3. MODELING
The model catalyst pellet is assumed as an isothermal and isobaric sphere for common simplicity, and a 1-dimensional steady state continuum model can be used to describe the reaction-diffusion process in the pellet.
The hierarchical pore structure prepared can be simplified as Figure 1 based on the preparation method of the meso-macroporous SiO2 monoliths in section 2.1. At typical industrial conditions of cobalt-based FTS, the hydrocarbons products (especially waxes) could condense and accumulate in pores and form a thin liquid layer on the external surface of the pellet. Therefore, the syngas must solubilize in the liquid film first and further diffuse into the pores and reach the internal active sites, accompanying with FTS reaction5,7-9,14,38,42-44. The external diffusion limitations in the gas and liquid films were assumed to be negligible because of the high gas linear velocity employed in industrial reactors45. Since the mesopores formed by ammonia solution etching were mainly dispersed on the internal surface of the macropores and diffusion length in mesopores was relatively short, the internal diffusion limitations in such mesopores was not considered in the pellet modeling.
The following Eq.(1) was adopted to describe the overall stoichiometry of FTS on cobalt catalyst, according to the assumption by Mandic5 to the average molecular weight of hydrocarbon products for their kinetics experiments with 0.48%Re-25%Co/Al2O3 catalyst.
\(CO+\left(\frac{13}{6}\right)H_{2}=H_{2}O+(\frac{1}{6})C_{6}H_{14}\)(1)
Fick’s law has been widely used in describing the internal transport process in FTS catalyst pellet 5,8,14,45. For an isothermal spherical pellet, the generalized reaction–diffusion continuity equation at steady state can be expressed as:
\(\frac{1}{x^{2}}\bullet\frac{d}{\text{dx}}\left(x^{2}\bullet\frac{d\Psi_{i}}{\text{dx}}\right)+\upsilon_{i}\cdot(\phi_{i}^{{}^{\prime}})^{2}\cdot{\overline{R}}_{\text{CO}}=0\)(2)
The boundary conditions are given as:
\(x=0:\ \frac{d\Psi_{i}}{\text{dx}}=0\) (3)
\(x=1:\Psi_{i}=\ \frac{c_{i}^{s}}{c_{\text{CO}}^{s}}\) (4)
Herein, \(x\) is the dimensionless distance to the center of the pellet\(,\ x=\frac{r}{r_{p}}\).\(\Psi_{i}=\ \frac{c_{i}}{c_{\text{CO}}^{s}}\ \)represents the local dimensionless concentration of component i (i = CO, H2, H2O and C6H14).\({\overline{R}}_{\text{CO}}=\ \frac{R_{\text{CO}}}{R_{\text{CO}}^{s}}\)is the dimensionless reaction rate of CO. \(\upsilon_{i}\ \)is the stoichiometric coefficient, and equals -1, -13/6, 1 and 1/6 respectively for CO, H2, H2O and C6H14. The dimensionless group\(\phi_{i}^{{}^{\prime}}=r_{p}{(\frac{\rho_{P}\left(-R_{\text{CO}}^{s}\right)}{(D_{e,i}\cdot C_{\text{CO}}^{S})})}^{1/2}\)is comprised of the radius of the pellet, \(r_{p}\), the pellet density,\(\rho_{p}\), CO consumption rate at the external surface of the pellet,\(R_{\text{CO}}^{s}\), the effective diffusion coefficient of componenti ,\(\ D_{e,i}\), and CO concentration at the external surface,\(C_{\text{CO}}^{S}\).