## 3 Mathematical Modeling
### Exergoeconomic Analysis  
Exergoeconomic analysis incorporates the exergy into the economic concepts to offer the exergetic costs of each stream and the system unit product, and it also reveals the methods to reduce the system product cost. The component investment costs (\(Z_{k}\)) are shown in Table 1.  
The exergoeconomic balance and the basic equations of the \(k\)th component are defined as [44]:  
\[\sum\dot{\text{C}}_{\text{o,k}}+\dot{\text{C}}_{\text{W,k}}=\sum\dot{\text{C} }_{\text{i,k}}+\dot{\text{C}}_{\text{Q,k}}+\dot{Z}_{\text{k}} \tag{11}\]  
\[\dot{\text{C}}_{\text{k}}=c_{\text{k}}\cdot\dot{E}_{\text{k}} \tag{12}\]  
\[c_{\text{F,k}}=\frac{\dot{\text{C}}_{\text{F,k}}}{\dot{E}_{\text{F,k}}} \tag{13}\]\[c_{\rm P,k}=\frac{\dot{\rm C}_{\rm P,k}}{\dot{\rm E}_{\rm P,k}} \tag{14}\]  
\[\dot{\rm C}_{\rm D,k}=c_{\rm F,k}\cdot\dot{\rm E}_{\rm D,k} \tag{15}\]  
where indexes i and o denote the input and the output stream, subscripts W and Q denote the output power and the input heat energy, where \(\dot{\rm C}\) and \(c\) are the exergetic cost rate of each stream and per unit stream, where \(c_{\rm F,k}\) and \(c_{\rm P,k}\) are the average cost rate per unit fuel and product of the \(k\)th component, where \(\dot{\rm C}_{\rm D,k}\) is the exergetic loss cost rate of the \(k\)th component.  
The investment cost rate of components can be acquired by [44]:  
\[\dot{\rm Z}_{\rm k}=\varphi\cdot\left(\frac{CRF}{N}\right)\cdot Z_{\rm k} \tag{16}\]  
where \(\varphi\) means the maintenance factor, and \(N\) means the annual running hours.  
\(CRF\) denotes the capital recovery factor, which is given in Equation (17), where \(i\) and \(n\) represent the interest rate and the system life span [45].  
\[CRF=\frac{i\cdot\left(i+1\right)^{n}}{\left(i+1\right)^{n}-1} \tag{17}\]  
The exergoeconomic performance criteria are the exergoeconomic factor (\(f_{\rm k}\)) and the relative cost difference rate (\(r_{\rm k}\)) for the \(k\)th component, and the system unit cost per product (\(c_{\rm P,tot}\)). \(f_{\rm k}\) represents the investment cost proportion of operation cost for system components, revealing that the \(k\)th component has either high investment cost or low efficiency. It is defined as [36; 45]:  
\[f_{\rm k}=\frac{\dot{\rm Z}_{\rm k}}{\dot{\rm Z}_{\rm k}+\dot{\rm C}_{\rm D,k}} \tag{18}\]  
\(r_{\rm k}\) reveals the exergetic cost difference between the fuel and the product, and the lower value is preferred. It could be acquired by [46]:  
\[r_{\rm k}=\frac{c_{\rm P,k}-c_{\rm F,k}}{c_{\rm F,k}} \tag{19}\]  
\(c_{\rm P,tot}\) represents the system total exergy cost per unit product and it is defined as [37]:  
\[c_{\rm P,tot}=\frac{\dot{\rm Z}_{\rm k}+\dot{\rm C}_{\rm W,c}}{\dot{\rm W}_{ \rm T}} \tag{20}\]  
where \(\dot{\rm C}_{\rm W,c}\) is the exergetic cost rate of compressor input power.  
Since the capital investment costs are cited at the reference year, all of them should be updated to 2023 by the use of the cost indicator, which can be found in the Chemical Engineering Plant Cost Index (CEPCI), and the specific function is shown as [38]:  
\[Z_{\rm 2023}=Z_{\rm origin}\cdot\left(CEPCI_{\rm 2023}/CEPCI_{\rm origin}\right) \tag{21}\]  
The values of the related parameters for the functions above are shown in Table 2. The detailed exergoeconomic balance functions and the auxiliary equations of each component are shown in Table 3.