## 3 Modeling Activity  
### Model Framework  
A detailed computer model for a pilot plant FT-SBCR (D = 0.1 m, H = 2.5 m) for all the key components of FT synthesis reactor was developed. A set of Partial Differential Equations (PDEs) of transport species including mass transfer and kinetics was successfully coupled with hydrodynamic parameters. In general, 10 equations for the gas phase (single bubble class) and 10 equations for the liquid phase need to be solved simultaneously. The reactor model was established with these assumptions: (1) an Axial Dispersion Model (ADM) in the form of a convection-diffusion phenomenon with a single class of gas bubble diameter is considered (ADM-SBCD); (2) the gas-liquid mass transfer resistance is positioned on both the gas and liquid side; (3) based on the non-dimensional form of Peng-Robinson equation-of-state (PR-EOS), the compressibility factor of the gas phase corresponds to near unity (\(Z\approx 1\)). The PR-EOS is a suitable fluid model for FT systems to predict vapor-liquid \begin{table}
\begin{tabular}{c c} \hline \hline
**Reaction Characteristics** & **Relations** & **Constants, Parameters and** \\  & & **Paraffin Reaction Form** \\ \hline Reaction Kinetics & \(-\mathrm{R}_{\mathrm{co}}=\frac{\alpha\mathrm{r}_{\mathrm{o}}\mathrm{r}_{ \mathrm{H}_{2}}}{(1+\mathrm{b}\mathrm{r}_{\mathrm{o}})^{2}}\) & a = 1.59064 \(\times\) 10\({}^{-12}\) \\  & b = 7.99389 \(\times\) 10\({}^{-6}\) \\ \hline Chain grow probability & \(\alpha=\left(\mathrm{A}_{\mathrm{y}_{\mathrm{o}}}\frac{\mathrm{Y}_{\mathrm{o}} }{\mathrm{Y}_{\mathrm{H}_{2}}\mathrm{Y}_{\mathrm{o}}}+\mathrm{B}\right)\left[10.0039(\mathrm{T}-533)\right]\) & A = 0.2332 \(\mathrm{B}=0.633\) \\  & \(Y_{\mathrm{co}}=0.2\) & \(\mathrm{Y}_{\mathrm{H}_{2}}=0.4\) \\ \hline [PERSON] & \(\mathrm{W}_{\mathrm{n}}=\mathrm{n}\left(1-\alpha\right)^{2}\).a\({}^{\mathrm{n-1}}\) & \(\log\frac{\mathrm{W}_{\mathrm{n}}}{\mathrm{n}}=\mathrm{n}\log(\alpha)+\log \frac{(1-\alpha)^{2}}{\alpha}\) \\ \hline Rate of paraffin & \(\mathrm{r}_{\mathrm{i}}=\mathrm{R}_{\mathrm{co}}\alpha^{\mathrm{n-1}}\) & \(\mathrm{n}\,\mathrm{CO}+\left(2\mathrm{n}+1\right)\mathrm{H}_{2}\to \mathrm{C}_{\mathrm{n}}\mathrm{H}_{2\mathrm{n}+2}+\mathrm{n}\mathrm{H}_{2}\mathrm{O}\) \\ distribution. & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Kinetic Characteristics of Fischer–Tropsch Slurry Bubble Column reactor (FT-SBCR): reaction parameters, product distribution. \begin{table}
\begin{tabular}{c c} \hline \hline
**Operating Condition** & \\ \hline Reactor Temperature & 503 K \\ Reactor Pressure & 20 bar \\ H\({}_{2}\)/CO & 2 \\ Reactor diameter & 0.1 m \\ Reactor height & 2.5 m \\ Volumetric Flow Rate (loads) & 3.5, 5, 7.5 m\({}^{3}\)/h \\ \hline \hline
**Liquid Phase Properties** & \\ \hline Liquid Density & 715 kg/m\({}^{3}\) \\ Surface tension & 0.023 N/m \\ Liquid Viscosity & 3 \(\times\) 10\({}^{-3}\) Pa.s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Operating conditions and liquid properties of lab-scale Fischer–Tropsch Slurry Bubble Column. compositions and flow rates inside the slurry reactor [23; 24]; (4) considering a low pressure drop in the slurry reactor, the operating pressure is assumed to be constant along the reactor height; (5) the reactor operates under isothermal condition; (6) by assuming constant pressure and temperature, the overall continuity balance occurs at the inlet and outlet of the reactor (Equation (3)); (7) liquid-solid mass transfer resistance may be ignored and consequently solid suspension in liquid is simulated as a single pseudo-homogeneous slurry phase in this work; (8) chemical reaction through Langmuir Hinshelwood Hougen Watson (LHHW) kinetics in liquid phase is considered.