## 3 Modeling Activity
### Model Framework
It is applicable in a wide range of operating conditions which leads to a good prediction of gas-liquid mass transfer [27]:  
\[\underbrace{\frac{\partial\big{(}\epsilon_{\rm g}C_{\rm g,i}\big{)}}{\partial t }}_{\rm Accumulation}=\frac{\partial}{\partial z}\bigg{(}D_{\rm g}\ \epsilon_{\rm g}\frac{\partial C_{\rm g,i}}{\partial z}\bigg{)}-\underbrace{ \frac{\partial}{\partial z}\big{(}\epsilon_{\rm g}U_{\rm g}C_{\rm g,i}\big{)} }_{\rm Convection}-\underbrace{(K_{\rm i}a)_{\rm g,i}\epsilon_{\rm l}\bigg{(} \frac{C_{\rm g,i}}{A_{\rm i}}-C_{\rm l,i}\bigg{)}}_{\rm Mass\ Transfer} \tag{1}\]4  
The superficial gas velocity is defined as the volumetric gas flow rate divided by the cross-sectional area of the reactor above the gas distributor [28]:  
\[\underbrace{\frac{\partial\big{(}\epsilon_{\rm g}C_{\rm g,i}\big{)}}{\partial t }}_{\rm Accumulation}=\frac{\partial}{\partial z}\bigg{(}D_{\rm g}\ \epsilon_{\rm g}\frac{\partial C_{\rm g,i}}{\partial z}\bigg{)}-\underbrace{ \frac{\partial}{\partial z}\big{(}\epsilon_{\rm g}U_{\rm g}C_{\rm g,i}\big{)} }_{\rm Convection}-\underbrace{(K_{\rm i}a)_{\rm g,i}\epsilon_{\rm l}\bigg{(} \frac{C_{\rm g,i}}{A_{\rm i}}-C_{\rm l,i}\bigg{)}}_{\rm Mass\ Transfer} \tag{1}\]5  
The initial bubble size depending on buoyancy forces and surface tension is derived by the theoretical Davidson and Schuler expression [29]: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Initial Condition (t = 0)** & **Reactor Inlet (z = 0)** & **Reactor Outlet (z = H)** \\ \hline \(C_{gi}=C_{g,in}\) & \(C_{gi}=C_{g,in}\) & \(\frac{\partial C_{g,i}}{\partial z}=0\) \\ \(C_{g,i0}=\frac{\partial V_{f}}{\partial z}\) & \(\frac{\partial C_{g,i}}{\partial z}=0\) \\ \hline \(C_{l}=C_{g,in}/A_{i}\) & \(\varepsilon_{l}\,D_{l}\left(\frac{\partial C_{l,i}}{\partial z}\right)_{x=0}=U _{l}\left(C_{l,i}-C_{g,i0/A_{i}}\right)\) & \(\frac{\partial C_{g,i}}{\partial z}=0\) \\ \hline \(U_{i}=U_{g,in}\) & \(U_{gi}=U_{g,in}\) & \(\frac{\partial U_{g}}{\partial z}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial and boundary conditions for the slurry reactor model. \[\mathrm{d_{B}}=\left[\frac{6\sigma\,\mathrm{d_{0}}}{g\,(\rho_{\mathrm{SL}}-\rho_{ \mathrm{G}})}\right]^{1/3} \tag{15}\]  
The gas hold up can be expressed as the volume of the gas phase divided by the reactor volume consisting of gas volume, liquid volume and volume of catalyst used [28]:  
\[\mathrm{\epsilon_{g}}=\frac{V_{gas}}{V_{gas}+V_{liquid}+V_{cat}} \tag{16}\]  
This is calculated to be 0.161 (m\({}_{\mathrm{G}}^{3}\) m\({}_{\mathrm{R}}^{-3}\)) and the catalyst volume fraction (\(\mathrm{\epsilon_{cat}}\)) is calculated to be 0.34 (m\({}_{\mathrm{cat}}^{3}\) m\({}_{\mathrm{L}}^{-3}\)). It should be noted that in the Equations (1) and (2), \(\mathrm{\epsilon_{L}}\) is the liquid holdup (m\({}_{\mathrm{L}}^{3}\) m\({}_{\mathrm{R}}^{-3}\)) and r\({}_{\mathrm{i}}\) denotes reaction rate of FT species (mol kg\({}_{\mathrm{cat}}^{-1}\) s\({}^{-1}\)). Moreover, the velocity of liquid inside the reactor is assumed to be 0.00089 m/s. In the slurry bubble column reactors, the gas bubble coalescence occurs in a short time with an increase of the column diameter, and the large bubbles collect around the center of the column in the operation of a churn turbulent regime. This is called heterogeneous flow, which usually occurs when the superficial gas velocity is greater than 0.05 m/s. When the gas-liquid mixture reaches the surface, the bubbles disengage, allowing the degassed liquid to recirculate. Therefore, the main cause of back-mixing and liquid dispersion in the lab-scale slurry FT reactor is attributed to downward velocity of the liquid in the wall region and upward direction (V\({}_{\mathrm{L}}\)(r)) in the central axis [30]. Based on Riquarts correlation [31] the magnitude of V\({}_{\mathrm{L}}\)(0) depends on the column diameter, superficial gas velocity and kinematic viscosity of the liquid phase as described in Equation (17).