## 4 Results and Discussion
### Case Study 1: Comparison with Non-Optimized Literature Work
Figure 7: Case 1 concentrator optical efficiencies at an incidence angle of 1.0\({}^{\circ}\) for the iterated parameters (optimum values from the algorithm are enlarged in each curve) of (**a**) SOE radius, \(r\), in mm, (**b**) receiver radius, \(FS\), in mm, and (**c**) SOE-receiver spacing, \(SRS\), in mm (shaded areas: left—higher concentration, right—better homogeneity). The algorithmically optimized value of the SOE radius, \(r\), returned the highest optical efficiency for a fixed receiver size and position, as seen in Figure 7a. Smaller SOEs could not catch all refracted rays from the POE, significantly reducing the efficiency. On the other hand, increasing the SOE radius maintained good efficiency but was gradually unable to effectively mitigate the focal shift, which eventually reduced the received energy again. Widening the receiver radius, \(FS\), in Figure 7b, which also means reducing the geometrical concentration ratio, \(CG\), increased the optical efficiency due to the widened energy capture area. This behavior continued until converging to a maximum value. Since the actual receiver dimensions can vary, this algorithmically optimized value of the receiver radius denotes the start of convergence for the \(FS\)-\(\eta_{o}\) curve at constant \(r\) and \(SRS\). As for the SOE-receiver spacing, \(SRS\), placing the receiver too far will result in a decreased optical efficiency for a constant \(CG\). It can be gleaned from Figure 7c that this distance peaks and maintains a high optical efficiency as the receiver gets closer to the SOE. It was also noted from the simulations leading to Figure 7c that carefully increasing the value of the \(SRS\) (within the blue-shaded range right of the optimum) will result in a more uniform focal irradiance, while only slightly increasing the f-number of the system. On the other hand, decreasing this spacing (within the red-shaded range left of the optimum) results in a higher concentration of rays and increases the peak focal irradiance. For a better understanding, Figure 8a-f illustrates the focal irradiance distribution at 1.0\({}^{\circ}\) incidence for SOE-receiver spacings of 0, 5, and 12 mm. Figure 8: Case 1 receiver’s irradiance distribution in W/m\({}^{2}\) at an incidence angle and an SOE–receiver spacing (SRS), respectively, of (**a**) normal and 0 mm, (**b**) 1.0\({}^{\circ}\) and 0 mm, (**c**) normal and 5 mm, (**d**) 1.0\({}^{\circ}\) and 5 mm, (**e**) normal and 12 mm, and (**f**) 1.0\({}^{\circ}\) and 12 mm. (Optical efficiencies are indicated below each figure; the f-number was taken with respect to the receiver.) Since the purpose of the dome SOE is to increase the CAP value [5], the optimized spacing, \(SRS_{opt}\), achieved the highest CAP with moderate irradiance homogeneity, as illustrated in Figure 8c,d. A highly concentrated focal flux, shown in Figure 8a,b, that peaks at 4000 suns with normal incidence can be achieved with no receiver-SOE gap at all. This is more suited for high-temperature thermal applications [21; 22] but will result in a 22% CAP decrease. A better homogeneity of the focal irradiance can be seen in Figure 8e,f, achieved by moving the receiver slightly farther from the SOE. A 12 mm gap, almost double the optimum, enhanced the uniformity of the focal flux to a plateau of about 40 suns at normal incidence and still decreased the CAP by 15%. This position is more applicable for concentrated photovoltaics.