Multi-Physical Field, Coupled, Mixed Lubrication Analysis of Hydraulic Reciprocating Vacuum Lip Seal

Abstract

Engineering practice has demonstrated that seal failure can result in severe leakage and wear, reducing the efficiency of hydraulic systems and even leading to major safety risks. Currently, analyses of the thermal aspect of seal interfaces are relatively limited, with most studies focusing on mechanical analysis. However, in actual applications, temperature has a significant impact on sealing performance. In this paper, nonlinear elastomechanics, viscous fluid mechanics, micro-contact mechanics, micro-deformation theory, and thermodynamics are coupled to establish a mixed lubrication model considering the thermal effect. The reliability of the mixed lubrication model is verified through experiments, and the temperature distribution of the oil film in the sealing area and the temperature distribution of the seal ring are simulated. Finally, the effects of the reciprocating speed, root mean square roughness, fluid medium pressure, and seal pre-compression on seal friction force and leakage are investigated. The results show that the heat generated in the sealing area accumulates at the bottom of the V-ring. Under the same conditions, compared with the instroke, the temperature-rise area of the outstroke is biased to the left and the increase in temperature is greater. In addition, the piston rod speed and the preliminary compression of the seal ring have a greater impact on the overall seal friction force and leakage. Under a lower seal pre-compression, the RMS roughness has a great influence on the leakage and friction in the outstroke, while the impact of the internal stroke is limited.

1. Introduction

With the development of the times, sealing structure systems are also constantly improving. The hydraulic reciprocating seal structure belongs to the formed packing seal and has been widely used in the aerospace, electronics, machinery, and chemical fields. The main function of the formed packing seal is to prevent the leakage of the sealed medium and the entry of pollutants, and to ensure the safe and stable operation of the hydraulic system.

In the hydraulic structure, the movement of the piston rod will cause the different pressures inside and outside the sealing device to create a pressure difference. Under the action of the pressure difference and the pulling force of the piston rod, the liquid medium will form a lubricating oil film with a micron-scale in the sealed area. The lubricating oil film can effectively reduce the contact between the sealing material and the piston rod, but also increases the leakage of the liquid medium [1,2,3]. In order to explore the lubrication mechanism in the working process of the sealing structure in detail, relevant scholars have carried out detailed theoretical analyses and experimental research [4,5,6].

In recent years, through in-depth research, some scholars have revealed the operating mechanism of the oil film to a great extent. However, during the reciprocating motion of the piston rod, the temperature at the sealing contact interface will increase, which will have a certain impact on the mechanical behavior of the sealing material, such as creep and fatigue. These factors will have a significant impact on the sealing performance and service life of the sealing material. However, in the current research, the mechanism of the thermal effect in the reciprocating hydraulic seal is rarely studied. According to the research results of relevant scholars regarding the rotary lip seal, the thermal effect will change the viscosity of the oil film in the sealed area, thereby affecting the sealing performance [7,8,9]. According to the experimental study of Pinedo et al. [10], during the working process of the reciprocating hydraulic seal, the temperature rise at the point of contact between the rubber and the piston rod is very obvious, which will cause the friction performance of the rubber to change. In addition, the temperature of the oil film in the sealed area will also affect the aging and swelling of the rubber. Therefore, it is of great significance to study the temperature field in the sealed area.

At present, there is some research devoted to studying the influence of thermal effects on reciprocating seal behavior. For example, Nikas et al. [11] established a theoretical model to study the influence of different parameters on the sealing performance of a rectangular elastomer. In this theoretical model, the influence of temperature factors on the mechanical properties of rubber is considered, but the viscous shear effect of the fluid medium in the sealed area is not considered.

The vacuum lip (VL) seal is a specialized structure designed for rod-sealing applications, characterized by its unique geometry that effectively isolates oil and air, ensuring optimal sealing properties. The design of the VL seal focuses on minimizing friction during contact with dynamic surfaces, which not only reduces the system’s energy consumption but also decreases heat generation. This, in turn, extends the service life of the seal itself and the related components. By reducing leakage and friction, the VL seal significantly enhances the efficiency and performance of the entire system.

Hydraulic cylinders are critical components in various industrial applications, and their performance often determines the overall efficiency and reliability of machinery. Seal failures are among the most common causes of hydraulic cylinder malfunctions, leading to increased maintenance costs and downtime. Despite the importance of effective sealing solutions, most current studies predominantly focus on other sealing structures, such as O-rings. This paper aims to address this gap by exploring a VL type sealing structure as the research object. By investigating the VL seal’s potential to mitigate common hydraulic cylinder failures, this study seeks to underscore its value in improving system reliability and operational efficiency.

This paper uses a VL sealing structure to study lubrication performance in the sealing environment. A comprehensive lubrication model is established, considering the surface roughness, sticky wedge effect, and thermal effect. The model is mainly used to calculate the distribution of the oil film temperature and is solved using the generalized Reynolds equation. Regarding the characteristics of the VL sealing structure in the mixed lubrication state, the model integrates many factors, such as the cavitation effect, surface roughness effect, and oil film adhesive wedge effect, as well as considering the thermal effect, to create a thermal elastohydrodynamic model. Furthermore, we refer to the quasi-3D energy equation proposed by Meng et al. [12] to predict the temperature distribution of oil membranes. Through this model, this study not only analyzes the mechanism of oil film temperature distribution, but also studies the law of friction force and changes in leakage under working conditions.

2. Model Establishment

The aircraft hydraulic actuator is a key component of the aircraft hydraulic system, responsible for converting hydraulic energy into mechanical energy to perform various flight control and operation tasks. It usually consists of a hydraulic cylinder, piston, piston rod, and associated seals. The hydraulic actuator pushes the piston in motion through the pressurized hydraulic oil provided by the hydraulic pump, thus creating a linear or rotational motion. This movement is used to control the aircraft’s elevator, ailerons, and rudder, and the expansion of the landing gear. Hydraulic actuators have high efficiency and reliable and responsive characteristics, and can work with a high load and under high-pressure conditions. The design’s durability and safety must be considered to ensure it has stable energy under various flight conditions. Regular maintenance and inspection are essential to ensure the normal operation of the hydraulic actuators and to extend their service life. The red circle in Figure 1 denotes the aircraft hydraulic system.

During the working process of the hydraulic reciprocating seal device (Figure 2a), when the piston rod moves outward (i.e., the outstroke), the piston rod moves to the air side, and a small amount of hydraulic oil will be brought out by the piston rod during this process; when the piston rod moves inward (i.e., the instroke), the piston rod moves into the inside of the hydraulic cylinder and will return a small amount of hydraulic oil from the air side to the inside of the hydraulic cylinder. After a stroke cycle is completed, the amount brought out during the outstroke minus the amount brought in during the instroke is the actual leakage amount of a working cycle of the hydraulic cylinder. Figure 2b shows the schematic diagram of the VL model sealing structure model. According to the research of relevant scholars, the hydraulic reciprocating sealing structure is always in a mixed lubrication state during work [1,2]. Therefore, it is indicated that there is both an oil film and contact between the sealing material and the piston rod in the sealing area (Figure 2c). In order to accurately obtain the friction characteristics and leakage rate of the sealing structure, it is necessary to study and discuss the characteristics of the lubricating oil film thickness and the oil film pressure.

2.1. Hypothesis of the Thermal Elastohydrodynamic Model for Hydraulic Reciprocating Seals

The calculation of the hydraulic reciprocating seal oil film is solved using the most basic equations of fluid lubrication theory, namely the Reynolds equation derived from the motion equation and the continuity equation. Before establishing the model, the following assumptions need to be made:

(1)

There is no sliding at the interface of the fluid.

(2)

The effect of volume forces (such as gravity) is ignored.

(3)

The pressure change in the fluid in the direction along the thickness of the lubricating film is ignored.

The above assumptions are for general fluid problems. To simplify the problem, the following assumptions are introduced:

(1)

The lubricating fluid is a Newtonian fluid.

(2)

The flow of the oil film is laminar, without an eddy current and turbulence.

(3)

The inertia force of the fluid is ignored.

(4)

The viscosity value of the fluid remains constant in the direction along the thickness of the lubricating film.

Based on all the above assumptions, the Reynolds equation is derived from the Navier–Stokes equation and the continuity equation using the infinitesimal element analysis method. The specific steps are as follows: First, the velocity distribution of the fluid along the thickness direction of the film is obtained based on the force balance condition of the fluid’s infinitesimal elements. Secondly, the fluid flow rate is obtained by integrating the fluid along the thickness direction of the lubricating film. Finally, the Reynolds equation is derived based on the continuous condition of the flow rate. In order to minimize the error under the hypothesis conditions, the HIDALISS type 32 hydraulic oil produced in China was selected as the experimental hydraulic oil, with a viscosity of 32 mm²/s.

2.2. Nonlinear Elastic Mechanics Analysis

When using the elastic hydrodynamic lubrication method, due to the extremely thin film thickness, the mixed lubrication contact pressure distribution in the sealing contact area under static conditions is almost the same as the dry contact pressure distribution. Therefore, the finite element method can be used to determine the dry contact pressure in the sealing area. The studied VL seal structure consists of an O-ring and a V-ring, forming a combined seal structure. Therefore, during the calculation process, it can be converted into a two-dimensional axisymmetric model to reduce the amount of calculation required. In the VL seal structure, the elastic modulus of rubber and PTFE materials is much smaller than that of steel, so it can be assumed that the piston rod and the housing have no deformation process. The material of the O-ring is mainly NBR material, and its stress–strain behavior can be described by the Mooney–Rivlin model. The model parameters in this paper are taken from the literature [13]: C₁₀ = 0.202, C₀₁ = 6.858. The Mooney–Rivlin model describes the nonlinear behavior of materials on the basis of the strain energy density function; this is a widely recognized model with high analytical accuracy. The strain energy function of the M-R native model of the rubber material is as follows:

$$W=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+{\left(J-1\right)}^2/d_0$$

(1)

where I₁ and I₂ are the first and second invariants of the strain tensor, respectively, while C₁₀ and C₀₁ are the M-R constant, reflecting the stress and strain properties of the material; d₀ and J are the parameters reflecting the elastic deformation degree of the material, and W is the strain energy density.

The material of the V-ring is mainly polytetrafluoroethylene (PTFE), and its mechanical properties can be described using the isotropic yield model of the Mises yield surface theory. In this paper, the pre-tightening force is applied to the seal structure to determine the compression of the seal structure, and the fluid infiltration load is used to determine the oil side’s pressure boundary conditions. The dry contact pressure distribution under different working conditions is calculated using the finite element analysis method.

As shown in Figure 2c, the VL sealing structure is affected by the combined effect of three forces: the preload force formed by the seal installation force, the piston rod, the support force of the sealing material, and the fluid force of the sealing structure. The three forces can be viewed as equivalent to three contact pressure: static contact pressure p_s, rough peak contact pressure p_c and oil film pressure p_f, respectively. Among them, the rough peak contact pressure p_c and the oil film pressure p_f act on the surface of the sealing lip and move vertically upward, while the static contact pressure p_s of the seal ring acts on the sealing lip in a direction perpendicular to the downward direction. Under the combined action of these three pressures, the seal lip of the seal ring will deform, thereby changing the oil film thickness h. The change in the oil film thickness h will, in turn, change the oil film pressure p_f and the rough peak contact pressure p_c. Eventually, when the three pressures in the seal ring area reach a balance, the oil film thickness h will no longer change, the entire numerical solution process will converge, and the final oil film pressure p_f, rough peak contact pressure p_c, oil film thickness h, friction force F, and leakage q can be obtained.

2.3. Viscous Fluid Dynamics Analysis

At present, when the traditional elastic fluid dynamic lubrication model uses the influence coefficient matrix method to calculate the microelastic deformation of the sealing element, the extraction of the influence coefficient matrix requires a lot of simulation calculations using the finite element software. In order to facilitate the convergence of the flow–solid coupling iterative calculations and reduce the computation time as much as possible, the contact area of the seal ring and the plunger is often established using a local grid, which greatly increases the time needed to extract the influence coefficient matrix using finite element software. The fluid dynamic lubrication method is calculated, or the measured static contact pressure is assumed to be approximately equal to the fluid pressure. Using the Renault equation, the thickness of the oil membrane can be determined, and then the sealing performance parameters of the friction force and the roughness and the thickness of the oil membrane can be obtained. Therefore, we first obtain an approximate initial film thickness distribution using the hydrodynamic inverse solution method, which is then corrected using the elastic hydrodynamic lubrication method.

Firstly, according to the literature [14], the Reynolds equation can be simplified into the following one-dimensional steady-state form:

$${\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{h^3}{\mu }}{\displaystyle \frac{d{p}_f}{dx}}\right)=6u{\displaystyle \frac{dh}{dx}}$$

(2)

In the formula, h is the average oil film thickness in the sealing area, μ is the fluid dynamic viscosity, pf is the fluid pressure in the sealing area, and u is the reciprocating speed of the piston rod. Integrating the above formula leads to the following equation:

$${\displaystyle \frac{d{p}_f}{dx}}=6\mu u{\displaystyle \frac{h-h_0}{h^3}}$$

(3)

where h₀ is the oil film thickness at the maximum pressure, dp/dx = 0. Differentiating both sides of Formula (3) leads to the following:

$$h^3{\displaystyle \frac{d^2{p}_f}{d{x}^2}}+3{\displaystyle \frac{dh}{dx}}\left(h^2{\displaystyle \frac{d{p}_f}{dx}}-2\mu u\right)=0$$

(4)

At the inflection point a of the fluid pressure, d²p/dx² = 0; therefore, the oil film thickness at this point can be expressed as follows:

$$h_a=\sqrt{{\displaystyle \frac{2\mu u}{{\left(\frac{d{p}_f}{dx}\right)}_{\max }}}}$$

(5)

Substituting Equation (5) into Equation (3) provides the following:

$$h_0={\displaystyle \frac{2}{3}}{h}_a={\displaystyle \frac{1}{3}}\sqrt{{\displaystyle \frac{8\mu u}{{\left(\frac{d{p}_f}{dx}\right)}_{\max }}}}$$

(6)

The following pressure gradient G and oil film thickness H′ can be defined as follows:

$$G={\displaystyle \frac{h_0^2}{6\mu u}}{\displaystyle \frac{d{p}_f}{dx}}$$

(7)

$$H^{\prime }(x)={\displaystyle \frac{h(x)}{h_0}}$$

(8)

Equations (7) and (8) can be substituted into Equation (2); after simplification, a cubic equation for H can be obtained:

$$G\cdot {H^{\prime }}^3(x)-H^{\prime }(x)+1=0$$

(9)

By calculating the fluid pressure distribution function p_f(x) and solving Equations (6), (7) and (9), the average oil film thickness distribution in the sealing area is obtained. The above content shows the process of calculating the film thickness using the hydrodynamic inverse solution method. Next, the elastic hydrodynamic lubrication method is used to correct the thickness of the film. According to the research [15] of Salant et al., using the dimensionless Reynolds equation considering cavitation and surface roughness, the pressure distribution under a fixed film thickness is obtained. Since the fluid in the sealing area cavitates when the fluid pressure is less than the cavitation pressure p_cav, this paper uses the Jakobsson–Floberg–Olsson (JFO) cavitation theory for its description [16]. “When the piston or piston rod is rapidly moving fast in the cylinder, there may be a local low-pressure area near the air side of the sealing lip. If the pressure is lower than the evaporative pressure of the hydraulic oil, the hydraulic oil may evaporate to form bubbles, creating a cavitation effect”. According to the JFO cavitation theory, the sealed lubrication region can be divided into a fully fluid region and a gas–liquid coexistence region, and mass conservation is observed in the fully fluid region, the cavitation region, and the interface between the two. By assuming the previous conditions, the sealing structure can be simplified into a one-dimensional axisymmetric model to reduce the calculation time. The Reynolds equation considering fluid cavitation effects is obtained from Elrod [17], Payvar [18], and Xiong [19]:

$${\displaystyle \frac{\partial }{\partial x}}\left[{\phi }_{xx}{\displaystyle \frac{h^3}{\mu }}{\displaystyle \frac{\partial \left(F\Phi \right)}{\partial x}}\right]={\displaystyle \frac{6u}{p_a-p_{cav}}}{\displaystyle \frac{\partial }{\partial x}}\left\{\left[1+\left(1-F\right)\Phi \right]\cdot \left(h_t+\sigma \cdot {\phi }_{scx}\right)\right\}$$

(10)

h is the average oil film thickness, h_t is the truncated true film thickness. p_a is the atmospheric pressure, and p_cav is the cavitation pressure; the pressure flow factors in the x direction and the correction factor of the velocity component cause the medium to flow. u is the rod speed, F is the cavitation factor, and Φ is the average density function, obtained from Reference [4]. The boundary conditions are as follows:

$$\left\{\begin{array}{l}\Phi \ge 0, F=1 and \overline{p}=\Phi \mathrm{in} \mathrm{the} \mathrm{liquid} \mathrm{zone}\\ \Phi <0, F=0 and \overline{p}=0, \overline{\rho }=1+\Phi \mathrm{in} \mathrm{the} \mathrm{cavitation} \mathrm{zone}\end{array}\right.$$

In Equation (10), μ is obtained from the viscosity–temperature equation [20,21]:

$$\mu =\epsilon \exp \left\{\left(\ln {\mu }_0+9.67\right)\left[-1+{\left(1+5.1\times 10-9p\right)}^{z_0}{\left({\displaystyle \frac{T-138}{T_0-138}}\right)}^{-s_0}\right]\right\}$$

(11)

$$z_0={\displaystyle \frac{\alpha }{5.1\times {10}^{-9}\left(\ln {\mu }_0+9.67\right)}}$$

(12)

$$s_0={\displaystyle \frac{\beta \left(T_0-138\right)}{\ln {\mu }_0+9.67}}$$

(13)

$$\epsilon =1+{\displaystyle \frac{0.6\times {10}^{-9}p}{1+1.7\times {10}^{-9}p}}-0.0007\left(T-T_0\right)$$

(14)

In Equation (11), μ₀ is the viscosity of fluid at normal pressure, and α is the viscosity–pressure coefficient. T₀ is the reference temperature, T is the film temperature of the model fluid, and β is the viscosity–temperature correction coefficient. The local thickness of the oil film is as follows:

$$h_T(x,y)=h(x,y)+{\delta }_1(x,y)+{\delta }_2(x,y)$$

(15)

When two rough surfaces come into contact, the oil film is truncated so that the local oil film thickness h_T at that location is equal to 0; when solving the contact deformation problem of two rough surfaces, this is generally equivalent to a rough surface and a smooth surface coming into contact, and the comprehensive rough peak deviation from the centerline amplitude δ(x, y) of the equivalent rough surface is as follows:

$$\delta (x,y)={\delta }_1(x,y)+{\delta }_2(x,y)$$

(16)

Using probability and statistics methods, assuming that the rough peak of the rough surface follows the probability density function distribution of f(δ), the average oil film thickness can be written as follows:

$$h_T(x,y)={\displaystyle {\int }_{-h}^{\infty }\left[h(x,y)+\delta (x,y)\right]f\left(\delta \right)}d\delta$$

(17)

Assuming that the surface rough peak follows a Gaussian function distribution, Equation (17) can be calculated using the following formula:

$$h_T={\displaystyle \frac{h}{2}}+{\displaystyle \frac{h}{2}}erf\left({\displaystyle \frac{h}{\sqrt{2}}}\right)+{\displaystyle \frac{1}{\sqrt{2\pi }}}\exp \left({\displaystyle \frac{-h^2}{2}}\right)$$

(18)

2.4. Microscopic Contact Mechanics Analysis

Hydraulic reciprocating seals work in such a way that there is a sealing surface in the contact area between the seal ring and the piston rod, which includes not only the physical contact between the two, but also the contact relationship between the hydraulic oil (Figure 2c). Therefore, this paper uses the G-W contact model to describe the interaction between the roughnesses of the sealing friction pair surfaces [22]. This model assumes that each vertex of the roughness is a sphere with the same radius, each roughness is calculated using Hertz contact theory, and the roughnesses are independent of each other and do not affect each other. The dynamic contact pressure p_c is calculated as follows:

$$p_c={\displaystyle \frac{F_a}{A_n}}={\displaystyle \frac{4}{3}}\rho R^{1/2}{E}^{\prime }{\displaystyle {\int }_d^{\infty }{(h-h_t)}^{3/2}dz}$$

(19)

In Formula (19), F_a is the total contact load, A_n is the density of rough peaks on the nominal contact area, ρ is the roughness density, R is the roughness radius, $E^{\prime }$ is the equivalent elastic modulus of the two surfaces, calculated by Formula (19), and h is the probability density function of the peak.

$${\displaystyle \frac{1}{E^{\prime }}}={\displaystyle \frac{1-{\upsilon }_1^2}{E_1}}+{\displaystyle \frac{1-{\upsilon }_2^2}{E_2}}$$

(20)

In Formula (20), $E^{\prime }$ is the equivalent elastic modulus; E₁, E₂, ${\upsilon }_1$ and ${\upsilon }_2$ are the elastic modulus and Poisson’s ratio of the two contact surface materials, respectively. Since the elastic modulus E₁ of the rigid material plunger is much larger than the elastic modulus E₂ of the flexible material seal ring, the flexible material elastic factor can be ignored, and the equivalent elastic modulus is simplified as follows:

$$E^{\prime }={\displaystyle \frac{E_1}{1-{\upsilon }_1^2}}$$

(21)

$$p_c={\displaystyle \frac{4}{3}}\rho R^{1/2}{\displaystyle \frac{E_1}{1-{\upsilon }_1^2}}{\displaystyle {\int }_h^{\infty }\frac{1}{\sqrt{2\pi }{\sigma }_S}}{e}^{{\displaystyle \frac{-h_t^2}{2{\sigma }_S}}}{\left(h_t-h\right)}^{3/2}dz$$

(22)

In Formula (22), σ_s represents the standard deviation of the peak.

2.5. Microscopic Deformation Analysis

According to Section 2.2, the oil film thickness generated by the hydrodynamic pressure effect is much smaller than the radial deformation in the sealing element during installation and pressurization.

During a reciprocating motion, the compression and rebound contact force of the sealing element are approximately equal to the static contact force. Therefore, the micro-elastic deformation H_def of the sealing element in the sealing area during a reciprocating motion is caused by the pressure difference between the supporting force, composed of the fluid pressure pf and the roughness contact pressure p_c, and the rebound contact pressure p_s, which causes the radial deformation of the sealing material and the piston rod:

$$\Delta p=p_f+p_c-p_s$$

(23)

The oil film thickness is at the micron scale, and the small deformation linear superposition theory can be used. The change in the oil film thickness at any point in the sealing area is equal to the sum of the displacements generated by all nodes in the contact area at that point. The elastic deformation generated by the pressure Δp_j at node j in the sealing area at x_i is as follows:

$$\Delta H_{def}(x_i)=k_{ij}\Delta p_j$$

(24)

If k_ij represents the deformation amount of node i under unit load at node j, then the oil film thickness at any point in the sealing area can be expressed as follows:

$$H_{def}(i)={\displaystyle \sum _{j=1}^n{k}_{ij}\Delta p_j}$$

(25)

2.6. Thermodynamic Analysis

In this paper, it is assumed that the liquid film temperature follows a functional distribution in the direction of the film thickness. The problem is simplified to derive the average energy equation of the average temperature of the liquid film, and the oil temperature T of the oil film in the sealing area is assumed to be a fourth-order function of the first position z in the direction of h.

$$T=a{z}^4+b{z}^3+c{z}^2+dz+e$$

(26)

The liquid membrane temperature will be affected by the fluid viscosity shear and the microbulge contact shear friction, which are calculated by the following 3D energy equation:

$${\nabla }_3\cdot \left(k\nabla T\right)-{\nabla }_3\left(\rho cuT\right)+\Phi +{\displaystyle \frac{f_c{p}_c\left|U\right|}{h}}-{\displaystyle \frac{\partial \left(\rho cT\right)}{\partial t}}=0$$

(27)

The first term represents the liquid film heat transfer, the second term represents the liquid film convection heat transfer, the third term represents the liquid film viscosity shear heat, the fourth term represents the micro-convex body shear heat, the fifth term represents the transient heat source, ρ represents the oil film density, u represents the x, y, z directions for the liquid film velocity vector, and Φ represents the energy dissipation density function, as follows:

$$\Phi ={\displaystyle \frac{\mu U^2}{h^2}}+{\displaystyle \frac{{(h-2z)}^2}{4\mu }}\nabla p\cdot \nabla p+{\displaystyle \frac{U}{h}}(h-2z)\nabla p$$

(28)

Assuming that the specific heat capacity c of the oil film is a fixed value, the thermal conductivity k is independent of the film thickness direction, and the heat conduction of the oil film in the axial and circumferential directions is ignored. By introducing a continuity equation, the Equation (27) can be expressed as follows:

$$k{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}-\rho c\left(u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}+w{\displaystyle \frac{\partial T}{\partial z}}\right)+\Phi +{\displaystyle \frac{f_c{p}_c\left|U\right|}{h}}-\rho c{\displaystyle \frac{\partial T}{\partial t}}=0$$

(29)

As mentioned above, the hydraulic ring in this study can be reduced to a one-dimensional axisymmetric model, so the average energy equation of the sealed interface oil film, considering fluid viscous dissipation, microbulgity contact shear friction, and the heat exchange of fluids with the sealing ring and the piston rod, can be expressed as follows:

$$\nabla \cdot \left(kh\nabla T_h\right)-\rho ch{u}_h\cdot \nabla T_h+{\Phi }_{h}h+f_c{p}_c\left|u\right|-q_n-q_b=0$$

(30)

where c is the specific heat of the lubricant and f_c is the dry friction coefficient. q_n and q_b represent the local heat fluxes entering the V-shaped seal ring and the piston rod, respectively. u_h is the velocity vector of the hydraulic oil, which can be calculated using Formula (31). k is the thermal conductivity of the oil film, which is the combined thermal conductivity of the oil–air mixture components due to cavitation, and can be calculated using Formula (32). Φ_h is the average power dissipation density function of the entire fluid film, which can be obtained by the following expression in Formula (33):

$$u_h={\displaystyle \frac{U}{2}}-{\displaystyle \frac{h^2}{12\mu }}{\displaystyle \frac{\partial p}{\partial x}}$$

(31)

$$k=k_L{\displaystyle \frac{\rho }{{\rho }_L}}+\left(1-{\displaystyle \frac{\rho }{{\rho }_L}}\right)k_G$$

(32)

where k_L and k_G are the thermal conductivities of hydraulic oil and air, respectively. ρ and ρ_L are the fluid cavitation mixing density and the complete fluid density, respectively.

$${\Phi }_m={\displaystyle \frac{\mu U^2}{h^2}}+{\displaystyle \frac{h^2}{12\mu }}\nabla p\cdot \nabla p$$

(33)

According to Equation (26), the five coefficients a~e control the temperature field distribution of the whole sealing region; hence, the five governing equations for the temperature field function of the oil film are needed. In the sealed area, the local Cartesian coordinate system is adopted in Oxyz (x is axial, y is circumferential, and z is membrane thickness). Suppose the piston rod surface (z = 0) temperature is fixed Trod, the seal ring surface temperature is T_Seal, and at z = 0 in Equation (29), u = U and u = v = 0. The following five controlling equations involving the temperature field function of the liquid film are used:

$$T\left(0\right)=T_{Rod}$$

(34)

$$T\left(h\right)=T_{Seal}$$

(35)

$${\displaystyle {\int }_0^{h}Tdz}=T_m\cdot h$$

(36)

$$T_{rod}^{″}=-{\displaystyle \frac{1}{k}}\left[\Phi \left(0\right)+{\displaystyle \frac{f_c{p}_c\left|U\right|}{h}}\right]$$

(37)

$$T_{Seal}^{″}=-{\displaystyle \frac{1}{k}}\left[\Phi \left(h\right)+{\displaystyle \frac{f_c{p}_c\left|U\right|}{h}}\right]$$

(38)

where Φ(0) and Φ(h) represent the energy dissipation function of the oil film in the seal area on the piston rod and the seal ring surface, respectively, expressed using Equation (28):

$$\Phi (0)={\displaystyle \frac{\mu U^2}{h^2}}+{\displaystyle \frac{h^2}{4\mu }}\nabla p\cdot \nabla p+U\nabla p$$

(39)

$$\Phi (h)={\displaystyle \frac{\mu U^2}{h^2}}+{\displaystyle \frac{h^2}{4\mu }}\nabla p\cdot \nabla p-U\nabla p$$

(40)

In Equations (28) and (34)–(38), five coefficients a~e of the oil film temperature field T in the sealing area are expressed as follows:

$$a={\displaystyle \frac{1}{h^4}}\left[5T_m-{\displaystyle \frac{5}{2}}(T_{Seal}+T_{Rod})+{\displaystyle \frac{5}{24}}{h}^2(T_{Rod}^{″}+T_{Seal}^{″})\right]$$

(41)

$$b={\displaystyle \frac{1}{h^3}}\left[-10T_m+5(T_{Rod}+T_{Seal})-{\displaystyle \frac{7}{12}}{h}^2{T}_{Rod}^{″}-{\displaystyle \frac{1}{4}}{h}^2{T}_{Seal}^{″}\right]$$

(42)

$$c={\displaystyle \frac{1}{h^2}}\left({\displaystyle \frac{1}{2}}{h}^2{T}_{Rod}^{″}\right)$$

(43)

$$d={\displaystyle \frac{1}{h}}\left(5T_m-{\displaystyle \frac{7}{2}}{T}_{Rod}-{\displaystyle \frac{3}{2}}{T}_{Seal}-{\displaystyle \frac{1}{8}}{h}_2{T}_{Rod}^{″}+{\displaystyle \frac{1}{24}}{h}_2{T}_{Seal}^{″}\right)$$

(44)

$$e=T_{Rod}$$

(45)

The coefficients a~e are used to represent the functions of variables T_h, T_Rod, T_Seal, $T_{Seal}^{″}$, and $T_{Rod}^{″}$. Except for T_h, the other parameters can be regarded as constant in the temperature solution process, so the solution for the temperature field distribution T of the oil film in the sealing area can be simplified to determine the average temperature of the oil film T_h. Further, to solve average energy Equation (30), it is necessary to clarify the energy exchange between the oil film and the seal ring and the piston rods q_n and q_b. The basic formula of heat conduction can be used to establish the following equations:

$$q_n={\displaystyle \frac{k}{h}}\left[5T_h-{\displaystyle \frac{7}{2}}{T}_{Rod}-{\displaystyle \frac{3}{2}}{T}_{Seal}-{\displaystyle \frac{1}{8}}{h}^2{T}_{Rod}^{″}+{\displaystyle \frac{1}{24}}{h}^2{T}_{Seal}^{″}\right]$$

(46)

$$q_b={\displaystyle \frac{k}{h}}\left[5T_h-{\displaystyle \frac{3}{2}}{T}_{Rod}-{\displaystyle \frac{7}{2}}{T}_{Seal}+{\displaystyle \frac{1}{24}}{h}^2{T}_{Rod}^{″}-{\displaystyle \frac{1}{8}}{h}^2{T}_{Seal}^{″}\right]$$

(47)

When the hydraulic cylinder is working, the piston rod reciprocates, the surface sealing area of the piston rod can change at any time, and the sealing area in the same position needs a cycle to return. During this period, the heat of the piston rod will be fully exchanged with the fluid medium, so the surface temperature of the piston rod is equal to the temperature of the fluid medium. The following two-dimensional heat conduction method is used to derive the temperature field distribution of the sealing ring. This study considers the roughness effect of Gu et al. [23] and Stefani et al. [14]:

$$\nabla \cdot \left(k_n\nabla T_h\right)=0$$

(48)

In Formula (48), k_n and T_n represent the thermal conductivity and temperature of the V-shaped seal ring, respectively.

2.7. Boundary Conditions

By resolving the Reynolds equation and energy equation with the following boundary conditions, it is assumed that both the oil side and the air side are completely submerged entrances. It should be noted that this study assumes that both the oil side and the air side are under completely submerged conditions, and does not consider the influence of fluid pressure disturbance on the oil film velocity at the entrance and exit of the sealing area.

$$\left\{\begin{array}{l}U(z=0)=u, \mathrm{on} \mathrm{the} \mathrm{rod} \mathrm{surface}\\ U(z=h)=0,\mathrm{on} \mathrm{the} \mathrm{seal} \mathrm{surface}\end{array}\right.$$

(49)

$$\left\{\begin{array}{l}p(x=0)=p_{sealed},\mathrm{oil} \mathrm{side}\\ p(x=L_x)=p_a,\mathrm{air} \mathrm{side}\end{array}\right.$$

(50)

In the thermal analysis, the thermal boundary condition is an important part of the overall sealing system. In this paper, based on the research of Chungu [22] and Gu et al. [15], the following thermal boundary conditions are adopted for the fluid film:

$$\left\{\begin{array}{l}T=T_{oil} (\mathrm{x}=0),\mathrm{at} \mathrm{inflow} \mathrm{on} \mathrm{inlet} \mathrm{side}\\ dT/dx=0(x=0),\mathrm{at} \mathrm{outflow} \mathrm{on} \mathrm{inlet} \mathrm{side}\\ dT/dx=0(x=L_x),\mathrm{at} \mathrm{oulet} \mathrm{side}\end{array}\right.$$

(51)

In order to predict the temperature field more accurately, this paper makes a series of simplifications to solve the 3D energy equation, and divides the boundary conditions into three types:

• Convection boundary (S1): This kind of boundary mainly describes the contact interface between the sealing part and the air, which is manifested as the heat convection between the sealing area and the outside air.
• Indiabatic boundary (S2): The boundary conditions represent the contact interface between the sealing part and the shell; heat is not transmitted through this interface, which is regarded as the adiabatic state.
• Thermostatic interface (S3): This refers to the interface between the sealing part and the fluid medium; the temperature at the interface remains constant and is equal to the temperature of the fluid medium.

Figure 3 is boundary condition calculation domain. Therefore, the boundary conditions between the piston rod and the seal portion can be clearly defined based on the above classification:

$$\left\{\begin{array}{l}{\left.-k_{Rod}{\displaystyle \frac{\partial T_{Rod}}{\partial n}}\right|}_{z=0}=q_n\\ {\left.-k_{Seal}{\displaystyle \frac{\partial T_{Seal}}{\partial n}}\right|}_{z=h}=q_b\\ {\left.-k_h{\displaystyle \frac{\partial T_h}{\partial n}}\right|}_{S1}=k_{air}\left(T_h-T_0\right)\\ {\left.{\displaystyle \frac{\partial T}{\partial n}}\right|}_{S2}=0\\ {\left.T\right|}_{S3}=T_{oil}\\ T(z=0)=T_{Rod}\end{array}\right.$$

(52)

2.8. Overall Calculation Process

Considering the temperature effect, the mixed lubrication model is shown in Figure 4. The calculation steps are as follows:

(1)

Use the finite element software ABAQUS 2016 to establish the geometric model of the VL-ring, and conduct a macro-mechanical analysis of the VL-ring under a given working condition to obtain the static contact pressure p_s, the length L_x of the sealing area, and the influence factor matrix I_n.

(2)

Substitute the static contact pressure p_s into the EHL theoretical calculation model to obtain the initial film thickness h₀.

(3)

Input the film thickness h₀ into the IHL theory and G-W model to obtain the oil film pressure p_f and the uniform contact pressure p_c.

(4)

If $\left|p_s-p_f-p_c\right|<c$ and $\left|h_{\min }^{new}-h_{\min }^{old}\right|/h_{\min }^{old}\le \epsilon$, go to step (5) and calculate the friction force F_rod and leakage q. Otherwise, recalculate the new film thickness h by substituting the calculated results into the influence factor algorithm, and then repeat step (3) for calculation.

(5)

After the convergence of step (4), the calculated results are used to determine the oil film temperature T_h and the temperature T_n of the V-ring through thermal analysis.

After the calculation converges, the various parameter indicators of the seal structure during the working process are obtained, revealing the sealing mechanism of the V-ring. The friction force F_rod on the rod surface and the leakage q of the seal structure are calculated by Formulas (53) and (56):

$$F_{rod}=\pi D{\displaystyle {\int }_0^L\left({\tau }_v+{\tau }_c\right)}dx$$

(53)

where ${\tau }_v$ is the friction force composed of the viscous shear stress of the fluid and ${\tau }_c$ is the contact shear stress of the surface roughness, which are calculated using the following formulas [24,25]:

$${\tau }_v=-{\displaystyle \frac{\mu u}{h}}\left({\phi }_f-{\phi }_{fs}\right)+{\phi }_{fp}{\displaystyle \frac{h}{2}}{\displaystyle \frac{\partial p_f}{\partial x}}$$

(54)

$${\tau }_c=-f\cdot p_c\cdot {\displaystyle \frac{u}{\left|u\right|}}$$

(55)

$$q=-{\phi }_{xx}\left({\displaystyle \frac{h^3}{12\mu }}{\displaystyle \frac{d{p}_f}{dx}}\right)+{\displaystyle \frac{u}{2}}\left\{[1+(1-F)\Phi ]\cdot (h_t+\sigma \cdot {\phi }_{scx})\right\}$$

(56)

The calculation process-related parameters are shown in

Table 1. The calculation process-related parameters.

O-ring material	NBR
V-ring material	PTFE
Rod material	45#
Elastic modulus of PTFE, Es	411 MPa
Poisson’s ratio of PTFE, υs	0.45
Rod diameter, d	25 mm
Stroke length	110 mm
Reference viscosity, μ0	0.0387 Pa·s
Pressure-viscosity coefficient, α	2 × 10−8 Pa-1
Viscosity-temperature coefficient, β	3.17908 × 10−2 K-1
Dry friction coefficient, fc	0.1
RMS roughness, σ	1 μm
Asperity radius of V-ring, R	4 μm
Asperity density of V-ring, η	1 × 1011
Specific heat capacity of rod, c	460 J/(kg·°C)
Specific heat capacity of oil, coil	2000 J/(kg·°C)
Density of rod, ρ	7850 kg/m3
Density of oil, ρoil	820 kg/m3
Thermal conductivity of rod, λ	46 W/(m·K)
Thermal conductivity of in liquid state, λoil	0.123 W/(m·K)
Thermal conductivity in gas state, kG	0.01 W/(m·K)
Thermal conductivity of seal, kS	0.25 W/(m·K)
Heat convection coefficient of air, hair	15 W/(m2·K)
Reference temperature, T0	308.15 K
Oil temperature	308.15 K
Air temperature	308.15 K

.

2.9. Hydraulic Reciprocating Seal Test Verification

The entire experimental system consists of a power drive unit, a control unit, a data acquisition and analysis unit, and other parts. The test bench is shown in Figure 5 and Figure 6. The main structures include a motor, a tension and compression sensor, a manual hydraulic pump, a hydraulic cylinder, a collector, and a computer. The working process is as follows: disassemble the hydraulic cylinder; install a VL-type seal ring and reconnect it to the device; open the check valve; press the manual pump to allow the hydraulic oil to soak the entire cylinder until the hydraulic oil can flow out evenly from the check valve. Then, close the check valve, start pressing the manual pump, and observe the pressure gauge on the manual pump. When the target pressure is reached, stop pressing. Then, open the computer analysis software to start recording the friction force of the reciprocating rod, adjust the motor speed regulator to achieve the target reciprocating speed, and start collecting experimental data. After the collection is completed, organize the test instruments for future use.

The RMS roughness was measured using an Olympus OLS4100 confocal optical microscope produced in Tokyo, Japan. The non-contact method allows for the observation and measurement of the sample’s surface in three dimensions. With a resolution of up to 10 nm, images can be obtained quickly and conveniently; see Figure 7. The microscopic surface contour morphology of the VL-sealed PTFE slip ring was obtained using the optical microscope. The sampling size was 1280 μm, and the surface morphology of the PTFE slip ring is shown in Figure 8. The PTFE slip ring and surface morphology have an uneven, non-smooth surface. Through the rough contour height of a typical section of the PTFE sliding ring, the sliding ring surface can be calculated using the software calculation with the optical microscope. The surface roughness of the piston rod is much less than that of the PTFE sliding ring, so the surface roughness of the piston rod can be ignored.

To ensure the correctness of the test, the average accuracy and standard deviation of the friction force of the piston rod reciprocating velocity prepress were calculated, as shown in

Table 2. Tests of the average accuracy and standard deviation of friction in 10 samples.

Sample	1	2	3	4	5
Friction force/N	246.6	245.7	248.9	244.2	249.4
6	7	8	9	10	Test accuracy/%
245.5	250.7	246.1	247.3	251.8	247.62 ± 2.34

. The working conditions are as follows: the reciprocating speed of the piston rod was 0.2 m/s, the fluid pressure was 28 MPa, the sealing pre-compression was 0.1 mm, and the rms roughness was 1 μm.

In order to ensure the correctness of the test, the average accuracy and standard deviation of the friction force of the same piston rod’s reciprocating speed pre-compression are calculated.

As shown in Table 2, although the results of 10 friction tests fluctuated within a small range, the standard deviation was only 2.34%, indicating that the bench can effectively measure the friction of reciprocating seals.

Figure 9 shows the friction force of the VL seal structure under five different fluid pressures when the speed of the piston rod is 0.1 m/s, and a comparison is made between the experimentally measured friction force and the friction force calculated by the mixed lubrication model. It can be seen from the figure that as the fluid pressure increases, the friction force of the instroke seal ring and the outstroke seal ring both show an approximately linear increasing trend. The instroke friction force calculated based on the mixed lubrication model is slightly larger than the outstroke friction force, which is similar to the conclusion obtained by Xiang et al. [26]. In addition, the total friction force in the contact areas of the two seal rings increases with the increase in seal pressure, and the friction force law calculated using numerical simulation is basically the same as the experimental results. The friction force calculated based on the fluid dynamic inverse solution method has an average error of only about 3.9% compared with the experiment, which indicates that the numerical lubrication model used to study the friction and sealing characteristics of the VL seal structure has high reliability.

3. Results and Analysis

3.1. Temperature Distribution

As shown in Figure 10, simulation results were obtained under various working conditions with a surface roughness of 0.5 μm. At a fluid pressure of 14 MPa and a reciprocating rod speed of 0.1 m/s, the calculated results are depicted in Figure 10. Due to the pretension force upon seal installation, the V-ring is radially compressed. During the instroke process, the VL seal structure is further compressed because of the fluid pressure, leading to an increased contact area between the V-ring and the piston rod.

Figure 10a,b indicate that the temperature in the sealing area remains relatively high, with the temperature distribution concentrating in the middle during the instroke, while it shifts to the left during the outstroke. The overall temperature during the outstroke is higher than that during the instroke, which is consistent with Wang et al.’s research [12]. During the instroke and outstroke, the V-ring’s temperature increase differs: the temperature is about 6 K during the instroke and about 4 K during the outstroke. This indicates that during the instroke, the fluid dynamic pressure effect is stronger. Under elastic lubrication conditions, the contact area between the micro-asperities of the V-ring and the piston rod is larger, resulting in a more significant temperature rise. However, during the outstroke, the fluid pressure is low, and only the pretension force acts radially, reducing micro-asperity contact under mixed lubrication conditions. Consequently, the degree of V-ring deformation and the seal pressure are reduced, leading to a lower temperature rise in the sealing area compared to the instroke.

In Figure 10a,c, the influence of reciprocating rod speed on temperature distribution is shown to be more pronounced during the instroke than the outstroke. According to the energy conservation, determined using, Equation (21), heat generation directly correlates with the reciprocating rod speed, influencing the fluid dynamic pressure effect at the sealing contact interface. This results in decreased friction and less heat generation at the sealing contact interface, with an insignificant change in heat. However, Figure 10d shows that temperature generation is less during the instroke due to the increased hydrodynamic effect at a certain fluid pressure, leading to lower temperatures.

Figure 10a,e demonstrates that fluid pressure significantly affects the temperature rise in the sealing area. At a fluid pressure of 28 MPa, the V-ring and O-ring are further compressed, generating higher contact pressure and increasing the micro-asperity contact area in the sealing zone. This results in a notable increase in outstroke temperature. During the instroke, the geometric shape of the V-ring causes a lower pressure to occur at the sealing interface entrance compared to the outstroke, though it remains higher than that during the 14 MPa fluid pressure instroke. The enhanced fluid dynamic pressure effect leads to a temperature rise.

Figure 11 shows the oil film temperature distribution under different working conditions. From Figure 11a,b, when the piston rod speed is 0.1 m/s, the overall surface of the heating zone presents a spoon-shaped profile, approaching the fluid pressure side, while the heating profile is nearly uniformly distributed in the middle position during the instroke. As shown in Figure 11c,d, when the rod speed increases from 0.1 m/s to 0.3 m/s, the hydrodynamic effect is enhanced, and the temperature rise relies on the strong viscous shear effect of the fluid, significantly increasing the region of the temperature rise, indicating that the oil film temperature is greatly influenced by the rod speed at this time. When the piston rod speed is 0.1 m/s and the fluid pressure increases from 14 MPa to 28 MPa, the temperature rise region further expands. This is because, according to the law of the conservation of energy, the oil film thickness at higher pressures is smaller, the fluid velocity is larger, and the oil film temperature is not only affected by the rod speed but is also related to the seal pressure. Due to the increase in fluid pressure, the seal pressure further increases, thereby affecting the contact pressure on the sealing surface, and the fluid velocity of the oil film further increases, resulting in a further expansion of the temperature rise range. In addition, the temperature at the bottom of the oil film is lower than that at the top. This is because, during the reciprocating motion of the piston rod, the piston rod brings heat into the cylinder body, and the internal hydraulic oil has a higher specific heat. The heat carried by the piston rod surface is instantly absorbed by the hydraulic oil, resulting in little change in the temperature at the bottom of the oil film.

3.2. Study of the Influence of Fluid Pressure and Reciprocating Speed on Friction Force and Leakage

Figure 12 shows the influence of fluid sealing pressure and plunger reciprocating speed on the dynamic reciprocating seal performance. It can be seen from the figure that the friction force during the outstroke of the VL seal structure increases with the increase in sealing pressure and slightly decreases with the increase in reciprocating speed. This phenomenon can be explained as follows: the mixed lubrication friction force is composed of two parts—fluid viscous shear stress and roughness contact pressure. The static contact pressure increases with the increase in reciprocating speed. Generally, the oil film thickness formed by the hydrodynamic effect increases with the increase in reciprocating speed. Therefore, when the piston rod speed is low, the formed oil film is smaller than the roughness height, and the friction force at this time mainly derives from the interaction between the piston rod and the housing. During the instroke of the piston rod, the reciprocating seal friction force does not change with the increase in sealing pressure, and is always equal to the friction force at low pressure. This is because the sealing pressure during the instroke is atmospheric pressure, which is independent of the fluid pressure. Similarly, the friction force during the instroke slightly decreases with the increase in reciprocating speed, and the friction force during the outstroke is significantly greater than that during the instroke.

At a temperature of 308.15 K, the variation law of reciprocating seal leakage with seal pressure and plunger movement speed is shown in Figure 13. As shown in Figure 13a, with the increase in the fluid pressure and reciprocating speed of the piston rod, the leakage of the hydraulic cylinder during the outstroke of the hydraulic cylinder increases significantly. This is because with the increase in fluid pressure, the pressure gradient in the sealing area also increases, resulting in an increase in leakage with the increase in fluid pressure. According to the previous study, when the piston rod speed is high, the generated oil film is thicker, which also increases the leakage. During the instroke of the hydraulic cylinder, the leakage also increases with the increase in reciprocating speed. During the instroke of the hydraulic cylinder, the fluid sealing pressure is approximately equal to the atmospheric pressure. At this time, the fluid flow in the sealing area mainly relies on the shear flow caused by the movement of the piston rod, and the influence of fluid pressure during the instroke is limited.

3.3. Study of the Influence of Interference and Reciprocating Speed on Friction Force and Leakage

At a temperature of 308.15 K, the influence of interference and reciprocating speed on seal friction force is shown in Figure 14. It can be seen from the figure that in the discharge stroke of the plunger pump, the reciprocating seal friction force increases with the increase in preliminary compression and the decrease in plunger speed. A larger installation interference causes a larger extrusion deformation of the seal ring, resulting in a relative increase in contact pressure, thereby leading to a larger friction force. Conversely, when the preliminary compression is relatively small, the deformation of the seal ring produced is also small. Therefore, during the instroke of the piston rod, the reciprocating seal friction force decreases significantly with the decrease in preliminary compression.

At a temperature of 308.15 K, the influence of interference fit and reciprocating speed on the leakage of the sealing device is shown in Figure 15. As can be seen from the figure, during the instrokes and outstrokes of the piston rod, the leakage of the seal ring decreases with the increase in preliminary compression. This is because the increase in preliminary compression leads to an increase in the contact pressure in the sealing area, thereby enhancing the sealing effect. In addition, when the reciprocating speed of the plunger increases, the leakage of the hydraulic cylinder increases regardless of whether it is an instroke or outstroke. In addition, we can also see that with the increase in plunger speed, reduced interference fit will result in more leakage. Therefore, when the speed of the piston rod increases, the preliminary compression should be appropriately increased to minimize leakage as much as possible.

3.4. Study of the Influence of Root Mean Square Roughness and Reciprocating Speed on Friction Force and Leakage

At 308.15 K temperature, the variation laws of reciprocating seal friction with the root mean square roughness of the seal ring and the speed of the piston rod during the outer and instrokes are shown in Figure 16. Increasing the surface roughness significantly increases the friction force during the outstroke, and when the reciprocating speed of the piston rod is large, the friction force is more sensitive to changes in roughness. The friction force increases slightly with the increase in roughness during the instroke, because a small part of the shear flow generated by the movement of the piston rod during the instroke produces a limited fluid hydrodynamic effect.

Figure 17 shows the leakage distribution law with roughness during the instrokes and outstrokes of the piston rod reciprocating motion at a temperature of 308.15 K. As can be seen from the figure, the leakage of both the instrokes and outstrokes increases with the increase in plunger speed. In addition, with the increase in the root mean square roughness of the seal ring surface, the increase in the leakage of the outstroke of the piston rod at the same rod speed is greater than that of the instroke. The increase in seal pressure will increase the pressure of the fluid in the sealed area, thereby increasing the density and flow rate of the fluid, resulting in an increase in the action of the inertial force and viscous shear force, enhancing the fluid hydrodynamic effect and increasing the leakage.

4. Conclusions

-

A mixed lubrication model considering the influence of temperature was established to predict the sealing behavior of the VL-type reciprocating hydraulic rod seal. The model considers macro-solid mechanics, fluid mechanics, micro-contact mechanics, and micro-deformation mechanics, and couples them to obtain performance parameters such as the average temperature in the sealing area, the film pressure, and the leakage.

-

A hydraulic reciprocating seal test bench was built, and the reliability of the mixed lubrication model was verified by comparing the total friction force. Subsequently, the oil film temperature, temperature of the V-shaped ring seal surface, leakage, and friction performance were predicted and discussed.

-

The thermal effect has a significant influence on the oil film temperature distribution and seal ring temperature distribution in the sealing area. Moreover, according to the temperature distribution in the seal ring, the viscous wedge effect caused by the oil film temperature distribution significantly affects the sealing performance of the entire sealing structure.

-

An increase in temperature reduces the viscosity of the fluid film, thereby reducing the fluid hydrodynamic effect and increasing the friction force at the seal interface.

-

Regarding the seal leakage and friction force under different working conditions at 308.15 K temperature, it was found that the piston rod speed and seal ring preliminary compression had a greater impact on the overall seal friction force and leakage, while the root mean square roughness had a limited impact at low interference.

It is important to note that this model reveals the temperature distribution in the VL-type seal structure under steady-state conditions, but only for seal structures operating at normal temperatures. Many seal structures operate under extreme temperatures, so future research should consider a wide temperature range in the model. In addition, after the temperature rises, the performance of the sealing material will change, for example through creep and aging, which will make the calculation of the internal temperature of the material more complex. To be more in line with the actual situation, the material properties, such as high-temperature creep and aging factors, should be considered in the mixed lubrication model to further improve the mixed lubrication model.