# Performance Prediction and Optimization of Multi-Mirror Combined Solar Dish Collector
## 2 Method: The Optical Efficiency Calculation Model and Optimization Method for Combined Disk System
### Optical Efficiency Calculation Model for Combined Dish System  
We previously proposed a method to calculate the optical efficiency of a rectangular spherical mirror with a rectangular receiver [29]. The model developed in this work converts the square receiver to a circular receiver. The calculation principle is similar, as briefly introduced below. For a reflection point P of the dish system, the reflected light intensity distribution is \(B_{\mathit{eff}}\left(\theta\right)\). As shown in Figure 2, the central solar ray from the sun is reflected at point P to intersect with the receiver at point O. The center solar ray OP is set as the \(z\)-axis. When the light cone reaches the reference plane U which is perpendicular to the \(z\)-axis and passes point O, a circular light spot is formed. If the angle between the light ray and the central solar ray is \(\theta\), the radiation energy reflected from point P absorbed by the receiver can be expressed as follows [30]:  
\[I_{p}=f_{at}*f_{rec}\iint_{D}B_{eff}(\theta)d\varphi\sin\theta d\theta, \tag{4}\]  
where \(f_{at}\) is the reflectivity of the mirror, and \(f_{rec}\) is the absorptivity of the receiver. D is the integral range which is determined by the range of the receiver projection on the reference plane. The distribution of the reflected solar intensity is obtained by the convolution of the actual solar intensity distribution and Gaussian distribution of the optical error. If \(B\) (\(\theta\)) is the normalized reflected radiation distribution, then  
\[ \begin{array}{ll}\eta_{p}&=f_{at}*f_{rec}\int_{0}^{\theta p}\int_{0}^{\varphi _{p}(\theta)}d\varphi B(\theta)\sin\theta d\theta\\ &=f_{at}*f_{rec}\int_{0}^{\varphi_{p}}2\pi*g(\theta)*B(\theta)\sin\theta d \theta\end{array} , \tag{5}\]  
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Figure 2: Efficiency calculation from the reflection point of a mirror when the radial distribution of brightness for an effective source is applied. where \(\eta_{p}\) is the optical efficiency at point P of the heliostats, which is defined as the ratio of the energy absorbed by the receiver to the energy reflected from the point P. \(\theta_{p}\) is the maximum angle of \(\theta\) where the ray at the circle can be partly intercepted by the receiver. As shown in Figure 3, \(\theta_{p}=\arctan\left(\text{OE}/\text{OP}\right)\), where O is the intersection point of the reflected central solar ray on the receiver plane, and OE and OP are the distances between the points O and E, and P, respectively. Here, the ray tracing method was applied to calculate the value of \(g\left(\theta\right)\). It represents the percentage of light intercepted by the receiver. When the radial angle of the reflected light is \(\theta\) and the azimuth angle is \(\varphi\), the vector coordinate of the light in the coordinate axis is \(\left(\text{sin}\theta\text{cos}\varphi,\text{sin}\theta\text{sin}\varphi, \text{cos}\theta\right)\). Therefore, whether the reflected light is received by the receiver can be estimated, and the intersection of the reflected light ray on the receiver plane can be calculated. As shown in Figure 3, there are two cases of O inside and outside the receiver. The first case is that the point O is in the area of the receiver, as shown in Figure 3 (left); thus,  
\[\eta_{P}=f_{at}*f_{rec}\left[\int_{0}^{\theta_{T}}2\pi B(\theta)\sin\theta d \theta+\int_{\theta_{T}}^{\theta_{p}}2\pi g(\theta)B(\theta)\sin\theta d \theta\right], \tag{6}\]  
\[\gamma_{p}=\int_{0}^{\theta_{T}}2\pi B(\theta)\sin\theta d\theta+\int_{\theta_ {T}}^{\theta_{p}}2\pi g(\theta)B(\theta)\sin\theta d\theta. \tag{7}\]  
\(\theta_{\text{T}}\) is determined by the minimum distance to the center of the receiver, and \(\theta_{\text{T}}=\arctan(\text{OT}/\text{OP})\), where OT is the distance between the point O and T. If \(\theta\) is less than or equal to \(\theta_{\text{T}}\), all reflected rays are intercepted, i.e., \(g\left(\theta\right)=1\).