468
Ind. Eng. Chem. Res. 1995,34, 468-473
A Methodology for the a Priori Selection of Catalyst Particle Models? K. Venkat Redd9 and C. V. S. Murty* Modelling and Simulation Group, Chemical Engineering Division, Indian Institute of Chemical Technology, Hyderabad 500 007, India
Numerical simulation with catalytic reactor models demands the use of computationally efficient intraparticle models because of the need for repetitive evaluation of the effectiveness factors. Some of the most widely used models of this type are the flux and the dusty-gas models, based on the Stefan-Maxwell equation, and the effective diffusivity model. But these are known to have varying degrees of rigour and computational speed, and in the absence of any conclusive evidence about the relative efficacy of these models, one often tends to use complex models in situations when simpler models suffice. The present work addresses this problem and suggests a procedure for the a priori selection of a suitable intraparticle model in a given situation. This is based on the interactions among the various constituents making up the reaction system and the type of regime prevailing within the catalyst pellet. With the help of this methodology, it has been shown that the effective diffusivity model, in spite of its shortcomings, could be safely used even when the intraparticle regime is diffusion-controlled, if the flux interactions among the species are low. Its use should however be restricted to cases with reaction-rate-controlled regimes, if there are strong interactions.
Introduction In fixed bed catalytic reactors, intraparticle resistances to heat and mass transfer often play a significant role, affecting the overall rate processes. In such cases, a rigorous mathematical treatment of the phenomena occurring in catalyst particles should form an essential part of the overall reactor modeling. Catalyst particle modeling has long been an area of intense research activity, and considerable effort has gone into the model development, application aspects, and newer solution algorithms. The most notable of the intraparticle models with an increasing degree of complexity are the effective diffusivity model, the Stefan-Maxwell equation-based flux model, and the dusty-gas model. In a recent paper, Krishna (1993) gave an excellent review of the various types of diffusion occurring in porous media and presented a unified model combining all of these. He has also stressed the need for using complex models as, he felt, the effective diffusivity type models are inadequate, for they cannot even represent the phenomena in a mechanistic fashion. Salmi and Warna (1991) have made a comparative study of the effective diffusivity model and the StefanMaxwell model in two case studies involving methanol synthesis and the water-gas shift reaction. Although they noted considerable differences in intraparticle concentrations predicted by the two models for highly diffusion-controlled regimes, the effectiveness factors were nearly the same in both cases. They advocate, however, the use of the Stefan-Maxwell model in reactor simulations only if major differences between the two models are observed for the single pellet case. Analytical solution for the intraparticle models mentioned above is possible only in some simple or hypothetical cases (Skrzypek, 1984)and is rarely feasible in real situations involving highly nonlinear rates, thus
* Author t o whom correspondence
should be addressed. IICT Communication No. 3384. t Present address: I F F W S A T , Post Box No. 25862, Abu Dhabi, U.A.E. +
necessitating the use of numerical methods of solution. For efficient reactor simulation studies, the catalyst particle model employed should, therefore, be computationally fast. The effective diffusivity model is easily the fastest method since it involves the solution of far fewer differential equations than the other two models and is, therefore, a natural choice. But the central question is how good are its predictions in different situations particularly when compared to those of the other methods. When could one safely use it, when should one steer clear of it, and is there any trade-off between accuracy and computational time while using it are some of the other pertinent questions. Salmi and Warna (1991) have not provided any clear cut answers nor did Krishna (1993). In fact, Krishna discourages the use of the Fickian type models because they do not even qualitatively represent the diffusion process in multicomponent systems. But based on our experiences in the modeling of industrial reactors, we feel that the effective diffusion coefficient approach, though it fails in the mechanistic sense, is nevertheless adequate in several situations and could be safely used for reactor simulation studies in such cases. The basic problem, however, is how t o identify these a priori. The present work has been undertaken with the view of formulating a procedure for defining such situations. The basic difference between the effective diffusivity approach and the other two models is that the latter take into account the effect of interactions among the various species on the diffusion of all the components, while the former ignores these interactions. So it is legitimate to assume that the effective diffusivity model will be adequate if the conditions are such that the interactions are negligible. If a way could be found to ascertain a priori whether the interactions are significant or not, it could be used to determine whether there is any need for complex models. Fortunately, such a procedure is available and is described by Krishna (1993). The modus operandi we intend t o pursue now is to determine the type of interactions present among the species in some systems with the help of this procedure and also the type of intraparticle regime
0888-5885/95/2634-0468$09.00/0 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 469 prevailing and then use this knowledge together with the information obtained from an evaluation of the different models in those cases to find out if there exists any correlation between the two, so that a methodology could be established for the a priori selection of an intraparticle model. The reaction systems chosen for investigation are ammonia synthesis and the steam reforming of methane, and the models tested are the effective diffusivity model (EDM), the Stefan-Maxwell flux model (SFM), and the dusty-gas model (DGM). The dusty-gas model is known to be the mast rigorous of the three models and hence can serve as a standard against which the performance of the other two models can be tested. The selection of the two cases for the present study is guided by the fact that these two industrially important reactions offer an ideal testing ground for the intraparticle models, as they represent widely varying regimes within the catalyst pellet. The reforming reactions are known to be highly diffusion-controlled,while in the case of the ammonia synthesis reaction, both pore diffusion and reaction are known to play equally important roles.
Model Description Important features of the three models are described in this section. The catalyst particle is assumed t o be slab-shaped and also isothermal. Dusty-Gas Model. For a set of N isothermal reactions with M components taking place inside a porous catalyst, the flux relations of the dusty-gas model for a component i take the form (Kaza and Jackson, 1980)
-
(i = 1, ..., M) (1)
D' The above equations can be transformed, after some mathematical manipulation, into
(i = 1,..., M) (2) where
Flux Model. This is a special case of the dusty-gas model with the pressure being invariant inside the pellet. For this condition, the dusty-gas model equations are reduced to the familiar Stefan-Maxwell equations:
(i = 1, ...,M - 1) (7) The material balance equations given earlier hold here also. Effective Diffusivity Model. The flux model can be further simplified by defining an effective diffusivity for component i, such that the model equations are transformed to d%i dz2
De,i-= rP2Rrrt,
(i = 1,...,MI
(8)
The effective diffusivity has been calculated for the ammonia synthesis case on the basis of Stefan-Maxwell equations, using the procedure suggested by Krishna (1989). On the other hand, Wilke's approximation has been used for the reforming of methane (Bird et al., 1960). Appropriate boundary conditions are used in all cases for the solution of the equations.
Numerical Solution The model equations encountered in all three models are of the two-point boundary value type. Global spline collocation involving Jacobi polynomials has been employed for the solution of the equations. The rate term has been linearized using Taylor's expansion, and a mild under-relaxation has been employed at times for the concentrations. The details of the solution procedure used are not given here as they are available elsewhere (Finlayson, 1980; Murty and Reddy, 1992). One could get a rough idea about the computational speeds of the different models from the fact that the effective diffusivity model requires the solution of just one differential equation as against three in the case of the StefanMaxwell model and four in the case of the dusty-gas model, when only one reaction is taking place.
Application of the Models Case Study 1: Ammonia Synthesis. Ammonia is synthesized according to
+
'IzN2 3/2H2--L NH,
and
For a complete description of the problem, material balance equations are needed and these are given by
(6)
The above reaction is carried out on the industrial scale using an iron catalyst. The regime inside the catalyst pores is known t o be diffusion-controlled in some parts of the bed and reaction-rate-controlled in others. As the reactants nitrogen and hydrogen diffuse into the pores in the presence of inerts made up of methane and argon, the reaction product ammonia diffuses out of the pores. The method described by Krishna (1993) is now used t o determine the nature of interactions among the various species. In an M-component ideal system, the concentration gradients may be related t o the diffusion fluxes by the Stefan-Maxwell equation (Bird et al., 1960):
470 Ind. Eng. Chem. Res., Vol. 34,No. 2, 1995
dYi -
dz
5
(Naj-N’i)
j=lj#i
(i = 1, ...,M - 1)
cD,
(10) with
-
22 .o
-
20.0
-
f
Y
r
where
.-
M
N, =
18.0-
ul
.-0 5 .-
Ni i=l
16.0-
>
0
Equation 10 can be rearranged and written in matrix notation as
0
14.0
12.0
where the elements of the B matrix are defined as (12)
~
2 .o
1 .o
3 .O Thiele
4 .O
5 .O
6 0
modulus
Figure 1. Variation of deviations in effectiveness factors with Thiele modulus. Table 1. Operating Data for the Synthesis Converter data set
and
1
2
3
4
3.2 20.6 65.1 7.3 3.8 663.0 207.0
10.7 18.4 59.0 7.8 4.1 785.0 207.0
14.2 16.1 57.4 7.9 4.4 775.0 207.0
18.3 16.1 53.0 8.1 4.5 712.0 207.0
bulk gas composition (mol %) NHs Nz
Because of the Gibbs-Duhem restriction, the order of the matrices in equation 11 is reduced to M - 1. It follows then from eq 11that
[J] = -c[B]-’Vy
(14)
The diffusion fluxes J for the M - 1 independent components in a system can thus be explicitly expressed in terms of concentration gradients of all the components. An analysis of these fluxes will then indicate how the diffusion flux of any component is influenced by the concentration gradients of several other components present in the system. In a typical situation encountered in the synthesis converter involving an ideal mixture of ammonia (11, nitrogen (21, hydrogen (3), and inerts (4) at a temperature of 712 K and pressure of 207 atm with y1= 0.183, y2 = 0.161, y3 = 0.53,and y4 = 0.126, the fluxes of the individual components can be written, using the procedure described above, as
I:] 20
1
-6
;!]
VYl
[
= - c[ (Il t 8 x x Yh] (15) It is evident from the above equation that both ammonia and nitrogen have a reasonably good interaction with hydrogen with the result that their fluxes are considerably influenced by the concentration gradient of hydrogen. On this basis, one could expect considerable variations in the values of fluxes and effectiveness factors predicted by the effective diffusivity model and the other two models. The three intraparticle models are now tested using the plant data (see Table 1)reported by Singh and Saraf (1979). The reaction rate used is a modified version of the Temkin-Pyzhev equation. Other pertinent data like the thermodynamic data, the equilibrium constant, etc. have been taken from the work of Shah et al. (1967). The results of the model evaluation are presented as
Hz CH4
Ar temperature (K) pressure (atm)
Table 2. Model Predictions for Ammonia Production effectiveness factor data set no. EDM SMF DGM Thiele modulus 1 2 3 4
0.3385 0.2021 0.2646 0.5973
0.4381 0.2860 0.3710 0.7296
0.4108 0.2610 0.3386 0.6955
2.27 4.90 3.90 1.59
effectiveness factors in Table 2. The magnitude of effectiveness factors confirms that diffusion and reaction are rate controlling in different regions of the bed. A comparison of the three models reveals that the differences in effectiveness factors predicted by the effective diffusivity model and the other two models are tending to increase with decreasing effectivenessfactors or with increasing Thiele moduli (Table 2) i.e., as the regime in the particle is becoming increasingly diffusion controlled. This is evident from the variation of the percentage deviations of the effectiveness factors, calculated on the basis of the dusty-gas model, with Thiele moduli as shown in Figure 1. For getting a clearer picture, the reaction rate has been varied arbitrarily several fold for a specific data set (data set no. 1). The simulation results obtained for each case are shown in Table 3. The deviations in effectiveness factors are again plotted against Thiele moduli (also shown in Table 3) in Figure 2, which confirms the trend shown in Figure 1. The predictions made by the effective diffusivity model are deviating more and more from those of the dusty-gas model as the regime becomes more and more diffusion-controlled. What this means in physical terms is that if pore diffusion is dominating, the reaction rates are comparatively higher, the fluxes are higher, and the effects of flux interactions become more pronounced. On
Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 471 Table 4. Operating Data for the Reformer data set bulk gas composition (mol %) CH4
coz co
Hz HzO temperature (K) pressure (atm)
1
2
3
4
5
13.0 7.4 0.3 12.0 67.3 873.0 31.1
10.3 8.6 0.9 19.5 60.7 910.0 30.22
8.2 9.0 1.8 25.1 55.9 980.0 29.36
6.3 9.2 3.0 29.2 52.1 1012.0 28.26
4.8 9.0 4.2 32.5 49.5 1036.0 26.34
Table 5. Model Predictions of Effectiveness Factors for CJ& and C02 data set no. 0.0
I
I
1
1
component CH4
2
CHI
coz
c02 Thiele
modulus
Figure 2. Variation of deviations in effectiveness factors with Thiele modulus.
1 2 3 4 5 6 7
100.0 50.0 10.0 1.0 0.1 0.01 0.001
0.038 0.054 0.123 0.338 0.693 0.939 0.993
0.058 0.081 0.173 0.438 0.794 0.968 0.997
0.054 0.074 0.159 0.411 0.769 0.962 0.996
22.70 16.05 7.18 2.27 0.72 0.23 0.07
the other hand, as we move toward the reaction-ratecontrolled regime with increasing effectiveness factors, the fluxes are lower and the effects of interaction are less. The flux interaction effects are seen to assume a high or low profile depending on whether the intraparticle regime is diffusion-controlled or reaction-ratecontrolled. Thus, it is apparent that even when flux interactions are considerable, the effective diffusivity model is adequate for situations with rate-controlled regimes. Once the interactions among the species are known to be strong as indicated by the B matrix of Krishna (19931, the Thiele modulus characterizing the type of regime prevailing within the pellet can guide the choice of the model to be employed. The simulations with the dusty-gas model show that on average there is a pressure drop of about 0.4 atm within the particle, due t o a decrease in the number of moles because of the reaction. Whatever differences are noticed in the predictions of the Stefan-Maxwell model and the dusty-gas model may be attributed t o the intraparticle pressure gradients. Case Study 2: Steam Reforming of Methane. The reforming reactions are represented by
+ H20 CH4 + 2H,O CO + 2H,O CH4
--L
--L
CO
+ 3H2
+ 4H, CO, + H2
CO,
(16) (17) (18)
These reactions are carried out industrially on nickelon-alumina catalysts and are k n o w n t o be highly diffusion-controlled inside the catalyst pores. The interactions among the various components present are
CH4
4
CH4
coz
c02
5
Table 3. Simulation Results for Ammonia Production (Based on Hypothetical Reaction Rates) simulation multiplication effectiveness factor Thiele no. factor for the rate EDM SMF DGM modulus
3
CH4 c02
EDM
SMF
DGM
0.0239 -0.0526 0.0199 0.0036 0.0216 0.0030 0.0208 0.0054 0.0209 0.0062
0.0244 -0.0555 0.0202 0.0033 0.0215 0.0033 0.0213 0.0052 0.0212 0.0059
0.0243 -0.0558 0.0202 0.0032 0.0214 0.0029 0.0212 0.0052 0.0212 0.0059
now calculated in the same manner as mentioned earlier. For an ideal mixture of methane (l),carbon dioxide (21, carbon monoxide (31, hydrogen (4), and water (5), at a temperature of 1036 K and a pressure of 26.3 atm, with y1 = 0.048, yz = 0.09, y3 = 0.042, y4 = 0.325, and y5 = 0.495, the fluxes of the (M- 1) components may be written as (Krishna, 1993) IJI'
J2 J3
hJ4,
- -C
'235 2 1 ,13
4 193 4 17
3 7 207 15
-15' -33 17 533.
10-4
'VY 1 VY2 vy3 pY4.
(19) Except for a mild interaction of hydrogen with carbon dioxide, the system may be termed as interaction-free. It will be interesting to study the performance of the intraparticle models in this type of situation. The operating data used as input for testing the models has been taken from the simulation results of a reactor model of a commercial reformer. Five sets of data have been used altogether, taken at different positions along the length of the reformer, and these are given in Table 4. The kinetic expressions and data for the rate constants, equilibrium constants, etc. required for the simulation have been taken from Xu and Froment (1989). The results obtained for the three models are presented in terms of effectiveness factors (Table 5). The magnitude of the effectiveness factors indicates that the regime within the catalyst pellet in all the cases is diffusion-controlled. The Thiele modulus calculated for all the data sets is seen to hover around 50 and is in conformity with the magnitude of the effectiveness factors. But what is more important is that there is virtually no difference in the predictions made by the different models. One might therefore conclude that in the absence of flux interactions among the constituent species, the type of regime prevailing within the pellet does not influence the choice of the intraparticle model. In all such cases, the simple effective diffusivity model can be expected to yield results as good as those from the complex models. Also, it has been noticed that the pressure is almost invariant within the pellet, because of which, it
472 Ind. Eng. Chem. Res., Vol. 34, No. 2,1995
is presumed, there are no differences in the predictions made by the Stefan-Maxwell model and the dusty-gas model. The procedure outlined above has been applied to the two cases dealt with by Salmi and Warna (1991) so that it would serve as further validation of the methodology. In the first case of the water-gas shift reaction there is no need to calculate the interaction matrix, because the high values of effectiveness factors reported (about 0.9) rule out any diffusion limitations. In such a clear cut reaction-rate-controlled regime, the effect of any interactions present on intraparticle diffusion will be played down and hence could be neglected, as already discussed. The effective diffusivity model and the StefanMaxwell model should, therefore, yield nearly identical results, which is precisely the case (Salmi and Warna, 1991). The second example taken up for analysis is the methanol synthesis. The B matrix formulated for the conditions described by Salmi and Warna (1991) indicates that there are considerable interactions between hydrogen and carbon monoxide which make the flux of carbon monoxide a strong function of the hydrogen concentration gradient. The effectiveness factors reported for this case are of the order of 0.27, which indicates a lightly diffusion-controlled regime. Using the criteria developed earlier, considerable differences in the predictions of the two models can be expected. This is indeed the case, as wide variations in intraparticle concentrations have been reported for some of the species.
Conclusions The basic thrust of the present work has been directed at showing how a procedure described by Krishna (1993) for calculation of flux interactions among constituent species in a system could be used, together with the Thiele modulus, as a methodology for the a priori selection of a suitable intraparticle model in a given situation. The ideas have been fortified by the results obtained from model evaluation studies conducted on two different systems. Further validation of the applicability of the procedure came from the work of Salmi and Warna (1991). The usefulness of the proposed methodology can be better appreciated from the following guidelines evolved on the basis of the case studies: 1. If the reaction system is characterized by negligible interactions among the constituent species, the effective diffusivity model can be safely used even when the intraparticle regime is diffusion-controlled. 2. On the other hand, if the interactions are strong, the effective diffusivity model can be used only if the intraparticle regime is reaction-rate-controlled. 3. If there are strong interactions among the species and also if the intraparticle regime is expected to be lightly-to-highly diffusion-controlled, as indicated by a large Thiele modulus, the Stefan-Maxwell model should be used. 4. If, in addition to what is given in 3, there is a considerable increase or decrease in the number of moles due to the reaction, the dusty-gas model is recommended. These have been summarized in Figure 3 in the form of a qualitative picture of the operable regions for the three different models. Although a somewhat quantitative measure of the effectof the interactions among the various components on the component fluxes is available from eq 14, this
yr
t
.-0
T hiele modulus
Figure 3. Rough contours for operable regions for different models.
information is still qualitative in the sense that the strength of the interactions is yet to be correlated with the deviations in the predictions of different intraparticle models. Furthermore, only ideal systems have been considered in the present work but system nonidealities could be severe sometimes. In such cases, the diffusional fluxes will depend not only on the StefanMaxwell diffusivities but also on some thermodynamic factors characterizing the nonideal nature of the system. It is thus clear that further investigations are necessary to generate new information, which should, hopefully, facilitate a sharper definition of the contours in Figure 3.
Nomenclature B = matrix of inverted Maxwell-Stefan diffusivities Bo = permeability of the pellet c = molar concentration of the mixture D,,i = effective diffusivity of the component i DP = effective Knudsen diffusion coefficient for species i D i = effective binary diffusion coefficient for species i and j J = difision flux relative to the molar average reference velocity Ni = molar flux of species i P = total pressure pi = partial pressure of species i R = universal gas constant R = reaction rate term rp = equivalent radius of the catalyst particle T = temperature y = mole fraction z = dimensionless axial distance 7 = effectiveness factor Ai = flux associated with reaction i p = viscosity of the gas mixture vu = stoichiometric coefficient for ith component in jth reaction
Literature Cited Bird, R. B.; Stewart, W. E.; Lightfoot,E. N. Transport Phenomena; John Wiley: Tokyo, 1960. Finlayson, B. A. Nonlinear Analysis in Chemical Engineering; McGraw-Hill: New York, 1980. Kaza, K. R.; Jackson, R. Diffision and reaction of multicomponent gas mixtures in isothermal porous catalysts. Chem. Eng. Sci. 1980,35,1179. Krishna, R. Comments on simulation and optimization of an industrial ammonia reactor. Ind. Eng. Chem. Res. 1989,28, 1266.
Ind. Eng. Chem. Res., Vol. 34,No.2,1995 473 Krishna, R. Problems and pitfalls in the use of the Fick formulation for intraparticle diffusion. Chem. Eng. Sei. 1993,48,845. Murty, C . V. S.; Venkat Reddy, K. Modelling of diffusion and multiple reversible reactions in a porous catalyst. Comput. Chem. Eng. 1992,16,621. Salmi, T.; Warna, J. Modelling of catalytic packed-bed reactorsComparison of different diffusion models. Comput. Chem. Eng. 1991,15,715. Shah, M.J. Control simulation in ammonia production. Znd. Eng. Chem. 1967,59,72. Singh, C. P. P.; Saraf, D. N. Simulation of ammonia synthesis reactors. Znd. Eng. Chem. Process Des. Dev. 1979,18(3), 364. Skrzypek, J.; Grzesik, M.; Szopa, R. Theoretical analysis of twoparallel and consecutive reactions in isothermal symmetrical
catalyst pellets using the dusty-gas model. Chem. Eng. Sei. 1984,39,515. Xu, J.;Froment, G. F. Methane-steam reforming, methanation and water-gas ShiR: I. Intrinsic kinetics. MChE J. 1989,35,97. Received for review April 14, 1994 Revised manuscript received October 11, 1994 Accepted October 27, 1994@
I39402473 Abstract published in Advance ACS Abstracts, January 1, 1995. @
