# Polynomial Expressions for the Thermal Efficiency of the Parabolic Trough Solar Collector
## 3 Results
### Total Analysis
So, the use of the third power term seems to be a very good idea for the proper approximation of the PTC performance. In the equations family with three terms, the best fitting has been found for the following formula:  
\[\eta_{cal}=0.73116-1.2402\cdot 10^{-6}\cdot\frac{\Delta T^{3}}{G_{b}}-5.4012 \cdot 10^{-5}\cdot\Delta T,\ (\text{R2 }=99.92\%\ -\ \text{MAPE }=0.13\%) \tag{33}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Terms** & \(a_{0}\) & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) & \(b\) & \(\text{R2 }(\%)\) & **MAPE** \\ \hline \multirow{4}{*}{2 terms} & 0.74137 & \(-1.5091\cdot 10^{-1}\) & & & & & 83.74 & 1.85\% \\ \cline{2-8}  & 0.73177 & & \(-4.7848\cdot 10^{-4}\) & & & & 97.74 & 0.60\% \\ \cline{2-8}  & **0.72547** & & & \(-\)**1.4155\(\cdot 10^{-6}\)** & & & **99.11** & **0.47\%** \\ \cline{2-8}  & 0.72123 & & & \(-\)**4.0835\(\cdot 10^{-9}\)** & & & 96.64 & 0.93\% \\ \cline{2-8}  & 0.74828 & & & & & \(-\)3.0396\(\cdot 10^{-4}\) & 73.60 & 2.27\% \\ \hline \multirow{4}{*}{3 terms} & 0.72579 & 6.5758\(\cdot 10^{-2}\) & \(-6.6344\cdot 10^{-4}\) & & & 99.03 & 0.43\% \\ \cline{2-8}  & 0.72773 & \(-1.6142\cdot 10^{-2}\) & & \(-\)1.2901\(\cdot 10^{-6}\) & & & 99.29 & 0.38\% \\ \cline{2-8}  & 0.72930 & \(-4.7559\cdot 10^{-2}\) & & \(-\)3.0666\(\cdot 10^{-9}\) & & 98.96 & 0.48\% \\ \cline{2-8}  & 0.74828 & \(-1.0610\cdot 10^{-1}\) & & & & \(-\)1.1801\(\cdot 10^{-4}\) & 87.45 & 1.69\% \\ \cline{2-8}  & 0.72733 & & \(-1.2798\cdot 10^{-4}\) & \(-\)1.0450\(\cdot 10^{-6}\) & & & 99.32 & 0.37\% \\ \cline{2-8}  & 0.72786 & & \(-2.7576\cdot 10^{-4}\) & & \(-\)1.8163\(\cdot 10^{-9}\) & & 99.31 & 0.36\% \\ \cline{2-8}  & 0.73484 & & \(-4.3829\cdot 10^{-4}\) & & & \(-\)3.5114\(\cdot 10^{-5}\) & 98.03 & 0.66\% \\ \cline{2-8}  & 0.72672 & & & \(-\)1.8891\(\cdot 10^{-6}\) & 1.3942\(\cdot 10^{-9}\) & & 99.28 & 0.39\% \\ \cline{2-8}  & **0.73116** & & & \(-\)**1.2402\(\cdot 10^{-6}\)** & & & **-5.4012\(\cdot 10^{-5}\)** & **99.92** & **0.13\%** \\ \cline{2-8}  & 0.73131 & & & & \(-\)3.3020\(\cdot 10^{-9}\) & \(-\)8.6225\(\cdot 10^{-5}\) & 99.02 & 0.42\% \\ \hline \multirow{4}{*}{4 terms} & 0.72722 & 3.1334\(\cdot 10^{-3}\) & \(-\)1.5039\(\cdot 10^{-4}\) & \(-\)1.0045\(\cdot 10^{-6}\) & & & 99.32 & 0.37\% \\ \cline{2-8}  & 0.72715 & 1.5569\(\cdot 10^{-2}\) & \(-\)3.5634\(\cdot 10^{-4}\) & & \(-\)1.4867\(\cdot 10^{-9}\) & & 99.33 & 0.35\% \\ \cline{2-8}  & 0.72912 & 6.8755\(\cdot 10^{-2}\) & \(-\)6.2450\(\cdot 10^{-4}\) & & & \(-\)4.1393\(\cdot 10^{-5}\) & 99.44 & 0.37\% \\ \cline{2-8}  & 0.72747 & \(-\)1.0444\(\cdot 10^{-2}\) & & \(-\)1.5393\(\cdot 10^{-6}\) & 6.0320\(\cdot 10^{-10}\) & & 99.30 & 0.38\  
the aforementioned comparison, Figure 7 has been added in order to directly compare the model (\(a_{0}\), \(a_{3}\), \(b\)) with the models (\(a_{0}\), \(a_{1}\), \(a_{2}\)), (\(a_{0}\), \(a_{1}\), \(a_{4}\)) and (\(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{4}\)). In this figure, the results from the ESS for \(G_{b}=1000\) W/m\({}^{2}\) are plotted with the found curves for the three models. It can be said that the model (\(a_{0}\), \(a_{3}\), \(b\)) is close to the found points while the other models present some deviations especially in low and high values of (\(\Delta T\)). In the future, there is a need for performing similar detailed and systematic approximating studies by taking into account the wind speed (especially for bare tube cases), the solar angle (through the incident angle modifier) and the flow rate. Moreover, studies about different PTCs with a non-evacuated tube, non-selective absorber, different concentration ratios, different working fluid and different materials can also be studied.