## 3 Modeling of the System
### Solar Collector
Therefore, the convective heat transfer between the absorber and glass cover caused by free movement of the molecules should be little. The heat transfer between the absorber and glass envelope can be given as:  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]1  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]2  
where \(D_{\rm mo}\) is the outer diameter of absorber, \(T_{\rm mo}\) is the average temperature of outer surface of absorber, \(T_{\rm gi}\) is the average temperature of the inner surface of the glass envelope, \(\sigma\) is Boltzmann constant, which is \(5.67\times 10^{-8}\)W/(m\({}^{2}\cdot\)K\({}^{4}\)), \(\epsilon_{\rm mo}\) is the emissivity of outer surface of absorber, \(\epsilon_{\rm gi}\) is the emissivity of inner surface of glass envelope, and \(hc_{\rm mo-gi}\) is the convection heat transfer coefficient of the air in the interlayer, which can be calculate by [35]:  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]3  
with  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]4  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]5  
where \(k_{\rm std}\) is the thermal conductivity of air under standard conditions, \(b\) is the interaction coefficient, \(a\) is the thermal adaptation coefficient, which is usually taken as 1, \(\gamma\) is the heat capacity ratio of air which is 1.4, \(\lambda\) is the mean free path of molecular, \(T_{\rm mo-gi}\) is the average temperature of absorber and glass cover, \(P_{\rm mo-gi}\) is the pressure of the area approximated as 0.0001 mmHg, \(\delta\) is molecular diameter of air taken as \(3.53\times 10^{-8}\) cm. The heat transfer from the outer surface of the glass envelope to the environment can be calculated as:  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]6  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]7  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]8  
where \(D_{\text{go}}\) is the outer diameter of glass envelope, \(hc_{\text{air}}\) is the convection heat transfer coefficient of air in the atmosphere, \(T_{\text{go}}\) is the average temperature of outer surface of glass envelope, \(T_{\text{atm}}\) is the temperature of atmosphere, \(k_{\text{air}}\) is the thermal conductivity of air when temperature is 0.5(\(T_{\text{go}}\)+ \(T_{\text{atm}}\)), \(Nu_{\text{air}}\) is the Nusselt number of air when temperature is 0.5(\(T_{\text{go}}\)+ \(T_{\text{atm}}\)), \(\varepsilon_{\text{go}}\) is the emissivity of outer surface of glass envelope, \(T_{\text{sky}}\) is sky temperature, which can be calculated by [36]:  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]9  
The Nusselt number can be calculated by the correlation [34]:  
\[Q_{\text{gi-go}}=Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}} \tag{3}\]0  
with:  
\[\left\{ \begin{array}{l}C=0.75,\ m=0.4,\text{ for }\text{Re}_{\text{go}}=1-40\\ C=0.51,\ m=0.5,\text{ for }\text{Re}_{\text{go}}=40-1000\\ C=0.26,\ m=0.6,\text{ for }\text{Re}_{\text{go}}=1000-200000\\ C=0.076,\ m=0.7,\text{ for }\text{Re}_{\text{go}}=200000-1000000\\ \end{array} \right. \tag{20}\]  
And  
\[\left\{ \begin{array}{l}n=0.37,\text{ for }\text{Pr}_{\text{atm}}\leq 10\\ n=0.36,\text{ for }\text{Pr}_{\text{atm}}>10\\ \end{array} \right. \tag{21}\]  
where subscript \(atm\) is the atmosphere and subscript \(go\) is the outer surface of glass envelope. The parameters used in this the parabolic trough solar collector are shown in Table 1. According to the parameters in the table and the above formula, the required collector area under different conditions can be calculated.