# Polynomial Expressions for the Thermal Efficiency of the Parabolic Trough Solar Collector
## 3 Results
### Total Analysis  
The next stage of the present analysis is the investigation of more detailed approximation models with more terms. It is very interesting to see the approximation statistics with different linear combinations of the terms \"\(a_{m}\cdot\Delta T^{m}/G_{b}\)\" with and without the term \"\(\Delta T\)\". Figure 5 shows the results of the different linear combinations without the \"\(\Delta T\)\", while Figure 6 shows results with the term \"\(\Delta T\)\". Figure 5: Evaluation of the thermal efficiency equations of the form \({}^{\prime\prime}a_{0}+\sum(a_{m^{\prime}}\Delta T^{\prime\prime}/G_{b})^{\prime\prime}\) (**a**) [\(a_{0}\), \(a_{1}\)] (b) [\(a_{0}\), \(a_{1}\), \(a_{2}\)] (c) [\(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\)] (d) [\(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\)] (d) [\(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\)]  
Figure 6: Evaluation of the thermal efficiency equations of the form \({}^{\prime\prime}a_{0}+\sum(a_{m^{\prime}}\Delta T^{\prime\prime}/G_{b})+b \cdot\Delta T^{\prime\prime}\) (**a**) [\(a_{0}\), \(a_{1}\), \(b\)] (**b**) [\(a_{0}\), \(a_{1}\), \(a_{2}\), \(b\)] (**c**) [\(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(b\)] (**d**) [\(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(b\)]  
The general formula that is used in Figure 5 results can be written as \"\(\eta_{col}=a_{0}+\sum(a_{m}.\Delta T^{m}/G_{b})\)\". Figure 5 indicates that there is a need for using at least three terms in order for the points to be close to the red line and so to have a relatively accurate approximation. Also, the use of more terms leads to higher R\({}^{2}\) and to lower MAPE, something that leads to better fitting. However, the difference between the cases with and without the term \"\(\Delta T^{4}/G_{b}\)\" is extremely low (R\({}^{2}\) = 99.35% with this term and R\({}^{2}\) = 99.32% without this term). This result indicates that there is a need for using the third-order term, while the fourth-order term has a very small impact on expression accuracy. The next step is the implementation of the extra term \"\(\Delta T\)\" in the previous expressions and so the general formula is \"\(\eta_{col}=a_{0}+\sum(a_{m}.\Delta T^{m}/G_{b})+b\cdot\Delta T\)\". Figure 6 indicates that the use of this term leads to very accurate models for all the cases except the case with the first-order term \"\(\Delta T/G_{b}\)\", where the R\({}^{2}\) is only 87.45%. The cases with the order with m = 2, m = 3 and m = 4 lead to R\({}^{2}\) at 99.44%, 99.94% and 99.94%, respectively. The MAPE is found to be found 1.69%, 0.37%, 0.12% and 0.11% for the cases with m = 1, m = 2, m = 3 and m = 4, respectively. Practically, the use of the fourth power term is useless because it leads to approximately the same results with the third term results. To conclude about the results of Figures 5 and 6, the use of the third-order term is very important and also the use of the \"\(\Delta T\)\" is a very useful weapon for the proper prediction of the PTC thermal efficiency. The next step in this analysis is to present all the results about the coefficients of approximation and the evaluation indexes about the 31 examined cases. Table 3 includes all these data and this table summarizes all the found results of this work. In this table, the 31 different approximations are included. The evaluation indexes (R\({}^{2}\) and MAPE), as well as the values of the coefficients (\(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{4}\), \(b\)) are given. In the cases with empty cells, there is no utilization of the respective term and so the empty cells indicate that the respective value is zero. In the left column of Table 3, the number terms of the used formulas are given and the equations are classified in categories with 2 to 6 terms. Among the equations with two terms, the best formula is the following:  
\[\eta_{cal}=0.72547-1.4155\cdot 10^{-6}\cdot\frac{\Delta T^{3}}{G_{b}},\ (\text{R2 }=99.11\%\ -\ \text{MAPE }=0.47\%) \tag{32}\]  
Equation (29) is a very simple formula that leads to relatively accurate results and it is a promising choice.