# An Innovative Parabolic Trough Collector Design with a Twin-Cavity Receiver
## 2 Materials and Methods
### Analysis of Equations  
In this section, the optical, thermal, and flow analysis equations are provided [38; 39]. The thermal efficiency of the PTC is defined as:  
\[\eta_{th}=\frac{Q_{u}}{Q_{s}}=\frac{Q_{u}}{A_{a}\cdot G_{bT}} \tag{1}\]  
The optical efficiency of the PTC is defined as:  
\[\eta_{opt}=\frac{Q_{abs}}{Q_{s}} \tag{2}\]  
Figure 3: Mesh grid inside the flow tube in both cases. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline
**Examined Mesh** & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline
**Mesh Cells** & 0.49 \(\times\) 10\({}^{6}\) & 0.68 \(\times\) 10\({}^{6}\) & 0.88 \(\times\) 10\({}^{6}\) & 1.11 \(\times\) 10\({}^{6}\) & 1.37 \(\times\) 10\({}^{6}\) & **1.62 \(\times\) 10\({}^{6}\)** & 1.95 \(\times\) 10\({}^{6}\) & 2.25 \(\times\) 10\({}^{6}\) \\ \hline
**T\({}_{e}\)** (\({}^{\circ}\)**C** & 109.8 & 111.5 & 118.2 & 118.4 & 118.6 & **118.6** & 118.7 & 118.7 \\ \hline
**\(\eta\)th** & 74.18\% & 74.04\% & 73.44\% & 73.41\% & 73.39\% & **73.37\%** & 73.36\% & 73.36\% \\ \hline
**Final selection** & & & & & & & & \\ \hline \multicolumn{8}{c}{**PTC-F**} \\ \hline
**Examined Mesh** & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline
**Mesh Cells** & 0.57 \(\times\) 10\({}^{6}\) & 0.75 \(\times\) 10\({}^{6}\) & 0.98 \(\times\) 10\({}^{6}\) & 1.18 \(\times\) 10\({}^{6}\) & 1.39 \(\times\) 10\({}^{6}\) & 1.66 \(\times\) 10\({}^{6}\) & **1.94 \(\times\) 10\({}^{6}\)** & 2.25 \(\times\) 10\({}^{6}\) \\ \hline
**T\({}_{e}\)** (\({}^{\circ}\)**C** & 113.9 & 115.9 & 116.2 & 116.4 & 116.5 & 116.5 & **116.6** & 116.6 \\ \hline
**\(\eta\)th** & 67.79\% & 67.70\% & 67.67\% & 67.68\% & 67.66\% & 67.69\% & **67.61\%** & 67.63\% \\ \hline
**Final Selection** & & & & & & & **Selected mesh** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sensitivity analysis regarding the mesh selection for the PTC-TC and PTC-F. The solar irradiation on the collector surface is calculated as:  
\[Q_{s}=A_{a}\cdot G_{bT} \tag{3}\]  
The useful heat production is calculated as:  
\[Q_{u}=\dot{m}\cdot C_{p}\cdot(T_{o}-T_{i})=h_{f}\cdot A_{s}\cdot\left(T_{s}-T_{f, m}\right)=Q_{abs}-Q_{L} \tag{4}\]  
The mean fluid temperature is estimated as:  
\[T_{f,m}=\frac{T_{o}+T_{i}}{2} \tag{5}\]  
The thermal losses can be calculated as:  
\[Q_{L}=\left[h_{m}\cdot\left(T_{g}-T_{\alpha}\right)+\varepsilon_{g}\cdot \sigma\cdot\left(T_{g}^{4}-T_{\alpha}^{4}\right)\right]\cdot A_{g,o} \tag{6}\]  
Equation (1) provides two different expressions for thermal efficiency, and Equation (2) indicates that the optical performance is a function of the absorbed solar power (\(Q_{abs}\)) and the available solar power on the PTC (\(Q_{s}\)). The latter is expressed in Equation (3), while the useful output is available in Equation (4). Equation (5) is used for the calculation of the average temperature of the working fluid. Last but not least, the total thermal losses are expressed by Equation (6). Apart from the optical and thermal analysis equations, the respective equations for the flow analysis part are equally significant, and thus it was necessary to present them too [40]. The Reynolds number, which is given in Equation (7), is one of the most important parameters for flow analysis, since it constitutes an indication of the flow regime intensity. The critical Reynolds number in Equation (8) provides us with information on whether or not the flow regime transitions from the laminar to the turbulent [41]. This number is calculated as a function of the Darcy friction factor, which is described in Equation (9) [42]. In Equation (10), the Nusselt number is expressed as the product between the heat convection coefficient inside the flow tube and the inner diameter of the tube divided by the thermal conductivity, which is determined considering the average fluid temperature according to Equation (5). The heat transfer coefficient (\(h_{f}\)) is calculated via Equation (4).