## 3 Mathematical Model Description
### Thermal Fluid Specifications  
This section is concerned with the equations and the main correlations that are found in the literature to describe the thermal properties of obtained hybrid nanofluids. In detail, the classical models in the following equations were used to describe the thermal properties of hybrid nanofluid flow inside of an absorber tube; this model is based on assuming that the suspension of nanoparticles and fluid can be described as effective material properties [56]. In addition, to simplify all the equations and correlations, different symbols were used to describe the obtained thermal properties. In this research, hybrid nanoparticles of alumina and tungsten oxide, which are described by \(n_{p1}\) and \(n_{p2}\), were added to Therminol VPI as a base fluid, described by \(by\) to define a modified hybrid nanofluid that is described by \(h_{nf}\). The thermal efficiency was examined under the conditions of the dynamic radiation intensity in Budapest, and the effect of using hybrid nanofluids for different total volume fractions (\(\varphi_{tot}\)) varied from 0 to 4%.  
The total volume fraction combination obtained by inserting tungsten oxide and alumina oxide is defined by Equation (25) [57].  
\[\varphi_{tot}\ =\ \varphi_{np1}+\varphi_{np2} \tag{25}\]Equation (26) expresses the hybrid nanofluid density (kg/m\({}^{3}\)) [58].  
\[\rho_{lnf}\ =\ \varphi_{np1}\cdot\rho_{np1}+\varphi_{np2}\cdot\rho_{np2}+(1- \varphi_{tot})\cdot\rho_{bf} \tag{26}\]  
A large part of the literature used the specific heat capacity (J/kg\(\cdot\)K) formula represented in Equation (27); this is because of its precision for a wide range of volume concentrations [58].  
\[C_{p,lnf}\ =\ \frac{\varphi_{np1}\cdot\rho_{np1}\cdot C_{p,np1}+\varphi_{np2} \cdot\rho_{np2}\cdot C_{p,np2}+(1-\varphi_{tot})\cdot\rho_{bf}\cdot C_{p,bf}}{ \rho_{lnf}} \tag{27}\]  
The thermal conductivity of the hybrid nanofluid (W/m\(\cdot\)K) was obtained using the Bruggeman correlation, as shown in Equation (28); this is because it is valid over an extensive range of volume fractions and temperatures for nanofluids obtained by mixing nanoparticles with oil, as in References [37; 59; 60].  
\[k_{lnf}\ =\ 0.25[(3\varphi_{tot}-1)k_{p}+(2-3\varphi_{tot})k_{bf}+\Delta^{ \frac{1}{2}} \tag{28}\]  
where  
\[k_{P}\ =\ \left(\frac{k_{np1}\varphi_{np1}+k_{np2}\varphi_{np2}}{\varphi_{ tot}}\right) \tag{29}\]  
\[\Delta\ =\ \left[(3\varphi_{tot}-1)k_{p}+(2-3\varphi_{tot})k_{bf}\right]^{2 }+8k_{bf}k_{p} \tag{30}\]  
Finally, the Brinkman model correlation was used to obtain the nanofluid's dynamic viscosity, as represented by Equation (31) [61].  
\[\mu_{nf}\ =\ \mu_{bf}\frac{1}{(1-\varphi)^{2.5}} \tag{31}\]  
In this study, Therminol VP1 was used as the base fluid due to its ability to work safely at high temperatures without any change in phase; regression analysis of the datasheet for different thermal properties was obtained as polynomial functions with temperatures, as represented by the following equations (Equations (32)-(36)) [62].  
\[Cp_{bf}\ =\ 4.394*10^{-8}T^{4}-7.7663*10^{-5}T^{3}+0.049862T^{2}-11.017T+2125 \tag{32}\]  
\[\rho_{bf}\ =\ -2.379*10^{-6}\ T^{3}+0.002737T^{2}-1.871\ T+1439 \tag{33}\]  
\[k_{bf}\ =\ 1.062*10^{-11}T^{3}-1.9367*10^{-7}T^{2}+2.035*10^{-5}T+0.1464 \tag{34}\]  
\[\mu_{bf}\ =\ -7.723*10^{-6}T^{3}+8.3409*10^{-3}T^{2}-3.0154T+3.661*10^{2} \tag{35}\]  
For 285.15 K \(\leq T\leq\) 373.15 K.  
\[\mu_{bf}\ =\ 1.6543*10^{-10}T^{4}-3.9844*10^{-7}T^{3}+3.617*10^{-4}T^{2}-0. 1476T+23.165 \tag{36}\]  
For 373.15 K \(\leq T\leq\)698.15 K.  
In this research, the examined nanoparticles' thermal properties were obtained from the literature, as shown in Table 5 [42; 46]. Both proposed nanoparticle types in our model had a high purity and a spherical shape with diameter sizes equal to 90 nm and 20 nm for WO\({}_{3}\) and Al\({}_{2}\)O\({}_{3}\), respectively. At the same time, the spherical shape supported the usage of the Bruggeman model (Equation (28)) in our approach since it depends on a spherical solid-fluid mixture [26].