## 2 Materials and Methods
### Modeling Methodology
#### 2.2.3 Methanol Synthesis and Water Gas Shift Reaction Process  
After syngas post-treatment, the syngas (stream 14) was introduced into the methanol synthesis reactor. At this step, CO\({}_{2}\) hydrogenation to methanol and water gas shift reaction over a commercial Cu/ZnO/Al\({}_{2}\)O\({}_{3}\) catalyst occur simultaneously [33]. As the methanol synthesis is exothermic and volumetric reduction reaction, the productivity can be increased by rising pressure and reducing the reaction temperature. Therefore, the methanol synthesis reactor was assumed to operate at 250-350 \({}^{\circ}\)C and 50-150 bar [34]. The main reactions in the methanol synthesis unit can be expressed as described in the following Equations:  
\[ \begin{array}{cc}\text{CO}_{2}+3\text{H}_{2}\rightarrow\text{CH}_{3}\text{ OH}+\text{H}_{2}\text{O}&-49.5\text{kJ/kmol}&\text{Methanol synthesis (R-9)}\\ \text{CO}+\text{H}_{2}\text{O}\leftrightarrow\text{CO}_{2}+\text{H}_{2}&-41.2 \text{kJ/kmol}&\text{Water gas shift (R-10)}\end{array} \]  
The methanol synthesis model is based on several assumptions, i.e.,  
1. Pressure loss in the methanol synthesis reactor is not considered. 2. Syngas is preheated sufficiently and the temperature distribution in the methanol synthesis reactor is assumed to be uniform. Table 4 lists the blocks that were applied in methanol synthesis system. The post-treatment syngas (stream 11) obtained from Rectisol was heated up to 250 \({}^{\circ}\)C by H-1 and compressed to 50 bar by COMPR block. The methanol synthesis reactor is simulated by the R5 block where CO\({}_{2}\) hydrogenation and water gas shift reaction occur. The Langmuir-Hinshelwood-Hougen-Watson (LHHW) kinetic model is applied in the format of the reactant fugacity of the hydrogenation of CO\({}_{2}\) and water gas shift. The kinetic parameters applied in the LHHW block are obtained from the William experimental results [35]. Finally, a SEP-3 block was used to separate residual syngas from the mixture of methanol and water. The LHHW expression is presented in Equation (2):  
\[\text{R}=(\text{kinetic term})\frac{(\text{driving}-\text{force term})}{( \text{adsorption term})} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Flow Sheet Block ID** & **Aspen Plus Block ID** & **Description** \\ \hline H-1 & Heater & Used to heat up syngas to 250 \({}^{\circ}\)C \\ \hline COMPR & MCompr & Used to compress syngas to 50 bar \\ \hline R5 & RPlug & Used to simulate methanol synthesis reactor \\ \hline C-2 & Heater & Used to cool down the outlet syngas of methanol synthesis reactors \\ \hline SEP-3 & Flash 2 & Used to separate produced methanol and residual syngas \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Description of Aspen plus block used in methanol synthesis and operating condition. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Flow Sheet Block ID** & **Aspen Plus Block ID** & **Description** \\ \hline SEP-2 & SSplit & Used to separate ash from the syngas. Operation at 800 \({}^{\circ}\)C. \\ \hline C-1 & Heater & Used to decrease syngas temperature to 80 \({}^{\circ}\)C. \\ \hline P-1 & Pump & Used to increase syngas pressure to 40 bar \\ \hline RECTISOL & Radfrac & Used to remove acid syngas and partial carbon dioxide \\ \hline SEP-4 & Flash 2 & Used to separate part of CO\({}_{2}\) and impurities syngas from methanol and H\({}_{2}\)O \\ \hline SEP-5 & Sep2 & Used to separate H\({}_{2}\)O from methanol \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description of Aspen plus blocks used in Rectisol. The reaction rate to produce methanol from carbon dioxide hydrogenation is given in Equation (3). \[\mathrm{R_{CH_{3}OH}(kmol/kgcat\cdot s)}=\left(k_{\mathrm{MeOH}P_{CO_{2}}P_{H_{2}}} \right)\frac{[1-\frac{1}{K_{E1}}(\frac{P_{H_{2}O}P_{CO_{2}P_{H_{2}}^{3}}}{P_{CO_{ 2}P_{H_{2}}^{3}}})]}{[1+k_{3}\bigg{(}\frac{P_{H_{2}O}}{P_{H_{2}}}\bigg{)}+k_{1} \sqrt{P_{H_{2}}}+k_{2}P_{H_{2}O}]}^{3} \tag{3}\]  
The reaction rate for the water gas shift reaction is given in Equation (4).