## 3 Methodology
### System Analysis of Energy and Exergy
It takes into account the heat input that is brought into the system and used by the system to produce work [39; 40; 41]:  
\[\dot{\mathrm{Q}}+\dot{\mathrm{W}}=\sum\dot{\mathrm{m}}_{\mathrm{out}}\mathrm{ h}_{\mathrm{out}}-\sum\dot{\mathrm{m}}_{\mathrm{in}}\mathrm{h}_{\mathrm{in}} \tag{4}\]  
The system's overall exergy balance can be expressed as follows [41; 42]:  
\[\sum\Bigl{(}\dot{\mathrm{m}}\mathrm{e}\mathrm{x}\Bigr{)}_{\mathrm{in}}+\dot{ \mathrm{E}}_{\mathrm{in},\mathrm{W}}+\dot{\mathrm{E}}_{\mathrm{in},\mathrm{Q} }-\dot{\mathrm{E}}_{\mathrm{D}}=\sum(\mathrm{m}\mathrm{e}\mathrm{x})_{\mathrm{ out}}+\dot{\mathrm{E}}_{\mathrm{out},\mathrm{W}}+\dot{\mathrm{E}}_{\mathrm{out}, \mathrm{Q}} \tag{5}\]  
\[\dot{\mathrm{E}}_{\mathrm{Q}}=\dot{\mathrm{Q}}\biggl{(}1-\frac{\mathrm{T}_{0} }{\mathrm{T}_{\mathrm{s}}}\biggr{)} \tag{6}\]  
\[\dot{\mathrm{m}}_{\mathrm{in}}\;\mathrm{S}_{\mathrm{in}}\;+\left(\frac{\dot{ \mathrm{Q}}}{\mathrm{T}}\right)+\dot{\mathrm{S}}_{\mathrm{gen}}\;=\dot{ \mathrm{m}}_{\mathrm{out}}\;\mathrm{S}_{\mathrm{out}} \tag{7}\]  
The parabolic solar collector's energy equations are shown below [42; 43]. These formulas take into consideration two types of energy (thermal energy): input of solar energy \(\dot{\mathrm{Q}}_{\mathrm{Solar}}\) derived from the total solar radiation and useful solar energy \(\dot{\mathrm{Q}}_{\mathrm{u}}\) expressed as the heat transferred to the system. These energies are computed using the following equations:  
\[\dot{\mathrm{Q}}_{\mathrm{Solar}}\;=\mathrm{A}_{\mathrm{A}}\cdot\mathrm{F}_{ \mathrm{R}}\cdot\mathrm{S}\cdot\mathrm{n}_{\mathrm{cp}}.\mathrm{n}_{\mathrm{ cs}} \tag{8}\]  
\[\dot{\mathrm{Q}}_{\mathrm{u}}=\mathrm{n}_{\mathrm{cp}}\mathrm{n}_{\mathrm{cs} }\mathrm{F}_{\mathrm{R}}[\mathrm{S}\mathrm{A}_{\mathrm{a}}-\mathrm{A}_{ \mathrm{r}}\mathrm{U}_{\mathrm{L}}(\mathrm{T}_{\mathrm{ri}}-\mathrm{T}_{ \mathrm{o}})] \tag{9}\]  
In these equations, \(\mathrm{n}_{\mathrm{cp}}\), \(\mathrm{n}_{\mathrm{cs}}\), \(\mathrm{A}_{\mathrm{a}}\), and \(\mathrm{A}_{\mathrm{r}}\) denote the collector's both parallel and series numbers, as well as the parabolic solar collector's aperture and receiving areas, respectively. \(\mathrm{F}_{\mathrm{R}}\), \(\mathrm{S}\), and \(\mathrm{U}_{\mathrm{L}}\) symbolize the heat gain factor of the collector, the absorbed solar radiation amount, and the collector's heat loss coefficient, respectively. They are calculated using the following equations:  
\[\mathrm{F}_{\mathrm{R}}=\frac{\dot{\mathrm{m}}_{\mathrm{c}}\mathrm{C}_{ \mathrm{p,c}}}{\mathrm{A}_{\mathrm{r}}\mathrm{U}_{\mathrm{L}}}\biggl{[}1- \exp\left(-\frac{\mathrm{U}_{\mathrm{L}}\mathrm{F}^{\prime}\mathrm{A}_{ \mathrm{r}}}{\dot{\mathrm{m}}_{\mathrm{c}}\mathrm{C}_{\mathrm{p,c}}}\right) \biggr{]} \tag{10}\]  
\[\sum\dot{\mathrm{E}}_{\mathrm{in}}=\sum\dot{\mathrm{E}}_{\mathrm{out}} \tag{2}\]0  
\[\sum\dot{\mathrm{E}}_{\mathrm{in}}=\sum\dot{\mathrm{E}}_{\mathrm{out}} \tag{2}\]1  
\[\sum\dot{\mathrm{E}}_{\mathrm{in}}=\sum\dot{\mathrm{E}}_{\mathrm{out}} \tag{2}\]2  
\[\sum\dot{\mathrm{E}}_{\mathrm{in}}=\sum\dot{\mathrm{E}}_{\mathrm{out}} \tag{2}\]3  
The collector modeling incorporates various parameters such as \(\gamma\) (reflection coefficient), \(\tau\) (transmission coefficient of the glass cover), \(\alpha\) (receiver's absorption coefficient), and \(\mathrm{K}\) (incidence angle modifier). These parameters are detailed in Table 2, which provides the input data for the collector. Though solar radiation energy entering the system is computed using the above-mentioned formulas, The following formula can be used to calculate the total solar collecting area [42]:  
\[\sum\dot{\mathrm{E}}_{\mathrm{in}}=\sum\dot{\mathrm{E}}_{\mathrm{out}} \tag{2}\]4  
The exergy destruction \(\mathrm{Ex_{dest}}\) of all components will be quantified utilizing the exergy equilibrium calculation [44].