Yan-liang DONG，Kai-chao ZHONG

（School of Mechanical and Electrical Engineering，Harbin Institute of Technology，Harbin 150008，China）

Abstract：A numericalmodel to evaluate the sealing performance of the hydraulic combined dynamic seal is introduced in this paper.Based on thermal elastohydrodynamic lubrication（TEHL）theory，the impact of thermal effect on dynamic sealing oil film is emphatically considered in thismodel besides pressure and linear velocity.The relations between the flow field parameters and the sealing performance are established under a three-dimensionalmodel，then a numerical solution is reached and the result is analyzed.

Key words：Combined dynamic seal，TEHL theory，Thermal effect，Sealing performance

1 Introduction

Combined seal iswidely used in all fields，which is made up of a PTFE sealing ring and a rubber ring.Due to the effect of various factors such as surface friction，there is always a little oil in the sealing gap and the pressure oilwill lead the seal ring to be separated from the sealing surface［1］.The shear force and the pressure will cause the film temperature rising during the contact process between the sealing ring and themotor cylinder，which will significantly influence the viscosity，even exert some effects on the density and other parameters.Therefore，the thermal effect is an assignable factor when establish the numericalmodel.

In the study of the dynamic sealmodel，numerical calculation and finite element simulation of hydrodynamic seal are widely studied both at home and abroad.White and Denny researched on reciprocating seals and carried out experiments，which made the sealing research had a great breakthrough between 1944 and 1947［2］.The research made by the Nikas team mainly focuses on sealing the elastic deformation calculation［3］.Xu Namodeled the effect of the temperature field on the sealing performance simply and obtained reasonable results［4］.

The paper mainly focuses on the influence of thermal effect on the dynamic sealing performance.A reasonable numericalmodel of dynamic seal based on the theory of TEHL is established while considering the complete temperature field.The numerical solution is based on MATLAB，and then the result is analyzed.

2 Num ericalm odel of dynam ic seal

The model mainly focuses on the dynamic seal at the blades of electrohydraulic swingmotorwhich shows in Fig.1.The seal is composed essentially of an O-ring and a PTFE ring，which can overcome the shortcomings of traditional O-ring seal that are easy extruded and easy tofail.The oil will be pulled into the sealing gap between the cylinder and PTFE ring when the relative movement exists.This phenomenon will become even more serious in the case that the oil pressure is very high.

Fig.1 SeaIstructure of the com bined sea I

The previousmodel was simplified as a two-dimensionalmodel and ignored the influence along theOyaxis，which is suitable for the situation that the width of the seal is long enough.As the width of the seal is very small，its effect cannot be ignored.This paper establishes a three-dimensional model，which can analysis the problem more entirely and accurately.

2.1 Reynolds equation

A modified Reynolds equation is derived in case of considering the thermal effect.Wen and Yang［5］derived a generalized Reynolds equation which is effective to analysis the thermal effect.In this paper this generalized Reynolds equation is applied.

Where

Where，ρis the oil density，his the oil thickness，η is the oil viscosity，pis the oil pressure，Vis the oil velocity along thexdirection.

According to the real working condition，The boundary conditions are：

2.2 Film thickness equation

The oil thickness in sealing gap can be calculated from themechanical deformation of the seal，the cylinder and the vane.

Where，ubodyzis the deformation of the cylinder，usealis the deformation of the seal，upre-loadingzthe inner surface displacement of the cylinder at the temperature of installation owing to pre-loading.The mathematical model of the deformation is as follow.

2.2.1 The inner surface displacement of the cylinder owing to pre-loading upre-loadingz

In the process of installation，The O-ring mounted in a seal groove is subjected to deformation.The force of the contact pressure and the distribution of contact pressure are determined by the Hertz solution，which was verified experimentally by Karaszkiewicz in 1979［6］.

The contact line of the elastomerwill become a narrow surface.When the O-ring mounted in a seal groove is subjected to deformation.The distribution of contact pressure is very similar to the semi-ellipse distribution［7］

Where，p0is the contact pressure owing to pre-loading，pHis themaximum contact pressure.

Therefore，the inner surface displacement of the cylinder owing to pre-loading is as follow.

Where，νbodyis the Poisson’s ratio of the cylinder，Ebodyis the elasticmodulus of the cylinder.

2.2.2 Normal deformation of the cylinder

According to the Elastic Mechanics，the deformation of the inner surface of the cylinder can be calculated as follow.

2.2.3 Normal deformation of the seal useal

Compared with the cylinder，the seal is so smooth that the way mentioned before cannot be suitable.In this paper，themethod used by Xu Na is quoted.The following equation can be derived according to the force balance of theOzaxis.

2.3 Roelands equation and density equation

The simplified Roelands equation is chosen to describe change of the viscosity with pressure and temperature.

Where，η0is the ambient viscosity of the oil.

Meanwhile，an empirical formula shows that of the density.

Where，ρ0is the ambient density of the oil.

2.4 Temperature field model

2.4.1 The film energy equations

The heat is dissipated because of heat conduction and convection effects，and a steady-state temperature field would form after balancing.The temperature field is determined by the energy equation and its boundary conditions.The film energy equation would be as below without considering the thermal radiation.

2.4.2 The solid energy equations

This paper only analyzes the stead state and ignores the heat transfer along theOxandOyaxis isd.Then it can derive the solid energy equations as follow.

Where，c1，c2is respectively the Specific heat of the PTFE and the cylinder.k1，k2is respectively the thermal conductivity of the PTFE and the cylinder.ρ1，ρ2is respectively the density of the PTFE and the cylinder.z1，z2is respectively the coordinate along theOzaxis of the PTFE and the cylinder.

2.4.3 The continuity equation at the interfaces

Even though there are 2 temperature control equations，the temperature field is unique.The temperature of the interface between the film and solid must be controlled by the continuity equation.

Where，d2is the thickness of the PTFE.

2.4.4 The oil film velocity equation

The simplified Navier-Stokes is utilized here.The oil film velocity equation can be integrated directly so that the velocity and the derivative of the velocity along theOxaxis can be given.

In a similarway，the derivative of the velocity along theOyaxis can be given.

2.5 Leakage and the friction force

Themass leakage of an elementary rectangle of fluid with dimensionsdydzin the sealing contact isdˉm·=ρufdydz，integrating the elementary leakage forOzbetweenoandh.Theoretically，since the fluid continuity equation is satisfied at all points in the contact，leakage need be computed at any one pointx，which could be the centre of the contact.Meanwhile，it is theoretically better to compute the“average”leakage to avoid this arbitrary selection.

Moving on to the calculation of friction，the frictional force on the seal from hydrodynamic viscous shear is then computed by numerical integration from.

3 Solution and analysis

The equations are all nonlinear equations，which can only be solved by numerical analysis.A series of nonlinear equationswith pressure，film thickness，and temperature can be obtained through discrediting the control equations.The pressure field is iteratively updated using the relaxation iterative scheme，while the temperature field is updated using the sequential sweepingmethod.

A numerical solution is reached by MATLAB.The data used for calculation examples are shown in Table 1.

Tab Ie 1 MATLAB so Iution param eters

The temperature distribution in the film under the thermal condition is shown in Fig.2.It is easy tofind that the distribution of temperature is similar to that of pressure.As being shown in Fig.2（a），the temperature changes little along theOyaxis in thexyplane，while the temperature increases gradually along theOxaxis，and themaximum temperature rise is at the exit of the film，which is corresponding to the minimum film thickness.Since the smallest thickness of the film located at the exit，whichmeans the shear stress is much larger than that of the entrance，more friction heatwas generated at the exit.

As is shown in Fig.2（b），the temperature of the moving surface is the lowest in thexzplane，and the temperature increases along thezaxis；however，the temperature rise on the moving surface is so small that it is negligible.The temperature on the moving vane is lower than that of static cylinder surface，which is because the points on the surface went through the sealed zonewith relatively high speed and the heat transferred from film to PTFE in very limited time is considerably little.The heat from the film in high temperature and high pressure enters through the interface to the static cylinder，which heats the cylinder to thermal equilibrium.

Fig.2 The isotherm a Idiag ram of the tem perature fie Id

The distributions of viscosity and density have been changed because of the unbalanced distribution of the temperature，which bringsmuch influence to the distributions to the thickness and pressure of the film.Taking the influence of temperature into account，the distributions of the thickness and pressure of the film in three-dimensions are shown in Fig.3（a）and（b）.

As is shown in Fig.3（c），the film pressure increased before it drops along thexaxis，and the trend is similar to the contact stress distribution of the seal before the seal leaks.There is a pressure peak about 18.565 MPanear the seal contact inlet position，and the pressure drops sharply near the seal outlet position.The thickness of the film decreased at the beginning and remained almost the same afterward and is roughly inversely to that of pressure，which is consist-entwith the results of thermal-EHL theory.

As is shown in Fig.3（d），along the y axis，the values in the figure are the average value along theOXaxis for ease of analysis.The average film pressure almost keep the same value 11.32 MPa in themost areas along theOyaxis，while there is a pressure peak about11.65 MPa at the each side of the end.Similarly，the average film thickness almost keep the same value 1.71μm in the most areas along theOyaxis，while there is a thickness necking about 1.69μm at the each side of the end.With limited seal ring length，obvious pressure peak and thickness necking behavior can be found at the end region.Atboth sides of the seal，the clearance along the ends is diffusing，where the pressure dropped significantly.This leads to bulges on the seal.Overall，the film thickness and pressure changed so little that it can be considered as one dimensionalmodel.

Fig.4 shows the effect of temperature on the oil film pressure and the pressure thickness.It can be found from Fig.4（a）that the thickness of the oil film in the case of considering the temperature field is significantly lower about0.25μm than that in isothermal condition，which shows that film will produce a lot of heat in the case of high-speed motor running.Meanwhile，the oil film thickness decreases under the thermal condition，which exacerbate friction and abrasion.

Fig.3 The distribution of p ressure and fiIm thickness under the therma Ieffect

Fig.4 The distribution of p ressure and fiIm thickness under the therma Ieffect

As is shown in Fig.4（b），the pressure distribution at the pressure peak is very different，which can reach 17.5 MPa under thermal condition compared with 16.8 MPa under the isothermal condition，while in other positions differs little.Therefore，the effect of thermal effects cannot be neglected in the case of high-speed situation.

Fig.5 shows the distribution of the motor leakage，viscous frictional force under isothermal conditions and thermal condition.As is shown in Fig.5（a），compared with isothermal condition，the leakage is averagely 100 mg/h higher than the leakage under the thermal condition，mainly caused by the effect of oil density and velocity synthesis.At the same time，as is shown in Fig.5（b），the viscous frictional force under thermal condition is averagely 0.5 N lower than that under isothermal condition，which ismainly caused by viscosity decreased with the temperature falling.This suggests that the thermal effect can significantly influence the leakage of seal and viscous friction force distribution.

Fig.5 The m ass Ieakage and friction force under different Iinear veIocity

Fig.5 also shows the comparison of the leakage and friction force under the condition of isothermal and temperature field situation.As shown in Fig.4，with the increase of linear velocity，average mass leakage will increase significantly.The mass leakage will increase from 98.86mg/h to 913.72mg/h when the linear velocity increases from 100 mm/s to 1 000 mm/s.At the same time，with the increase of the linear velocity，viscous friction force at the sealing contact decrease significantly.

Keep the linear velocity of themotor，according to Fig.6，the leakage increased significantly from 808 mg/h to 1 102mg/h when the sealing pressure increases from 8 MPa to 16 MPa，mainly caused by the increase of oil film thickness.At the same time，with the increase of the sealing pressure，viscous friction force at the sealing contactwill increase significantly.Under the thermal effect，the leakage is higher than that under isothermal condition，which is consistent with the previous conclusion.

Fig.6 The mass Ieakage under different oiIp ressure

4 Conclusions

The following conclusions can bemade after analysis：

（1）the oil film shear force and the pressure will cause the oil film temperature rising during the contact process，themaximum temperature rise is at the exit of the film，which is correspond to theminimum film thickness，the temperature of themoving surface is the lowest.

（2）Compared with themodel under the isothermal condition，the oil film pressure is larger and the film thickness is thinner under the thermal condition；the leakage is higher，mainly caused by the effect of oil density and velocity synthesis；the frictional force under the thermal condition is lower.

（3）Compared with themodel under the isothermal condition：with the increase of the sealing pressure and linear velocity，the leakage and the friction force both will increase significantly.