## 3 System Modelings  
### Thermodynamic Model  
The thermodynamic model for the combined cycle is developed according to the mass and energy balance equations of each component. Thermodynamic relations and the effectiveness for HTR and LTR are expressed as Equations (1)-(4),  
\[h_{6}-h_{7}=h_{4}-h_{3} \tag{1}\]  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]  
\[\varepsilon_{HTR}=\frac{T_{6}-T_{7}}{T_{6}-T_{3}} \tag{3}\]  
\[\varepsilon_{LTR}=\frac{T_{7}-T_{8}}{T_{7}-T_{2}} \tag{4}\]  
where \(x\) is the mass separation ratio of stream 3b to stream 3, and \(\varepsilon_{HTR}\) and \(\varepsilon_{LTR}\) are the effectiveness of HTR and LTR, respectively. For turbine, main compressor and recompression compressor, the isentropic efficiencies are calculated with Equation (5)-(7),  
\[\eta_{\rm tur}=\frac{h_{5}-h_{6}}{h_{5}-h_{6s}} \tag{5}\]  
\[\eta_{\rm mcom}=\frac{h_{2s}-h_{1}}{h_{2}-h_{1}} \tag{6}\]  
\[\eta_{\rm rcom}=\frac{h_{3s}-h_{8}}{h_{3}-h_{8}} \tag{7}\]  
where \(\eta_{\rm tur}\), \(\eta_{\rm mcom}\), and \(\eta_{\rm rcom}\) are the isentropic efficiencies for the turbine, main compressor, and recompression compressor, respectively. The thermodynamic relation for the cooler is calculated using Equation (8),  
\[(1-x)m_{\rm CO_{2}}(h_{10}-h_{1})=m_{cw1}(h_{12}-h_{11}) \tag{8}\]  
where \(m_{\rm CO2}\) and \(m_{cw1}\) are the mass flow rates of CO\({}_{2}\) and the cooling water of the cooler, respectively. The power output of the turbine and the consumptions of two compressors can be derived as Equations (9)-(11),  
\[W_{\rm tur}=m_{\rm CO_{2}}(h_{5}-h_{6}) \tag{9}\]  
\[W_{\rm mcom}=(1-x)m_{\rm CO_{2}}(h_{2}-h_{1}) \tag{10}\]  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]0  
For the sCO\({}_{2}\)/AHP system, the energy conservation equation of DHE is Equation (12),  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]1  
while that for the sCO\({}_{2}\)/DH system is Equation (13),  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]2  
where \(m_{hw1}\) is the mass flow rate of the heating water flowing through the DHE. \(Q_{\rm DHE}\) is the heat absorption of DHE from the topping cycle. In the sCO2/AHP system, the conservation equations for the AHP are expressed as follows. The thermodynamic relation for the generator is Equation (14),  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]3  
where \(Q_{\text{Gen}}\) is the heat absorption amount of generator from the topping cycle. Assuming \(y\) as the concentration of LiBr or ammonia in the water, the concentration balance equation is obtained as Equation (15),  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]4  
The conservation equations for the condenser, evaporator, and absorber are obtained as Equations (16)-(18),  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]5  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]6  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]7  
where \(m_{lm2}\) is the mass flow rate of the heating water flowing through the AHP. The conservation equation for SHE is obtained as Equation (19),  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]8  
For the pump, the isentropic efficiency and power consumption are defined according Equations (20) and (21),  
\[(h_{3}-h_{2})(1-x)=h_{7}-h_{8} \tag{2}\]9  
\[\varepsilon_{HTR}=\frac{T_{6}-T_{7}}{T_{6}-T_{3}} \tag{3}\]0  
The net output power of the combined cycle can be calculated as Equation (22),  
\[\varepsilon_{HTR}=\frac{T_{6}-T_{7}}{T_{6}-T_{3}} \tag{3}\]1  
The heat amount supplied to the heat users, \(Q_{h,users}\), and the heat absorption from the topping cycle for the purpose of heating, \(Q_{h,absorb}\) were then defined for both combined systems.