A mixed thermal elastohydrodynamic lubrication model of rotary lip seal

Abstract

Rotary lip seal is used in various applications where the rotation shaft needs to be sealed, such as hydraulic pumps, fuel pumps, camshafts, crankshafts, and so on. Many thermal elastohydrodynamic lubrication models of rotary lip seal have been introduced, and most of these models neglect the asperity contact. This article proposes a mixed thermal elastohydrodynamic lubrication model of rotary lip seal, in which the microstructure of sealing lip surface, influence of temperature on fluid viscosity, and deformation of lip surface, as well as the asperity contact, are taken into consideration. Simulation study is carried out, and the results show that the asperity contact should not be neglected for analyzing the sealing performance of the rotary lip seal. The influence of speed on the sealing performance is also analyzed based on the proposed model.

Introduction

Rotary lip seals are usually installed in rotating equipment at the end of rotary shaft, which isolate the transmission components from the output components to prevent leakage of lubricating oil. Rotary lip seal has the advantages of simple structure, small installation position, good sealing performance, long service life, low cost, good adaptability to the vibration of the machine, and the eccentricity of the spindle, so it is widely used in aviation industry, automobile industry, construction machinery, mining machinery, and various other mechanical equipment. ¹ When the shaft rotates, a thin layer of lubricating oil film exists between the shaft and the seal lip, which can not only reduce the friction but also prevent leakage of lubrication. Figure 1 shows the schematic of rotary lip seal. A rotary lip seal usually consists of oil resistant rubber, steel frame, and garter spring.

Figure 1.

Schematic of a typical rotary lip seal.

Based on the experimental results, Guo et al. ² found that under the condition of normal operation, there is a continuous lubricating oil film between the seal lip and shaft. Later they found that the friction torque between the seal lip and shaft under dry friction is much larger than that under lubricated condition, and the relationship between velocity and friction coefficient was analyzed.^3,4 Fowell et al. ⁵ measured the thickness of the lubricant film directly on the rotary lip seal of the lip contact area. Fatu and Hajjam ⁶ found out that the thickness of the lubricant film is about several microns in normal operation condition.

The theory of reverse pumping effect has been widely accepted. ⁷ In the experiment, it has been observed that all the lip surfaces of the rotary lip seal are irregular. It was assumed that the shear force in the oil film causes the corrugation deformation. When the shaft rotates, the seal surface works like a viscous shear pump, and the oil is pumped from air side to oil side. Jia et al. ⁸ found that the reverse pumping effect was related to seal lip surface roughness. Frölich et al.^9,10 tested normal and failed rotary shaft seals with hollow glass shaft, and found that sealing performance was related to surface roughness distribution. Yang et al. ¹¹ established a simplified analytical model, in which it was assumed that the shear force in the lubricating oil film causes deformation of the seal, and the maximum circumferential displacement was closer to the oil side. Salant and Flaherty ¹² confirmed that there was an asymmetric tangential deformation on the seal lip surface.

Many lubrication models of rotary lip seal did not consider the thermal effect. However, thermal effect has a great impact on the performance of rotary lip seal. ¹³ Nakanishi et al. ¹⁴ established a finite element model, which assumed that the lip heat was transferred through the shaft and the thermal conductivity of rubber was neglected. Kang et al. established a thermodynamic model, in which the oil film thickness was assumed to be constant. The temperature of the shaft was calculated using a two-dimensional axisymmetric model and the heat transferred to the seal was neglected. ¹⁵ Fatu et al. ¹⁶ analyzed the influence of micro fluctuation of seal lip on the performance. The combination method of finite element analysis and elastic fluid dynamics analysis was proposed, and the power consumption, reverse pumping rate and temperature of rotary lip seal were analyzed. The results showed that the power consumption of the rotary lip seal was proportional to the square of the shaft speed. The leakage of rotary lip seal was negative, which confirmed the pumping effect, and the pumping rate increased when shaft speed increased. The temperature of the rotary lip seal also increased with the increased of shaft speed. Then Hajjam and Bonneau established a heat balance model of rotary lip seal, which considered the roughness of rotary lip seal. In this research, it was assumed that the heat was produced by the friction between the shaft and the seal lip, and was lost through the shaft and the surrounding oil by heat convection. ¹⁷ The results showed that the influence of different roughness on the oil film thickness was greater than that of the rotating speed. The heat was proportional to the speed of the shaft, and the pumping rate was proportional to the shaft speed.

Most of these thermal elastohydrodynamic (TEHD) lubrication models can calculate the lubrication status of the contact surface for rotary lip seal considering thermal effect, and can get comprehensive results. However, the abovementioned models did not take the micro contacts between lip surface and shaft surface into account. Guo et al. ¹³ built a mixed elastohydrodynamic lubrication model of rotary lip seal at constant temperature, and demonstrated that the micro contacts had certain influence on the accuracy of calculation results and should not be neglected. This article builds a mixed TEHD lubrication model of rotary lip seal, in which the micro contacts between lip surface and shaft surface are taken into consideration. A rough surface simulation method based on a two-dimensional digital filter is used to simulate the rough surfaces of shaft and sealing lip. Fluid mechanics analysis, contact mechanics analysis, as well as deformation mechanics analysis are proceeded to build the mixed TEHD model for rotary lip seal.

The rest of the article is organized as follows. In Section 2, first the rough surfaces of shaft and sealing lip are simulated based on a two-dimensional digital filter, then the mixed TEHD model for rotary lip seal is built based on fluid mechanics analysis, contact mechanics analysis, as well as deformation mechanics analysis. In Section 3, simulation study is carried out, and asperity contact influence and the effect of shaft speed on sealing performance are discussed. The conclusions are in Section 4.

Numerical model and analysis

First, the following assumptions are made:

1. Shaft surface is perfectly smooth, and seal lip surface is rough.

2. Because the stiffness of the shaft is much larger than that of the seal, so the shaft is considered as rigid body, and the lip seal is regarded as elastic body. The shaft is rotating, and the lip seal is static.

3. The influence of oil film curvature is neglected; therefore, the translation speed is used instead of the rotation speed.

4. The shear deformation of the asperities on the rough lip surface does not affect the normal macroscopic deformation of the lip seal, so the lip seal can be considered as asymmetrically macroscopic.

5. In order to calculate the reverse pumping rate, the air side is assumed to be filled with fluid all the time.

Numerical simulation of rough surfaces

It had been proved that the exponential and exponential cosine autocorrelation function can express many random phenomena very precisely. ¹² The experimental results of Whitehouse and Archard showed that the height distribution of many engineering surfaces can be described by the Gauss distribution, and the surface profile has an exponential autocorrelation function. ¹⁸ In this article, a rough surface simulation method based on two-dimensional digital filter is used to simulate the 3D rough surfaces of shaft and sealing lip, as shown in Figure 2. The exponential autocorrelation function can be given by

$$R({\tau }_x,{\tau }_y)={\sigma }^2\exp (-2.3{({\left(\frac{{\tau }_x}{{\lambda }_{\mathrm{avg}-x}}\right)}^2+{\left(\frac{{\tau }_y}{{\lambda }_{\mathrm{avg}-y}}\right)}^2)}^{1/2})$$

(1)

where σ is root mean square roughness (RMS) of the surface, λ_{avg_x} represents correlation length in the direction of x, and λ_{avg_y} is correlation length in the direction of y. If the surface is isotropic, λ_{avg_x} = λ_{avg_y}.

Figure 2.

Surface topography of the seal.

Fluid mechanics analysis

When the hydrodynamic oil film lubricates the contact zone, a pumping action which absorbs oil from the air side to the liquid side will occur, which is called the pumping effect. Due to the combined influence of the interference and the spring force, a contact pressure distribution is generated by the elastic deformation from the elastomeric ring. Profile of the distribution is asymmetrical, and when the shaft rotates, the asymmetrical profile of the distribution will generate the pumping action. The pumping rate is one of the most important performance characteristics of the rotary lip seal, and needs to be calculated in the establishment of TEHD model. When the lip seal is installed on the shaft with interference, because the stiffness of the shaft is much larger than that of the seal lip surface, deformation of the asperity on the seal lip surface will occur. Once the shaft rotates, a hydrodynamic oil film is formed on the contact surface between the seal lip and shaft. The dynamic pressure oil film is dynamically distributed on the contact zone, and the elastic deformation of each point on the lip surface is continuously changing.

Therefore, this article assumes that the average distance between the two surfaces of the seal lip and shaft is the average film thickness when the shaft does not rotate. The film thickness between any point on the seal lip and shaft surfaces can be given by

$$h(x,y)=h_{\mathrm{avg}}+h_{\mathrm{seal}}(x,y)$$

(2)

where h_avg is lip roughness, and h_seal represents the seal surface height distribution. As illustrated in Figure 1, here x is circumferential direction, and y is axial direction.

Once the shaft is rotating, a hydrodynamic lubricant film will be generated, and the oil film thickness can be expressed as

$$h(x,y)=h_{\mathrm{avg}}+h_{\mathrm{seal}}(x,y)+d_z(x,y)$$

(3)

where d_z is the elastic deformation of the lip surface under the oil film pressure.

Due to the existence of surface roughness, the two relatively sliding surfaces will be separated from the micro wedge. According to the basic theory of fluid lubrication, a discrete hydrodynamic region will be generated. If the hydrodynamic pressure is large enough to form hydrodynamic lubrication, the governing equations can be expressed by the Reynolds equation at hydrodynamic lubrication zone, as given by

$$\frac{\partial }{\partial x}\left(h^3\frac{\partial p_f}{\partial x}\right)+\frac{\partial }{\partial y}\left(h^3\frac{\partial p_f}{\partial y}\right)=6\mu U\frac{\partial h}{\partial x}$$

(4)

where p_f is the film pressure, U indicates the speed of shaft, h is the film thickness distribution, and μ is the dynamic viscosity of lubrication.

The Reynolds boundary condition adopted in this article is given by

$$\{\begin{array}{c}p(x,y_{\mathrm{in}})=p_{\mathrm{seal}},p(x,y_{\mathrm{out}})=p_e\hfill \\ \multicolumn{1}{c}{p(0,y)=p(l,y)}\\ \multicolumn{1}{c}{p(x,y)\ge 0}\end{array}$$

(5)

where p_e is the environmental pressure, l indicates the calculated area width in circumference direction. In this article, the finite volume method is used to discretize the Reynolds equation. After the pressure distribution and lubrication film thickness distribution is obtained, the viscosity shear stress and pumping rate can be given by

$${\tau }_f=\frac{h}{2}\frac{\partial p}{\partial x}+\mu \frac{U}{h}$$

(6)

$$q_y=-\frac{h^3}{12\mu }\frac{\partial p_f}{\partial y}$$

(7)

Contact mechanics analysis

According to the theory of mixed lubrication, when the ratio of oil film thickness over surface roughness is less than three, there will be a rough contact, and the influence of contact stress cannot be ignored. The contact pressure of asperities on seal lip surface is calculated by Greenwood-Williamson contact model. The radius of each contact asperity is regarded as R, and the contact is Hertz contact in Greenwood-Williamson contact model. Furthermore, it is assumed that all rough deformation is elastic deformation in the Hertz contact. Greenwood-Williamson contact model is a statistical based model, and the contact stress and contact area can be given as

$$A_e=\pi \eta A_n{R}_e{\int }_{h\left(y\right)/\sigma }^{\infty }\varphi \left(z\right)(z-\frac{h_{\mathrm{avg}}\left(y\right)}{\sigma })\mathrm{d}z$$

(8)

$$p_c=\frac{F_c}{A_n}=\frac{4}{3}\eta {R_e}^{\frac{1}{2}}{E}_e{\int }_{h\left(y\right)/\sigma }^{\infty }\varphi \left(z\right){(z-\frac{h_{\mathrm{avg}}\left(y\right)}{\sigma })}^{\frac{3}{2}}\mathrm{d}z$$

(9)

where p_c is the contact stress, A_e is the real contact area, R_e is the average radius of curvature, η is the asperities density, A_n is the nominal contact area, F_c is the total contact force, and E_e is the elastic modulus. In addition, $\varphi (z)$ is the distribution of surface height, which can be expressed as

$$\varphi \left(z\right)=\frac{1}{\sqrt{2\pi }\sigma }\exp (-\frac{{\left(z\right)}^2}{2{\sigma }^2})$$

(10)

where σ is the surface roughness.

The dry friction shear stress can then be given by

$${\tau }_c\left(y\right)=-f{p}_c\left(y\right)$$

(11)

with f indicates the friction coefficient in dry friction condition.

Deformation mechanics analysis

In the deformation analysis, the normal deformation and shear deformation of lip seal can be obtained through the influence coefficient method. Then the height distribution of lubrication fluid is determined. It should be noted that when using the influence coefficient method, one needs to follow the following three assumptions. First, the stiffness of the sealing ring remains constant in the whole calculation process. Second, the deformation at any position of the sealing contact area is linear with the applied load, that is, the theory of small deformation is applied to the seal lip deformation caused by oil film. Finally, micro morphology does not affect the macroscopic deformation, so the elastic deformation of seal lip can simplify the two-dimensional problem. Therefore, the formula for calculating the normal deformation of any node on the seal lip is given by

$${\left({\delta }_z\right)}_i=\sum _{k=1}^m{\left(I_z\right)}_{\mathrm{ik}}{[p_f+p_c]}_k$$

(12)

The shear deformation of any node is calculated as

$${\left({\delta }_x\right)}_i=\sum _{k=1}^m{\left(I_x\right)}_{\mathrm{ik}}({\tau }_c+{\tau }_f)$$

(13)

where influence coefficient matrix (I_z) _ik , (I_x) _ik and the static contact pressure p_sc are obtained by ANSYS finite element analysis off-line. Here plane 183 is selected as solid element, target 169 is selected as contact target element, and contact 172 is selected as contact element. τ_f represents the fluid viscous force and τ_f indicates dry friction. The normal influence coefficient matrix (I_z) _ik represents the ith axial node deformation caused by kth axial node unit normal load. The shear influence coefficient matrix (I_x) _ik represents the ith axial node deformation caused by kth axial node unit shear load. The material of lip seal is rubber, so Mooney-Rivlin model is selected as the constitutive model.

Lip surface thermal model

The friction of the sealing area will generate a lot of heat, which will cause temperature rise in the sealing surface. One of the most important parameters to characterize the lubricating oil is viscosity, which varies with operating temperature. The effect of temperature on the performance of lip seal cannot be neglected. However, as the sealing area is very small, the temperature change in this area is not large, so the analysis of the sealing lip temperature focuses on its average value. With the operation of lip seal, the temperature of sealing surface rises, as given by

$$T-T_{\mathrm{ref}}=\frac{\mathrm{\Phi }}{Q_{\rho }{C}_p+h_{c}2\pi \mathrm{RL}}$$

(14)

where h_c indicates heat exchange coefficient of oil, R represents shaft radius, L is the length of shaft, T_ref represents reference temperature, that is, environmental temperature, Φ indicates the total heat production of seal surface, Q_ρ represents leakage rate, and C_p is specific heat capacity of oil.

The thermal effect of rotary lip seal is that friction generates heat and temperature of contact zone increases when the shaft rotates. The temperature increment is closely related to static contact pressure caused by pre-tightening force of spring and interference between shaft and lip seal.

In the process of theoretical analysis, the seal lip static contact load is generally used. The relationship between the axial force and static contact load of seal lip can be expressed as

$$F=\frac{G}{D}$$

(15)

where F is static contact load of seal lip, G is the axial force, and D is diameter of shaft.

The total heating power of seal lip can be given by

$$\mathrm{\Phi }=\frac{{\pi }^2\mathrm{fn}{D}^{2}F}{60}=\frac{{\pi }^2\mathrm{fnDG}}{60}$$

(16)

where f is friction factor of seal lip in mixed lubrication condition, and n is the rotating speed of shaft.

Once the average temperature of the sealing lip is obtained, the liquid viscosity can be obtained by the relationship between the viscosity of liquid and the temperature. Roelands viscosity–temperature equation and Reynolds viscosity–temperature equation are two of the most frequently-used viscosity–temperature equations. Roelands’ viscosity–temperature equation is given by

$$\mu ={\mu }_0\exp \{(\ln {\mu }_0+9.67)[{\left(\frac{T-138}{T_0-138}\right)}^{s_0}-1]\}$$

(17)

where μ₀ is viscosity at temperature T₀, μ is viscosity at temperature T, and s₀ is viscosity temperature coefficient.

Reynolds viscosity–temperature equation is given by

$$\mu ={\mu }_0\exp [-a(T-T_0)]$$

(18)

where μ₀ is viscosity at temperature T₀, μ is viscosity at temperature T, and a is viscosity temperature coefficient.

Compared with the Reynolds viscosity–temperature equation, the Roelands viscosity–temperature equation is more complex. The Reynolds viscosity–temperature can be considered as the simplification of the Roelands viscosity–temperature equation when the range of temperature is limited. Because the variation range of seal lip temperature is wide, the Reynolds viscosity–temperature equation is applied to predict the viscosity of the oil at the lubricated zone in this work.

Computational procedure

The solution of lip seal mixed thermal lubrication model is essentially a strong coupling solution of fluid mechanics, contact mechanics, thermal, and deformation. Figure 3 shows the flow chart of the numerical solution. It should be noted that, although only one loop step is shown in the diagram, in fact, there is a nested loop step in solving the Reynolds equation. When the two cycles converge, the parameters such as pumping rate and friction torque can be obtained.

Figure 3.

Flow chart of numerical calculation.

Result and discussion

The mixed TEHD lubrication model is applied to a certain type of rotary lip seal. To analyze the influence of the micro contact between seal lip and shaft and shaft speed on lip seal performance, the numerical calculation of TEHD lubrication model is conducted under different rotating speed. The Reynolds equation is solved by finite difference method. The influence of micro contact between seal lip and shaft is discussed. Average film pressure, maximum film pressure, minimum film thickness, heat productivity, average temperature of the contact area, and friction torque are calculated for different operating conditions. The parameters are first set, as shown in Table 1. In this section, the influence of the asperity contact is first discussed, and the necessity of the consideration of asperity contact is proved. Then the influence of rotation speed on the sealing performance is studied based on the proposed model.

Table 1.

Values of the main simulation parameters.

Parameter	Meaning	Value
Tref	Reference temperature	30°C
havg	Average liquid film thickness	3 μm
ps	Environmental pressure	0.1 MPa
R	Shaft radius	60 mm
λx	Autocorrelation length	1.667 μm
λy	Autocorrelation length	5.0 μm
E	Elastic modulus of seal lip	9.8 MPa
ν	Poisson ratio	0.5
f	Friction coefficient	0.1
RMS	Roughness of seal lip	1 μm

RMS: root mean square.

Asperity contact influence

The Greenwood and Williamson contact model (G-W model) is used to analyze the contact condition between the shaft surface and sealing lip surface from a statistical point of view. Then the contact pressure and dry friction shear stress caused by contact between the two surfaces are calculated, and the contact pressure in axial direction compared with fluid pressure in axial direction are shown in Figure 4. It can be seen from Figure 4 that both contact pressure and fluid pressure experience several large fluctuations. The maximum value of fluid pressure is about 1.5 MPa, and the maximum value of contact pressure is about 0.8 MPa. The maximum value of fluid pressure is about twice as that of contact pressure. Hence, contact between the two surfaces cannot be ignored from the view of pressure supporting.

Figure 4.

Fluid pressure and contact pressure in axial direction.

Fluid torque in axial direction compared with contact torque in axial direction is illustrated in Figure 5, with U = 1 m/s, σ = 1 μm. It can be seen from Figure 5 that the plot of fluid torque is generally smooth, and only a few small fluctuations exist. However, for the plot of contact torque, several large fluctuations exist. Furthermore, although the fluid shear stress is much bigger than dry friction shear stress, as shown in Figure 5, the deviation of dry friction shear stress is not negligible compared with fluid shear stress. Hence, the asymmetry of tangential deformation mainly depends on the friction shear stress distribution, that is, dry friction has significant impact on pumping effect. As mentioned before, pumping effect is one of the most important performance characteristics of rotary lip seal and affects the sealing performance; hence, the asperity contact cannot be neglected when analyzing the sealing performance of the rotary lip seal.

Figure 5.

Fluid torque and contact torque in axial direction.

Effect of shaft speed on sealing performance

Figure 6 shows the relationship between the average oil film thickness of the calculation zone and the rotation speed. It can be seen that the average oil film thickness is significantly affected by the rotation speed, and the average oil film thickness increases with the increase of the rotation speed. This is because that higher rotation speed can provide higher hydrodynamic pressure, and higher oil film thickness is obtained. Hence, it can be seen that higher speed is helpful to maintain the lubrication of the sealing zone for rotary lip seal.

Figure 6.

Relationship between average film pressure and rotation speed.

Figures 7 and 8 show the influence of rotation speed of shaft on the maximum and minimum hydrodynamic pressure in the sealing zone, respectively. As shown in Figures 7 and 8, the maximum film pressure will decrease when rotation speed increases and the minimum film pressure will increase when rotation speed increases. The maximum film pressure is significantly affected by rotation speed when rotation speed is low, and the maximum film pressure will reach the minimum value when rotation speed is over 2500 rpm. The minimum film pressure is linearly related to the rotation speed. It can be seen that the hydrodynamic pressure distribution of the sealing zone will become smooth with the increase of rotation speed.

Figure 7.

Relationship between maximum film pressure and rotation speed.

Figure 8.

Relationship between minimum film thickness and rotation speed.

Figure 9 shows the relationship between leakage rare and rotation speed. As shown in Figure 9, the leakage is negative when the sealed fluid pressure is lower than ambient pressure, which means that the leaking fluid will be reversely pumped to the fluid side when the shaft rotates (pumping action). Furthermore, the pumping rate will increase with the increase of rotation speed. Hence, it can be seen that the high rotation speed is helpful for both the lubrication of the sealing zone and the sealing performance of the rotary lip seal.

Figure 9.

Relationship between leakage rate and rotation speed.

Figures 10 and 11 show the influence of rotation speed on the heat production and average temperature of sealing zone. As shown in Figures 10 and 11, the heat productivity is approximately linearly related to the rotation speed. Similarly, the average temperature of the sealing zone will also increase when rotation speed increases, and the relationship is also approximately linear. As rotary lip seal is mostly made of rubber, aging will also be accelerated at higher temperature. When temperature increases, the compression set of rotary lip seal will increase dramatically, and the stress-strain ratio and friction torque will also increase. ¹³ The pumping rate, which is one of the most important performance characteristics of rotary lip seal, will gradually decrease when temperature increases. ¹³ All these phenomena show that the aging speed is bigger at higher temperature. It can be concluded that although higher rotation speed of shaft is helpful for the sealing performance of the rotary lip seal, the higher temperature caused by higher rotation speed will increase the aging speed, and thus reduce the operation lifetime.

Figure 10.

Relationship between heat productivity and rotation speed.

Figure 11.

Relationship between average temperature and rotation speed.

Figure 12 shows the relationship between the friction torque and rotation speed. As shown in Figure 12, the friction torque of the rotary lip seal increases with the increase of rotation speed. The increasing rate of the friction torque will decrease when the rotation speed increases.

Figure 12.

Relationship between friction torque and rotation speed.

Conclusion

A mixed TEHD lubrication model of rotary lip seal is introduced in this work. The microstructure of sealing lip surface, asperity contact, influence of temperature on fluid viscosity, and deformation of lip surface are taken into consideration in this model. Simulation results show that although the fluid shear stress is more considerable than dry friction shear stress, the deviation of dry friction shear stress is large compared with fluid shear stress, so the asymmetry of tangential deformation mainly depends on the friction shear stress distribution. The influence of shaft speed on the seal performance is also studied by the proposed model. Simulation results show that high rotation speed of shaft will improve the sealing performance of the rotary lip seal, but the temperature of sealing zone will also increase and the aging of the seal will be accelerated in this situation. Future work should focus on simulating the rough surfaces of shaft and sealing lip with more accurate models, including other types of autocorrelation functions. Meanwhile, the calculation efficiency of the proposed model should also be assessed.