## 4 Influence of Convection Conditions on the Temperature  
Convection is the dominant mechanism for heat flow from bearing components to the environment. Other heat flow mechanisms, such as radiation, have less impact on the thermal condition of the bearing, and will therefore not be taken into account.  
Convection, in addition to modeling heat sources and the accepted contact model of balls with raceways, has a decisive influence on the temperature distribution.  
The bearing, from the point of heat transfer with the environment, is a specific assembly because some of its components are stationary (e.g., the body of the seat of bearing) and the other part is movable (the shaft on which the bearing rests). This is important from the point of view of the convection mechanism (Figure 3a).  
Figure 3: Illustrative drawing of a headstock with angular contact bearings (**a**) and geometric model of the bearing for modeling with the FEM (**b**).  
Figure 2: Exemplary visualization of the contact areas: (**a**) Bonded, for the unloaded contact of the ball with the inner race, (**b,c**) Bonded, for the loaded contact of the outer race with the ball by the preload force.  
There are generally two types of convection, the so-called free and forced. The first concerns elements that are stationary or moving at lower speeds. The heat flow is due to the temperature difference. The second is characteristic of moving parts.  
The amount of heat due to free convection is proportional to the size of the surface and the so-called convection coefficient \(\alpha_{s}\), which can be determined from the relationship [14]  
\[\alpha_{s}=c(T-T_{out})^{n} \tag{4}\]  
where \(T\), \(T_{out}\)--surface and ambient temperate, and \(c\), \(n\)--constants depending on the multiplication of Grashof number Gr and Prandtl Pr.  
For the considered test bearing (for a solid body of the stand in the form of a cylinder) \(c=1.7\) and \(n=0.33\)[14]. Such coefficients will be taken into account in the FEM simulation tests.  
If we assume that the temperatures in the bearing are within the range of 25 \({}^{\circ}\)C-100 \({}^{\circ}\)C, the convection coefficient \(\alpha_{s}=(2.44-7.31)\) W/m\({}^{2}\).  
The amount of heat of forced convection is proportional to the size of the surface and the convection coefficient \(\alpha_{k}\), which can be determined from the relationship [14]  
\[\alpha_{k}=5.9\cdot d^{0.05}\cdot\left(\frac{0.5\cdot V^{2}}{d}+\frac{9.81}{ \frac{T_{out}}{T-T_{out}}+0.5}\right)^{0.35}\approx 0.612d^{0.35}n^{0.68} \tag{5}\]  
where \(V\)--linear speed of the moving element [ms\({}^{-1}\)], \(n\)--rotational speed of the moving element [rpm] and \(d\)--diameter of the rotating element [m].  
The author showed that the influence of the temperature difference between the movable element and the environment is small, hence the \(\alpha_{k}\) coefficient may finally be presented as a function of the rotational speed and the diameter of the movable element. In the considered example of the test bearing, the \(\alpha_{k}\) coefficient in the speed range 1000 to 9000 rpm is within the range of 26.7 to 119 Wm\({}^{-2}\)T\({}^{-1}\), which is significantly greater than the coefficient \(\alpha_{s}\). This means that, with comparable heat convection surfaces, much more heat will flow into the environment through shaft than through the seat body in which the bearing rests. This can significantly differentiate the bearing outer and inner ring temperatures.