# Polynomial Expressions for the Thermal Efficiency of the Parabolic Trough Solar Collector
## 1 Introduction
\[\eta_{col}=a_{0}+a_{1}\cdot\frac{(T_{in}-T_{am})}{G_{b}}+a_{2}\cdot\frac{(T_{ in}-T_{am})^{2}}{G_{b}}+a_{3}\cdot\frac{(T_{in}-T_{am})^{3}}{G_{b}} \tag{4}\]  
They also tried to approximate the results with an alternative formula, as below:  
\[\eta_{col}=a_{0}+a_{1}\cdot\frac{(T_{in}-T_{am})}{G_{b}^{0,33}} \tag{5}\]  
A complex polynomial expression has been suggested by [PERSON] and [PERSON] [20] which incorporates the wind speed (\(V_{w}\)) as an extra parameter which includes a third-order term (\(G_{b}\cdot T_{in}\cdot V_{w}\)), as below:  
\[\eta_{col}=c_{0}+c_{1}\cdot G_{b}+c_{2}\cdot G_{b}^{2}+c_{3}\cdot T_{in}+c_{4 }\cdot T_{in}^{2}+c_{5}\cdot V_{w}+c_{6}\cdot V_{w}^{2}+c_{7}\cdot G_{b}\cdot T _{in}+c_{8} \tag{6}\]  
The next part of the literature studies includes fourth-order terms. These formulas try to take into account the radiation thermal losses in a proper way because they are dependent on the fourth power of the absorber temperature level. [PERSON] et al. [21] tried to approximate the thermal losses with an expression with a fourth power and then the following expression for the thermal efficiency has been found:  
\[\eta_{col}=c_{0}+\frac{c_{1}\cdot\left(T_{in}-T_{am}\right)+c_{2}\cdot\left(T_{ in}-T_{am}\right)^{4}}{G_{b}\cdot\cos(\theta)} \tag{7}\]  
They stated that this expression is in accordance with the standard IEC 62727 [22]. The use of the \"\(\cos\) (\(\theta\))\" in the denominator is justified due to the approximation of the thermal losses for specific heat input. Practically, the thermal losses have to be normalized in every case in the absorbed irradiation, and the use of the cosine is a good choice for the approximation of the incident angle modifier. The thermal efficiency was calculated by reducing the thermal losses from the absorbed heat in every case. [PERSON] et al. [23] have used a similar formula but they incorporated the wind speed inside the efficiency expression:  
\[\eta_{col}=a_{0}+\left(a_{1}+c_{1}\cdot V_{w}\right)\cdot\frac{\left(T_{in}-T _{am}\right)}{G_{b}}+a_{4}\cdot\frac{\left(T_{in}-T_{am}\right)^{4}}{G_{b}} \tag{8}\]  
One alternative formula has been suggested also by [PERSON] et al. [24] which includes a fourth-order term and an order which takes into account the transient phenomena. The authors have expressed this formula in terms of \"\(Q_{ul}/A_{a}\)\", but below, it is presented in terms of efficiency for having a uniform presentation style in the paper:  
\[\eta_{col}=a_{0}+a_{4}\cdot\frac{\left(T_{in}-T_{am}\right)^{4}}{G_{b}}+c_{1} \cdot\frac{dT_{in}}{dt}\cdot G_{b}^{-1} \tag{9}\]  
In another work, [PERSON] et al. [25] used a formula with an extra term, as below:  
\[\eta_{col}=a+a_{1}\cdot\frac{\left(T_{in}-T_{am}\right)}{G_{b}}+a_{4}\cdot \frac{\left(T_{in}-T_{am}\right)^{4}}{G_{b}}+c_{1}\cdot\frac{dT_{in}}{dt}\cdot G _{b}^{-1} \tag{10}\]  
The previous formula is associated with the IEC 62862-3-2 standard which also suggests the use of a second-order term instead of the fourth-order term. More specifically, the previous formula has been found by eliminating terms from the general formula according to DIS/ISO 9806 (2017) [26], which indicates approximately the following:  
\[ \begin{array}{c}\frac{Q_{R}}{A_{a}}=c_{0,b}\cdot G_{b}+c_{0,d}\cdot G_{d}+c _{1}\cdot\left(T_{in}-T_{am}\right)+c_{2}\cdot\left(T_{in}-T_{am}\right)^{2} +c_{3}\cdot V_{w}\cdot\left(T_{in}-T_{am}\right)\\ +c_{4}\cdot\left(E_{L}-\sigma\cdot T_{in}^{4}\right)+c_{5}\cdot\frac{dT_{in} }{dt}+c_{6}\cdot V_{w}\cdot\left(G_{b}+G_{d}\right)+c_{7}\cdot V_{w}\\ \left\{E_{L}-\sigma\cdot T_{in}^{4}\right)+c_{8}\cdot\left(T_{in}-T_{am} \right)^{4}\end{array}  \tag{11}\]  
where the \"\(\text{E}_{L}\)\" is the longwave irradiance with wavelength over 3 \(\upmu\)m. Moreover, it has to be said that [PERSON] et al. [25] compared Equation (10) with Equation (12). They stated that formula (10) is slightly better than Formula (12) but the difference is negligible.