# A Solar-Thermal-Assisted Adiabatic Compressed Air Energy Storage System and Its Efficiency Analysis
## 3 Thermodynamic Analysis Model
### Energy Analysis Model
#### 3.2.2 Stc  
The STC system includes PTC and a heat storage tank. The PTC thermal model is shown in Figure 2. Its collecting thermal efficiency, \(\eta_{c}\), is defined as the ratio of useful energy \(Q_{u}\) absorbed by the HTF to energy incident on the parabolic reflecting mirror, which is outlined as follows [27]:  
\[\eta_{C}=\frac{Q_{u}}{Q_{s}}, \tag{11}\]  
where the incident energy, \(Q_{s}\), is proportional to the PTC area and direct normal insolation per unit of collector area, and is computed by  
\[Q_{s}=I_{D}A, \tag{12}\]  
where \(I_{D}\) is the direct normal irradiation and \(A\) is the effective collection area. A part of the solar energy is delivered to the receiver, and the amount of that energy is a function of the parabolic trough mirror optical efficiency. Equation (13) shows the relationship between the optical efficiency and the incident energy. We assumed a constant value of 0.9 for the optical efficiency of the collector. The reflecting heat power \(Q_{R}\) by parabolic reflecting mirror is  
\[Q_{R}=\eta_{op}Q_{s}, \tag{13}\]  
where \(\eta_{OP}\) is the optical efficiency. The useful heat energy \(Q_{u}\) obtained by the absorber tubes is computed by  
\[Q_{u}{=}\ Q_{R}-Q_{L}. \tag{14}\]  
Figure 2: Parabolic trough collector thermal model. To obtain the useful heat absorbed by the receiver, its heat loss should be calculated. In general, there are three types of heat dissipation, namely conduction, convection, and radiation. Therefore, the total heat loss \(Q_{L}\) of the receiver can be expressed as [28]:  
\[Q_{L}=Q_{cond}+Q_{conv}+Q_{rad}. \tag{15}\]  
To simplify the analysis, the thicknesses of the glass envelope and absorber tube are not considered, and the heat conduction loss between the absorber tube and the glass tube is ignored. The heat convection loss, based on [PERSON]'s law, is as follows:  
\[ \begin{split} Q_{conv}&=Q_{conv1,2}+Q_{conv2,0},\\ Q_{conv1,2}&=h_{d}(T_{1}-T_{2})A_{ab},\\ Q_{conv2,0}&=h_{c}(T_{2}-T_{0})A_{g},\end{split}  \tag{16}\]  
where \(Q_{conv1,2}\) is the heat convection loss between the absorber tube surface and inner glass tube. \(Q_{conv2,0}\) is the heat convection loss between the inner glass tube and the external environment. \(A_{ab}\) and \(A_{g}\) are the surface areas of the absorber and glass tube, respectively. \(h_{d}\) is the convection heat transfer coefficient of the annular space. \(h_{c}\) is the convection heat transfer coefficient of the glass tube and environment, and \(T_{1}\) and \(T_{2}\) are the surface temperatures of the absorption tubes and glass tubes, respectively. \[Q_{rad}=Q_{rad1,2}+Q_{rad2,0}. \tag{17}\]  
The operation characteristics of the absorber tube show that the temperature of absorber tube is much higher than that of the glass tube. The heat radiation between them is calculated by  
\[Q_{rad1,2}=\frac{\sigma(T_{1}^{4}-T_{2}^{4})}{1/\varepsilon_{1}+(1-\varepsilon _{2})D_{1}/\varepsilon_{2}D_{2}}A_{ab}, \tag{18}\]  
where \(\sigma\) is the Stefan-Boltzmann constant (W/m\({}^{2}\)-K\({}^{4}\)) and D\({}_{1}\) and D\({}_{2}\) are the outer absorbers and inner glass tube diameters, respectively. \(T_{1}\) is the outer absorber surface temperature, and \(T_{2}\) is the inner glass tube surface temperature. \(\varepsilon_{1}\) is the absorber selective coating emissivity, and \(\varepsilon_{2}\) is the glass tube emissivity. The heat radiation between the glass absorber and the environment is calculated via  
\[Q_{rad2,0}=\sigma\varepsilon_{2}\Big{(}T_{2}^{4}-T_{0}^{4}\Big{)}A_{g}. \tag{19}\]  
The thermal energy tanks are modeled by dynamic mass and energy balances for two tanks [29]. The mass balance for a tank is  
\[\rho_{HTF}\frac{dV_{HTF}}{dt}=m_{in}-m_{out}, \tag{20}\]  
where \(V_{HTF}\) is the volume of HTF in the tank, and \(\rho_{HTF}\) refers to the density of HTF.