## 2 Materials and Methods
### Mathematical Formulation  
This subsection provides the formulations utilized for thermal and optical analyses. Firstly, the electrical efficiency in terms of electricity production is affected by the temperature of the PV cells, as it is described in Equation (1) [24; 25]. \[\eta_{el}=\eta_{0}\cdot\left(1-\beta_{c}\cdot\left(T_{pv,m}-T_{0}\right)\right) \tag{1}\]  
Another alternative way for the electrical efficiency calculation is given in Equation (2) [26]. \[\eta_{el}=\frac{\mathrm{Q}_{el}}{G_{pv}\cdot A_{pv}} \tag{2}\]  
In the above equation, \(\eta_{0}\) represents the cell efficiency considering standard testing conditions (STCs) (\(STC\), \(G=1000\frac{W}{m^{2}}\), \(T_{0}=25\)\(\,\mathrm{\SIUnitSymbolCelsius}\)). \(\beta_{c}\) denotes the coefficient of temperature with regards to the photovoltaic (PV) module. A mean value for the PV was taken into consideration. By combining Equations (1) and (2), the electrical production for the cells in the absorber upper side is calculated using Equation (3) [24]. \[Q_{el,up}=A_{pv,up}\cdot G_{T}\cdot\eta_{0}\cdot\left(1-\beta_{c}\cdot\left(T _{pv,up,m}-T_{0}\right)\right) \tag{3}\]  
The available solar irradiance that reaches the absorber down side is calculated using Equation (4) [24]. \[G_{cpc}=G_{T}-\frac{1}{C}\cdot G_{d} \tag{4}\]  
where \(G_{d}\) represents the intensity of diffuse radiation from the sun, \(G_{T}\) denotes the solar radiation that directly falls on the collector aperture perpendicularly and \(C\) represents the ratio of concentration, which indicates the ratio between the glass area and the area of the hybrid absorber [18]. \[C=\frac{A_{g}}{A_{r}} \tag{5}\]  
As a result of the above equations, the electrical production of the cells located on the bottom side of the receiver is defined using Equation (6) [24]. \[Q_{el,down}=A_{pv,down}\cdot G_{cpc}\cdot\left(\frac{\mathrm{A_{pv,tot}}}{ \mathrm{A_{pv,down}}}\mathrm{C}-\frac{\mathrm{A_{pv,up}}}{A_{pv,tot}}\right) \cdot\eta\cdot\left(1-\beta_{c}\cdot\left(T_{pv,down,m}-T_{0}\right)\right) \tag{6}\]  
The effective solar radiation, which is perpendicularly oriented on the collector aperture, can be written using Equation (7) [18]:  
\[G_{eff,T}=G_{b,T}+\frac{G_{d}}{C} \tag{7}\]  
The efficiency in terms of thermal operation is determined using Equation (8) [27]. \[\eta_{th}=\frac{m\cdot Cp\cdot\left(T_{out}-T_{in}\right)}{A_{g}\cdot G_{eff,T}} \tag{8}\]  
where the numerator is the useful heat, which is defined by multiplying the mass flow rate of heat transfer fluid (\(m\)) with the specific heat capacity of the medium (\(Cp\)) and the difference between the outlet and inlet temperature (\(T_{out}-T_{in}\)). The denominator is the result of the surface area of the solar unit and the effective radiation of the sun, which is defined using Equation (7). The sky temperature, as a function of the ambient temperature, is calculated using Equation (9) [28]. \[T_{sky}=0.0552\cdot T_{a}^{1.5} \tag{9}\]Equation (10) is employed to determine the coefficient of the heat transfer between the outer collector surfaces and the surrounding air [25]. \[h_{wind}=\frac{8.6\cdot u^{0.6}}{L^{0.4}} \tag{10}\]  
The parameter \(u\) represents the velocity of the wind in meters per second (m/s), while \(L\) denotes the collector's length measured in meters (m). Reynolds number is calculated using Equation (11) [18]:  
\[\eta_{el}=\frac{\mathrm{Q}_{el}}{G_{pv}\cdot A_{pv}} \tag{2}\]0  
Parameter \(u\) is the mean velocity of the water in the channel, \(v\) is the kinematic viscosity and \(D_{h}\) represents the hydraulic diameter, which is expressed using Equation (12) [29]:  
\[\eta_{el}=\frac{\mathrm{Q}_{el}}{G_{pv}\cdot A_{pv}} \tag{2}\]1  
Parameter \(A\) is the cross-sectional area of the channel and \(P\) is the channel perimeter. The mean fluid velocity is calculated with the following equation:  
\[\eta_{el}=\frac{\mathrm{Q}_{el}}{G_{pv}\cdot A_{pv}} \tag{2}\]2  
\(A_{h}\) is the cross section of the hydraulic diameter.