Study on lubrication characteristics of rotary combination seals under stress relaxation

In this paper, a two-dimensional stochastic process conforming to the Gaussian distribution is jointly generated through a function toolbox of numerical software and filter of computer. On this basis, the autocorrelation function as well as the Fourier transform and Fourier inverse transform are used to finally generate the surface roughness distribution function z(x,y) for the sealing area [23, 24], and the function is applied to generate the roughness matrix for RMS roughness of 0.4 and 0.8. Finally, the EHL model is numerically analyzed and calculated using the roughness matrices. The parameters related to the roughness are: the contact area length of the initial slip ring is 2 mm, and the RMS roughness is 0.4 μm. Figures 3(a) and (b) show the RMS roughness distributions for 0.4 μm and 0.8 μm, respectively.

Zoom In Zoom Out Reset image size

Based on the lubrication characteristics of the rotating combination seal, the conventional Reynolds equation [25] is simplified and the following assumptions are made:

• (1)  
  The fluid does not slide at the interface, i.e., the oil layer sticks to the interface at the same speed as the interface;
• (2)  
  The variation of pressure along the direction of lubrication film thickness is not considered;
• (3)  
  The surface of the slip ring is rough and the surface of the main shaft is smooth;
• (4)  
  The density of the fluid medium is constant and incompressible;
• (5)  
  The fluid in the contact area is considered as a Newtonian fluid. Based on the above assumptions, the Reynolds equation can be simplified by neglecting the stretching and squeezing effects as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial x}\left(\displaystyle \frac{{h}^{3}}{\eta }\displaystyle \frac{\partial p}{\partial x}\right)+\displaystyle \frac{\partial }{\partial y}\left(\displaystyle \frac{{h}^{3}}{\eta }\displaystyle \frac{\partial p}{\partial y}\right)=6v\displaystyle \frac{\partial h}{\partial x}\end{eqnarray} \tag{ 3 }$$

Where h is the oil film thickness of the sealing gap (m); η is the lubricant viscosity of the sealing gap (Pa.s); p is the pressure of the lubricant in the sealing gap (Pa); v is the rotating linear velocity (m/s).

According to EHL theory, when the oil film pressure reaches a certain large value, the oil film will break naturally in the diffusion region. When Reynolds boundary conditions are used to solve the numerical solution of Reynolds equation, each row in the grid area is calculated point by point from the starting edge to the ending edge. If the pressure at a point is calculated to be negative, cavitation is present at that point. The location of this point can be used as an approximate location of the natural rupture boundary of the oil film. The pressure at each point after this point is taken to be zero for each iteration. As the number of iterations increases, the approximate location of the rupture boundary will gradually approach the natural rupture boundary, making the pressure distribution of the entire oil film of the rotating combination seal close to the pressure distribution of the actual Reynolds boundary conditions.

Based on the EHL theory, the oil film thickness mainly consists of the deformation of the seal groove and the slip ring in contact with the oil film under the action of oil pressure. In addition, the influence of the pre-pressure of the O-ring on the deformation of the inner surface of the sealing groove and the influence of the roughness on the sealing oil film are also considered. In summary, the constraint equation of the oil film thickness is shown in equation (4).

$$\begin{eqnarray}&&h={u}_{h}+{u}_{{\rm{p}}}-{u}_{0}+R\end{eqnarray} \tag{ 4 }$$

Where u_h is the deformation of the contact surface of the shaft. Considering that the stiffness of the shaft is much larger than that of the slip ring, u_h is assumed to be zero; u_p is the deformation of the slip ring under fluid pressure; u₀ is the deformation of the slip ring caused by pre-pressure; R is a function of the roughness of the seal contact surface.

The temperature distribution of the lubricant film has a significant impact on lubrication performance. The viscosity of lubricating oil decreases significantly with the increase of temperature, which affects the distribution of oil film pressure and oil film thickness, as well as the bearing capacity of the oil film. To make the calculation more convenient and effective, the oil film temperature T, oil film pressure p, and oil film viscosity n are considered to be consistent in the oil film thickness direction, and the density is not affected by temperature. The general expression of the general energy equation [26] is integrated along the oil film thickness direction to obtain a simplified form of the energy equation:

$$\begin{eqnarray}&&{q}_{x}\displaystyle \frac{\partial T}{\partial x}+{q}_{y}\displaystyle \frac{\partial T}{\partial y}=\displaystyle \frac{\eta {U}^{2}}{J\rho {c}_{\rho }h}+\displaystyle \frac{{h}^{3}}{12\eta J\rho {c}_{\rho }}\left[{\left(\displaystyle \frac{\partial p}{\partial x}\right)}^{2}+{\left(\displaystyle \frac{\partial p}{\partial y}\right)}^{2}\right]\end{eqnarray} \tag{ 5 }$$

Where, ${q}_{x}=\displaystyle \frac{Uh}{2}-\displaystyle \frac{{h}^{3}}{12\eta }\displaystyle \frac{\partial p}{\partial x},$ ${q}_{y}=-\displaystyle \frac{{h}^{3}}{12\eta }\displaystyle \frac{\partial p}{\partial y};$ ${c}_{\rho }$ is the specific heat capacity of the lubricating oil, and J is the heat equivalent.

If the effect of temperature on the viscosity of the lubricant is considered, the viscosity of the lubricant is η₀ at an initial temperature of T₀. Rolelands' equation [27] is used to investigate the problem of viscosity and temperature:

$$\begin{eqnarray}&&\eta ={\eta }_{0}\exp \left\{(ln{\eta }_{0}+9.67)\left[{\left(\displaystyle \frac{T-138}{{T}_{0}-138}\right)}^{-1.1}-1\right]\right\}\end{eqnarray} \tag{ 6 }$$

Micro-deformation analysis is performed on the main seal surface of the slip ring to calculate the oil film thickness in the seal zone. According to the small deformation theory, it is assumed that the normal deformation at any position within the seal zone is linearly related to the applied load, so the normal deformation within the seal zone can be obtained the influence coefficient method [28]. At the same time, since the variation of oil film thickness (normal deformation) is in the order of microns, it can be assumed that the macroscopic deformation of the main seal surface is not affected by the microscopic deformation. Based on the above assumptions, the increment of film thickness at any point in the sealing zone within the slip ring area can be calculated by the following equation (7):

$$\begin{eqnarray}&&{\left({u}_{p}\right)}_{i}=\displaystyle {\sum }_{j=1}^{m}{\left({I}_{n}\right)}_{ij}{\left({p}_{f}-{p}_{r}\right)}_{j}\end{eqnarray} \tag{ 7 }$$

Where m is the number of axial grid nodes. I_n is the normal deformation coefficient matrix, i.e., the displacement generated at the i-th node after a unit radial force is applied to the j-th node. p_r is the oil film pressure and p_f is the rough contact pressure. Figures 4(a) and (b) show the influence coefficient matrices for the cases without and with stress relaxation, respectively. It can be seen that the film thickness of the slip ring in the seal area becomes larger after the stress relaxation. In both cases, the peak deformation of the main sealing surface occurs at the location where the load is applied.

Zoom In Zoom Out Reset image size

The nonlinear equations of oil film thickness, oil film pressure and oil film temperature were obtained by discretizing the control equations of the flow and temperature fields in the main seal area. The solution process is shown in figure 5.

• (1)  
  First, the basic parameters such as deformation matrix, contact length, initial contact pressure, and initial displacement obtained through ABAQUS are entered into the script and interpolated. Then, dimensional parameters such as shaft diameter, calculated length and width, and operational parameters such as initial temperature, shaft speed and pressure are input into the main program. Finally, the roughness matrix and interpolation results are passed to the main program.
• (2)  
  The Reynolds equation is solved by the finite difference method, and then the obtained results are coupled with the deformation matrix to solve the elasticity equation.
• (3)  
  Convergence criterion of oil film pressure:

  $$\begin{eqnarray}&&\displaystyle \frac{\displaystyle \sum _{j=2}^{n}\displaystyle \sum _{i=2}^{m}| {P}_{i,j}^{k}-{P}_{i,j}^{(k-1)}| }{\displaystyle \sum _{j=2}^{n}\displaystyle \sum _{i=2}^{m}| {P}_{i,j}^{k}| }\leqslant error1\end{eqnarray} \tag{ 8 }$$

  Allowable error: error1 <9 × 10⁻⁵, if the condition is met, perform the next step; otherwise, the film thickness should be adjusted when recalculating.
• (4)  
  Convergence criterion for load balancing:

  $$\begin{eqnarray}&&\displaystyle \frac{\displaystyle {\sum }_{j=2}^{n}\displaystyle {\sum }_{i=2}^{m}| {P}_{i,j}{\rm{x}}| {A}_{i,j}-{F}_{0}| }{{F}_{0}}\leqslant error2\end{eqnarray} \tag{ 9 }$$

  Allowable error:error2<1e⁻⁵, if the conditions are met, output the oil film pressure P and oil film thickness h, and iterate the temperature field; otherwise, the initial pressure of the oil film should be adjusted when recalculating.
• (5)  
  Calculate the temperature field and output the result when both the temperature and pressure fields converge. Otherwise, the oil film thickness should be adjusted and recalculated until convergence is achieved.

Zoom In Zoom Out Reset image size

Based on the mathematical model of TEHL, the mathematical model for dynamic analysis of rotary combination seal is established by considering the temperature, speed and surface roughness of the seal, and the program for solving the oil film pressure and oil film thickness distribution is written by using MATLAB. The initial working conditions are listed in table 2.

Table 2. Initial working condition.

Symbol and parameter	Value and units
${p}_{f}:$Fluid pressure	5 MPa
${\eta }_{0}:$Lubricant oil viscosity	0.072 Pa.s
${\rho }_{L}:$Lubricant oil density	890 kg m−3
${c}_{\rho }:$Lubricant oil specific heat capacity	1870 J/(kg.K))
${C}_{O}:$O-ring specific heat capacity	1050 J/(kg.K))
${C}_{p}:$PTFE-ring specific heat capacity	1700 J/(kg.K))
The cross-sectional diameter of O-ring	5.3 × 10−3m
The diameter of shaft	50 × 10−3m
The width of groove	7.6 × 10−3m
The depth of groove	11.7 × 10−3m
The height of combination seal	7.1 × 10−3m

When the spindle of a rotary combination seal is rotating, the frictional force is mainly influenced in the circumferential direction. The velocity distribution in the circumferential direction can be obtained from the Reynolds equation:

$$\begin{eqnarray}&&{V}_{x}=-\displaystyle \frac{1}{12\eta }\displaystyle \frac{\partial p}{\partial x}{h}^{3}\end{eqnarray} \tag{ 10 }$$

Then, equation (10) is integrated along the perimeter and the area between z = 0 and z = h to obtain the expression for the leakage:

$$\begin{eqnarray}&&Q=\displaystyle \frac{\pi r}{6\eta }\displaystyle {\int }_{0}^{b}{h}^{3}\displaystyle \frac{\partial {\rm{\Delta }}p}{\partial x}dx\end{eqnarray} \tag{ 11 }$$

In addition, the viscous shear force and friction force of the rotary combination seal can be obtained from equations (12) and (13), and then the friction torque of the rotary combination seal can be obtained accordingly.

$$\begin{eqnarray}&&{\tau }_{{\rm{x}}}=\eta \displaystyle \frac{\partial {V}_{{\rm{x}}}}{\partial z}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&F=\displaystyle \iint {\tau }_{x}{| }_{z=0}dA\end{eqnarray} \tag{ 13 }$$

Figure 6 shows the contact pressure cloud and curve of the main sealing surface at different fluid pressures. It can be observed that the contact pressure increases with the increase of fluid pressure when the contact length is less than 1.2 mm. In addition, the contact pressure on the oil side is much higher than that on the drilling fluid side throughout the contact length, which is due to the increase of fluid pressure on the outside of the seal with the increase of drilling depth. When the fluid pressure reaches a certain level, it causes the separation of the spindle as well as the slip ring part near the drilling fluid side. Also, when the fluid pressure on both sides is 5 and 4.5 MPa, respectively, the contact pressure for the contact length between 1.2 and 1.8 mm is not zero. This is due to the relatively low internal and external pressure values and the small separation between the main seal surface near the drilling fluid side and the spindle under low pressure conditions. In addition, the contact pressure of the seal increases and then decreases, mainly due to the influence of the fluid pressure and the deformation of the rubber seal, which gradually increases the contact pressure of the seal. The gradual decrease in external contact pressure is due to the pressure of external drilling fluid that is pushing the seal surface apart. According to the finite element analysis, the variation curve of the maximum contact pressure of the seal surface with time for different fluid pressures is shown in figure 6(b). It can be observed that the maximum contact pressure of the combination seal increases with the increase of relaxation time and fluid pressure, and the curve gradually becomes flat. Since all the potential energy is almost released after a certain degree of relaxation, the stress does not change basically. Moreover, the maximum contact pressure at the main sealing surface is greater than the maximum fluid pressure on both sides of the seal, so a good sealing effect can be achieved. As can be seen from the above results, the oil side of the combination seal is in a high contact pressure state, making that side of the contact closer, so it reduces leakage. But with the increase in oil pressure, the main seal surface friction torque may gradually increase, which has a certain impact on the life of the seal surface.

Zoom In Zoom Out Reset image size

The maximum contact pressure relaxation curves of the main sealing surface under the combined effect of stress relaxation and different compression rates is shown in figure 7. It can be seen that the maximum contact pressure decreases with the increase of compression, which is due to the increase of the contact length of the main seal surface caused by the increase of compression. As the relaxation time increases, the maximum contact pressure increases at a slower rate and finally reaches an almost constant value. These results analyze the existence of an inverse relationship between the maximum contact pressure at the main sealing surface and the compression rate, and it can be speculated that a certain degree of increase in the leakage rate and a decrease in the friction torque may occur at high compression rates.

Zoom In Zoom Out Reset image size

The relaxation curves of the maximum contact pressure and maximum Mises stress on the main sealing surface under the combined effect of stress relaxation and different temperatures are shown in figure 8.

Zoom In Zoom Out Reset image size

It can be seen that when stress relaxation is considered, the maximum contact pressure and the maximum Mises stress on the main sealing surface increase with time. Also, the maximum contact pressure and the maximum Mises stress gradually increase as the temperature increases. This is because the increase in temperature causes an increase in volume, which leads to an increase in the pre-pressure of the seal, causing an increase in the contact pressure and Mises stress of the seal. Based on the above results, it can be thought that the temperature rise will cause a decrease in viscosity and a faster flow rate of lubricating fluid, which in turn makes the viscous shear force decrease and may cause an increase in the leakage rate of the combination seal as well as a decrease in the friction torque.

Figure 9(a) shows the distribution curves of seal circumferential oil film thickness at different temperatures. In this paper, the temperatures of 303 K and 343 K are selected for numerical simulation. It can be seen that the circumferential oil film thickness of the seal decreases with the increase of temperature.

Zoom In Zoom Out Reset image size

Figure 9(b) shows the axial oil film pressure distribution curve of the seal at different temperatures. It can be seen that the higher the temperature is, the greater the oil film pressure which increases first and then decreases along the Y dimensionless length direction, for the reason that the oil at the entrance is blocked to form a peak pressure, meanwhile, the accuracy of the dynamic pressure effect is reflected.

It can be seen from figure 9 that the lubricant viscosity decreases as the temperature increases, which leads to an increase in oil film pressure and a decrease in oil film thickness. With these results, it can be thought that the increased fluidity and gradual loss of lubricating fluid due to increased temperature may cause an increase in the leakage rate, while the thinning of the film thickness may eventually increase the risk of wear.

Figure 10(a) shows the distribution curve of circumferential oil film thickness with and without stress relaxation, and figure 10(b) shows the distribution curves of axial oil film pressure with and without stress relaxation. From figure 10, it can be seen that the oil film thickness increases significantly and the oil film pressure decreases significantly after considering stress relaxation. This is due to the fact that the stress within the main seal surface gradually decreases with the increase of stress time. Further reflection reveals that after the relaxation of the seal occurs, the membrane thickness increases to make leakage more likely, and the seal becomes more unstable with lower membrane pressure, which may further accelerate the process of leakage, thus causing the seal to become less effective.

Zoom In Zoom Out Reset image size

Figure 11 shows the distribution of oil film thickness under different rotary linear velocity. It can be seen that the up and down fluctuations are more obvious when the velocity is 1.5 m s⁻¹, while quite stable when the velocity is 0.6 m s⁻¹. In addition, as the rotating linear velocity increases, the oil film thickness of the seal generally shows an increasing trend. We can see the reason of which: With the increase of rotary linear velocity, the dynamic pressure effect occurs near the sealing contact area, and the nearby more lubricating oil is sent to the contact area, thus forming a thicker lubricating oil film.

Zoom In Zoom Out Reset image size

Figure 12 shows the axial oil film pressure distribution curve at different rotating linear speeds. The overall oil film pressure of the seal increases with the increase of rotational speed. In addition, the oil film pressure differs significantly at two different rotating linear speeds, indicating that the increase of rotating linear speed causes the dynamic pressure effect.

Zoom In Zoom Out Reset image size