# Development of a Thermal Analysis Tool for Linear Machines
## 2 Overview of the Thermal Model  
_Simulink_ contains a toolbox that is prepared for LP thermal network modeling. Thus, it is a very appropriate software package for a fast implementation of this kind of model. Figure 1 shows the general view of the implemented network.  
Figure 1: General view of the thermal model. (**a**) Simplified geometry. (**b**) Modeled elements. (**c**) General view of the thermal LP network.  
The toolbox has predefined elements for the three main heat transfer mechanisms: conduction, convection and radiation, and also thermal capacitances. The conductive thermal resistance \(R_{cond}\) is calculated with  
\[R_{cond}=\frac{l}{k\cdot A}, \tag{1}\]  
where \(l\), \(k\), and \(A\) are, respectively, the thickness of the material in the direction of the heat-flow, the thermal conductivity of the material and the area of the plane of the material that is perpendicular to the heat-flow. In _Simulink_, the thermal resistance is calculated by the thermal resistor block itself, and the user must specify the values of the thickness, the area and the thermal conductivity.  
However, the block does not allow to change the thermal conductivity of an element in function of the temperature of the material. Thus, certain elements have been modeled with variable thermal resistors instead of bare conductive thermal resistances. As an example of the variation of the thermal conductivity of materials, Figure 2 displays the evolution of the thermal conductivity of air in function of its temperature.  
All the thermal resistances that model some kind of air volume have been consequently modeled with the aforementioned variable resistors.  
The same issue has been detected with the convective heat transfer blocks of the library. They only allow a single value for the heat transfer coefficient. Thus, variable resistors have also been used for modeling all the convective heat-flow paths. The convective resistance \(R_{conv}\) is calculated with  
\[R_{conv}=\frac{1}{h\cdot A}, \tag{2}\]  
where \(h\) is the convective heat transfer coefficient, and \(A\) is the total surface area of the body in contact with the surrounding fluid. The value of \(h\) is dependant of the shape, size, and temperature of the analyzed object and also the thermal properties of the surrounding fluid, air in this case. Thus, it is not possible to use a single value to model the thermal behavior of the machine precisely enough.  
Regarding radiative heat transfer, the variation of the heat-flow is only driven by the temperature difference of the elements in the heat-exchange. Thus, the pre-defined block from the library of Simulink is valid. The block establishes a heat flow \(Q\) based on  
\[Q=k_{rad}\cdot A\cdot\Big{(}T_{s}^{4}-T_{\infty}^{4}\Big{)}. \tag{3}\]  
Here, \(A\), \(T_{s}\), and \(T_{\infty}\) are the emitting area, the temperature of the emitting body and the temperature of the surrounding surfaces. \(k_{rad}\) is the radiation coefficient. This  
Figure 2: Thermal conductivity of air at 1 atm in function of the temperature, adapted from Reference [26].  
last parameter is also changed by the shape of the objects. However, the formula for parallel plates,  
\[k_{rad}=\frac{\sigma}{\frac{1}{\varepsilon_{1}}+\frac{1}{\varepsilon_{2}}-1}, \tag{4}\]  
has been used for all the radiating elements to simplify the calculation process.  
In this way, the only configuration parameters are the area of the body and the emissivities of the emitting \(\varepsilon_{1}\) and receiving materials \(\varepsilon_{2}\). \(\sigma\) is the Stefan-Boltzmann constant, \(5.6704\times 10^{-8\text{ W}/m^{2}k^{4}}\).  
To ensure universality between different devices, the model only consists of three parts: end windings, primary core, and slot. The secondary is not contemplated by the model because, as demonstrated in Reference [22], the temperature of the secondary side of the machine is not affected by the losses if the stroke length is large enough.