## 3 Discussion
### Summary and Practical Application  
A summary showing the procedure to obtain the temperature of an SC cell will be hereafter presented. The input data that define the SC will be the ones given by the manufacturer or those obtained in laboratory normalized tests: electrical resistance, R, capacitance, C, thermal resistance, R\({}_{\text{TH}}\), and thermal capacitance, C\({}_{\text{TH}}\). These are used together with the values of other variables, such as ambient temperature T\({}_{\text{amb}}\) (assumed invariable), the initial value of the internal voltage, U\({}_{0}\), initial value of the cell temperature, T\({}_{0}\), and the charge or discharge power must be selected by the user, P.  
To obtain the temperature of the cell vs. time, the process is the following:  
1. By means of the values of R, C, U\({}_{0}\), and P, the function g(t), described in the Equation (2), must be defined.
2. Once g(t) is obtained, the function g\({}_{1}\)(t) must be calculated by means of Equation (1).
3. By using the values of R, C, R\({}_{\text{TH}}\), and C\({}_{\text{TH}}\), the constant \"a\" will be defined according to Equation (16).
4. The initial thermal jump, \(\theta_{0}\), will be calculated as the difference between the initial temperature of the cell, T\({}_{0}\), and the ambient temperature, T\({}_{\text{amb}}\); \(\theta_{0}\) = T\({}_{0}\) - T\({}_{\text{amb}}\).
5. Once \(\theta_{0}\), R\({}_{\text{TH}}\), P, a, and g\({}_{1(0)}\) (g\({}_{1}\)(t) for t = 0) are known, the value of the constants k\({}_{\Theta 1}\) and k\({}_{\Theta 2}\), according to the Equations (29) and (30), will be calculated. The function f(a, g\({}_{1(0)}\), g\({}_{1}\)) will be defined as well, as stated in Equation (28), g\({}_{1}\)(t) being the function previously defined in the point 2 of this summary.
6. The evolution of the temperature of the cell vs. time will be then the following:  
Figure 10: Evolution of the cell temperature as a function of the external voltage.  
Figure 9: Evolution of the cell temperature as a function of the variable g\({}_{1}\).  
\[T_{\rm cell}(t)=k_{01}\cdot\left(\frac{g_{1(0)}}{g_{1}(t)}\right)^{a}\cdot\exp(a \cdot g_{1}(t))+k_{02}\cdot f\left(a,g_{1(0)}g_{1(t)}\right)+T_{\rm amb} \tag{39}\]  
To express the temperature of the cell as a function of any other electrical variable, i.e., internal or external voltage, dissipated energy, etc., the process is nearly the same. Steps 1 and 2 should be omitted and \(g_{1}\) should be defined as a function of the selected variable. If for the sake of clarity the desired variable is called \"y\", the procedure is as simple as replacing \(g_{1}\) by \(g_{1}(y)\); \(f(a,g_{1(0)},g_{1})=f(a,g_{1(0)}(y),g_{1}(y))\) and with the f function; \(f(a,g_{1(0)}(y),g_{1}(y))\), the evolution of the temperature of the cell vs. \(y\) will be as follows:  
\[T_{\rm cell}(y)=k_{01}\cdot\left(\frac{g_{1(0)}(y)}{g_{1}(y)}\right)^{a}\cdot \exp(a\cdot g_{1}(y))+k_{02}\cdot f\left(a,g_{1(0)}(y),g_{1}(y)\right)+T_{\rm amb}. \tag{40}\]  
This result has been obtained in the same way it was done in Equation (36) to calculate the temperature of the cell as function of the external voltage, \(u_{\rm co}\).