## 3 Results and Discussion
### Numerical Simulations  
In order to determine the heat flux that was dissipated from the motor housing, an approach was developed employing the comparison of the results of measurements that were performed in the experimental setup with these of FEM-based computer simulations. As a simulation tool, COMSOL Multiphysics (3.5, COMSOL AB., Burlington, MA, USA) package was applied. This package was validated in the field of heat transfer by [PERSON] [17]. The following mathematical model of the steady-state heat transfer with convection and conduction is used in COMSOL:  
\[\bigtriangledown(-\mathrm{k}\bigtriangledown\mathrm{T})=Q-\rho C_{p}v\bigtriangledown T \tag{3}\]  
where \(T\)--temperature, \(k\)--thermal conductivity, \(Q\)--heat generation per unit volume.  
The geometrical form of the finned motor housing was numerically modelled in two-dimensional (2-D) using CAD tools of COMSOL Multiphysics package. Regarding the boundary conditions, the following assumptions were adopted:  
- thermal conductivity for aluminum: 180 W/(mK),  
- ambient temperature: 295 K,  
Figure 7: Values of the convective heat transfer coefficients.  
external housing surface exposed to the ambient air is cooled by forced convection described by the expression:  
\[-k\frac{\partial\mathrm{T}}{\partial n}=h(T-\mathrm{T}_{t}) \tag{4}\]  
where, \(n\)--unit vector normal to the studied surface, \(T_{f}\)--ambient temperature.  
The interaction between air and external surface housing was modeled by convective heat transfer taken into account. The value of coefficient was calculates by using Equation (1). The values for the coefficient for each considered case are shown in Figure 7.  
Heat losses are function of heat flux transferred onto the inner surface of motor housing. The flux was one of the boundary conditions in numerical model. The value of heat flux was being changed so that the results of average outside temperature of housing were as close as possible to the results of thermographical measurement. A series of simulations were performed for the various current frequencies. The distribution of housing temperature in cross-section for the heat value of 26.95 W is shown in Figure 8. It is the case of current frequency of 50 Hz and mean value of housing surface temperature of 306.10 K (equal to the mean value obtained from the thermographic studies).  
The validation of the numerical results was carried out by comparing the obtained values \(T_{simul}\) with temperature measurements \(T_{measured}\) in the corresponding conditions. A numerical model is considered, as well-validated if a relative difference of aforementioned results is less than 10% [18; 19]. The relative difference is calculated as:  
\[\varepsilon=\frac{\left|\Delta T_{measured}-\Delta T_{simul}\right|}{\Delta T_{ measured}}\times 100\% \tag{5}\]  
Average temperature values simulated/measured at fin base and fin head are given in Table 3, where the value of relative difference \(\varepsilon\%\) is also shown.  
The results clearly demonstrate that the relative difference is within a 10% margin.  
Figure 8: Temperature distribution in cross-section of the housing.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frequency of the Supply Current (Hz)** & **40** & **45** & **50** & **55** \\ \hline base/head of fin, \(T_{measured}\) (K) & 314.60/313.20 & 311.80/310.82 & 306.67/305.90 & 315.04/313.99 \\ base/head of fin, \(T_{simul}\) (K) & 314.57/313.28 & 311.81/310.91 & 306.64/305.92 & 315.06/313.91 \\ \(\varepsilon\%\) & 9.28 & 8.16 & 6.49 & 9.52 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Numerical results and their comparisons with experimental results.