## 2 Research Methodology
### Mathematical Model  
The reference mathematical model used for battery convective cooling is shown in Figure 10. This model analyzes the convective heat transfer in a single transversal section of a battery cell and it has already been used and validated in several research studies [24; 32; 33; 34; 35; 36; 37]. However, such a model is most suitable for cylindrical battery cells in a vertical position. This paper proposes a new mathematical model using a thermal resistance network for cylindrical battery cells placed in a horizontal position, where the thermal  
Figure 10: Mathematical model of a cell’s transversal section. Figure 9: Heat transfer coefficient variation with generated heat obtained from CFD simulations. characteristics of a battery cell are not considered as a whole, but as an interaction between several sections of the cell. This approach allows us to consider the differences of convective thermal resistance on different surface areas on the cell, independently of the temperature between different zones inside the cell, according to Figure 11.  
The convective heat transfer rate on a cell's surface is defined by Equation (4). \[Q_{con,i,j}=\frac{T_{s,i,j}-T_{f,i,j-1}}{R_{con,i,j}} \tag{4}\]  
The value of convective heat transfer rate depends on the temperature difference between the cell surface and the air (fluid) coming from the previous cell section or the convective thermal resistance, which is calculated based on the convective heat transfer coefficient value obtained from the CFD simulations. The changes in the temperatures in the core and the surface of the cell or the temperatures of the air are represented by a system of differential equations (Equations (5)-(8). It is observed that the variation in the core temperature is influenced by the heat generated inside the cell and the temperature on the surface. At the same time, given the division of the batteries, the heat transfer by conduction between these sections must be considered, according to Figure 11.  
For the two sections at the end of a cell, Equation (5) is used, where C\({}_{\text{c}}\) is the battery core heat capacity, R\({}_{\text{c}}\) is the core thermal resistance in radial direction and R\({}_{\text{c-c}}\) represents the core thermal resistance in axial direction. For the other three middle sections, Equation (6) is used:  
\[C_{c}\frac{dT_{c,i,j}}{d\tau}=Q+\frac{T_{s,i,j}-T_{c,i,j}}{R_{c}}+\frac{T_{c,i,j\pm 1}-T_{c,i,j}}{R_{c-c}} \tag{5}\]  
\[C_{c}\frac{dT_{c,i,j}}{d\tau}=Q+\frac{T_{s,i,j}-T_{c,i,j}}{R_{c}}+\frac{T_{c,i,j-1}+T_{c,i,j+1}-2T_{c,i,j}}{R_{c-c}} \tag{6}\]  
The variation of the cell surface temperature is influenced by both the core temperature and the temperature of the air that comes into contact with the battery cell, according to Equation (7):  
\[C_{s}\frac{dT_{s,i,j}}{d\tau}=\frac{T_{f,i,j-1}-T_{s,i,j}}{R_{con,i,j}}-\frac{ T_{s,i,j}-T_{c,i,j}}{R_{c}} \tag{7}\]  
where C\({}_{s}\) is the battery cell surface heat capacity. Figure 11: Proposed mathematical model for battery cells placed in horizontal position. Coolant, in this case the air, is heated after passing next to each cell section, with this heating being influenced by the temperature of the cooled surfaces and the flow characteristics, according to Equation (8):  
Figure 10:0  
The convective heat transfer resistance, defined in Equation (9), depends on the characteristics of the air flow and the number of transfer units:  
Figure 10:1  
The number of transfer units (NTU), representing the dimensionless size of the heat transfer, is defined according to the convective heat transfer area, the heat transfer coefficient and the air flow characteristics:  
Figure 10:2  
where the convection heat transfer coefficient \(h\) is obtained from CFD simulations.