## 3 High-Voltage Battery and Air Cooling System Modeling
### Single Temperature Lumped-Parameter Model and Battery Cooling System
#### 3.2.2 Battery Thermal Model  
The heat generated by the battery due to Joule effect (\(Q_{joule}\)) can take three different paths: a portion \(Q_{bat,heat}\) remains stored in the battery and contributes to increasing its temperature, while a quantity \(Q_{conv,nat}\) is dissipated by means of natural convective phenomena with the surrounding air. This latter is expected to remain at a constant temperature whose values refer to the cabin temperature \(T_{cab}\). The final heat path \(Q_{conv,\ AC}\) is different from zero only when the battery cooling system is in operation. This represents the portion of heat transferred from the battery pack to the air flowing at cabin temperature \(T_{cab}\) thanks to Fan2 being activated. When the _HVAC_ system is not in operation, the cabin temperature \(T_{cab}\) has been considered as equal to the ambient temperature. On the other hand, when the HEV passengers activate the air conditioning, cabin temperature has been assumed to be constantly at 20 \({}^{\circ}\)C. The thermal balance for the battery pack can thus be obtained as in Equation (18):  
\[Q_{joule}=Q_{bat,heat}+Q_{conv,nat}+Q_{conv,\ AC} \tag{18}\]  
which in turn can be written by detailing each term as follows:  
\[\begin{array}{l}R_{bat}\cdot{I_{bat}}^{2}=m_{bat}\cdot c_{bat}\cdot\frac{d{T_{ bat}}}{dt}+h_{side,air}\cdot S_{bat}\cdot(T_{bat}-T_{cab})+\\ h_{cooling,air}\cdot S_{cooling}\cdot(T_{bat}-T_{cab})\cdot(Bat_{cooling}>0)\end{array} \tag{19}\]  
where \(m_{bat}\), \(c_{bat}\), \(T_{bat}\), and \(S_{bat}\) refer respectively to battery mass, specific heat, temperature, and exchange surface with the surrounding air. \(Bat_{cooling}\) is a binary flag and its value is set to 1 when the battery air cooling system is in operation (i.e., Fan2 is activated). All the thermal parameters have been reported in Table 2. It should be noted that \(Q_{joule}\) in  
Figure 4: Schematic diagram of the cabin and battery air cooling system.  
Equation (19) has been considered to be related to Joule's heating only. Battery temperature variation due to entropy changes was not considered here in the same way as normally assumed in the literature [43].  
It should be admitted that a few assumptions were made for the illustrated battery thermal model. First, the formulation for \(Q_{conv,\ AC}\) used in Equation (19) assumes that the air flow remains at a constant temperature equal to \(T_{cab}\) during the heat exchange with the battery pack. Moreover, assuming a constant value for the convective coefficient \(h_{cooling,air}\) is not rigorous. Indeed, the convective coefficient is not an intrinsic property of the considered fluid, rather it also depends on the geometry of the object immersed in the fluid flow and on the flow rate magnitude [35]. As consequence, the evaluation of \(h_{cooling,air}\) is rather complex and it is normally obtained by means of empirical correlations available in the literature. These correlations relate to forced convection and they consider the a-dimensional coefficients of Reynolds, Prandtl, and Nusselt [42]. However, these correlations normally refer to very simple cases and they might lead to important errors if applied to more complicate configurations. Although these approximations are not realistic, they aim to avoid introducing further modeling inaccuracy and they could therefore be retained for the purposes of this paper.