Interface Leakage Theory of Mechanical Seals Considering Microscopic Forces

Abstract

The fluid flow in the small pore throat is a nonlinear flow, and the microscopic force between the fluid and the wall cannot be ignored. However, the previously established theories about the leakage between sealing interfaces have not considered the influence of microscopic forces. Based on contact mechanics and percolation theory, the void characteristics of the sealing interface were clarified, and the influence of microscopic force on fluid flow in porous medium was analyzed. Combined with the capillary force, the concept of a critical void radius between the mechanical seal interfaces is proposed. The fluid flow resistance model and leakage rate calculation equation of the sealing interface considering the van der Waals force are established, and the leakage judgment criterion of the sealing interface is provided. Through numerical calculation and experiments, the effect of microscopic force is verified in terms of the fluid flow law and macroscopic leakage rate. The results show that van der Waals forces have an important influence on the fluid flow between the sealing interfaces. As the microchannel size decreases, the van der Waals forces between solid and liquid increase, and the influence of these van der Waals forces on the fluid flow between the sealing interfaces cannot be ignored. The calculation model of the sealing interface leakage rate proposed in this paper shows little difference with the results of the Persson model, and is in good agreement with the experimental results; the maximum relative error is 8.7%, the minimum relative error is only 3.8%.

1. Introduction

Mechanical seals play an important role in preventing fluid leakage and are widely used in various fluid machinery and equipment [1]. For a mechanical end seal intended to prevent leakage between the power input shaft and the pump casing, performance is related to the normal operation and shutdown safety of the equipment [2]. In order to ensure the safe and long-life operation of the mechanical seal, in certain circumstances where the working environment is harsh, such as in the case of mechanical end face seals for nuclear main pumps, it is even required that the mechanical seal be able to achieve zero leakage and zero escape during operation, that is, to achieve zero leakage on the end face of the mechanical seal [3]. The problem of leakage at the sealing end face remains the main cause of seal failure [4]. To this end, there is a need to understand the causes behind leakage of mechanical seals through continuous research in order to explain the leakage mechanism of the sealing interface, accurately predict the leakage rate, formulate effective leakage prevention measures, and reduce the waste caused by premature replacement of mechanical seals and the material loss and environmental pollution caused by overdue service. Mayer [5] proposed the “fluid exchange flow theory” through a large number of experiments, which states that there are contact and non-contact areas between the rotating and stationary ring end faces; when the rotating and stationary rings rotate relative to each other, the fluid is transferred from the void on the high-pressure side of the sealing surface to the void on the low-pressure side. Lebeck [6] proposed the “waviness theory”, which states that the waviness effect generated by the heat and pressure of the sealing end face is the cause of hydrodynamic pressure and leakage. Persson [7,8,9,10,11] explained the leakage mechanism of the sealing interface based on Persson’s contact mechanics theory and two-dimensional percolation theory, suggesting that when the relative area ratio of the sealing surface A_r/A₀ (where A_r is the real contact area and A₀ is the nominal contact area) is about 0.42 [12], a leakage channel will occur at the sealing interface, and the fluid will flow from the high-pressure side to the low-pressure side. Sun Jianjun [13], and Ji Zhengbo et al. [14] discussed the percolation characteristics of sealing interfaces under different grid layers based on percolation theory, put forward the conditions for macroscopic leakage of liquid working fluid and gas working fluid at the sealing interface, and established the macroscopic leakage criterion of the percolation channel. When Zheng Wei [15] studied the pressure coefficient of mechanical seals based on percolation theory, he suggested that there was a complete sealing line at the mechanical sealing interface in a non-percolation state, which prevented the occurrence of end face fluid leakage. Although the percolation theory has brought new ideas to explain the leakage mechanism of mechanical seals, it cannot explain the situation in which there is a leakage channel at the sealing interface and zero leakage in the percolation state.

In recent years, with the development of microelectromechanical systems, it has been found that when liquids flow in channels with a diameter of less than a few hundred microns they behave as nonlinear flow. This means that the original linear seepage theory is no longer applicable, and that the microscopic force between the pipe wall and the fluid is the main cause of the problem [16]. When the contact mechanical seal is working, the end face gap is 0.5~2 μm [17] and the fluid flow between the end faces is inevitably affected by microscopic force. However, neither the “fluid exchange flow theory” nor the “waviness theory” considers the influence of microscopic force on mechanical seal leakage, nor does the leakage mechanism of mechanical seals based on the percolation theory.

This paper intends to analyze the effect of microscopic forces on fluid flow in porous media starting from the void characteristics of the sealing interface. The fluid flow resistance model and leakage rate calculation formula of the sealing interface are established considering the van der Waals force. Through numerical simulation and experiment, the effect of microscopic force is verified from the two aspects of fluid flow law and macroscopic leakage rate. In addition, the leakage judgment criterion of the sealing interface is provided. This paper can provide a basis for improving the theory of mechanical seal leakage.

2. Void Characteristics of the Sealing Interface

Figure 1 shows the microscopic morphology of the rotating and stationary rings end faces under the microscope. From the figure, it can be seen that both the rotating and stationary rings have uneven surfaces. In the contact process of the rotating and stationary rings, the sealing interface is composed of a contact area and a void part. Under the working state, the rotating and stationary rings change with the force on the end face, the asperities between the sealing interfaces are deformed, and the void characteristics of the sealing surface change.

2.1. Contact Mechanics Model of the Sealing Interface

Research has shown that the contact between two rough surfaces can be equivalent to the contact between a rough surface and a rigid smooth plane [18]. The Weierstrass Mandelbrot function (W-M function) can be used to characterize rough surfaces [19,20], and the contour curve of a single asperity can be expressed using the following equation [21]:

$$z\left(x\right)=G^{D-1}{l}^{2-D}\cos \left(\frac{\pi x}{l}\right), -\frac{l}{2}<x<\frac{l}{2}$$

(1)

where D is the fractal dimension, G is the scale coefficient, l is the diameter of the bottom surface of the asperity profile, and x is the position coordinate of the profile curve.

For the soft ring and hard ring surfaces with fractal parameters D₁, G₁, D₂, and G₂, the fractal dimension D and scale coefficient G of the equivalent rough contact surface can be calculated by Equations (2) and (3), respectively:

$$D=D_1-{\left(1-\frac{D_1{D}_2}{D_1+D_2}\right)}^3\left(2-D_2\right)\left(D_1-1\right)$$

(2)

$$G=\sqrt{G_1^2+G_2^2}$$

(3)

The root mean square roughness of the equivalent surface, soft ring, and hard ring surface can all be obtained by Equation (4) [22]:

$$\sigma =\frac{1}{2{\left[\left(2-D\right)\ln \gamma \right]}^{1/2}}{G}^{D-1}{L}_{\mathrm{s}}^{2-D}$$

(4)

where L_s is the standard length of the scanned sample and L_s = 0.8 mm for the polished surface [23]. In addition, for a certain rough surface the height h of the asperities can be obtained according to the surface integrity coefficient k and the average roughness [5,24] h = σ/k.

Figure 2 shows a contact model of a single asperity. The asperity is deformed under the action of the face contact pressure; according to the geometric relationship in the figure, the actual compression amount δ of the asperity can be calculated by the following equation:

$$\delta =G^{D-1}{l}^{2-D}(1-\cos \frac{\pi \sqrt{a}}{2l})$$

(5)

where a is the actual contact area of the asperity and a = 4x². In addition, the actual compression amount δ is determined by the deformation of the asperity (elastic deformation, elastoplastic deformation or plastic deformation of the asperity under the action of external force). The calculation method of the actual compression amount of the asperity at different face contact pressures can be obtained from [13].

According to [13,25], the real contact area of the sealing interface can be calculated under the action of the end face contact pressure load F_c. Regardless of the deformation properties of the asperities, the real contact area A_r on the calculated nominal area can be expressed as:

$$A_{\mathrm{r}}=\frac{D}{2-D}{\psi }^{\left(2-D\right)/2}{a}_{\mathrm{L}}$$

(6)

where ψ is the correction coefficient of the ratio A_r/a_L between the real contact area A_r and the maximum micro contact area a_L [26].

When the rotating and stationary rings produce compression δ under the action of F_c, and the maximum contact area of the asperity is a_L, then A_r is the real contact area; when $a_{\mathrm{L}}=\frac{\pi }{4}{l}^2$, A_r is equal to the calculated nominal area A_n [26]:

$$A_{\mathrm{n}}=\frac{D}{2-D}{\psi }^{\left(2-D\right)/2}\frac{\pi }{4}{l}^2$$

(7)

2.2. Void State

The porosity ϕ₀ of the sealing interface in the initial state can be expressed by the following equation [13]:

$${\varphi }_0=1-\frac{2-D}{2}{a}_{\mathrm{L}}^{\frac{D}{2}-1}{\displaystyle {\int }_0^{a_{\mathrm{L}}}{a}^{1-\frac{D}{2}}}\sin \frac{\sqrt{\pi a}}{l}\mathrm{d}\frac{\sqrt{\pi a}}{l}$$

(8)

Under the action of face contact pressure, the porosity of the sealing interface changes; according to the geometric relationship of the asperity, the porosity ϕ after loading can be calculated by the following equation [13]:

$$\varphi =\frac{h{\varphi }_0-\delta }{h-\delta }$$

(9)

According to the percolation theory [27], according to the size relationship between the porosity ϕ and the percolation threshold ϕ_c (known from references [13,14,15] as ϕ_c = 0.312), the void state of the sealing interface can be divided into three types: the percolation state, percolation point state, and non-percolation state:

(1) Non-percolation state: at this time, ϕ < ϕ_c, the void group of the sealing interface does not penetrate both sides of the sealing interface, there is no microchannel formation on the sealing surface, and the fluid cannot pass through the sealing surface to form leakage.

(2) Percolation point state: at this time, ϕ = ϕ_c, the void group of the sealing interface penetrates both sides of the sealing interface for the first time, and a single leakage microchannel appears on the sealing surface, at which time the fluid may pass through the sealing surface to form microleaks.

(3) Percolation state: at this time, ϕ > ϕ_c, the sealing interface is completely percolating, that is, there are not just a few percolation points, but a large void group, and the sealing interface has multiple microchannels formed; at this time, the fluid may pass through the sealing surface to form leakage.

The void state between the sealing interfaces depends on the size of the face contact pressure p_c; increasing the face contact pressure can reduce the porosity of the sealing interface, and can even compact the sealing surface to make the porosity equal to 0. Figure 3 shows the void distribution at the rotating and stationary rings sealing interfaces under different percolation states. The white cubes represents the void areas [27].

As can be seen from Figure 3, when the porosity of the sealing interface reaches the percolation threshold, microchannels appear at the sealing interface through the sealing surface. When the porosity is less than the percolation threshold, the sealing interface is in a non-percolation state and no microchannels are present. However, whether the sealing interface in the percolation state must have fluid flow depends on whether the pushing force on both sides of the microchannel is greater than the resistance of the fluid flow.

Figure 4 is a schematic diagram of the fluid flow at the sealing interface of the rotating and stationary rings. From the above analysis, it can be seen that leakage occurs at the sealing interface only when the following relationships are satisfied:

$$\Delta pS=\left(p_{\mathrm{a}}-p_{\mathrm{b}}\right)S>f$$

(10)

where Δp is the pressure difference between the two sides of the sealing surface, p_a is the pressure on the high-pressure side of the sealing surface, p_b is the pressure on the low-pressure side of the sealing surface, S is the cross-sectional area of the microchannel, and f is the total resistance of fluid flow in the microchannel.

Equation (10) provides the prerequisites for fluid flow at the sealing interface, which is different from the explanation in [10,11,27] of the fluid flow at the sealing interface, which holds that as long as there is a leakage channel at the sealing interface, the fluid on the sealing surface will flow from the high-pressure side to the low-pressure side and leakage will occur.

3. Microscopic Force

3.1. Function and Classification

In the working process of a contact mechanical seal, the scale of the end face gap is generally between 0.5~2 μm and the fluid flow between the end faces belongs to the microscale flow range. Due to the small size of the sealing gap feature, when the fluid flows in the sealing interface, the fluid flow is affected by the microscopic forces between the microchannel wall and the fluid, meaning that the previous explanation of the leakage mechanism of the fluid at the sealing end face can no longer be fully applied to the leakage law of the fluid in the sealing interface. Microscopic forces mainly include the van der Waals force, electrostatic force, spatial configuration force, and surface tension [28]; the magnitude and range of the various forces are analyzed below.

3.1.1. Van der Waals Force

The van der Waals force can be divided into the short-range van der Waals force and long-range van der Waals force. The van der Waals force between individual molecules is called the short-range van der Waals force, which consists of an alignment force, an induction force, and a dispersion force. When molecular aggregates overlap with each other, the van der Waals force action is called the long-range van der Waals force. For example, the van der Waals force between a molecular aggregate with a radius of R and a solid wall belongs to the long-range van der Waals force, as shown in Figure 5. At this time, its magnitude can be calculated by the following equation [29]:

$$V=-\frac{A_{\mathrm{sw}}R}{D_{\mathrm{f}}}$$

(11)

where V is a long-range van der Waals force, D_f is the distance from the molecular aggregate to the tube wall, and A_sw is the Hamaker constant between two substances.

The flow law of fluids in microchannels is generally believed to follow Poiseuille’s law. However, when a fluid flows in a microchannel, it produces additional viscosity due to the influence of the long-range van der Waals force between the solid surface and the fluid molecular aggregate, which increases the resistance of the fluid flow. The action range of the long-range van der Waals force is much greater than 0.1 μm, and can reach a few microns. Therefore, the flow of fluid in the sealing interface is inevitably be affected by van der Waals forces, which in turn affect the leakage of the fluid at the sealing interface.

The van der Waals force mainly reflects the interaction between molecular aggregates at the microscopic scale. It can be seen from Equation (11) that its size has nothing to do with the height of the microchannel; instead, it is related to the distance to the tube wall. When the molecular aggregate is farther away from the solid tube wall, the van der Waals force between the fluid molecules and the solid wall molecules is smaller, while when the molecular aggregate is closer to the solid tube wall, the van der Waals force between the fluid molecules and the solid wall molecules is greater. This leads to the layering phenomenon in the flow process of fluid in the pipe. When the pipe size is smaller, the effect of the van der Waals force is more obvious.

3.1.2. Electrostatic Force

When the electrolyte solution flows in the microchannel after the solid wall surface comes into contact with the solution containing charged ions, a net charge is formed in the solution and an electric potential is formed on the solid wall surface. Therefore, the charged solid wall surface and the net ionic layer in the solution near it form a double electric layer, a compact layer, and a diffusion layer, as shown in Figure 6. The magnitude of the electrostatic force can be calculated by the following equation; its operating distance is longer than the van der Waals force, which is most important at distances less than 0.1 μm, and has an effect at 10 μm:

$$F=k_{\mathrm{e}}\frac{Qq}{r_{\mathrm{d}}^2}$$

(12)

where F is the electrostatic force, Q and q are the charged amounts of different charged particles, r_d is the distance between two particles, and k_e is the electrostatic constant.

When the solution is driven by the pressure difference, the net charge in the double electric layer moves with the solution, resulting in a flow-induced electric field, and the electric field makes the net charge on the microchannel double electric layer move in the opposite direction of the solution flow, slowing down the positive flow rate of the solution, which is equivalent to increasing the viscosity of the fluid.

3.1.3. Spatial Configuration Force

As shown in Figure 7, in a liquid containing chain-like molecules, one end of the chain-like molecule is attached to the surface and the other end extends into the flowing liquid to swing freely; when the liquid flows in the microchannel, the long-chain molecules attached to the solid wall surface cause resistance to the flowing liquid, resulting in the spatial configuration force which hinders the flow of the fluid. In liquid flows with a large number of long-chain molecules, the spatial configuration force has the most significant effect on fluid flow. Its acting distance is greater than 0.1 μm, which can be calculated using the following equation [29]:

$$U_{\mathrm{R}}^{\mathrm{S}}=U_{\mathrm{R}}^{\mathrm{e}}+U_{\mathrm{R}}^{\mathrm{E}}+U_{\mathrm{R}}^{\mathrm{O}}+U_{\mathrm{R}}^{\mathrm{H}}$$

(13)

where $U_{\mathrm{R}}^{\mathrm{S}}$ is the spatial configuration force, $U_{\mathrm{R}}^{\mathrm{e}}$ is the entropy repulsion potential energy generated by the loss of structural entropy of polymer molecules, $U_{\mathrm{R}}^{\mathrm{E}}$ is the elastic repulsion potential energy, $U_{\mathrm{R}}^{\mathrm{O}}$ is the osmotic repulsive potential energy, and $U_{\mathrm{R}}^{\mathrm{H}}$ is the enthalpy repulsive potential energy.

3.1.4. Surface Tension

Surface tension is caused by the imbalance in the gravitational pull of molecules in the surface layer of a liquid. The molecules at the liquid interface are different from the molecules in the liquid body, and the surface molecules store excess free energy than the internal molecules; lift the molecules inside to the surface, it is necessary to pay energy in order to perform the work, and this energy is converted into surface free energy. The surface free energy tends to be minimal; the liquid surface tends to shrink automatically, and the shrinkage force is the surface tension, which can be expressed as [30]:

$${\sigma }_{\mathrm{b}}={\left(\frac{\partial U}{\partial A}\right)}_{T,p,n_{\mathrm{b}}}$$

(14)

where σ_b is the surface tension, U is the free energy of the system, A is the new surface area added, and T, p, and n_b represent the temperature, pressure, and number of components of the system, respectively.

Surface tension is related to temperature and interface properties. For the flow of fluid in microchannels, the role of surface tension is mainly reflected in the influence of capillary force on the fluid flow; the greater the surface tension, the greater the capillary force [28].

In summary, it can be seen that the range of action of the four microscopic forces is greater than 1 μm, and that the electrostatic force and the spatial configuration force only appear in special media solutions, while the surface tension and van der Waals force are the two microscopic forces present in all microscale flows. Therefore, the van der Waals force and surface tension are selected as our main considerations, and their effects on the fluid flow at the sealing interface are analyzed to illustrate the new sealing mechanism.

3.2. Critical Void Radius

The capillary force exists in the sealing interface, and its effect varies depending on the material used in the rotating and stationary rings. As shown in Figure 8a, for impermeable sealing interfaces, the capillary force formed by the sealing medium in the microchannel prevents leakage of the sealing medium [31]. To ensure that the sealing medium exists at the sealing interface for lubrication and does not leak out of the microchannel, it is necessary to control the size of the microchannel aperture. Increasing the face contact pressure can reduce the porosity of the sealing interface, reduce the radius of the pore capillary, increase the capillary force, and increase the leakage resistance of the sealing interface. According to the Young–Laplace equation, the relationship between the capillary force of the sealing interface microchannel and the microchannel radius r can be calculated using the following equation [32]:

$$p_{\mathrm{ca}}=\frac{2{\sigma }_{\mathrm{b}}\cos \phi }{r}$$

(15)

where p_ca is the capillary force and ϕ is the contact angle.

When p_ca takes the medium pressure, r_c is the critical void radius when there is no leakage. The porosity maintained by a sealing interface with a critical void radius microchannel is the critical porosity when there is no leakage. When the aperture of the microchannel of the sealing interface is less than the critical void radius r_c, the pressure difference between the two sides of the sealing surface is less than the capillary force; thus, the sealing interface has no leakage and has the ability to prevent leakage.

As shown in Figure 8b, for a sealing interface that can be wetted, the capillary force will push the sealed medium to leak. When the sealing interface is in a non-percolation state, there is no leakage channel formed at the sealing interface, and at this time there is no leakage at the sealing interface; When the sealing interface is in a percolation state, whether or not leakage occurs at the sealing interface depends on the relationship between the fluid flow resistance and leakage driving force. It can be seen from Equation (10) that when the leakage driving force composed of the pressure difference on both sides of the sealing surface and the capillary force is greater than the fluid flow resistance at the sealing interface,

$$\Delta pS+F_{\mathrm{ca}}=\left(p_{\mathrm{a}}-p_{\mathrm{b}}+p_{\mathrm{ca}}\right)S>f$$

(16)

where F_ca is the driving force formed by the capillary force, F_ca = p_caS. At this time, the sealing interface will leak.

3.3. Fluid Flow Resistance

To facilitate analysis of flow resistance, we can considered that the microchannels at the sealing interface are obtained by combining a large number of voids into a combination of pore segments, as shown in Figure 9. For contact mechanical seals, the total flow resistance f of fluid flows from the high-pressure side (outer diameter) to the low-pressure side (inner diameter) is equal to the sum of the total flow resistance hf of the straight pipe of each hole segment and the total local resistance $h_{\mathrm{f}}^{\prime }$ of sudden expansion or contraction of the hole channel:

$$f=h_{\mathrm{f}}+h_{\mathrm{f}}^{\prime }$$

(17)

The total flow resistance hf of the straight pipe of each hole segment and the total local resistance $h_{\mathrm{f}}^{\prime }$ of sudden expansion or contraction of the hole channel can be calculated separately by the following equation [33]:

$$h_{\mathrm{f}}={\displaystyle \sum _{i=1}^n{\chi }_i}\frac{L_i{u}_i^2}{2d_i}$$

(18)

$$h_{{}_{\mathrm{f}}}^{\prime }={\displaystyle \sum _{i=1}^{n-1}{\zeta }_i}\frac{u_i^2}{2}$$

(19)

where u_i is the flow rate, L_i is the length of each hole segment, d_i is the equivalent diameter of the flow section of each hole segment, ζ_i is the reduced resistance coefficient of the hole segment transition from a large cross-section to a small cross-section, and χ_i is the resistance coefficient of the straight pipe.

Due to the van der Waals force between the fluid and the wall of the microchannel at the sealing interface, the molecular gravity between the fluid increases, as does the fluid flow resistance, which is reflected in the increase of the viscosity of the fluid. Assuming that the initial viscosity of the fluid without taking into account the van der Waals force is μ₀, the viscosity of the fluid under the van der Waals force is μ [34]:

$$\mu ={\mu }_0+b\frac{\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)}{c}$$

(20)

$$A_{\mathrm{w}}={\left(\frac{\pi {\rho }_{\mathrm{w}}{N}_{\mathrm{A}}}{M_{\mathrm{w}}}\right)}^2{\beta }_{\mathrm{w}}$$

(21)

$$A_{\mathrm{s}}={\left(\frac{\pi {\rho }_{\mathrm{s}}{N}_{\mathrm{A}}}{M_{\mathrm{s}}}\right)}^2{\beta }_{\mathrm{s}}$$

(22)

where b is the viscosity increase factor, c is the distance to the pipe wall, A_s and A_w are the Hamaker constants for solids and fluids, respectively, ρ_s and ρ_w are the densities of solids and fluids, respectively, N_A is the Avogadro constant, M_S and M_w are the relative molecular masses of solids and fluids, respectively, and β_s and β_w are the van der Waals force acting factors for solids and fluids, respectively.

For a continuous-flow medium, the fluid flow rate of the medium has the following relationship with viscosity:

$$Re=\frac{d_i{u}_i\rho }{\mu }$$

(23)

Therefore, substituting Equations (18)–(20) and (23) into Equation (17), the total flow resistance f of fluid flowing through the leakage microchannel at the sealing interface under the action of van der Waals forces can be expressed as follows.

$$f={\displaystyle \sum _{i=1}^n{\chi }_i}\frac{L_{i}R{e}^2{\left[{\mu }_{0}c+b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)\right]}^2}{2d_i^3{\rho }^2{c}^2}+{\displaystyle \sum _{i=1}^{n-1}{\zeta }_i}\frac{R{e}^2{\left[{\mu }_{0}c+b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)\right]}^2}{2d_i^2{\rho }^2{c}^2}$$

(24)

4. Leakage Rate Calculation Model

Figure 10 shows a schematic diagram of the pore throat structure of the sealing interface microchannel; the z-axis is perpendicular to the sealing surface and the flow is in the x-axis direction, with u_x = u, u_y = 0, and u_z = 0. It is worth noting that in order to simplify the analysis process the pore throat structure of the sealing interface is assumed to be a rectangular structure [35]. According to the N-S equation, we have the following.

$$u\frac{du}{dx}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\frac{\mu }{\rho }\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial z^2}\right)$$

(25)

$$-\frac{1}{\rho }\frac{\partial p}{\partial y}=0$$

$$-\frac{1}{\rho }\frac{\partial p}{\partial y}=0$$

When z = 0, u = U₀ (i.e., a lower sealing surface slide). When z = h, u = 0 (i.e., the upper sealing surface is stationary), the flow velocity distribution in the z-axis direction can be obtained as follows:

$$u=-\frac{1}{2\mu }\frac{\mathrm{d}p}{\mathrm{d}x}\left(h-z\right)z+U_0\left(\frac{h-z}{h}\right)$$

(26)

For the surface of the stationary ring, U₀ = 0 and the fluid velocity distribution in the channel is

$$u=-\frac{1}{2\mu }\frac{\mathrm{d}p}{\mathrm{d}x}\left(h-z\right)z$$

(27)

Because the pressure p only changes in the x direction, and because it is assumed the sealing gap between the two sealing surfaces is parallel and the sealing gap h is unchanged, p falls uniformly along the x-axis direction:

$$\frac{\partial p}{\partial x}=\frac{\mathrm{d}p}{\mathrm{d}x}=-\frac{p_1-p_2}{L}$$

(28)

where p₁ and p₂ are the pressures on both sides of the pore throat.

Then, the fluid velocity distribution is

$$u=\frac{1}{2\mu }\frac{p_1-p_2}{L}\left(h-z\right)z$$

(29)

For the leakage microchannel on the unit space of the sealing end surface, we assume that the length of the pore throat after loading is L_min, the width is W_min, and the height is h − δ. Then the leakage rate Q_Th can be obtained as follows.

$$Q_{\mathrm{Th}}=W_{\min }{\displaystyle {\int }_0^{h-\delta }u\mathrm{d}z}=\frac{W_{\min }{\left(h-\delta \right)}^3}{12\mu }\frac{p_1-p_2}{L_{\min }}$$

(30)

If the middle diameter of the sealing interface is D_m, the width is B, the side length of the calculation unit area is L, and the width of the pore throat section is W_min, then after loading, the sealing interface is expanded along the circumference to become a rectangular channel with length B, width πD_m, and height h − δ. Here, we consider that the sealing interface is composed of two parts, namely, a void channel and a solid skeleton, where the size of the void channel is a rectangular microchannel with length B, width πD_mW_min/L, and height h − δ; then, the leakage rate Q of the sealing interface is as follows.

$$Q=\frac{\pi D_{\mathrm{m}}{W}_{\min }}{L}\frac{{\left(h-\delta \right)}^3}{12\mu }\frac{\Delta p}{B}=\frac{\pi D_{\mathrm{m}}{W}_{\min }{\left(h-\delta \right)}^3\Delta pc}{12LB\left[{\mu }_{0}c+b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)\right]}$$

(31)

5. Leakage Theory Verification

Due to the limitations of the existing technology and experimental conditions, in this paper we cannot directly measure the microscopic force between the sealing interface. Therefore, the correctness of the leakage rate calculation model established in this paper is verified based on the two aspects of fluid flow law and macroscopic leakage rate.

5.1. The Influence of Microscopic Forces on Fluid Flow

To facilitate the verification of the influence of the van der Waals force on the fluid flow law in the microchannel and simplify the analysis process, the following assumptions can be made: (1) the pore throat structure of the microchannel in Figure 10 is a circular cross-sectional structure; (2) the sealing medium is a Newtonian fluid, and its circumferential and radial velocity components are both 0; (3) the pressure gradient on the cross-section along the y direction in Figure 10 is constant. Based on the above assumptions, according to the N-S equation and Equation (20) the fluid viscosity in the microchannel can be solved to obtain the velocity of the fluid in the microchannel as follows:

$$u=-\frac{1}{{\mu }_0}\frac{dp}{dx}\left[\begin{array}{c}\frac{1}{4}\left({\frac{\left(h-\delta \right)}{4}}^2-z^2\right)+\frac{1}{2}\frac{b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)\left(\frac{h-\delta }{2}-z\right)}{{\mu }_0}\hfill \\ +\frac{1}{2}\frac{b\frac{h-\delta }{2}\left(\sqrt{A_{\mathrm{s}}}/\sqrt{A_{\mathrm{w}}}-1\right)\ln \frac{{\mu }_0\left(\frac{h-\delta }{2}-z\right)+b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)}{b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)}}{{\mu }_0}\hfill \\ +\frac{1}{2}\frac{b^2{\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)}^2\ln \frac{{\mu }_0\left(\frac{h-\delta }{2}-z\right)+b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)}{b\left(\sqrt{A_{\mathrm{s}}{A}_{\mathrm{w}}}-A_{\mathrm{w}}\right)}}{{\mu }_{{}_0}^2}\hfill \end{array}\right]$$

(32)

The parameters are as follows: fluid viscosity μ = 1 mPa·s, pressure gradient dp/dx = 1 × 10⁵ Pa/m, A_w = 4.2 × 10⁻²⁰ J for fluid water, and Hamaker constant A_s of the microchannel tube wall taken as 4.2 × 10⁻²⁰, 4.8 × 10⁻²⁰, and 5.4 × 10⁻²⁰ J, respectively. For ease of calculation, the microchannel radius was taken from 1, 10, 100, and 1000 μm, respectively. Through numerical simulation, the velocity distribution of the fluid in different pipe diameter microchannels shown in Figure 11 was obtained.

It can be seen from Figure 11 that when the radius of the micro circular tube is 1~10 μm, considering the van der Waals force between the microchannel wall and the fluid, the velocity distribution of the fluid in the microchannel is significantly lower than that of the Poiseuille flow, moreover, it can be seen that with a smaller pipe diameter there is a greater velocity deviation from the Poiseuille flow. The larger the Hamaker constant on the solid wall, the greater the velocity deviation from the Poiseuille flow. With increasing pipe diameter, the influence of the van der Waals force gradually decreases and the fluid velocity moves closer to the Poiseuille flow; when the pipe diameter reaches 1000 μm, the difference between the velocity distribution under van der Waals force and the velocity of the Poiseuille flow is almost zero. From the above results, it can be seen that the smaller the size of the microchannel, the greater the influence on the fluid flow of the microscopic force between the fluid and the solid wall. This is because, the force between fluid molecules and solid molecules becomes more and more significant at the microscale. When the mechanical seal is in normal operation, the end face gap is generally 0.5~2 μm, which shows that the influence of the van der Waals force on the fluid flow between the sealing interfaces cannot be ignored.

5.2. Calculation Example

In order to verify the correctness of the leakage model proposed in this paper, five groups of sealing interface rotating and stationary rings with different leakage channel sizes were established, and the theoretical calculation values of the leakage rate calculation Equation (31) obtained by considering the van der Waals force in this paper were compared with the leakage rate model results in several references [5,10,27,31]. To obtain leakage channels with different cross-sectional dimensions, five kinds of metallographic sandpaper with mesh numbers P600, P1000, P1500, P2000, and P2500 were used to grind the end faces of the stationary ring specimen; five kinds of stationary ring specimens with different rough surfaces were obtained, as shown in Figure 12. The structural parameters and material performance parameters of the sealing ring [36] are shown in

Table 1. Rotating and stationary ring structurale and material performance parameters.

Physical Parameters		Hard Ring	Soft Ring
Material		YG8	M106K
Elastic modulus E (Gpa)		600	20
Yield strength σy (Mpa)		/	50
Poisson’s ratio		0.24	0.29
Fractal dimension D, Scale coefficient G (m)	Pair 1 (P600)	2.607, 2.27 × 10−11	2.326, 5.34 × 10−10
	Pair 2 (P1000)		2.350, 4.30 × 10−10
	Pair 3 (P1500)		2.394, 3.69 × 10−10
	Pair 4 (P2000)		2.424, 3.07 × 10−10
	Pair 5 (P2500)		2.426, 2.48 × 10−10
Inner diameter ri (mm)		62	68
Outer diameter ro (mm)		82	79

; the working parameters are 0 MPa on the inner diameter side of the sealing surface and 0.3 MPa on the outer diameter side of the sealing surface, the spring specific pressure p_sp = 0.14 MPa, and the speed is 3000 r/min [5,36].

Figure 13 shows the calculation of the leakage rate under different pairs of rotating and stationary rings. It can be seen from the figure that the calculation results of the leakage rate model proposed in this paper are almost the same as the Persson model, with a maximum relative error of 8.7%, minimum relative error of only 3.8%, and similar declining trend, which proves that the leakage rate model proposed in this paper is correct. In addition, it can be seen that the calculated value of the Persson and Bottiglione model is always smaller than that of the model proposed in this paper. This is because the former uses a percolation threshold for leakage at the sealing interface of 0.593, which is greater than the percolation threshold used in this paper. In fact, when the porosity of the sealing interface reaches 0.593, the sealing interface has already leaked. In addition, it can be seen from the figure that the calculation results of the Mayer model are the largest among all models, while the calculation results of the Sun model are the smallest. This may be because the Mayer model does not take into account whether percolation occurs at the sealing interface; on the other hand, the distribution function of the contact point area of the sealing interface used in the Sun model is not suitable for description by n(a), resulting in too few leakage channels.

5.3. Leakage Rate Experiment

Through the experimental method, the leakage rate of the mechanical seal can be measured under actual working conditions. The theoretical calculation value of the leakage rate calculation model in this paper was compared with the experimental results to further verify the correctness of the leakage rate model proposed in this paper. However, it should be pointed out that because the measurement of the van der Waals force needs to be carried out in a microscopic environment, and this experiment is a macroscopic experiment, it aims to explore whether the calculation results of the leakage rate model in this paper are consistent with the change in macroscopic experimental values, meaning that it is not possible to actually measure the van der Waals force.

5.3.1. Test Rig

The leakage rate experiment was performed on a self-designed mechanical seal experimental device; its overall structure is shown in Figure 14. The experimental device is mainly composed of a synchronous conversion motor, transmission shaft, and a test unit that can realize real-time monitoring of film thickness, film pressure, end face temperature, mobile phase transition, and friction status between sealing surfaces under various operating conditions. In the experiment, water was used as the sealing medium and the working environment was room temperature. The rotating and stationary ring specimens used in the experiment are shown in Figure 12. During the experiment, the leaking sealing medium was collected by a measuring cup with a measuring range of 25 mL and an accuracy of 0.5.

5.3.2. Experimental Procedure

After the rotating and stationary ring specimens were prepared, the experimental operation was started. The specific process was as follows:

(1) Install the rotating and stationary rings, and check whether the sealing chamber leaks during the installation process. Let the sealing medium fill the sealing chamber first, then check whether there is any leakage at the end cover, the sealing surface of the rotating and stationary rings, or the threaded holes on the sealing chamber. For the first group of tests, we selected the static ring ground with P600 sandpaper.

(2) Open the liquid supply system, read the pressure of the medium in the sealing chamber, and bring the medium pressure to 0.3 MPa [5].

(3) Run the motor to increase the speed from 0 to 3000 r/min within 30 s, run it stably for 30 min, collect the leakage medium, and record the experimental data [36].

(4) Replace the stationary rings ground by P1000, P1500, P2000, and P2500 sandpaper in sequence, repeat the above steps (1)~(3), test again, and record the test results.

5.3.3. Experimental Results

After the experiment, the leakage rate under different rotating and stationary ring pairs is shown in Figure 15. It can be seen from the figure that the calculated value of the leakage rate model in this paper is in good agreement with the change trend of the experimental value, which further shows that the leakage rate calculation model in this paper is correct. With a smoother end face of the stationary ring (D is larger, G is smaller), the leakage rate is smaller, which is because the smoother end face, results in a smaller leakage channel height h − δ. Because the leakage channel pore throat size is smaller, the leakage rate is smaller as well. In addition, it can be seen from Figure 15 that the theoretical value of the leakage rate is always smaller than the experimental value; this is because the mutual reinforcement between the asperities is ignored in the theoretical derivation, making the theoretical compression amount of the asperity larger than the actual compression amount under the given contact pressure. This results in the theoretical value of the leakage channel interface height h − δ being less than the actual value, in turn resulting in the experimental leakage rate being greater than the theoretical leakage rate. In addition, the processing error and installation error of the rotating and stationary rings and the self-excited vibration of the sealing system affect the leakage of the sealing interface during the test.

6. Leak Judgment Criterion

In summary, microscopic forces have an important effect on fluid leakage at the sealing interface. For the sealing interface composed of rough rotating and stationary ring sealing surface contact, whether leakage occurs can be determined by the following criteria:

(1) When the porosity of the sealing interface ϕ < ϕ_c = 0.312, there is no leakage microchannel running through the sealing interface and no leakage occurs at the sealing interface.

(2) When the porosity of the sealing interface ϕ > ϕ_c = 0.312, there is a leakage microchannel running through the sealing interface; whether leakage occurs at the sealing interface is related to the critical void radius r_c. At this point, the following determination needs to be made.

When (h − δ)/2 < r_c: for non-wettable sealing surfaces, the capillary force prevents leakage at the sealing interface, in which case no leakage occurs and the sealing interface is self-sealing. For wettable sealing surfaces, the capillary force pushes the fluid flow into the leaking microchannel, and whether leakage occurs depends on the relationship between the thrust force formed by the pressure difference between the two sides and the resistance of the fluid flow when the fluid flows. When Equation (16) is satisfied, leakage of the sealing interface occurs; otherwise, no leakage occurs.

When (h − δ)/2 > r_c: regardless of whether the sealing surface is wettable or non-wettable, whether leakage occurs at the sealing interface at this time is independent of the capillary force, instead depending on the relationship between the thrust force formed by the pressure difference between the two sides and the resistance of the fluid flow when the fluid flows. When Equation (10) is satisfied, leakage of the sealing interface occurs, otherwise no leakage occurs.

7. Conclusions

Explaining the leakage mechanism of mechanical sealing interfaces has important practical significance for the design and engineering application of mechanical seals. In this paper, starting from the contact theory and percolation theory, the leakage mechanism of the sealing interface under the action of microscopic forces is studied. The main conclusions are as follows:

(1) A calculation model of the leakage rate of the sealing interface considering the van der Waals force was established. The correctness of the model was verified through numerical calculations and macroscopic leakage rate experiments.

(2) The van der Waals force between solid and liquid have an important influence on the flow law of fluids in microchannels. The smaller the end face gap, the greater the difference between the velocity distribution of the medium fluid flow and the Poisson flow. The influence of the van der Waals force on the fluid flow between the sealing interfaces cannot be ignored.

(3) The concept of a critical void radius between mechanical sealing interfaces is proposed, and the leakage judgment criteria of sealing interfaces are provided. In addition, further experimental research on the influence of the van der Waals force on fluid flow in microchannels will be carried out in the future.