# A Realistic Approach of the Maximum Work Extraction from Solar Thermal Collectors
## 4 Results--Discussion
### The Application of the Model in a Real Solar Collector  
The more mature technology in concentrating solar collectors is the parabolic trough collectors. A well-established solar collector, the IST-PTC (Industrial Solar Technology Corporation product), is the examined example (Figure 5). Equation (24) gives its measured efficiency by SANDIA Reports [18; 20]. This analysis is a simple validation of the developed model (Equations (16) and (23)). \[\eta_{coll}=0.762-0.2125\cdot\left(\frac{T_{r}-T_{amb}}{G}\right)-0.001672\cdot G \cdot\left(\frac{T_{r}-T_{amb}}{G}\right)^{2} \tag{24}\]  
It has assumed that the receiver temperature is equal to the fluid temperature in order to achieve ideal conditions, without exergy losses between the receiver and working medium. By multiplying the Equation (24) to the respective Carnot efficiency, the system efficiency can be calculated. Equation (25) presents the system efficiency using the thermal efficiency curve by SANDIA Reports. This efficiency will be called the \"real system efficiency\", while the system efficiency of the suggested model will be called the \"model system efficiency\". \[\eta_{sys,real}=\left[1-\frac{T_{amb}}{T_{r}}\right]\cdot\left[0.762-0.2125 \cdot\left(\frac{T_{r}-T_{amb}}{G}\right)-0.001672\cdot G\cdot\left(\frac{T_{r }-T_{amb}}{G}\right)^{2}\right] \tag{25}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Collector** & \(C\) & \(\eta_{opt}\) & \(\epsilon_{r}\) & \(T_{A,opt}\) (K) & \(\eta_{sys}\) \\ \hline Flat plate collector & 1 & 0.8 & 0.25 & 401 & 0.1449 \\ Evacuated tube collector & 1 & 0.7 & 0.10 & 451 & 0.1777 \\ Compound parabolic collector & 3 & 0.6 & 0.10 & 527 & 0.2079 \\ Parabolic trough collector & 30 & 0.75 & 0.10 & 884 & 0.4400 \\ Solar dish collector & 100 & 0.75 & 0.10 & 1110 & 0.5011 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimum operation for typical solar collectors [9; 10; 18; 19]. Figure 5: Depiction of a parabolic trough collector. Table 2 includes the basic data of this collector as it is given in literature [16; 18]. Figure 6 gives the system efficiency as a function of the receiver temperature or the operational temperature. The curve of \"real system\" is calculated according to Equation (25), while the \"model\" curves according to the developed model of Equation (16). These quantities regard the system efficiency (thermal collector and thermal engine), and so they can be compared directly in Figure 6. The curves are close to each other. The model results give a greater efficiency of about 3%, in high-temperature levels. The first reason for these results is the negligence of convection heat losses in the developed model. The other reason is dependent on the temperature range that the real collector curve has been developed. More specifically, this curve can be adopted to temperature levels up to 300 \({}^{\circ}\)C.  
The optimum temperature according to the developed model is 595 K, while according to the real system approximation the optimum temperature is about 563 K. This difference can be explained by the assumption of no convection losses in the developed model. This obstacle can be addressed by increasing the emittance by 0.1, an empirical rule. The new curve, green curve, is closer to the \"real system curve\" and is maximized at the same temperature (563 K). In this way, it is proved that the increase in the emittance can lead to accurate results. Another useful comparison that can be made is between the previous optimum temperature values and the optimum temperature from Torres-Reyes model [17], according to Equation (7). The problem in this model is the need for knowledge of heat loss coefficient U\({}_{L}\). In this case, this parameter was calculated from Equation (24), as Equation (26) shows:  
\[U_{L}=0.2125-1.672\cdot\left(\frac{T_{r}-T_{am}}{G}\right) \tag{26}\]  
The results gave optimum temperature equal to 630 K, a value that is greater enough than the real optimum.