## 2 Mathematical Modeling
### Exergy Balances  
Exergy analysis as a potent tool provides information about inefficiencies in a system that can lead to performance enhancement. Generally, exergy includes four parts; physical, chemical, potential, and kinetic. Specific exergy can be formulated as follows [50]:  
\[ex=(h-h_{0})-T_{0}(s-s_{0})+\frac{V^{2}}{2}+gz+\sum x_{i}ex_{che}+T_{0}\sum x_{ i}R_{i}\;lny_{i} \tag{27}\]  
Here, h, T, and s are specific enthalpy, temperature, and specific entropy. \(x_{i}\) and \(y_{i}\) are mass and molar fraction. \(ex_{che}\) defines the chemical exergy, and \(R\) denotes the gas constant. Subscript 0 refers to ambient conditions. Other parameters are described earlier. As mentioned, the potential and kinetic exergies were ignored in this study, and only the physical and chemical exergy was considered. The equations of exergy balance and exergy destruction rate for the GT-HRSG integrated system's components are listed in Table 5.  
In Table 6, symbols and subscripts are similar to Table 5, which was earlier presented. Similar to the exergy balance of the GT-HRSG combined cycle, for Rankine and gas cycles of system (b), GT-HRSG-HSR exergy efficiency equations and exergy destruction rates can be formulated. For the HSR cycle, Table 6 can be used for exergy efficiency and exergy destruction rate equations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Component** & **Exergy Efficiency** & **Exergy Destruction rate** \\ \hline \multicolumn{4}{c}{**Gas cycle**} \\ \hline
1 & Combustion Chamber & \(\frac{\dot{m}_{0}ex_{3}}{\dot{m}_{0}ex_{2}+m_{11}ex_{11}}\) & \(\dot{m}_{2}ex_{2}+m_{11}ex_{11}-\dot{m}_{3}ex_{3}\) \\ \hline
2 & Gas Turbine & \(\frac{\dot{W}_{GT}}{\dot{m}_{0}ex_{3}-\dot{m}_{0}ex_{4}}\) & \(\dot{m}_{3}ex_{3}-\dot{m}_{4}ex_{4}-\dot{W}_{GT}\) \\ \hline
3 & Compressor & \(\frac{\dot{m}_{2}(ex_{2}-ex_{1})}{\dot{W}_{\rm{comp}}}\) & \(\dot{m}_{1}ex_{1}+\dot{W}_{comp}-\dot{m}_{2}ex_{2}\) \\ \hline
4 & Booster Compressor & \(\frac{\dot{m}_{10}(ex_{11}-ex_{10})}{\dot{W}_{\rm{BC}}}\) & \(\dot{m}_{10}ex_{10}+\dot{W}_{BC}-\dot{m}_{11}ex_{11}\) \\ \hline \multicolumn{4}{c}{**Rankine cycle**} \\ \hline
5 & HRSG & \(\frac{\dot{m}_{6}(ex_{7}-ex_{6})}{\dot{m}_{4}(ex_{4}-ex_{3})}\) & \(\dot{m}_{4}(ex_{4}-ex_{5})-\dot{m}_{6}(ex_{7}-ex_{6})\) \\ \hline
6 & Turbine & \(\frac{\dot{W}_{T}}{\dot{m}_{7}ex_{7}-\dot{m}_{6}ex_{8}}\) & \(\dot{m}_{7}ex_{7}-\dot{m}_{8}ex_{8}-\dot{W}_{T}\) \\ \hline
7 & Condenser & \(\frac{\dot{Q}_{\rm{comp}}\Big{(}1-\frac{T_{0}}{\tau_{\rm{comb}}}\Big{)}}{\dot{m} _{8}(ex_{8}-ex_{9})-\dot{Q}_{\rm{comb}}\Big{(}1-\frac{T_{0}}{\tau_{\rm{comb}} }\Big{)}}\) \\ \hline
8 & Pump & \(\frac{\dot{m}_{8}(ex_{6}-ex_{9})}{\dot{W}_{p}}\) & \(\dot{m}_{9}(ex_{6}-ex_{9})-\dot{W}_{p}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Equations of exergy balance and exergy destruction rate for the GT-HRSG integrated system [31; 32].  
In Table 7, \(\dot{E}x_{Q\text{s}}\) defines the exergy rate of solar energy input given as follows [30; 51; 52]:  
\[\dot{E}x_{Q\text{s}}=G_{b}A_{ap}\bigg{[}1-\frac{4}{3}\bigg{(}\frac{T_{0}}{T_{Sun} }\bigg{)}+\frac{4}{3}\bigg{(}\frac{T_{0}}{T_{Sun}}\bigg{)}^{4}\bigg{]} \tag{28}\]  
In which the \(T_{0}\) and \(T_{Sun}\) are the temperature of the environment and the sun.  
The exergy efficiencies of GT and the combined cycle can be expressed as follows:  
\[\eta_{xergy,\ GT}=\frac{\dot{\text{W}}_{net,GT}}{m_{10}ex_{10}} \tag{29}\]  
\[\eta_{xergy,\ CC}=\frac{\dot{\text{W}}_{net,CC}}{m_{10}ex_{10}} \tag{30}\]  
The above-mentioned equations can be considered in whether to use the HSR cycle or not. The solar radiation was considered in the analysis. The following equation can be proposed for energy efficiency of the GT-HRSG-HSR system:  
\[\eta_{xergy,\ HSR-GT-HRSG}=\frac{\dot{\text{W}}_{net,CC}}{m_{10}ex_{10}+G_{b} A_{ap}\bigg{[}1-\frac{4}{3}\bigg{(}\frac{T_{amb}}{T_{Sun}}\bigg{)}+\frac{4}{3} \bigg{(}\frac{T_{amb}}{T_{Sun}}\bigg{)}^{4}\bigg{]}} \tag{31}\]