# Development of a Thermal Analysis Tool for Linear Machines
## 3 Description of the Thermal Networks
### Thermal Network of the Primary Core  
As it can be seen in Tables 1 and A1, the thermal conductivity of the primary core (steel) is much higher than that of the impregnation and the air. Thus, the conductive thermal resistance of the primary core has been neglected. This means that the model assumes that the temperature of the core of the machine is the same along the whole primary. Due to this simplification, the temperature of the copper is also assumed to be the same in all the slots of the same type because they are exposed to the same losses and the same thermal resistances. The same happens with the influence of the end effect. In the reality, the amplitude of the airgap flux density is different in the edges of linear machines when compared to that in their central part. This generally creates a non-uniform distribution in the core losses. However, the influence of these non-uniform distribution of losses over the temperature is generally small. Hence, assuming a uniform temperature in the whole core of the machine, is a good approximation that helps to the simplification of the LP model. The thermal network of the primary, which is shown in Figure 4, consists of the thermal capacity of the primary core, and convective and radiative thermal resistances to model the heat-exchange with the surrounding elements. The primary core has been separated in different elements to allow an easier calculation of the convective heat transfer coefficient and the heat exchange areas. These elements are the top and bottom surfaces, the \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Material** & **Thermal Conductivity** & **Specific Heat Capacity** & **Density** & **Source** \\ \hline Steel & 30 & 460 & 7650 & [28] \\ Wedge & 0.2 & 1200 & 1000 & [28] \\ Impregnation & 0.21 & 1700 & 1400 & [28] \\ Liner & 0.21 & 1000 & 700 & [28] \\ Copper & 401 & 385 & 8933 & [28] \\ Plywood & 0.12 & 1200 & 550 & [29] \\ Aluminium & 180 & 963 & 2950 & [28] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermal conductivity and specific heat capacity of typical materials. left and right sides, and the front and rear faces. Figure 1a shows the simplified geometry that has been assumed in the construction of the model. The first step when calculating the convective heat transfer coefficient is to determine if it is a natural convection environment or a forced convection situation. In the examples of this article, the thermal tests for the validation of the model were made statically. Hence, natural convection correlations have been applied. Then, the thermal properties of the fluid are calculated at operating temperature,  
\[T_{f}=\frac{T_{s}+T_{\infty}}{2}. \tag{8}\]  
The temperature for the calculation of the properties of the fluid is \(T_{f}\), and \(T_{s}\) and \(T_{\infty}\) are the temperatures of the surface and the ambient in Kelvin degrees. As both the ambient temperature and the surface temperature can change while the machine is operating, in the developed tool, the user has to specify the range of ambient and surface temperatures for the simulation. The tool then generates a matrix of \(T_{f}\) for every possible combination. Then, the respective matrices are calculated for the properties of the fluid, based on Table 1.  
Once that the properties of the surrounding fluid are obtained, the next step is to calculate the thermal volumetric expansion coefficient of the fluid \(\beta\). For ideal gases, this follows  
\[\beta=\frac{1}{T_{f}}. \tag{9}\]  
Then, the Grashof number \(Gr_{L}\) and the Rayleigh \(Ra_{L}\) number are obtained from  
\[Gr_{L}=\frac{\beta\cdot\beta\cdot(T_{s}-T_{\infty})\cdot L_{c}^{3}}{v^{2}}, \tag{10}\]  
and  
\[Ra_{L}=Gr_{L}\cdot Pr, \tag{11}\]  
Figure 4: Thermal network for the primary core. where \(g\) is the gravity in m/s\({}^{2}\), \(L_{c}\) is the characteristic length of the surface, \(v\) is the kinematic viscosity of the fluid at the operating temperature, and \(Pr\) is the Prandtl number. Later, the Nusselt number is calculated from empirical correlations.