Effects of the cavitation and fluid inertia on the performance in dimpled mechanical seal

To develop a hydrodynamic lubrication model of mechanical seals considering cavitation and fluid inertia and investigate the effects of cavitation and fluid inertia on the sealing performance of a textured mechanical seal, a finite element numerical procedure combined with a Newton-downhill scheme is developed. The numerical model was validated by comparing the results of the Reynolds equation with JFO theory (RE-JFO) and CFD simulation. The effects of fluid inertia and cavitation on the sealing performance are numerically compared and analyzed. Compared with the RE-JFO, the present model gives a better description of cavitation and fluid inertia. The results show that the centrifugal inertia can produce a throttling effect, and thus the leakage rate is reduced. The occurrence of cavitation improves the load-carrying capacity and reduces the coefficient of friction. The textures have a dominant influence on the sealing performance.


Introduction
Surface texturing has been well accepted as a feasible means to enhance the tribological properties of mechanical components in recent decades.Therefore, it has been widely applied in industries, such as mechanical seals, bearings, piston rings, and magnetic storage devices [1].
Fluid inertia is a universal phenomenon of fluid mechanics in hydrodynamic lubrication.Billy et al. [2][3] indicated that fluid inertia can cause fluid recirculation and affect the fluid pressure buildup, resulting in a net load-carrying capacity (LCC) for textured surfaces.Cupillard et al. [4] pointed out that it may also reduce the LCC, which is closely related to the texture geometric parameters.In the above research, Navier-Stokes (NS) equations were solved by the CFD technology.To study fluid inertia conveniently, Constantinescu and Galetuse [5] put forward a hypothesis that fluid inertia does not affect velocity profile and only affects its magnitude.Based on the assumption, Brunetière and Tournerie [6] investigated the inertia flow of hydrostatic seals and found that it can partly reduce seals' leakage.Syed and Sarangi [7] also applied the assumption to study the fluid inertia for the textured surfaces and the results show that it can affect the fluid flow.
Cavitation is not considered in the above research, but it is often encountered due to the presence of surface textures.According to Braun and Hannon [8], there are three cavitation forms: vapor cavitation, gaseous cavitation, and pseudo-cavitation.Cross et al. [9] found that cavitation can improve tribological properties.Cross et al. [9] and Meng et al. [10] found that the impact of cavitation on the hydrodynamic effect and LCC is closely dependent on the geometrical parameters of dimples.Chen et al. [11] and Ma et al. [12] revealed that cavitation can affect the fluid flow between sealing faces.Cavitation is often solved by the JFO theory, which includes the film rupture and regeneration process and meets mass-conservation.However, cavitation pressure is forcibly set as a constant, which deviates from the experimental results [8].In recent years, the Rayleigh-Plesset equation (RPE) model has been developed.In the model, it is assumed that there is a certain amount of microbubble in the liquid medium originally, and thus the growth and collapse process of microbubbles is used to approximate the film rupture and regeneration process.Remarkably, the RPE model takes into account the physical process of cavitation.Therefore, similar cavitation models have been widely applied in studies of hydrofoils, piston motors, bearings, etc.
In the present study, a hydrodynamic lubrication model of mechanical seals considering cavitation and fluid inertia (ICM) is proposed based on the assumption [5] and RPE.The numerical results are compared with the results of CFD and RE-JFO.The influences of cavitation and fluid inertia on the film pressure and performance of mechanical seals are investigated.

Geometric model
A geometrical model of circle-dimpled face seals is shown in Figure 1.The rotor rotates along the clockwise direction with an angular velocity ω.The circle dimples are evenly distributed at the stator face.The film thickness is given as: The geometric and operational parameters are listed in Table 1.

Mathematical model
The following hypotheses are made in the ICM.
The sealed medium is Newtonian and its flow regime between sealing faces is laminar.
The fluid inertia only affects the fluid velocity magnitude and does not affect its profile [5].
The sealed medium is a homogenous mixture of liquid and vapor, and the mixture is assumed to be isothermal and incompressible.The liquid density is a constant and the vapor is assumed as an ideal gas, which shares the flow rates and pressure.

Governing equations.
For the thin film lubrication, the continuity equation and momentum equations can be written as: The following velocity profiles of inertial flow are derived.
where the unknowns qx and qy are the flow rates of unit length along the x and y directions, and u0 and v0 are the velocity components of the rotor.Equation ( 2) is integrated along the film thickness and the following integral forms are obtained based on Equation (3).
where Ixx, Ixy, and Iyy are the integrals of the products of velocity components.

∋ (
Equations (3)(4)(5) are obtained based on [6].The flow in the sealing gap is governed by the following single-phase mass transfer equations.(6) where FL and FG are the volume fractions of liquid and vapor, and S is the cavitation source term in the RPE model, namely the phase change rate.sgn( ) where Ŕb is the first-order approximation of RPE, ρG is the vapor density, C is the mass transfer coefficient, and N is the number of initial bubbles.The integral forms of Equation ( 3) can be obtained.
Integrating Equation ( 6) along the film thickness, the following two-phase flow mass transfer equation is derived based on Equation (8).
The derivation process of Equation ( 9) is seen in the reference, and only the lubrication equations are different.

Boundary conditions.
To solve Equations ( 4) and (9), Equations (10) and (11) (11) where r is an arbitrary radius at the sealing face, and α is the angle between the radius and the x positive-direction.

Numerical approach
The computational region is meshed with triangular elements and Equations ( 4) and ( 9) are solved with the finite element method.For the ICM, they are converted into the nonlinear equation groups with the variables qx, qy, p, and FL at every node.The Newton-downhill method is applied to compute the unknowns.
The load-carrying capacity (LCC), coefficient of friction (COF), and leakage rate can be computed by the following equations.∋ ( where lx and ly are the unit vectors, and l is the integral curve.

Validation
To conveniently compare the different numerical models including the ICM, RE, RE-JFO, and NS equations, a simpler geometrical model with only one dimple in the radial direction is applied, where ro=31.5 mm, ri=30 mm, and ps=0.11MPa.The other parameters are listed in Table 1.
Figure 2 presents the pressure distribution at r=30.75 mm when the cavitation is not included.The pressure profiles outside of the dimple deviated from each other and the inside of the dimple coincided with each other.ICM's results are closer to those of the CFD model.
Figure 3 shows the liquid volume fraction and pressure distribution at r=30.75 mm when the cavitation is considered.There are three models including cavitation methods, namely NS-Schnerr Sauer model (NS-SS), ICM, and RE-JFO.The NS-SS is the default cavitation model in ANSYS FLUENT.The theoretical basis of cavitation for the NS-SS and ICM is the RPE and homogeneous two-phase flow theory, and thus they all consider the physical process of cavitation.In the NS-SS, the vapor density is assumed to be a constant.To better approximate the correlation between vapor density and pressure, in the ICM, the vapor density is determined by the ideal gas equation of state.In the thin film flow, solid interface, flow channel volume, and pressure difference significantly affect the rupture and regeneration of liquid film.Therefore, compared with the NS-SS, the modified vapor and liquid mass transfer coefficients are used in the ICM due to the above characteristics.The RE-JFO does not take into account the physical process of cavitation, and cavitation pressure is forcibly set as a given pressure pc.Viewed from Figure 3 (a), all the models yield the same film rupture location, which is the dimple inlet.However, there is a difference in the liquid film regeneration location.The cavitation region and liquid volume fraction given by the ICM are closer to the results of NS-SS. Figure 3 (b) shows that the pressure distribution of the three models is consistent with each other.In the film rupture zone, the RE-JFO shows a similar pressure to the NS-SS, while in the film regeneration zone, the ICM gives closer pressure to the NS-SS.At the outside of the dimple, ICM's results are more consistent with the NS-SS.The improvements in cavitation treatment make ICM's pressure distribution smoothly change in the cavitation region.
Table 2 tabulates the LCC, COF, and leakage rate.The LCC simulated by the three models is very close to each other.However, for the COF and leakage rate, the results are different.The values of ICM are closer to the results of CFD.For example, the RE-JFO overestimates the COF and the error is about 10%, and the RE and RE-JFO seriously overestimate the leakage rate, and the error is about 16% and 38%, respectively.The main reason is that the ICM includes the fluid inertia and physical process of cavitation.

Influences of the cavitation and fluid inertia on the sealing performance
The mechanical seal in Figure 1 is numerically simulated.Figure 4 shows the film pressure p, liquid volume fraction FL, and cavitation source term S given by the ICM. Figure 5 presents p and FL at the given radii.Figure 4 (a) and Figure 5 (a) show that a low-pressure region is formed at the inlet of each dimple due to the abrupt increase in film thickness.The high-pressure region is built up at the outlet of each dimple.The local maximum pressure is closely dependent on the radial location of each dimple.As shown in Figure 5 (b), there is a quarter-moon region at the low-pressure one of every dimple inlet, in which FL<1.The fluid film is a mixture of vapor and liquid, which indicates that the cavitation occurs here.The cavitation areas are larger for the inside dimples.At the edge of each quarter-moon region, the change of FL is extremely significant.FL in the quarter-moon region tends to be constant.As shown in Figure 4 (c), S<0 can be found at the inlet of every dimple, which means that the liquid film ruptures and begins to convert into vapor in the region.The dumbbell-shaped region can also be found in each dimple, in which S>0.It shows the location of the liquid film regeneration boundary, where the bubbles collapse and the vapor begins to convert into the liquid.S is close to 0 within the cavitation region, and there is almost no net vapor-liquid mass transfer, so the stable cavitation region is formed.According to Brunetière and Tournerie [6], for the untextured seals, the radial reduced Reynolds number (Re) ReP * indicates the magnitude of convective inertia terms, and the circumferential reduced Re ReC * indicates the magnitude of centrifugal inertia terms.As shown in Figure 6, with the increasing rotation speed, RePo * is always less than 1.5×10 -5 .However, ReCo * increases rapidly.This indicates that the centrifugal inertia increases rapidly and the fluid is hardly affected by the convective inertia in the process.It also indirectly indicates that, for the textured face seals, the fluid is mainly affected by the centrifugal inertia.
Figure 7 shows that, for the untextured seals (Plane), with the increasing rotation speed, ICM's LCC decreases slightly.However, it is a constant for the RE-JFO.Mainly the centrifugal inertia partially balances the pressure flow.For the textured seals (Texture), when the speed ω>1000, the LCC increases rapidly.With its increase, the hydrodynamic effect increases, and the liquid cavitation enhances the LCC, as shown in Figure 5 (a).ICM's LCC is also smaller than that of the RE-JFO due to the centrifugal inertia.
As seen in Figure 8, compared with the untextured seals, the cavitation occurs at the inlet of each dimple for the textured seals.There is a large amount of vapor in the cavitation region, as a result, the COF decreases significantly.The RE-JFO tends to overestimate the liquid volume fraction in the cavitation region, as seen in Figure 3 (a).Therefore, for cases with cavitation regions, the RE-JFO gives a greater COF than the ICM.In the present study, the excessive estimation of COF |fICM -fRE-JFO|/ fICM can reach 10%.
As shown in Figure 9, the leakage rate of the untextured seals given by the RE is little affected by the rotation speed.However, when the fluid inertia is considered (ICM), the leakage rate gradually decreases with increasing speed, which is the contribution of centrifugal inertia.Figure 10 shows that the fluid midplane velocities u and v along the line (x=0, 30 ≤ y ≤ 36) are given by the RE and ICM, where u and v denote the circumferential and radial speeds, respectively.When ω=50, u and v obtained by the two models are identical to each other.Mainly the centrifugal inertia is very small for the small speed, as shown in Figure 6.When ω=4000, u from the two models is also the same.However, ICM gives a much smaller v than RE.This indicates that the centrifugal inertia produces a throttling effect and reduces leak in the radial direction.Therefore, RE overestimates the leakage rate and |QICM -QRE-JFO|/QICM is up to about 33% for the untextured seals, as shown in Figure 9.For the textured seals, the leakage rate of RE-JFO first decreases and then increases, but the variation is very small, which means that the speed also has little effect on the leakage rate.The ICM gives the same trend of leakage rate as the RE, but the variation is greater.Mainly the dimples and centrifugal inertia affect the fluid velocity distribution simultaneously, and the throttling effect caused by the centrifugal inertia partly balances the downstream pumping effect of the dimples.The RE-JFO gives an excessively large leakage rate and the value of |QICM -QRE-JFO|/QICM can reach 18%.In addition, the dimples have the most significant impact on the leakage rate, and the values of textured seals are about 2 times those of the untextured seals.The results show that compared with the RE and RE-JFO, the ICM gives a more precise cavitation prediction and sealing performance evaluation, which are closer to the results of CFD simulation.The occurrence of cavitation improves the load-carrying capacity of liquid film and partly reduces the coefficient of friction.The centrifugal inertia partly weakens the pressure flow between sealing faces.Therefore, it generates a throttling effect, which can effectively reduce seals' leakage.Compared with the RE-JFO, the ICM takes into account the physical process of cavitation, and the predicted coefficient of friction is smaller when the cavitation phenomenon is significant.The textures at the sealing face have a dominant influence on the sealing performance.

Figure 6 .
Figure 6.Reduced Re versus for untextured seals.Figure 7. Effect of rotation speed on Fo.

Figure 7 .
Figure 6.Reduced Re versus for untextured seals.Figure 7. Effect of rotation speed on Fo.

Figure 8 .
Figure 8.Effect of rotation speed on f.Figure 9.Effect of rotation speed on Q.

Figure 9 .
Figure 8.Effect of rotation speed on f.Figure 9.Effect of rotation speed on Q.

Figure 10 .
Figure 10.Influence of fluid inertia on the fluid velocity distribution for untextured seals.
model of mechanical seals considering cavitation and fluid inertia (ICM) was proposed based on the assumption of Constantinescu and Galetuse and the Rayleigh-Plesset equation.A finite element numerical procedure combined with a Newton-downhill scheme was developed to solve the model.The numerical model was validated by comparing with the results of the Reynolds equation (RE), the Reynolds equation with JFO cavitation theory (RE-JFO), and the CFD simulation.The effects of fluid inertia and cavitation on the pressure distribution and performance of mechanical seals were investigated.

Table 1 .
The geometric and operating parameters.
are applied.

Table 2 .
Sealing performance parameters comparison.