## 4 Results and Discussion  
Equations (1)-(31) describing the present numerical model have been solved considering the heat and mass transfer mechanisms during the solar PV-thermal energy conversion process under magnetic field forces and nanofluids. The abovementioned equations were coded with finite-difference formulations and solved as per the logical flow chart depicted in Figure 2. In addition, for validation purposes, the predicted simulated results for PV-thermal solar panels were compared to the data published in the literature under various conditions. Figure 2: Logical flow diagram. In the following sections, \"we present analysis and discussions of the numerical results predicted as well as validations of the proposed simulation model. The simulations were performed with various temperature differences across the thermal tubes of the heat exchanger welded beneath the PV solar panels, however, only results for the temperature difference of 15 \({}^{\circ}\)C across the thermal tube are presented and analyzed. It is worthwhile noting that the numerical simulation presented here was conducted under different conditions, such as PV cell temperatures from 10-38 \({}^{\circ}\)C, ambient temperatures from 10 \({}^{\circ}\)C through 38 \({}^{\circ}\)C, solar radiations of 500, 750, 1000, and 1200 w/m\({}^{2}\), different nanofluids (Al\({}_{2}\)O\({}_{3}\), Fe\({}_{3}\)O\({}_{4}\), SiO\({}_{2}\), and CuO) at various concentrations (5%, 10%, and 20%), and under a magnetic force that varied from 127 to 3000 Gauss. However, in the following, we present discussion and analysis for the nanofluid concentration of 5% for practical reasons to avoid the high-pressure drop and losses in the PV-thermal solar collector loop in practical applications\". The PV panel characteristics under consideration in this study were obtained from [PERSON] et al. [35]. \"The parameters adopted in this study were: total surface area of the PV module (SP) is 0.617 m\({}^{2}\), total surface area of cells in module (Sc) is 0.5625 m\({}^{2}\), module efficiency of 12% at the reference temperature (298 K), the overall absorption coefficient is 0.73, and temperature coefficient is 0.0045 K\({}^{-1}\). Readers interested in the full range of values of the other parameters are advised to consult [PERSON] et al. [35] and [PERSON] [31]\". It was also assumed in this simulation that the whole panel was covered in PV cells, with no packing material (the material used to fill in gaps between the cells on a panel). The PV cells are commercial grade monocrystalline silicon cells with an electrical efficiency of 12% and have a thermal coefficient of 0.54% (1/K) [31]. As per the equations outlined above, solar radiation increased the PV cell temperature and resulted in an increase in the back cell temperature and, consequently, the heat transport fluid temperature due to the heat transfer from solar energy by conduction and convection as well as radiation, respectively. It has been demonstrated by [PERSON] [31] that the higher the cell temperature, the higher the back cell and fluid temperatures, and the higher the solar radiations, the higher the PV cell temperature and, consequently, the higher the temperature of the cell until it reaches the design temperature. For further details, interested readers are advised to consult [PERSON] [31]. \"The effects of the PV panel operating temperature on the output efficiency have been well documented in the literature [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32], where the increasing temperature of the PV cell decreased the amount of power available. However, it is important to note that the changes in the PV cell temperature caused by solar radiation have a dynamic nature. The PV panel heats up and cools down gradually depending upon the changes in solar radiation in dynamic response and consequently the power output from the PV panel.