodrigues, A. E. Dynamic Behaviour of Fixed Bed Reactors with Large Pore Catalysts: A
Bidimensional Heterogeneous Diffusion Convection Model. Comput. Chem. Eng. 1992,16,721-751. Rodrigues, A. E. Letter to the Editor. AZChE J. 1991,37,1117-118.
Rodriguee,A.;Ahn, B.; Zoulalian,A. Intrapartice Forced Convection Effect in Catalyst Diffusivity Measurements and Reactor Design. AZChE J. 1982,28,541-546. Rodrigues, A.; Loureiro, J.; Derreiera, R. Intraparticle Convection Revisited. Chem. Eng. Commun. 1991a,107,21-33. Rodrigues, A.; Zuping, L.; Loureriro, J. M. Residence Time Distribution of Inert and Linearly Adsorbed Species in a Fixed Bed Containing "Large Pore" Supports: Applications in Separation Engineering. Chem. Eng. Sci. 1991b,46, 2765-2773. Weekman, V. W., Jr.; Gorring, R. L. The influence of volume change on gas phase reactions in porous catalysts. J . Catal. 1965,4,260270.
Received for review September 17, 1993 Accepted October 27, 19930
* Abstract published in Advance ACS Abstracts, January 1, 1994.
