## 3 Mathematical Model Description
### Thermal Model  
This section aims to solve the energy balance equation, employing the analytical expression used in literature, by describing the thermal model inside the receiver tube of the PTC at different nodes [42; 50]. In this section, heat gain, heat losses, and temperatures are presented to obtain the thermal efficiency enhancements under the radiation intensity conditions of Budapest and the effect of using hybrid nanofluids at various concentrations. The solution was plotted and expressed using MATLAB symbolic code for a dynamic radiation intensity and different concentrations to cover a broad range of conditions. The assumption of this solution is based on simplifying the heat losses (convection and radiation) from the outlet diameter of the glasses cover in Equation (6) to be as in Equation (7); this is achieved by assuming the cover temperature to be close to ambient temperature, which leads to writing the fourth-order temperature using the Taylor series, as in Equation (8). This assumption has proved to have accurate results, as published in different studies in the literature [42; 50]. Moreover, comparing the thermal efficiency and temperature results of the model with the available experimental results showed a minor variation, which supports the use of this assumption in this type of research. \[Q_{loss}\ =\ A_{co}\cdot\epsilon_{c}\cdot\sigma\cdot\left(T_{co}^{4}-T_{am}^{4} \right)+A_{co}\cdot h_{out}\cdot\left(T_{co}-T_{am}\right) \tag{6}\]\[Q_{loss}\ =\ \Big{\{}A_{co}\cdot e_{c}\cdot\sigma\cdot 4\cdot T_{am}^{3}+A_{co} \cdot h_{out}\Big{\}}\cdot(T_{co}-T_{am}) \tag{7}\]  
\[T_{co}^{4}-T_{am}^{4}\cong 4\cdot T_{am}^{3}\cdot(T_{co}-T_{am}) \tag{8}\]  
To simplify the definitions of the output results in Equations (9)-(14), various symbols, called \(K\), with values from 1 to 5, are used. Those symbols are based on collecting the known parameters between brackets, as shown in Equation (7) and summarized in Table 4 [42]. Thermal efficiency for the PTC can be determined as:  
\[\eta_{th}\ =\ K_{4}-K_{5}\cdot\frac{T_{in}^{4}-T_{am}^{4}}{A_{a}*G_{b}} \tag{9}\]  
Thermal losses (\(Q_{loss}\)) are expressed as:  
\[Q_{loss}\ =\ \big{(}\eta_{opt}-K_{4}\big{)}\cdot Q_{s}+K_{5}\cdot\Big{(}T_{in}^{4 }-T_{am}^{4}\Big{)} \tag{10}\]  
Moreover, the equations below express the temperatures of the fluid (outlet and mean), the receiver, and the cover glass. \[T_{out}\ =\ T_{in}+\frac{K_{4}}{\dot{m}\cdot C_{P}}\cdot Q_{s}-\frac{K_{5}}{ \dot{m}\cdot C_{P}}\cdot\Big{(}T_{in}^{4}-T_{am}^{4}\Big{)} \tag{11}\]  
\[T_{fm}\ =\ T_{in}+\frac{K_{4}}{2\cdot m\cdot C_{P}}\cdot Q_{s}-\frac{K_{5}}{2 \cdot m\cdot C_{P}}\cdot\Big{(}T_{in}^{4}-T_{am}^{4}\Big{)} \tag{12}\]  
\[T_{r}\ =\ T_{in}+\frac{K_{4}}{K_{3}}\cdot Q_{s}-\frac{K_{5}}{K_{3}}\cdot\Big{(}T _{in}^{4}-T_{am}^{4}\Big{)} \tag{13}\]  
\[T_{c}\ =\ T_{am}+\frac{\eta_{opt}-K_{4}}{K_{1}}\cdot Q_{s}+\frac{K_{5}}{K_{1}} \cdot\Big{(}T_{in}^{4}-T_{am}^{4}\Big{)} \tag{14}\]  
Some parameters should be known to determine the thermal behavior of the PTC. The parameters that are expressed in Equations (15)-(18) are the heat transfer coefficient (\(h\)) and various dimensionless numbers, such as Prandtl (\(Pr\)), Reynolds (\(Re\)), and Nusselt (\(Nu\)) [50]. \[h\ =\ \frac{Nu\cdot k}{d_{ri}}. \tag{15}\]  
\[Q_{loss}\ =\ \Big{\{}A_{co}\cdot e_{c}\cdot\sigma\cdot 4\cdot T_{am}^{3}+A_{co} \cdot h_{out}\Big{\}}\cdot(T_{co}-T_{am}) \tag{7}\]0  
\[Q_{loss}\ =\ \Big{\{}A_{co}\cdot e_{c}\cdot\sigma\cdot 4\cdot T_{am}^{3}+A_{co} \cdot h_{out}\Big{\}}\cdot(T_{co}-T_{am}) \tag{7}\]1  
\[Q_{loss}\ =\ \Big{\{}A_{co}\cdot e_{c}\cdot\sigma\cdot 4\cdot T_{am}^{3}+A_{co} \cdot h_{out}\Big{\}}\cdot(T_{co}-T_{am}) \tag{7}\]2