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,\(C_{\text{CO}}^{S}\).