# New Physically-Based Mathematical Model and Experiments for a Recently Invented Solar Pot
## 3 Mathematical Modelling
### Exact Calculation of the Equilibrium  
Equations (3) and (4) represent the determining formulas for the equilibrium values of the modelled temperatures (\(\mathrm{T_{m,out}}\) and \(\mathrm{T_{t}}\)) in accordance with that the derivatives (at the left hand side of Equations (1) and (2) should be zero in the equilibrium. The equilibrium values of \(\mathrm{T_{m,out}}\) and \(\mathrm{T_{t}}\) are denoted by \(\mathrm{T_{m,out,eq}}\) and \(\mathrm{T_{t,eq}}\), respectively.  
\[0=\frac{\mathrm{v}}{\mathrm{V_{m}}}\big{(}\mathrm{T_{m,in}}-\mathrm{T_{m,out, eq}}\big{)}+\frac{\mathrm{A_{m,e}U_{p,e}}}{\mathrm{pcV_{m}}}\big{(}\mathrm{T_{p,e}}- \mathrm{T_{m,out,eq}}\big{)}+\frac{\mathrm{A_{p,in}U_{p,in}}}{\mathrm{pcV_{m}}} \big{(}\mathrm{T_{t,eq}}-\mathrm{T_{m,out,eq}}\big{)}, \tag{3}\]  
\[0=\frac{\mathrm{A_{p,in}U_{p,in}}}{\mathrm{pcV_{t}}}\big{(}\mathrm{T_{m,out,eq} }-\mathrm{T_{t,eq}}\big{)}+\frac{\mathrm{A_{t,e}U_{p,e}}}{\mathrm{pcV_{t}}} \big{(}\mathrm{T_{p,e}}-\mathrm{T_{t,eq}}\big{)}, \tag{4}\]  
from which the equilibrium temperatures can be expressed explicitly as follows:  
\[\mathrm{T_{m,out,eq}}=\frac{\mathrm{A_{p,in}U_{p,in}vpcT_{m,in}}+\mathrm{A_{p, in}U_{p,in}}\mathrm{A_{m,e}U_{p,e}T_{p,e}+A_{m,e}U_{p,e}^{2}A_{te}T_{p,e}+A_{p,in}U_{p,in}A_{ te}U_{p,e}T_{p,e}+A_{te}U_{p,e}vpcT_{m,in}}}{\mathrm{A_{p,in}U_{p,in}vpc+A_{te}U_{p,e}vpc+A_{ p,in}U_{p,in}A_{me}U_{p,e}+A_{me}U_{p,e}^{2}A_{te}+A_{p,in}U_{p,in}A_{te}U_{p,e}}, \tag{5}\]  
\[\mathrm{T_{t,eq}}=\frac{\mathrm{A_{p,in}U_{p,in}vpcT_{m,in}}+\mathrm{A_{p,in}U _{p,in}}\mathrm{A_{m,e}U_{p,e}T_{p,e}+A_{m,e}U_{p,e}^{2}A_{te}T_{p,e}+A_{p,in}U _{p,in}A_{te}U_{p,e}T_{p,e}+A_{t,e}U_{p,e}vpcT_{p,e}}}{\mathrm{A_{p,in}U_{p,in} vpc+A_{t,e}U_{p,e}vpc+A_{p,in}U_{p,in}A_{me}U_{p,e}+A_{m,e}U_{p,e}^{2}A_{te}+A_{p,in}U_{p,in}A_{ te}U_{p,e}}. \tag{6}\]  
After substituting the above parameter values (\(\mathrm{A_{m,e}}=0.19\mathrm{m^{2}}\), \(\mathrm{A_{t,eq}}=0.041\mathrm{m^{2}}\), \(\ldots\)) into Equations (5) and (6), 77.9 \({}^{\circ}\)C and 78.9 \({}^{\circ}\)C is resulted in case of 100 \(\frac{\mathrm{L}}{\mathrm{h}}\) and 200 \(\frac{\mathrm{L}}{\mathrm{h}}\), respectively, for \(\mathrm{T_{m,out,eq}}\), and 76.8 \({}^{\circ}\)C and 77.8 \({}^{\circ}\)C is resulted in case of 100 \(\frac{\mathrm{L}}{\mathrm{h}}\) and 200 \(\frac{\mathrm{L}}{\mathrm{h}}\), respectively, for \(\mathrm{T_{t,eq}}\). Taking into account that the simulated final tank temperature was 76.5 \({}^{\circ}\)C and 77.7 \({}^{\circ}\)C in case of 100 \(\frac{\mathrm{L}}{\mathrm{h}}\) and 200 \(\frac{\mathrm{L}}{\mathrm{h}}\), respectively, at the end of Section 3.2, we can state that the (estimated) numerical (simulated) temperature values are well in accordance with the exact equilibrium values. (It is normal that the monotonically increasing simulated values are slightly lower since the equilibrium could be reached in infinitely long time, in principle.) Considering that the initial temperature values in the simulations (25 \({}^{\circ}\)C) was rather far from the exact equilibrium values, it is suggested that the equilibrium is (globally asymptotically) stable.