4324
Ind. Eng. Chem. Res. 2001, 40, 4324-4335
Modeling of Catalyst Pellets for Fischer-Tropsch Synthesis Yi-Ning Wang, Yuan-Yuan Xu, Hong-Wei Xiang, Yong-Wang Li,* and Bi-Jiang Zhang State Key Laboratory of Coal Conversion, Institute of Coal Chemistry, Chinese Academy of Sciences, P.O. Box 165, Taiyuan 030001, People’s Republic of China
The present work focuses on the transfer and reaction phenomenon in a catalyst pellet for Fischer-Tropsch synthesis. It is considered that the pores of catalyst pellets are filled with liquid wax under Fischer-Tropsch synthesis conditions. The reactants in the bulk gas phase dissolve in the wax at the external surface of the pellet, and the dissolved components diffuse through the wax inside the pellet and react on the internal surface of the pellet. The thermodynamic equilibrium between the gases in the bulk and the liquid wax in the pores is correlated by a modified SRK equation of sate (MSRK EOS). On the basis of the phenomena observed from experiments, a comprehensive pellet model is suggested for catalyst design simulation, in which detailed Fischer-Tropsch mechanistic kinetics is properly imbedded. The reaction and diffusion interaction in a catalyst pellet and its effect on the product selectivity are further investigated. The potential of using eggshell-type catalyst pellets is explored as a means of decoupling the severe transport restriction and, hence, enhancing the overall selectivity of desired products. 1. Introduction The Fischer-Tropsch synthesis (FTS), in which syngas is primarily converted into a wide spectrum of hydrocarbons ranging from methane to heavy wax, has drawn renewed attention as an option for the production of clean transportation fuels and chemical feedstocks. Although a wealth of research has been conducted in the course of the development of fixed-bed-based Fischer-Tropsch technology, information concerning detailed descriptions of the reactivity and selectivity at the pellet level, which is of vital importance for reactor scale-up and for catalyst development, is inadequate or even scarce in the literature so far, because of the inherent complexity of FTS system. The importance of the diffusion of reactants in Fischer-Tropsch synthesis has been realized for a long time.1 Typical industrial FTS processes with fixed-bed reactors normally produce complex mixtures consisting of hydrocarbons with carbon numbers ranging from 1 to hundreds. The catalyst pores are, therefore, often filled with a stagnant liquid phase formed by the heavy waxy products, as a result of diffusion limitations and capillary condensation.2-6 Slow diffusion rates of reactant and product molecules through liquid-filled pores can be a determining factor for the overall reaction rate and can therefore substantially influence the product selectivity when the particle diameter is larger than about 1 mm.3,7,8 Intraparticle reactant gradients have been neglected in some previous simulation studies,9,10 even though transport limitations prevail in pellets.3,11,12 More recent studies3,12 have considered the diffusion and reaction process for packed beds but include only hydrogen concentration gradients. These models describe only the diffusion effect on the rate of syngas conversion and cannot predict the diffusion effect on the product selectivity. Most recently, Iglesia et al.4,5,8,13 have put forward a comprehensive study on the diffusion and * To whom correspondence should be addressed. E-mail:
[email protected] (or
[email protected]).
reaction process in Co and Ru catalyst pellets, leading to an olefin readsorption model.13 In their models, the CO hydrogenation model and the olefin readsorption model are separated, mainly because of a lack of well unified kinetics for the FTS system investigated. The latest critical review of FTS kinetics studies pointed out the lack of comprehensive kinetics models in which rate expressions for syngas consumption, hydrocarbon formation, and the water-gas shift (WGS) reaction should properly be unified.14 To elucidate transfer and reaction behavior and its effect on reactivity and selectivity, it is of great importance to combine a self-consistent intrinsic kinetics model with the transfer-reaction model for catalyst pellets. Although considerable efforts have been devoted to research on Fischer-Tropsch kinetics, most of the existing kinetics models, because of the complex nature of the reaction, involve the lumping of species, so that information on only the overall consumption rate of syngas is included, and the product distributions are usually described with an additional semiempirical product distribution model on the basis of chain growth mechanism assumptions.15-17 Normally these lumping kinetics models can only be used to predict overall conversion relations and temperature profiles in reactors by combining them with reactor models. Detailed kinetics will definitely be needed to obtain more information, for example, about the selectivity or product quality, from reactor modeling. A detailed kinetics model for the Fischer-Tropsch reaction has been reported by Lox and Froment18 for a precipitated iron catalyst. The kinetics can be used to predict the product distribution with variations in the operating conditions of the Fischer-Tropsch synthesis. It should be noted that olefin readsorption is commonly believed to have a significant influence on the Fischer-Tropsch product distribution.4-6,8,13 However, the effect of secondary reactions of the olefins was not included and was even not considered in the kinetics model proposed by Lox and Froment.18 More recently, Wang19 developed, on the basis of olefin co-feeding kinetics experiments over an
10.1021/ie010080v CCC: $20.00 © 2001 American Chemical Society Published on Web 09/08/2001
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4325
industrial Fe-Cu-K catalyst, a detailed kinetics model in which the effect of olefin readsorption was taken into account. In view of the fact that catalyst pores are filled by waxy hydrocarbons in Fischer-Tropsch synthesis, the solubilities of gases in the liquid wax are essential to the construction of a comprehensive model for the diffusion and reaction in an FTS catalyst pellet. However, the available experimental data on gas-liquid solubilities for Fischer-Tropsch synthesis are very limited. Although some simple correlations have been developed in the literature,20,21 these correlations are not flexible enough to adapt to variations in the carbon number of the waxy solvents, which can differ greatly in real FTS operations. Recently, a generalized gasliquid equilibrium correlation for Fischer-Tropsch synthesis was developed on the basis of a modified SRK equation of state (MSRK EOS) by Wang et al.22 This model can be used for systems with a wide range of solutes, including CO, H2, CO2, CH4, C2H4, C2H6 , and heavy wax solvents from C20 to C61, and it has been verified that the model has acceptable accuracy for some systems without experimental data. As a continuing part of our previous efforts,19,22 the objective of this work is to establish a comprehensive catalyst pellet model for the complex Fischer-Tropsch system, into which the detailed mechanistic kinetics19 and the modified SRK EOS correlation22 special for gas-liquid equilibrium in FTS are imbedded. The multicomponent diffusion-reaction model is then applied to analyze the interaction between diffusion and reaction and to simultaneously determine the reactivity and selectivity at the pellet level. Orthogonal collocation on finite elements23 is adopted as the discretization approach to solve the resulting differential equations.
Figure 1. Catalyst pellet with liquid wax in pores as a continuum phase.
2.2. Mathematical Description of Pellet Model. According to the above model assumptions, for a single catalyst pellet in which multiple reactions occur, the mass and energy balances at steady state can be expressed by the following differential equations
( ) ∑ ( )
NR 1 d 2 dcs,i Deff,i r ) -Fp RijR ˆ j (i ) 1, NPG) dr j)1 r2 dr NR 1 d 2 dTs r ) -Fp (-∆Hj)R ˆj Keff dr j)1 r2 dr
(2)
Boundary conditions dTs dcs,i ) 0; )0 dr dr
r ) 0: 2. Catalyst Pellet Model 2.1. Assumption of Physical-Chemical Environments. Observations1,2,4-6,8,13,24 have confirmed that, after a short startup period, the pores of catalyst pellets or particles become completely filled with liquid wax during Fischer-Tropsch synthesis, from which very heavy hydrocarbons are produced. The microflux field in the liquid phase is certainly very complex because reactants dissolve in the wax at the pore entrance and move and react inside the pellet while products move out by diffusion through the liquid wax in the pores and are evaporated at the external surface of the pellet as illustrated in Figure 1. The real state of the bulk containing a very small amount of heavy hydrocarbons (wax) could be in a saturated gas state with wax or in a dispersed colloid state of wax fog. At the interface between the catalyst pellet and the bulk gas phase, gas-liquid equilibrium can be assumed to be approached if the effect of the gas film on mass transfer can be neglected. In general, the treatment of gas film effects is, according to Froment et al.,25 reasonable for fixed-bed reactors. For heat transfer between the catalyst pellet and the bulk, however, the resistance of the gas film is dominant compared with the resistance of the heat conduction through the pellet media. For mass transport through the liquid-filled pores, it can be assumed that each component dissolving in the liquid wax phase in a very small portion moves through the liquid by molecular diffusion, and in this case, Fick’s law can be applied.
∑
(1)
r ) Rp: yi )
φLi
xi; φVi
hf dTs )(T - T) dr Keff s
(3)
(4)
Normally, it is convenient to use the dimensionless form for the numerical solution of the pellet model shown in eqs 1-4. For this purpose, some dimensionless variables can be introduced as follows
cs,i ci ; Ys,i ) s cT c
Yi )
(5)
s,T
NC
cT )
ci; ∑ i)1
NC
css,T )
s cs,i ∑ i)1
Ts r ; θs ) Rp T
s)
(6)
(7)
Rewriting the pellet model in eqs 1-4 using these dimensionless variables leads to
1 d
( )
s2 ds
s2
dYs,i ds
NR
) -φ2Di
1 d s2 ds
RijFj(Y,θs) ∑ j)1
( ) s
2
dθs ds
(i ) 1, NPG) (8)
NR
) -β
HjFj(Y,θs) ∑ j)1
(9)
4326
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001
R ˆ CH4 )
with the corresponding boundary conditions
dYs,i dθs ) 0; )0 ds ds
s ) 0: s ) 1: Yi )
φLi Ys,i; φVi
dθs ) -βh(θs - 1) ds
1+ 1+ (11)
It should be noted that, for the catalyst pellet simulation, the effectiveness factor is defined here on the basis of the conditions at the external surface of the pellet. The effectiveness factor on the basis of component i can thus be calculated from
∫01∑Ri,jRˆ j(Y,θs)s2 ds ηi )
(
(10)
K2K3K4 P 2 H2
∫01∑Ri,jRˆ j(Y,θs)s2 ds j
∫0 ∑RCO,jRˆ j(Y,θs)s
(13) 2
K3K4 PH2
K4
(
PH2O
1
K2K3K4 P 2 H2
1
1
+
+
K3K4 PH2
φ )
Deff,Icss,T
K4
(-∆H) )
∑j
R ˆj Deff,I Di ) R ˆJ Deff,i
(-∆Hj) Hj )
Bh )
i
N
(∏Rj) ∑ i)1 j)1
n
(
1+ 1+
ds
(-∆H)FpR ˆ JRp2 β) TKeff
Fj(Ys,i,θs) )
)
1
(n g 2) (19)
k6(1 - βn)
where the dimensionless parameters and variables are defined by
FpR ˆ JRp2
i
N
(∏Rj) ∑ i)1 j)1
n
Rj ∏ j)1
PH2O
1
1
+
K2K3K4 P 2 H2
Rj ∏ j)1 1
+
K3K4 PH2
∆Hj ∆H
hfRp Keff
(14)
K4
(17)
2.3. Activity Distribution Model. With a view toward the realistic feasibility of the shaping technique for Fe-Cu-K catalyst, a step-distribution function (instead of theoretical Dirac delta function26) is employed to describe the activity distribution of nonuniformly distributed catalyst pellet. Specifically, the activity zone is located within the bounds of [r1,r2] (here, 0 e r1 e r2 e Rp). This provides a convenient pathway to realize the numerical analysis of some typical distributions of active components (including eggyolk, eggwhite, eggshell, and uniform ones).
kv(PCOPH2O/PH20.5 - PCO2PH20.5/Kp) 1 + KvPCOPH2O/PH20.5
(21)
In this kinetics model, the chain growth factor for a carbon number of 1, R1, can be calculated from
R1 )
k1PCO (n ) 1) k1PCO + k5MPH2
(22)
The overall chain growth factor (Rn) for a carbon number of n (n g 2) is a combination of the Anderson-SchulzFlory (ASF) chain growth factor (RA) and olefin readsorption factor (βn), indicating that overall chain growth factor (Rn) is intrinsically dependent on the carbon number (n) and is no longer a constant as defined in the conventional ASF formalism. The expressions for R1, Rn, and βn (n g 2) are
Rn )
βn )
k1PCO k1PCO + k5PH2 + k6(1 - βn)
/[
k1PCO PCnH2n RAn-1 + k6 k1PCO + k5PH2
k-6
k-6
n
k1PCO + k5PH2 + k6
(
RAi-2PC ∑ i)2
(n g 2) (23)
)]
(n-i+2)H2(n-i+2)
(n g 2) (24)
3. Kinetic and Physicochemical Properties 3.1. Detailed Kinetics and Application. The olefin readsorption kinetics model from Wang,19 in which the overall consumption rate of syngas and the product distribution model are unified as compared with the conventional models available, is applied here.
i
N
(∏Rj) ∑ i)1 j)1
(n g 2) (20)
R ˆ CO2 )
(15)
(16)
)
1
j
2
)
1
(n ) 1) (18)
The selectivity of component i is defined by using the CO-based selectivity,18 viz.
1
+
R ˆ CnH2n )
j
Si,CO )
1
1
+
k5PH2
(12)
∫01∑Ri,jRˆ j(Y,1)s2 ds
PH2O
1
R ˆ CnH2n+2 )
1+ 1+
j
k5MPH2R1
RA )
k1PCO k1PCO + k5PH2 + k6
(25)
The equilibrium constant of the water-gas shift reac-
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4327 Table 1. Corresponding Kinetic Parameters for Detailed Kinetics Model parameter k1 k5M,0 E5M k5,0 E5 k6,0 E6
units g-1
s-1
value bar-1
mol mol g-1 s-1 bar-1 kJ mol-1 mol g-1 s-1 bar-1 kJ mol-1 mol g-1 s-1 kJ mol-1
parameter
10-5
2.23 × 4.65 × 103 92.89 2.74 × 102 87.01 2.66 × 106 111.04
units g-1
kv,0 Ev k-6 Kv K2 K3 K4
s-1
value
bar-1.5
mol kJ mol-1 mol g-1 s-1 bar-1 bar-0.5 -
15.70 45.08 2.75 × 10-5 1.13 × 10-3 1.81 × 10-2 4.68 × 10-2 0.226
Table 2. Conversion Paths of Fischer-Tropsch Reaction reaction path reactant product
CO + H2O CO2 + H2
CO + 3H2 CH4 + H2O
2CO + 4H2 C2H4 + 2H2O
2CO + 5H2 C2H6 + 2H2O
nCO + 2nH2 CnH2n + nH2O
nCO + (2n + 1)H2 CnH2n+2 + nH2O
CO H2 CO2 H2O CH4 C2H4 C2H6 ‚‚‚ CnH2n CnH2n+2
-1 1 1 -1 0 0 0 ‚‚‚ 0 0
-1 -3 0 1 1 0 0 ‚‚‚ 0 0
-2 -4 0 2 0 1 0 ‚‚‚ 0 0
-2 -5 0 2 0 0 1 ‚‚‚ 0 0
-n -2n 0 n 0 0 0 ‚‚‚ 1 0
-n -(2n + 1) 0 n 0 0 0 ‚‚‚ 0 1
tion, Kp, can be calculated using
Kp )
5078.0045 - 5.897 208 9 + 13.958 689 × T 10-4T - 27.592 844 × 10-8T2 (26)
All of the other parameters in the kinetic model are listed in Table 1. In addition, the macro reaction steps defining the conversion relationship between the reactants and products are listed in Table 2. For the purpose of applying the kinetics model developed under the gas-phase mode, the virtual pressure Pi of component i in the liquid wax is then calculated with its corresponding equilibrium concentration
Pi )
PxiφLi φVi
(27)
The calculations of the fugacity coefficients for the gas and liquid phases in the FTS wax are performed using the MSRK EOS.22 It should be added that the coupling diffusion effects of the reactants, CO, H2, CO2, and H2O, and the dissolved light products, C1-C10, are taken into account simultaneously in our calculation, whereas the effects of other organic gases dissolved in wax on the diffusion of reactants are included in the wax lumps. 3.2. Physical and Chemical Properties. According to NMR tests,19 the FTS wax in our cases was characterized with the molecular formula, n-C28H58. By fitting the reported diffusivities27 of the gaseous solutes, CO, H2, and CO2, in the liquid solvent, n-C28H58, we obtained the following simple correlations
DCO,B ) 5.584 × 10-7 exp(-1786.29/T)
(28)
DH2,B ) 1.085 × 10-6 exp(-1624.63/T)
(29)
DCO2,B ) 3.449 × 10-7 exp(-1613.65/T)
(30)
Because the direct experimental data are not available for the other relevant components, their molecular
diffusivities in the liquid wax are, with the aid of diffusivity correlations in infinitely dilute solutions,28 estimated according to the equation
Di,B ) DCO,B(VCO/Vi)0.6
(31)
The mole volume Vi is calculated by the Le Bas’s additive method.28 The corresponding effective diffusivity of component i in the catalyst pores is calculated by correcting the molecular diffusivity with the porosity and tortuosity
Deff,i )
pDi,B τ
(32)
Assuming that the enthalpy variation effect of the reactant solubility balances with that of product evaporation, the overall enthalpy difference can then be approximated by the formula
∆Hj ) ∆Hj,298K +
T ∆Cp,j dT ∫298
(33)
The basic data are taken from the thermodynamic database available in the book by Reid et al.28 4. Numerical Solution Model eqs 8 and 9, together with the boundary conditions given by eqs 10 and 11, lead to a two-point boundary-value problem (BVP). By applying standard discretization methods (e.g., finite difference, orthogonal collocation, etc.),13,23,26,29 the solution of the boundaryvalue problem can be transformed into the solution of a set of nonlinear algebraic equations. Orthogonal collocation on finite elements (OCFE)23,30 is chosen here because it has the salient advantage of combining the rapid convergence of the orthogonal collocation method with the convenient location of the geometrical regions along the pellet radius by finite difference methods. The dimensionless domain (0 e s e 1) is, according to the OCFE method,23 first divided into NE elements by placing the dividing points at sl (l ) 1, ..., NE + 1)
4328
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001
with s1 ) 0.0 and sNE+1 ) 1.0. Imposing normalization on each element leads to
ul ) (s - sl)/∆sl, ∆sl ) sl+1 - sl
(34)
By means of the usual procedures of orthogonal collocation,29 the pellet model in eqs 8-11 can then be discretized, resulting in a set of nonlinear algebraic equations derived from three parts described as follows: (1) The interior discretization (jth interior collocation point on the lth element) leads to
1
NCL+2
∆sl2
m)1
∑
BkmYlm
1
2
+
∑
sl + ukl∆sl ∆sl
AkmYlm
)
∆sl
1
2 sl +
ulk∆sl
(35)
NCL+2
∆sl
HjFj(Y,θs)]lk ∑ j)1
(36)
(l ) 1, ..., NE; k ) 2, ..., NCL + 1) (2) The boundary condition discretization leads to NCL+2
∆s1 1 ∆s1
A1mY1m ) 0 ∑ m)1
(s ) 0)
(37)
(s ) 0)
(38)
NCL+2
1 A1mθs,m )0 ∑ m)1
NE ) Y0 (s ) 1) YNCL+2
1
NCL+2
∆sNE
m)1
∑
(39)
NE NE ANCL+2,mθs,m + Bh(θs,NCL+2 - 1) ) 0
(s ) 1) (40) (3) The requirement of flux continuity between two neighboring elements leads to l ) Yl+1 YNCL+2 1 NCL+2
1 ∆s1
l+1
A1,mYl+1 ∑ m ) 0 m)1
l l+1 θs,NCL+2 ) θs,1
1 ∆sl
NCL+2
∑ m)1
ANCL+2,mθlm -
1 ∆sl+1
(41)
NCL+2
1
ANCL+2,mYlm ∑ ∆s m)1
(42) (43)
NCL+2
A1,mθl+1 ∑ m ) 0 m)1 (l ) 1, ..., NE - 1) (44)
where
Y0 )
(
φL1
)
Si,CO )
l Wm∑Ri,jR ˆ j(Ylm,θs,m )(∆slum + sl)2∆sl ∑ ∑ l)1 m)1 j NE NCL+2
(47)
l ) ∑ Akmθs,m m)1
-[β
(46)
l )(∆slum + sl)2∆sl ∑ ∑ Wm∑j RCO,jRˆ j(Ylm,θs,m l)1 m)1
NR
1
∑j RijRˆ j(Y0,1) NE NCL+2
NR
l + ∑ Bkmθs,m 2 m)1
ηi )
m)1
RijFj(Y,θs)]lk ∑ j)1
NCL+2
NE NCL+2
l )(∆slum + sl)2∆sl ∑ ∑ Wm∑j Ri,jRˆ j(Ylm,θs,m l)1 m)1
3
NCL+2
-[φ2Di 1
Collecting eqs 35-44, the final system has (NPG + 1) × [NE × (NCL + 1) + 1] nonlinear equations with (NPG + 1) × [NE × (NCL + 1) + 1] unknowns. The effectiveness factor and selectivity of component can, accordingly, be discretized as follows
φL2 φLNPG Y , Y , ..., YNPG ; 1 2 φV1 φV2 φVNPG Y ) (Ys,1, Ys,2, ..., Ys,NPG) (45)
Finally, the numerical solution of the resulting nonlinear algebraic equations is carried out by employing an appropriate routine, HYBRD1, which is based on a modification of the Powell hybrid algorithm.31 It should be added that the solution of the above nonlinear equations is straightforward, but the convergence of this complex system might be difficult in some cases. For this reason, our effort here is devoted to improving the initialization of the unknowns. In practice, the surface liquid concentrations equilibrated with gas bulk concentrations are set, as guess values, to the unknown concentration variables. If the nonlinear solver fails to converge in some cases, the radius of catalyst pellet is halved continuously until convergence is approached. Evidently, the extreme case will lead to a catalyst pellet of infinitesimal size, suggesting that the convergence solution of the unknowns is equal to the surface equilibrium values readily determined from gas-wax equilibrium calculations. In reality, when the pellet size is reduced to a finite radius, convergence often occurs. By updating the guessed values with current solutions and recovering the current radius to its original value with 20 uniform steps, final convergence is generally obtained. It seems that this solution strategy is effective for the present nonlinear system. 5. Simulation and Discussion The basic parameters used in the pellet simulation are listed in Table 3, and are regarded to have their default values if no explicit declaration is made. In our simulation, the default parameters NCL ) 3 and NE ) 5 are set for the OCFE-based discretization. 5.1. Intrapellet Concentration Profiles. It is of interest to investigate the interaction between diffusion and reaction in a Fischer-Tropsch catalyst pellet, which is rather complex compared to cases with simple reactions. Figure 2 shows the concentration profiles of key components in wax-filled catalyst pores under typical conditions. Because hydrocarbon-forming reactions generally prevail in the Fischer-Tropsch system, it is not surprising that the concentration of the accompanying product, H2O, exhibits a steadily increasing trend along the pellet dimension. The resulting relatively high liquid concentration of H2O means that the WGS reaction will be driven gradually from a startup state to a fully developing state. Meanwhile, CO is consumed not only
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4329 Table 3. Basic Parameters Involved in Pellet Simulations catalyst physical properties density of pellet, Fp porosity of pellet, p tortuosity of pellet,τ
(g/m3)
a
value 1.95 × 0.51 2.6a
106
heat transfer parameters m-2
value s-1
K-1)
heat transfer coefficient, hf (J conductivity coefficient, Keff (J m-1 s-1 K-1)
100.0 6.65
Estimated according to Hugo’s correlation (see p 55 of ref 32).
Figure 2. Concentration profiles of key components in catalyst pellet.
Figure 3. Variation of H2/CO ratio along the pellet dimension.
by the FTS but also by the WGS reaction, leading to the rapid consumption of CO. It can therefore be found from the concentration gradient of CO that, the main reaction zone is located within [0.8, 1.0], corresponding to a 0.2Rp shell thickness, under current typical reaction conditions and for a pellet size (3 mm in diameter) of significance for industrial applications. This thickness is in good agreement with the value (about 0.24Rp) reported in the literature.33 At the position of about 0.85Rp, the CO transferred from the bulk cannot even satisfy the requirements of the FTS reactions, because of the more severe diffusion limitation of CO than of H2. However, the concentrations of CO2 and H2 stay relatively high at this position; thus, the reversible WGS reaction will be progressively triggered to produce CO for promotion of the hydrocarbon-forming reactions. These phenomena can be judged from the decreasing trend of the CO2 concentration after it reaches its peak (see Figure 2), suggesting that the WGS reaction is dominant over the FTS reaction when it passes over this turning point. Because of the low CO concentration, the rate of the FTS reaction in the interior region is relatively lower than that in the outer region of pellet. From the variation of the ratio of the H2 and CO fugacities along the pellet dimension (as shown in Figure 3), we can see that, the ratio at the central point approaches a very high level (ca 200-300 times higher than that at the external surface). According to the chain growth formula (eqs 22 and 23), it can then be derived that the termination probability for this polymerization reaction will be significantly increased in the internal region because of the high H2 concentration. Therefore, the lighter and more paraffinic hydrocarbons are more easily formed in the internal region, and the heavier hydrocarbons are more easily formed in outer shell region. From an analysis of the concentration profiles, we can see that a serious intraparticle diffusion limitation exists, especially for the reactant CO.4 An improvement
of diffusion effects in the catalyst pellet would be expected to improve the whole conversion performance of the catalyst pellet and also to increase the selectivity to heavy hydrocarbons. 5.2. Reactivity and Selectivity Analysis. 5.2.1. Temperature Effect. For the current simulation, our focus is on understanding the effects of intraparticle transportation on catalytic reaction performance. The external gas film is considered to contribute mainly to heat transfer, whereas intraparticle diffusion contributes to mass transfer, as assumed in section 2.1 and according to ref 25. With the assumed external heat transfer coefficient hf ) 100.0 J m-2 s-1 K-1, the simulation results show that the differences between the bulk temperature T and the temperature at the external surface of the pellet are within 2 K, whereas the temperature differences between the external surface and the center of the pellet are less than 0.02 K because of the excellent heat conductivity of the FeCu-K catalyst, resulting in good agreement with the facts relevant to the current system.25 The variations of the CO-based effectiveness factor with temperature are shown in Figure 4. It can be seen from this figure that, with increasing temperature, the effectiveness factor first decreases to a minimum and then slightly increases at a certain temperature (502 K in this case). After passing through the maximum value, the effectiveness factor continues decreasing with further increases in temperature. This phenomenon is, on one hand, reasonable in the case of complex kinetics expressions.26 For Fischer-Tropsch synthesis, on the other hand, the CO conversion rate at the external surface of catalyst pellets should be different from that inside the pellet, because CO can be converted to CO2 as a result of the higher water fugacity inside, but not at the external surface of, the pellet. With increasing temperature, the diffusion of reactants gradually becomes relatively slow compared with the intrinsic reaction rates inside the catalyst, leading to a monotonically decreasing trend at the lower temperature range be-
4330
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001
Figure 4. Variation of effectiveness factor with operation temperatures at different pressures.
Figure 6. Product distribution of hydrocarbons at different temperatures.
Figure 5. Variation of effectiveness factor with Thiele modulus.
Figure 7. Selectivities of CO2 and C5+ products at different temperatures.
cause of the diffusion limitations. The slight increasing trend appearing at the higher temperature range reflects an improvement of the matching relationship between the diffusion and reaction behaviors inside the pellet. These changes result not only from the increasing diffusivities of the reactants in the wax at higher temperatures but also from the effect of the WGS reaction on CO conversion. In fact, the latter plays a more important role in the unconventional variation trends of the effectiveness factor. Figure 4 also indicates that the reaction pressure shows only a slight influence on the reactivity, mainly because of the small effects of pressure on the diffusivities of the reactants in the waxy products. The variation of the effectiveness factor with the Thiele modulus is illustrated in Figure 5, corresponding to trends in the case of the effectiveness factor variation with temperature. Figures 6 and 7 show the effect of temperature on selectivity, indicating that the product selectivities are very sensitive to temperature changes. An increase in temperature, which is related to an increase in the chain termination probability, leads to a remarkable increase in the CH4 selectivity (Figure 6) and, accordingly, to a suppression of the formation of C2+ products. Furthermore, it can be found from Figure 7 that the selectivities
of C5+ and CO2 go through maximum values with increased bulk temperature. From these results, it can be concluded that low temperatures are much more favorable for suppressing the selectivity of the undesired products (CH4 and CO2). In addition, an optimal selectivity of C5+ products can be approached by properly selecting the operating temperature. 5.2.2. Pellet Size Effect. Figure 8 shows the variations of the effectiveness factor with pellet size at different temperatures. It can clearly be seen that, for very small catalyst particles, the CO-based effectiveness factor is close to unity. With increasing pellet size, the effectiveness factor decreases continuously from a value of about 1 to a relatively low value. In particular, for industrial pellets with diameters of about 2-4 mm, the effectiveness factor is within a range of 0.14-0.28 in our case, corresponding to severe diffusion limitations. The calculated range of the effectiveness factor is comparable to the results from an FTS pilot plant reported by Jess et al (i.e., catalyst diameter ) 2.5 mm, effectiveness factor ≈ 0.2 at 523.2 K).10 It is worthy mentioning that the diffusion limitation exists up to a pellet size of about 0.15 mm (100 ASTM mesh), which corresponds to a Thiele modulus of 1.58 (Figure 9). This suggests that very small particles should be used in FTS intrinsic
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4331
Figure 8. Variation of effectiveness factor with pellet size.
Figure 9. Variation of effectiveness factor with Thiele modulus.
Figure 11. Selectivity variations of C2+ and C5+ products with pellet radius.
increasing pellet size, leading to general decreasing trends of the selectivities of C2+ and C5+ products (Figure 11). According to the reactivity and selectivity analysis, it can be deduced that catalysts of larger size play a negative role in improving the reactivity and selectivity of desired products because of strong diffusion effects. To enhance the reactivity and especially the selectivity, small particles are preferred. For realistic industrial applications, however, it is clear that the catalyst pellets should have a proper size, normally 2-4 mm in diameter, to maintain a low pressure drop in the reactor and provide effective heat removal.7,32 That means that it is impossible to reduce the pellet size in industrial cases, presenting a contradiction between the requirement for better catalyst performance and the pressure drop constraint.34 5.3. Catalyst Design Consideration: Eggshell and Uniform. In view of industrial practice, it has been reported that eggshell catalysts remain a very attractive solution for relieving intrapellet transport restrictions in packed-bed FTS reactors.5,35-37 If the catalyst density over the active layer of the eggshell-type pellet is the same as that in the uniformly distributed catalyst pellet, the volume of the active part of the eggshell catalyst pellet is
4 Veggshell ) π(r23 - r13) 3
(48)
and the active volume of the uniform catalyst pellet is
4 Vuniform ) πr23 3
(49)
Veggshell r23 - r13 ) Vuniform r3
(50)
Thus
Figure 10. Selectivity variations of CO2 and CH4 with pellet radius.
kinetic experiments if liquid wax is present in the catalyst pores. Figures 10 and 11 show the effect of pellet size on product selectivities. It can be seen from Figure 10 that the selectivities of the undesired products increase with
2
This ratio is less than 1.0 for eggshell catalyst pellets, indicating that the total activity of the eggshell catalyst pellet is less than that of the uniform catalyst pellet. Hence, the design objective for eggshell catalyst pellets is to find the optimal size of the inert core of a certain size of catalyst pellet to enhance the selectivity while
4332
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001
Figure 12. (a-d) Comparison of intraparticle concentrations of key components for eggshell-type and uniform catalysts.
keeping the activity of the whole pellet at a proper level compared to that of the uniform pellet. The simulation results indicate that the intraparticle concentration profiles of key components are different between the eggshell and uniform catalyst pellets, as shown in Figure 12. It is worth mentioning that the inert core of eggshell pellets in Figure 12 is assumed to be 0.85Rp. It is clear from this figure that, because of the introduction of an inert core, the eggshell catalyst pellet can avoid the extreme depletion of CO and thus lower the high H2/CO ratio imposed on the internal surface of the pellet, as compared with the uniform pellet. Therefore, the eggshell pellet can suppress the relatively fast chain termination reaction in the deep zone of the catalyst pellet and should normally be able to produce more heavy hydrocarbons than the uniform pellet (Figure 13). From Figure 13, it is shown that, as the inert core increases, the C5+ selectivity increases, a phenomenon consistent with the experiment results of Iglesia et al.5 Associated with the improvement to the diffusion restriction, the WGS reaction also shows completely different behavior for the two catalyst types (see Figure 12c). For the eggshell catalyst, the WGS reaction mainly produces CO2 in the thin active layer. For the uniform catalyst pellet, however, CO2 is produced within the outer layer with a thickness of 0.2Rp but is consumed continuously in the deep zone of the catalyst pellet because of the high H2/CO ratio. From Figure 14, we can see that, when the inert core is away
Figure 13. Effect of inert core radius on C5+ selectivity.
from the above-mentioned main reaction zone (i.e., r1 e 0.80Rp), the CO2 selectivity is insensitive to the variation of the inert core. The tiny change of CO2 selectivity reflects a very weak WGS reverse reaction taking place in the deep zone for the uniform case, which is mainly due to the inhibiting effect of high intraparticle H2O fugacity (see Figure 12d). However, when the inert core touches the main reaction zone (for instance, 0.90Rp), an essential change in the CO2 selectivity
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4333
Figure 14. Effect of inert core radius on CO2 and CH4 selectivities.
properties of gases and Fischer-Tropsch wax, which makes the pellet simulation feasible. Orthogonal collocation on finite elements is used to discretize the pellet model and proves to be an efficient numerical technique to solve the catalyst pellet model. The simulation results show that severe intraparticle diffusion restrictions in the wax-filled pores, which introduce concentration gradients of the reactants and products, not only lower reactant conversion rates, but also lead to significantly different product selectivities. In view of practical applications, large catalyst pellets with diameters of few millimeters are required for FTS in packed-bed reactors, suggesting that a compromise of catalyst performance and reactor operation demand has to be found to decouple the serious transport limitations. In terms of catalyst design considerations, numerical simulation shows that eggshell catalyst pellets can be used to enhance the selectivities of heavier hydrocarbons and, at the same time, allow a significant increase of the reactor volume to be avoided, providing that the active layers are properly controlled. Acknowledgment Financial support from the Chinese Academy of Sciences and the National Natural Science Foundation of China (Project 29673054) is gratefully acknowledged. Nomenclature
Figure 15. Effect of inert core radius on effectiveness factor.
(Figure 14) will occur, and decreases in the selectivities of both CH4 and CO2 will contribute directly to an increase in the selectivity of the C5+ products. Therefore, with a view toward selectivity enhancement, the use of eggshell-type catalysts is feasible, because of its positive improvement of intraparticle diffusion limitations. Figure 15 shows the effect of the inert core radius on the effectiveness factor. It can be seen that the activity decreases are not remarkable when the inert core is less than 0.80Rp. This means that the eggshell catalyst, with only a small fraction of the active volume of the uniform catalyst (e.g., 49% for 0.80Rp), can provide almost the same overall reactivity or activity over a wide range of operating temperatures. This indicates that using eggshell catalyst pellets can avoid a significant increase of the reactor volume to obtain the same productivity as compared to uniform catalyst pellets if a proper active layer is adopted. Hence, eggshell-type catalysts can provide better properties than uniform catalysts, especially when large catalyst pellets must be used. 6. Conclusions On the basis of detailed reaction kinetics, a comprehensive catalyst pellet model is developed and shown to be a reasonable description of diffusion and reaction in a Fischer-Tropsch catalyst pellet. In this model, a modified SRK EOS is successfully used to correlate the
Akm ) coefficient matrix for first derivative Bkm ) coefficient matrix for second derivative Bh ) Biot number for heat transfer ci ) bulk gas concentration of component i, mol/m3 cs,i ) liquid concentration of component i, mol/m3 s cs,i ) liquid concentration of component i at pellet surface, mol/m3 css,T ) total concentration of liquid components, mol/m3 cT ) total concentration of bulk gas phase, mol/m3 ∆Cp,j ) residual heat capacity of jth reaction, J mol-1 K-1 Di,B ) liquid diffusivity coefficient of component i in wax, m2/s Deff,i ) effective diffusivity coefficient of key component i, m2/s Deff,I ) reference diffusivity coefficient, m2/s Di ) dimensionless diffusivity coefficient of component i E5 ) activation energy for paraffin formation, kJ mol-1 (n g 2) E5M ) activation energy for methane formation, kJ mol-1 E6 ) activation energy for olefin formation, kJ mol-1 (n g 2) Ev ) activation energy for WGS reaction, kJ mol-1 Fj ) dimensionless rate of jth reaction hf ) heat transfer coefficient of external film, J m-2 s-1 K-1 -∆H ) overall reaction heat, J/mol Hj ) dimensionless reaction heat of jth reaction -∆Hj ) reaction heat of jth reaction, J/mol k ) kth collocation point k5 ) rate constant of paraffin formation, mol g-1 s-1 bar-1 k5,0 ) preexponential factor of paraffin formation (n g 2), mol g-1 s-1 bar-1 k5M ) rate constant of olefin desorption reaction, mol g-1 s-1 bar-1 k5M,0 ) preexponential factor of rate constant of methane formation, mol g-1 s-1 bar-1 k6 ) rate constant of olefin desorption reaction, mol g-1 s-1 k6,0 ) preexponential factor of rate constant of olefin desorption reaction, mol g-1 s-1
4334
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001
kv ) rate constant of CO2 formation, mol g-1 s-1 bar-1.5 kv,0 ) preexponential factor of rate constant of CO2 formation, mol g-1 s-1 bar-1.5 k-6 ) rate constant of olefin readsorption reaction, mol g-1 s-1 bar-1 K2, K3, K4 ) equilibrium constants in kinetics expression Keff ) effective conductivity coefficient, J m-1 s-1 K-1 Kp ) equilibrium constant of WGS reaction Kv ) group of constants in WGS reaction, bar-0.5 l ) lth element m ) mth collocation point N ) maximum carbon number of the hydrocarbons involved NE ) element number NC ) total number of components involved NCL ) number of interior collocation points NPG ) number of key components involved NR ) total number of reactions involved Pi ) virtual pressure of component i, bar P ) total pressure, bar r ) pellet dimension, m r1 ) left bound of active layer, m r2 ) right bound of active layer,m R ˆ CnH2n ) formation rate for olefin, mol g-1 s-1 R ˆ CnH2n+2 ) formation rate for paraffin, mol g-1 s-1 R ˆ WGS ) formation rate for CO2, mol g-1 s-1 R ˆ j ) rate of jth reaction, mol g-1 s-1 R ˆ J ) reference reaction rate, mol g-1 s-1 Rp ) pellet radius, m s ) pellet dimension ∆sl ) element band of lth element sl ) axis of lth element Si,CO ) CO-based selectivity of component i T ) bulk temperature of gas phase, K Ts ) pellet temperature, K ul ) local dimensionless position of lth element Veggshell ) active volume of the eggshell catalyst pellet, m3 Vuniform ) active volume of the uniform catalyst pellet, m3 Vi ) liquid molar volume of component i, m3/mol Wm ) integral weight of mth collocation point xi ) molar fraction of component i in liquid wax yi ) molar fraction of component i in bulk gas phase yl ) working variable Yi ) dimensionless concentration of component i in bulk gas phase Ys,i ) dimensionless liquid concentration of component i Y0 ) equilibrium concentration vector of key component in liquid phase Y ) concentration vector of key component in liquid phase Ylm ) concentration vector of key component on mth collocation point in lth element Greek Symbols R1 ) chain growth factor for carbon number of 1 Rij ) stoichiometric coefficient of component i in jth reaction Rn ) chain growth factor for carbon number of n (n g 2) RA ) chain growth probability in the Anderson-SchulzFlory distribution β ) Prate number for heat transfer βn ) readsorption factor of 1-olefin with carbon number of n p ) porosity of catalyst pellet φ ) Thiele modulus φLi ) fugacity coefficient of component i in liquid wax φVi ) fugacity coefficient of component i in gas phase ηi ) effectiveness factor of component i l θs,m ) dimensionless pellet temperature of pellet on mth collocation point in lth element θs ) dimensionless pellet temperature
τ ) tortuosity of catalyst pellet Fp ) density of catalyst pellet, g/m3 Superscripts and Subscripts i ) index indicating reactions j ) index indicating components n ) number of carbon atoms
Literature Cited (1) Anderson, R. B.; Seligman, B.; Schultz, J. F.; Kelly, R.; Elliott, M. A. Fischer-Tropsch Synthesis: Some Important Variables of the Synthesis on Iron Catalysts. Ind. Eng. Chem. 1952, 44, 391. (2) Lox, E. S.; Marin, G.; Graeve, De; Bussiere, P. Characterization of a Precipitated Promoted Iron Catalyst for Fischer-Tropsch Synthesis. Appl. Catal. 1988, 40, 197. (3) Sie, S. T.; Senden, M. M. G.; Van Wechem, H. M. H. Conversion of Natural Gas to Transportation Fuels via the Shell Middle Distillate Synthesis Process. Catal. Today 1991, 8, 371394. (4) Madon, R. J.; Iglesia, E. Hydrogen and CO Intrapellet Diffusion Effects in Ruthenium-Catalyzed Hydrocarbon Synthesis. J. Catal. 1994, 149, 428-437. (5) Iglesia, E.; Soled, S. L.; Baumgartner, J. E.; Reyes, S. C. Synthesis and Catalytic Properties of Eggshell Cobalt Catalysts for the Fischer-Tropsch Synthesis. J. Catal. 1995, 153, 108-122. (6) Kuipers, E. W.; Vinkenburg, I. H.; Oosterbeek, H. Chain Length Dependence of 1-Olefin Readsorption in Fischer-Tropsch Synthesis. J. Catal. 1995, 152, 137-146. (7) Sie, S. T.; Krishma, R. Fundamentals and Selection of Advanced Fischer-Tropsch Reactors. Appl. Catal. 1999, 186, 5570. (8) Iglesia, E.; Reyes, S. C.; Madon, R. J. Transport-Enhanced 1-Olefin Readsorption Pathways in Ru-Catalyzed Hydrocarbon Synthesis. J. Catal. 1991, 129, 238-256. (9) Bub, G.; Baerns, M. Prediction of the Performance of Catalytic Fixed Bed Reactor for Fischer-Tropsch Synthesis. Chem. Eng. Sci. 1980, 35, 348-355. (10) Jess, A.; Popp, R.; Hedden, K. Fischer-Tropsch Synthesis with Nitrogen-rich Syngas: Fundamentals and Reactor Design Aspects. Appl. Catal. 1999, 186, 321-342. (11) Zimmerman, W. H.; Rossin, J. A.; Bukur, D. B. Effect of Particle Size on the Activity of Fused Iron Fischer-Tropsch Catalyst. Ind. Eng. Chem. Res. 1989, 28, 406-413. (12) Post, M. F. M.; van’t Hoog, A. C.; Minderhoud, J. K.; Sie, S. T. Diffusion Limitations in Fischer-Tropsch Catalysts. AIChE J. 1989, 35, 1107-1114. (13) Iglesia, E.; Reyes, S. C.; Soled, S. L. Reaction-Transport Selectivity Models and the Design of Fischer Tropsch Catalysts. In Computer-Aided Design of Catalysts and Reactors; Becker, E. R., Pereira, C. J., Eds.; Marcel Dekker: New York, 1993; p 199. (14) Van Der Laan, G. P.; Beenackers, A. A. C. M. Kinetics and Selectivity of the Fischer-Tropsch Synthesis: A Literature Review. Catal. Rev.-Sci. Eng. 1999, 41, 255-318. (15) Bell, A. T. Catalytic Synthesis of Hydrocarbons over Group VIII Metals. A Discussion of the Reaction Mechanism. Catal. Rev.Sci. Eng. 1981, 23, 203-232. (16) Dixit, R. S.; Tavlarides, L. L. Integral Method of Analysis of Fischer-Tropsch Synthesis Reactions in a Catalyst Pellet. Chem. Eng. Sci. 1982, 37, 539-544. (17) Liu, Z.-T.; Li, Y.-W.; Zhou, J.-L.; Zhang, B.-J. Intrinsic Kinetics of Fischer-Tropsch Synthesis over an Fe-Cu-K Catalyst. J. Chem. Soc., Faraday Trans. 1995, 91, 3255-3261. (18) Lox, E. S.; Froment, G. F. Kinetics of the Fischer-Tropsch Reaction on a Precipitated Promoted Iron Catalyst. 2. Kinetic Modeling. Ind. Eng. Chem. Res. 1993, 32, 71-82. (19) Wang, Y.-N. Modelization and Simulation of Fixed-Bed Fischer-Tropsch Synthesis: Kinetics, Pellet and Reactor. Ph.D. Dissertation, Institute of Coal Chemistry, Chinese Academy of Sciences, Taiyuan, China, 2001. (20) Kuo, C. W. Two-Stage Process for Conversion of Synthesis Gas to High Quality Transportation Fuels; Final Report, DOE Contract DE-AC22-83PC60019; U.S. Department of Energy: Washington, D.C., 1985. (21) Caldwell, L.; Van Vuuren, D. S. On the Formation and Composition of Liquid Phase in Fischer-Tropsch Reactors. Chem. Eng. Sci. 1986, 41, 89-96.
Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4335 (22) Wang, Y.-N.; Li, Y.-W.; Bai, L.; Zhao, Y.-L.; Zhang, B.-J. Correlation for Gas-Liquid Equilibrium Prediction in FischerTropsch Synthesis. Fuel 1999, 78, 911-917. (23) Finlayson, B. A. Nonlinear Analysis in Chemical Engineering; McGraw-Hill: New York, 1980. (24) Raak, H. A Pressurized Reactor with Integrated Load Cell for Measuring the Activities of Fischer-Tropsch Synthesis: Their Loading with Liquid Reaction Products (Wax). Erdol-Erdgas-Kohle 1998, 114, 251. (25) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; Wiley: New York, 1990. (26) Aris, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts; Clarendon Press: London, 1975. (27) Akgerman, A. Diffusivities of Synthesis Gas and FischerTropsch Products in Slurry Media. In Proceedings of the Indirect Liquefaction Contractors’ Review Meeting, Pittsburgh, PA, 1987; Federal Energy Technology Center (FETC), U.S. Department of Energy: Washington, D.C., 1987; p 354. (28) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (29) Villadsen, J.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice Hall: Englewood Cliffs, NJ, 1978. (30) Carey, G. F.; Finlayson, B. A. Orthogonal Collocation on Finite Elements. Chem. Eng. Sci. 1975, 30, 587-596.
(31) More´, J.; Garbow, B.; Hillstrom, R. User Guide for MINPACK-1; Report ANL-80-74; Argonne National Laboratories: Argonne, IL, 1980. (32) Wijngaarden, R. J.; Kronberg, A.; Westerterp, K. R. Industrial Catalysis: Optimizing Catalysts and Processes; Wiley: New York, 1998. (33) Jess, A.; Hedden, K. Fischer-Tropsch Synthesis with Nitrogen-Rich Syngas: An Alternative to Common Process. Prepr. Pap. Am. Chem. Soc. Div. Pet. Chem. 2000, 45, 204. (34) Sie, S. T. Process Development and Scale-up: IV. Case History of the Development of Fischer-Tropsch synthesis Process. Rev. Chem. Eng. 1998, 14, 109-157. (35) Van Erp, W. A.; Nanne, J. M.; Post, M. F. M. Process for Preparation of Catalyst. U.S. Patent 4,637,993, 1987. (36) Maudlin, C. H.; Riley, K. L. Process for the Preparation of Surface Impregnated Dispersed Cobalt Metal Catalysts. U.S. Patent 4,977,126, 1990. (37) Iglesia, E.; Vroman, H.; Soled, S. L.; Baumgartner, J. E.; Fiato, R. A. Selective Catalysts and Their Preparation for Catalytic Hydrocarbon Synthesis. U.S. Patent 5,036,032, 1991.
Received for review January 25, 2001 Accepted July 6, 2001 IE010080V
