148
I d . Eng. Chem. Res. 1995,34, 148-157
Flow Field and Non-Isothermal Effects on Diffusion, Convection, and Reaction in Permeable Catalysts Jose C. B. Lopes,* Madalena M. Dias, Vera G. Mata, and Alirio E. Rodrigues Laboratory of Separation and Reaction Engineering, Faculdade de Engenharia d a Universidade do Porto, 4099 Porto Codex, Portugal
This paper analyzes non-isothermal effects on the efficiency of large-pore catalyst particles, where intraparticle convection may be important. The particle permeability is taken into account by studying the flow field within the particle simultaneously with the mass and the energy transport processes. Two particle geometries, slab and sphere, are considered. Non-isothermal effects are observed on the occurrence of multiple steady states due to reaction ignition. It is found that, when the flow field equations are taken into account, the maximum values of the efficiency and the maximum temperatures reached inside the particle are lower than the ones estimated considering just the mass and energy transport equations. Introduction Large-pore materials (with pore diameters above 500 A) are finding various applications in separation and reaction engineering such as catalysts, adsorbents, HPLC packings, ceramic membranes, building materials, supports for mammalian cell culture or biomass growth, etc. It has been shown by various studies (Nir and Pismen, 1977; Rodrigues et al., 1982; Cresswell, 1985; Stephanopoulos and Tsiveriotis, 1989; Afeyan et al., 1990; Carta et al., 1992) that intraparticle forced convection is a mass transport mechanism which cannot be neglected in these large-pore materials. These materials can play an important role in industrial reactive processes like in the oxidation of ethylene, an exothermic catalytic reaction, in the steam reforming process, an endothermic catalytic reaction usually carried out at high pressures, and in membrane reactors. Nir and Pismen (1977) studied the effect of intraparticle convection on the performance of large-pore catalysts, for an irreversible isothermal first-order reaction in different particle geometries (slab, spherical, and cylindrical) with uniform internal flow. They observed an appreciable enhancement of the catalyst effectiveness in the region where the reaction rate becomes diffusion controlled. Komiyama and Inoue (1974) solved the problem of intraparticle convection, diffusion, and firstorder reaction in a finite cylinder, parallel to the flow, and provided a method to estimate the intraparticle velocity. Rodrigues and Quinta Ferreira (1988) extended this study to convective flow inside non-isothermal catalysts of slab geometry, and analyzed the effect of intraparticle convection on the measurements of effective diffusivities of large-pore supports. Rodrigues et al. (1992) developed an analogy between the slab and the sphere geometries for the problem of diffusion and convection in permeable isothermal particles. Lu et al. (1993) extended this analogy to isothermal reactive particles, and Rodrigues et al. (1993) studied the effect of intraparticle convection on the conversion of isothermal fured bed reactors. Direct experimental evidence of the effect of intraparticle convection has been referenced by several authors. Cogan et a l . (1982) studied the depolymerization of paraldehyde, catalyzed by nickel sulfate supported on cylindrical pellets of porous silica gel, in a basket-type reactor and verified that the effective reac-
* Author to whom correspondence should be addressed. E-mail:
[email protected].
tion rate increased with the intensity of the convective flux within the porous pellets. Cresswell (1985) presented an experimental study of simultaneous intrapellet diffusion and convection upon various a-alumina catalyst supports with bimodal (comprising pores with diameters larger than 20 pm) and unimodal pore size distributions (with virtually no pores in this range). The main conclusion of this study was that while the pore structure did not affect the effective diffusivity, it exerted a dramatic effect on the permeability. In the more permeable supports, the material flux through the pellet increased by a factor of 2-3 in the presence of relatively small total pressure gradients. Cheng (1986) studied the hydrogenation and oxidation of ethylene and found a relationship between the apparent and the true diffusivities through the intraparticle convective flux. For isothermal reactions the extent of intraparticle convection can be characterized by the intraparticle mass Peclet number, d,, whose magnitude has been reported in various studies. Cresswell (1985), working with a permeable bimodal a-alumina catalyst support with an estimated permeability of 1.9 x m2, assuming a total pressure gradient of 2.5 cm HzO/cm pellet as being fairly typical for turbulent flow in a packed bed, and using a gas viscosity of 1.9 x Paos, estimated the intraparticle velocity as being 0.3 c d s . The calculated intraparticle mass Peclet number was equal to 0.5 and 10 for operating pressures of 1 and 20 bar, respectively. Cresswell concluded that for nearatmospheric processes further development of the pore structure was needed in the catalytic supports to achieve a 1 order of magnitude increase in the catalyst permeability and that intraparticle forced convection is best directed a t those processes operating at moderate to high pressures. Rodrigues et a l . (1982) studied the partial oxidation of butene to maleic anhydride in a highly porous catalyst and reported experimental values of d, from 1 to 15. Rodrigues and Quinta Ferreira (1990) considering, as a typical reference case, the oxidation of o-xylene to phthalic anhydride catalyzed by vanadium pentoxide in a tubular reactor at atmospheric pressure estimated a value of 10 for the intraparticle mass Peclet number. Rodrigues and Quinta Ferreira (1988) considered the gas velocity t o be constant inside a non-isothermal slab geometry particle. In the present study, the intraparticle flow field varies with the internal particle conditions of temperature and pressure. Furthermore, an external pressure gradient is imposed on the surface of
0888-5885/95/2634-0148$09.00/0 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 149 Table 1. Dimensionless Parameters name
Slab
def~tion vsa
intraparticle mass Peclet number
I, =gees
intraparticle heat Peclet number
Ih =-
@&""P Le
J$
Thiele modulus
+a=a
dimensionless maximum pressure drop
/3, = -
u
s
p50
(-AH)
Prater number
'=
Arrhenius number
y==
heat capacities ratio
a=P
IeTs
E
Sphere
C
C"
ps
where E is the intraparticle gas superficial velocity vector (mass average) and P is the intraparticle pressure field. It is assumed that the fluid trajectory is unaffected by the orientation of the pores, the particle permeability, B , is isotropic, and the gas viscosity is a function of temperature and pressure, p = p(T, P). Intraparticle Mass and Energy Conservation Equations. Inside the particle the following conservation equations are observed: (1) The overall mass balance
R
t
V@ = 0
2
where e is the gas density. (2) The mole balance to reactant species A:
Figure 1. Slab and sphere coordinates and surface pressure distribution.
the catalyst particle, which implies the use of a constant reactant mole fraction on the particle surface rather than constant reactant concentration as in Rodrigues and Quinta Ferreira (1988). In this paper the inter-relationship between the velocity, pressure, and temperature fields for slab and sphere catalyst particles is studied, the interaction between momentum, heat, and mass transfer is analyzed, and its impact on the overall performance of the catalyst particle (i.e., the catalyst effectiveness factor) is presented.
Model Formulation Flow Field Equations. The permeable catalyst particle, slab or sphere, is considered t o be immersed in a gas where a stationary, pressure-driven interparticle flow field, representative of the flow inside a fixed bed reactor, is present (Figure 1). The flow in the fixed bed reactor is essentially parallel (in a spatial average sense), and the pressure gradient outside the particles is considered as locally constant and can be determined by Ergun's equation (Ergun, 1952). The flow field inside the particles is governed by Darcy's law for fluid flow in porous media
+v = - - VBP P
(2)
V R A
+ rA(cA,T) = 0
(3)
where CA is the intraparticle reactant concentration and Tis the intraparticle temperature. It is considered that, in the gas phase, there is a diluted reactant, A, which undergoes a heterogeneous first-order reaction, A B, at a local given chemical rate of disappearance per particle unit volume, ~ C A 2')., The vector, NA,represents the molar flux of A, including the effective intraparticle diffusion term (Mason and Evans, 1969) and the convective transport term:
-
R A
= CAZ -
(4)
GeCVYA
where C is the local total mole concentration, YA is the reactant mole fraction, and the effective intraparticle diffusivity is a function of temperature and pressure, @e = GJT, P). (3) The energy balance V'Q
+ Pv'Z -
(-hH)rA(cA,T)
=0
(5)
where P is the intraparticle pressure. The first term represents the rate of heat transport by conduction and convection Zj = (ecvrrj)- AeVT
(6)
where ,le is the effective particle thermal conductivity and cv is the gas heat capacity at constant volume. The other two terms in eq 5 take into account the rate of work done by the fluid against the pressure forces and
150 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 Table 2. Slab and Sphere Coordinates, Operators, and Boundary Conditions slab
sphere Characteristic Dimension a=R Dimensionless Spatial Coordinate 6 = rlR Dimensionless Differential Operators
a = 112 x = zll
(VS),
as
=-
ax
Boundary Conditions
ap *A e = O , n a n d O r r I R -=-=--
ae
ae
aT-O
ae
Boundary Conditions in Dimensionless Variables x = 0 )f = 1) T" = 1, ly = 1
C = 1and 0 I8 I n
x = 1)f = 1)T = 1)ly = 0
q = 0.5(1- cos e)
f=l,T=l
8=O,nandOsEIl * = g = % = O
ae ae
T
T
100 0.1
the rate of heat generated by chemical reaction, respectively. "he reaction is considered to be exothermic, AH -= 0. Viscous dissipation is neglected. The followingadditional constitutiverelationships are considered: (1) The gas is assumed to be ideal, and the pressure, concentrations, and temperature fields are related by
P m c=-m
YAP CA = -*
(7)
(2) The readon rate is described as a first order reaction rA(cA,T) = k(T)cA
(8)
where k(T) is the reaction velocity constant, whose
100'0.1
temperature dependence fOllOWS the Arrhenius equation
k ( T )= k ,
(9)
Particle Boundary Conditions. It is assumed that the yield of a single particle is negligible compared the overall reactor yield, i.e., the mole fraction of reactant, y&, and the temperature, T,,are assumed to be uniform a t the particle surface. The total pressure at the particle external surface decreases according to an linear Pressure gradient in the fixed bed (Figure 1). In this way, both the gas compressibility and the intraparticle flow, driven by the interparticle pressure drop, are taken into account. Dimensionless Equations and Parameters. We now introduce the appropriate dimensionless spatial
Ind. Eng. Chem. Res., Vol. 34, No. 1,1995 151
....* . .
tl
hm
I 0.0
0.5
0.0 1.2-
0.5
0.0
0.5
"
1.0
{0.1 0.1
1
Q
10
Figure 4. Slab effectiveness factor, 7, versus the Thiele modulus, 4, for various values of the intraparticle mass Peclet number, A,, and = 0. Present work, solid lines; Rodrigues and Quinta Ferreira (1988),dashed lines.
A
V*
variables with respect to the geometry-dependent particle characteristic length, a, and the following dimensionless dependent variables
where us = (B/p)(AP$2a)is the average intraparticle interstitial velocity; the following system of differential equations is obtained
(11)
v**21,(1+ pp& v*q + v*T*
[
u
I+
where v* is the appropriate dimensionless nabla operator for each geometry. The dimensionless effective diffisivity and viscosity are defined by
@(T *,q)= G?Je(TS)/GJe(T,SJ = Q&Zes (14a) and p
*
*
(T ) = p(TQYp(T,Q60) = p/p
(14b)
Table 1gives the definition of the relevant dimensionless parameters. Table 2 shows the dimensionless coordinates, operators, and boundary conditions €or the slab and sphere geometries. In the case of the sphere
X
1
.o
Figure 5. Dimensionless mole fraction, f , temperature, T *, and velocity, u*, slab profiles in the multiple steady-state region for = 0, A, = 25 and 4 = 0.5. Present work, solid lines; Rodrigues and Quinta Ferreira (1988),dashed lines.
it is considered that the problem is axis symmetric in the direction of the overall gas flow. The above nonlinear differential equations were solved using numerical methods for a case study in which the Prater number, the Arrhenius number, and the heat capacity ratio take the values that are typical of gas/ solid reactions: ,B = 0.1, y = 20, and a = 1.4 (Rodrigues and Quinta Ferreira, 1988). The dimensionless maximum pressure drop is fixed at Pp = 0.001, except when compressibility effects are analyzed. As it will be shown, the effect of the pressure and/or temperature dependence of the effective intraparticle diffisivity and the viscosity on the simulations results is very small in most cases. Except where indicated, these physical properties were kept constant and equal to the surface properties, that is, G8* = 1 and p* = 1. The numerical solution of the slab geometry equations, a one-dimensionalproblem, was carried out using finite differences based on an irregular point-distributed spatial grid (usually, around 60 intervals) and on an upstream weighting scheme (see, for example, Aziz and Settari, 1979). The sphere problem, a two-dimensional problem, was solved numerically by using a finite elements technique, using six-node triangles (up t o 250 triangles, depending on the difficulty of the problem) with quadratic basis functions (Strang and Fix, 1973) and an adaptive spatial mesh distribution (Pereyra and Sewell, 1975).
152 Ind. Eng. Chem. Res., Vol. 34, No. 1,1995 10
."
h, = 1
rl
rl
h, = 5 1
1
0.001
0.1
0.1
0.5 0.01 0.01
0.1
1
100
10
0.01
fm
0.1
$
1
10
100
0.01 0.01
0.1
10
1
100
4
0
(a) (b) (C) Figure 6. Slab effectiveness factor, q ,versus the Thiele modulus, 4, for various values of the maximum dimensionless pressure drop, Pp. (a) = 1 and Am = 5 , b) = 1 and Am = 10, (c) Ah = 0.1 and I , = 10. The dashed lines show the effect of pressure and/or temperature dependence on the effective diffusivity, Ue = T 3/2P-l, and the viscosity, p = T 1/2. I .oo
f
1.o
0.5
0.99
0.0
I .o
0.5
5EI 0.001
0.001
0.5 o.l
I
Y
-." ,
0.0
0.5
1.0
0.0
0.5
1.o
l ?
1.oo
J0.95 -4 0.0
0.5
X
I .o
I .5
Figure 7. Dimensionless mole fraction, f, and temperature, T *,
u*
slab profiles for various values of the maximum dimensionless pressure drop, bp,Ah = l , A m = 5 , and 4 = 0.1. 1 .o
Effect of Intraparticle Convection on the Catalyst Effectiveness Factor The overall performance of the catalyst particle can be well characterized by its effectiveness factor
4 0.001
\L 0.1 \
0.5 0.0
which, in dimensionless form, and according to the boundary conditions used, can be shown to be for the slab
3 1 + P,w for the sphere 11 = -J" f'f 2+PP0 O T" X e-y[(l/T)-ll
62 sin 0 d6 de
(17)
The effect of intraparticle convection on the catalyst
L-
0.5
0.5 X
1
.o
Figure 8. Dimensionless mole fraction, f, temperature, T *, and velocity, u*, slab profiles for various values of the maximum dimensionless pressure drop, BPI&, = 1, A, = 5 , and 4 = 1.
effectiveness factor is analyzed next, for both the slab and the sphere geometries. The Slab Geometry. Figures 2 and 3 show the effect of intraparticle convection on the slab effectiveness factor, 7. Three-dimensional representations where chosen in order to simultaneously show the effects of the intraparticle Peclet numbers and the Thiele modulus. Figure 2 shows that, for a given value of Ah, a maximum is observed around q5 = 1 and A, I5 and a zone of multiple steady states is observed for Am > 5 and 4 1. Three states are possible: a nonignited
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 153
1
,
;
:
O
0
+
r\ l
10
1
h,= 1 0.1 0.1
10
1
4)
(b) Figure 9. Sphere effectiveness factor, q-, versus the !l'hiele modulus, 4, for various values of the intraparticle mass Peclet number, . , 1 (a) Ah = 1 and (b) = 10. - VP 4
f
T*
1
1.5
0.8
1.4
a 0.6
1.3
11
'i
. .. ....... .. .. .. ....... .. .. ....... .. . . .
= ~ .
0.4
1.2
0.2
1.1
0
1
Figure 10. Sphere contour maps for a case where thermal effects are important, Ah = 1, 1 , = 30, and 4 = 2. (a) Dimensionless temperature, T *, showing the streamlines and (b) Dimensionless mole fraction, f , showing the molar flux field.
particle with q close to 1, an ignited state with large values of q, and an unstable state with intermediate values of q. Figure 3 shows that for values of Ah less than unity, the effectiveness factor curve is independent of Ah. Similar qualitative results had been obtained by Quinta Ferreira and Rodrigues (1988) who assumed constant intraparticle gas velocity. In Figure 4, the values of the q obtained from the present model equations (solid lines) are compared with Quinta Ferreira and Rodrigues (1988)results (dashed lines) for the case where the temperature effects are more pronounced, Ah = 0. The same overall features are observed in both cases, the main difference being that the maximum values of q obtained in the present work are considerably smaller than the ones obtained by Quinta Ferreira and Rodrigues (1988). The explanation for these differences can be drawn from Figure 5, which shows the dimensionless reactant mole fraction, temperature and gas velocity profiles, for a case with multiple states (Ah = 0, A, = 25, and 9 = 0.5). In such cases, where temperature effects in the catalyst are important, the
internal flow field plays a decisive role in the concentration and temperature profiles. The following is observed: (1)The results for the nonignited state from both models are similar, and this observation can be generalized to any set of model parameters. (2) For the ignited state, the results for this work show higher reactant mole fractions and lower temperatures. This result can be rationalized by comparing the temperature and the velocity profiles inside the particle for the two models. For 0 < x < 0.2, the gas expands and faces increasing temperatures. Part of its energy is spent on the work done against the pressure forces, resulting in a decrease of the temperature rise rate, thus lowering the chemical conversion. The maximum in temperature is now lower and deeper inside the particle. For 0.2 < x 0.5, the temperature profiles are similar in both models, but since the gas velocity is higher than the mean interstitial velocity (considered constant in the previous work), there is a significant decrease in the gaskatalyst contact time and once again the reaction conversion decreases. In the region 0.5 < x 0.9, the catalyst region is depleted of reactant and for x > 0.9, the countercurrent diffusion plays its role because the gas velocity is now lower than the mean interstitial velocity. (3) For the unstable state, the temperature profiles are symmetric, and the results of this work show higher temperatures, lower reactant mole fractions, and lower velocities in the reactant enriched peripheral regions of the catalyst, when compared with the isothermal case. The extent of the effect of the internal pressure drop is characterized by the dimensionless maximum intraparticle pressure drop parameter, Pp. Usually, this parameter takes values of the order of the ratio between the particle and the reactor characteristic dimensions, that is, around or lower. The results of simulations show that the solution is rather insensitive to the values of PPup to 10-l. On the other hand, this parameter can take larger values in the cases of reactors operating at very low 'pressures or membrane reactors. Figure 6 shows that the effect of Pp on the slab effectivenessfactor is higher when 9 < 2 and Pp > 0.1, in which case q decreases with increasing PP,and it can take values
154 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 10,
10
I
*rl
*rl
1
0.1
1
l--z-l --c
0.1
0.001 0.1 0.1
10
1
0.5
-e-
0.1
--t
0.001
1
4
(a)
--c
4
(b)
Figure 11. Sphere effectiveness factor, q , versus the Thiele modulus, @, for various values of the maximum dimensionless pressure drop, = 30. (a) Ah = 1 and (b) Ah = 10.
pp, and Am
1""
u Am/+
0.8 0.01
0.1
1
(4
J
10
10
100
4
0.01 0.01
0.1
1
(b)
10
loo
4
Figure 12. (a) Slab highest dimensionless temperature, T Im, versus the Thiele modulus, 6,for various values of the intraparticle mass Peclet number, Am, and ,&,/Ah = 10. (b) Corresponding plot for the effectiveness factor, q.
lower than unity for small values of the Thiele modulus. The reason for this is 2-fold: the first is a direct consequence of the definition of (eq 12); the second results from the actual and correct model prediction of the possibility of gas cooling inside the particle due to the gas expansion, as can be observed in Figure 7. The effect of PP in the particle profiles is illustrated in Figure 8. The main observation is the change of regime for large values of &, which can be seen in the gas velocity profile. For Bp I0.1 the velocity profile is dictated mainly by the temperature profile, while for PP.= 0.5, u* increases monotonically along the particle axis due to the predominance of the gas expansion by pressure decrease inside the particle. Finally, the issue of the importance of the temperature andor pressure dependence of the effective diffisivity and the viscosity is addressed. The dashed lines
in Figure 6b show the results obtained assuming that @e = T312P-l (see Bird et al., 1960) and p T1l2 (independent of pressure for low reduced pressures; see Reid et aZ.,1977). The main conclusion is that only in the case of a large external pressure drop, PP= 0.5, does the temperature andlor pressure dependence of the physical properties become important, in which case the effectiveness factor shows lower values than in the case of constant physical properties, but the functional trends are still maintained. The Spherical Geometry. Although the spherical geometry introduces, compared to the slab geometry, two-dimensionality to the conservation equations, it is reasonable to expect that the overall performance for both geometries is similar, since the governing equations are formally the same. Nevertheless, as it is shown in Figure 9, in the region where thermal effects 0~
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 166
*
lo000
T m
3
0.01 0.01
0.1
1
10
4 (b) Figure 13. (a) Slab maximum dimensionless temperature, T *m,versus the Thiele modulus, 6, for various values of the intraparticle Peclet numbers ratio, A&,, and 1 , = 60. (b) Corresponding plot for the effectiveness factor, r].
I
T*"
I
'mAh
2.5
T*mm
I
2.5
2 2
I .5
1.5 '
I
-
0
1
20
I
I
I
I
40
60
80
100
4n
(a)
1 0
5
10
15
20 Am%
Figure 15. Slab and sphere dimensionless maximum temperature, T *", versus the intraparticle Peclet numbers ratio, &&, in the plateau region. 1.5
*
I
I
I
!
0
50
100
I50
(b)
'm
Figure 14. Maximum dimensionless temperature, T versus the intraparticle mass Peclet number, A,,,, and various values of the intraparticle Peclet numbers ratio, n&h. (a) Slab and (b) sphere
are important, the maximum values found for the sphere effectiveness'factor, 7, are considerably smaller than the ones found for the slab. Moreover the occurrence of multiple steady states is now limited to a narrower range of "hiele modulus values, and it occurs
only for very high values of the intraparticle heat Peclet number, Am. This significant change in behavior, from the slab to the sphere, of the overall performance of the catalyst is mainly due to geometric factors. First, in the sphere, the external surface area per unit volume of the catalyst is six times larger than in the slab. Second, in the sphere, the gas can move in two directions, contrary to the one-dimensional movement in the slab. This is illustrated in Figure 10, where temperature and mole fraction contour maps as well as the reactant molar flux field and the gas streamlines are shown for a case where thermal effects are important. The reactant molar flux field vector is given by eq 4. The steady-state streamlines are identical to the trajectories obtained by integrating the velocity vector components, &ldt = u,(xy) and dyldt = u,(xy), for various initial conditions t = 0; z = X O ; y = yo. Figure 10a shows the temperature map inside the particle and the gas streamlines, represented
156 Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 I
I
f
I
f
f
0.5
0.5
0.5
80
E h,/h,=
10
100
n
0I
0 0.5
0
0
1
0.5
X
1.8
1.8
T*
X
1 1
I
T*
14
I 0
0.5
1
0
O5
x
I
0
0.5
1 X
(C) (a) (b) Figure 16. Dimensionless mole fraction, f , and temperature, T', slab profiles in the plateau region for various values of 1,. (a) l m / A h = 5, (b) l m / A h = 6.7,and (C) Am/& = 10.
as solid lines. These lines show a slight two-dimensional behavior resulting from the gas first moving away from the hot spot in the catalyst, and then moving back to its original lateral position and eventually getting closer t o the axis of the particle. This is a consequence of the thermal gas expansion and results in a net gas acceleration along the particle. This gas acceleration exists in particle regions where the reactant is already depleted, as can be observed in Figure lob. Due to the gas acceleration, fresh reactant is sucked in from the particle surface, as it is shown by the reactant molar flux vectors. This, in turn, results in higher reactant concentrations and lower effectiveness factors. Therefore, the gas acceleration due to thermal effects coupled with the sphere's larger specific area explains the smaller values found for the sphere particle efficiency. The dependence of the sphere effectiveness factor on Pp is shown in Figure 11, and it is consistent with the results found for the slab.
Effect of Intraparticle Convection on the Catalyst Temperature An important issue in the design and operation of catalytic reactors is the prediction of the maximum temperature that can be achieved inside the catalyst particles. In the absence of intraparticle convection,the maximum dimensionless temperature resulting from the competition of reaction and diffusion is given by T *ma = 1 + p (Prater, 1956) and is independent of the Thiele modulus. When intraparticle convection is important, the highest temperature inside the particle becomes dependent not only on the Thiele modulus, 4, but also on the intraparticle heat and mass Peclet numbers ratio, &/Ah. Figure 12a shows the dependence of the highest dimensionless temperature observed inside a slab particle, T, ,* as a function of the Thiele modulus, 4. For comparison purposes, Figure 12b shows the corresponding values of the effectiveness factor, q. As expected, for low values of the intraparticle mass Peclet, Am = 1,T ,* increases for low values of 4, and for high values of 4, it reaches a constant maximum value of T * m a = 1 + /3 = 1.1. For higher values of Am, T ,* goes through a maximum corresponding to the
highest effectiveness factor value. The influence of the ratio on T ,* when intraparticle convection is important, A, = 60, is shown in Figure 13a, again with the corresponding plot of q in Figure 13b. When A,,,/& 5 1.0, T,* behaves very much as in the case where A, is small, reaching a constant high value of T *ma = 1 P = 1.1. As A,,& decreases, T *m,increases, going through a maximum value corresponding again to the highest effectiveness factor value. If the maximum dimensionless temperature, T *ma, for each pair of 1 , and Am/&, is plotted against the intraparticle mass Peclet number, Am,the plots of Figure 14 are obtained. These values of T * m m correspond to the maximum temperature that can be achieved in a catalyst independently of its kinetics. One observes that for a given value of &,&, T*,,, increases with Am, reaching a constant value plateau for sufficiently high A,. Similar results are obtained in both the slab and spherical particles, the main difference being that, in the sphere, the maximum temperatures reached are lower than in the slab. For both slab and sphere, the plateau value is a linear function of &/Ah, for Am/Ah up to 20, as it is shown in Figure 15. Figure 16 shows the mole fraction and temperature profiles in this plateau region. One can observe that both mole fraction and temperature tend to constant profiles as A, increases, but while for A,,,/& = 5, the profiles move in the direction of the external pressure gradient, for &,/Ah = 10 the profiles move in the opposite direction, and an inversion is seen around ,?,/Ah = 6.7.
+
Conclusions This paper has shown that, when intraparticle convection is important, non-isothermal effects play an important role in the intraparticle flow field of catalyst particles. This can significantly change the calculated catalyst effectiveness factors in comparison with the values obtained when the gas velocity is assumed constant. Furthermore, it was shown that, for the sphere geometry, the thermal effects are not as severe as in the case of the slab and for large values of the intraparticle heat Peclet number there is no occurrence of multiple states in the catalyst.
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 157 In the range of the physical parameters studied, the slab and the sphere show similar behavior. Pressuredriven compressibility effects, characterized by the pp parameter, are negligible up to values of 0.1, but can be significant for higher values of ,Bp. Finally it has been shown that the maximum temperature achieved inside the catalyst particle, an important design and operation variable, is a function not only of the Prater number (as is the case when intraparticle convection is not important) but also of the intraparticle heat and mass Peclet numbers, and in particular of their ratio, &/Ah. When intraparticle convection dominates, the maximum temperature that can be achieved inside the catalyst particle increases with &/Ah but does not depend on the intraparticle mass Peclet number alone.
Notation a = particle geometry characteristic B = catalyst permeability, m2
dimension, m
C = total mole concentration, malm-3 CA =
particle reactant concentration field, m ~ l - m - ~
c, = gas heat capacity (constant pressure), JSkg-l-K-l
cv = gas heat capacity (constant volume), J*kg-l*K-l a, = effective intraparticle diffusivity, m2s-l E = activation energy, J-mol-l f = dimensionless mole fraction (eq 10) -AH = specific heat of reaction, J-mol-1 k = reaction kinetic constant, s-l I,= slab thickness, m NA = reactant molar flux vector, malm-%-' P = particle pressure field, Pa = heat flux vector, J.m-'.s-l r = sphere radial coordinate, m = chemical rate of disappearance of reactant per particle unit volume, moim-3-s-l R = sphere radius, m 5% = ideal gas constant, Jomol-l.K-l s = scalar field T = particle temperature field, K T * = dimensionless temperature (eq 10) T *m = highest dimensionless temperature T *max = maximum dimensionless temperature si = vector field ii = intraparticle gas superficial velocity vector, ms-l us = intraparticle gas superficial average velocity, m-s-l u* = dimensionless velocity (eq 10) V = particle volume, m3 x = dimensionless slab spatial coordinate Y A = reactant mole fraction in the gas phase z = slab spatial coordinate, m Greek Symbols
a = gas heat capacities ratio (Table 1)
p = Prater number (Table 1) pp = dimensionless maximum intraparticle pressure drop (Table 1) 7 = catalyst effectiveness factor (eq 14) 8 = sphere angular coordinate 1 h = intraparticle heat Peclet number (Table 1) Am = intraparticle mass Peclet number (Table 1) y = Arrhenius number (Table 1) 1, = effective particle thermal conductivity, J-m-l-s-l*K-l p = gas viscosity, N*m-l*s-l 6 = dimensionless sphere radial coordinate e = gas density, kgm-3 = dimensionless pressure (Table 1) 4 = Thiele modulus (Table 1) Subscripts 0 = denotes parameter values at position z = 1 o r r = R and 8 = n
s = denotes particle surface conditions
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Received for review November 23, 1993 Revised manuscript received August 1, 1994 Accepted September 19,1994@
IE930600P @
Abstract published in Advance ACS Abstracts, December
1, 1994.
