## 3 Modeling of the System
### Solar Collector  
Figure 4 shows the energy balance at the cross section of the parabolic trough solar collector used in this study. The outer blue ring represents the glass envelope while the inner yellow ring represents the metal heat-absorbing tube in the figure. The equations of energy can be written as [33]:  
\[Q_{\text{fluid}}=Q_{\text{mo-mi}} \tag{1}\]  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]  
\[Q_{\text{gi-go}}=Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}} \tag{3}\]  
\[Q_{\text{conv,go-atm}}+Q_{\text{rad,go-atm}}=Q_{\text{gi-go}}+Q_{\text{go}} \tag{4}\]  
where \(Q\) is quantity of heat, subscript fluid indicates the working fluid, subscript mo-mi the transmitting process from the outer surface of the metal absorber to the inner surface, subscript mo indicates the outer surface of absorber. Subscript conv,mo-gi indicates the convection heat transfer process from absorber to the inner surface of glass envelope, subscript rad,mo-gi indicates the radiation heat transfer process between absorber to the inner surface of glass envelope, subscript mo-loss indicates the loss from absorber to the environment through the bracket, which is at the end of absorber. Subscript gi-go indicates the conducting process from the inner surface to the outer surface of the glass cover, subscript conv,go-atm indicates the loss converted from the outer wall of glass cover, subscript rad,go-atm indicates the loss caused by radiation from the outer wall of glass cover, and subscript go indicates the outer wall of the glass cover. The solar radiation absorbed by the metal absorber is not only related to the properties of the absorber, but also the properties of the trough reflector and glass envelope. It can be calculated by:  
\[Q_{\text{mo}}=G\cdot\pi\cdot D_{\text{mo}}\cdot l\cdot CR\cdot\rho\cdot\tau_{ \text{g}}\cdot\alpha_{\text{m}} \tag{5}\]  
Moreover, the solar radiation absorber by the glass envelope can be calculated by:  
\[Q_{\text{go}}=G\cdot\pi\cdot D_{\text{go}}\cdot l\cdot CR\cdot\rho\cdot\alpha_ {\text{g}} \tag{6}\]  
where \(G\) is solar radiation intensity, \(D_{\text{mo}}\) is the outer diameter of absorber, \(l\) is the length of the collector, \(CR\) is the concentration ratio of the collector, \(\rho\) is the reflectivity of trough reflector, \(\tau_{\text{g}}\) is the  
Figure 4: The energy balance for the solar receiver. transmittance of glass envelope, \(\alpha_{\rm m}\) is the absorptivity of absorber, \(D_{\rm go}\) is the outer diameter of glass envelope, \(\alpha_{\rm g}\) is the absorptivity of the glass envelope. The heat transfer between the fluid and the inner surface of the absorber can be regarded as a forced convective heat transfer, which can be calculated by:  
\[Q_{\rm fluid}=\pi\cdot D_{\rm mi}\cdot l\cdot hc_{\rm fluid}\cdot(T_{\rm mi}-T _{\rm fluid}) \tag{7}\]  
\[h_{\rm fluid}=N\it{u}_{\rm fluid}\frac{k_{\rm fluid}}{D_{\rm mi}} \tag{8}\]  
where \(D_{\rm mi}\) is the inner diameter of absorber, \(hc_{\rm fluid}\) is the convective heat transfer coefficient of fluid, \(T_{\rm mi}\) is the average temperature of the inner surface of the absorber, \(T_{\rm fluid}\) is the average temperature of the working fluid, \(N\it{u}_{\rm fluid}\) is the Nusselt number of the working fluid, \(k_{\rm fluid}\) is the thermal conductivity of the working fluid. The Nusselt number is calculated by the Gnielinski correlation [34]:  
\[Nu_{\rm fluid}=\frac{(f/8)(Re_{\rm fluid}-1000)Pr_{\rm fluid}}{1+12.7\sqrt{f/8 }(Pr_{\rm fluid}^{2/3}-1)}(\frac{Pr_{\rm fluid}}{Pr_{\rm mi}})^{0.11} \tag{9}\]  
where \(Re_{\rm fluid}\) is the Reynolds number of working fluid, \(Pr_{\rm fluid}\) is the Prandtl number of fluid, and \(Pr_{\rm mi}\) is the Prandtl number of the inner surface of the absorber, \(f\) is the friction coefficient of fluid:  
\[Q_{\text{mo}}=Q_{\text{mo-mi}}+Q_{\text{conv,mo-gi}}+Q_{\text{rad,mo-gi}}+Q_{ \text{mo-loss}} \tag{2}\]0  
The area between the glass envelope and the absorber keeps a vacuum to reduce heat loss.