## 2 3D Thermal Model
### Heat Transfer Coefficient Boundary Conditions
#### 2.2.5 The Heat Transfer Coefficient of Water Jacket  
Considering the larger of the motor losses produced in the stator portion, the cooling type of totally-enclosed water-cooled is used in the SRM for better cooling. The stator core of the SRM is unfolded by the WJ with a spiral type water channel, as shown in Figure 5. A partial sectional view of the 3D thermal model is shown as Figure 5 including stator and rotor core, shaft spoke, stator windings and housing with water channels. The cooling system removes heat in order to maintain SRM operation within the desired temperature range. The cooling system has a direct influence on the process and service life of the SRM. Presume all the losses produced in the SRM can be dissipated by the cooling water according to [PERSON]'s law of cooling [2]. Hence, the maximum wall temperature of the WJ can be given as:  
\[P_{loss}=h_{cv}A_{s}(T_{max}-T_{f}). \tag{10}\]  
Moreover, the expression of the total quantity of heat generated in the SRM, that is lost due to the cooling water as given as:  
\[P_{loss}=\dot{m}C_{p}(T_{outlet}-T_{inlet}), \tag{11}\]  
Figure 5: The sections of water jacket (WJ) with a spiral type water channel with the SRM. where \(P_{loss}\) is total loss, [W], \(C_{p}\) is specific heat at constant pressure of coolant, [J/kg\({}^{o}\)C], \(\dot{m}\) is the mass flow rate of coolant, [kg/s], \(T_{max}\) is the maximum wall temperature of the WJ, [\({}^{\circ}\)C], and \(T_{f}\) is the mean fluid temperature between the coolant inlet and outlet temperature, [\({}^{\circ}\)C]. \(T_{outlet}\) and \(T_{inlet}\) are the inlet and an outlet temperature of the fluid WJ, [\({}^{\circ}\)C]. Then, the value of the maximum temperature in the wall at operating point conditions is obtained as:  
\[T_{max}=T_{f}+\frac{P_{loss}}{A_{s}h_{cv}}, \tag{12}\]  
The product of \(h_{cv}\) and \(A_{s}\) surface heat transfer area can be determined as \(h_{cv}\)\(A_{s}\) convection heat transfer factor for estimating the cooling capability. It is obvious that the bigger \(h_{cv}\)\(A_{s}\), the smaller the temperature-rise occuring in the SRM under the condition that the total losses of the SRM are constant. The relation between \(\dot{m}\), mass flow rate [kg/s], \(Q\), volumetric flow rate [m\({}^{3}\)/s], \(V\) velocity of the fluid [m\({}^{2}\)/s], \(D_{h}\), hydraulic diameter [m\({}^{2}\)], and \(Re\), Reynolds number are given as follows:  
\[Q=\frac{\dot{m}}{\rho}=VD_{h}, \tag{13}\]  
\[D_{h}=\frac{2 HW}{(H+W)}, \tag{14}\]  
\[Re=\frac{\rho VD_{h}}{\mu}=\frac{2 VHW}{v(H+W)}, \tag{15}\]  
where \(H\), \(W\) are the height and width of the water channel. The \(h_{cv}\) strongly depends on \(Re\). Thus, the \(h_{cv}A_{s}\) value of WJ can be estimated based on empirical formulations for laminar \(Re<2300\), for transition \(2300<Re<4000\) and for turbulent \(Re>4000\) flow respectively as [32; 36; 37]:  
\[h_{cv}A_{s}= \begin{cases}\frac{3.66k_{f}LN(H+W)^{2}}{\frac{HW}{ 0.012k_{f}LN(H+W)^{2}}}&Re\leq 2300,\\ \frac{0.012k_{f}LN(H+W)^{2}}{HW}(Re^{0.87}-280)(P_{r_{f}})^{0.4}(\frac{P_{r_{ f}}}{P_{r_{w}}})^{0.11}[1+(\frac{(2 HW)}{H+W)L})^{2/3}]&2300<Re<10^{4},\\ \frac{0.023k_{f}LN(H+W)^{2}}{HW}Re^{0.8}P_{r_{f}}^{0.4}&Re\geq 10^{4}.\end{cases}  \tag{16}\]  
The surface heat transfer area for rectangular channels is defined as:  
\[A_{s}=2 NL(H+W), \tag{17}\]  
where \(N\) is series number of the water channel in the WJ. The thermal entrance length can be estimated from [38]:  
\[\frac{L}{D_{h}}= \begin{cases}0.05 Re&Re\leq 2300,\\ 0.033 ReP_{r_{f}}&2300<Re<10^{4},\\ 60&Re\geq 10^{4}.\end{cases}  \tag{18}\]  
When fluid flows inside a pipeline, the friction occurs between the moving fluid and the stationary pipe wall. Fluid head losses are commonly the outcome of two mechanisms: friction along the pipe walls, head losses alongside the channel wall are called friction losses due to friction, and losses due to turbulence within the bulk fluid are called minor losses, which can be calculated by the Darcy-Weisbach equation respectively.