## 3 System Modelings
### Exergoeconomic Model
The electric power consumed by the pump in the AHP is assumed to come from the power production of the turbine. In addition, the capital investment cost at the present year is obtained according to the cost indices, namely the Chemical Engineering Plant Cost Index (CEPCI). The cost at present year can be obtained with Equation (39) [32],  
\[\text{Cost at present year}=\text{Original}\ \cos t\ \times\ \frac{\text{Cost index for present year}}{\text{Cost index for original year}}. \tag{39}\]  
The Gauss-Seidel method is utilized to solve the linear system of equations in the Table 5 for two combined systems. After the solution, the cost rates for all exergy streams can be obtained. Table 5 shows the cost balance equations and auxiliary equations for the combined systems. \begin{table}
\begin{tabular}{c c c} \hline
**Components** & **sCO2/DH System** & **sCO2/AHP System** \\ \hline Reactor & \(\hat{C}_{5}=\hat{C}_{4}+\hat{Z}_{\text{reactor}}+\hat{C}_{\text{fuel}}\) & \(\hat{C}_{5}=\hat{C}_{4}+\hat{Z}_{\text{reactor}}+\hat{C}_{\text{fuel}}\) \\ \hline sCO2 turbine & \(\hat{C}_{6}+\hat{C}_{W_{\text{max}}}=\hat{C}_{5}+\hat{Z}_{\text{tur}}\) & \(\hat{C}_{6}+\hat{C}_{W_{\text{max}}}=\hat{C}_{5}+\hat{Z}_{\text{tur}}\) \\  & \(\frac{\hat{C}_{6}}{\hat{C}_{5}}=\frac{\hat{C}_{4}}{\hat{E}_{6}}\) & \(\frac{\hat{C}_{5}}{\hat{E}_{5}}=\frac{\hat{C}_{6}}{\hat{E}_{6}}\) \\ \hline HTR & \(\hat{C}_{4}+\hat{C}_{7}=\hat{C}_{3}+\hat{C}_{6}+\hat{Z}_{\text{HTR}}\) & \(\hat{C}_{4}+\hat{C}_{7}=\hat{C}_{3}+\hat{C}_{6}+\hat{Z}_{\text{HTR}}\) \\  & \(\frac{\hat{C}_{6}}{\hat{E}_{6}}=\frac{\hat{C}_{7}}{\hat{C}_{3}}=\hat{C}_{34}+ \hat{C}_{3b}\) & \(\frac{\hat{C}_{6}}{\hat{E}_{6}}=\frac{\hat{C}_{7}}{\hat{E}_{7}},\hat{C}_{3}= \hat{C}_{3u}+\hat{C}_{3b}\) \\ \hline LTR & \(\hat{C}_{8}+\hat{C}_{3u}=\hat{C}_{2}+\hat{C}_{7}+\hat{Z}_{\text{LTR}}\) & \(\hat{C}_{8}+\hat{C}_{3u}=\hat{C}_{2}+\hat{C}_{7}+\hat{Z}_{\text{LTR}}\) \\  & \(\frac{\hat{C}_{7}}{\hat{E}_{7}}=\frac{\hat{C}_{8}}{\hat{E}_{8}},\hat{C}_{8a}= \hat{C}_{8}(1-x)\) & \(\frac{\hat{C}_{7}}{\hat{E}_{7}}=\frac{\hat{C}_{8}}{\hat{E}_{8}},\hat{C}_{8a}= \hat{C}_{8}(1-x)\) \\ \hline Main compressor & \(\hat{C}_{2}=\hat{C}_{1}+\hat{C}_{W_{\text{max}}}+\hat{Z}_{\text{mcom}}\) & \(\hat{C}_{2}=\hat{C}_{1}+\hat{C}_{W_{\text{max}}}+\hat{Z}_{\text{mcom}}\) \\  & \(\frac{\hat{C}_{W_{\text{max}}}}{W_{\text{mcom}}}=\frac{\hat{C}_{W_{\text{max}}} }{W_{\text{max}}}\) & \(\frac{\hat{C}_{W_{\text{max}}}}{W_{\text{mcom}}}=\frac{\hat{C}_{W_{\text{max}}} }{W_{\text{max}}}\) \\ \hline Recompression compressor & \(\hat{C}_{3b}=\hat{C}_{8b}+\hat{C}_{W_{\text{max}}}+\hat{Z}_{\text{rcom}}\) & \(\hat{C}_{3b}=\hat{C}_{8b}+\hat{C}_{W_{\text{recom}}}+\hat{Z}_{\text{rcom}}\) \\  & \(\frac{\hat{C}_{W_{\text{max}}}}{W_{\text{max}}}=\frac{\hat{C}_{W_{\text{max}}} }{W_{\text{max}}},\hat{C}_{8b}=x\hat{C}_{8}\) & \(\frac{\hat{C}_{W_{\text{max}}}}{W_{\text{max}}}=\frac{\hat{C}_{W_{\text{max}}} }{W_{\text{max}}},\hat{C}_{8b}=x\hat{C}_{8}\) \\ \hline DHE & \(\hat{C}_{10}+\hat{C}_{\text{H}6}=\hat{C}_{8a}+\hat{C}_{\text{H}4}+\hat{Z}_{ \text{DHE}}\) & \(\hat{C}_{10}+\hat{C}_{\text{H}5}=\hat{C}_{9}+\hat{C}_{\text{H}4}+\hat{Z}_{ \text{DHE}}\) \\  & \(\frac{\hat{C}_{6a}}{\hat{E}_{8a}}=\frac{\hat{C}_{10}}{\hat{E}_{8b}}\) & \(\frac{\hat{C}_{9}}{\hat{E}_{8}}=\frac{\hat{C}_{10}}{\hat{E}_{10}},\hat{C}_{\text {H}6}=\hat{C}_{15}+\hat{C}_{\text{H}3}\) \\ \hline Cooler & \(\hat{C}_{1}+\hat{C}_{12}=\hat{C}_{10}+\hat{C}_{11}+\hat{Z}_{\text{cooler}}\) & \(\hat{C}_{1}+\hat{C}_{12}=\hat{C}_{10}+\hat{C}_{11}+\hat{Z}_{\text{cooler}}\) \\  & \(\hat{C}_{11}=0,\hat{C}_{12}=0\) & \(\hat{C}_{11}=0,\hat{C}_{12}=0\) \\ \hline Generator & / & \(\hat{C}_{A7}+\hat{C}_{A8}+\hat{C}_{9}=\) \\  & \(\frac{\hat{C}_{8a}}{\hat{E}_{8}}=\frac{\hat{C}_{9}}{\hat{E}_{9}},\frac{\hat{C}_{ 9a}\hat{C}_{9a}\hat{C}_{9a}\hat{C}_{9a}\hat{C}_{9a}\hat{C}_{9a}\hat{C}_{9a}\hat{C}_{9a }\hat{C}_{