# New Physically-Based Mathematical Model and Experiments for a Recently Invented Solar Pot
## 3 Mathematical Modelling
### Simulation Results  
Equations (1) and (2) were fed into the Matlab/Simulink programming environment. The computer realization of the model can be seen in Figure 4.  
To run the model, the values of the solar pot's parameters need to be provided. A\({}_{\text{m,e}}\) = 0.19 m\({}^{2}\), A\({}_{\text{t,e}}\) = 0.041 m\({}^{2}\), A\({}_{\text{p,in}}\) = 0.112 m\({}^{2}\), V\({}_{\text{m}}\) = 3.691 L(\(=3.691\cdot 10^{-3}\) m\({}^{3}\)), and V\({}_{\text{t}}\) = 4.418 L(\(=4.418\cdot 10^{-3}\) m\({}^{3}\)) are geometric parameters of the physically manufactured solar pot (see Figure 1 in Section 2). The density (\(\rho\)) and specific heat capacity (c) of water are also needed. These values are 1000 \(\frac{\text{kg}}{\text{m}^{3}}\) and 4200 \(\frac{\text{I}}{\text{kgK}}\), respectively. The determinations of U\({}_{\text{p,e}}\) = 20.4 \(\frac{\text{W}}{\text{m}^{2}\text{K}}\) (overall heat loss coefficient) and U\({}_{\text{p,in}}\) = 351 \(\frac{\text{W}}{\text{m}^{2}\text{K}}\) (overall heat transfer coefficient) are made by calculations based on recommendations from the literature. U\({}_{\text{p,e}}\) is the overall heat loss coefficient between the mantle and the solar pot's environment. It is determined by considering the following:  
1. Convective heat transfer between the inside of the mantle and the inner surface of the mantle's wall. (This wall is adjacent to the environment.) Taking into account the liquid in the mantle (water) and the material of the mantle's wall (stainless steel), and assuming forced convective heat transfer along the inner surface of the mantle's wall, the heat transfer coefficient is 850 \(\frac{\text{W}}{\text{m}^{2}\text{K}}\), which is an average value according to [34]. 2. Thermal conduction inside the mantle's wall. Taking into account the material of the mantle's wall, which is stainless steel (with a thickness of 0.002 m), the thermal conductivity coefficient is 15 \(\frac{\text{W}}{\text{m}\text{K}}\), which is an average value according to [34]. 3. Convective heat transfer between the outer surface of the mantle's wall and the solar pot's environment. Taking into account the material of the mantle's wall (stainless steel) and the air surrounding the pot from outside, and assuming free convective heat transfer along the surface of the wall, the heat transfer coefficient is 21 \(\frac{\text{W}}{\text{m}^{2}\text{K}}\), which is an average value according to [34]. The above details were fed into the [35] online program, which made the necessary calculations for the overall heat loss coefficient. The result is U\({}_{\text{p,e}}\) = 20.4 \(\frac{\text{W}}{\text{m}^{2}\text{K}}\). Figure 4: Computer realization of the mathematical model. In the value of A\({}_{t,e}\) (which is the responsible surface regarding the heat loss of the cooking tank to the environment), the surface area of both the cover and that part of the tank's wall which is above the mantle are included. Furthermore, U\({}_{p,e}\) is applied to this surface when determining the heat loss of the cooking tank to the environment. Of course, this approach contains certain simplifications (which is clearly advantageous when one would like to apply the model in the practice) since it does not take into account that there is some air between the tank's fluid and the cover. Nevertheless, it must underestimate the pot's real heat performance a bit since the air has some (advantageous) thermal insulating effect. It also means that we do not provide even a slight advantage for the solar pot when presenting its potential applicability. In addition, we carried out simulations (not detailed here) where we tried to model the special effect of the air under the cover. In fact, the difference was so slight (within 1% regarding the modelled temperature values) that we think they are not worth dealing with in the present model. U\({}_{p,in}\) is the overall heat transfer coefficient between the mantle and the cooking tank. It is determined by considering the following:  
1.