# Polynomial Expressions for the Thermal Efficiency of the Parabolic Trough Solar Collector
## 2 Material and Methods
### Followed Methodology for the Approximation Methods  
In this work, the developed model in the Engineering Equation Solver (EES) [30] was applied in order to create a dataset for comparing the different formulas for the PTC thermal efficiency. EES is a flexible program for solving quickly and with accuracy numerous non-linear equations with numerical techniques [30]. It also includes libraries for using the thermal properties of different working fluids. Moreover, this program gives the possibility of conducting optimization studies for the maximization or the minimization of an objective function. In this work, the solar irradiation (\(G_{b}\)) was examined from 300 W/m\({}^{2}\) up to 1000 W/m\({}^{2}\) with a step of 50 W/m\({}^{2}\), while the inlet fluid temperature (\(T_{in}\)) from 25 \({}^{\circ}\)C up to 375 \({}^{\circ}\)C with a step of 50 \({}^{\circ}\)C. In this work, the temperature difference (\(\Delta T=T_{in}-T_{am}\)) is important parameters which are examined from 0 \({}^{\circ}\)C up to 350 \({}^{\circ}\)C with a step of 50 \({}^{\circ}\)C, which is an equivalent parameter with the (\(T_{in}\)). The working fluid is Sylthern 800 which can operate up to 400 \({}^{\circ}\)C [37], something that has been taken into consideration in this work. The thermal properties of this fluid have been taken from the literature [37] and they are dependent on the fluid temperature level. Totally, 120 set points have been found and they are given in Appendix A. About the solving procedures, for every case, the inputs in the program are the following: inlet temperature (\(T_{in}\)), oil flow volumetric rate (\(V\)), solar beam irradiation (\(G_{b}\)), ambient temperature (\(T_{am}\)), and the wind speed (\(V_{w}\)). Moreover, in this work, there is a zero-incident angle and so the optical \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \(G_{b}\) & \(V_{w}\) & \(T_{am}\) & \(T_{in}\) & \(V\) & \multicolumn{4}{c}{**Outlet Temperature**} & \multicolumn{4}{c}{**Collector Thermal Efficiency**} \\  & **(W/m\({}^{2}\))** & **(m/s)** & **(\({}^{\circ}\)C)** & **(\({}^{\circ}\)C)** & **(L/min)** & **Ref [18]** & **Model** & **Deviation** & **Ref [18]** & **Model** & **Deviation** \\ \hline
**1** & 933.7 & 2.6 & 21.2 & 102.2 & 47.7 & 124.0 & 124.1 & 0.58\% & 0.7251 & 0.7209 & 1.38\% \\
**2** & 968.2 & 3.7 & 22.4 & 151.0 & 47.8 & 173.3 & 173.6 & 0.89\% & 0.7090 & 0.7153 & 2.29\% \\
**3** & 982.3 & 2.5 & 24.3 & 197.5 & 49.1 & 219.5 & 219.9 & 0.88\% & 0.7017 & 0.7079 & 1.83\% \\
**4** & 909.5 & 3.3 & 26.3 & 250.7 & 54.7 & 269.4 & 269.4 & 0.95\% & 0.7025 & 0.6958 & 0.76\% \\
**5** & 937.9 & 1.0 & 26.2 & 297.8 & 55.5 & 316.4 & 317.0 & 0.01\% & 0.6798 & 0.6799 & 0.85\% \\
**6** & 880.6 & 2.9 & 28.8 & 299.0 & 55.6 & 317.2 & 316.9 & 1.70\% & 0.6892 & 0.6775 & 2.66\% \\
**7** & 903.2 & 4.2 & 27.5 & 355.9 & 56.3 & 374.0 & 374.3 & 3.88\% & 0.6234 & 0.6476 & 1.18\% \\
**8** & 920.9 & 2.6 & 31.1 & 379.5 & 56.8 & 398.0 & 398.3 & 0.86\% & 0.6382 & 0.6327 & 2.12\% \\
**Mean** & - & - & - & - & - & - & - & 1.22\% & - & - & 1.68\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Validation of the present model with experimental data from the Reference [18]. efficiency has its maximum value according to Table 1. The program uses these inputs and calculates the outputs which are mainly the oil outlet temperature, the useful heat production, the thermal losses and the collector efficiency. The solution procedure is performed by combing the Equations (13) to (29) all together and by solving this system as a matrix with all the proper unknown parameters. Practically, it is a system with 17 equations and the following 17 unknown parameters: (\(\eta_{col}\)), (\(Q_{lu}\)), (\(Q_{s}\)), (\(Q_{abs}\)), (\(Q_{loss}\)), (\(T_{out}\)), (\(T_{fm}\)), (\(T_{r}\)), (\(T_{c}\)), (\(T_{sky}\)), (\(h\)), (\(h_{out}\)), (\(Nu\)), (\(Re\)), (\(Pr\)), (\(m\)), (\(\varepsilon_{r}\)). The developed model in EES is solved iteratively in order for all these equations to be converged in the final values.