Figure 1. Illustrations of the meso-macropore networks.
3.1 kinetics and chain growth factor model
CO consumption rate with Langmuir-Hinshelwood kinetics by Yates and Satterfield 41 was used in this study. The values of reaction rate constant (k ) and adsorption equilibrium constant (a ) listed in Table 1S (see Supporting Information ) are from the regression of experimental data with 0.48%Re-25%Co/Al2O3 catalyst by Mandic et al. 5. The correction factor (f ) in Eq.(5) was used to correct reaction activity based on our experimental data in the absence of diffusion resistance.
\(\left(-R_{\text{CO}}\right)=\frac{\text{f\ k}\ P_{\text{CO}}P_{H_{2}}}{{(1+a\ P_{\text{CO}})}^{2}}\)(5)
A chain growth factor model related to temperature and concentration ratio of H2 and CO reported by Vervloet,42 was adopted to calculate the product selectivity within the pellet.
\(\alpha=\frac{1}{1+k_{\alpha}{(\frac{c_{H_{2}}}{c_{\text{CO}}})}^{\beta}exp(\frac{E_{\alpha}}{R}(\frac{1}{493.15}-\frac{1}{T}))}\)(6)
3.2 Physical properties
The concentrations of species CO, H2, H2O and C6H14 dissolved in the wax phase assumed to be in equilibrium with those in the gas phase can be described by Henry’s law 5,8,45. Henry’s law constants for gases and light hydrocarbons in n-paraffins were calculated from Marano and Holder’s correlation 46.
Wax accumulation in FTS catalyst pores is a non-negligible factor leading to strong internal diffusion limitations because of the low diffusivity of reactants and products in liquid wax7,43,44. Considering the huge diffusivity difference in liquid phase and gas phase, the diffusivity in pellet is closely related to the wax filling degree in pores. In the case of pores with wax fully filled, the liquid phase molecular diffusion dominates the mass transfer restrictions while Knudsen diffusion limitations could be neglected. The diffusivity in liquid wax, \(D_{i,liq},\) thought to be related to the composition of wax and temperature, can be estimated from Akgerman’s correlation 47,48. Although the assumption of the pellet with wax fully filled has been applied for simplicity in almost all numerical modeling studies of FTS catalyst pellet5,7-9,14,38,42-44. The studies by Jess et al.43,44,49 indicated there might exist only partly filled state in pores under practical industrial FTS conditions. Furthermore, the introduction of macropores in pore structure is in favor of a reduction in the filling degree of liquid wax. Despite this, accurate calculation or measurement of filling degree F at a certain condition is still a complicated and tough task, because of the complex interactions among catalyst apparent reactivity, diffusivity in pores, product distribution, and liquid phase accumulation rate in pores. Since the focus of this study is to compare the performances at various porosity and macropore size, F is assumed as constant with series of values (0.6, 0.7, 0.8, 0.9, 1.0) for simplicity in the following. In the case of the pellet pores with wax partially-filled (F <1), both Knudsen and molecular diffusion contribute to the mass transfer process in the pores, and Eq. (10) proposed by Jess50 based on the random pore model 51can be used to calculate the effective diffusivity in the whole pellet scale.
\(D_{eff,i}={(1-F)}^{2}\varepsilon^{2}D_{i,g}+\frac{F^{2}\varepsilon^{2}(1+3(1-F)\varepsilon)}{1-(1-F)\varepsilon}D_{i,liq}\)(7)
Herein, ɛ is the porosity of the catalyst pellet.Di,g , the gas phase diffusivity in pores, can be described by the Wilke-Bosanquet model 45:
\(\frac{1}{D_{i,g}}=\frac{1}{D_{i,m}}+\frac{1}{D_{i,k}}\) (8)
Di,m , the molecular diffusivity of componenti in the gas mixture of reactants and products in the pore, can be calculated from the binary gas diffusivityDi,j and the molar contentyi 50.
\(D_{i,m}=\frac{1-y_{i}}{\sum_{j=1,i\neq j}^{n}{(y_{i}/D_{i,j})}}\)(9)
The binary gas diffusivity Di,j can be calculated by the Fuller Method52 with the temperature T, pressure P, molecular weight of each components Mi, and diffusion volume Σv. The value of Σv for components used in the present study can be obtained from the literature52.
\(D_{i,j}=\frac{0.00143T^{1.75}}{PM_{i,j}^{\frac{1}{2}}{[\left(\Sigma_{v}\right)_{A}^{\frac{1}{3}}+\left(\Sigma_{v}\right)_{B}^{\frac{1}{3}}]}^{2}}\)(10)
\(M_{i,j}=2\left(\frac{1}{M_{i}}+\frac{1}{M_{j}}\right)^{-1}\)(11)
The Knudsen diffusivity Di,k can be calculated based on the kinetic theory of gases 52 with the radius of pore Rp, temperature T and molecular weight Mi.
\(D_{i,k}=97R_{P}{(\frac{T}{M_{i}})}^{\frac{1}{2}}\) (12)
Parameters used in simulation of the 2 mm pellet were listed in Table 1.
Table 1 Parameters for 2 mm pellet simulations