## 2 Mathematical Modeling
### Mass and Energy Balances
\begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Component** & **Mass Balance** & **Energy Balance** \\ \hline \multicolumn{4}{c}{**Gas cycle**} \\ \hline
1 & Combustion chamber & \(m_{2}+m_{11}=m_{3}\) & \(m_{2}h_{2}+m_{11}h_{11}=m_{3}h_{3}\) \\ \hline
2 & Gas turbine & \(m_{3}=m_{4}\) & \(\dot{W}_{GT}=\dot{m}_{3}(h_{3}-h_{4}))_{GT}\) \\ \hline
3 & Compressor 1 & \(m_{1}=m_{2}\) & \(\dot{W}_{comp}=\frac{\dot{m}_{3}(h_{3}-h_{1})}{\dot{m}_{comp}}\) \\ \hline
4 & Booster compressor & \(m_{10}=m_{11}\) & \(\dot{W}_{BC}=\frac{\dot{m}_{3}(h_{3}-h_{10})}{\dot{m}_{BC}}\) \\ \hline \multicolumn{4}{c}{**Rankine cycle**} \\ \hline
5 & HRSG & \(\dot{m}_{6}=\dot{m}_{7}\) ; \(\dot{m}_{4}=\dot{m}_{5}\) & \(\dot{m}_{4}(h_{4}-h_{5})=\dot{m}_{6}(h_{7}-h_{6})\) \\ \hline
6 & Turbine & \(m_{7}=m_{8}\) & \(\dot{W}_{T}=\dot{m}_{7}(h_{7}-h_{8})\eta_{T}\) \\ \hline
7 & Condenser & \(m_{9}=m_{8}\) & \(\dot{Q}_{cond}=m_{8}(h_{9}-h_{8})\) \\ \hline
8 & Pump & \(\dot{m}_{9}=\dot{m}_{6}\) & \(\dot{W}_{p}=\frac{\dot{m}_{3}(h_{9}-h_{9})}{\dot{m}_{P}}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Equations of mass and energy balances for GT-HRSG combined system [31; 32]. In which, subscript GT is the gas turbine, and \(\dot{\mathrm{W}}_{comp}\) and \(\dot{\mathrm{W}}_{BC}\) define the power requirement of the compressor and the booster compressor. For the combined cycle, the net power production is achieved as follows:  
\[\dot{\mathrm{W}}_{CC,net}=\dot{\mathrm{W}}_{GT}+\dot{\mathrm{W}}_{T}-\dot{ \mathrm{W}}_{comp}-\dot{\mathrm{W}}_{BC}-\dot{\mathrm{W}}_{P} \tag{6}\]  
For system (b), GT-HRSG-HSR, the energy, and mass balance equations of the gas and Rankine cycles are similar to Table 2 presented for the GT-HRSG system except for the change in streams' number. The mass and energy balances for the HSR system are presented in Table 3.  
First and foremost, for solar radiation modeling, solar time can be achieved from the following equation [36]:  
\[\textit{Solar time}=\textit{Standard time}+E-4(L_{st}-L_{loc}) \tag{7}\]  
In which, \(L_{st}\) and \(L_{loc}\) are respectively local zone time standard meridian, and location longitude. Moreover, the parameter E from equation above is given as [36]:  
\[E=229.2(0.000075+0.001868\textit{Cos}\beta-0.04089\textit{Sin}2\beta-0.014615 \textit{Cos}-0.032077\textit{Sin}\beta) \tag{8}\]  
\[\beta=\frac{(n-1)360}{365} \tag{9}\]  
Here, \(n\) is equal to 1 on the first of January. Sunset hour angle is another key parameter that can be formulated as follows [36]:  
\[\omega=\arccos(tan(\varphi)tan(\delta)) \tag{10}\]  
Here, \(\varphi\) and \(\delta\) denote latitude and deflection angles, respectively. The deflection angle is calculated using the following equation [37]:  
\[\delta=23.45\textit{sin}\left(\frac{360(284+n)}{365}\right). \tag{11}\]  
In which, n is equal to 1 on the first of January. The solar direct beam irradiation can be obtained as follows [37]:  
\[G_{b}=\textit{Acos}(\theta_{z})\textit{exp}\left(\frac{-B}{\textit{cos}( \theta_{z})}\right) \tag{12}\]  
Here, A and B are constants that can be found in Reference [37], and \(\theta_{z}\) represents a solar zenith angle. Detailed information can be found in Reference [37; 38]. For the heliostat field solar receiver, the following equation is formulated for calculation of total heat absorbed by the solar receiver [39]:  
\[\dot{\mathrm{Q}}_{net}=\dot{\mathrm{Q}}_{S}-\dot{\mathrm{Q}}_{loss} \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Component** & **Mass Balance** & **Energy Balance** \\ \hline \multicolumn{4}{c}{**HSR Cycle**} \\ \hline
1 & Receiver & \(m_{2}=m_{3}\) & \(\dot{\mathrm{m}}_{1}(h_{3}-h_{2})=\dot{\mathrm{Q}}_{S}\) \\ \hline
2 & Pump & \(\dot{\mathrm{m}}_{1}=\dot{\mathrm{m}}_{2}\) & \(\dot{\mathrm{W}}_{P}=\frac{\dot{\mathrm{m}}_{1}(h_{3}-h_{1})}{\dot{\mathrm{q}}_ {P}}\) \\ \hline
3 & Storage Tank & \(m_{3}+m_{5}=m_{1}\;+m_{6}\) & \(\dot{\mathrm{m}}_{1}(h_{3}-h_{1})=\dot{\mathrm{m}}_{6}(h_{6}-h_{5})\) \\ \hline \hline \end{tabular}
\end{table}