Reliability Evaluation of a Dynamic-Pressure Mechanical Seal Based on Liquid Film Vaporization Phase Transition

Abstract

Aiming at the problem of researching the reliability of dynamic-pressure mechanical seals, this paper proposes a reliability evaluation method for dynamic-pressure mechanical seals based on the Monte Carlo method. Based on the influence of the mass transfer coefficient on vaporization phase transition, a liquid film vaporization model of a hydrodynamic mechanical seal’s end face is established, and the working condition parameters and groove structure parameters are designed using the experimental design method. The vaporization characteristics of the liquid film under various parameters are analyzed, and the functional functions of the vaporization characteristics are obtained by fitting. Combined with the maximum vapor phase volume fraction when the dynamic-pressure mechanical seal changes from the liquid miscible phase to the vapor miscible phase, the limit state equation of the vapor phase volume fraction is obtained. Finally, based on the Monte Carlo simulation method, the sealing reliability under specific groove structure parameters is calculated. Our research shows that this method has practicability and effectiveness for the reliability evaluation of mechanical seals with different working conditions and different groove structures.

1. Introduction

Improving the reliability of mechanical seal operation is a critical problem perplexing mechanical engineers and equipment maintenance personnel. Thus, a large number of investigations have been presented in the literature aiming to improve the reliability of mechanical seals. He et al. [1] analyzed the reliability of mechanical seals used in coal mine mechanical equipment, taking into account the sealing principle, part design and precision control, end-face friction and wear, and lubrication. Ojile et al. [2] studied the failure causes of mechanical seals by using Weibull and Weibyes technologies, and the feasibility of this method in the reliability analysis of mechanical seals was verified. Wei [3,4] and Zhan [5] demonstrated that the failure data of mechanical seals for petrochemical pumps is in line with the Weibull distribution. The parameter values of the corresponding Weibull distribution can be obtained by the parameter estimation method. The corresponding reliability can thus be calculated if the working life of the mechanical seal has been determined. In a connected study, Hao et al. [6] analyzed the failure of the mechanical seal of a centrifugal pump by using the fuzzy comprehensive evaluation method, and proved that the method can be used to judge the reliability of a mechanical seal according to the risk matrix. Liao et al. [7] defined the reliability parameter Λ according to the film stiffness, minimum film thickness, and contact force, and optimized the design parameters of mechanical seals based on this. Chen et al. [8] took the friction torque and leakage rate as the performance indexes of the seal degradation process, based on the binary correlation analysis method, established a reliability evaluation model of the mechanical seal, and verified that the model showed good performance throughout the degradation experiment.

At present, most of the investigations into the reliability of mechanical seals are aimed at contact mechanical seals. Based on the long-term use of seals, it is important to evaluate whether their end-face wear, leakage rate, or structural strength [9] exceeds the specified value as the failure criterion. For typical dynamic-pressure-type non-contact mechanical seals, under the action of a groove, a thin liquid film will be affected between the sealing interface of the dynamic and static rings during operation. In addition, the dynamic and static rings will be gradually separated from the bonding state under the action of the dynamic ring. The formed liquid film can effectively reduce the end-face wear and control leakage [10], and it is thus difficult to obtain failure data for reliability investigation. Yang et al. established a thermal elastohydrodynamic lubrication theoretical analysis model for textured end-face mechanical seals under mixed lubrication conditions, and found that the frictional heat of the end face had a significant impact on the liquid film [11]. Li et al. analyzed and studied the failure mechanism of mechanical seals during operation, and found that when the temperature between the sealing faces increased, the liquid film vaporized, leading to seal failure [12]. Overall, dynamic-pressure mechanical seals are often used in the petrochemical industry. The liquid film is very vulnerable to the influence of temperatures and pressures on vaporization and phase change, leading to the seal entering the vapor–liquid two-phase state [13]. The vapor–liquid two-phase state can be broken down into two states: liquid-like miscible and vapor-like miscible. When the sealing is in a liquid-like miscible state, the liquid film has good stability, but when the sealing is in a vapor-like miscible state, the liquid film is extremely unstable, which can easily lead to seal failure [14]. Therefore, the vapor-like miscible state can be used as the criterion for evaluating seal failure. Moreover, the seal state can be assessed by the vapor phase volume fraction during seal operation.

At present, the reliability analysis methods for dynamic-pressure mechanical seals are mainly divided into qualitative analysis and quantitative analysis. Qualitative analysis methods mainly involve making judgments on the nature, characteristics, and patterns of change of the research object based on the researcher’s intuition, experience, and the continuation status of the research object in the past and present, as well as the latest information. Because qualitative analysis requires a high level of researcher experience and effective research data, it can easily lead to the problem of unreliable analysis results. Quantitative analysis methods involve the use of statistical methods to analyze the research object, establish a mathematical model based on the relevant statistical data, and obtain all indicators and corresponding data of the research object within the calculation range of the mathematical model. This article adopts a quantitative analysis method and uses the operating condition of dynamic-pressure mechanical seals as a mathematical model to calculate and analyze the reliability impact of vaporization phase change on seal operation based on the average vapor volume fraction as the judgment standard.

Based on the review presented and the challenges highlighted, this paper proposes a reliability evaluation method for hydrodynamic mechanical seals based on liquid film vaporization phase transition. Combined with a finite element analysis of liquid film phase transition in dynamic-pressure mechanical seals, the function of the structural parameters of the end-face groove and the average vapor phase volume fraction are established. The limit state equation is constructed according to the limiting average vapor phase volume fraction of the transformation from liquid-like miscible seals to vapor-like miscible seals. Then, the Monte Carlo method is used to accurately estimate and evaluate the reliability of the face seal.

2. Vaporization-Phase Transition Model

2.1. Theoretical Analysis

When domestic and foreign scholars use computational fluid dynamics (CFD) methods to simulate phase transition processes, they generally use the Lee model [15] and the Thermal LB model [16]. The Lee model is widely used in numerical simulations of boiling, condensation, and other problems. Its equation form is simple and easy to implant into CFD commercial software for use. The definition of its equation determines that the source term is not limited to the generation of gas–liquid interface regions, which means that in a simulation, if there is overheating in the liquid mesh, phase transition will occur immediately [17]. Therefore, in this study, the Lee model was selected to characterize the phase transition process in the computational fluid dynamics analysis. Equation (1) shows the evaporation term of the mass transfer equation, while the condensation term is shown in Equation (2).

$${\dot{m}}_{l\to v}={\lambda }_c{\alpha }_l{\rho }_l\frac{T_l-T_{sat}}{T_{sat}}\left(T_l>T_{sat}\right)$$

(1)

$${\dot{m}}_{v\to l}={\lambda }_c{\alpha }_v{\rho }_v\frac{T_v-T_{sat}}{T_{sat}}\left(T_v<T_{sat}\right)$$

(2)

In the above equations, ${\dot{m}}_{l\to v}$ and ${\dot{m}}_{v\to l}$ represent the mass transfer rate due to evaporation and condensation. T_sat, T_l, and T_v are the saturation temperature, liquid-phase temperature, and vapor-phase temperature, in K. λ_c is the phase change mass transfer coefficient, and α_l and α_v are the volume fractions of the liquid and vapor phases. Furthermore, ρ_l and ρ_v denote the respective densities of the liquid and the vapor phases, in kg/m³.

The phase change mechanism of a spiral-groove liquid-film seal is very complex. To simplify the calculation, the following simplified assumptions are made [18]:

(1) The fluid medium belongs to Newtonian fluids;

(2) The fluid flow state is laminar flow, and the temperature and viscosity do not change with time;

(3) The sealing end face is smooth, ignoring the influence of its roughness on fluid flow;

(4) The film is very thin, and the pressure and density remain unchanged in the thickness direction;

(5) The end-face material is not affected by temperature and does not deform;

(6) There is no relative slip between the fluid and the sealing end face;

(7) The disturbance and vibration effects of the system during the working process are ignored.

2.2. Geometric Model

Dynamic-pressure mechanical seals have various groove structures due to their different application fields. Therefore, to evaluate the reliability of a mechanical seal, it is very important to establish the geometric model of the liquid film between the dynamic and static ring sealing surface according to their respective groove structures. As a typical groove structure, this study takes the spiral-groove mechanical seal as an example. Its rotating-ring seal-face groove structure is shown in Figure 1. Here, R_o, R_i, and R_g represent the outer radius, inner radius, and groove root radius of the dynamic-ring end face, in mm. θ₁ and θ₂ represent the angles corresponding to the center of the spiral-groove area and the weir area, in °.

The angle between the tangent of any point on the helix and its circular tangent is the helix angle, expressed by θ. For the convenience of the analysis, the groove diameter ratio β is the ratio of the difference between the groove root radius and the inner radius to the difference between the outer radius and the inner radius. The groove weir ratio γ is the ratio of θ₁ to the sum of θ₁ and θ₂. This is shown in Equations (3) and (4) [19].

$$\beta =\left(R_g-R_i\right)/\left(R_o-R_i\right)$$

(3)

$$\gamma ={\theta }_1/\left({\theta }_1+{\theta }_2\right)$$

(4)

It can be seen from Figure 1 that the N_g grooves on the seal face of the rotating ring are distributed along the circumferential direction and follow a periodic law. Therefore, it can be assumed that each groove area and weir area have the same flow field movement state. To improve the computing efficiency and reduce the computational cost, Figure 2 shows that the 1/N_g liquid-film model was selected as the computational domain. For example, when the number of grooves (N_g) in Figure 1 is 12, Figure 2 shows a groove, which is 1/12 of the dynamic-ring end-face groove structure shown in Figure 1.

2.3. Meshing

As the vaporization phase transition model involves fluid calculations, an unstructured grid was used for division. It is also good to note that the liquid film forming between the groove and the seal face of the dynamic-pressure mechanical seal is infinitesimally small, while other size parameters are a bit larger, with a difference of several orders of magnitude. Therefore, for greater calculation accuracy, this study defines the number of nodes on the boundary of the liquid-film model [20].

Table 1. Boundary condition settings.

Boundary	Boundary Type
Surface ABHG	Pressure—inlet
Surface CDFE	Pressure—outlet
Surface cdfe	Pressure—outlet
Surface ABDC, EFHG	Periodic boundary
Surface BDFH, aceg	Interface
Surface abdc, abhg, efhg	Moving wall
Surface bdfh	Moving wall
Surface ACEG	Stationary wall

shows the boundary conditions of the computational domain.

2.4. Solver Settings

The two-dimensional incompressible N–S equation and continuity equation are shown in Equation (5) [21]:

$$\{\begin{array}{c}\frac{\partial u}{\partial t}+u\cdot \nabla u=\frac{1}{\rho }\nabla p+v{\nabla }^{2}u\\ \nabla \cdot u=0\end{array}$$

(5)

An explanation is needed for the variables in the above equation; u represents the flow velocity of the fluid in the x direction, v represents the flow velocity of the fluid in the y direction, ρ represents the fluid density, and p represents pressure. For the governing equation, it was discretized based on the FVM, the mixture model was selected as the polynomial flow model, and the pressure-based double-precision solver was chosen. In addition, the evaporative condensation model was chosen as the phase transition model.

The SIMOLEC algorithm is based on the SIMPLE algorithm and introduces a correction function for surface flux to solve the problem of the difficulty in obtaining solutions for pressure–velocity coupling methods, thereby improving the convergence speed. So, the SIMPLEC algorithm was selected as a solver in the computational fluid dynamics while the PRESTO! format was used for the pressure discrete item. The momentum and energy terms were considered as the second-order upwind style, where the volume fraction term was used as the first-order upwind style. Furthermore, the pressure relaxation factor was set to 0.3 and the convergence accuracy was set to 10⁻⁶.

3. Limit State Equation Establishment

The limit state equation is composed of the functional function of the vapor phase volume fraction and the average vapor phase volume fraction α_mk (the vapor phase volume fraction at the maximum film pressure coefficient) when the liquid-film state at the seal face of the dynamic-pressure mechanical seal changes from liquid-like miscible to vapor-like miscible.

3.1. The Function

To establish the limit state equation, the function needs to perform multiple regression analysis on the simulation results. So, experimental testing can be carried out on the structural parameters of the groove and the parameters of the working conditions to better analyze the data [22]. The test index is the gas phase volume fraction calculated through simulation. In addition, the test factors are tank structure parameters and operating condition parameters that will affect the calculated gas phase volume fraction. The levels were divided according to the state of the experimental factors, and the level combination table of the experimental factors is then constructed.

The specific parameters corresponding to the factor level in each test number are substituted into the equations in Section 2 to establish the geometric model of the liquid film on the seal face of the dynamic-pressure mechanical seal and set up the solver. The results of the vaporization simulation analysis are reported for all tests, and the data are analyzed using multiple regression. Considering the vapor phase volume fraction as the dependent variable, and the groove structure parameters and working condition parameters as the independent variables, the functional function of the vapor phase volume fraction can thus be obtained. The resulting function is shown in Equation (6).

$$\begin{array}{ll}g\left({\alpha }_v\right)& ={\beta }_0+{\beta }_1{x}_1+{\beta }_1{x}_1+\cdots \cdots +{\beta }_n{x}_n\\ & +{\beta }_{11}{x}_1{x}_1+{\beta }_{12}{x}_1{x}_2+\cdots \cdots +{\beta }_{1n}{x}_1{x}_n\\ & +{\beta }_{22}{x}_2{x}_2+{\beta }_{23}{x}_2{x}_3+\cdots \cdots +{\beta }_{2n}{x}_2{x}_n\\ & ⋮\\ & +{\beta }_{n-1n-1}{x}_{n-1}{x}_{n-1}+{\beta }_{n-1n}{x}_{n-1}{x}_n\\ & +{\beta }_{nn}{x}_n{x}_n\end{array}$$

(6)

Here, β₀, β₁…… β_n–1n, and β_nn are the constant terms of the equation and the value of the coefficient before each variable. Moreover, x₁, x₂…… x_n−1, and x_n are groove structure parameters and working condition parameters, where n is the total number of groove structure parameters and working condition parameters.

3.2. Limit State Equation

The liquid-film states of the seal face can be divided into four types: full liquid film, liquid-like miscible phase, vapor-like miscible phase, and full vapor phase [14]. When the seal starts to run, the liquid film on the seal face is a full liquid film. With the generation of the phase transition, the state of the liquid film changes from a full liquid film to a liquid-like miscible phase. Friction within the fluid leads to a further increase in the degree of phase transition. Once the state of the liquid film becomes a vapor-like miscible phase, the stability of the liquid film will decrease sharply, resulting in seal failure. Thus, the vapor phase volume fraction when the liquid-film state on the seal face changes from liquid-like miscible to vapor-like miscible is calculated using the groove parameters and the working condition parameters.

The film pressure coefficient and vapor phase volume fraction of the vapor–liquid mixed-phase mechanical seal can be determined by Equations (7) and (8) [14].

$$\begin{array}{c}Km=\left[3v_g{p}_B\left(p_2^2-p_B^2\right)+2v_l\left(p_B^3-p_1^3\right)\right]\\ { \left[6v_g{p}_B\left(p_2-p_B\right)+3v_l\left(p_B^2-p_1^2\right)\right]}^{-1}/p_s\end{array}$$

(7)

$$\alpha =\frac{{\left(R_i+\left(6v_l\left(p_B^2-p_1^2\right)\right){\left(v_l\left(p_B^2-p_1^2\right)+2v_g{p}_B\left(p_2-p_B\right)\right)}^{-1}\right)}^2-R_i^2}{R_o^2-R_i^2}$$

(8)

In the above equations, Km represents the membrane pressure coefficient of gas–liquid mixed-phase mechanical seals and p_B is the saturated vapor pressure, in Pa. Moreover, v_g and v_l are the kinematic viscosities of saturated steam and saturated liquid, in m²/s. Furthermore, p₁ and p₂ are the respective pressures at the inner and outer diameters of the sealing surface, and p_s is the pressure difference between the inside diameter and outside diameter, all in Pa.

By substituting the operational conditions and other parameters of the dynamic-pressure mechanical seal into Equations (7) and (8), the maximum vapor phase volume fraction of the liquid-film state changing from liquid-like miscible to vapor-like miscible under these conditions can be obtained. The calculated fraction in combination with the function allow the establishment of the limit state equation, as shown in Equation (9).

$$Z={\alpha }_{mk}-g\left(\alpha \right)$$

(9)

A simple explanation is needed for the parameters in the above equation; α_mk is the gas phase volume fraction when the liquid-film state changes from a liquid-like miscible to a vapor-like miscible state.

According to the limit state equation Z ≥ 0, it is considered that the liquid-film phase transition does not lead to seal failure, and thus the seal face seal is deemed to be reliable. On the other hand, when Z < 0, it is considered that the phase change of the liquid film will cause the instability of the liquid film and sealing failure, and the seal face will thus be unreliable.

4. Reliability Evaluation Based on Liquid Film Vaporization Phase Transition

4.1. Reliability Evaluation and Assessment

Employing the obtained limit state equation highlighted in the previous section, the structural parameters of the grooves were sampled using the Monte Carlo method, and a reliability evaluation was then carried out according to the sampling results. Figure 3 shows the reliability evaluation process.

Figure 4 shows the calculation flow of the Monte Carlo method. From the figure, it can be seen that when using the Monte Carlo method for calculation, it is first necessary to determine the number of simulation times, N. This is because the calculation accuracy of the Monte Carlo method will increase with the increase in simulation time. When the calculation accuracy has improved to a certain level, a very large number of calculations are required to further improve the accuracy. One of the well-demonstrated solutions to this issue is parallel computing [23]. Parallel computing is carried out using the Monte Carlo method to calculate the loops independently, without interfering with each other, and the calculation results are thus not connected. Combined with the parallel computing capabilities of multi-core CPUs, computing efficiency is improved without affecting computing accuracy. With the current development level of computer hardware, the number of calculations is no longer a difficult problem perplexing the Monte Carlo method. An appropriate number of simulation times, N, can be selected for calculation while meeting the requirements of calculation accuracy.

Computational accuracy can also be improved by enhancing the sampling method [24]. The purpose of the improvement to the sampling method is to improve sampling efficiency. The sampling center is concentrated near the key points to ensure that the probability of sampling in the failure area is large, thereby reducing the variance of the failure rate estimation. In this study, the distribution law is determined according to the distribution range parameters of the groove structure and operating conditions during the operation of the dynamic-pressure mechanical seal. This is the basis on which the parameter sampling is carried out.

By substituting the sampled parameters into Equation (9), the distribution of simulation results can be calculated and the corresponding reliability can be obtained. After N simulation calculations, a reliability evaluation can be obtained for each calculation, and the average of all reliability estimations is considered as the final calculation result of the Monte Carlo method.

4.2. Case Study Application

In this study, the spiral-groove mechanical seal was considered as a case study application, aiming to implement the methodology presented and calculate the corresponding reliability.

Table 2. Structural parameters of the seal-face groove and parameters of operating conditions.

Parameter Name	Parameter Value
Liquid-film outer radius, Ro/mm	31
Groove-root circle radius, Rg/mm	28.75
Liquid-film inner radius, Ri/mm	26
Helix angle, θ/°	23
Groove diameter ratio, β	0.55
Groove weir ratio, γ	0.3
Groove depth, hg/μm	9
Liquid-film thickness, h/μm	3
Number of grooves, Ng	12
Inlet pressure, pi/MPa	1.1
Inlet temperature, T/K	413
Outlet pressure, po/MPa	0.1
Rotating speed, n/rpm	1500

shows the structural parameters and operational parameters of the grooves of the sealing surface of the rotating ring.

The vaporization phase transition model was established by employing the parameters in Table 2. The model was then verified using the previous literature, and Figure 5 shows the verification results. The figure shows that when the mass transfer coefficient is set at 0.1, the maximum error of the average vapor volume fraction calculated by the two models will not exceed 10%, meeting the application requirements.

When Cao et al. [25] studied the phase transition phenomenon of double-row spiral-groove liquid-film seals, the mass transfer coefficient was set as the default coefficient of 0.1. However, Da [26] and Yang et al. [27] believed that the mass transfer coefficient was considered an empirical value. Qiu [28] claimed that when CFD calculates the phase transition problem, the calculation results will become more accurate with the increase in the mass transfer coefficient. In a previous study [29], only the influences of the structural parameters of a spiral-groove mechanical seal on the gas phase volume fraction under different mass transfer coefficients were considered. On this basis, the effects of operating parameters on the gas phase volume fraction with different mass transfer coefficients were considered in this study. The results are shown in Figure 6.

From Figure 6a–c, it can be seen that within the given range, the average vapor volume fraction increases with the increase in inlet temperature, and decreases with the increase in rotational speed and pressure difference. Meanwhile, the change in the mass transfer coefficient results in a change in the calculated average vapor volume fraction under the same operating conditions. Regardless of changes in temperature, speed, or pressure difference, the average vapor volume fraction will gradually increase with the increase in the mass transfer coefficient, and when the mass transfer coefficient increases to a certain extent, the average vapor volume fraction will remain stable.

When the speed varies between 500 and 5500 rpm (Figure 6a), the temperature varies from 373 to 413 K (Figure 6b), and the pressure difference is in the range of 0.1–0.9 MPa (Figure 6c). In addition, the average vapor volume fraction will increase with the increase in the mass transfer coefficient, and the average vapor volume percentage will stabilize at the mass transfer coefficient of 100. According to the results, when the mass transfer coefficient reaches 48, the growth rate of the average vapor phase volume fraction obtained by each calculation is less than 1%. Considering the computational efficiency and computational cost, the mass transfer coefficient of 48 was then selected for simulation calculation.

To reduce the calculation cost and improve the calculation accuracy, the average vapor phase volume fractions were compared and analyzed when the surface mesh size was set to 0.1, 0.08, 0.06, 0.05, and 0.04. Figure 7 shows that when the surface mesh size was 0.05, the relative error of the mean gas phase volume fraction was 1.5%. It was shown that the average vapor volume fraction is not affected to a large extent by the mesh size. Therefore, the decision was made to use a mesh size of 0.05 as the mesh partition standard to consider the influence of surface mesh size on the calculation cost [29].

The considered case study example exhibits a reliability analysis under certain working conditions. Only the structural parameters of the groove (spiral groove, groove diameter ratio, groove weir ratio, and groove depth) need to be used as the test factors for the uniform test design. The experimental design specifications are shown in

Table 3. Experimental design specifications.

Test Number	Column Number
	θ/°	β	γ	Hg/μm
1	1	10	14	15
2	2	3	11	13
3	3	13	8	11
4	4	6	5	9
5	5	16	2	7
6	6	9	16	5
7	7	2	13	3
8	8	12	10	1
9	9	5	7	16
10	10	15	4	14
11	11	8	1	12
12	12	1	15	10
13	13	11	12	8
14	14	4	9	6
15	15	14	6	4
16	16	7	3	2
17	17	17	17	17

. In addition, a multiple regression analysis was carried out on the simulation results, and the functional function obtained is shown in Equation (10) [29].

$$\begin{array}{ll}g\left(\alpha \right)& =-0.0264\theta +1.1999\beta -0.3349\gamma -0.0552h_g\\ & +0.0002{\theta }^2-0.0091\theta \times \beta +0.0068\theta \times \gamma +\\ & 0.0023\theta \times h_g-0.9658{\beta }^2-0.0897\beta \times \gamma -\\ & 0.0062\beta \times h_g+0.0250{\gamma }^2+0.0720\gamma \times h_g-\\ & 0.0004{h_g}^2+0.46961\end{array}$$

(10)

Figure 8 and Figure 9 were obtained considering the specifications listed in Table 1 and the results from Equations (7) and (8). It was found that when the temperature is less than 373 K or α is equal to 0, the seal is in the full liquid-phase region, and the film pressure coefficient is equal to 0.615. When the temperature is higher than 455 K or α is equal to 1, the seal is in the full vapor phase and the film pressure coefficient is equal to 0.778. Moreover, when the temperature reaches 444 K or α is equal to 0.288, the film pressure coefficient is equal to 0.871, and when the temperature is higher than 373 K and lower than 444 K, the seal is in the liquid-like miscible region. Finally, as the temperature is increased above 444 K but is still below 455 K, the seal is in the vapor-like miscible region.

According to the obtained maximum membrane pressure coefficient α_mk and the function, the limit state equation is established as shown in Equation (11).

$$\begin{array}{ll}Z& ={\alpha }_{mk}-g\left(\alpha \right)\\ & =0.0264\theta -1.19995\beta +0.33491\gamma +0.05522h_g\\ & -0.00024{\theta }^2+0.00914\theta \times \beta -0.00686\theta \times \gamma -\\ & 0.00229\theta \times h_g+0.96579{\beta }^2+0.08968\beta \times \gamma +\\ & 0.00617\beta \times h_g-0.02501{\gamma }^2-0.07205\gamma \times h_g+\\ & 0.00037{h_g}^2-0.18161\end{array}$$

(11)

Errors will occur when machining a spiral-groove mechanical-seal-end. In this study, it was assumed that the parameters of the groove structure meet the normal distribution type, and the sampling calculation was then carried out. The dimensional tolerance for the approximate calculation of the spiral-groove structural parameters is 0.01 times this dimension, and the standard deviation is one-third of the dimensional tolerance.

Table 4. Distribution of design parameters.

Number	Parameter Name	Mean Value	Standard Deviation	Distribution Type
1	θ/°	23	0.077	Normal distribution
2	β	0.55	0.00183	Normal distribution
3	γ	0.3	0.001	Normal distribution
4	hg/μm	9	0.03	Normal distribution

shows the distribution of the groove structure parameters.

The distribution of the parameters of the groove structure obtained by sampling is shown in Figure 10a–d. As can be seen from the figure, the sampled data are consistent with the actual situation and follow the normal distribution.

In this study, 1000 Monte Carlo samplings were conducted. The reliability of each sampling was calculated, and the mean distribution of reliability was provided. The mean distribution is shown in Figure 11.

Figure 11 shows that when the sampling times are low, the change in the mean reliability is more obvious. In addition, it is shown that the final calculated reliability of the 500 samples is very different each time they are sampled. With the increase in simulations, the mean reliability value tends to converge. It was also shown that the 1000 Monte Carlo simulations had met the calculation accuracy. The final reliability was calculated to be 0.956. According to the calculated reliability, the optimization design can be carried out in the seal-face groove structure of the spiral-groove mechanical seal to make the seal more stable.

It should be noted that this study is based on the analysis of simulation results. Although the corresponding literature [28] provides a comparison analysis for the key index of the average gas phase volume fraction, the reliability evaluation results still need to be verified by further tests. This provides a basis for additional improvements and upgrades to the current study in the future.

5. Conclusions

(1) A three-dimensional model of the liquid film on the end face of a spiral-groove mechanical seal was established, and the flow state of the liquid film on the end face was analyzed. The N–S equation was simplified and assumed, and boundary conditions, solvers, ICEM mesh division, and mesh independence verification were completed. The feasibility of the calculation model in this study was also verified.

(2) A simulation analysis was conducted on the influence of the end-face-groove structural parameters and operating conditions on the vaporization characteristics and sealing performance of mechanical seals. The effects of factors such as the helix angle, groove diameter ratio, groove weir ratio, groove depth, temperature, speed, and pressure difference on the vaporization characteristics and sealing performance of the liquid film under single variable conditions were analyzed. The main conclusions obtained are as follows: the average vapor volume fraction increases with the increase in the helix angle, groove to weir ratio, groove depth, and temperature; it first increases and then decreases with the increase in groove diameter ratio, and then decreases with the increase in speed and pressure difference.

(3) Based on the mechanical-seal liquid film vaporization model, four variable factors including the helix angle, groove diameter ratio, groove weir ratio, and groove depth were selected. A uniform experiment with four factors and 17 levels was designed using the uniform experimental design method. Through fluid simulation analysis and multiple regression analysis, a regression equation was established for the structural parameters of the spiral-groove mechanical-seal end-face groove with respect to the average vapor volume fraction.

(4) A finite element reliability calculation of dynamic-pressure mechanical seals based on the Monte Carlo method was proposed. Based on the operating conditions of dynamic-pressure mechanical seals, the relationship curves between the membrane pressure coefficient and temperature, as well as the membrane pressure coefficient and average vapor volume fraction, were calculated. Combined with the regression equation regarding the average vapor volume fraction, the limit state equation required for the reliability analysis of the end-face liquid-film phase change was jointly constructed. Using the reliability analysis module and the Monte Carlo sampling method, the reliability of the end-face seal was calculated, which has certain reference significance for the research and design of gas–liquid two-phase-flow mechanical seals.