2. Lubrication model of the multi-layer sintered material

The physical configuration of the multi-layer bearing system is shown in Figure 1. As shown in Figure 1, the porous disk bearing (f34 � 4 mm) prepared by powder metallurgy is fixed. And the counterpart rotates at the angular velocity ω. T<sup>1</sup> and T<sup>2</sup> are the thicknesses of the surface and bottom layers. Similarly, k<sup>1</sup> and k<sup>2</sup> are the permeability of the two layers respectively. The o-x-y-z coordinate system is built on the surface of the porous bearing. Assuming that the counterpart surface is smooth. While the bearing surface has a sinusoidal roughness. The fluid film thickness h can be shown as

$$h = h\_0 + \delta\_0 \sin\left(2b\pi r\right)\sin\left(c\theta\right) \tag{1}$$

where the minimum film thickness h0 equal to 10 μm, and the height of the roughness asperity δ<sup>0</sup> equal to 2 μm. The characters b and c represent rough peaks in radial and circumferential directions, respectively.

Figure 1. Two rotary parallel disc samples.

originally by Morgan and Cameron, who gave a solution for a short bearing based on Darcy model [3]. Later, Darcy's equations were extensively used in the study of the lubrication characteristic of oil bearing. The unsteady state, non-Newtonian effect and rough surface were coupling to the lubrication model to improve the numerical accuracy [4–9]. These studies were all focused on single-layer bearing materials, without considering the change porosity in the thickness direction. Meurisse [10] and Usha [11] found that reducing the porosity can prevent the oil leaking into the porous medium and improve the bearing strength and hydrodynamic capacity. But the decreased porosity leads to the decrease of oil content and then will deteriorates the self-lubrication performance. Therefore, it can be concluded that the coexistence of high strength and good lubrication characteristics of the porous bearing are difficult to achieve. This is the biggest problem that oil bearing has encountered in the industrial application. Hence, adjusting the permeability of bearing reasonably is the key to improve the lubrication property. Based on the above studies, Naduvinamani [12] and Rao [13] discussed the effect of the multi-layer structure parameters on the lubrication property, which promoted the theoretical development of the oil bearing. This multiple-layer structure is useful, as it would not only increase the load capacity of the bearing because of reduced oil seepage into its wall but would also help to bring oil between the surfaces, thereby improving the bearing performance when saturated with oil Inadequately. But for now, there is no systematic research on the multi-layer oil bearing materials compared with the single-layer sintered materials. Especially, most researchers ignored the surface Darcy flow to simply the boundary condition in the previous work. It did not coincide with the homogeneity and isotropy hypothesis, which will undoubtedly have a bad effect on the analysis accuracy. In this paper, the multi-layer oil bearing composites with different porosities were made to achieve the unification of high strength and good lubrication property. Hydrodynamic lubrication model of the porous bearing in polar coordinate system was established based on Darcy's law. The effect of surface Darcy flow and porous structure on the lubrication property were also discussed. In the end, the tribology experiments of the multi-layer materials were presented on the end face tribo-

tester to verify the simulation results.

114 Lubrication - Tribology, Lubricants and Additives

film thickness h can be shown as

directions, respectively.

2. Lubrication model of the multi-layer sintered material

The physical configuration of the multi-layer bearing system is shown in Figure 1. As shown in Figure 1, the porous disk bearing (f34 � 4 mm) prepared by powder metallurgy is fixed. And the counterpart rotates at the angular velocity ω. T<sup>1</sup> and T<sup>2</sup> are the thicknesses of the surface and bottom layers. Similarly, k<sup>1</sup> and k<sup>2</sup> are the permeability of the two layers respectively. The o-x-y-z coordinate system is built on the surface of the porous bearing. Assuming that the counterpart surface is smooth. While the bearing surface has a sinusoidal roughness. The fluid

where the minimum film thickness h0 equal to 10 μm, and the height of the roughness asperity δ<sup>0</sup> equal to 2 μm. The characters b and c represent rough peaks in radial and circumferential

h ¼ h<sup>0</sup> þ δ<sup>0</sup> sin 2ð Þ bπr sin ð Þ cθ (1)

If take b equal to 2 and take c equal to 8, the film thickness is show in Figure 2.

Suppose the porous matrix is homogeneous and isotropic. That means the permeability is equal in any coordinate direction. The flow in porous matrix is governed by the Darcy's law.

$$\mathcal{U}l\_0 = \frac{k\_1}{\eta} \frac{\partial p}{\partial x}; V\_0 = \frac{k\_1}{\eta} \frac{\partial p}{\partial y}; W\_0 = -\frac{k\_1}{\eta} \left(T\_1 + \frac{k\_2}{k\_1} T\_2\right) \left(\frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2}\right) \tag{2}$$

where η is the fluid viscosity, and p is the fluid pressure in porous matrix.

Owing to the fluid continuity in the porous bearing, the pressure p satisfies the Laplace equation

$$
\frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} + \frac{\partial^2 p}{\partial z^2} = 0 \tag{3}
$$

Integrating the continuity Eq. (3) over the fluid film thickness and using the Eq. (2) as the velocity conditions, the general Reynolds equation can be obtained. By the coordinate transform technology, the Reynolds equation under the polar coordinate shown as

$$\frac{\partial}{\partial r}\left[r\lambda\frac{\partial p}{\partial r}\right] + \frac{\partial}{r\partial \theta}\left[\lambda\frac{\partial p}{\partial \theta}\right] = -6\eta\omega r \frac{dh}{d\theta} \tag{4}$$

where <sup>λ</sup> <sup>¼</sup> <sup>h</sup><sup>3</sup> � <sup>6</sup>hk<sup>1</sup> � <sup>12</sup>k1T<sup>1</sup> � <sup>12</sup>k2T2. And the surface Darcy flow in the three coordinate directions are all considered.

Similarly, the Reynolds equation without surface Darcy flow shown as

$$\frac{\partial}{\partial r}\left[r\xi\frac{\partial p}{\partial r}\right] + \frac{\partial}{r\partial\theta}\left[\xi\frac{\partial p}{\partial\theta}\right] = -6\eta\omega r \frac{dh}{d\theta} \tag{5}$$

where <sup>ξ</sup> <sup>¼</sup> <sup>h</sup><sup>3</sup> � <sup>12</sup>k1T<sup>1</sup> � <sup>12</sup>k2T2.

As we all know, the internal powder particles are sintered at high temperature during the preparation of the oil bearing by powder metallurgy technology. The pores between the spherical particles are connected with each other to form the porous channels of the oil bearing, which is consistent with the modeling idea of the Kozeny–Carman equation. So the

Figure 2. Film thickness.

relationship between pore structure and fluid pressure drop is often represented by Kozeny– Carman equation. Supposing that the porous bearing material is composed of tiny spherical particles with an average diameter of Dc, the permeability of the bearing can be described as

$$k\_i = \frac{D\_c^2 \varphi\_i^3}{180(1 - \varphi\_i)^2} \tag{6}$$

where w<sup>i</sup> is the porosity of surface and substrate layers of the porous bearing.

The boundary condition shown as.

$$\left.p\right|\_{r=a} = 0; p\_{\theta=0} = p\_{\theta=2\pi} \tag{7}$$

pressure also presents a sine form, similarly to the film thickness distribution. The maximum pressure occurs at the center of the porous circle surface, and the minimum occurs at the outer circle. In addition, the oil film pressure increases with the increase of speed, and the pressure is higher if take the rough surface Darcy flow into consideration in every instantaneous speed. As shown in Figure 3(d), the pressure amplification increases with the increase of speed along the radius direction. And it is more obvious that the effect of the surface Darcy flow on the pressure amplification with the increase of speed. The maximum pressure amplifications in

Figure 3. Effect of surface flow on pressure and its amplification in varied speeds. (a) Pressure with no surface flow (b)

Lubrication and Friction of Porous Oil Bearing Materials http://dx.doi.org/10.5772/intechopen.72620 117

Figure 4 illustrates the variation of the load capacity and friction coefficient with the speeds. It is observed that the surface Darcy flow has obvious significance to the lubrication property. The property turns better when considering the surface flow. And its effect becomes obvious with the increase of speed. For example, when the speed is 1500 r/min, the load capacity caused by surface Darcy flow increases by 11.64%. Moreover, the lubrication property of single

, 5.78 104 Pa, respectively.

, 3.84 104

Pressure with surface flow (c) Pressure distribution (d) The pressure amplification.

three speeds are 1.92 104

where a is the outer radius of the disc bearing.
