2. Oil-jet cooling mechanism and experimental setup

Lubrication and cooling are important for the rolling bearing fatigue life extension. It is good for the bearing temperature control. The fluid jet is an effective cooling method which has been used widely in heat dissipation. In the rolling bearing operation, the oil-jet method is used for very high-speed conditions. The number of nozzles, the jet velocity, and the oil flow rate are significant to satisfy the bearing lubrication requirement. In order to overcome the detrimental centrifugal effects in high-speed applications, the lubricant oil under pressure is directed at the side of the bearing. The jet velocity should be sufficiently high since the air surrounding the bearing causes the oil to be deflected from the inner ring. Further, a sufficient amount of lubricant oil should be supplied to the bearing. However, an excessive amount of lubricant oil will increase the bearing temperature unnecessarily.

Figure 1(a) shows the schematic configuration of the oil-jet lubrication ball bearing under investigation. The cooling oil is injected into the bearing raceway through a small-diameter nozzle to lubricate and cool the bearing and ensure the safe and reliable operation of the rolling bearing. The high-speed oil-jet atomizes into droplets which then mix with the surrounding air to from a two-phase mixture. Besides, the two-phase mixture is working in an open space environment that has a direct contact with air of the outlets. In this way, coexistence of oil and air twophase flow is formed inside the ball bearing.

In the high-speed operation of bearings, a large amount of heat generated by friction between moving pairs will lead to bearings and their adjacent parts. A sharp increase of friction heat will lead to a significant increase in the working temperature. If the heat not effective discharged, it will inevitably lead to the failure of bearings, which will damage the service life of the bearings. The three major heat dissipation methods inside the ball bearing are shown in Figure 1(b), including the conduction through solid structures, the convection from solid structures to fluids, and the radiation to surrounding media. The latter one is not taken into consideration since it dissipates a negligible part of heat in the bearing case. When the bearing is in a state of thermal equilibrium, the main factors that impact the bearing temperature are bearing load, speed, oil-in temperature, and oil volume fraction inside the bearing, which indicates that the flow field and heat transfer characteristics between the two-phase flow and solid components have a considerable influence in the bearing temperature. Thus, it is essential to clarify the correlations between the fluid flow and thermal behaviors in oil-jet cooling bearing, in order to

Figure 1.

Configuration of the oil-jet cooling ball bearing and heat transfer method: (a) configuration of jet cooling mechanism and (b) heat dissipation methods.

make precise assessment of cooling method and optimize the design of cooling devices.

Experiment apparatuses for testing the flow pattern and temperature distribution of the ball bearing have been built up, as shown in Figure 2. No load is applied in the flow pattern test, as shown in Figure 2(a). The shaft of the tested ball bearing is horizontal. The maximum tested speed is 4500 r/min which is the rated speed of the driving motor. The parameters of the tested ball bearing are given in Table 1. The exact value of the oil volume fraction distribution is difficult to process, since the flow field inside the bearing is compact. Thus, the photograph of the flow pattern is presented to validate the simulated results.

Further, a tested apparatus for temperature test of ball bearing has been built up, as shown in Figure 2(b). The outer ring temperatures of different positions are measured by three temperature sensors. r is the radial coordinates and ω is the circumferential azimuth angle. The technical data of the experimental apparatus is presented in Table 2. The measured bearing temperature distribution can be used for computational model validation.

#### Figure 2.

Test apparatuses: (a) rolling bearing flow pattern test apparatus and (b) rolling bearing temperature distribution test apparatus.


#### Table 1. Specifications of the test ball bearing.

Flow and Heat Transfer in Jet Cooling Rolling Bearing DOI: http://dx.doi.org/10.5772/intechopen.84702


Table 2.

Technical data of the test apparatus.

### 3. Mathematical modeling

Since the two-phase flow changes constantly in high-speed rotating rolling bearings, the development of a computational fluid dynamics (CFD) model with a two-phase flow method to determine the amount of lubricant oil is of practical value as a substitution of empirical correlations. The volume of fluid (VOF) model is selected to model the two-phase interactions inside the bearing for its recognized ability in tracking the air-oil interface. The two-phase coexistence of the bearing is in the turbulent state. The RNG k-ε model is an economical turbulent model for the rotating or swirling flows, so it is chosen as the turbulent governing equations.

#### 3.1 Governing equations

The oil is defined as the primary phase in the two-phase calculation, and its volume fraction in each cell is denoted as φoil, with φoil = 1 representing a pure oil phase and φoil = 0 representing a pure air phase. If 0 < φoil < 1, the cell of interest is in the air-oil two-phase state. If the air volume fraction is similarly denoted as φair, then the sum of the volume fractions of the two phases in the flow domain should meet the follow the constraint:

$$
\rho\_{\rm oil} + \rho\_{\rm air} = \mathbf{1} \tag{1}
$$

The properties of an air-oil two-phase flow in the VOF method are treated as the volumetric average of that of the individual phase. Thus, the density, dynamic viscosity, and thermal conductivity are expressed as

$$
\rho = \rho\_{\text{oil}} \rho\_{\text{oil}} + \rho\_{\text{air}} \rho\_{\text{air}} \tag{2}
$$

$$
\mu = \varphi\_{\rm oil} \mu\_{\rm oil} + \varphi\_{\rm air} \mu\_{\rm air} \tag{3}
$$

$$k = k\_{\text{oil}} a\_{\text{oil}} + k\_{\text{air}} \mu\_{\text{air}} \tag{4}$$

where ρoil is the oil density, ρair is the air density, μoil is the oil dynamic viscosity, μair is the air dynamic viscosity, koil is the oil thermal conductivity, kair is the air thermal conductivity, and kf represents the effective thermal conductivity of the two-phase flow.

The governing equations for the entire computational domain are as follows: Continuity equation

$$\nabla \cdot \left(\rho \overrightarrow{\nu}\right) = \mathbf{0} \tag{5}$$

Momentum equation

$$\nabla \cdot \left( \rho \stackrel{\rightarrow}{\nu} \vec{\nu} \vec{\nu} \right) = \nabla \cdot \left( \mu \left( \nabla \vec{\nu} + \left( \nabla \vec{\nu} \right)^{T} \right) \right) - \nabla \cdot p + \rho \vec{\mathbf{g}} + \vec{F} \tag{6}$$

Energy equation for the fluid

$$\nabla \cdot \left[ \overrightarrow{\nu} \left( \rho E \right) \right] = \nabla \cdot \left( k\_f \nabla T \right) \tag{7}$$

where

$$E = \frac{\alpha\_{\rm oil} \rho\_{\rm oil} E\_{\rm oil} + \alpha\_{\rm air} \rho\_{\rm air} E\_{\rm air}}{\alpha\_{\rm oil} \rho\_{\rm oil} + \alpha\_{\rm air} \rho\_{\rm air}} \tag{8}$$

Energy equation for the solid

$$\nabla \cdot (\mathbf{k}\_\sharp \nabla T) = \mathbf{0} \tag{9}$$

! ! ! where υ is the velocity vector, p is the pressure, F is the external force, g is the gravity acceleration, T is the temperature, E is the energy, and ks is the thermal conductivity of the solid structure which varies with different material types.

In order to model fluid turbulence of high-speed swirl inside the bearing, the RNG k-ε model is selected for it is a recognizable model for rotating or swirling flow. The RNG k-ε turbulence model is a model derived from the transient N-S equation using the renormalization group method. Its k equation and ε equation have the similar form with the standard k-ε turbulence model but are more accurate in calculating the flow field with larger velocity gradient and strong rotational flow by increasing an additional term that is more responsive to the rapid curvature of strain and streamlines.

The transport equation of the turbulence kinetic energy k is

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho k u\_i)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_i} \left[ a\_k \mu \frac{\partial k}{\partial \mathbf{x}\_i} \right] + G\_k + \rho \varepsilon \tag{10}$$

The transport equation of the turbulence dissipation rate equation ε is

$$\frac{\partial(\rho\varepsilon)}{\partial t} + \frac{\partial(\rho\varepsilon u\_i)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_i} \left[ a\_e \mu \frac{\partial \varepsilon}{\partial \mathbf{x}\_i} \right] + \frac{\mathbf{C}\_{1x}\varepsilon}{k} \mathbf{G}\_k - \mathbf{C}\_{2x}\rho \frac{\varepsilon^2}{k} \tag{11}$$

where Gk is the production term of the turbulent kinetic energy caused by the average velocity gradient. xi and ui represent the coordinate directions and the velocity components, respectively. C1<sup>ε</sup>, C2<sup>ε</sup>, Cμ, α<sup>k</sup> and αε are the turbulence model constants.

#### 3.2 Grid meshing

Before the numerical solution of the governing equations, it is important to determine the computational domain and mesh the domain. In order to improve mesh quality and reduce the computer consumption, the computational domain should be simplified and reasonably divided before meshing. Figure 3 shows the relevant grid schematic diagram of the computational domain, which contains a fluid domain and a solid domain. The fluid domain comprises the internal area of the nozzle, the bearing, and the outlets as shown in Figure 3(a). The flow field

Flow and Heat Transfer in Jet Cooling Rolling Bearing DOI: http://dx.doi.org/10.5772/intechopen.84702

Figure 3.

Three-dimensional bearing heat transfer grid system: (a) mesh of the fluid domain and (b) mesh of the solid domain.

inside the bearing is the space enclosed by the inner surface of the bearing and the outer surface of the raceway, which is responsible for the most part of the interactions between the fluid flow and the solid elements.

Figure 3(b) is the solid domain, which includes the shaft, the rolling elements, the inner ring, the outer ring, the bearing seat, and the cage. The structured hexahedral mesh is used in the areas with regular shapes, such as the inner ring, the outer ring, the bearing seat, and the flow area inside the nozzle, and both of the outlets, etc., in contrast, for areas with relatively complex shapes, such as the flow field inside the bearing, was divided by a tetrahedral unstructured mesh.

The rotation of air-oil flow around the bearing rings is imposed by the rolling elements and the cage, which is rotating at a constant speed, which can be calculated by

$$m\_{\rm b} = \frac{1}{2}n\left(1 - \frac{D\_{\rm b}\cos a}{d}\right) \tag{12}$$

where nb is the rotary speed of the rolling elements, n is the rotary speed of the inner ring, Db is the ball diameter, α is the bearing contact angle, and d is the pitch diameter of the bearing.

When setting boundary conditions, the nozzle inlet is initialized with a constant mass flow rate and an initial flow temperature. Flow outlet is set at the end faces of the two outlets and is set to be a pressure outlet boundary condition. The ambient environment is set as 0 Pa gauge pressure and 298 K, respectively. The operating pressure is 101,325 Pa. The viscosity of the oil phase changes with the oil temperature, as shown in Table 3. The multiple reference frame (MRF) method is used to describe the rotation of the rolling elements and the flow field inside the bearing. The sliding mesh planes are defined to deal with the interferences between the stationary and rotating computational domains. The heat source was applied to the contact area between the rolling elements and the raceway. Based on the experiment condition, the heat source power was calculated according to Harris's method [13]. Regarding the heat transfer between the fluid and solid structures, both the air and oil heat transfer properties are considered. Coupled heat transfer wall condition


Table 3. Variation of the oil viscosity with the temperature.

was set at the solid-fluid interface. The convection coefficients were calculated by following the energy conservation equation in the computation.

### 3.3 Solution methods

The ANSYS FLUENT software platform [17] is used to perform the simulations. The solution format is set as a high-order solution mode. For the VOF air-oil twophase model, the air phase is set to be the main phase. All the governing equations are discretized by finite volume (FV) method with the second-order upwind scheme and solved by the SIMPLEC method.

The residuals and the flux conservation on boundaries, for example, the mass flow rate on inlet and outlet boundaries, are monitored to detect the convergence of the governing equations. The conservation standard is set to be 0.01; that is to say, the computation is considered as convergence, whenever the net mass flow rate between the inlet and outlet boundaries drops to 1% of the mass flow rate on the inlets. Other convergence criteria of residual, such as the volume of fluid function and each velocity component, are all set to be 10�<sup>5</sup> , except for that of the turbulent kinetic energy and the turbulent kinetic energy dissipation rate which is 10�<sup>3</sup> .

The criteria are given as follows:

$$\frac{|m\_{\text{inlet}} - m\_{\text{outlet}}|}{|m\_{\text{inlet}}|} < \varepsilon\_m \tag{13}$$

$$\frac{\left|T\_{\text{oj}+5000} - T\_{\text{oj}}\right|}{T\_{\text{oj}+5000}} < \varepsilon\_T \tag{14}$$

where minlet is the oil mass flow at the inlet, moutlet is the oil mass flow at the outlet, TO is the outflow temperature, and ε<sup>m</sup> and ε<sup>T</sup> are the tolerance of oil mass flow and temperature, respectively. The subscript j is the iteration number.

In numerical simulation, the mesh density has a great influence on the accuracy and correctness of the calculation results. Choosing the appropriate number of meshes not only save the workload of the computer but also increase the reliability of the calculation results. Therefore, grid independence verification is an indispensable task in the process of numerical computation. To perform the grid independence verification, a set of numerical calculation is carried out on a computer platform with the flowing configuration: Processor-Intel E5540 � 2@2.53 GHz CPU, RAM-16 GB (4GB � 4), Hard Disk-2 TB, Graphic Card e AMD RADEON HD 6670–2 GB, and Operating system-Window 7 Ultimate 64-bit.

To determine the appropriate number of grids, three different sets of meshes with 135,929, 287,117, and 554,992 cell faces have been tested, and the outlet oil mass flow rate and the average oil volume fraction of the two-phase flow are obtained, as shown in Table 4. The numerical calculations show that with the increase of the number of grids, the variations of the calculated outlet oil mass flow rate and the average oil volume fraction are less than 2%. Considering both


Table 4. Technical data of the test apparatus.

calculation time and accuracy, the mesh with 135,929 cell faces is selected as a suitable number of grids.
