2029
Ind. Eng. Chem. Res. 1994,33, 2029-2033
Geometry Effects on the Concentration and Reaction Rate Profile in Methanol Production Catalyst. 1. Neglecting External Diffusion Y. Duan' and Di Wu Institute for Coordination Chemistry, Inner Mongolia Teachers' University, Hohhot, Inner Mongolia 010022, China
W . B. Earl and C. J. Williamson Department of Chemical and Process Engineering, University of Canterbury, Christchurch, New Zealand
In the low-pressure synthesis of methanol from carbon oxides and hydrogen over copper-based catalysts, the effect of catalyst pellet geometry has been investigated. A model for studying reaction within any sufficiently symmetrical porous catalyst particle has been also provided. Fugacity coefficients and compositions are calcd. by using the Redlich-Kwong-Soave state equation, and the kinetics model announced by Villa et al. has been used for computing the reaction rates. The reaction is considered to proceed within isothermal and isobaric catalyst pellets. External diffusion has been neglected.
Introduction
(3)
I t has long been recognizedthat catalyst pellet geometry will effect reaction performance within the catalyst particle (Do and Weiland, 1981; Morbidelli et al., 1982; Yuan et al., 1983;Skrzypek et al., 1984;Vayenas and Pavlou, 1987). Previous developments in the field were mainly concerned with infinite slab, infinite cylinder, and sphere catalyst pellets for simple reaction kinetics (i.e., zero-order and first-order kinetics). In fact, most of the catalyst pellets used in the chemical industry are alwaysbeyond this scope (i.e,,the finite cylindrical catalyst pellet). The model here presented will be used to solve this kind of problem. It can be used for any symmetrical catalyst pellets for arbitrary kinetics. Results have been calculated by the model to test the stability of the numerical method.
boundary conditions:
where there are two parameters for the catalyst pellet geometry ( n for external shape andxo for size of the catalyst pellets; n takes on the values 0, 1, 2 for the infinite slab, infiiite cylinder, and sphere, respectively), x is the distance from the geometry center of a symmetrical catalyst pellet, and is the formation rate of methanol according to reaction 1, which is
Description of Methanol Synthesis Methanol synthesis can be carried out by hydrogenating carbon oxideson suitable catalysts. This model has mainly relied on the following two reactions and the copper-based catalyst. The reactions considered are CO CO,
where
A = exp[2.45012 - 109.814(1/T- 1/506)1
+ 2H, F! CH,OH
(1)
B = exp[1.40823 - 39647.5(1/T- 1/506)1
+ H, e CO + H,O
(2)
C = exp[2.82261+ 12268.7(1/T = 1/506)1
The important parameters for the copper-based catalyst are the following. Chemical composition (wt %): CuO, 54.6; ZnO, 19.0; A1203,9.1; COz, 8.9. Catalytic particle: porosity (ep), 0.5; density (pp), 1980 kg/m3; surface area (S',J,61 m2/g.
Mathematical Model It is known (Cappelli et al., 1972) that, under the temperature conditions used in industrial reactors for methanol synthesis, the conversionreaction rate of reaction 2 is very high. I t is therefore assumed that equilibrium conditions are reached at the external surface of the catalytic particles, and therefore the material balance for only methanol within the isothermal and isobaric catalyst pellets is derived. 0888-5885f 94f 2633-2029$04.50/ 0
D = exp[1.05957 + 5449.78(1/T- 1/506)1 K,, = 3.27 X lo-'' exp(11.628/T) K,, = 117 exp(-4827/T)
Algorithm Effective Diffusivity. The molecular diffusivity is calculated using Wilke-Lee's method, the Knudsen diffusivity is directly calculated according to molecular dynamic theory, and the combined diffusivity is calculated from Bird's method. The effective diffusivity can be calculated from the combined diffusivity, the porosity, and the tortuosity ( 7 = 5 assumed). rij = (ri
+ rj)/2
0 1994 American Chemical Society
2030 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
+
c = c,e"
e+ e4 + e+
tij = (CiCj)1'2
E = kT/eij
yi = ZRTCJP = Ysi + pilxl 1- 2x1
f(E) = 0.762633 - 0.319947 ln(E) + 0.0407552[1n(E)I2
x1 = ysco - Yco 1- 2Yco
(10)
Numerical Method. Using the well-known Newton backward difference formula and taking its first few terms (in fact, the first seven terms were taken here), eq 4 can be transformed into a corresponding numerical formula as following. j#i
x = xo
DE = 97rh(T/Mi)'/2
Parameters at the External Surface of t h e Catalyst Pellets. Since the equilibrium conditions of reaction 2 have been reached at the external surface of the catalyst pellets, the mole fraction and the formation rate of reaction 1can be iteratively calculated according to the SRK state equation (Soave, 1972) and Villa's kinetic model (Villa et al., 1985,1987;see also eq 4) (XI = 0 assumed as the initial value):
+ ah
(a= 0 to-rn)
(11)
Substituting eq 12 into eq 3 for rl and integrating twice under the boundary conditions (Duan, 19911,the formula for the concentration of methanol at different locations within the catalyst pellets can be derived.
+ ...I
aVf(xo)
[n + l
+
l a ( x o ah) -
(n+ l)(n + 2) f(x,)
fi
1
da d a
+
= YiJiP
Initial Values. The solutions in an infinite slab for first-order kinetics can be taken as the initial values for the mole fraction profiles and the reaction rate profile for reaction 1 within the catalyst pellets using carbon monoxide as the reference component. Therefore, the initial values for the profiles within the catalyst pellets can be easily calculated (7 = 0.7 assumed as the initial value):
+ ah)3 - x; 3(n + l)(n + 2)h (x,
1
Vf(x,)
1
+ ...
Taking a uniform distribution of the formation rate of methanol through the catalyst particle for reaction 1as a first approximation value, the process of an iteration can be carried out to calculate the concentration of methanol at different locations within the catalyst particle. Hence, the corresponding mole fraction of methanol can be calculated from eq 10, the mole fraction of the other components can also be calculated according to the
Ind. Eng. Chem. Res., Vol. 33, No. 9,1994 2031 Table 1. TO= 207 "C;PO= 80 atm; xo= 0.0005 ma n=O
n = 0.4
n = 0.8 ~
Bk 0 -1 -2 -3 -4
-5 -6
16.5000 16.5649 16.5013 16.4493 16.4090 16.3802 16.3630 16.3573
71.5000 71.4351 7.13807 71.3363 71.3019 71.2773 71.2626 71.2577 0.9776
9 0
8.2000 8.1351 8.1506 8.1632 8.1730 8.1800 8.1842 8.1856
16.5000 16.5649 16.5191 16.4818 16.4528 16.4321 16.4197 16.4155
71.5000 71.4351 71.3960 71.3641 71.3393 71.3216 71.3110 71.3075 0.9862
8.2000 8.1351 8.1462 8.1553 8.1624 8.1674 8.1704 8.1714
16.5000 16.5649 16.5292 16.5000 16.4773 16.4612 16.4515 16.4483
71.5000 71.4351 71.4046 71.3796 71.3603 71.3465 71.3382 71.3354 0.9906
~~
8.2000 8.1351 8.1438 8.1509 8.1564 8.1603 8.1627 8.1636
Bk means the position in the bulk fluid.
CY
CY
Figure 1. Concentration of methanol versus location within the catalyst pellet. TO= 207 "C; PO= 80 atm; xo = 0.0005 m. CM X 1@, mo1.L-1.
Figure 3. Concentration of methanol versus location within the catalyst pellet. TO= 207 "C; PO = 80 atm; xo = 0.003 m.
Fl r1
1.1
&
1.10:
................0
8 0
..................
-----______. 0 1.Mb
t
I
I
,
, -6
0.6 '*a
-1 -2 -3 -4 -6 Figure 2. Formation rate of methanol versus location within the catalyst pellet. To = 207 "C;PO= 80 atm; xo = 0.0005 m. rl X l(r, mol.gl(catalyst).min-l.
stoichiometric coefficients of reaction 1,the fugacities of every components a t different locations within the catalyst particle can be calculated from eq 7,and the rate profile of reaction 1 within the catalyst pellets can be calculated from eq 4. Error Criterion. Taking the six points within the catalyst pellets as the reference ones, iterate until the sum of the relative iteration errors of the six points is less than 0.005.
Results Bulk Fluid Conditions. Temperature 207 and 237 "C; pressure 80atm, gas composition (molefraction) (Islam
1
0
-1
-2
-3
-4
-6
I +
-6
a
Figure 4. Formation rate of methanol versus location within the catalyst pellet. TO= 207 "C; PO= 80 atm; xo = 0.003 m.
and Earl, 1990): CO, 0.165; COS, 0.082;Hz, 0.715;CH4, 0.029;HzO, 0.001; Nz, 0.008;CH30H, 0.00. Computation Scope. n = 0, 0.2,0.4,0.6, 0.8,1,2;xo = 0.0005-0.004. Part of the results have been graphed in Figures 1-8 and listed in Tables 1-4.
Discussion There exists the geometric center for the symmetrical pellet, infinite plane for infinite slab, infinite line for infinite cylinder, and a point for sphere, respectively. The finite cylinder's geometric center is a finite surface, so the value of n for the finite cylinder should be between 0 and 1. The value of n for a regular polyhedron should be between 1 and 2.
2032 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 = 0.003 m Table 2. To = 207 O C ; PO= 80 atm; n=O a YCo x 102 )'HI x 102 yc& 10' Yco 102 16.5000 8.2000 71.5000 Bk 16.5000 16.5649 8.1351 16.5649 71.4351 0 15.3604 8.4986 70.1589 -1 15.0711 8.7759 14.4373 13.9314 69.1852 -2 13.7545 8.9802 13.0916 68.4677 -3 13.2846 9.1205 12.5149 67.9750 -4 13.0094 9.2027 12.1772 67.6865 -5 12.9188 9.2297 67.5918 -6 12.0663 0.5603 7
Table 3. TO= 237 O C ; a
yCOx102
YHnX1O2
yCOax102
yCOx102
Bk
16.5000 16.6934 16.1936 15.7886 15.4760 15.2540 15.1214 15.0772
71.5000 71.3066 70.8760 70.5270 70.2577 70.0665 69.9522 69.9142 0.9282
8.2000 8.0066 8.1267 8.2241 8.2992 8.3526 8.3845 8.3951
16.5000 16.9634 16.3283 16.0319 15.8030 15.6402 15.5429 15.5106
-5 -6 9
Table 4.
16.5000 16.6934 12.6876 9.9099 8.0372 6.8402 6.6333 5.9492
0 -1 -2 -3 -4
-5 -6 7
I
YCOX1@
YkX102
16.5000 16.5649 15.5556 14.7784 14.2019 13.8041 13.5708 13.4939
71.5000 71.4351 70.5728 69.9089 69.4163 69.0764 68.8771 68.8114 0.7587
ycqX102 8.2000 8.1351 8.3807 8.5698 8.7100 8.8068 8.8636 8.8823
n = 0.4 YH, x 10' 71.5000 71.3066 70.9920 70.7367 70.5394 70.3992 70.3154 70.2875 0.9556
n = 0.8 '
)'Con
x 102
8.2000 8.0066 8.0943 8.1656 8.2206 8.2597 8.2831 8.2909
yC0 x 102 16.5000 16.6934 16.4060 16.1725 15.9920 15.8636 15.7868 15.7613
YHa
x lo2
71.5000 71.3066 71.0589 70.8578 70.7023 70.5917 70.5255 70.5035 0.9696
yC&
102
8.2000 8.0066 8.0757 8.1318 8.1752 8.2060 8.2245 8.2306
To = 237 O C ; PO= 80 atm; XO = 0.002 m
71.5000 71.3066 67.8555 65.4625 63.8492 62.8181 62.6398 62.0504 0.3527
n = 0.8
n = 0.4
n=O
Bk
n = 0.8
ycoI '02 8.2000 8.1351 8.4282 8.6528 8.8189 8.9332 9.0002 9.0222
PO= 80 atm; xo = 0.0005 m n=O
0 -1 -2 -3 -4
n = 0.4 '02 71.5000 71.4351 70.4061 69.6174 69.0340 68.6326 68.3975 68.3201 0.6811
8.2000 8.0066 8.9695 9.6373 10.0874 10.3752 10.4249 10.5894
16.5000 16.6934 13.1858 10.9068 9.0212 7.9400 7.3311 7.1423
71.5000 71.3066 68.2847 66.1491 64.6970 63.7655 63.2409 63.0783 0.5332
8.2000 8.0066 8.8498 9.4457 9.8509 10.1108 10.2572 10.3025
16.5000 16.6934 13.5836 11.3475 9.7982 8.7911 8.2242 8.0420
71.5000 71.3066 68.6274 66.7010 65.3663 64.4988 64.0104 63.8534 0.6439
8.2000 8.0066 8.7541 9.2917 9.6641 9.9062 10.0425 10.0863
-0
@n=O o n = 0.4
...........
__.___-C----.
_,c
.-.-. -.-.-..
0
0 n=O - 0 0 0 - 0
'~.
',.
-.
Figure 5. Concentration of methanol versus location within the catalyst pellet. TO= 237 "C;PO= 80 atm; xo = 0.0005 m.
Figure 6. Formation rate of methanol versus location within the catalyst pellet. TO= 237 "C; PO = 80 atm; xo = 0.0006 m.
The larger the parameter, n,for the external shape of the catalyst pellet, the larger the effectiveness factor (see Tables 1-4). The larger the radius or the half-thickness of the pellet, the smaller the effectiveness factor (see Figures 2, 4, 6, and 8,and Tables 1-4). As for the reversible reactions in the catalyst particle with a homogeneousactivity profiie, the greater the degree of the reaction, the less the distance from equilibrium and the smaller the rate of the reaction. The average reaction rate on a catalyst particle will gradually become larger since the value of n increases from 0 to 2 because the
activity amount of a catalyst pellet with the same thickness decreases more quickly. (see Figures 2, 4, 6, and 8 and Tables 1-4). (The average reaction rate on a catalyst particle can be estimated on the basis of its effectiveness factor.) Tables 1-4 have been provided for the mole fraction profiles of the reactants CO, H2,and C02 within catalyst pellets. The values obtained are reasonable.
Nomenclature A, B = parameters in the SRK state equation, dimensionless
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2033
PO= pressure in the bulk fluid, atm r1 = formation rate for methanol synthesis of reaction 1,
g-mol.g'(catalyst)-min-l ri = radius of a molecule, m rij = mean radius of two molecules, m r b = mean pore radius of a porous catalyst pellet, m TO= temperature in the bulk fluid, "C or OK x = dimensional distance, m xo = radius or half-thickness of pellet, m yo;, ysi, yi = mole fraction of ith component in bulk fluid, at
external surfaceof a catalyst pellet, within a catalyst pellet, dimensionless Z = compressibility factor of a mixture, dimensionless Greek Letters ~i = Lennard-Jones energy of a molecule, J tij = interaction Lennard-Jones energy of two collision Figure 7. Concentration of methanol versus location within the catalyst pellet. TO= 237 "C; Po = 80 atm; xo = 0.002 m.
rl 9.4
molecules, J = catalyst particle void fraction, dimensionless 4 = effectiveness factor, dimensionless pp = density of catalyst particle, kgm3 4 = Thiele modulus, dimensionless & = fugacity of ith component in a mixture, dimensionless V = backward difference symbol tp
Literature Cited
2.3
b
__.''":"-..........,.,, -6 -6 I...
-1
-2
-8
-4
Figure 8. Formation rate of methanol versus location within the catalyst pellet. TO= 237 "C; Po = 80 atm; xo = 0.002 m.
bi = substance constant in the SRK state equation for a pure
component, Lsmol-I b = substance constant in the SRK state equation for a mixture L-mol-' CM = concentration of methanol, g-mo1.L-l C ~ =M concentration of methanol at the external surface of catalyst pelleb, g-mol.L-' Ci = concentration of the ith component at different locations within the catalyst pellet, g-mo1.L-l Di, = molecular diffusivity of two molecules, m2.s-1 D,i = molecular diffusivity of the ith component through a mixture, m2.s-1 Dg = Knudsen diffusivity of the ith component through the pore of a porous catalyst pellet, m2w1 D,i = combined diffusivity of the ith component, m2w1 D,n = effectivediffusivityof methanolwithin the gas mixture through the catalyst pellet, m2w1 E = kT/tij, variable of the collision function, dimensionless {(E)= collision function, dimensionless h = equal-interval step length taken, m Kbi, Kpi,K,i = chemical equilibrium constants, dimensionless Mi = weight of the ith component n = parameter for the external shape of the catalyst pellets, dimensionless
Cappelli,A.; Dente, M. Mathematical Model for Simulating Behavior of Fauser-Montecatini Industrial Reactors for Methanol Synthesis. Ind. Eng. Process Des. Deu. 1972,ll (2),184-190. Do, D. D.; Weiland, R.H. Self-poisoning in Single Catalyst Pellets. Ind. Eng. Chem. Fundam. 1981,20,34-41. Duan, Y. "Computer Aided Simulation of a Heterogeneous Methanol Synthesis Reactor"; Research Report, Department of Chemical and Process Engineering, University of Canterbury, Christchurch, New Zealand, 1991. Islam, K. A.; Earl, W. B. Deactivation of IC1 Low Temperature Methanol Catalyst in a Industrial Reactor. Can. J. Chem. Eng. 1990,68,702-704. Juang, H.; Weng, H. Performance of Catalyst with Nonuniform Activity Profiles. 2. Theoretical Analysis for Nonisothermal Reactions. Znd. Eng. Chem. Fundam. 1983,22,224-230. Morbidelli, M.; Servida, A. Optimal Catalyst Activity Profiles in Pellets. Znd. Eng. Chem. Fundam. 1982,21,278-289. Soave, G. Equilibrium constants from a modified Redlich-Kwong Equation State. Chem. Eng. Sci. 1972,27,1197. Skrzypek, J.; Grzesik, M.; Szopa, R. Theoretical Analysis of Two Parallel and Consecutive Reactions in Isothermal Symmetrical Catalyst Pellets Using the Dusty-Gas Model. Chem. Eng. Sci. 1984,39,515-521. Vayenas, C. G.; Pavlou, S. Optimal Catalyst Distribution for Selectivity Maximization in Pellets: Parallel and Consecutive Reactions. Chem. Eng. Sci. 1987,42,1655-1666. Villa, P.; Forzatti, P.; Guido, B.; Guido, G.; Pasquon, I. Synthesis of Alcohols from Carbon Oxides and Hydrogen. 1. Kinetics of the Low-Pressure Methanol Synthesis. Znd. Eng. Chem. Process Des. Deu. 1985,24,12-19. Villa, P.; et al. Response to Comments on 'Synthesis of Alcohols from Carbon Oxides and Hydrogen. 1. Kinetics of the LowPressure Methanol Synthesis. Znd. Eng. Chem. Res. 1987,26, 400-402. Yuan, Q.; Huang, B.; Li, J. Isothermal Effectiveness Factors for Nonuniform Active Catalyst. J. Chem. Znd. Eng. (China) 1983, 4, 327-334. Receiued for review May 9, 1994 Accepted May 31,1994. e Abstract published in Aduance ACS Abstracts, July 15,1994.
