## 3 Modeling Activity
### Model Framework
Based on these model assumptions, the mass balances of each component in gas and liquid phases can be derived as follows:  
Gas phase equations:  
\[\underbrace{\frac{\partial\big{(}\epsilon_{\rm g}C_{\rm g,i}\big{)}}{\partial t }}_{\rm Accumulation}=\frac{\partial}{\partial z}\bigg{(}D_{\rm g}\ \epsilon_{\rm g}\frac{\partial C_{\rm g,i}}{\partial z}\bigg{)}-\underbrace{ \frac{\partial}{\partial z}\big{(}\epsilon_{\rm g}U_{\rm g}C_{\rm g,i}\big{)} }_{\rm Convection}-\underbrace{(K_{\rm i}a)_{\rm g,i}\epsilon_{\rm l}\bigg{(} \frac{C_{\rm g,i}}{A_{\rm i}}-C_{\rm l,i}\bigg{)}}_{\rm Mass\ Transfer} \tag{1}\]  
Liquid phase equations:  
\[\underbrace{\frac{\partial(\epsilon_{\rm l}C_{\rm l,i})}{\partial t}}_{\rm Accumulation }=\underbrace{\frac{\partial}{\partial z}\bigg{(}\epsilon_{\rm l}D_{\rm ax, l}\frac{\partial C_{\rm l,i}}{\partial z}\bigg{)}}_{\rm Axial\ Dispersion}-\underbrace{\frac{\partial}{\partial z} \big{(}\epsilon_{\rm l}U_{\rm l}C_{\rm l,i}\big{)}}_{\rm Convection}+\underbrace{ (K_{\rm i}a)_{\rm g,i}\epsilon_{\rm l}\bigg{(}\frac{C_{\rm g,i}}{A\rm i}-C_{ \rm l,i}\bigg{)}}_{\rm Mass\ Transfer}-\underbrace{\epsilon_{\rm l}\rho_{\rm cat }\epsilon_{\rm cat}\Gamma_{\rm i}}_{\rm Reaction} \tag{2}\]  
Total concentration:  
\[\sum_{\rm i}^{\rm n}C_{\rm g,i}=C_{\rm tot,in}=\frac{\rm P}{\rm ZRT}=C_{\rm tot,out} \tag{3}\]  
In the above equations, \(A_{\rm i}\) denotes the [PERSON]'s constant (\(H_{\rm i}\)) for syngas and light hydrocarbons (CO, \(H_{2}\), \(H_{2}\)O, \(C_{1}\) and \(C_{2}\)). By employing [PERSON]'s law solubility factor \((P_{\rm i}=x_{\rm i}\times H_{\rm i}^{\infty})\), the [PERSON]'s constant for each reactant gas was calculated based on the heat of solution for each component and operating temperature [25]. \[A_{\rm i}=H_{\rm i}/\rm RT \tag{4}\]  
\[H_{\rm i}=H_{\rm i}^{*}\exp\bigg{(}-\frac{\Delta H_{\rm s,i}}{\rm RT}\bigg{)} \tag{5}\]  
where, the value of parameters \(H_{\rm i}^{*}\) and \(\Delta H_{\rm s,i}\) are listed in reference [25]. For heavier components (\(C_{4}^{+}\)), \(Ai\) was calculated using [PERSON]'s law for gas-liquid phase at the equilibrium as follows:  
\[A_{\rm i}=H_{\rm i}^{\infty}/\rm RT.1/C_{\rm tot}^{L} \tag{6}\]  
\[C_{\rm tot}^{L}=\rho_{L}/\rm MW_{L,avg} \tag{7}\]  
where, \(C_{\rm tot}^{L}\) is the total concentration of liquid components, \(\rho_{L}\) is the liquid density, \(\rm MW_{L,avg}\) is the average molecular weight of liquid components and \(H_{\rm i}^{\infty}\) is the [PERSON]'s constant at infinite dilution which is defined as follow:  
\[H_{\rm i}^{\infty}=\gamma_{\rm i}^{\infty}\ P_{\rm i,sat} \tag{8}\]  
where \(\gamma_{\rm i}^{\infty}\) is the activity coefficient for heavier components and \(P_{\rm i,sat}\) is the vapor pressure of component i which is calculated from asymptotic behavior correlations (extension of the Antoine equations). By supposing ideal behavior of FT mixture due to long-chain \(n\)-paraffins, the [PERSON]'s constant for the mixture can be expressed by:  
\[LnH_{\rm i,mix}^{\infty}=\sum_{\rm j}x_{\rm j}Ln\ H_{\rm j}^{\infty} \tag{9}\]where j is the components in the solvent. Since \(\rm LnH_{i}^{\infty}\) and \(\rm Ln\gamma_{i}^{\infty}\) is asymptotically linear with solute carbon number (m), the formula for calculating the infinite-dilution activity coefficient becomes as follow:  
\[\underbrace{\frac{\partial\big{(}\epsilon_{\rm g}C_{\rm g,i}\big{)}}{\partial t }}_{\rm Accumulation}=\frac{\partial}{\partial z}\bigg{(}D_{\rm g}\ \epsilon_{\rm g}\frac{\partial C_{\rm g,i}}{\partial z}\bigg{)}-\underbrace{ \frac{\partial}{\partial z}\big{(}\epsilon_{\rm g}U_{\rm g}C_{\rm g,i}\big{)} }_{\rm Convection}-\underbrace{(K_{\rm i}a)_{\rm g,i}\epsilon_{\rm l}\bigg{(} \frac{C_{\rm g,i}}{A_{\rm i}}-C_{\rm l,i}\bigg{)}}_{\rm Mass\ Transfer} \tag{1}\]0  
where n is the carbon number of solvent (n-paraffin) and r denotes the carbon number of reference solute (n-\(\rm C_{6}H_{14}\)) in the same solvent. They are described in more detail in references [24; 26].