MIXED CONVECTION HEAT TRANSFER OF SIO2-WATER AND ALUMINA-PAO NANO-LUBRICANTS USED IN A MECHANICAL BALL BEARING

In this study, the mixed convection heat transfer in a mechanical ball bearing filled with nano-lubricants were investigated theoretically. In our case, the bearing including eight balls revolving in counter clockwise while the inner shaft rotates in clockwise direction and the inner and outer walls of bearing were kept at constant hot and cold temperatures, respectively. Two kinds of nano-lubricants SiO2-water and AluminaPolyalphaolefin (PAO) with different shapes of nanoparticles were considered. The governing equations including velocity, pressure, and temperature formulation were solved based on the Galerkin finite element method. The governing parameters such as nanoparticle volume fraction, Reynolds and Rayleigh numbers, etc., were discussed. It turns out that the average Nusselt number increases by increasing the nanoparticle volume fraction (averagely 15% for each 0.02 increase) and the oil-based nano-lubricant has greater Nusselt number than the water based one. More importantly, the Nono-rod Alumina was found to show much greater heat transfer performance (averagely 5%) than the spherical alumina nanoparticles and nano-rod Alumina-PAO has the best performance and maximum Nusselt numbers for the heat transfer.

In this study, it is aimed to find the mixed heat transfer treatment of two kinds (water and oil based) of nanolubricant used in a ball bearing. Also, the effect of nanoparticles shape on the heat transfer will be investigated to find the best nano-lubricant from the heat transfer view point for this interesting application.

PROBLEM DESCRIPTION
As shown in Fig. 1, a mechanical ball bearing is considered with eight heated balls. It is assumed that the space between the balls is filled with nano-lubricants such as SiO2-water and alumina-PAO nano-oil. The outer wall of bearing is kept at constant Tc, while the inner wall, due to shaft rotation, is in Th high temperature. As seen in Fig.  2, detailed boundary conditions are presented while the balls temperature is 2Th and they rotate in counter-clockwise while the inner shaft rotates in clock-wise directions. Tables 1 and 2 present the thermal properties of applied nanolubricant. It is tried to find the mixed heat transfer of these nano-lubricants, so the following non-dimensional parameters should be defined to change the dimensional governing equations [29][30] The 2D mixed convection flow in the problem using conservation of mass, momentum, and energy can be written as the following dimensionless form [29][30] The equations of thermal properties of two different considered nano-lubricants are determined as the following sections based on the literature. a) Farooq et al. [29] b) Current code

SIO2 Water Based Nano-Lubricant
Since the lubricating mechanism of nanoparticles is complicated and is related to SiO2 suitable characteristics such as rolling friction mechanism, thin film lubrication mechanism, boundary lubrication layers and etc., it is thus used in hot rolling lubrication machinery [31][32]. Ajeel et al. [33] used the following relations for the SiO2-water thermal properties as summarized in Table 1&2.
The nanofluid density and its heat capacity can be calculated as follows [33]: To compute the effective thermal conductivity, the empirical correlation has been adopted which takes into account the influence of Brownian motion as shown below [33] = + where K is the Boltzmann constant and, Also, β for the SiO2 nanoparticles is presented as [33] The effective dynamic viscosity of nanofluid is given as [34] = ( 1 × −1.03 where, equivalent diameter of based molecule is, While Jumpholkul et al. [35] used the following relations from the literature for their SiO2-water modeling:

Alumina-PAO Based Nano-Lubricant
Polyalphaolefin (PAO) is the most common major synthetic base oil used in industrial and automotive lubricants. PAO has more Newtonian treatment than other oil lubricants [36]. So, based on this assumption, Hajmohammadi [37] applied alumina-PAO as a nano-lubricant in a rotary system such as between two cylinders. The following correlations were used for its thermal properties [38] A short review of Alumina-polyalphaolefin thermal properties is performed by Yu et al. [39] = (1 − ) + for Nano-rods nanoparticles (22) For solid-liquid mixtures, the relative thermal conductivity can be estimated by the Hamilton-Crosser model [40] ( ) where the shape factor is n = 3/ψ, and ψ is the sphericity defined as the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle. For spherical particles, ψ= 1. Yu et al. [39] proposed that: For spherical nanoparticles (24) = 1 + 9.4539 For nano-rods nanoparticles (25)  where r is the radial direction. The average Nusselt numbers on the bearing outer and inner walls are named as Nu1 and Nu2, respectively as:

METHODOLOGY OF SOLUTION
In this study, the governing equations beside the boundary conditions are analysed numerically by Galerkin weighted residual along with finite element methods. The finite element analysis of the momentum equations (3) and (4) is showing by the following procedure: Firstly, we employ the penalty finite element method by eliminating the pressure ( ) with a penalty parameter ( ) as the following:


Selection of the interpolation functions for providing an approximation for the velocity distribution and temperature distribution as: .
The nonlinear residual equations for the momentum equations that obtained from the Galerkin weighted residual finiteelement method are: where the superscript is the approximate index, subscripts , and are the residual number, node number and iteration number, respectively. To simplify the nonlinear terms in the momentum equations, a Newton-Raphson iteration algorithm was used. The iteration of the present study is assumed to be convergence solution when the corresponding error of each variable is equal or less than 10 -5 .

RESULTS AND DISCUSSION VALIDATION STUDY
To validate the current FEM code based on Galerkin weighted residual method, the initial code results is compared to Farooq et al. [29] outcomes as depicted in Fig. 3  inner walls for SiO2-water nano-lubricant when Ra=10000 and φ=0.08 in different Reynolds numbers

Mesh Independency
As depicted in Fig. 2, mesh generation on the geometry is made using triangles while over the three boundaries (outer, inner and balls walls) boundary mesh layer is applied for better accuracy which are in quad shapes. In this section, eight grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and ensure that the 145 results are grid independent as depicted in Table 3 for SiO2-water at Re=25, Ra=10000 and φ=0.08. As seen for both Nu1 and Nu2 results, the G7 grid is the most suitable grid size from both accuracy and time computation study.

SIO2-Water Nano-Lubricant
As mentioned in introduction section, water-based nano-lubricant is more often used for the rolling machinery for cooling and lubricating purposes. In this section, it is tried to see its performance in ball bearing lubricant application and compare the results with the oil-based nano-lubrcant which is presented in next section. Figs. 4-9 are depicted for the SiO2-water nano-lubricant with φ=0.08 to show the effect of Reynolds and Rayleigh numbers on the results. Fig. 4 and Fig. 5 demonstrate the effect of Ra on the temperature and stream lines contours, respectively. As seen for the Ra=8000 case, due to natural convection effect, the temperatures value between the balls is greater than the other cases, while for the streamline, due to low velocity of Re=25, there is not a significant change between the graphs and just one case is presented as sample case. The effects of the Ra numbers on the local Nu1 and Nu2 (for outer and inner walls, respectively) are depicted in Fig. 6. For both walls, Ra=10000 has the maximum values of Nusselt numbers and also have the maximum range of variations, while the Ra=1000 has the minimum values of variations which is approximately 1 on the inner wall length. Figs. 7 & 8 show the effect of Reynolds numbers on the temperature and streamline contours, respectively when Ra=10000, φ=0.08. As seen for the Re=500, the maximum temperatures between the balls occurs. Therefore, this case is considered to have more heat transfer performance than the others. This fact is depicted in Fig. 9 for the local Nusselt numbers. The maximum local Nusselt numbers (for both walls) occurs for the Re=500 and maximum variation for the outer wall occurs for this case while for the inner case maximum variation of Nusselt numbers happens for the Re=25. These variations have a significant effect on the average Nusselt numbers which will be fully discussed in section 4.5.

Alumina-PAO Nano-Lubricant
In this section the effect of alumina-PAO nano-lubricant on the mixed heat transfer mechanism is investigated through Figs. 10-19 for two kinds of nanoparticle shapes, i.e., spherical and nano-rods. Figs. 10-15 show the effect of spherical alumina nanoparticles on the results. Fig. 10 presents the Ra effects on the temperature and streamline contours. By increasing the Ra, the temperature distribution in the domain is much greater due to better heat transfer through the natural convection mechanism. However, in the stream lines there is not a significant difference due to high viscosity of PAO and low Reynolds numbers (Re=25). Based on this definitions Fig. 11 confirms that Ra=10000 has the maximum values of local Nusselt numbers for the both walls under study. Fig. 12 shows that by increasing the Re number (against the Ra increasing) the streamline contours varies significantly and vortexes between the balls deformed from symmetry (for Re=25) to asymmetry (for Re=500) due to nano-lubricant flow in higher Re numbers. This effect of Re numbers on the local Nusselt numbers are depicted via Fig. 13 which confirms that Re=500 has the maximum values and variations of Nusselt numbers. To see the effect of spherical nanoparticles volume fractions on the heat transfer mechanism, Figs. 14 & 15 are depicted for Ra=10000, Re=25. It is completely clear that increasing the nanoparticles volume fraction make an enhancement in the heat transfer and improvements in local Nusselt numbers, consequently. In this case, as seen in Fig. 16, Ra=1000 and 3000 has greater maximum values of local Nusselt number of outer wall while for the inner wall maximum values occurs for the Ra=5000. When Ra becomes=10000, it has the minimum peak values in Nusselt numbers. The results could be attributed to the difference between viscosity and thermal conductivity of these two nanoparticle shapes as presented in section 2.2. Based on defined equations alumina nano-rods has greater viscosity and thermal conductivity than spherical nanoparticles. Figs. 17 and 18 shows the effect of Re number on the outcomes Re=50 has the maximum values of Nusselt numbers as seen in the graphs. Finally, the effect of nanoparticles volume fraction on the local Nusselt numbers is depicted in Fig. 19. Increasing this parameter in nano-rods makes an improvement in Nusselt numbers as well as the spherical nanoparticles treatments.

Average Nusselt Numbers
To have a comparison between three described nano-lubricants, the average Nusselt numbers (Nu1 and Nu2) are presented in Tables 4-9 to also show the effect of Rayleigh, Reynolds and nanoparticles volume fractions on the average Nusselt numbers. From these figures it can be visible that water-based nano-lubricant has the lowest Nusselt numbers, while the Nano-rod alumina-PAO has the maximum values for the Nusselt numbers and can be introduced as the most efficient nano-lubricant in this application. Table 4 and 5 reveal that in most cases increasing the Rayleigh number makes and increase in Nusselt number due to more natural convection heat transfer while Tables 6 and 7 exhibits that Reynolds increments have different treatments for water based and oil-based nano-lubricants which increase the Nu for the water based and decrease it for the oil-based nano-lubricants, averagely. As the last parameter  Tables 8 and 9 show the effect of nanoparticles volume fraction on the Nusselt numbers. It is evident that greater values lead to higher thermal conductivity and consequently increase the Nusselt numbers.

CONCLUSION
In this paper, the mixed convection inside a mechanical ball bearing with the outer cold fixed wall and inner hot rotating wall and treated with nano-lubricants (SiO2-water and Alumina-PAO) was studied numerically using COMSOL Multiphysics code built on a finite element method. The influence of Rayleigh number, Reynolds number, nanoparticles volume fraction, and shapes of nanoparticles on the heat transfer mechanism is investigated and it is found that Rayleigh number increament enhances the heat transfer process as well as the nanoparticles volume fraction, averagely, while the Reynolds increasing has different treatments. Also, the Nono-rod Alumina was found to show much greater heat transfer performance than the spherical alumina nano-particles. It was recommended that nano-rod Alumina-PAO has the best performance and maximum Nusselt numbers for the heat transfer in these applications.