# Multi-Field Coupling Numerical Analysis and Experimental Validation of Surface-Textured Metal Seals in Roller Cone Bits

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Geometric Model

_{d}~r

_{i}) on the inner diameter side, the sealing zone (radial range of r

_{i}~r

_{o}) in the middle, and the chamfering zone (radial range of r

_{o}~r

_{s}) on the outer diameter side. Within the sealing zone of the stator, five typical micro-texture shapes, namely, circle, ellipse I (horizontal), ellipse II (vertical), triangle, and chevron [25,26], are adopted, as shown in Figure 2. The micro-textures with the same size are uniformly distributed in the sealing zone in a circular array of T columns. Each column comprises N

_{t}micro-textures along the radius with a depth, h

_{p}, and an area ratio, S

_{p}. The texture parameters are shown in Table 2.

#### 2.2. Multi-Field Coupling Model

#### 2.2.1. Fluid Lubrication Analysis

_{0}is the sealing gap, h

_{p}is the texture depth and $\delta $ is the deformation of the sealing end faces.

_{r}and ϕ

_{θ}are pressure flow factors, expressed as

_{s}is the shear flow factor, expressed as

_{1}and σ

_{2}are the standard deviations of the surface roughness of the rotor and stator, respectively, and Φ

_{s}can be calculated as [28]

_{1}, α

_{2}, α

_{3}, and α

_{4}are constants.

_{c}is the contact factor, which can be expressed as

_{c}) is introduced to consider the cavitation effect, where ρ is the density of the sealing medium and ρ

_{c}is the density of the sealing medium at the cavitation zone. Equation (1) can be written as

_{c}is the cavitation pressure.

_{r}nodes along the radial direction and divided into the same unit node length along the circumferential direction to obtain fully structured grids [33,34]. According to the number of textures N

_{t}, each column can be regarded as a composition of N

_{t}evenly distributed units along the radial direction. Given the initial film thickness h

_{0}and texture shape criterion r

_{p}, the center of the unit is taken as the origin to determine whether the position of each node in each unit is within the texture boundary. Thus, the initial grid nodes of the textured column are generated. Furthermore, the grid refinement and independence verification of the computational domain are performed to enhance the texture boundary treatment. The numerical models of each texture column with the number of radial nodes N

_{r}ranging from 50 to 450 are selected for independence verification. The results show that the error of the numerical results is within 0.5% when the number of radial nodes N

_{r}reaches 300. Therefore, this study adopts the numerical model with 300 radial nodes to conduct the simulation, considering the calculation time and accuracy, as shown in Figure 3b.

#### 2.2.2. Mechanical Analysis

_{open}) and closing force (F

_{close}) between the end faces of the stator and rotor are in balance, and a stable liquid film forms in the sealing gap. Since the metal seals are contact seals, the opening force (F

_{open}) comprises the above bearing force of liquid film (F

_{p}) and the contact force of the asperities (F

_{c}). The contact force (F

_{c}) can be characterized using the plastic contact model [36,37] as follows:

_{c}is the contact pressure, H is the flow stress (yield stress or compressive strength) of the softer face material, and z is the axial coordinate.

_{close}) is produced through the support of the O-ring and backup ring to the floating stator, as well as the influence of the inner and outer pressure. It can be determined by integrating the static contact pressure (p

_{sc}) on the end faces of metal seals. Considering the significant nonlinearity and complex structural arrangement of the rubber O-ring and backup ring, a two-dimensional axisymmetric finite element numerical model of metal seals is established via ANSYS 15.0 software to conduct the mechanical analysis, as shown in Figure 4.

^{5}MPa and Poisson’s ratio ν = 0.3 [38]. The rubber O-ring and backup ring use the typical hyperelastic material fluororubbers (FKM). The two-parameter Mooney–Rivlin hyperelastic model is adopted to describe the strain energy function of the rubber, given as [39,40]

_{10}and C

_{01}are the parameters, with C

_{10}= 1.444 and C

_{01}= 0.0165 for the O-ring and C

_{10}= 1.833 and C

_{01}= −0.003 for the backup ring.

_{y}= Y) is applied to the journal, and the radial and axial fixed constraint boundary conditions (U

_{y}= 0, U

_{X}= 0) are applied to the outer diameter side and back of the rotor, respectively. Step 2 entails applying the lubricant oil pressure and mud pressure on the inner and outer sides of the metal seals. After the numerical solution, the static contact pressure (p

_{sc}), close force (F

_{close}), and initial deformation of the end faces on the metal seals can be obtained under environmental pressure.

#### 2.2.3. Thermal Analysis

_{0}is the dry friction coefficient [42,43].

_{1}and S

_{2}are the corresponding surfaces, h

_{c}is the convective heat transfer coefficient, T

_{0}is the environmental temperature, and n

_{c}is the exterior normal direction.

#### 2.2.4. Deformation Analysis

_{m}) and thermal deformation (δ

_{t}). The microscopic deformation of the stator and rotor can be solved using the influence factor method [41].

_{i}is the initial deformation at radial node i, and DR

_{i}and DS

_{i}are the deformations of the rotor and stator at radial node i, respectively, expressed as

_{j}is the force acting on the node j, q

_{j}is the heat generation rate, $M{{R}_{\mathrm{i}}}_{\mathrm{j}}$ and $M{{S}_{\mathrm{i}}}_{\mathrm{j}}$ are the influence factor matrices of the mechanical deformation of the rotor and stator, respectively, and $T{{R}_{\mathrm{i}}}_{\mathrm{j}}$ and $T{{S}_{\mathrm{i}}}_{\mathrm{j}}$ are the influence factor matrices of the thermal deformation of the rotor and stator, respectively.

#### 2.3. Performance Parameters

_{f}, leakage rate Q, and frictional force F

_{f}, can be expressed as [44]

_{m}is the average radius of the sealing zone.

## 3. Calculation Procedure

_{m}) to satisfy the force balance between the end faces of the rotor and stator. The outer cycle adjusts the interface temperature (T) on the end faces. Finally, the end face deformation, film thickness, and interface temperature are converged. Then, the fluid film characteristics and sealing performance parameters of surface-textured metal seals can be obtained.

## 4. Results and Discussions

#### 4.1. Liquid Film Pressure on End Faces

_{oi}= 3 MPa, pressure difference Δp

_{i}= 0.5 MPa, rotational speed n

_{i}= 200 rpm, the texture depth h

_{p}= 3 μm, and the area ratio S

_{p}= 0.2. Different from the gradual decline in the liquid film pressure on the untextured flat end face, the presence of micro-textures on metal seals results in the formation of textured convergent and divergent zones along the rotation direction. This strong hydrodynamic effect leads to higher peak values of liquid film pressure (p

_{max}) in the textured convergent zone, reaching approximately 3.9 to 5.62 times the environmental pressure. This magnitude far exceeds that observed in untextured metal seals. Hence, the textured end face can provide a specific bearing capacity of liquid film. As a consequence of the rotational shear exerted on the end faces, the fluid density and pressure in the divergent zone of the textured liquid film significantly decrease, leading to the occurrence of cavitation at the textured zones. The extent of cavitation, which varies depending on the shape of the textures, accounts for approximately 5% to 27% of the overall texture area.

_{max}) among the five shape textures. In comparison to the circle- and ellipse-shaped textures, the triangle- and chevron-shaped textures exhibit a pronounced angularity in the direction of the fluid flow, thereby facilitating a more significant hydrodynamic effect in the convergent zone of the liquid film. For line 2, the liquid film pressure (p) at the bottom of these textures diminishes to zero. The distributions of liquid film pressure in the divergent zone of the circle-, ellipse I- and ellipse II-shaped textures are comparable, with their film pressure values surpassing those of the triangle and chevron textures. The sharp-angle texture on the end faces of metal seals promotes the concentration and divergence of the liquid film pressure.

#### 4.2. Influence of Operating Conditions on Sealing Performance

_{f}), maximum interface temperature rise (ΔT

_{max}), leakage rate (Q), and frictional force (F

_{f}), respectively, under different rotational speeds (n = 100~500 rpm) and environmental pressures (p

_{o}= 3~69 MPa), corresponding to a formation depth of l = 260~6000 m.

_{f}) of the textured end faces gradually increase, and the leakage rates (Q) increase significantly due to the increase in film thickness. In contrast, the liquid film bearing coefficient (K

_{f}) and leakage rate (Q) of the untextured end face undergo minimal alterations. Furthermore, the liquid film bearing capacities of the triangle- and ellipse II-textured end faces are greater than those of the other three textured end faces. However, the leakage rate (Q) of the ellipse II-textured end face exhibits the highest value among the five different textured end faces. The maximum temperature rise (ΔT

_{max}) of both the untextured and textured end faces increase approximately linearly with the rotational speed (n), while the variation in frictional force (F

_{f}) on the end faces differs greatly with the rotational speed (n). The untextured end face generates substantial contact pressure and frictional heat due to its minimal film thickness, resulting in significantly larger values of maximum temperature rise and frictional force (ΔT

_{max}, F

_{f}) compared to those of the textured end faces. The order of the texture shapes, namely, ellipse I, circle, ellipse II, chevron, and triangle, correspond to the decreasing maximum temperature rise and frictional force (ΔT

_{max}, F

_{f}). Specifically, when the rotational speed (n) is 500 rpm, the frictional force (F

_{f}) of the triangle-textured end face decreases to 52.7% and to 74.5% regarding the untextured and ellipse I-textured end faces, respectively. This indicates that the triangle texture enhances the bearing capacity of the liquid film, making it easier to open the sealing end face and resulting in reduced contact frictional heat.

_{f}) and leakage rates (Q) of the untextured end face increase as a result of the radial redistribution of the film thickness, leading to a notable reduction in frictional heat between the sealing rings. The liquid film bearing coefficients (K

_{f}) of the textured end faces are larger than that of the untextured end face and gradually increase up to 98.9%. It is noteworthy that the hydrodynamic effect of the textured end faces diminishes gradually when the environmental pressure is p ≥ 30 MPa, resulting in liquid film bearing coefficients that are no longer superior to that of the untextured end face. This can be attributed to the substantial alterations in film thickness and contact pressure, leading to a fluctuation in the leakage rates (Q) of the circle-, triangle- and chevron-textured end faces. The leakage rates (Q) initially decrease and then increase with the environmental pressure (p), ultimately reaching the minimum values at p = 11 MPa. The maximum temperature rise and frictional force (ΔT

_{max}, F

_{f}) of these three textured end faces decrease with the environmental pressure (p), consistently remaining lower than those of the untextured end face. It can effectively weaken the wear of the sealing end faces. The leakage rates (Q) of the ellipse I- and ellipse II-textured end faces gradually decrease to a stable value with the environmental pressure (p). However, except for the low-pressure condition, the ellipse I- and ellipse II-textured end faces exhibit higher maximum temperature rise and frictional force (ΔT

_{max}, F

_{f}) compared to the untextured, circle-, triangle-, and chevron-textured end faces. This can be attributed to the significant shear effect of the liquid film on the ellipse-textured end faces, resulting in an inferior sealing performance under high-pressure conditions. Therefore, the influence of surface textures on hydrodynamic pressure is crucial for enhancing the sealing efficacy in low-pressure conditions. As the environmental pressure increases, the hydrodynamic pressure effect induced by the textures weakens, while the circle-, triangle- and chevron-shaped textures still help to reduce the contact pressure and frictional heat of the sealing rings.

#### 4.3. Influence of Texture Parameters on Sealing Performance

_{p}) and area ratio (S

_{p}) are further investigated.

_{p}= 2.6~8.6 μm) and area ratios (S

_{p}= 0.1$~$0.4), respectively. With the increase in the texture depth (h

_{p}), the hydrodynamic pressure effect resulting from the fluid extrusion in the textured convergent zone and the cavitation area caused by the fluid shear in the textured divergent zone of the liquid film both weaken. The pressure distribution of the fluid film on the triangle-textured end face becomes more uniform with the increase in h

_{p}. Additionally, the peak value of liquid film pressure (p

_{max}) at h

_{p}= 8.6 μm is 71.6% lower compared to h

_{p}= 2.6 μm. Consequently, the liquid film bearing coefficient (K

_{f}) and the opening degree of the sealing end face decrease significantly. In particular, the leakage rate (Q) of metal seals reduces rapidly to less than 0.1 mL·h

^{−1}. However, the maximum temperature rise and frictional force (ΔT

_{max}, F

_{f}) of the triangle-textured end face increase and gradually stabilize after h

_{p}≥ 4.3 μm.

_{p}) increases from 0.1 to 0.4, the textured convergent zone and divergent zone of the liquid film continue to expand. This expansion is accompanied by a heightened hydrodynamic effect and shear effect, resulting in a 41.9% increase in the peak pressure (p

_{max}) and a 17.4% increase in the bearing coefficient of the liquid film (K

_{f}) on the triangle-textured end face. Accordingly, the maximum temperature rise and frictional force (ΔT

_{max}, F

_{f}) decrease with the attenuated contact area. It is noteworthy that the leakage rate (Q) of the metal seals increases first and then decreases with the area ratio (S

_{p}). Eventually, a phenomenon known as ‘negative leakage’ emerges. Specifically, when S

_{p}increases from 0.3 to 0.35, the leakage rate (Q) changes from positive to negative, and the ‘zero leakage’ of metal seals can be obtained theoretically at S

_{p}= 0.34 and h

_{p}= 3 μm. Increasing the texture area ratio makes the textures gradually approach the outer diameter side of the end faces, resulting in a certain degree of directional pumping effects. This leads to the invasion of the external drilling mud and accelerates the wear failure on the end faces of metal seals.

#### 4.4. Experimental Validations

_{p}= 0.2 are adopted and compared with the untextured flat end face of metal seals. The circumferential and radial arrangements of the textures in the experiment are consistent with those in the numerical calculation.

## 5. Conclusions

- A comprehensive multi-field coupling numerical model for the surface-textured metal seals of roller bits is established, taking into account the influence of surface roughness, asperity contact, micro-deformation, and cavitation. Under the initial condition of p
_{oi}= 3 MPa and n_{i}= 200 rpm, the textured convergent zones and divergent zones of the liquid film are formed along the rotational direction on end faces with different shape textures. The sharp-angle textures, including triangle and chevron textures, can produce a stronger hydrodynamic pressure effect on the end faces of metal seals. The cavitation area of different shape textures accounts for about 5% to 27% of the total texture area. - The liquid film bearing coefficient, leakage rate, and maximum temperature rise of the metal seals with five shape textures increase with an increase in the rotational speed (n) from 100 rpm to 500 rpm. The temperature rise and frictional force of the textured end faces under different rotational speeds are significantly improved when compared to the untextured flat end face, particularly in the case of the triangle-textured end face. As the environmental pressure increases, the hydrodynamic pressure effect induced by the textures weakens, although the textures continue to contribute to a reduction in the contact pressure and frictional heat of the sealing rings. However, when the environmental pressure is p ≥ 30 MPa, the textured end face gradually loses its advantage in the sealing performance of metal seals.
- The liquid film characteristics and sealing performance of triangle-textured end faces are significantly affected by texture parameters, specifically the depth and area ratio. The hydrodynamic pressure effect and cavitation phenomenon intensify with a decrease in the depth (h
_{p}) from 8.6 μm to 2.6 μm and with an increase in the area ratio (S_{p}) from 0.1 to 0.4. Consequently, the maximum temperature rise and frictional force (ΔT_{max}, F_{f}) decrease with the area ratio (S_{p}). The effect of depth (h_{p}) on the sealing parameters is significantly weakened when h_{p}≥ 4.3 μm. The leakage rate (Q) changes from positive to negative in the range of S_{p}= 0.3~0.35, and the optimal texture parameters of S_{p}= 0.34 and h_{p}= 3 μm can achieve an ideal state of ‘zero leakage’. - A sealing performance test bench of surface-textured metal seals is constructed to validate the accuracy of multi-field coupling simulation. The application of surface textures to metal seals offers advantages in terms of reducing temperature rise, friction and wear on the end faces. The untextured and textured end faces are in a semi-dry friction and mixed friction state during the test, respectively, resulting in the occurrence of abrasive wear and adhesive wear on the end faces of the rotor and stator. The triangle texture has proven to have the best hydrodynamic lubrication and wear resistance among the five shape textures.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**Sealing performance of metal seals under different rotational speeds. (

**a**) Fluid film bearing coefficient; (

**b**) Maximum temperature rise; (

**c**) Leakage rate; (

**d**) Frictional force.

**Figure 10.**Sealing performance of metal seals under different environmental pressures. (

**a**) Fluid film bearing coefficient; (

**b**) Maximum temperature rise; (

**c**) Leakage rate; (

**d**) Frictional force.

**Figure 11.**Contours of fluid film pressure under different texture parameters. (

**a**) Texture depth; (

**b**) Area ratio.

**Figure 12.**Sealing performance under different texture parameters. (

**a**) Texture depth; (

**b**) Area ratio.

**Figure 17.**Worn surface morphology of rotor and stator with different shape textures. (

**a**) Untextured; (

**b**) Circle; (

**c**) Ellipse I; (

**d**) Ellipse II; (

**e**) Triangle; (

**f**) Chevron.

Structural Parameters | Values | Operating Parameters | Values |
---|---|---|---|

Inner diameter of stator r_{d} (mm) | 29 | Environmental pressure p_{o} (MPa) | 3~69 |

Outer diameter of rotor r_{s} (mm) | 34.5 | Pressure difference Δp (MPa) | 0.5 |

Sealing inner diameter r_{i} (mm) | 31 | Environmental temperature T_{o} (°C) | 8~180 |

Sealing outer diameter r_{o} (mm) | 34.3 | Rotational speed n (rpm) | 100~500 |

Wedging angle α (°) | 5 | Density of lubricant oil (kg/m^{3}) | 861 |

Incline angle β (°) | 65 | Density of drilling mud (kg/m^{3}) | 1742 |

Surface roughness σ (μm) | 0.2 | Viscosity of lubricant oil (Pa·s) | 0.002~0.189 |

Dry friction coefficient f_{o} | 0.08 | Viscosity of drilling mud (Pa·s) | 0.02~0.03 |

Type | 3-D Shape | Definition [25] | Parameter | Value |
---|---|---|---|---|

Circle | ${S}_{\mathrm{p}}=\frac{\pi {r}_{\mathrm{p}}^{2}}{{l}_{\mathrm{c}}^{2}}$ | Texture depth h_{p} (μm) | 2.6~8.6 | |

Ellipse I | ${S}_{\mathrm{p}}=\frac{\pi \mathit{ab}}{{l}_{\mathrm{c}}^{2}}$ | Texture control unit l_{c} (μm) | 550 | |

Ellipse II | ${S}_{\mathrm{p}}=\frac{\pi \mathit{ab}}{{l}_{\mathrm{c}}^{2}}$ | Column number T | 300 | |

Triangle | ${S}_{\mathrm{p}}=\frac{3\sqrt{3}{r}_{\mathrm{p}}^{2}}{4{l}_{\mathrm{c}}^{2}}$ | Texture area ratio S_{p} | 0.1~0.4 | |

Chevron | ${S}_{\mathrm{p}}=\frac{3\sqrt{3}{\left(1-{K}_{\mathrm{p}}\right)r}_{\mathrm{p}}^{2}}{4{l}_{\mathrm{c}}^{2}}$ | Texture quantity N_{t} | 6 |

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Ma, Y.; Li, Z.; Yuan, Z.; Meng, X.; Peng, X.; Jiang, J.
Multi-Field Coupling Numerical Analysis and Experimental Validation of Surface-Textured Metal Seals in Roller Cone Bits. *Lubricants* **2024**, *12*, 15.
https://doi.org/10.3390/lubricants12010015

**AMA Style**

Ma Y, Li Z, Yuan Z, Meng X, Peng X, Jiang J.
Multi-Field Coupling Numerical Analysis and Experimental Validation of Surface-Textured Metal Seals in Roller Cone Bits. *Lubricants*. 2024; 12(1):15.
https://doi.org/10.3390/lubricants12010015

**Chicago/Turabian Style**

Ma, Yi, Ziang Li, Ziyang Yuan, Xiangkai Meng, Xudong Peng, and Jinbo Jiang.
2024. "Multi-Field Coupling Numerical Analysis and Experimental Validation of Surface-Textured Metal Seals in Roller Cone Bits" *Lubricants* 12, no. 1: 15.
https://doi.org/10.3390/lubricants12010015