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Effects of Catalyst Activity, Particle Size and Shape, and Process Conditions on Catalyst Effectiveness and Methane Selectivity for Fischer−Tropsch Reaction: A Modeling Study Miloš Mandić,† Branislav Todić,† Ljiljana Ž ivanić,†,∥ Nikola Nikačević,‡ and Dragomir B. Bukur*,†,§ †

Chemical Engineering Program, Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar Faculty of Technology and Metallurgy, University of Belgrade, Karnegijeva 4, Belgrade 11000, Serbia § Artie McFerrin Department of Chemical Engineering, Texas A&M University, MS 3122, College Station, Texas 77843-3122, United States ‡

ABSTRACT: We investigate effects of catalyst activity, catalyst particle shape (sphere, slab, and hollow cylinder), size (i.e., diffusion length), catalyst distribution (uniform vs eggshell type distribution for a spherical particle), and process conditions (temperature, pressure, syngas composition, and conversion level) on catalyst effectiveness factor and methane selectivity inside the catalyst pellet. In numerical simulations we utilize kinetic parameters for CO consumption rate and CH4 formation rate determined from experiments with a highly active Co/Re/γ-Al2O3 catalyst. It is found that the use of small spherical particles (0.2−0.5 mm) or eggshell distribution for larger spherical particles with catalyst layer thickness less than approximately 0.13 mm is needed to avoid negative impact of diffusional limitations on CH4 selectivity under typical Fischer− Tropsch synthesis operating conditions. For monolith reactors with wash-coated catalyst, diffusional limitations can be avoided by using a catalyst layer thickness less than 0.11 mm at base case conditions (473 K, 25 bar, and H2/CO molar ratio of 2).

1. INTRODUCTION Fischer−Tropsch synthesis (FTS) is a heterogeneous reaction used to convert synthesis gas into a range of hydrocarbon products. This reaction is a key step in the gas-to-liquid (GTL) process in which natural gas is converted into liquid fuels and value-added chemicals. Low-temperature FTS is conducted commercially in two types of reactors: slurry bubble column reactors and multitubular fixed bed reactors.1−3 The overall goal of fixed bed reactor design is to maximize productivity and selectivity to desired products (low methane and high selectivity to liquid hydrocarbons−C5+ hydrocarbons) while minimizing pressure drop and costs. To achieve these objectives, a judicious choice for the catalyst particle size and shape, as well as process conditions, is required. To avoid high pressure drop, the use of relatively large particles (1−3 mm) is required,1,2 whereas to achieve high productivity in a given reactor volume, one has to use very active catalysts. These requirements lead to intraparticle diffusional limitations resulting in lower reaction rate (decrease in catalyst effectiveness) and decrease in selectivity to liquid hydrocarbons.4−8 Slurry reactors utilize small catalyst particles (less than 100 μm), and in this case, the intraparticle diffusional limitations are not expected under normal operating conditions. Microreactors for FTS have recently received a great deal of attention because of increased heat removal at moderate pressure drop.9,10 Several types of reactors have been considered for this purpose, including (a) reactors with microstructured catalysts (e.g., monoliths and foams), (b) coated microchannel reactors, © XXXX American Chemical Society

and (c) micro- and milli-fixed bed reactors in which small catalyst particles are loaded into the packed bed.3 These types of reactors in general minimize negative effects of intraparticle diffusional limitations on activity and selectivity but are characterized by low reactor productivity due to low amount of catalyst in the reactor.11,12 The latter can be compensated by using highly active FTS catalysts such as those developed by Oxford Catalysts Group8,13 and milli-fixed bed reactors loaded with small particles. The interplay between chemical reaction and intraparticle diffusion for FTS applications (Co- and Fe-based catalysts) has been studied in the literature either on a single-particle scale or as a part of modeling of fixed bed reactors and has received increased attention in recent years. In a majority of previous studies a simplified kinetics (first-order in hydrogen or nth-order in carbon monoxide) was utilized to calculate catalyst effectiveness factor as a function of Thiele modulus, and selectivity inside the particle was not considered.14−22 The first quantitative analysis of chemical reaction with diffusion problem for FTS reaction was published by Dixit and Tavlarides23 who used a form of Langmuir−Hinshelwood (LH) kinetics derived from experimental data with 0.5%Ru/γ-Al2O3 catalyst for FTS.24 This form of rate equation was later utilized by a number of Received: Revised: Accepted: Published: A

January 5, 2017 February 20, 2017 February 23, 2017 February 23, 2017 DOI: 10.1021/acs.iecr.7b00053 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

ASF distribution. In particular, the actual CH4 selectivity is significantly higher than that predicted from the ideal ASF distribution; selectivity of C2 hydrocarbons is lower than predicted value; and α varies with chain length.36−38 Thus, predicted values of C5+ selectivity using the proposed approach would not be very accurate. Also, the authors have used kinetic parameters for syngas consumption from Yates and Satterfield25 experimental data with a catalyst different than the catalyst used to obtain the expression for variation of α with temperature and syngas composition. In general, one should use kinetic parameters for activity and product distribution obtained from an experimental study with the same catalyst. In this study we investigate effects of particle shape (sphere, slab, and hollow cylinder), size (i.e., diffusion length), catalyst distribution (uniform vs eggshell type distribution for a spherical particle), and process conditions (temperature, pressure, syngas composition, and conversion level) on catalyst effectiveness factor and methane selectivity inside a particle. In numerical simulations we utilize kinetic parameters for CO consumption rate and CH4 formation rate determined from experiments with a highly active Co/Re/γ-Al2O3 catalyst, which is more representative of current Co-based catalysts used in industrial applications than the catalyst used by Yates and Satterfield.25 Models of diffusion−reaction interaction are helpful in determining an optimal catalyst layer thickness that maximizes catalyst effectiveness and improves FTS product selectivity and provide guidance for catalyst design and choice of process conditions.

research groups and became known as Yates and Satterfield kinetics.25 Dixit and Tavlarides23 conducted a parametric study (in terms of dimensionless parameters) to investigate the relationship between the effectiveness factor (η) and Thiele modulus (ϕ) and found that the η versus ϕ curve passes through a maximum and can reach values exceeding 1. Also, they reported that for some values of model parameters there exists a narrow range of ϕ values over which multiple steady-state solutions are possible. Selectivity aspects were not considered in this study. Researchers at Exxon26−28 have introduced selectivity aspects into mathematical modeling. Their model included continuity equations for both fluid phase (plug flow reactor) and the catalyst particle. Reaction rates for CO consumption and CH4 formation were described by LH kinetics, and a separate model was used for modeling hydrocarbon selectivity based on 1-olefin readsorption concept. They found that selectivity of liquid-phase hydrocarbons (C5+ hydrocarbons) increases initially with increase in diffusional limitations due to diffusion-enhanced 1-olefin readsorption resulting in formation of high molecular weight hydrocarbons and then begins to decrease because of diffusional limitations by reactants, resulting in high H2/CO ratio inside the pellet which favors chain termination reactions that favor lower molecular weight hydrocarbons. To overcome these problems and satisfy pressure drop requirements, Iglesia et al.4 utilized an eggshell catalyst distribution where the catalyst is deposited only in a thin layer near the outer surface of a relatively large particle. A method of preparation of eggshell Co-based FTS catalysts was described, and its beneficial effects on product selectivity were illustrated with experimental data. This type of analysis was carried one step further by Wang et al.5 who utilized a comprehensive kinetic model for the catalyst pellet in which both the overall syngas consumption and the hydrocarbon product distribution are unified. This model was developed from kinetic studies with Fe-based FTS catalyst29 which utilized the olefin readsorption concept. The advantages of eggshell catalyst distribution on product distribution have been investigated through numerical simulations. A similar approach was used recently in analyzing performance of a bench scale fixed bed reactor with Co-based FTS catalyst (15%Co/γ-Al2O3).30 A comprehensive kinetic model of Todic et al.31 was used in numerical simulations, which included a continuity equation for the catalyst particle. An alternative, and numerically significantly less demanding, approach was used by Vervloet et al.7 to study the effect of process conditions on catalyst effectiveness and C5+ productivity and provide guidance for optimal reactor operation. The interplay between reaction and diffusion in a single catalyst particle was analyzed using the LH rate expression of Yates and Satterfield25 for CO consumption rate and a variable chain growth parameter (α) dependent on temperature and syngas composition (H2/CO ratio). The same approach (the use of Yates and Satterfield25 kinetic parameters and variable α parameters from Vervloet et al.7) was used by Becker et al.6,32 as well as Gardezi and Joseph.33 There are some issues associated with the approach used by Vervloet et al.7 Variation of α with temperature and H2/CO ratio was determined from experimental data of De Deugd34 for methane selectivity. Local α values were calculated from reactant concentrations in the particle, and average value of α was obtained by numerical integration. Selectivities of C1−C4 and C5+ hydrocarbons were then calculated assuming the ideal Anderson−Schulz−Flory (ASF) distribution.28,35,36 However, it is well-known that actual product distribution on Co FT catalysts deviates from the ideal

2. MODEL DESCRIPTION The overall stoichiometry of FTS on cobalt catalyst was described by the following equation: CO + (13/6)H 2 = H 2O + (1/6)C6H14

(1)

which is representative of an average molecular weight of hydrocarbons produced and H2/CO consumption ratio for experiments with 0.48%Re−25%Co/Al2O3 catalyst.31 In general, H2/CO consumption ratio varies with product selectivity and lies between 2 (production of very long chain molecules, α = 1) and 3 (production of CH4 from syngas). It is assumed that the catalyst particle is filled with high molecular weight hydrocarbons (wax) and that external massand heat-transfer resistances are negligible, so that concentrations of species and temperature at the particle surface are the same as those in the bulk. The reactants and products are dissolved in wax at the surface and throughout the particle. For steady-state conditions, assuming Fick’s law of diffusion and isothermal pellet, the general reaction−diffusion continuity equation is given as 1 d ⎛ g dyi ⎞ · ⎜x · ⎟ + υi ·(ϕ′i )2 ·R̅ CO = 0 x g dx ⎝ dx ⎠

(2)

where x is the dimensionless distance and superscript g stands for geometry (g = 2, 1, and 0 for sphere, cylinder and slab, respectively); yi (Csi /CsCO) is dimensionless concentration (i = CO, H2, H2O, and n-hexane), νi stoichiometric coefficient (νi = −1, −13/6, 1, and 1/6 for CO, H2, H2O, and n-hexane, respectively), R̅ CO the dimensionless CO consumption rate (R C O /R Cs O ), and ϕ′ i the dimensionless parameter (ϕ′i = Lg

s ρp ·(−R CO ) s De ,i ·CCO

). This parameter includes catalyst density

(ρp), geometric characteristic length (Lg), surface concentration B

DOI: 10.1021/acs.iecr.7b00053 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research of CO (CsCO) in the liquid medium, and effective diffusivity coefficient of species i (De,i). The assumption of uniform temperature distribution in a pellet (isothermal conditions) was verified by numerical solution of mass and energy balance equations under a conservative set of conditions to maximize temperature gradients inside the pellet: relatively large spherical particle (dp = 4 mm); low value of effective thermal conductivity, keff = 0.1 W/(m·K); and high value of heat of reaction, −173 kJ/mol. It was found that the maximum temperature difference in the particle is 0.15 K, which has no impact on concentration profiles, effective diffusivity, and CH4 selectivity. Vervloet et al.7 reached the same conclusion by using relevant dimensionless groups for estimation of the maximum ΔT in a pellet. The second-order differential equation (eq 2) was solved for several particle shapes (sphere with uniform catalyst distribution and eggshell catalyst distribution, flat plate, and hollow cylinder), as shown in Figure 1. The geometric characteristic length (Lg) for

x = 0 (sphere or slab):

dyi dx

=0

(2b)

s x = ri /ro (hollow cylinder): yi = C is/CCO

x = 1 − δ /R (eggshell distribution):

dyi dx

(2c)

=0

(2d)

2.1. Kinetics. CO consumption rate per unit mass of catalyst is described by LH kinetics:23,25 ( −R CO) =

k·PCO·PH2 (1 + a ·PCO)2

(3)

Numerical values of kinetic rate constant (k) and adsorption constant (a) have been reported by Yates and Satterfield25 at two temperatures (220 and 240 °C) from regression of experimental data with 21.4% Co/3.9% Mg/SiO2 catalyst, which will be referred to as YS catalyst. Maretto and Krishna39 estimated activation energy and adsorption enthalpy of k and a for YS catalyst. Temperature dependence for these two parameters is given by the following equations: ⎡ ⎛ 1 1 ⎞⎤ k(T ) = 8.8533 × 10−3· exp⎢4494.41⎜ − ⎟⎥ ⎝ 493.15 ⎣ T ⎠⎦ mol/(s kgcat bar 2)

(4)

⎡ ⎛ 1 1 ⎞⎤ a(T ) = 2.226· exp⎢ − 8236⎜ − ⎟⎥1/bar ⎝ 493.15 ⎣ T ⎠⎦

(5)

It should be noted that Yates and Satterfield25 and Maretto and Krishna39 obtained numerical values of k and a from data for syngas (H2 + CO) consumption rate rather than CO consumption rate. It appears that several research groups used eq 4 as the rate constant for CO consumption rate,6,7,33 without correction for stoichiometry, in modeling the reaction with diffusion in a single particle. We have estimated parameters k and a in eq 3 from experimental data with a catalyst synthesized at Center for Applied Energy Research (CAER) at the University of Kentucky having composition 0.48%Re−25%Co/Al2O3. Experimental data used to estimate kinetic parameters can be found in Table 1 of Todic et al.37 Numerical values of kinetic parameters used in simulations for both CAER and YS catalysts are summarized in Table 1. Numerical value of k in eq 4 for YS catalyst was divided by (1 + 13/6) to the obtain rate constant for CO consumption. It is worth noting that CAER catalyst is an order of magnitude more active under typical FTS operating conditions compared to the YS catalyst,25 and as such is more representative of modern Co catalysts used in industrial applications. The key aspect of FTS catalyst selectivity is methane selectivity. It is the most undesirable product, and reduction in the amount of methane is of utmost importance. Methane formation rate was calculated using a recently proposed expression:40

Figure 1. Representation of particle shapes and dimensions: (a) spherical particle with uniform catalyst distribution, (b) spherical particle with eggshell catalyst distribution, (c) hollow cylinder particle, and (d) slab (flat plate).

each particle shape is as follows: radius (R) for a sphere, outer radius (ro) for a hollow cylinder, and thickness (L) for a slab (flat plate). Dimensionless distance x, in eq 2, for different pellet shapes is x = r/R (sphere), x = r/ro (hollow cylinder), or x = z/L (slab). Dirichlet boundary conditions are used for surfaces exposed to the surrounding fluid (x = 1), and Neumann (zero flux) conditions are used at x = 0 (sphere with uniform catalyst distribution and slab) or at x = 1 − δ/R (eggshell distribution in a spherical particle): s x = 1 (outer surface of the particle, all four geometries): yi = C is/CCO

(2a)

Table 1. Kinetic Parameters for Rate of CO Disappearance According to Equation 3

catalyst k (mol/(s·kg·bar2)) a (1/bar)

Yates and Satterfield (YS)

CAER

21.4% Co/3.9% Mg/SiO2 26·exp(−4494.4/T) 1.24 × 10−7·exp(8236/T)

25% Co/0.48% Re/Al2O3 4.2 × 105·exp(−8742/T) 6.45 × 10−2·exp(1295.3/T)

C


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Industrial & Engineering Chemistry Research aM bM R CH4 = kM PCO P H2 /(1 + mM ·PH2O/PH2)

kM = AM e−EM / R gT

Table 2. Reaction Conditions and Parameters Used in Numerical Simulations

(6) (6a)

where numerical values of kinetic parameters were estimated from the same set of experimental data as those in eq 3 (Table 1 of Todic et al.37). Estimated values of numerical parameters in eq 6 are aM = −0.99, bM = 1.28, mM = 0.58, AM = 2.25 × 1011 (mol/ (kg·s·bar0.29)), and EM = 140 (kJ/mol). Substituting numerical values for constants aM, bM, and mM into eq 6, we obtain the following expression for CH4 formation rate: R CH4 =

kM(T) ·(PH2 /PCO)0.99 ·PH2 0.29 1 + 0.58·(PH2O/PH2)

(mol/(kg·s)) (7)

Equation 7 predicts that CH4 formation rate increases with temperature, partial pressure of H2, and H2/CO ratio and is inhibited by partial pressure of water. The inhibiting effect of water on CH4 selectivity is well-established from studies with both indigenous and externally added water,40−47 whereas the underlying causes for this behavior are still not completely understood.41,48,49 2.2. Physical Properties. We consider the case in which the catalyst particle is completely filled with hydrocarbon wax, the physical properties of which are assumed to be those of noctacosane (C28H58). Diffusivities of species in wax were estimated from Akgerman’s correlation.50,51 This correlation has been used previously,6,21 and its predictions are similar to those obtained using correlations of Wang et al.5 and Makrodimitri et al.52 The effective diffusivity of the component was calculated from ε De,i = Dwax,i (8) τ

ref

temperature pressure syngas ratio in the bulk catalyst porosity catalyst tortuosity catalyst particle density catalyst particle diameter catalyst layer thickness (slab) eggshell layer thickness CO diffusivity in wax H2 diffusivity in wax

T P H2/CO ε τ ρp dp L δ Dwax,CO Dwax,H2

473−493 K 20−30 bar 1.4−2.0 0.5 2 1200 kg m−3 0.085−4 mm 0.035−0.300 mm 0.075−0.300 mm (1.24−1.44) × 10−8 m2 s−1 (3.13−3.62) × 10−8 m2 s−1

50 50

H2O diffusivity in wax

Dwax,H2O

(2.10−2.42) × 10−8 m2 s−1

50

C6H14 diffusivity in wax

Dwax,C6H14

(5.12−5.98) × 10−9 m2 s−1

50

Henry’s coefficient for CO Henry’s coefficient for H2

HCO HH2

345.1−356.6 bar 442.3−472.6 bar

53 53

Henry’s coefficient for H2O

HH2O

41.9−49.3 bar

53

Henry’s coefficient for C6H14

HC6H14

9.88−12.3 bar

53

density of wax (n-octacosane)

ρw

678.1−691.4 kg m−3

50

SCH ̅ 4=

1 · V

∫0

V

1 ρp R CH4 dV / · V

∫0

V

ρp ( −R CO)dV

V

=

(9)

∫0 R CH4 dV V

∫0 (−R CO)dV

(12)

It should be noted that CH4 formation rate (eq 7) is not bounded, and for large H2/CO ratios it can assume a very high value. Also, as the CO gets depleted inside the particle, RCO approaches zero. If either one or both of these two possibilities occur, the local CH4 selectivity would tend to infinity, whereas its value physically cannot exceed 100%, i.e., CH4 formation rate cannot exceed CO consumption rate (RCH4 ≤ (−RCO)). Thus, if at some point in the pellet it is found that RCH4 > (−RCO), the condition RCH4 = (−RCO) is imposed in calculation of local and average CH4 selectivities (eqs 11 and 12) from this point onward. 2.4. Numerical Methods. Solution was obtained using Matlab’s bvp4c function, which is used to solve boundary value problems for ordinary differential equations. The function uses finite difference method, which implements a colocation formula. For each component, eq 2 was written as a system of two first-

V

∫0 (−R CO)ρp dV s V ·( −R CO )ρp

value

which is valid for Co catalysts because of their low water gas shift activity, i.e., low CO2 formation. Then, average CH4 selectivity can be obtained by integration of local reaction rates as follows:

where Ci is molar concentration of species i (mol/m3), CL the total liquid-phase concentration (mol/m3), Pi partial pressure (bar), and Hi Henry’s law constant (bar). The parameter values used in model simulations are summarized in Table 2. 2.3. Catalyst Performance Indicators. Catalyst efficiency and CH4 selectivity are the two key factors describing catalyst performance. The efficiency is expressed in terms of effectiveness factor, which represents a ratio of average reaction rate in the pellet divided by rate at the surface conditions: η=

symbol

For normal reactions (rate decreases as conversion increases), the effectiveness factor is ≤1, and is a measure of utilization of catalyst volume for reaction. However, in our case, the CO reaction rate (eq 3) passes through a maximum before it starts decreasing with decrease in reactant concentrations, and it is possible to have η > 1 for some conditions. Solution of eq 2 provides partial pressures needed to evaluate methane formation rate (eq 7) and calculate local methane selectivity at any point inside the catalyst as R CH4 SCH4 = ·100% ( −R CO) (11)

where ε is particle porosity and τ is tortuosity factor. Henry’s law constants for reactants and products in noctacosane were calculated from Marano and Holder’s correlation.53 This correlation is based on experimental data with fluids (different types of FTS waxes and high molecular weight hydrocarbons) and process conditions (temperature and pressure) that are typical for FTS (Marano54). Liquid-phase concentration of a species i is related to partial pressure of i via Henry’s law constant Pi = Hi(C i /C L)

description

(10)

where V represents the volume of the region in which the catalyst is located. D


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Figure 2. Effect of catalyst activity (CAER vs YS catalyst) on (a) CO and H2 concentration, (b) H2/CO ratio, (c) CO reaction rate, and (d) local CH4 selectivity (CAER catalyst). Conditions: dp = 2 mm, T = 473 K, P = 25 bar, XCO = 0%, and H2/CO = 2.

order differential equations, resulting in a system of eight firstorder differential equations with conditions specified at two different points. Spacing between grid points was not fixed because of the nature of bvp4c algorithm, because the method iteratively adjusts spacing in order to achieve convergence tolerance. Initial guesses used depended on expected profiles. For conditions which would result in small values of ϕ′i (low diffusional limitations), boundary conditions on the surface were taken as initial guesses for all points, while for larger ϕ′i conditions, guesses were chosen by trial and error. Local CH4 selectivity was calculated algebraically using previously calculated concentration profiles of CO, H2, and H2O. Average CH4 selectivity and effectiveness factor were calculated using trapezoidal rule (Matalab’s function trapz). This method is accurate enough because of the high density of mesh over which the integration is done.

representative of sizes (1−3 mm) used in industrial reactors to avoid a high pressure drop.1,2,4 Results are shown for conditions near the reactor inlet where the CO conversion is very small (XCO = 0%). These conditions are the least favorable for achieving low methane selectivity because of the absence of water at the particle surface. Steep gradients are observed for reactant liquid-phase concentrations in the case of CAER catalyst, whereas concentrations decrease slowly for YS catalyst. For CAER catalyst, the concentration of CO approaches zero at approximately 0.2 mm from the particle surface (Figure 2a) because of lower rate of diffusion of CO compared to H2. This in turn causes very rapid increase in H2/CO ratio inside the particle (Figure 2b). We show results only up to H2/CO = 20, but this ratio tends to infinity deeper inside the particle. On the other hand, H2/CO ratio for the YS catalyst increases slightly from 2 at the surface of the particle to ∼2.4 at the center. CO consumption rate for the CAER catalyst increases rapidly in the region near the surface to reach a maximum value at approximately 0.11 mm from the surface and then decreases to essentially zero value at about 0.27 mm from the surface (Figure 2c). This behavior is caused by change in effective reaction order from negative at high CO concentration to positive at low CO concentrations (see eq 3). For YS catalyst, the CO consumption rate increases slightly from the surface to the center of the particle, which is consistent with moderate decrease in CO concentration shown in Figure 1a. It is important to note the difference in scale for the two ordinate axis. Consumption rate of CO is much lower with the YS catalyst relative to the CAER

3. RESULTS AND DISCUSSION Even though the governing differential equations were solved numerically in their dimensionless form, we present results in terms of dimensional variables to facilitate better physical understanding and relate them to previous studies and industrial applications. 3.1. Effect of Catalyst Activity. Results from simulations with CAER catalyst (high-activity catalyst) and YS catalyst (lowactivity catalyst) are shown in Figure 2 for conditions typical of industrial practice (473 K, 25 bar, H2/CO ratio of 2). Simulations are for a spherical particle 2 mm in diameter, which is E


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Figure 3. Effect of particle size on (a) CO concentration, (b) H2/CO ratio, (c) CO reaction rate, and (d) local CH4 selectivity. Conditions: T = 473 K, P = 25 bar, XCO = 0%, and H2/CO = 2.

were calculated assuming reaction stoichiometry described by eq 1. Calculated values of the effectiveness factor were 0.46 (CAER) and 1.06 (YS), whereas CH4 selectivity was 13.8%. Initial (at the surface) reaction rates are higher when XCO = 0%, but the impact on effectiveness factor is small, whereas CH4 selectivity is significantly lower when XCO = 50%. The latter is due to the fact that water is present at the outer surface and it inhibits CH4 formation rate (see eq 7). However, the calculated value of CH4 selectivity is too high for industrial applications. The above results show that the use of YS rate equation with original parameter values obtained from experiments with Co/ Mg/SiO2 catalyst25 underestimates the extent of diffusional resistance in the case of more active Co FTS catalysts. Some researchers7,8 have simply multiplied YS rate by a constant value (up to 10) to simulate performance of more active catalysts, but this approach does not account correctly for variation of rate with temperature and pressure. The ratio of reaction rates (CAER/YS catalyst) at the surface is a function of process conditions, and it varied between 14.9 (at 493 K, 20 bar, and XCO = 50%) and 37 (at 473 K, 30 bar, and XCO = 0%). For each catalyst, one needs to determine the correct values of activation energy and enthalpy of adsorption experimentally, as well as reaction orders with respect to reactants. In the remaining sections of this paper we present results using kinetic parameter values for CAER catalyst. 3.2. Effect of Particle Size. Although the use of small particles (less than 1 mm) is not feasible for a large-scale industrial fixed bed reactors because of excessive pressure drop, this is not as significant constraint for microchannel structures

catalyst. The ratio of reaction rates (CAER/YS) at the catalyst surface for these process conditions is 36. Observed differences in concentration profiles and H2/CO ratios with the two catalysts are caused by respective differences in the catalyst activity. Intraparticle diffusional limitations are much stronger with the more active catalyst. Local CH4 selectivity (Figure 2d) increases exponentially, mirroring trends in H2/CO ratio (Figure 2b), and reaches 100% at approximately 0.16 mm from the surface. This is in accordance with the rate equation for CH4 formation (eq 7), which predicts increase in rate with increase in H2/CO ratio, and the definition of local selectivity, which has the rate of CO disappearance in the denominator (eq 11). The calculated value of local CH4 selectivity tends to infinity with increasing distance from the surface of the particle, but physically selectivity cannot exceed 100%; this is shown as a broken horizontal line in Figure 2d. We do not have rate parameters for CH4 formation for the YS catalyst; thus, there are no results for local CH4 selectivity for this catalyst. The catalyst effectiveness values are 0.52 (CAER) and 1.05 (YS), whereas the average CH4 selectivity, calculated from eq 12, for CAER catalyst is 19.4%. The latter clearly indicates a negative impact of intraparticle diffusional resistance on the catalyst performance (high CH4 and consequently low C5+ selectivity). Simulations were performed for these process conditions and particle size for external conditions corresponding to location in a reactor where CO conversion is 50%. Liquid-phase concentrations of species (reactants and products) at the catalyst surface F


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Figure 4. Effect of catalyst shape and catalyst distribution on (a) CO concentration, (b) H2/CO ratio, (c) CO reaction rate, and (d) local CH4 selectivity. Conditions: L = δ = ro − ri = 0.15 mm, T = 473 K, P = 25 bar, XCO = 0%, and H2/CO = 2.

For a given set of conditions (473 K, 25 bar, H2/CO = 2, and XCO = 0%) the use of spherical particle of 0.4 mm in diameter would be an optimal choice because it results in high effectiveness factor and low CH4 selectivity. Compared to the 0.2 mm particle, increased diffusional resistance is beneficial in that it gives a slightly higher effectiveness factor without an adverse effect on CH4 selectivity. The use of 0.4 mm particles in microchannel structure may be an upper limit for channels up to about 5 mm, whereas a smaller particle size would be preferable for smaller channels. The use of larger particle 0.6 mm in diameter results in even higher effectiveness factor, but CH4 selectivity is higher than with the two smaller particles, which makes it less suitable for commercial applications. 3.3. Effects of Particle Shape and Catalyst Distribution. One way to minimize negative impact of intraparticle diffusional limitations on hydrocarbon selectivity (high CH4 and low C5+ selectivity) while maintaining pressure drop requirements is to use eggshell catalyst distribution where cobalt is located near the outer pellet surface.4,33,55−57 Reduction in diffusion length can also be accomplished using wall coated monolith reactors for FTS reaction, and this has received considerable attention in recent years.12,58−62 This geometry can be approximated by that of an insulated slab (flat plate). Another way to minimize diffusion path of reactants is to utilize hollow cylinder particles of small thickness. Figure 4 shows results of simulations with these three geometric shapes (slab/plate, hollow cylinder, and sphere with eggshell distribution) having the same catalyst layer thickness of

which utilize shorter lengths. The use of small particles decreases intraparticle transport resistances, which is essential for maintaining low CH4 selectivity. Results of simulations (CAER catalyst) with small spherical particles (0.2, 0.4, and 0.6 mm) and comparison with a 2 mm spherical particle are shown in Figure 3. As expected, the CO concentration profile becomes steeper as the particle size increases (Figure 3a). H2/CO ratio (Figure 3b) for smaller particles (0.2 and 0.4 mm in diameter) is close to the bulk ratio of 2, whereas it increases rapidly for larger particles (0.6 and 2 mm). CO consumption rate increases gradually from the surface to the center for the smaller particles (Figure 3c) because of reduction in rate inhibition by CO. For larger particles, CO consumption rate passes through a maximum and then decreases. For dp = 0.6 mm, the reaction rate at the center of the pellet is about one-third of the rate at the surface, whereas for the largest particle (dp = 2 mm), the CO rate approaches zero at approximately 0.27 mm from the surface, and a large fraction of the particle volume is not utilized for reaction. Local methane selectivity increases with distance from the surface, but the increase is small for two small particles (0.2 and 0.4 mm) and very rapid for the larger particles (0.6 and 2 mm). In the latter case, local selectivity reaches 100% at 0.23 mm for 0.6 mm particle and at 0.16 mm for 2 mm particle (Figure 3d). Effectiveness factor values are 1.01 (0.2 mm), 1.04 (0.4 mm), 1.12 (0.6 mm), and 0.52 (2 mm), whereas the corresponding average values of CH4 selectivity are 5.6, 5.7, 7.6, and 19.4%, respectively. G


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Table 3. Effect of Process Conditions on Effectiveness Factor and Average Methane Selectivity of a Spherical Catalyst Particle effect of process conditions dp = 0.4 mm

effect of pressure effect of temperature effect of syngas ratio

effect of CO conversion

dp = 2 mm

effect of pressure effect of temperature effect of syngas ratio

effect of CO conversion a

P (bar)

T (K)

H2/CO (−)

XCO (%)

η (−)

S̅CH4 (%)

ϕCOa (−)

20 30 25 25 25 25 25 25 25 25 20 30 25 25 25 25 25 25 25

473 473 473 493 473 473 473 473 473 473 473 473 473 493 473 473 473 473 473

2 2 2 2 2 1.7 1.4 2 2 2 2 2 2 2 2 1.7 1.4 2 2

0 0 0 0 0 0 0 0 25 50 0 0 0 0 0 0 0 0 50

1.042 1.041 1.042 1.112 1.042 1.016 1.001 1.042 1.036 1.020 0.458 0.563 0.516 0.355 0.516 0.552 0.586 0.516 0.455

5.71 5.71 5.67 12.7 5.67 5.39 5.13 5.67 5.05 4.08 19.8 18.9 19.4 28.3 19.4 16.4 11.8 19.4 13.8

0.56 0.48 0.52 0.76 0.52 0.46 0.39 0.52 0.53 0.55 2.81 2.40 2.58 3.82 2.58 2.28 1.97 2.58 2.75

ϕCO defined by eq 13.

0.15 mm (L = δ = ro − ri = 0.15 mm in Figure 1, and ro = R = 1 mm). CO concentration declines monotonically with distance from the surface for slab and eggshell catalysts, whereas it passes through a minimum at approximately 0.075 mm for a hollow cylinder (Figure 4a). The observed trend for the latter is the consequence of the fact that a hollow cylinder has two surfaces exposed to the surrounding fluid, whereas the slab and the particle with the eggshell catalyst distribution have only one surface exposed to the fluid. The steepest gradient is observed with the slab layer, followed by the eggshell layer. As a result of differences in diffusion rates of H2 and CO in the particle, the H2/ CO ratio increases with the distance from the surface, with a trend opposite to that of CO concentration (Figure 4b). Internal H2/CO ratios for the slab and eggshell layer are appreciably higher than those at the surface, whereas this ratio is relatively constant for the hollow cylinder. Local CH4 selectivity follows the same qualitative trend as the H2/CO ratio (Figure 4c). We see from these data that intraparticle diffusional resistance increases in the following order: hollow cylinder < eggshell layer < slab, which is consistent with the order of increasing effective diffusional path (Lc = V/S) for these three geometries. Even though these three geometries have the same catalyst layer thickness, the corresponding effective diffusional paths are different. Effectiveness factor values are 1.216 (slab), 1.171 (eggshell), and 1.024 (hollow cylinder). As seen above for small spherical particles, a moderate intraparticle diffusional resistance results in higher effectiveness factor due to reduction in rate inhibition by CO (negative reaction order kinetics). However, CH4 selectivity is adversely affected by diffusional limitations because of increase in H2/CO ratio inside the pellet, and the average values of CH4 selectivity are 7.90% (slab), 6.53% (eggshell), and 5.62% (hollow cylinder). In this case, the hollow cylinder pellet would be the best choice from the performance perspective; however, it would not be suitable for industrial applications because of low mechanical strength. To avoid diffusional limitations in wall coated monolith reactors using CAER catalyst at these conditions (473 K, 25 bar, H2/CO = 2, XCO = 0%), one would need to use layers with thickness less than 0.11 mm, whereas the

eggshell thickness should be less than about 0.13 mm (see section 3.5). Several research groups have reported that low CH4 selectivity, comparable to that of powder catalysts, can be obtained with monolith reactors having layer thickness less than 0.05 mm.12,58 This is much more conservative than the estimated value for CAER catalyst at base case conditions. Iglesia et al.4 used eggshell catalyst distribution (thickness 0.1−0.2 mm) with spherical particles 2.2 mm in diameter at 473 K, 20 bar, H2/CO = 2.1, and CO conversion of 50−60%, whereas Fratalocchi et al.56 reported that eggshell catalyst with 0.075 mm thickness (0.6 mm particle diameter) has the same CH4 selectivity and slightly higher C5+ selectivity than that of the powder catalyst (0.075−0.10 mm diameter) at 503 K, 25 bar, and H2/CO = 1.7 and CO conversions between 34 and 42%. Our estimate is in good agreement with values reported by Iglesia et al.4 under similar reaction conditions and with a similar particle size, whereas Fratalocchi et al.56 reported smaller thickness but at significantly higher reaction temperature and with a smaller overall particle size. 3.4. Effect of Process Conditions. Effect of process conditions (pressure, temperature, syngas ratio on the surface, and CO conversion level) on effectiveness factor and methane selectivity of CAER catalyst is summarized in Table 3 for two spherical particles with diameters of 0.4 mm and 2 mm. We found (section 3.2) that the use of a small particle (0.4 mm) results in negligible diffusional limitations at the baseline conditions (473 K, 25 bar, H2/CO = 2, and XCO = 0%), whereas pore diffusion has a strong effect on catalyst performance in the case of the 2 mm particle. Results shown with these two particle sizes show influence of process conditions in kinetic regime (dp = 0.4 mm) and pore diffusion regime (dp = 2 mm). Change of total pressure from 20 to 30 bar has a negligible effect on η and SC̅ H4 for the small particle, whereas for 2 mm, the increase in pressure results in a small increase in η and a small decrease in S̅CH4, which is due to decrease in the value of the Thiele modulus. H


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Industrial & Engineering Chemistry Research Increase in temperature from 473 to 493 K results in 7% increase in the value of η and significant increase in methane selectivity (from 5.67 to 12.7%) for dp = 0.4 mm, whereas in the region of pore diffusion control (dp = 2 mm), η decreases and S̅CH4 increases because of increase in diffusional resistance (higher value of ϕCO). Effect of syngas ratio changes on η is relatively small, but decrease in the H2/CO ratio results in reduction of SC̅ H4 for both particle sizes. The effect on S̅CH4 is more pronounced in the case of larger particles. Vervloet et al.7 have shown through model simulations and optimization that space-time-yield of C5+ hydrocarbons can be significantly improved relative to typical conditions of T = 500 K, P = 30 bar, and H2/CO = 2 by increasing reaction temperature to 530 K and pressure to 36 bar while simultaneously decreasing bulk H2/CO ratio to 1. Robota et al.8 showed CO, H2, and H2/ CO profiles for different bulk H2/CO ratios ranging from 0.5 to 1.8 for a 0.25 mm diameter particle at 478 K and 24 bar using the YS rate equation with scaling factor of 10. For H2/CO bulk ratios less than 0.9, the internal H2/CO ratio decreases slightly from the surface to the center of the particles, whereas for bulk ratios greater than 1.1, the ratio increases with distance from the surface. It is clear from eq 7 that the rate of CH4 formation will decrease with decreasing H2/CO ratio; however, it is known that one of the mechanisms for catalyst deactivation is carbon deposition,63−65 and operation at low H2/CO ratios would accelerate the rate of deactivation. Therefore, it is not clear whether lowering of H2/CO feed ratio would be a feasible method of controlling CH4 selectivity under diffusion limited conditions. As the conversion along the reactor increases, the concentration of water outside the catalyst particle increases, and this has significant effect on CH4 selectivity. The effect of CO conversion on η is small with both particles because the value of the Thiele modulus does not change much with conversion (0− 50% range). However, significant reductions in S̅CH4 were observed as conversion increases from 0% (reactor inlet) to 50% because of increase in water concentration, which inhibits methane formation rate (see eq 7). 3.5. Effectiveness Factor and Methane Selectivity as a Function of Generalized Thiele Modulus. Results from simulations with different particle shapes (sphere, slab, hollow cylinder), different sizes (sphere with diameters ranging from 0.085 mm to 4 mm, catalyst layers of different thicknesses ranging from 0.035 mm to 0.5 mm), form of catalyst distribution (uniform and eggshell), and process conditions (T = 473−493 K, P = 20−30 bar, H2/CO = 1.4−2) are shown in Figures 5 and 6. Abscissa values represent generalized Thiele modulus based on characteristic length for diffusion (Lc = V/S) as proposed by Aris.66 The formula for generalized Thiele modulus is ϕ = ϕCO = Lc

Figure 5. Effectiveness factor as a function of Thiele modulus. (a) Slab, sphere, eggshell and hollow cylinder particle (range of conditions: T = 473−493 K, P = 20−30 bar, CAER and YS catalyst, XCO = 0−50%, dp = 0.085−4 mm, L = 0.035−0.300 mm, δ = 0.075−0.300 mm, ro − ri = 0.150−0.500 mm). (b) Spherical particle (CAER catalyst, range of conditions: T = 473−493 K, P = 20−30 bar, XCO = 0−50%, dp = 0.085−4 mm).

Results in Figure 5a show η versus ϕ data for all shapes, sizes, and process conditions investigated except for data corresponding to H2/CO ratios of 1.4 and 1.7. These data are excluded because the H2/CO ratio has effect on both boundary condition (eq 2a) for H2, as well as on dimensionless reaction rate in the differential equation (eq 1). Changes in total pressure have no effect on the boundary conditions and have a small effect on reaction rate for the range of pressures investigated (20−30 bar); thus, data for all pressures in this range are shown in Figure 5a. Temperature has a small effect on the boundary condition (eq 2a) for H2 (through variation of solubility with temperature), and dimensionless reaction rate is not very sensitive to temperature (473−493 K) at a constant value of ϕ. Therefore, data at both temperatures (473 and 493 K) are included. The largest scatter of points occurs in the region 0.7 < ϕ < 1, because different shapes have different values of η at similar values of ϕ, and differences in kinetic parameters (CAER vs YS catalyst) have impact on results in this range of ϕ values. In general, values of η for spherical particles become greater than 1.06 in the region 0.7 < ϕ < 1, whereas for other shapes this takes place at higher values of ϕ (0.99 < ϕ < 1.26). For ϕ values less than about 0.6, η ≈ 1 (1− 1.06), which is the region of kinetic control, whereas for ϕ > 1.6, the effectiveness factor decreases with increasing ϕ and is less than 1 for all particle shapes. This is the region where intraparticle diffusional resistance becomes increasingly important. For the YS catalyst, the maximum value of ϕ was 0.94

ρp ·( −R COs) s De,CO·CCO

(13)

where Lc = V/S (V = volume in which the catalyst is located; S = surface area exposed to surrounding fluid). Characteristic diffusion lengths for different pellet shapes and forms of catalyst distribution as shown in Figure 1 are as follows: Lc = R/3 (sphere with uniform catalyst distribution), Lc = (R/3)· [1 − (1 − δ/R)3] (sphere with eggshell catalyst distribution), Lc = (ro − ri)/2 (hollow cylinder), and Lc = L (plate/slab). I


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6b (slab, eggshell, and hollow cylinder). Methane selectivity does not depend only on ϕ, but also on temperature, H2/CO ratio outside the particle, and conversion level in the reactor (i.e., on partial pressure of water at the catalyst surface); therefore, data were grouped into several categories. Legends in Figure 6 show which condition was changed relative to base case value, while the remaining conditions are kept at the corresponding base case values. For ϕ < 0.7, average methane selectivity is nearly constant and then starts increasing rapidly with increasing value of the Thiele modulus (Figure 6a). Decrease in the H2/CO ratio results in lower CH4 selectivity compared to the base case (H2/CO = 2), but the impact is small in the kinetically controlled region (ϕ < 0.6−0.7), whereas the effect is much more pronounced as ϕ increases. Increase in CO conversion results in significant reduction of CH4 selectivity (XCO = 50% vs XCO = 0%) even in the kinetically controlled region. As stated earlier, this is caused by higher partial pressure of water which inhibits methane formation rate (see eq 7). Increase in temperature to 493 K leads to increase in CH4 selectivity for all values of ϕ. At both temperatures, CH4 selectivity starts increasing rapidly for ϕ > 0.7. Changes in pressure (20 or 30 bar) have small effect in both kinetic (ϕ = 0.48−0.56) and diffusion control (ϕ = 2.4−2.8) regions. Selected CH4 selectivity data for different shapes (slab and hollow cylinder) and eggshell distribution with spherical particle with dp = 2 mm are shown in Figure 6b for ϕ values less than 1.3. The qualitative trends regarding the influence of H2/CO ratio, CO conversion, temperature, and pressure are the same as in the case of spherical particles with uniform catalyst distribution. In this case, the kinetically controlled region extends to ϕ ≈ 0.9. Both higher CO conversion and lower H2/CO feed ratio have beneficial effect on CH4 selectivity, and the combined effect (H2/ CO = 1.7 and XCO = 50%) results in significant reduction in CH4 selectivity relative to the base case conditions. From data shown in Figure 6 we can estimate the maximum particle size (sphere) or thickness of catalyst layer for slab and eggshell catalyst distribution for which the intraparticle diffusion limitations may be considered to have negligible effect on CH4 selectivity at base case conditions for the CAER catalyst. We define negligible effect on CH4 selectivity as a value of ϕ where the increase in CH4 selectivity is less than 6% of the value corresponding to minimum CH4 selectivity at the base case conditions and beyond which CH4 selectivity increases more rapidly. The minimum value of CH4 selectivity for CAER catalyst at the base conditions for all shapes considered is 5.60%. Using the maximum 6% increase in CH4 selectivity as the criterion, this gives the following values for the magnitude of ϕ: 0.65 for spherical particle with uniform catalyst distribution and 0.90 for all others (slab, hollow cylinder, and eggshell distribution). In terms of particle size or catalyst layer thickness, we obtain dp = 0.50 mm for a spherical particle with uniform catalyst distribution; L = 0.116 mm (slab/flat plate thickness); δ = 0.132 mm (eggshell thickness in 2 mm spherical particle); and ro − ri = 0.232 mm (hollow cylinder with the outer radius of 1 mm). It should be noted that in all these cases the catalyst effectiveness factor is greater than 1 (η = 1.08−1.12).

Figure 6. Methane selectivity as a function of Thiele modulus (a) for spherical particle under different conditions (shown in legend) and (b) for different shapes and conditions (shown in legend). Base case: T = 473 K, P = 25 bar, H2/CO = 2, XCO = 0%. Other conditions: Legend shows numerical value of condition that was changed from its base case value, whereas the remaining conditions are kept at the corresponding base case values.

(corresponding to dp = 4 mm) for the range of conditions investigated because of low activity of this catalyst compared to the CAER catalyst. Most of the points for the YS catalyst are in the region of negligible diffusional resistance (ϕ < 0.46). In the region 0.8 < ϕ < 1, the effectiveness factor for the YS catalyst is higher than that for CAER catalyst (spherical geometry, H2/CO = 2). Our results are in qualitative agreement with previous studies on the effect of catalyst shape on the effectiveness factor as a function of generalized Thiele modulus. Rester and Aris67 reported that maximum deviation between flat plate and spherical particle for a first-order reaction is 14−16% and occurs for 1.3 < ϕ < 1.7. Knudsen et al.68 showed that deviations could be higher (up to 27.2%) for some forms of LH kinetics in the region of ϕ values between 1 and 1.5. Figure 5b shows data for CAER catalyst and spherical particle (P = 20−30 bar, T = 473−493 K, H2/CO = 2, XCO = 0−50%). As can be seen, the data scatter is much less, in comparison to Figure 5a, when the effects of particle shape and catalyst type are removed. Minor scatter is caused by differences in pressure, temperature, and/or conversion level. The maximum value of η is 1.15 for ϕ = 0.71 (dp = 0.4 mm at 493 K, 30 bar and XCO = 0%), whereas the maximum value of η at base case conditions (473 K, 25 bar, XCO = 0%) is 1.12 at ϕ = 0.77 (dp = 0.6 mm). Methane selectivity as a function of generalized Thiele modulus is shown in Figure 6a (spherical particle) and Figure

4. CONCLUSIONS We have utilized a simple one-dimensional catalyst particle model which consists of four second-order differential equations for reactants (H2 and CO) and product species (H2O and C6H14) and an algebraic equation for methane formation rate in terms of concentrations of H2, CO, and H2O inside the particle. J


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Industrial & Engineering Chemistry Research Notes

Numerical solution of this model enables one to calculate two key performance parameters: catalyst effectiveness factor and methane selectivity (both local and average). Kinetic parameters for the rate of CO consumption and rate of CH4 formation were determined from experimental data with a state-of-the art 25% Co/0.48% Re/γ-Al2O3 catalyst (CAER catalyst). It was shown that activity of CAER catalyst is 15−37 higher (depending on process conditions) than that of Yates and Satterfield catalyst. Kinetic parameters for YS catalyst were used in several recent modeling studies of diffusion with chemical reaction. The use of kinetic parameters derived from original experimental data with YS catalyst severely underestimates the importance of diffusional limitations in FTS under conditions representative of industrial practice. From numerical simulations with kinetic parameters for CAER catalyst, the following conclusions can be made: • The use of small spherical particles (0.2−0.5 mm) or eggshell distribution with a larger spherical particle (dp = 2 mm) with catalyst layer thickness less than approximately 0.13 mm is needed to avoid negative impact of diffusional limitations on CH4 selectivity. The catalyst effectiveness factor under these conditions is slightly higher than 1 because of negative reaction order with respect to CO at high CO concentrations (i.e., low CO conversion inside the particle). • For monolith reactors with wash-coated catalyst, diffusional limitations can be avoided by using catalyst layer thickness less than 0.12 mm at base case conditions (473 K, 25 bar, H2/CO = 2, XCO = 0%). • Methane selectivity decreases with increase in overall CO conversion along the reactor because of presence of water which inhibits CH4 formation rate. • The use of substoichiometric H2/CO feed ratios (H2/CO = 1.7 and 1.4 in our study) leads to improvement in lower values of CH4 selectivity (lower H2/CO ratio inside the pellet). This is consistent with results reported by Vervloet et al.7 • Total pressure (20−30 bar) has a small effect on the catalyst effectiveness factor and CH4 selectivity. • Temperature (473−493 K) has a small effect on the effectiveness factor, but CH4 selectivity increases significantly with increase in temperature. • Graphs of η versus ϕ presented in the paper can provide rough estimates of η for other Co-based FT catalysts provided that sufficient information is available to calculate the value of ϕ. The biggest errors would occur in the region 0.5 < ϕ < 1.5. To capitalize on some of these findings in practical applications, one would need to develop preparation methods which increase catalyst loading in a layer near the surface (eggshell catalyst distribution) and develop Co catalysts that are stable during long-term operation at low H2/CO feed ratios.

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The authors declare no competing financial interest. ∥ Deceased.

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ACKNOWLEDGMENTS This research was made possible by a grant (NPRP Grant 7-5592-211) from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.

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NOTATION a = adsorption coefficient for CO reaction rate (bar−1) aM = reaction order of CO in methane formation reaction AM = pre-exponential factor for CH4 reaction rate coefficient (mol kg−1 s−1 bar−0.29) bM = reaction order of H2 in methane formation reaction Ci = concentration of species i in liquid phase (mol m−3) CL = total liquid phase concentration (mol m−3) dp = sphere particle diameter (mm) De,i = effective diffusivity coefficient of species i in particle (m2 s−1) Dwax,i = diffusivity coefficient of component i in wax (m2 s−1) EM = activation energy for CH4 reaction (J mol−1) Hi = Henry’s law constant for component i (bar) k = reaction rate coefficient for CO consumption reaction rate (mol kg−1 s−1 bar−2) kM = CH4 formation reaction rate constant (mol kg−1 s−1 bar−0.29) keff = effective thermal conductivity of catalyst particle (W m−1 K−1) L = slab thickness (mm) Lc = characteristic diffusion length (mm) Lg = geometric characteristic length (mm) mM = water effect coefficient in CH4 formation equation P = total pressure (bar) Pi = partial pressure of component i (bar) r = variable radius within sphere and hollow cylinder (mm) ri = hollow cylinder inner radius (mm) ro = hollow cylinder outer radius (mm) R = sphere radius (mm) RCH4 = reaction rate of CH4 (mol kg−1 s−1) RCO = reaction rate of CO (mol kg−1 s−1) R̅ CO = dimensionless CO reaction rate Rg = universal gas constant (J mol−1 K−1) S = surface area of catalyst exposed to surrounding fluid (m2) SCH4 = local methane selectivity (%) SC̅ H4 = average methane selectivity (%) T = temperature (K) V = volume of catalyst loaded region (m3) x = dimensionless length XCO = CO conversion (%) yi = dimensionless concentration of species i in liquid phase z = variable length within slab (mm)

Greek Letters

AUTHOR INFORMATION

α = chain growth parameter δ = eggshell layer thickness (mm) ε = pellet porosity η = effectiveness factor νi = stoichiometric coefficient of species i ρp = catalyst density (kg m−3) ρw = wax density (kg m−3) τ = pellet tortuosity

Corresponding Author

*Chemical Engineering Program, Texas A&M University at Qatar 219S Texas A&M Engineering Building, Education City, PO Box 23874, Doha, Qatar. Tel.: +974 4423 0134. Fax: +30 2310 996184. E-mail: [email protected]. ORCID

Dragomir B. Bukur: 0000-0002-5065-4163 K


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Industrial & Engineering Chemistry Research ϕ′i = dimensionless Thiele modulus based on geometric characteristic length of a particle ϕ = generalized Thiele modulus based on characteristic length for diffusion

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Superscripts

s = value at the surface

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