## 2 3D Thermal Model
### Heat Transfer Coefficient Boundary Conditions
#### 2.2.5 The Heat Transfer Coefficient of Water Jacket
\[H_{f}=\xi\frac{LV^{2}}{2 gD_{h}}, \tag{19}\]  
\[H_{J}=\zeta\frac{V^{2}}{2g}, \tag{20}\]  
where \(\xi\) and \(\zeta\) are the frictional and local head loss coefficient, which depend on \(Re\) and relative roughness. They are very important since the \(V\) velocity and \(h_{cv}\) of the WJ is influenced by these design parameters. Figure 6 shows the investigation into the influence of the series number of the water channel on the hydraulic diameter of the WJ, for \(N\) in the range (1 to 21), at a constant height of the water channel. It is evident that the hydraulic diameter of the WJ decreases, when the \(N\) of the water channel increases. Moreover, the width of the water channel is defined as:  
\[W_{N}=\frac{W_{t}-(N-1)S}{N}, \tag{21}\]  
where \(S\) is space between channels, \(W_{t}\) is the total width of WJ. The dimensions of the water channel (\(H\) and \(W\)) can be calculated through Equations (13) to (21), hence, the results are shown in Figures 7 and 8. The fluid velocity of WJ increases as the hydraulic diameter of the WJ decreases, as shown in Figure 7a. Figure 7b-d show that the increase in velocity leads to the Reynolds number, the convection heat transfer coefficient, and fluid head losses increasing remarkably. It is evident with volumetric flow increase in the range 0.006 L/s to 0.6 L/s; all values are increased, such as the fluid velocity and the head losses. Moreover, the total head loss of the fluid can exceed the constraint (10 kPa) for the \(N\) greater than 17 at the volumetric flow rate is 0.6 L/s. Figure 8 illustrates the investigation into the influence of the WJ height (40 mm to 49 mm), and a number of water channels (1 to 21), on convection heat transfer analysis of WJ at constant flow rate 0.6 L/s. Considering the maximum convection heat transfer and design constraint on fluid losses with restriction limited to (10 kPa), the dimensions of the water channel (\(H\) and \(W\)) can be obtained, and the result is shown in Figure 8a-d.  
As can be seen in Figure 8d, the WJ has a height of more than 40 mm and (\(N\)) may be less than 21 channels. The total water head loss can exceed the constraint (10 kPa) at the volumetric flow rate of 0.6 L/s, when (\(N\)) is greater than 17. Furthermore, as shown in Figure 8c, the range of \(h_{cv}\) for a number of water channels (1 to 21) can be identified. The proposed structure of the 72/48 SRM in Figure 1 shows the WJ has one inlet and one outlet configuration, on two different sides. This means that the WJ must have a configuration with odd channels. Thus, the approximate range of \(h_{cv}\) for odd  
Figure 6: The hydraulic diameter of WJ with the series number of the water channel. channel (\(N\)) is presented in Table 5. From Table 5, the convection heat transfer coefficient (\(h_{cv}\)) value for 17 channels is the highest realistic value, whilst in 19 channels the fluid loss would be significantly higher. For comparison, it can be seen that the minimum of \(h_{cv}\) for 17 channels is approximately twice and quarter of the value of 9 and 5 channels, respectively. For this reason, 5, 9 and 17 channels are chosen, also 1 channel is also chosen as a benchmark. Based on previous descriptions and calculations, the limitations of configuration dimensions of WJ are proposed and presented in Table 6. The four proposal structures are simulated using ANSYS-CFD. The (\(h_{cv}\)) at various convective surfaces inside the motor for the four proposed structures of WJ were obtained using the CFD method.