# Numerical Calculation Method of Multi-Lip Seal Wear under Mixed Thermal Elastohydrodynamic Lubrication

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## Abstract

**:**

## 1. Introduction

## 2. Geometric Structure

## 3. Theoretical Analysis

#### 3.1. Hydrodynamics

_{T}is the average truncated film thickness, U is the piston rod speed, $\widehat{\alpha}$ is the pressure–viscosity coefficient, F is the cavitation index, Φ is the fluid pressure or the density function and ϕ

_{xx}and ϕ

_{s.c.x}are the flow factor [24].

_{s}is the sealed pressure in the high-pressure fluid side, P

_{i}

_{1}and P

_{i}

_{2}are the pressure in the first and second inter-lip zone and P

_{l}is the fluid pressure in the low-pressure fluid side.

_{f}, ϕ

_{fs}and ϕ

_{fp}are flow factors [25].

#### 3.2. Contact Mechanics

_{c}is the friction coefficient.

#### 3.3. Deformation Mechanics

_{s}is the initial film thickness and K and P

_{sc}are the deformation coefficient matrix and static contact pressure obtained using ANSYS software, respectively.

#### 3.4. Thermal Mechanics

_{0}is the ambient temperature, k

_{c}is the thermal conductivity of the cylinder wall, ρ

_{c}is the density of the cylinder wall, c

_{c}is the specific heat capacity of the cylinder wall, L is the contact length of the seal lip and Q is the heat production rate of the cylinder wall.

#### 3.5. Coupling Mechanics

_{0}is the corresponding viscosity under the ambient pressure p

_{0}and temperature T

_{0}, α is the Barus viscosity–pressure coefficient and β is the Reynolds viscosity–temperature coefficient.

#### 3.6. Wear Mechanics

_{n}is the dimensionless wear coefficient, ∆V

_{w}is the seal wear volume, H

_{B}is the seal material hardness, F

_{n}is the normal force and ∆L is the relative sliding distance.

_{n}= K

_{n}/H

_{B}as the seal material wear coefficient, so Equation (10) can be transformed into

_{w}is wear depth and p

_{n}is normal contact pressure.

_{n}of the cylinder wall to the seal equals the static contact pressure p

_{sc}of the seal against the cylinder wall, and p

_{sc}consists of film fluid pressure p

_{f}and asperity contact pressure p

_{c}, that is, p

_{sc}= p

_{f}+ p

_{c}.

_{n}is the wear coefficient.

_{stroke}into n parts to calculate the average asperity contact pressure ${\overline{p}}_{c}$, and then the Archard equation is modified. Finally, the modified Archard equation can be written as

_{c}is the inner diameter of the cylinder wall.

_{t}is used to describe the wear volume of a unit of time, and the wear distance rate (WDR) δ

_{L}is used to describe the wear volume of the unit distance, which are

#### 3.7. Numerical Calculation Procedure

- In ANSYS software, the finite element analysis (FEA) of seal static solid mechanics is performed based on the characteristics of the seal, fluid and operating conditions.
- The static contact pressure P
_{sc}, L_{x}, K obtained via FEA and the static initial film thickness H_{s}calculated by P_{sc}are input into the MATLAB program and initialized. - The film fluid pressure P
_{f}and the asperity contact pressure P_{c}are calculated using the fluid mechanics model and the contact mechanics model, respectively. Then, compare the sum of P_{f}and P_{c}with P_{sc}to obtain the pressure difference under dynamic conditions; the pressure difference is applied to calculate the new film thickness. - If the film thickness converges, the film temperature can be determined with the thermal mechanics model, which can also determine the fluid viscosity at this temperature convergence via the coupling mechanics model.
- If the fluid viscosity converges, compare whether the fluid flow rate of the three seal lips is equal. If not equal, the pressure in the first and second inter-lip zone is adjusted until the fluid flow rate is balanced.
- If the flow rate is equal, judge whether the stroke is reached. If not, carry on simulating the subsequent time step. The wear mechanics model is used to output the results once all the time steps have been performed.

## 4. Results and Discussion

#### 4.1. Static Sealing Performance

_{s}, indicating that the contact zone can seal the high-pressure fluid and ensure good sealing performance.

#### 4.2. Model Validation

#### 4.3. Dynamic Sealing Performance

#### 4.3.1. Effect of Sealed Pressure

#### 4.3.2. Effect of Piston Rod Speed

_{s}= 4 MPa and σ = 0.6 μm, there is an effect of different extension speeds on the seal lubrication characteristics. The average asperity contact pressure ratio between the main lip and the right lip shows a significant decreasing trend, as seen in Figure 6a. This is because as the sealed pressure is constant, the static contact pressure remains unchanged. When the piston rod speed increases, the hydrodynamic effect increases the film pressure, but the sum of the film pressure and the asperity contact pressure remains unchanged, so the proportion of the asperity contact pressure of the corresponding roughness of the main lip and the right lip decreases. The asperity contact pressure ratio of the left lip displays a slight convex shape as the extension speed is increased from 0.1 m/s to 0.5 m/s. When the speed is less than 0.3 m/s, the asperity contact pressure ratio remains positively correlated with the speed. The asperity contact pressure gradually decreases with increasing speed as speed increases over 0.3 m/s. This indicates that there is a critical speed near the piston rod extension speed of 0.3 m/s, which changes the variation trend of the asperity contact pressure on the left lip. Figure 6b demonstrates that the sealing zone is in a mixed lubrication state since the ratio of the average film thickness to the seal roughness of the three lips at different piston rod speeds is less than 3. The average film thickness of the main lip and the right lip in the sealing zone increases obviously as the extension speed rises, while the average film thickness of the left lip presents a slight concave shape, with the minimum film thickness near the speed of 0.3 m/s, which is consistent with the reason revealed in Figure 6a. At the same time, it further shows that there is strong coupling state in the working process of the multi-lip seal, and the lubrication characteristics of each seal lip are not consistent.

#### 4.3.3. Effect of Seal Roughness

#### 4.3.4. Effect of Fluid Viscosity

_{s}= 4 MPa, u = 0.1 m/s and σ = 0.6 μm, the influence of the asperity contact pressure ratio and average film thickness for various fluid viscosities is shown in Figure 8. The asperity contact pressure ratio of the three seal lips gradually decreases as the fluid viscosity gradually increases from 0.02 Pa·s to 0.10 Pa·s, as shown in Figure 8a. In general, the viscous shear effect becomes stronger with the rises in fluid viscosity, and the hydrodynamic pressure effect of the film fluid is enhanced accordingly. This conclusion is also given in Wang et al. [34]. According to Figure 8b, the average film thickness of the three seal lips in the sealing zone is proportional to the increase in fluid viscosity, and the variation range of the film thickness is 1.57~1.58 μm, 1.36~1.42 μm and 1.41~1.43 μm, respectively. This is because when the fluid viscosity increases, the interaction between the molecules within the fluid can hinder the fluidity of the fluid in the sealing zone, which increases the average film thickness and reduces the chance of asperity peak contact. Therefore, the selection of high-viscosity fluid is helpful to reduce the Coulomb friction force and seal wear caused by the seal roughness. On the contrary, the viscous friction force will increase accordingly.

#### 4.3.5. Effect of Ambient Temperature

_{s}= 4 MPa, u = 0.1 m/s and σ = 0.6 μm, the influence of asperity contact pressure ratio at various ambient temperatures is depicted in Figure 9a. As the ambient temperature rises, the asperity contact pressure ratio of the three seal lips in the sealing zone gradually increases. According to Equation (9), the fluid viscosity is a combined effect of temperature and pressure. The high-temperature environment will cause an increase in asperity contact pressure and a decrease in film fluid pressure, which will eventually lead to a decrease in film fluid load capacity and an increase in the asperity contact pressure ratio. The average film thickness of the three seal lips at various ambient temperatures is shown in Figure 9b. When the ambient temperature gradually increased from 273.15 K to 353.15 K, the film thickness gradually decreased. It is worth noting that when the ambient temperature is from 273.15 K to 293.15 K, the film thickness of the main lip and right lip decreases sharply. When the ambient temperature is greater than 293.15 K, the film thickness decreases slowly, which is mostly caused by the comprehensive effect of the ambient temperature and fluid pressure on the fluid viscosity. As can be noticed, the average film thickness of the left lip is maximum, while the average film thickness of the main lip is minimum. According to Figure 3d, when the seal is deformed, the static contact pressure of the main lip is the maximum, and the static contact pressure of the left lip is the minimum. The large static contact pressure makes the seal lip and the cylinder wall more tightly squeezed, so the average film thickness is the thinnest.

#### 4.4. Analysis of Seal Wear Characteristics

#### 4.4.1. Effects of Sealed Pressure

^{−4}~6.8548 × 10

^{−4}mm

^{3}/s, 0.0013~0.0047 mm

^{3}/s and 5.0752 × 10

^{−4}~0.0036 mm

^{3}/s, respectively. Figure 10b illustrates the effect of sealed pressure on leakage. Significantly, the leakage reduces as the sealed pressure rises as the piston rod speed remains constant. This is due to the fact that, as the piston rod extends, the cylinder interior moves in the opposite direction. The Poiseuille flow caused by the pressure difference hinders the Couette flow caused by the speed, resulting in the gradual leakage of fluid from the high-pressure side to the low-pressure side. The greater the medium pressure, the more significant the effect of Poiseuille flow. At the same time, it can be seen from Figure 5 that the greater the sealed pressure, the smaller the film thickness. Therefore, it is a challenge to find a seal with a long life or good dependability when there is high sealed pressure at work.

#### 4.4.2. Effects of Piston Rod Speed

^{−4}mm

^{3}/s, 0.0021 mm

^{3}/s and 0.0011 mm

^{3}/s to 0.0015 mm

^{3}/s, 0.0052 mm

^{3}/s and 0.0044 mm

^{3}/s, while the WDR decreased from 2.9742 × 10

^{−6}mm

^{3}/mm, 2.0770 × 10

^{−5}mm

^{3}/mm and 1.1426 × 10

^{−5}mm

^{3}/mm to 2.9676 × 10

^{−6}mm

^{3}/mm, 1.0323 × 10

^{−5}mm

^{3}/mm and 8.8675 × 10

^{−6}mm

^{3}/mm, respectively. As mentioned above, the hydrodynamic effect is enhanced when the extension speed rises, so that the film thickness increases and the number of asperity contact peaks decreases, resulting in reduced asperity contact pressure. Therefore, the leakage rises as the extension speed increases, as seen in Figure 11b.

#### 4.4.3. Effects of Seal Roughness

_{s}= 4 MPa and u = 0.1 m/s, the effects of different seal surface roughness on seal wear characteristics are shown in Figure 12. With the roughness increasing from 0.6 μm to 1.4 μm, the WTR and WDR of the seal lip showed an increasing trend, as displayed in Figure 12a. When the roughness increased from 0.6μm to 0.8 μm, the WTR and WDR of the main lip and right lip increased sharply. When the roughness is larger than 0.8 μm, the wear rate increases slowly. Figure 12b depicts the impact of different seal roughness on leakage. As the roughness rises, it is evident that the leakage also increases. According to Figure 7, the difference in film thickness distribution caused by the increase in roughness is not caused by an enhancement in the hydrodynamic pressure effect but by an increase in the roughness peak height, which leads to an increase in the seal clearance gap and stored fluid, so the fluid leakage increases correspondingly.

#### 4.4.4. Effects of Fluid Viscosity

_{s}= 4 MPa and u = 0.5 m/s, Figure 13 illustrates the influence of different fluid viscosity on seal wear characteristics. Figure 13a shows that both the WTR and WDR of the seal lip decrease sharply when the fluid viscosity rises from 0.02 Pa·s to 0.10 Pa·s. The main reason is that the fluidity of high-viscosity fluid is poorer than that of low-viscosity fluid, which leads to the fluid being stored in the seal clearance gap, increasing the film thickness and eventually leading to the separation of the seal lip from the cylinder inner wall, thus reducing the Coulomb friction generated by the asperity contact between the seal surface rough peak and the cylinder inner wall. This result corresponds with the increase in fluid viscosity reducing the asperity contact pressure ratio, as seen in Figure 8. The variation trend of seal leakage with fluid viscosity is shown in Figure 13b. As the fluid viscosity rises, it is evident that leakage also rises. This is mainly due to the increase in fluid viscosity, resulting in greater viscous shear stress. When the sealed pressure and the piston rod speed remain constant, the hydrodynamic effect of viscous fluid is stronger, and more fluid is dragged into the seal clearance gap; thus, the leakage increases due to the Couette flow caused by the movement.

#### 4.4.5. Effects of Ambient Temperature

## 5. Conclusions

- Some characteristics of the single-lip seal are also reflected in the DAS multi-lip combined seal. The increase in sealed pressure decreases the sealed performance and increases the seal wear rate. When the piston rod speed is increased, the WTR also increases. The high-temperature environment leads to the deterioration of the lubrication characteristics in the sealing zone and increases the seal wear. Therefore, it is not difficult to find that high pressure, high speed and high temperature are still great challenges for the lip seal.
- The load on the sealing zone under mixed lubrication is mainly shared by asperity contact, and the asperity contact load is as high as 50%. When the roughness is greater than 0.6 μm, the seal wear rate clearly increases. Therefore, smaller surface roughness is more conducive in reducing the seal wear and fluid leakage.
- The lubrication characteristics of each seal lip in a DAS multi-lip combined seal largely depend on the working conditions. In the piston rod extension motion, the sealing behavior of each seal lip is not identical due to the existence of critical speed.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

c_{c} | Specific heat capacity of the cylinder wall |

D_{c} | Insider diameter of cylinder |

E | Elastic modulus |

F | Cavitation index |

F_{n} | Normal force |

f_{c} | Friction coefficient for asperity contact |

H | Dimensionless average film thickness, h/σ |

H_{B} | Hardness of the seal material |

H_{s} | Dimensionless static film thickness, h_{s}/σ |

H_{T} | Dimensionless average truncated film thickness, h_{T}/σ |

∆h_{w} | Seal wear depth |

K | Deformation coefficient matrix |

k_{c} | Thermal conductivity of the cylinder wall |

k_{n} | Seal material wear coefficient, K_{n}/H_{B} |

L | Contact length of sealing zone |

L_{stroke} | Cylinder stroke length |

∆L | Relative sliding distance |

P_{c} | Dimensionless asperity contact pressure, p_{c}/E |

P_{f} | Dimensionless fluid film pressure, p_{f}/p_{a} |

P_{i1}, P_{i2} | Pressure in the first and second inter-lip region |

P_{l} | Dimensionless pressure in low-pressure side, p_{l}/p_{a} |

P_{s} | Dimensionless sealed pressure, p_{s}/p_{a} |

P_{sc} | Dimensionless static contact pressure, p_{sc}/E |

p_{a} | Ambient pressure |

p_{n} | Normal contact pressure |

Q | Heat production rate of the cylinder wall, $\left({\widehat{\tau}}_{f}+{\widehat{\tau}}_{c}\right)Eu$ |

$\widehat{q}$ | Dimensionless flow rate, $12{\mu}_{0}qL/({p}_{a}{\sigma}^{3})$ |

R | Radius of asperities |

T | Fluid film temperature |

T_{0} | Ambient temperature |

U | Dimensionless piston rod speed, ${\mu}_{0}uL/({p}_{a}{\sigma}^{2})$ |

∆V_{w} | Seal wear volume |

$\widehat{x}$ | Dimensionless coordinate parallel to the fluid film thickness, x/L |

z | Dimensionless coordinate normal to the fluid film thickness |

α | Viscosity-pressure coefficient |

$\widehat{\alpha}$ | Dimensionless pressure-viscosity coefficient, αp_{a} |

β | Viscosity-temperature coefficient |

ξ | ${R}^{1/3}{\eta}^{2/3}{E}_{s}L/{P}_{a}$ |

Φ | Fluid pressure/density function |

ϕ_{f}, ϕ_{fs}, ϕ_{fp} | Shear stress factors |

ϕ_{s}._{c}._{x}, ϕ_{xx} | Flow factor |

η | Asperity density |

μ_{0} | Fluid viscosity at atmospheric pressure |

ρ_{c} | Density of the cylinder wall |

v | Poisson’s ratio |

ρ_{f} | Fluid density |

$\widehat{\rho}$ | Dimensionless density, ρ/ρ_{f} |

$\widehat{\sigma}$ | Dimensionless RMS roughness of the seal, σR^{1/3}η^{2/3} |

${\widehat{\tau}}_{f}$ | Dimensionless viscous shear stress, ${\tau}_{f}/E$ |

${\widehat{\tau}}_{c}$ | Dimensionless asperity shear stress, ${\tau}_{c}/E$ |

## Abbreviations

DAS | Double-acting seal |

FEA | Finite element analysis |

FEM | Finite element method |

M-EHL | Mixed elastohydrodynamic lubrication |

M-TEHL | Mixed thermal elastohydrodynamic lubrication |

NBR | Nitrile rubber |

PTFE | Polytetrafluoroethylene |

POM | Polyformaldehyde |

RMS | Root mean square |

TPE | Thermoplastic polyester elastomer |

WTR | Wear time rate |

WDR | Wear distance rate |

## Appendix A

_{s}, E

_{c}, v

_{s}and v

_{c}are the elastic modulus and Poisson’s ratio of the seal ring and cylinder, respectively.

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**Figure 3.**FEA of DAS-combined seal: (

**a**) FEA meshing model, (

**b**) the von Mises stress after assembly and pressurization, (

**c**,

**d**) the nephogram and curve of static contact pressure distribution.

**Figure 5.**Lubrication situation under different sealed pressure: (

**a**) average asperity contact pressure ratio, (

**b**) average film thickness.

**Figure 6.**Lubrication situation under different piston rod speeds: (

**a**) average asperity contact pressure ratio, (

**b**) average film thickness.

**Figure 7.**Lubrication situation under different seal roughness: (

**a**) average asperity contact pressure ratio, (

**b**) average film thickness.

**Figure 8.**Lubrication situation under different fluid viscosity: (

**a**) average asperity contact pressure ratio, (

**b**) average film thickness.

**Figure 9.**Lubrication situation under different ambient temperature: (

**a**) average asperity contact pressure ratio, (

**b**) average film thickness.

Parameter Type | Value |
---|---|

Ambient temperature, T_{0} | 273.15~313.15 K |

Ambient pressure, p_{0} | 0.1 MPa |

Fluid viscosity, μ_{0} | 0.02~0.10 Pa·s |

Viscosity-pressure coefficient, α | 2 × 10^{−8} Pa^{−1} |

Viscosity-temperature coefficient, β | 3.17908 × 10^{−2} K^{−1} |

Sealed pressure, p_{s} | 2~10 MPa |

Wear coefficient, k_{n} | 1.2 × 10^{−5} mm^{3}/Nm [31] |

Stroke length, L_{stroke} | 300 mm |

Insider diameter of cylinder, D_{c} | 63 mm |

Piston rod extension speed, u | 0.1~0.5 m/s |

Seal RMS roughness, σ | 0.6~1.4 μm |

Friction coefficient, f_{c} | 0.1 [32,33] |

Thermal conductivity of cylinder wall, k_{c} | 46 W/(m·K) |

Density of cylinder wall, ρ_{c} | 7850 kg/m^{3} |

Specific heat capacity of cylinder wall, c_{c} | 460 J/(kg·K) |

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## Share and Cite

**MDPI and ACS Style**

Cheng, D.; Gu, L.; Sun, Y.; Shi, Y.
Numerical Calculation Method of Multi-Lip Seal Wear under Mixed Thermal Elastohydrodynamic Lubrication. *Lubricants* **2023**, *11*, 248.
https://doi.org/10.3390/lubricants11060248

**AMA Style**

Cheng D, Gu L, Sun Y, Shi Y.
Numerical Calculation Method of Multi-Lip Seal Wear under Mixed Thermal Elastohydrodynamic Lubrication. *Lubricants*. 2023; 11(6):248.
https://doi.org/10.3390/lubricants11060248

**Chicago/Turabian Style**

Cheng, Donghong, Lichen Gu, Yu Sun, and Yuan Shi.
2023. "Numerical Calculation Method of Multi-Lip Seal Wear under Mixed Thermal Elastohydrodynamic Lubrication" *Lubricants* 11, no. 6: 248.
https://doi.org/10.3390/lubricants11060248