# Polynomial Expressions for the Thermal Efficiency of the Parabolic Trough Solar Collector
## 2 Material and Methods
### Basic Mathematical Formulation  
The present section includes the main equations which have been used in the developed model in EES for the thermal efficiency determination of the PTC. The collector thermal efficiency (\(\eta_{cal}\)) is the ratio of useful heat production (\(Q_{u}\)) to the solar energy on the collector aperture (\(Q_{s}\)):  
\[\eta_{cal}=\frac{Q_{u}}{Q_{s}} \tag{14}\]  
The solar irradiation in the collector aperture (\(Q_{s}\)) can be written as:  
\[Q_{s}=A_{a}\cdot G_{b} \tag{15}\]  
The useful heat production (\(Q_{u}\)) is calculated by the energy balance in the working fluid:  
\[Q_{u}=m_{oil}\cdot c_{p}\cdot(T_{out}-T_{in}) \tag{16}\]  
The inlet temperature (\(T_{in}\)), the outlet temperature (\(T_{out}\)), the oil mass flow rate (\(m_{oil}\)) and the oil specific heat capacity (\(c_{p}\)) are used in the previous formula. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbols** & **Values** \\ \hline PTC width & (\(W_{a}\)) & 5.0 m \\ PTC length & (\(L\)) & 7.8 m \\ Focal distance & (\(F\)) & 1.84 m \\ Collecting area & (\(A_{a}\)) & 39 m\({}^{2}\) \\ Concentration ratio & (C) & 22.7 \\ Inner diameter of the receiver & (\(D_{ri}\)) & 66 mm \\ Outer diameter of the receiver & (\(D_{ro}\)) & 70 mm \\ Inner diameter of the cover & (\(D_{ci}\)) & 109 mm \\ Outer diameter of the cover & (\(D_{ro}\)) & 115 mm \\ Cover emittance & (\(c_{c}\)) & 86\% \\ Solar angle & (\(\theta\)) & 0\({}^{\circ}\) \\ Ambient temperature & (\(T_{am}\)) & 25 \({}^{\circ}\)C \\ Volumetric flow rate & (\(V\)) & 100 L/min \\ Wind speed & (\(V_{w}\)) & 1 m/s \\ Maximum optical efficiency & (\(\eta_{part,max}\)) & 75.3\% \\ Transmittance & (\(\tau\)) & 95\% \\ Absorbance & (\(\alpha\)) & 96\% \\ Equivalent reflectance & (\(p\)) & 83\% \\ Intercept factor & (\(\gamma\)) & 99.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data of the studied parabolic trough solar collector (PTC) module [31; 32]. Moreover, the useful heat production (\(Q_{u}\)) can be expressed as the heat that the absorber transfers to the working fluid:  
\[Q_{u}=A_{ri}\cdot h\cdot\left(T_{r}-T_{fm}\right) \tag{17}\]  
The mean receiver temperature (\(T_{r}\)), the mean fluid temperature (\(T_{fm}\)), the internal tube area (\(A_{ri}\)) and the convection heat transfer coefficient between fluid and absorber (\(h\)) are used in the previous formula. The energy balance on the absorber is an important step in the present methodology and it is given below:  
\[Q_{abs}=Q_{u}+Q_{loss} \tag{18}\]  
The absorbed solar energy (\(Q_{abs}\)) can be found below:  
\[Q_{abs}=\eta_{opt}\cdot Q_{s} \tag{19}\]  
The thermal losses (\(Q_{loss}\)) of the absorber to the cover can be written as below [33]:  
\[Q_{loss}=A_{ro}\cdot\sigma\cdot\frac{T_{r}^{4}-T_{c}^{4}}{\frac{1-\varepsilon _{r}}{\varepsilon_{c}}\cdot\frac{A_{ri}}{A_{\omega}}} \tag{20}\]  
At this point, it has to be said that the convention thermal losses between absorber and cover are neglected due to the vacuum between them. Moreover, it is essential to state that the thermal losses from the absorber to the cover are the same as the losses from the cover to the ambient because the system is studied in steady-state conditions [33]. So, it can be said that:  
\[Q_{loss}=A_{co}\cdot\sigma\cdot\varepsilon_{c}\cdot\left(T_{c}^{4}-T_{sky}^{ 4}\right)+A_{co}\cdot h_{out}\cdot\left(T_{c}-T_{am}\right) \tag{21}\]  
The sky temperature (\(T_{sky}\)) can be estimated as below by using the temperature levels in kelvin units [34]:  
\[T_{sky}=0.0552\cdot T_{am}^{1.5} \tag{22}\]  
The heat transfer coefficient between cover and ambient (\(h_{out}\)) is found as below [35]:  
\[h_{out}=4\cdot V_{w}^{0.58}\cdot D_{co}^{-0.48} \tag{23}\]  
More specifically, the wind speed (\(V_{w}\)) is selected at 1 m/s, which is a typical value, and the (\(h_{out}\)) is found to be around 10 W/m\({}^{2}\)K [36].