# Polynomial Expressions for the Thermal Efficiency of the Parabolic Trough Solar Collector
## 1 Introduction  
The parabolic trough solar collector (PTC) is one of the most mature solar concentrating technologies [1] which can be used in numerous applications such as electricity production [2], industrial heat [3], refrigeration [4], space-heating [5], chemical processes [6] and polygeneration [7]. This collector is a linear concentrating technology that operates with a single-axis tracking system in the majority of the cases. There are only a few investigations with two-axis tracking systems but these designs are complex and they have a high cost. The most usual receiver is an evacuated tube absorber in order to operate in high temperatures with a satisfying efficiency [8]. The operation with thermal oil (e.g., Therminol VP-1) is able to reach up to 400 \({}^{\circ}\)C [9], while with the proper molten salts (e.g., nitrate salt), there is the possibility for operation up to 600 \({}^{\circ}\)C [10]. The use of gas working fluids (e.g., air or CO\({}_{2}\)) gives the possibility to operate in higher temperatures up to 1000 \({}^{\circ}\)C [11]. The literature includes numerous studies that incorporate PTC as the main or the secondary thermal energy source. Usually, the simulation of the PTC performance is conducted by using a thermal efficiency expression formula. The thermal efficiency of the solar collector is the ratio of the useful heat production to the available solar irradiation on the collector aperture (\(\eta_{col}=Q_{a}/Q_{s}\)). Numerous formulas in the literature can be found and so it is interesting to present them as below. A great percentage of the literature uses a simple linear expression for the determination of the efficiency. For example, the study of [PERSON] [12] approximates the PTC performance with a linear expression and it compares it with other linear formulas from the literature. This idea has also been used in newer studies such as in [PERSON] et al. [13]. Below, the linear expression is given:  
\[\eta_{col}=a_{0}+a_{1}\cdot\frac{(T_{in}-T_{am})}{G_{b}} \tag{1}\]  
The previous formula is usually used for the flat plate systems according to [PERSON] and [PERSON] [14]. Thus, this linear expression can be an accurate choice for PTC systems that operate to low-temperature levels up to 100 \({}^{\circ}\)C. Moreover, the previous formula is also known as the Hottel-Whillier-Bliss equation. The next part of the literature studies uses a second-order expression which is in accordance with the ISO 9806-1 [15]. The goal of this formula is to better approximate the operating points of higher temperature levels. \[\eta_{col}=a_{0}+a_{1}\cdot\frac{(T_{in}-T_{am})}{G_{b}}+a_{2}\cdot\frac{(T_{ in}-T_{am})^{2}}{G_{b}} \tag{2}\]  
This formula has been used by a great part of the recent literature studies. For example, [PERSON] et al. [16] selected an expression as the previous one for approximating the Eurotrough module performance. In another older work, [PERSON] and [PERSON] [17] conducted experimental work with PTC and they compared the approximation of the efficiency with a linear and a second-term formula. They concluded that both expressions are accurate and the second-order formula can be applied for the cases with higher temperatures. An alternative expression of the previous one has been introduced by [PERSON] et al. [18] in an experimental work about LS-2. They tried to add an extra term with (\(T_{in}-T_{am}\)) without the incorporation of the solar beam irradiation:  
\[\eta_{col}=a_{0}+a_{1}\cdot\frac{(T_{in}-T_{am})}{G_{b}}+a_{2}\cdot\frac{(T_{ in}-T_{am})^{2}}{G_{b}}+b\cdot(T_{in}-T_{am}) \tag{3}\]  
[PERSON] et al. [19] used a third-degree polynomial in order to approximate the PTC thermal efficiency and they found high values for the R\({}^{2}\). More specifically, they tested the SkyTrough module with a Schott PTR-80 receiver.