## 2 Materials and Methods
### Solar System  
The solar dish concentrator is optically and thermally investigated using SolTrace and Maple software. References [10; 17; 18], presented the numerical methodology for the hemispherical, cubical,  
Figure 1: A schematic of the solar organic Rankine cycle (ORC) system. and cylindrical cavity receivers using pure thermal oil. In the current paper, the optimized hemispherical, cubical and cylindrical cavity receivers are considered as the ORC's heat source using different nanofluids. Three different cavity receiver shapes, each with good thermal performance, are therefore compared. As the cavity aperture area and depth decreases, the optical efficiency of the cavity receiver decreases, while the thermal efficiency of the cavity receiver increases. Consequently, the investigated cavity receivers have optimal dimensions. The main features and dimensions of these cavity receivers are shown in Table 1. All the structural characteristics were selected based on the real built cavities from Refs. [10; 22]. For the optical simulation of the cavity receivers a ray-tracing software, SolTrace [8], is used. A pillbox sun-shape is considered. Furthermore, the tracking error and optical error of the solar dish are assumed as 1\({}^{\circ}\) and 10 mrad, respectively [24]. The receiver tube absorptivity and the reflectivity of the solar dish are accounted for in SolTrace. Each coil of the hemispherical cavity receiver is defined separately in the optical analysis (see Table 1). Figure 2 shows that the one side of the hemispherical cavity receiver has a higher heat flux distribution due to the 1\({}^{\circ}\) tracking error assumption. It should be mentioned that the optical analyses of the hemispherical cavity receiver were presented in Ref. [23] in detail. For the cubical and cylindrical cavity receivers similar distributions have been found [17]. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Cubical** & **Cylindrical** & **Hemispherical** \\ \hline Concentrator diameter (m) & 1.9 & 1.9 & 1.9 \\ Focal length (m) & 1.351 & 1.351 & 1.351 \\ Rim angle of paraboloid & 36.84\({}^{\circ}\) & 36.84\({}^{\circ}\) & 36.84\({}^{\circ}\) \\ Collector aperture area (m\({}^{2}\)) & 2.545 & 2.545 & 2.545 \\ Receiver tube outer diameter (mm) & 10 & 10 & 10 \\ Receiver tube inner diameter (mm) & 9 & 9 & 9 \\ Number of tube coils & 12 & 14 & 10 \\ Cavity inner diameter, \(D_{in}\) (m) & - & 0.140 & 0.140 \\ Cavity outer diameter, \(D_{outer}\) (m) & - & 0.160 & 0.160 \\ Outer aperture length of cubical cavity, \(a_{outer}\) (m) & 0.145 & - & - \\ Inner aperture length of cubical cavity, \(a_{in}\) (m) & 0.125 & - & - \\ Cavity depth, \(h_{rc}\) (m) & 0.125 & 0.14 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Receiver specifications [23]. Figure 2: Heat flux distribution per hemispherical cavity receiver coil (shown for dish diameter of 1.9 m and rim angle of 50.82\({}^{\circ}\)) [23]. The radiation heat loss of the hemispherical cavity receiver is calculated based on the Nusselt number. The Nusselt number of radiation (\(Nu_{tmd}\)) for a hemispherical cavity receiver is expressed as [27]:  
\[Nu_{rad}=9.650\ Gr_{D}^{0.068}(1+\cos\theta)^{0.001}(\varepsilon)^{0.546}(N_{rc} )^{0.478}\times\left[1-(T_{R})^{4}\right]^{8.768}\left(\frac{D_{rec,ap}}{D_{rec, \ outer}}\right)^{0.493} \tag{12}\]  
where  
\[N_{rc}=\frac{\sigma T_{S}^{4}\left(\frac{D_{rec,\ outer}}{2}\right)}{(T_{s}-T _{\infty})k_{a}} \tag{13}\]  
\[T_{R}=\frac{T_{\infty}}{T_{s}} \tag{14}\]  
\[Gr_{D}=\frac{g\dot{\beta}(T_{s}-T_{\infty})D_{rec,\ outer}}{g^{2}} \tag{15}\]  
The cavity receiver is coated with black chrome with emissivity of 0.1.