## 2 Mathematical Modeling
### Mass and Energy Balances  
Regarding the modeling of these configurations, the following assumptions were considered:  
1. The temperature of the environment was assumed to be 15 \({}^{\circ}\)C, and the pressure was 101.3 kPa. 2. The pressure loss and heat loss were respectively considered to be 3 and 5%. 3. In compressors and turbines, the thermodynamic process was polytropic. 4. Potential energy and kinetic energy were negligible. 5. According to the market, the efficiency of the combustion chamber was 92%. 6. The terminal pressure loss was presumed to be 3% through the pipe
7. Heat loss of components was assumed around 3% of the energy released by the hottest steam at these components. 8. According to the market, the efficiencies of the RC turbine and pump were considered to be 85%. 9. According to the market, the energy efficiencies of the condenser and evaporator were considered to be 85%. 10. According to the market, the efficiencies of the gas turbine, compressor, and booster compressor were 88%, 87%, and 87%, respectively. For each component of both systems, energy and mass conservation laws were applied, which are presented as follows [30]:  
\[\frac{\mathrm{d}m}{\mathrm{d}t}=\sum_{\mathrm{in}}m-\sum_{\mathrm{out}}m \tag{1}\]  
\[\dot{\mathrm{Q}}-W=\sum_{\mathrm{out}}m\left(\mathrm{h}+\frac{\mathrm{V}^{2}} {2}+\mathrm{g}Z\right)-\sum_{\mathrm{in}}m\left(\mathrm{h}+\frac{\mathrm{V}^{ 2}}{2}+\mathrm{g}Z\right) \tag{2}\]  
Here \(m\) and \(\mathrm{h}\) represent mass flow rate and specific enthalpy. \(\dot{\mathrm{Q}}\) and \(W\) are heat and work transfer rate. \(\mathrm{V}\), \(\mathrm{g}\), and \(Z\) are velocity, gravitational acceleration, and height. For the combustion chamber of the system (a), GT-HRSG, the following equation can be expressed:  
\[m_{2}\mathrm{h}_{2}+m_{11}\mathrm{LHV}=m_{3}\mathrm{h}_{3}+(1-\eta_{\mathrm{ CC}})m_{11}\mathrm{LHV} \tag{3}\]  
In which, LHV denotes the low heating value of natural gas, for which the molar compositions can be found in Table 1. Other symbols and subscripts are mentioned earlier. Figure 1: Layout diagram of the combined: (**a**) gas turbine (GT)-heat recovery steam generator (HRSG) and (**b**) GT-HRSG-heliostat solar receiver (HSR). Using Table 1, the calculations that occur in the combustion chamber can be obtained as follows:  
\[ \begin{split}\frac{1}{r_{a}}C_{x1}H_{y1}+(x_{o_{2}}O_{2}+x_{N_{2}}N_ {2})&\to y_{CO_{2}}CO_{2}+y_{N_{2}}N_{2}+y_{o_{2}}O_{2}+y_{H_{2}O}H_ {2}O\\ r_{a}&=\frac{n_{Air}}{n_{Fuel}}\\ y_{O_{2}}&=x_{O_{2}}-\frac{x_{1}}{r_{a}}-\frac{y_{1 }}{4r_{a}}\\ y_{N_{2}}&=x_{N_{2}}\\ y_{CO_{2}}&=(\ \frac{x_{1}}{r_{a}}+x_{CO_{2}})\\ y_{H_{2}O}&=x_{H_{2}O}+\frac{y_{1}}{2r_{a}}\\ \end{split}  \tag{4}\]  
In Equation (4), \(r_{a}\) denotes the air to fuel ratio, and \(y_{i}\) and \(x_{i}\) are a mole and mass fraction of \(i\). The mass and energy balance equations for components of the GT-HRSG combined system are listed in Table 2.  
In Table 2, \(\eta\) is efficiency, and subscripts T, GT, comp, BC, cond, and p respectively define the turbine, the gas turbine, the compressor, the booster compressor, the condenser, and the pump. For GC stand-alone, the net electricity production can be obtained as follows:  
\[\dot{W}_{GC,net}=\dot{W}_{GT}-\dot{W}_{comp}-\dot{W}_{BC} \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Component** & **Mixture (mole\%)** \\ \hline
1 & CH4 & 81 \\ \hline
2 & C2H6 & 7.9 \\ \hline
3 & C3H8 & 4.2 \\ \hline
4 & C4H10 & 4.7 \\ \hline
5 & CO2 & 1.2 \\ \hline
6 & N2 & 1.0 \\ \hline \end{tabular}
\end{table} 
Table 1: The natural gas molar composition [31].