## 2 Models and Methodology
### Energetic and Exergetic Efficiency Models  
Detail heat loss models of conventional HCEs (CHCEs) and NHCEs have been presented in our previous study [25], thus no more repetitive introductions are conducted in this paper. Here, energetic and exergetic efficiency models used to simulate overall performance of NHCEs in a real PTC system are established by employing a volume unit method. The heat collecting process in the 72 m PTC system is exhibited in Figure 4, 72 m loop is divided into \(n\) control volumes (\(n=144\)) for accurately calculating overall performance of the HCEs in the PTC system. For per control volume (\(i\)) under quasi steady state, the fluid temperature and absorber temperature are marked as \(T_{f,i}\) and \(T_{s,i}\) (\(i=1,\,\ldots,\,n\)), the heat loss (_HL_) from the HCE and the amount of convection heat transfer between the absorber tube and the HTF are expressed as \(Q_{HL,i}\) and \(Q_{com,s\
eq f,i}\). \(Q_{com,s\
eq f,i}\) is written as follows [26]:  
\[Q_{com,s-f,i}=h_{f}D_{s}\pi L_{PTC,i}(T_{s,i}-T_{f,i}),\ (i=1,\,\ldots,\,n), \tag{1}\]  
where \(L_{PTC,i}\) refers to the length of control volume \(i\), m. The heat gain of control volume \(i\) can be expressed as:  
\[Q_{gain,i}=m_{f}c_{p}(T_{f,i+1}-T_{f,i}),\ (i=1,\,\ldots,\,n). \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(T_{\rm s}\) (\({}^{\circ}\)C)} & \multicolumn{3}{c}{\(\lambda_{c,opt}\) (\(\mu\)m)} \\ \cline{2-4}  & **DNI = 400 W/m\({}^{2}\)** & **DNI = 600 W/m\({}^{2}\)** & **DNI = 800 W/m\({}^{2}\)** \\ \hline
200 & 2.50 & 2.50 & 2.50 \\
240 & 2.45 & 2.45 & 2.50 \\
280 & 2.30 & 2.40 & 2.40 \\
320 & 1.80 & 1.81 & 2.30 \\
360 & 1.80 & 1.80 & 1.80 \\
400 & 1.75 & 1.80 & 1.80 \\
440 & 1.35 & 1.70 & 1.75 \\
480 &According to the principle of conservation of energy, the heat gain of control volume \(i\) equals to amount of convection heat transfer between the absorber tube and the HTF, that is:  
\[Q_{gain,i}=Q_{conv,s-f,i}. \tag{3}\]  
Consequently, heat gain of each control volume can be obtained, and the total heat gain throughout the full-length PTC system can be expressed as:  
\[Q_{gain}=\sum_{i=1}^{n}Q_{gain,i},\ (n=144). \tag{4}\]  
As a result, the energetic efficiency of the PTC system is calculated by:  
\[\eta=\frac{Q_{\text{gain}}}{Q_{\text{solar}}}, \tag{5}\]  
where \(Q_{\text{solar}}\) is total available solar irradiance received by the full-length PTC system, W/m\({}^{2}\), which can be expressed as:  
\[Q_{solar}=A_{ap}\cdot DNI\cdot\cos\gamma, \tag{6}\]  
where \(\gamma\) is the incidence angle of solar irradiance in the aperture plane of the PTC.  
Exergy is an important indicator to evaluate the process quality, high exergetic performance means low irreversibilities and considerable high-quality energy output. For observing the quality of the heat source obtained by the PTC system with different HCEs, the exergetic performance of the PTC system is simulated in this paper. The exergetic output of system (\(E_{\text{gain}}\)) can be expressed as the difference of the useful heat gain and irreversibilities, as shown in Equation (7):  
\[E_{gain}=Q_{gain}-m_{f}\cdot c_{p}\cdot(T_{\text{a}}+273)\cdot\ln[\frac{T_{ \text{out}}+273}{T_{\text{in}}+273}]. \tag{7}\]  
Thus, the exergy efficiency (\(\chi\)) of an HCE can be calculated by the equation as follows:  
\[\chi=\frac{E_{gain}}{E_{solar}}, \tag{8}\]  
Figure 4: Heat collecting process in the PTC system.  
where \(E_{solar}\) is the exergy of solar irradiance, W/m\({}^{2}\), which is expressed as [27]:  
\[E_{\text{solar}}=Q_{\text{solar}}\cdot[1-\frac{4(T_{\text{a}}+273)}{3(T_{solar}+2 73)}+\frac{1}{3}.(\frac{T_{\text{a}}+273}{T_{solar}+273})^{4}], \tag{9}\]  
where \(T_{solar}\) represents the equivalent solar temperature.