## 3 Selection of the Contact Model--The Influence of Heat Conduction  
When using Finite Element Method (FEM) modeling, an equally important but difficult issue is the adoption of the ball-raceway contact model. This model has a significant impact on the heat transfer in the form of conduction between the bearing elements, i.e., balls and raceways. The intensity of heat conduction in solids is a function of the temperature difference between the points, the distance between them and the material properties that the thermal conductivity coefficient represents. The mechanism of heat conduction between two bodies in contact is somewhat more complex. In the literature [5], this issue is referred to as Thermal Contact Conductance (_TCC_). The intensity of heat conduction through the contact area can be represented as follows  
\[q_{TCC}=TCC(T_{t}-T_{c}) \tag{1}\]  
where \(q_{TCC}\)--heat flux per unit area [Wm\({}^{-2}\)], \(T_{t}\), \(T_{c}\)--temperatures of the contact points on the target (\(t\)) and contact (\(c\)) surfaces, \(TCC\) [Ws\({}^{-1}\)m\({}^{-2}\)K\({}^{-1}\)]. The Thermal Contact Conductance (_TCC_) is used to evaluate the heat flux resistance caused by the joint surface. The influential factors of the _TCC_ include surface roughness, contact pressure, thermal conductivity and material hardness. The _TCC_ value is input through a real constant, which is a function of temperature, pressure, time and initial contact detection point location [12]. The determination of the _TCC_ coefficient is a complex issue, and in the literature [5], one can find relationships that allow for their estimation. _TCC_ are function of the effective _RMS_ roughness, of the thermal conductivity \(k\), elastic modulus and flow stress of the material, of the contact pressure, strain and microhardness. [PERSON] [13] presented the analytical form of the dependence that allows to calculate the _TCC_ value for rolling bearings. This dependence, after the transformations made by the author, takes the form  
\[TCC=\frac{4\pi ak_{1}k_{2}}{(k_{1}+k_{2})\int\limits_{0}^{\pi/2}\frac{d\Theta} {\sqrt{1-\left(1-\frac{b^{2}}{a^{2}}\right)\sin^{2}\Theta}}} \tag{2}\]  
where \(k_{1}\), \(k_{2}\)--conductivity coefficients of the materials of both bodies, \(a\), \(b\)--axes of the contact deformation ellipse. The author has shown by means of numerical simulations that for angular contact rolling bearings for which the quotient \(b/a\) is within the range 0.0-0.20, the integral:  
\[\int\limits_{0}^{\pi/2}\frac{d\Theta}{\sqrt{1-\left(1-\frac{b^{2}}{a^{2}} \right)\sin^{2}\Theta}}=\pi^{2}\]  
Assuming that \(k_{1}=k=k\) (due to the same material), _TCC_ can be determined as  
\[TCC\approx\frac{2 ak}{\pi} \tag{3}\]  
The intensity of heat conduction for bodies in contact significantly depends on the initial contact model [12]. Therefore, the aim of this research is to verify this assumption. Commercial FEM systems such as, for example, ANSYS, offer two variants of defining the Thermal Contact Conductance coefficient _TCC_:  
* default, called Program Controlled, when the system automatically calculates the current _TCC_ value according to an undefined algorithm
* manual, when the user enters a numerical _TCC_ value. Figure 2a shows an example of the default contact area for the tested FAG B7013-E test bearing. The Bonded contact in this example covers the entire surface of the inner race and the ball. On the other hand, Figure 2b,c illustrate the areas of contact of the ball with the outer race as a result of the bearing preload. One can formulate the view that in a bearing subjected to external and internal loads, the area of contact deformation significantly differs from the default one (Figure 2a) and, importantly, it will undergo changes with the change of loads. Thus, the area of heat conduction between the balls and the raceways will depend on the actual contact area.