Ind. Eng. Chem. Res. 1997, 36, 4087-4093
4087
Effect of Catalyst Shape on the Hydrolysis of COS and CS2 in a Simulated Claus Converter Shimin Tong,† Ivo G. Dalla Lana,*,‡ and Karl T. Chuang‡ Hyprotech Ltd., Calgary, Alberta, Canada, and Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
The alumina catalyst employed in the modified Claus process for sulfur recovery is often partially deactivated from sulfation of its active surface. This decreased catalytic activity is particularly detrimental to the hydrolysis reactions of COS and CS2 because incomplete hydrolysis results even though their theoretical conversion limit is 100%. Simulation of the reactor performance at typical Claus plant-operating conditions was possible using experimentally obtained rate functions for the two simultaneous hydrolysis reactions and the H2S/SO2 reaction. Using these rate constants, rather small values of the effectiveness factor were predicted for the hydrolysis reactions. By increasing the value of the effectiveness factor, it should be possible to improve the hydrolysis conversions without altering the process conditions appropriate for good sulfur recoveries. This was achieved by changing the particle shape to increase the external surface area. The simulation of a Claus catalytic converter, based upon a plug-flow adiabatic fixed-bed computer model using various shapes for the catalyst particles, showed that improved performance results even when the catalyst surface is partially sulfated. Introduction In the processing of sour natural gas, the conversion of hydrogen sulfide into elemental sulfur is often carried out using the modified Claus sulfur recovery process. In this process, the hydrogen sulfide is partially oxidized according to reaction 1 and then the remaining hydrogen sulfide is converted to elemental sulfur with reactions 2 and 3.
H2S + 3/2O2 w SO2 + H2O
(1)
2H2S + SO2 S 1/2S6 + 2H2O
(2)
S6 S 6/nSn
where n ) 2, 4, 8
(3)
Side reactions occurring in the front-end furnace may generate COS and CS2, which are more difficult to convert to elemental sulfur. Some hydrolysis of these two components can be achieved within the catalytic converters by reactions 4 and 5
expense of lowered equilibrium conversions for reaction 2. An alternative approach to obtain better hydrolysis conversions without the use of elevated converter temperatures is suggested by altering the catalyst shape to improve mass transfer between the gas and solid catalyst phases. The effectiveness factor of the catalyst particle may be improved by increasing either its external area or its pore sizes. In the following analysis, we investigated the influence of changing the catalyst particle shape upon the effectiveness factor for the alumina-catalyzing reactions (2)-(5). The different particle shapes were described by using Aris’ characteristic particle dimension, Lp. In this approach, one may change the particle shapes of the alumina catalyst by altering the external shape and/or porosity. After determining the new characteristic dimension, one may examine the influence of this particle change upon the hydrolysis conversion through computer simulations of the catalytic converter performance at actual reaction temperatures.
Basis of Reactor Simulations
CS2 + 2H2O w CO2 + 2H2S
(4)
COS + H2O w CO2 + H2S
(5)
Alumina, the most widely used catalyst in Claus plants, may be deactivated in several ways: by sulfation, carbon deposition, surface adsorption, or capillary condensation of sulfur, thermal aging, physical attrition, etc. (Guo and Zhang, 1986; Pearson, 1981). Such deactivation lowers the conversions attainable for these two hydrolysis reactions, with the consequence of higher sulfur content in the plant tailgas and correspondingly increased tailgas cleanup costs. By operating the catalytic converters at more elevated temperatures, higher hydrolysis conversions are attainable but at the * To whom correspondence should be addressed. † Hyprotech Ltd. ‡ University of Alberta. S0888-5885(97)00277-7 CCC: $14.00
In the following section, we describe the reactor model, the intrinsic rate functions to be used simultaneously for reactions 2, 4, and 5 when catalyzed by alumina, the basis for mass- and heat-transfer corrections, and the numerical procedures used to solve the set of differential equations. Reactor Model. Typical Claus process catalytic converters employ catalyst beds of about 1-m depth and bed diameters exceeding five meters. Such beds, operated under adiabatic conditions (Q˙ ) 0), exhibit radial temperature gradients only in the immediate vicinity of the vessel wall. The use of a one-dimensional plugflow fixed-bed reactor model has been shown to provide a reliable description of the expected performance (Razzaghi, 1985). The one-dimensional model described by Razzaghi includes mass- and heat-transfer effects and has been adapted for use in these computer simulations. © 1997 American Chemical Society
4088 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997
The mathematical description employs a differential material balance equation for each independent component (j ) 1, 2, ..., 6).
d(VfCf,j) + akm,j(Cf,j - CS,j) ) 0 dz
(6)
Six such differential equations are used to relate the masses of the six independent chemical species in the reaction system, H2S, SO2, H2O, S6, COS, and CS2, to position along the bed axis. Heat transfer between the fluid phase and the solid catalyst phase is described by
( )
ahm d(VfFmTf) (T - Tf) ) 0 dz Cp,m S
(7)
along with the mass- and heat-transfer coupling equations between these two phases,
∑i ai,jηi(-RS,i)
akm,j(Cf,j - CS,j) ) -Fb
∑i [(-∆HR,i)ηi(-RS,i)]
ahm(TS - Tf) ) Fb
(8) (9)
where i represents one of the three reactions, H2S/SO2, CS2/H2O, and COS/H2O. Equations 6 and 8 define the conservation of mass for the adiabatic converter. Equations 7 and 9 define the conservation of energy for the adiabatic converter. Sulfur vapor is a mixture of sulfur polymers, S2-S8, but only the even-numbered species are present in significant amounts. To handle this complex gaseous component when dealing with eq 6 at j ) S6, the model assumes that the four even-numbered sulfur homologues are only present in their relative equilibrium distribution amounts as determined by thermodynamic predictions at the local temperature and total pressure; the latter is assumed to remain constant. This equilibrium assumption does not seriously affect the calculations since the combined partial pressures of sulfur vapor will not exceed ∼1 mol % in typical Claus catalytic converters. The conservation of mass relations for the four sulfur species, S2, S4, S6 and S8, are defined in
simplifies to the pseudo-first-order form,
-Ri ) ki′Pi
(15)
where ki′ ) ki/KH2O. A concentration of water in excess of 20 mol % is often encountered in the Claus process gas line downstream from the furnace and waste-heat boiler. The reversible catalytic reaction (2) is the main reaction carried out in the presence of alumina catalyst. Its intrinsic rate function is defined in the thermodynamically consistent form
-RH2S ) ks(PH2SPSO20.5 - (1/K)PH2OPS60.25)/(1 + KH2OPH2O)2 (16) where K ) xKE, as originally developed by Dalla Lana et al. (1976) and subsequently extended by Razzaghi and Dalla Lana (1984). The thermodynamic equilibrium constant, KE, and the adsorption constant for water vapor, KH2O, are always evaluated at the local temperature during the integration of the equations. The constants KH2O and KE and rate constants kCS2, kCOS, and kS may be expressed in their temperature-dependent forms when ∆Hi and Ei are constants,
Ki ) K0 exp(-∆Hi/RT)
(17)
ki ) k0 exp(-Ei/RT)
(18)
and
Cf,S ) 2Cf,S2 + 4Cf,S4 + 6Cf,S6 + 8Cf,S8
(12)
and their corresponding parameter values are listed in Table 1. Interparticle Transport Parameters. Interparticle transport resistance between the catalyst pellet surface and the fluid gas phase may be significant if the catalyst is highly active. In the Claus process, the H2S/SO2 reaction is a very fast reaction on the highly active alumina catalyst. Therefore, the temperature and concentration at the catalyst surface are expected to differ from their respective bulk quantities along the bed axis (before equilibrium conversion is established). In the one-dimensional adiabatic reactor model described above, the interparticle transport resistance has been included in the model within eqs 6-13. The significance of the interparticle transport resistances to mass and heat transfer is expressed in terms of the heat (hm) and mass (km,j) transfer coefficients. The heat- and mass-transfer coefficients are correlated by the following equations:
CS,S ) 2CS,S2 + 4CS,S4 + 6CS,S6 + 8CS,S8
(13)
Jm ) km,jFm/GSc2/3
(19)
Jh ) hm/(Cp,mGPr2/3)
(20)
d(VfCf,S) + akm,S6(Cf,S - CS,S) ) 0 dz
(10)
akm,S6(Cf,S - CS,S) ) -6aS6FbηH2S(-RS,H2S) (11)
Equations 6-13 define the mathematical model of the Claus converter. Kinetic Models. Intrinsic rate expressions are needed for the reversible reaction (2) and for the two irreversible reactions (4) and (5), all catalyzed by alumina. The kinetics for reactions 4 and 5 were presented earlier (Tong et al., 1992, 1995), with both rate functions being of the form
-Ri ) kiPiPH2O/(1 + KH2OPH2O)
(14)
where i ) CS2 or COS. When water exceeds 4 mol % of the inlet acid gas, the condition, KH2OPH2O . 1 is satisfied and eq 14
where Jm and Jh are functions of the Reynolds number, Re. The correlations used to estimate the values for km,j and hm are shown in the Appendix. Intraparticle Transport Resistance. The effect of any intraparticle transport limitation is generally expressed by the lumped parameter, “effectiveness factor”, η, introduced by Thiele. It is defined as the ratio of the observed rate of reaction to the rate evaluated at the condition on the external surface of the catalyst pellet. Razzaghi (1985) has estimated that the internal thermal effect in the small Claus catalyst particles is negligible; that is, an isothermal temperature within the Claus
Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 4089 Table 1. Values of Parameters Appearing in the Kinetics Rate Equations reaction
k0,j, kmol/ (kg‚h‚(kPa)1.5)
Ej, kJ/mol
K0,H2O, 1/kPa
∆HH2O, kJ/mol
2H2S + SO2 S 2H2O + 1/2S6 CS2 + 2H2O f CO2 + 2H2S COS + H2O f CO2 + H2S
6.91 19.75 2.30
30.77 40.41 25.27
0.338 3.43 1.25
98.10 83.22
catalyst pellet could be assumed. Thus, when computing the effectiveness factor, the mass balance equation is the only equation to be solved. The value of the effectiveness factor depends on the Thiele modulus, which is defined as
KE, 1/kPa1/2 9.502 × 10-7 exp(1.11 × 104/T)
Table 2. Parameters for the Different Catalyst Shapes
Φ ) Lp[(reaction rate at the surface condition)/ (DeffCS)]0.5 (21) where Lp, Aris’ (1957) characteristic dimension for an arbitrary particle, is defined by
Lp ) Vp/Sp ) volume of particle/external area of particle (22) For the first-order kinetics of CS2 and COS hydrolysis, eq 21 reduces to
Φi ) Lpxki/Deff
(23)
The value for the Thiele modulus influences the effectiveness factor somewhat inversely, as shown in
ηi ) 1/Φi(1/tanh(3Φi) - 1/3Φi)
(24)
For the Claus reaction (2), the rate constant, ks, in rate equation (16) is not defined for a first-order function. The Thiele modulus is calculated from the following differential equation for an isothermal spherical Claus catalyst pellet (Razzaghi and Dalla Lana, 1984):
d2Ψ/dy2 + (2/y) dΨ/dy - 9ΦH2S2Rr,H2S ) 0 (25) with boundary conditions of Ψ ) 1 at y ) 1 and dΨ/dy ) 0 at y ) 0, where Ψ ) Cr,j/CS,j and y ) 2r/Dp. By solving eq 25, they obtained the effectiveness factor in the form
∫
2
2
ηH2S ) 3 Rr,H2Sy dy ) 1/3ΦH2S (dΨ/dy)y)1
(26)
where ΦH2S is the Thiele modulus of the H2S/SO2 reaction. The method used to solve eq 25 to predict the local effectiveness factor, ηH2S, in the Claus converter has been described (Razzaghi, 1985). According to the definition of eqs 21, 24, and 26, the smaller the value of Lp, the larger the catalyst effectiveness factor becomes and, therefore, we are likely to improve the CS2 and COS conversions by using catalyst particles with smaller values of Lp. Five characteristic pellet shapes were used to show how the effectiveness factors may be altered through changes in the characteristic dimension, and these shapes are illustrated in Table 2. Computer Simulation of the Claus Reactor. The system of coupled differential equations defining the reactor model is now combined with the three kinetic models for the three separate reactions, enabling
numerical integration from the following initial conditions using a space velocity of 1000 h-1:
z ) 0:
Tin ) 230-330 °C
Pin ) 101 kPa
C°f,H2S ) 7 mol % C°f,SO2 ) 3.5 mol % C°f,H2O ) 25 mol % C°f,CS2 ) 0.08 mol % C°f,COS ) 0.1 mol % C°f,N2 ) 64.32 mol % The catalyst bed size was 5-m diameter × 1-m depth. This initial condition is based on a practical case from the Hanlan-Robb gas plant in Alberta. Five different shapes of the alumina catalyst were examined in reactor simulations integrated from the same initial condition. The numerical procedure for solving the mass and energy differential equations is described in the Appendix. Results from the Computer Simulation Effect of the Inlet Temperature. Figure 1 shows the predicted temperature profile along the bed axis when the catalyst is a nondeactivated alumina in the form of spherical particles. For this catalyst with a bed space velocity of 1000 h-1, the temperature increases along the axis, approaching a maximum and constant value as the limiting equilibrium conversion is reached. Increasing the inlet temperature increases the asymptotic upper equilibrium temperature level because the main reaction (2) is exothermic and the reactor is operated adiabatically. However, as shown in Figure
4090 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997
Figure 1. Temperature profile as a function of the catalyst bed depth. Inlet temperature ) 230, 280, and 330 °C. Pressure ) 101 kPa. Space velocity ) 1000 h-1. Catalyst: Kaiser 201.
Figure 3. Effect of feed temperature on simultaneous Claus and hydrolysis reaction conversions. Inlet temperature ) 230 and 330 °C. Pressure ) 101 kPa. Space velocity ) 1000 h-1. Catalyst: Kaiser 201.
Figure 2. Effect of inlet temperature on the performance of the simulated Claus converter. Inlet temperature ) 230, 280, and 330 °C. Pressure ) 101 kPa. Space velocity ) 1000 h-1. Catalyst: Kaiser 201.
Figure 4. Deactivation effect on CS2 and COS conversions. Inlet temperature ) 230 °C. Pressure ) 101 kPa. Space velocity ) 1000 h-1. Catalyst: Kaiser 201.
2, the higher the inlet temperature, conversely, the lower the equilibrium conversion attained. A typical result with an inlet temperature of 230 °C shows that within a 1-m depth catalyst bed, a conversion of H2S to sulfur of 82% is attained. At an inlet temperature of 330 °C, the conversion decreases to 72%, in which conversion level is attained at a bed depth near 0.5 m. Under the latter operating condition, i.e., at higher inlet temperatures, more than 50% of the bed depth serves no useful purpose because equilibrium is attained early in the bed. Figure 3 shows that, simultaneously with reaction 2, the conversions for the hydrolysis reactions (4) and (5) also increase continuously along the bed without equilibrium limitations. A conversion in excess of 90% is predicted for both hydrolysis reactions at an inlet temperature of 230 °C, and these conversions continue to increase with higher inlet temperatures. Since, in typical modified Claus process plants, the maximum bed temperature is reached in the first reactor, this reactor likely determines the maximum level of COS or CS2 conversions attainable. The downstream reactors are staged at increasingly lower inlet temperatures to ensure an adequate maximum conversion for reaction 2; but, at these lower inlet temperatures, the adiabatic bed temperature increases are smaller and the hydrolysis reaction rates are slower. The reactors following the first one do not add significantly to the total hydrolysis conversion attained.
The simulation results indicate that a nondeactivated spherical alumina catalyst will be more beneficial to the hydrolysis reactions the higher the inlet bed temperature. The hydrolysis conversions may be anticipated to be better than 90% at the test feed gas composition and at an inlet temperature of 230 °C. Deactivation of the Catalyst. Commercial alumina catalysts exhibit deactivation of their hydrolysis activity, generally attributed to sulfation (Sulfur, 1990; Pearson, 1981; Kerr et al., 1977). Plant experience suggests that to compensate for the deactivation of a moderately aged catalyst, typically exhibiting conversions around 5060%, the temperature must be increased in the first converter to, for example, 350 °C. Figure 4 shows simulation results in which deactivation is represented by assuming a percentage loss in the activity predicted using the two hydrolysis rate functions. Such a 90% “sulfated” catalyst and a bed inlet temperature of 230 °C corresponds to 50-60% conversions for the hydrolysis of COS and CS2. Increasing the inlet temperature to 350 °C is necessary (see Figure 5) before the hydrolysis conversions will exceed 84%. Such an increase in inlet temperature is impractical because of the adverse effect upon the more important equilibrium conversion of H2S to elemental sulfur via reaction 2. Extending the catalyst bed depth beyond 1 m would provide increased contact time and facilitate better hydrolysis conversions, but this approach is economically undesirable. Alternately, one may improve the
Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 4091
Figure 7. One-dimensional adiabatic reactor
(internal or external) does not change with alterations in the catalyst particle shape. For most extruded catalysts, this premise is reasonable. Conclusions Figure 5. Effect of feed temperature on CS2 and COS conversions for the partially deactivated catalyst bed. Inlet temperature ) 230, 270, and 350 °C. Pressure ) 101 kPa. Space velocity ) 1000 h-1. Catalyst: Kaiser 201. Deactivated degree ) 90%.
Figure 6. Effect of catalyst geometry on CS2 and COS conversions in a simulated Claus converter with a partially deactivated catalyst bed. Inlet temperature ) 230 °C. Pressure ) 101 kPa. Space velocity ) 1000 h-1. Catalyst: Kaiser 201. Deactivated degree ) 90%.
hydrolysis conversion by increasing the effectiveness factor of the catalyst pellets as we have already discussed. We now examine the influence of the different catalyst particle shapes listed in Table 2 upon the simulated hydrolysis performances of a Claus process catalytic reactor operating at constant inlet conditions. The simulations of reactor performance using catalysts with different characteristic dimensions are summarized in Figure 6. The change in Lp from 0.001 (spherical particles) to 0.000 22 m (see geometry in Table 2) shows that the global conversions of COS and CS2 should increase from 60% to nearly 100%, even though the alumina catalyst may be 90% deactivated. Porous catalysts with large surface areas generally encounter diffusion-limited rates of reaction, which result in low values for their effectiveness factors. By increasing the external surface area of the particle significantly, the access to pore openings is proportionally increased, thereby greatly increasing molecular transport into the pores. When the catalyst becomes deactivated, fewer of these active sites on the internal pore area of the catalyst particle remain. By increasing the availability of the total number of sites, more nondeactivated sites remain and the overall hydrolysis rates are increased. This analysis is valid when the number of active sites per unit surface area of catalyst
This analysis of the performance of modified Claus process catalytic converters has included the simultaneous kinetics of the three more important reactions, shown in eqs 2, 4, and 5, in the presence of a commercial alumina catalyst. The results are in agreement with industrial experience that higher converter temperatures are needed to achieve good hydrolysis conversions. In fact, catalyst deactivation through sulfation of alumina somewhat hinders the rates of the hydrolysis reactions. In practice, the use of higher converter inlet temperatures or deeper catalyst beds will improve the hydrolysis performance but, in the former case, only at the expense of lowering the H2S/SO2 equilibrium conversion and, in the latter case, at the expense of excessive pressure drop across the bed (this pressure drop occasionally limits the flow capacity through the sequential converters), both being impractical consequences. The simulations of reactor performance clearly demonstrate that increasing the effectiveness factor of the porous alumina catalyst provides an alternative means of improving the hydrolysis conversions. Such an improvement should permit a longer useful catalyst life (with improved hydrolysis conversions) since sulfation per unit area of catalyst surface either externally or within the pores of the various particle shapes should not differ substantially. In effect, the sulfation of active sites per unit catalyst surface area may still occur. The added accessible surface area provided by these new catalyst particle shapes should provide some compensation for the loss of activity encountered with existing spherical catalyst particles. The use of different catalyst shapes should also be beneficial in lowering the pressure drops across the bed of catalyst in a Claus converter. Acknowledgment The support of this research by the Canadian Natural Gas Processors’ Association and by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Nomenclature a ) effective film transfer area, m2/m3 aj ) stoichiometric coefficient of component j, j ) H2S, SO2, H2O, S6, CS2 or COS Cf,j ) bulk fluid concentration of component j, kmol/m3 C°f,j ) inlet fluid concentration of component j, kmol/m3 Cf,S ) bulk fluid sulfur concentration based on element S, kmol/m3 Cf,S2 ) bulk fluid S2 concentration, kmol/m3
4092 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 Cf,S4 ) bulk fluid S4 concentration, kmol/m3 Cf,S6 ) bulk fluid S6 concentration, kmol/m3 Cf,S8 ) bulk fluid S8 concentration, kmol/m3 Cp,m ) fluid specific heat capacity, kJ/(kg‚K) Cr,j ) concentration of component j in catalyst particle at position r, kmol/m3 CS,j ) catalyst surface concentration of component j, kmol/ m3 CS,S ) catalyst surface sulfur concentration calculated using element S, kmol/m3 CS,S2 ) catalyst surface S2 concentration, kmol/m3 CS,S4 ) catalyst surface S4 concentration, kmol/m3 CS,S6 ) catalyst surface S6 concentration, kmol/m3 CS,S8 ) catalyst surface S8 concentration, kmol/m3 Dp ) diameter of a spherical catalyst, m Deff ) effective diffusion coefficient, m/s Ei ) activation energy of reaction i, kJ/kmol G ) superficial mass velocity, kg/(s‚m2) hm ) film heat-transfer coefficient, kJ/(m2‚s‚K) ∆HH2O ) heat of adsorption of water on catalyst surface, kJ/kmol ∆HR,i ) heat of reaction for reaction i, kJ/kmol ki′ ) first-order rate constant, kmol/(kg‚s‚kPa), i ) CS2 or COS ki ) rate constant for the hydrolysis rate equations, kmol/ (kg‚s‚kPa), i ) CS2 or COS km,j ) film mass-transfer coefficient for component j, m/s kS ) rate constant for the H2S/SO2 reaction rate equation, kmol/(kg‚s‚(kPa)1.5) KE ) equilibrium constant for reaction 2, 1/kPa1/2 KH2O ) water adsorption equilibrium constant in the rate equation, 1/kPa L ) length of the reactor, m Lp ) Aris characteristic dimension, m, defined in eq 2 Pj ) partial pressure for component j, kPa r ) radius of the spherical catalyst particle, m Rr,H2S ) rate of change of H2S in the H2S/SO2 reaction in a catalyst particle at position r, kmol/(kg‚s) RS,j ) catalyst surface reaction rate for reaction i (i represents one of the three reactions, H2S/SO2, CS2/H2O, and COS/H2O), kmol/(kg‚s) Tin ) reactor inlet temperature, K Tf ) fluid temperature, K TS ) catalyst temperature, K Vf ) superficial velocity, m/s y ) 2r/Dp z ) axial position of the simulated reactor, m Greek Letter Fb ) catalyst bed density, kg/m3 Fm ) bulk fluid density, kg/m3 ηi ) catalyst effectiveness factor for the CS2/H2O or COS/ H2O reaction ηH2S ) catalyst effectiveness factor for the H2S/SO2 reaction ΦH2S ) Thiele modulus for the H2S/SO2 reaction Φi ) Thiele modulus for the CS2/H2O or COS/H2O reaction Ψj ) Cr,j/CS,j Subscript i ) 1, 2, 3 1 ) H2S/SO2 reaction 2 ) CS2/H2O reaction 3 ) COS/H2O reaction j ) 1, 2, 3, 4, 5, 6 1 ) H2S 2 ) SO2 3 ) H2O 4 ) S6 5 ) CS2 6 ) COS
f ) bulk phase m ) gas mixture s ) catalyst surface
Appendix Numerical Computation. The Claus reaction is a very fast reaction; therefore, the concentration gradient between the fluid gas phase and the catalyst surface is not negligible. In defining the differential equations for the simulation model, inter- and intra-particle diffusion resistances must be included. In the mathematical model described herein, the adiabatic one-dimensional reactor model involves solving the following mass and energy conservation equations to determine the H2S, CS2, and COS conversions:
d(VfCf,j) + akm,j(Cf,j - CS,j) ) 0 dz
∑i ai,jηi(-RS,i)
akm,j(Cf,j - CS,j) ) -Fb
( )
d(VfFmTf) ahm (T - Tf) ) 0 dz Cp,m S
∑i [(-∆HR,i)ηi(-RS,i)]
ahm(TS - Tf) ) Fb
(A.1) (A.2)
(A.3) (A.4)
where i ) 1, 2, and 3 represent the H2S/SO2, CS2/H2O, and COS/H2O reactions, respectively. For numerical solution of the above differential equations, the adiabatic one-dimensional reactor is divided into N elements along the axis as shown in Figure 7. In general terms, starting from the inlet condition of Tin 1 and C°f,j (j )1, 2, ..., 6), the values of T1f and Cf,j at the point n ) 1 (z ) n‚L/N) are calculated through iterative numerical integration of eqs A.1-A.4. Then, for the 1 next element, from n ) 1, T1f and Cf,j , predict T2f and 2 Cf,j at n ) 2. Continue this procedure through consecutive elements, one by one, to the outlet of the reactor. In more detail, the procedure for numerical integration from n ) m - 1 to n ) m is as follows: (1) To start, use the inlet concentration and the inlet fluid temperature ) inlet catalyst surface temperature m-1 for Cf,j and Tm-1 , and Tm-1 , respectively (subscripts f f s and s represent bulk gas phase and the catalyst surface, respectively; here, m - 1 is the inlet and m is section m-1 1). Solve eqs A.2 to obtain, Cs,j and Rm-1 s,i . (2) From these inlet values (m - 1), integrate eqs A.1 using the Runge-Kutta-Fehlberg method to obtain the m incremental changes ∆Cf,j, enabling Cf,j to be determined at n ) m. (3) The heat of reaction effect occurring in this element can now be estimated from the changes in component concentrations. Equations A.3 and A.4 can now be numerically integrated using the Runge-Kuttam Fehlberg method to obtain the values for Tm f and Ts . (4) The previous steps provided approximate solutions at n ) m because inlet values were used over the incremental section. An improved solution is obtained m-1 by using (Tm-1 + Tm + Tm f f )/2 and (Ts s )/2 as the average temperatures for the element between m - 1 and m. Steps 2 and 3 are repeated to obtain the new ( m m m m m Tm f )* and (Ts )*. If |(Tf )* - (Tf )| and |(Ts )* - (Ts )| m m exceed the accuracy requirements, let Tf ) (Tf )* and m Tm s ) (Ts )*, recalculate the average temperatures, and
Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 4093
repeat steps 2 and 3 using these corrected temperatures. The procedure is continued until the accuracy requirements are satisfied. Tm f ,
Tm s ,
(5) From these four steps, we obtain and m Cf,j , and now we can integrate over the next element, m to m + 1, etc. Estimation of the Physical Parameters. In the calculation of mixture properties, ideal gas behavior is assumed for gas-phase components. The subscripts i and j used in the following formula represent the combination of components i and j. Subscript m represents the gas mixture:
energy function for the molecular pair i and j.
film diffusion coefficient (km,i, hm) G 2/3 0.725 Sc Fm Re0.41 - 0.15
km,i )
1.10 hm ) Cp,mGPr2/3 0.41 Re - 0.15 where
Re ) density of the gas mixture (Fm)
Sc )
RT
∑i ∑j (yjφij)
λm )
yiλii
∑i ∑j (yiφij)
where
φij )
[1 + (Ai/Aj)1/2(Mj/Mi)1/4]2
x8(1 + Mi/Mj)1/2 φji ) (Aj/Ai)(Mi/Mj)φij
A ) µ or λ of pure component i.
molecular diffusion coefficient (Dm,j)
∑i (1/Dji(yi - yjai/aj)
1 ) Dm,j
∑i
1 - yj
i*j
ai/aj
where 1.5
Dji )
Cp,mµm λm
where rjp ) 3 nm, the average pore radius of alumina. The effective molecular diffusion coefficient in the catalyst pore is
viscosity (µm) and thermal conductivity (λm) of the gas mixture µm )
Pr )
DK,j ) 9.7 × 10-3rjpxT/Mj
where Mi is the molecule weight, yi is the mole fraction of gas component i, and R is the gas constant.
yiµi
µm FmDm,j
Knudsen diffusion coefficient (DK,j) and effective diffusion coefficient (Deff)
∑i (yiMi)
P Fm )
dpG µm
where ) 0.6604, the porosity of alumina. The tortuosity, τ ) 4, was selected. Literature Cited Aris, R. On the Shape Factors for Irregular Particles-I. Chem. Eng. Sci. 1957 6, 262-268. Dalla Lana, I. G.; Liu, C. L.; Cho, B. K. The Development of a Kinetic Model for Rational Design of Catalytic Reactors in the Modified Claus Process. Proc. 6th Euro./4th Int. Symp. Chem. Reaction Eng., DECHEMA, 1976; pp V 196-205. Guo, H.; Zhang, Q. The Sulfated Poisoning of Carbonyl Sulfide Hydrolysis Catalyst. J. Taiyuan U. Technol. 1986 4, 15-22. Kerr, R. K.; Paskall, H. G.; Ballash, N. Claus Process: Catalytic Kinetics Part 3sDeactivation Mechanism, Evaluation and Catalyst. Ener. Process. Can. 1977, Jan-Feb, 40-51. Pearson, M. J. Special Catalyst Improves C-S Compounds Conversion. Hydrocarbon Process. 1981, 60, 131-134. Razzaghi, M. Modeling of the Catalytic Claus Process. Ph.D. Dissertation, University of Alberta, Edmonton, Alberta, Canada, 1985. Razzaghi, M.; Dalla Lana, I. G. Calculation of Effectiveness Factor for Multiple Claus Reactions with Single-Step Rate-Controlling. Can. J. Chem. Eng. 1984, 62, 413-418. Sulfur. New Generation SRU Catalysts Are More Active and More Durable. Sulfur No. 211, 1990 Nov-Dec, 27-36. Tong, S.; Dalla Lana, I. G.; Chuang, K. T. Kinetic Modelling of the Hydrolysis of Carbonyl Sulfide Catalyzed by Either Titania or Alumina. Can. J. Chem. Eng. 1992, 71, 392-400. Tong, S.; Dalla Lana, I. G.; Chuang, K. T. Kinetic Modelling of the Hydrolysis of Carbon Disulfide Catalyzed by Either Titania or Alumina. Can. J. Chem. Eng. 1995, 73, 220-227,.
Received for review April 18, 1997 Revised manuscript received July 7, 1997 Accepted July 12, 1997X
0.5
BT (1/Mi + 1/Mj) PσijΩ
B ) [10.85 - 2.50(1/Mi + 1/Mj)0.5]10-4
Deff ) (/τ)DK,j
IE970277G
Ω ) kT/ij
σij and ij are constants in the Lennard-Jones potential
X Abstract published in Advance ACS Abstracts, September 1, 1997.
