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Gas lubrication equations
CHAPTER
Gas lubrication equations
2
The principle equations of gas lubrication analysis include the gas equation, the Reynolds equation, the energy equation, the solid heat conduction equation, the interface equation, and the surface deformation calculation equation. The problem of gas thermohydrodynamic lubrication often occurs in high-pressure, hightemperature, and high-speed bearings and seals, where the contact form of friction pairs is mainly low-side contact. In the subsequent analysis and discussion of this book, the deformation calculation of the friction surface is adopted by the finite element method. Because the finite element method is a very advanced analysis of solid deformation, the calculation equation of surface deformation and the solution method are not discussed in this book. The gas state equation was discussed in Chapter 1, Properties of gases, so this chapter explains the other fundamental equations of gas thermohydrodynamic lubrication, including the Reynolds equation, the energy equation, the solid heat conduction equation, and the interface equation.
2.1 Reynolds equation The derivation process of the Reynolds equation is briefly described by using the micro-element analysis method [1], whose main steps include (1) calculation of the velocity distribution of the flow along the film thickness bases on the stress equilibrium condition of the micro-element, (2) obtaining the flow rate by integration of flow velocity across film thickness, and (3) derivation of the universal form of the Reynolds equation according to flux continuity. The following fundamental assumptions are used in the derivation: 1. The mass force is ignored. 2. The fluid is not sliding in the boundary, that is, the flow velocity on the surface is the same as the surface velocity. 3. The pressure variation along film thickness is considered. 4. The influence of the lubrication film curvature is ignored, and the rational speed is replaced with the translation speed. 5. The fluid is Newtonian. 6. The flow is laminar, with no eddy current or turbulence in the lubrication film. 7. The viscosity value is constant along the film thickness. Gas Thermo-hydrodynamic Lubrication and Seals. DOI: https://doi.org/10.1016/B978-0-12-816716-8.00002-4 Copyright © 2019 Tsinghua University Press Limited. Published by Elsevier Inc. All rights reserved.
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CHAPTER 2 Gas lubrication equations
2.1.1 Derivation of Reynolds equation For the Newtonian fluid with viscosity η, set ux, uy, and uz are the velocity of the fluid along the x, y, and z directions, and ux and uy are the main velocity components. The size of the film thickness, z, is much smaller than x and y. Therefore, compared with the velocity gradient, @ux =@z and @uy =@z, the other velocity gradients are negligible. Now let us consider a unit in the lubrication regime, as shown in Fig. 2.1. In the direction x, the fluid is affected by the fluid pressure p and the viscous force τ. In the steady state, all forces acting on the unit in the direction x are equal, so the following expression is obtained: @τ @p dz dxdy 5 p 1 dx dydz 1 τdxdy pdydz 1 τ 1 @z @x
(2.1)
Eq. (2.1) can be simplified to @p @τ 5 @x @z
(2.2)
According to Newton’s law of viscosity, τ 5 ηð@ux =@zÞ, the following equation can be obtained from Eq. (2.2). @p @ @ux 5 η @z @x @z
(2.3)
In the same way, the following equation is obtained in the y direction. @p @ @uy 5 η @z @y @z
FIGURE 2.1 Unit in lubrication regime.
(2.4)
2.1 Reynolds equation
In addition, @p 50 @z
(2.5)
Because ux is a function of z, and because the fluid velocity at the interface is equal to the surface velocity, there are boundary conditions with assuming velocity of the two solid surfaces, which are ux;0 and ux;h , respectively, as follows: ux jz50 5 ux;0 ;
ux jz5h 5 ux;h
(2.6)
Because p, η, and ρ are not functions of z, integration of Eq. (2.3) twice across z gives ð @ux @p @p 5 dz 5 z 1 C1 @z @x @x ð @p @p z2 z 1 C1 dz 5 1 C1 z 1 C2 ηux 5 @x @x 2 η
(2.7) (2.8)
Based on the boundary condition Eq. (2.6), C1 and C2 are determined by the following expressions: C2 5 ηu0 C1 5 ðuh 2 u0 Þ
η @p h 2 h @x 2
(2.9) (2.10)
Thus, the flow velocity in the x direction is given by ux 5
z 1 @p 2 ðz 2 zhÞ 1 ux;h 2 ux;0 1 ux;0 2η @x h
(2.11)
In the same way, the flow velocity in the y direction is obtained as follows: uy 5
z 1 @p 2 ðz 2 zhÞ 1 uy;h 2 uy;0 1 uy;0 2η @y h
(2.12)
And, the flow velocity boundary in the y direction is uy jz50 5 uy;0 ;
uy jz5h 5 uy;h
(2.13)
where uy;0 and uy;h are the velocity of the two solid surfaces, respectively, in the y direction. Further, let us analyze the flow rate through a unit with film thickness h, as shown in Fig. 2.2. The mass flow rate on the unit width is mx and my, respectively, in the x and y directions, and the volume flow rate is qx and qy, so then there are mx 5
ðh 0
mmol ρux dz
(2.14)
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CHAPTER 2 Gas lubrication equations
FIGURE 2.2 Flow rate of micro-element [1].
my 5
ðh 0
qx 5
mmol ρuy dz
(2.15)
ðh ux dz
(2.16)
uy dz
(2.17)
0
qy 5
ðh 0
The following expressions are obtained by ignoring density variation along film thickness. mx 5 mmol ρqx
(2.18)
my 5 mmol ρqy
(2.19)
Let u 5 ux;h 2 ux;0 and v 5 uy;h 2 uy;0 , and institute the expressions of u and v into Eqs. (2.16) and (2.17), respectively, to obtain expressions of volume flow rate. qx 5 2
h3 @p uh 1 12η @x 2
(2.20)
qy 5 2
h3 @p vh 1 2 12η @y
(2.21)
According to the mass conservation law, the quality of the inflow microelement should be equal to the quality of the outflow. Hence, from Fig. 2.2, the following expression is given: mx dy dxdy 0 1 my dx 1 mmol 1 ρuz;0 0
1 @m @m x y dxAdy 1 @my 1 dyAdx 1 mmol ρuz;h dxdy 5 @mx 1 @x @y
(2.22)
2.1 Reynolds equation
where it can be considered that the two solid surfaces with velocity uz,0 and uz,h move upward, causing a change in the film thickness h. Here, (uz,0uz,h)dxdy represents the change of the volume, that is: ρuz;h 2 ρuz;0 5
@ρh @t
(2.23)
Simplification of Eq. (2.22) and institution into Eqs. (2.18) and (2.19) gives the Reynolds equation, as follows: @ h3 ρ @p @ h3 ρ @p @ðρhÞ @ðρhÞ @ðρhÞ 1 5 6u 1 6v 1 12 @x η @x @y η @y @x @y @t
(2.24)
2.1.2 Reynolds equation in the polar coordinate system In the same way as in the derivation of the rectangular coordinate system, in the polar coordinate system, the flow velocities uθ and ur in the θ and r directions, respectively, are obtained as follows: uθ 5
ωrz 1 @p 2 z 2 zh 1 2η r@θ h
(2.25)
1 @p 2 z 2 zh 2η @r
(2.26)
ur 5
Consider a unit with film thickness h, as shown in Fig. 2.3. The volume flow rate on the unit width qθ and qr in the θ and r directions, respectively, is obtained as follows: qθ 5 2
h3 @p ωrh 1 2 12η r@θ
(2.27)
rh3 @p 12η @r
(2.28)
qr 5 2
According to the mass conservation law, the Reynolds equation in the polar coordinate system is obtained. @ h3 ρ @p @ rh3 ρ @p @ðρhÞ @ðρhÞ 1 5 6ω 1 12 r@θ η r@θ r@r η @r @θ @t
(2.29)
2.1.3 Reynolds equation in the cylindrical coordinate system In the same way as in the derivation of the rectangular coordinate system, in the cylindrical coordinate system, the flow velocities uθ and uz in the θ and z directions, respectively, are obtained as follows: uθ 5
ωr0 r 1 @p 2 r 2 rh 1 2η r0 @θ h
(2.30)
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CHAPTER 2 Gas lubrication equations
FIGURE 2.3 Flow rate of micro-element in the polar coordinate system.
where ω is the rotational speed. uz 5
1 @p 2 r 2 rh 2η @z
(2.31)
Consider a unit with film thickness h, as shown in Fig. 2.4. The volume flow rate on the unit width qθ and qr in the θ and z directions, respectively, is obtained as follows: qθ 5 2
h3 @p ωr0 h 1 2 12η r0 @θ
(2.32)
h3 @p 12η @z
(2.33)
qz 5 2
According to the mass conservation law, Reynolds equation in the cylindrical coordinate system is obtained. 3 @ h ρ @p @ h3 ρ @p @ðρhÞ @ðρhÞ 1 5 6ω 1 12 r0 @θ η r0 @θ @z η @z @θ @t
(2.34)
2.1.4 Lubrication parameters 1. Load w
ðð w 5 pdxdy
(2.35)
2.1 Reynolds equation
FIGURE 2.4 Flow rate of micro-element in the cylindrical coordinate system.
2. Flow rate Q In the lubrication regime, the flow rate Qx and Qy respectively in the x and y directions are expressed as ð
Qx 5
qx dy
(2.36)
qy dx
(2.37)
ð Qy 5
3. Friction force F In the x direction, the friction force of the lubricating film on the solid surfaces can be calculated by the following formula: ðð Fx;0 5 τ x z50 dxdy
(2.38a)
ðð Fx;h 5 τ x z5h dxdy
(2.38b)
where Fx,0 and Fx,h are friction force, respectively, at surface z 5 0 and z 5 h in the x direction. The shear stress of the fluid is τx 5
1 @p uη ð2z 2 hÞ 1 2 @x h
(2.39)
In the y direction, the friction force is calculated in the same way as in the x direction.
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CHAPTER 2 Gas lubrication equations
4. Friction torque M For the rotating friction pair in the polar coordinate system, the friction torque of the lubricating film on the solid surfaces can be calculated by the following formula: ðð M0 5 τ z50 r 2 drdθ
(2.40a)
ðð Mh 5 τ z5h r 2 drdθ
(2.40b)
where M0 and Mh are friction torque on surface z 5 0 and z 5 h, respectively. The shear stress of the fluid is given by τ5
1 @p ωrη ð2z 2 hÞ 1 2 r@θ h
(2.41)
2.2 Energy equation In research on point and line contact of liquid thermal elastohydrodynamic lubrication on the background of bearing and gear design, surface elastic deformation of friction pair, viscositypressure effect and temperatureviscosity effect of the fluid are mainly taken into account, on the basis of isothermal hydrodynamic lubrication. In derivation of energy equation for steady-state liquid hydrodynamic lubrication, due to the application of liquid lubricated bearing as the background, it is believed that the heat exchange in fluid film is dominated by convective heat transfer, namely not considering heat exchange between lubricant film and friction pairs, that is fluid film is regarded as adiabatic in the lubrication regime [1]. For gas thermohydrodynamic lubrication on the background of gas bearing and gas seals, the contact form of friction pairs is mainly contact with low-side contact. In addition to considering the viscosity variation of the gas and the surface elastic deformation, it is necessary to consider the surface thermal deformation. So, the influence of the heat exchange on the temperature distribution of the solid surface needs to be considered. At the same time, because of requirements for gas face seal to ensure the stability of the gas film lubrication to ensure the stability of the sealing performance, so also should be considered in the energy equation of compression on the influence of the gas film lubrication temperature changes. In this section, the energy equation is further modified by the method of literature, and the variation of kinetic energy and potential energy is ignored during the derivation. The change of heat energy and mechanical work in the flow is analyzed as follows.
2.2 Energy equation
2.2.1 Chang of gas inner energy Consider a x-direction unit of width δx, with length 1 in the y direction and height h in the z direction as shown in Fig. 2.5, the gas inner energy flow in the x direction can be expressed as following. Hx 5 qx Tρcv
(2.42)
In the same way, the gas inner energy flow in the y direction can be expressed as follows: Hy 5 qy Tρcv
(2.43)
Let δx 5 1 and δy 5 1, the total heat inflowing into the unit, be given as @Hx @Hy @Hx @Hy δx 2 Hx 1 Hy 1 δy 2 Hy 5 1 Hx 1 @x @y @x @y
(2.44)
If the mechanical work performed in the microcylinder of the unit’s sectional area is represented by S, then the following relation can be obtained according to the principle of energy conservation. @Hx @Hy 1 5S @x @y
(2.45)
Substitution of Hx and Hy into Eq. (2.45) gives @ðqx TÞ @ðqy TÞ S 1 5 @x @y ρcv
(2.46)
The following equation can be obtained from the flow continuity: @qx @qy 1 50 @x @y
FIGURE 2.5 Gas inner energy flow in the x direction.
(2.47)
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CHAPTER 2 Gas lubrication equations
Substitution of Eq. (2.47) into Eq. (2.46) gives qx
@T @T S 1 qy 5 @x @y ρcv
(2.48)
2.2.2 External work on gas and energy loss The changes in gas internal energy induced by external work and heat exchange include four parts: flow work, interface friction loss, heat conduction loss, and surface extrusion heat. 1. Flow work Consider an x-direction unit of width δx, as shown in Fig. 2.6; the flow work in the x direction is expressed as @p @qx δx 2 pq qx 1 p 1 δx @x @x
When high-order small quantities are ignored, and δx 5 1, the flow work in the x direction is simplified as qx
@p @qx 1p @x @x
Hence, the total flow work wflow on the unit is obtained as follows when flows in both x and y directions are considered: wflow 5 qx
FIGURE 2.6 Fluid flow in the x direction.
@p @p @qx @qy 1 qy 1p 1 @x @y @x @y
(2.49)
2.2 Energy equation
Substitution of Eq. (2.47) into Eq. (2.49) gives wflow 5 qx
@p @p 1 qy @x @y
(2.50)
2. Friction loss When a lubricating interface is seen as the driving surface, the other surface is the friction surface. Thus, the friction loss consumed in the unit is obtained. wfric 5 ðτ 0 ux;h 1 τ h ux;0 Þ 1 ðτ 0 uy;h 1 τ h uy;0 Þ
(2.51)
where h @p ηðux;h 2 ux;0 Þ h @p ηu τ 0 ux;h 1 τ h ux;0 5 2 1 ðux;h 2 ux;0 Þ 5 2 1 u 2 @x h 2 @x h h @p ηðuy;h 2 uy;0 Þ h @p ηv 1 ðuy;h 2 uy;0 Þ 5 2 1 v τ 0 uy;h 1 τ h uy;0 5 2 2 @y h 2 @y h
(2.52) (2.53)
3. Interface heat conduction loss Because the fluid in the lubrication area flows at a high-speed relative to the friction interface, there is a forced convective heat transfer between the fluid film and the solid. So, the heat loss is expressed as follows: wcon 5 kgs1 ðTs1 2 TÞ 1 kgs2 ðTs2 2 TÞ
(2.54)
where ks1 and ks2 are the convection heat transfer coefficient, and Ts1 and Ts2 are the solid surface temperature in the lubricating regime. 4. Surface extrusion heat The solid surfaces work on the gas film because of their squeezing movement in the z direction. The work can be calculated by the following formula: wp 5
@ph @t
(2.55)
Substitution of Eq. (1.1) into Eq. (2.55) gives wp 5 Ru 5 Ru Th
@ρTh @t
(2.56)
@ρ @Th 1 Ru ρ @t @t
where the first term on the right presents the work of density compression, and the second term describes the gas extrusion heat induced by the squeezing movement of solid surfaces. Hence, surface extrusion heat generated in the gas film is wh 5 Ru ρ
@Th @t
(2.57)
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CHAPTER 2 Gas lubrication equations
Thus, the total work S consumed in the unit is obtained as follows: S 5 wflow 1 wfric 1 w0 con 1 wh
1 0 1 @p @p @ h @p ηuA h @p ηv u 1 @2 1 qy 1 2 1 1 Av 5 qx @x @y 2 @x h 2 @y h
(2.58)
@Th 1 kgs1 ðTs1 2 TÞ 1 kgs2 ðTs2 2 TÞ 1 Ru ρ @t
Substitution of qx and qy into Eq. (2.58) gives ηðu2 1 v2 Þ h3 2 S5 h 12η
" 2 # @p 2 @p @Th 1 kgs1 ðTs1 2 TÞ 1 kgs2 ðTs2 2 TÞ 1 Ru ρ 1 @x @y @t (2.59)
Further, substitution of Eq. (2.48) into Eq. (2.58) gives the gas energy equation as follows: 2 @T @T ηðu2 1 v2 Þ h3 4 1 qy 5 qx 2 @x @y hρcv 12ηρcv 1
!2 @p @x
!2 3 5 1 @p @y (2.60)
kgs1 kgs2 Ru @Th ðTs1 2 TÞ 1 ðTs2 2 TÞ 1 ρcv ρcv cv @t
In the same way, we obtain the gas energy equation in the polar coordinate system as follows: 2 @T @T ηω2 r 2 h3 4 1 qr 5 qθ 2 @x @y hρcv 12ηρcv 1
!2 @p r@θ
!2 3 5 1 @p @r
kgs1 kgs2 Ru @Th ðTs1 2 TÞ 1 ðTs2 2 TÞ 1 ρcv ρcv cv @t
(2.61)
2.3 Solid heat conduction equation and the interface equation For ordinary solids, the heat conduction equation, the Laplace equation, is given. kc @2 Ts @2 Ts @2 Ts @Ts 1 2 1 2 5 ρs cs @x2 @y @z @t
(2.62)
where Ts is solid temperature, kc is thermal conductivity, ρs is density of the rotor material, and cs is specific heat capacity of the rotor.
2.4 Numerical analysis method
For a friction pair of relative motion, the heat conduction equation can be changed as following because the lubrication interface has the velocity u in the x direction [2]. kc @2 Ts @2 Ts @2 Ts @Ts 5u 1 1 ρs cs @x2 @y2 @z2 @x
(2.63)
In the lubricating regime, the heat exchange condition between gas film and solid satisfies the following equations: @Ts 5 kgs1 ðTs1 2 TÞ @n s @Ts 5 ks2 ðTgs2 2 TÞ 2kc2 @n s
2kc1
(2.64a) (2.64b)
where kc1 and kc2 are thermal conductivity of two solid friction pairs, and kgs1 and kgs2 are convective heat transfer coefficient of the two interfaces, which can be calculated by following formula [3,4]: 1=3
kgs1 5 kgs2 5
3Qr kc h0
gas
(2.65)
where Qr is the Prandtl number. Qr 5
Cp η kc gas
(2.66)
where Cp is gas isobaric specific heat capacity and kc_gas is gas thermal conductivity.
2.4 Numerical analysis method From the derivation of the Reynolds equation, the balance of force and the conservation of flow follow. So, it is necessary to satisfy these two conditions when solving with numerical methods, so as to ensure that the numerical results are closer to the theoretical real value. The finite difference method is now widely used as a numerical method, and this section discusses solving the problem of the Reynolds equation using this method.
2.4.1 Finite difference method The finite difference method can be used to solve the gas lubrication Reynolds equation. However, iterative divergence often occurs in solving gas lubrication problems of large bearing number, such as hard disk magnetic head. Huang [5,6] discussed this problem and gave the finite difference scheme of the problem. For solving the energy equation, the “T” finite difference method can be adopted
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CHAPTER 2 Gas lubrication equations
[1,5]. Our calculation analysis [4] showed that the “T” finite difference method is also applicable to solving the gas lubrication energy equation. Because of the geometric structure of the lubricating surface, dynamic boundary conditions must be adopted, which is discussed in subsequent chapters. For solving the solid heat conduction equation, the seven difference equations can be used, as shown in Fig. 2.7. Generally, for one point (m, n, k) in the cylindrical coordinate system, the following difference expressions of the conduction equation can be acquired [4]: @Ts
m;n;k
@r @2 Ts m;n;k Ts 5 @θ2 @2 Ts m;n;k Ts 5 @r 2 @2 Ts m;n;k Ts 5 @z2
5
Ts
m11;n;k
m;n11;k
m;n;k11
m;n;k
2 Ts Δr
m;n21;k
1 Ts m21;n;k 2 2Ts ðΔθÞ2 1 Ts m;n21;k 2 2Ts ðΔrÞ2 1 Ts m;n;k21 2 2Ts ðΔzÞ2
(2.67a) m;n;k
(2.67b)
m;n;k
(2.67c)
m;n;k
(2.67d)
2.4.2 Flow conservation For fluid lubrication bearing, the capacity of the fluid film is the main analysis parameter, not involving the flow analysis problem; thus flow conservation problems tend to be ignored, which will not produce obvious error analysis in engineering practice. However, for fluid seal, leakage rate is a very important parameter as well as open force. In theory, the leakage rate is the flow rate in the radial direction for the face seal. So, in solving the Reynolds equation, accurately
FIGURE 2.7 Illustration of finite difference grids.
2.4 Numerical analysis method
calculating flow conservation will ensure reliable results. With the application of surface texturing technology to seal surface design, the importance of flow conservation is more crucial. For example, as shown in Fig. 2.8, there are two types of dimples according to the change of depth: the continuous-depth dimples and the step-depth ones. For the continuous-depth dimples surface, the calculation error of flow conservation can be decreased to a certain extent through increasing grid density. However, for the step-depth dimples surface, flow conservation cannot be guaranteed by means of the encryption grid, that is, the flow conservation depends on the difference equation in nature, but not on mesh grid density in the numerical analysis. Huang [5,6] pointed out the flow conservation problem of the Reynolds equation by using the finite difference method, and gave a concrete difference formula. Here, from the infinitely wide Rayleigh steps, as shown in Fig. 2.9, we further discuss the flow conservation problem of the Reynolds equation by the finite difference method [7]. It is well known that when gas flows driven by shear velocity, surface step will resist the flow of the gas. This in turn will induce pressure gradient at the step, and an equivalent pressure flow must exit to keep flow continuity at the step. So, in the following derivation of this section, only pressure flow is considered, where the pressure may or may not be induced by shear velocity. The gas film temperature is assumed to be constant. For the ideal gas, the gas film pressure is obtained by solving the Reynolds equation (2.68). @ @p2 h3 50 @x @x
(2.68)
where p2 means square of pressure p. The boundary conditions are p 5 p2 at x 5 0 and p 5 p1 at x 5 B, and the pressure and the flow rate are continuous at x 5 b. The explicit solution of Eq. (2.68) is expressed as follows:
p 5 2
8 2 ps 2 p22 > > x 1 p22 > > < b
ð0 # x # bÞ
p21 2 p2s > 2 > > > : B 2 b ðx 2 bÞ 1 ps
ðb # x # BÞ
(2.69)
where ps is the pressure at the step. Thus, the flow rate in the x-direction qx can be obtained as follows: 8 3 2 h2 ps 2 p22 > > > > < 2pa b
h3 p @p h3 @p2 5 qx 5 5 h31 p21 2 p2s > pa @x 2pa @x > > > : 2pa B 2 b
ð0 # x # bÞ (2.70) ðb # x # BÞ
So, the unit linear flow rate at the step can be obtained as follows: qs 5
h32 p2s 2 p22 h3 p2 2 p2s 5 1 1 2pa b 2pa B 2 b
(2.71)
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FIGURE 2.8 Laser textured geometry of seal face. (A) Continuous-depth dimples and (B) step-depth dimples.
2.4 Numerical analysis method
FIGURE 2.9 Infinitely wide Rayleigh step [7].
where p2s 5
h32 p22 ðB 2 bÞ 1 h31 p21 b h32 ðB 2 bÞ 1 h31 b
(2.72)
Another expression of the flow rate in the x-direction at the step is as follows: 3
q0s 5
hs p21 2 p22 2pa B
(2.73)
where hS is the equivalent clearance at the step, and the following equation is obtained: qs 5 q0s
(2.74)
Hence, 3
hs 5
h31 h32 B 3 h2 ðB 2 bÞ 1 h31 b
(2.75)
It should be noted that Eq. (2.75) is also accurate for continuous clearance. The definition of calculation grids is shown in Fig. 2.10. When a sealing surface has steps, discontinuous clearances exist. When shear velocity is considered, the Reynolds equation to solve pressure distribution is as follows: 2 @ @ph 3 @p h 5 12uη @x @x @x
(2.76)
Generally, to solve Eq. (2.76), the five-point finite difference method is often used [1]. And it may be assumed that the steps always are at the center of the two
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CHAPTER 2 Gas lubrication equations
FIGURE 2.10 Grid definition of a rectangular step regime [7].
adjacent grid points, as shown in Fig. 2.10, when grid density is high enough. Hence, difference equation at the point i can be obtained as follows:
2 2
@ @p2 1 3 3 2 2 h3 5 h p 2 p h p 2 p 2 i i21 i11=2 i11 i21=2 i @x @x ðΔxÞ2 i
(2.77)
Where hi21=2 and hi11=2 are the equivalent clearance at the center of two adjacent grid points, their expressions can be obtained by simplified Eq. (2.75) when b 5 0.5B. 3
hi11=2 5 3
hi21=2 5
2h3i h3i11 h3i 1 h3i11
(2.78a)
2h3i h3i21 h3i 1 h3i21
(2.78b)
Eqs. (2.78a) and (2.78b) can keep flow continuity in numerical calculation and is the same as Huang’s difference equations in the case of the equidistant difference method. For the mechanical face seal, the leakage rate is actually the flow rate in the radial direction. According to the flow continuity principle, the leakage rate should be consistent at any radial, and its expression is as follows: Q5
ð 2π 0
p h3 @p rdθ 2 pa 12η @r
(2.79)
where pa is ambient pressure, and the standard atmospheric pressure is often used in the calculation Numerical analysis of the leakage rate of the step-dimpled face mechanical seal [7] shows that the flow conservation of Reynolds equation can be guaranteed by using the difference formula Eqs. (2.78a) and (2.78b), and the correctness of the sealing leakage rate formula Eq. (2.79) is confirmed. Fig. 2.11 presents the pressure distribution and leakage rate on the step-dimpled seal face. Correspondingly, as shown in Fig. 2.12, the leakage rate from the inside diameter
2.4 Numerical analysis method
FIGURE 2.11 Pressure distribution of the step-dimpled face mechanical seal [7].
FIGURE 2.12 Leakage rate of the step-dimpled face mechanical seal [7].
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CHAPTER 2 Gas lubrication equations
to the outside diameter is calculated by using Eq. (2.79). The maximum deviation of the leakage rate from different radii is less than 3%.
2.4.3 Friction force balance As shown in Eqs. (2.38a) and (2.38b), and (2.40a) and (2.40b), different expressions of friction force or friction torque are often misleading in that the friction forces of the upper and lower surfaces are different. But, it should be pointed out that, this difference of friction expression is just a mathematical problem deduced by the coordinate system of the Reynolds equation, which does not affect the correct description of the Reynolds equation for the essential physical properties of fluid flow. In order to illustrate the problem further, the example of the stepdimpled face gas seal is still used. For the friction pairs of rotational motion, the friction balance problem is presented as the friction moment balance. The friction torque on solid surfaces is calculated by the following formula: M0 5
ð 2π ð ro 0
Mh 5 2
τ z50 r 2 drdθ
(2.80a)
ri
ð 2π ð ro 0
τ z5h r2 drdθ
(2.80b)
ri
where M0 and Mh are friction torques on surfaces of z 5 0 and z 5 h respectively. The shear stress of the fluid is τ5
1 @p ωrη ð2z 2 hÞ 1 2 r@θ h
FIGURE 2.13 Influence of the bearing number on friction torque.
(2.81)
References
Substitution of Eq. (2.81) into Eqs. (2.80a) and (2.80b) gives 1 @p ωrη 1 rdrdθ 2 r@θ h 0 ri ð 2π ð ro 1 @p ωrη h1 rdrdθ Mh 5 2 2 r@θ h 0 ri M0 5
ð 2π ð ro
2
(2.82a)
(2.82b)
The dimensionless method of friction torque is discussed in Chapter 3, Isothermal gas lubrication. Fig. 2.13 shows the calculation results of friction torque on the upper and lower surfaces of the seal friction pairs, as shown in Ref. [7]. It can be seen from the figure that the friction torque of the upper and lower surfaces is the same in value but opposite in direction. This means that friction or friction torque balance can also be used as a basis for the convergence of numerical results.
References [1] S. Wen, P. Huang, Theory of Tribology, second ed., Press of Tsinghua University, Beijing, 2002. [2] S. Wen, P. Yang, Theory of Elastohydrodynamic Lubrication, Press of Tsinghua University, Beijing, 1998. [3] A.L. San, T.H. Kim, Thermohydrodynamic analysis of bump type gas foil bearings: a model anchored to test data, J. Eng. Gas Turbines Power 132 (4) (2010) 042504. [4] S. Bai, X. Peng, Y. Meng, et al., Modeling of gas thermal effect based on energy equipartition principle, Tribol. Trans. 55 (6) (2012) 752761. [5] P. Huang, Lubrication Numerical Calculation Method, Higher Education Press, Beijing, 2012. [6] P. Huang, Y. Meng, H. Xu, Tribology Course, Higher Education Press, Beijing, 2008. [7] S. Bai, X. Peng, Y. Li, et al., Gas lubrication analysis method of step-dimpled face mechanical seals, J. Tribol. 134 (1) (2012) 011702.
35
Gas lubrication equations
2
The principle equations of gas lubrication analysis include the gas equation, the Reynolds equation, the energy equation, the solid heat conduction equation, the interface equation, and the surface deformation calculation equation. The problem of gas thermohydrodynamic lubrication often occurs in high-pressure, hightemperature, and high-speed bearings and seals, where the contact form of friction pairs is mainly low-side contact. In the subsequent analysis and discussion of this book, the deformation calculation of the friction surface is adopted by the finite element method. Because the finite element method is a very advanced analysis of solid deformation, the calculation equation of surface deformation and the solution method are not discussed in this book. The gas state equation was discussed in Chapter 1, Properties of gases, so this chapter explains the other fundamental equations of gas thermohydrodynamic lubrication, including the Reynolds equation, the energy equation, the solid heat conduction equation, and the interface equation.
2.1 Reynolds equation The derivation process of the Reynolds equation is briefly described by using the micro-element analysis method [1], whose main steps include (1) calculation of the velocity distribution of the flow along the film thickness bases on the stress equilibrium condition of the micro-element, (2) obtaining the flow rate by integration of flow velocity across film thickness, and (3) derivation of the universal form of the Reynolds equation according to flux continuity. The following fundamental assumptions are used in the derivation: 1. The mass force is ignored. 2. The fluid is not sliding in the boundary, that is, the flow velocity on the surface is the same as the surface velocity. 3. The pressure variation along film thickness is considered. 4. The influence of the lubrication film curvature is ignored, and the rational speed is replaced with the translation speed. 5. The fluid is Newtonian. 6. The flow is laminar, with no eddy current or turbulence in the lubrication film. 7. The viscosity value is constant along the film thickness. Gas Thermo-hydrodynamic Lubrication and Seals. DOI: https://doi.org/10.1016/B978-0-12-816716-8.00002-4 Copyright © 2019 Tsinghua University Press Limited. Published by Elsevier Inc. All rights reserved.
15
16
CHAPTER 2 Gas lubrication equations
2.1.1 Derivation of Reynolds equation For the Newtonian fluid with viscosity η, set ux, uy, and uz are the velocity of the fluid along the x, y, and z directions, and ux and uy are the main velocity components. The size of the film thickness, z, is much smaller than x and y. Therefore, compared with the velocity gradient, @ux =@z and @uy =@z, the other velocity gradients are negligible. Now let us consider a unit in the lubrication regime, as shown in Fig. 2.1. In the direction x, the fluid is affected by the fluid pressure p and the viscous force τ. In the steady state, all forces acting on the unit in the direction x are equal, so the following expression is obtained: @τ @p dz dxdy 5 p 1 dx dydz 1 τdxdy pdydz 1 τ 1 @z @x
(2.1)
Eq. (2.1) can be simplified to @p @τ 5 @x @z
(2.2)
According to Newton’s law of viscosity, τ 5 ηð@ux =@zÞ, the following equation can be obtained from Eq. (2.2). @p @ @ux 5 η @z @x @z
(2.3)
In the same way, the following equation is obtained in the y direction. @p @ @uy 5 η @z @y @z
FIGURE 2.1 Unit in lubrication regime.
(2.4)
2.1 Reynolds equation
In addition, @p 50 @z
(2.5)
Because ux is a function of z, and because the fluid velocity at the interface is equal to the surface velocity, there are boundary conditions with assuming velocity of the two solid surfaces, which are ux;0 and ux;h , respectively, as follows: ux jz50 5 ux;0 ;
ux jz5h 5 ux;h
(2.6)
Because p, η, and ρ are not functions of z, integration of Eq. (2.3) twice across z gives ð @ux @p @p 5 dz 5 z 1 C1 @z @x @x ð @p @p z2 z 1 C1 dz 5 1 C1 z 1 C2 ηux 5 @x @x 2 η
(2.7) (2.8)
Based on the boundary condition Eq. (2.6), C1 and C2 are determined by the following expressions: C2 5 ηu0 C1 5 ðuh 2 u0 Þ
η @p h 2 h @x 2
(2.9) (2.10)
Thus, the flow velocity in the x direction is given by ux 5
z 1 @p 2 ðz 2 zhÞ 1 ux;h 2 ux;0 1 ux;0 2η @x h
(2.11)
In the same way, the flow velocity in the y direction is obtained as follows: uy 5
z 1 @p 2 ðz 2 zhÞ 1 uy;h 2 uy;0 1 uy;0 2η @y h
(2.12)
And, the flow velocity boundary in the y direction is uy jz50 5 uy;0 ;
uy jz5h 5 uy;h
(2.13)
where uy;0 and uy;h are the velocity of the two solid surfaces, respectively, in the y direction. Further, let us analyze the flow rate through a unit with film thickness h, as shown in Fig. 2.2. The mass flow rate on the unit width is mx and my, respectively, in the x and y directions, and the volume flow rate is qx and qy, so then there are mx 5
ðh 0
mmol ρux dz
(2.14)
17
18
CHAPTER 2 Gas lubrication equations
FIGURE 2.2 Flow rate of micro-element [1].
my 5
ðh 0
qx 5
mmol ρuy dz
(2.15)
ðh ux dz
(2.16)
uy dz
(2.17)
0
qy 5
ðh 0
The following expressions are obtained by ignoring density variation along film thickness. mx 5 mmol ρqx
(2.18)
my 5 mmol ρqy
(2.19)
Let u 5 ux;h 2 ux;0 and v 5 uy;h 2 uy;0 , and institute the expressions of u and v into Eqs. (2.16) and (2.17), respectively, to obtain expressions of volume flow rate. qx 5 2
h3 @p uh 1 12η @x 2
(2.20)
qy 5 2
h3 @p vh 1 2 12η @y
(2.21)
According to the mass conservation law, the quality of the inflow microelement should be equal to the quality of the outflow. Hence, from Fig. 2.2, the following expression is given: mx dy dxdy 0 1 my dx 1 mmol 1 ρuz;0 0
1 @m @m x y dxAdy 1 @my 1 dyAdx 1 mmol ρuz;h dxdy 5 @mx 1 @x @y
(2.22)
2.1 Reynolds equation
where it can be considered that the two solid surfaces with velocity uz,0 and uz,h move upward, causing a change in the film thickness h. Here, (uz,0uz,h)dxdy represents the change of the volume, that is: ρuz;h 2 ρuz;0 5
@ρh @t
(2.23)
Simplification of Eq. (2.22) and institution into Eqs. (2.18) and (2.19) gives the Reynolds equation, as follows: @ h3 ρ @p @ h3 ρ @p @ðρhÞ @ðρhÞ @ðρhÞ 1 5 6u 1 6v 1 12 @x η @x @y η @y @x @y @t
(2.24)
2.1.2 Reynolds equation in the polar coordinate system In the same way as in the derivation of the rectangular coordinate system, in the polar coordinate system, the flow velocities uθ and ur in the θ and r directions, respectively, are obtained as follows: uθ 5
ωrz 1 @p 2 z 2 zh 1 2η r@θ h
(2.25)
1 @p 2 z 2 zh 2η @r
(2.26)
ur 5
Consider a unit with film thickness h, as shown in Fig. 2.3. The volume flow rate on the unit width qθ and qr in the θ and r directions, respectively, is obtained as follows: qθ 5 2
h3 @p ωrh 1 2 12η r@θ
(2.27)
rh3 @p 12η @r
(2.28)
qr 5 2
According to the mass conservation law, the Reynolds equation in the polar coordinate system is obtained. @ h3 ρ @p @ rh3 ρ @p @ðρhÞ @ðρhÞ 1 5 6ω 1 12 r@θ η r@θ r@r η @r @θ @t
(2.29)
2.1.3 Reynolds equation in the cylindrical coordinate system In the same way as in the derivation of the rectangular coordinate system, in the cylindrical coordinate system, the flow velocities uθ and uz in the θ and z directions, respectively, are obtained as follows: uθ 5
ωr0 r 1 @p 2 r 2 rh 1 2η r0 @θ h
(2.30)
19
20
CHAPTER 2 Gas lubrication equations
FIGURE 2.3 Flow rate of micro-element in the polar coordinate system.
where ω is the rotational speed. uz 5
1 @p 2 r 2 rh 2η @z
(2.31)
Consider a unit with film thickness h, as shown in Fig. 2.4. The volume flow rate on the unit width qθ and qr in the θ and z directions, respectively, is obtained as follows: qθ 5 2
h3 @p ωr0 h 1 2 12η r0 @θ
(2.32)
h3 @p 12η @z
(2.33)
qz 5 2
According to the mass conservation law, Reynolds equation in the cylindrical coordinate system is obtained. 3 @ h ρ @p @ h3 ρ @p @ðρhÞ @ðρhÞ 1 5 6ω 1 12 r0 @θ η r0 @θ @z η @z @θ @t
(2.34)
2.1.4 Lubrication parameters 1. Load w
ðð w 5 pdxdy
(2.35)
2.1 Reynolds equation
FIGURE 2.4 Flow rate of micro-element in the cylindrical coordinate system.
2. Flow rate Q In the lubrication regime, the flow rate Qx and Qy respectively in the x and y directions are expressed as ð
Qx 5
qx dy
(2.36)
qy dx
(2.37)
ð Qy 5
3. Friction force F In the x direction, the friction force of the lubricating film on the solid surfaces can be calculated by the following formula: ðð Fx;0 5 τ x z50 dxdy
(2.38a)
ðð Fx;h 5 τ x z5h dxdy
(2.38b)
where Fx,0 and Fx,h are friction force, respectively, at surface z 5 0 and z 5 h in the x direction. The shear stress of the fluid is τx 5
1 @p uη ð2z 2 hÞ 1 2 @x h
(2.39)
In the y direction, the friction force is calculated in the same way as in the x direction.
21
22
CHAPTER 2 Gas lubrication equations
4. Friction torque M For the rotating friction pair in the polar coordinate system, the friction torque of the lubricating film on the solid surfaces can be calculated by the following formula: ðð M0 5 τ z50 r 2 drdθ
(2.40a)
ðð Mh 5 τ z5h r 2 drdθ
(2.40b)
where M0 and Mh are friction torque on surface z 5 0 and z 5 h, respectively. The shear stress of the fluid is given by τ5
1 @p ωrη ð2z 2 hÞ 1 2 r@θ h
(2.41)
2.2 Energy equation In research on point and line contact of liquid thermal elastohydrodynamic lubrication on the background of bearing and gear design, surface elastic deformation of friction pair, viscositypressure effect and temperatureviscosity effect of the fluid are mainly taken into account, on the basis of isothermal hydrodynamic lubrication. In derivation of energy equation for steady-state liquid hydrodynamic lubrication, due to the application of liquid lubricated bearing as the background, it is believed that the heat exchange in fluid film is dominated by convective heat transfer, namely not considering heat exchange between lubricant film and friction pairs, that is fluid film is regarded as adiabatic in the lubrication regime [1]. For gas thermohydrodynamic lubrication on the background of gas bearing and gas seals, the contact form of friction pairs is mainly contact with low-side contact. In addition to considering the viscosity variation of the gas and the surface elastic deformation, it is necessary to consider the surface thermal deformation. So, the influence of the heat exchange on the temperature distribution of the solid surface needs to be considered. At the same time, because of requirements for gas face seal to ensure the stability of the gas film lubrication to ensure the stability of the sealing performance, so also should be considered in the energy equation of compression on the influence of the gas film lubrication temperature changes. In this section, the energy equation is further modified by the method of literature, and the variation of kinetic energy and potential energy is ignored during the derivation. The change of heat energy and mechanical work in the flow is analyzed as follows.
2.2 Energy equation
2.2.1 Chang of gas inner energy Consider a x-direction unit of width δx, with length 1 in the y direction and height h in the z direction as shown in Fig. 2.5, the gas inner energy flow in the x direction can be expressed as following. Hx 5 qx Tρcv
(2.42)
In the same way, the gas inner energy flow in the y direction can be expressed as follows: Hy 5 qy Tρcv
(2.43)
Let δx 5 1 and δy 5 1, the total heat inflowing into the unit, be given as @Hx @Hy @Hx @Hy δx 2 Hx 1 Hy 1 δy 2 Hy 5 1 Hx 1 @x @y @x @y
(2.44)
If the mechanical work performed in the microcylinder of the unit’s sectional area is represented by S, then the following relation can be obtained according to the principle of energy conservation. @Hx @Hy 1 5S @x @y
(2.45)
Substitution of Hx and Hy into Eq. (2.45) gives @ðqx TÞ @ðqy TÞ S 1 5 @x @y ρcv
(2.46)
The following equation can be obtained from the flow continuity: @qx @qy 1 50 @x @y
FIGURE 2.5 Gas inner energy flow in the x direction.
(2.47)
23
24
CHAPTER 2 Gas lubrication equations
Substitution of Eq. (2.47) into Eq. (2.46) gives qx
@T @T S 1 qy 5 @x @y ρcv
(2.48)
2.2.2 External work on gas and energy loss The changes in gas internal energy induced by external work and heat exchange include four parts: flow work, interface friction loss, heat conduction loss, and surface extrusion heat. 1. Flow work Consider an x-direction unit of width δx, as shown in Fig. 2.6; the flow work in the x direction is expressed as @p @qx δx 2 pq qx 1 p 1 δx @x @x
When high-order small quantities are ignored, and δx 5 1, the flow work in the x direction is simplified as qx
@p @qx 1p @x @x
Hence, the total flow work wflow on the unit is obtained as follows when flows in both x and y directions are considered: wflow 5 qx
FIGURE 2.6 Fluid flow in the x direction.
@p @p @qx @qy 1 qy 1p 1 @x @y @x @y
(2.49)
2.2 Energy equation
Substitution of Eq. (2.47) into Eq. (2.49) gives wflow 5 qx
@p @p 1 qy @x @y
(2.50)
2. Friction loss When a lubricating interface is seen as the driving surface, the other surface is the friction surface. Thus, the friction loss consumed in the unit is obtained. wfric 5 ðτ 0 ux;h 1 τ h ux;0 Þ 1 ðτ 0 uy;h 1 τ h uy;0 Þ
(2.51)
where h @p ηðux;h 2 ux;0 Þ h @p ηu τ 0 ux;h 1 τ h ux;0 5 2 1 ðux;h 2 ux;0 Þ 5 2 1 u 2 @x h 2 @x h h @p ηðuy;h 2 uy;0 Þ h @p ηv 1 ðuy;h 2 uy;0 Þ 5 2 1 v τ 0 uy;h 1 τ h uy;0 5 2 2 @y h 2 @y h
(2.52) (2.53)
3. Interface heat conduction loss Because the fluid in the lubrication area flows at a high-speed relative to the friction interface, there is a forced convective heat transfer between the fluid film and the solid. So, the heat loss is expressed as follows: wcon 5 kgs1 ðTs1 2 TÞ 1 kgs2 ðTs2 2 TÞ
(2.54)
where ks1 and ks2 are the convection heat transfer coefficient, and Ts1 and Ts2 are the solid surface temperature in the lubricating regime. 4. Surface extrusion heat The solid surfaces work on the gas film because of their squeezing movement in the z direction. The work can be calculated by the following formula: wp 5
@ph @t
(2.55)
Substitution of Eq. (1.1) into Eq. (2.55) gives wp 5 Ru 5 Ru Th
@ρTh @t
(2.56)
@ρ @Th 1 Ru ρ @t @t
where the first term on the right presents the work of density compression, and the second term describes the gas extrusion heat induced by the squeezing movement of solid surfaces. Hence, surface extrusion heat generated in the gas film is wh 5 Ru ρ
@Th @t
(2.57)
25
26
CHAPTER 2 Gas lubrication equations
Thus, the total work S consumed in the unit is obtained as follows: S 5 wflow 1 wfric 1 w0 con 1 wh
1 0 1 @p @p @ h @p ηuA h @p ηv u 1 @2 1 qy 1 2 1 1 Av 5 qx @x @y 2 @x h 2 @y h
(2.58)
@Th 1 kgs1 ðTs1 2 TÞ 1 kgs2 ðTs2 2 TÞ 1 Ru ρ @t
Substitution of qx and qy into Eq. (2.58) gives ηðu2 1 v2 Þ h3 2 S5 h 12η
" 2 # @p 2 @p @Th 1 kgs1 ðTs1 2 TÞ 1 kgs2 ðTs2 2 TÞ 1 Ru ρ 1 @x @y @t (2.59)
Further, substitution of Eq. (2.48) into Eq. (2.58) gives the gas energy equation as follows: 2 @T @T ηðu2 1 v2 Þ h3 4 1 qy 5 qx 2 @x @y hρcv 12ηρcv 1
!2 @p @x
!2 3 5 1 @p @y (2.60)
kgs1 kgs2 Ru @Th ðTs1 2 TÞ 1 ðTs2 2 TÞ 1 ρcv ρcv cv @t
In the same way, we obtain the gas energy equation in the polar coordinate system as follows: 2 @T @T ηω2 r 2 h3 4 1 qr 5 qθ 2 @x @y hρcv 12ηρcv 1
!2 @p r@θ
!2 3 5 1 @p @r
kgs1 kgs2 Ru @Th ðTs1 2 TÞ 1 ðTs2 2 TÞ 1 ρcv ρcv cv @t
(2.61)
2.3 Solid heat conduction equation and the interface equation For ordinary solids, the heat conduction equation, the Laplace equation, is given. kc @2 Ts @2 Ts @2 Ts @Ts 1 2 1 2 5 ρs cs @x2 @y @z @t
(2.62)
where Ts is solid temperature, kc is thermal conductivity, ρs is density of the rotor material, and cs is specific heat capacity of the rotor.
2.4 Numerical analysis method
For a friction pair of relative motion, the heat conduction equation can be changed as following because the lubrication interface has the velocity u in the x direction [2]. kc @2 Ts @2 Ts @2 Ts @Ts 5u 1 1 ρs cs @x2 @y2 @z2 @x
(2.63)
In the lubricating regime, the heat exchange condition between gas film and solid satisfies the following equations: @Ts 5 kgs1 ðTs1 2 TÞ @n s @Ts 5 ks2 ðTgs2 2 TÞ 2kc2 @n s
2kc1
(2.64a) (2.64b)
where kc1 and kc2 are thermal conductivity of two solid friction pairs, and kgs1 and kgs2 are convective heat transfer coefficient of the two interfaces, which can be calculated by following formula [3,4]: 1=3
kgs1 5 kgs2 5
3Qr kc h0
gas
(2.65)
where Qr is the Prandtl number. Qr 5
Cp η kc gas
(2.66)
where Cp is gas isobaric specific heat capacity and kc_gas is gas thermal conductivity.
2.4 Numerical analysis method From the derivation of the Reynolds equation, the balance of force and the conservation of flow follow. So, it is necessary to satisfy these two conditions when solving with numerical methods, so as to ensure that the numerical results are closer to the theoretical real value. The finite difference method is now widely used as a numerical method, and this section discusses solving the problem of the Reynolds equation using this method.
2.4.1 Finite difference method The finite difference method can be used to solve the gas lubrication Reynolds equation. However, iterative divergence often occurs in solving gas lubrication problems of large bearing number, such as hard disk magnetic head. Huang [5,6] discussed this problem and gave the finite difference scheme of the problem. For solving the energy equation, the “T” finite difference method can be adopted
27
28
CHAPTER 2 Gas lubrication equations
[1,5]. Our calculation analysis [4] showed that the “T” finite difference method is also applicable to solving the gas lubrication energy equation. Because of the geometric structure of the lubricating surface, dynamic boundary conditions must be adopted, which is discussed in subsequent chapters. For solving the solid heat conduction equation, the seven difference equations can be used, as shown in Fig. 2.7. Generally, for one point (m, n, k) in the cylindrical coordinate system, the following difference expressions of the conduction equation can be acquired [4]: @Ts
m;n;k
@r @2 Ts m;n;k Ts 5 @θ2 @2 Ts m;n;k Ts 5 @r 2 @2 Ts m;n;k Ts 5 @z2
5
Ts
m11;n;k
m;n11;k
m;n;k11
m;n;k
2 Ts Δr
m;n21;k
1 Ts m21;n;k 2 2Ts ðΔθÞ2 1 Ts m;n21;k 2 2Ts ðΔrÞ2 1 Ts m;n;k21 2 2Ts ðΔzÞ2
(2.67a) m;n;k
(2.67b)
m;n;k
(2.67c)
m;n;k
(2.67d)
2.4.2 Flow conservation For fluid lubrication bearing, the capacity of the fluid film is the main analysis parameter, not involving the flow analysis problem; thus flow conservation problems tend to be ignored, which will not produce obvious error analysis in engineering practice. However, for fluid seal, leakage rate is a very important parameter as well as open force. In theory, the leakage rate is the flow rate in the radial direction for the face seal. So, in solving the Reynolds equation, accurately
FIGURE 2.7 Illustration of finite difference grids.
2.4 Numerical analysis method
calculating flow conservation will ensure reliable results. With the application of surface texturing technology to seal surface design, the importance of flow conservation is more crucial. For example, as shown in Fig. 2.8, there are two types of dimples according to the change of depth: the continuous-depth dimples and the step-depth ones. For the continuous-depth dimples surface, the calculation error of flow conservation can be decreased to a certain extent through increasing grid density. However, for the step-depth dimples surface, flow conservation cannot be guaranteed by means of the encryption grid, that is, the flow conservation depends on the difference equation in nature, but not on mesh grid density in the numerical analysis. Huang [5,6] pointed out the flow conservation problem of the Reynolds equation by using the finite difference method, and gave a concrete difference formula. Here, from the infinitely wide Rayleigh steps, as shown in Fig. 2.9, we further discuss the flow conservation problem of the Reynolds equation by the finite difference method [7]. It is well known that when gas flows driven by shear velocity, surface step will resist the flow of the gas. This in turn will induce pressure gradient at the step, and an equivalent pressure flow must exit to keep flow continuity at the step. So, in the following derivation of this section, only pressure flow is considered, where the pressure may or may not be induced by shear velocity. The gas film temperature is assumed to be constant. For the ideal gas, the gas film pressure is obtained by solving the Reynolds equation (2.68). @ @p2 h3 50 @x @x
(2.68)
where p2 means square of pressure p. The boundary conditions are p 5 p2 at x 5 0 and p 5 p1 at x 5 B, and the pressure and the flow rate are continuous at x 5 b. The explicit solution of Eq. (2.68) is expressed as follows:
p 5 2
8 2 ps 2 p22 > > x 1 p22 > > < b
ð0 # x # bÞ
p21 2 p2s > 2 > > > : B 2 b ðx 2 bÞ 1 ps
ðb # x # BÞ
(2.69)
where ps is the pressure at the step. Thus, the flow rate in the x-direction qx can be obtained as follows: 8 3 2 h2 ps 2 p22 > > > > < 2pa b
h3 p @p h3 @p2 5 qx 5 5 h31 p21 2 p2s > pa @x 2pa @x > > > : 2pa B 2 b
ð0 # x # bÞ (2.70) ðb # x # BÞ
So, the unit linear flow rate at the step can be obtained as follows: qs 5
h32 p2s 2 p22 h3 p2 2 p2s 5 1 1 2pa b 2pa B 2 b
(2.71)
29
FIGURE 2.8 Laser textured geometry of seal face. (A) Continuous-depth dimples and (B) step-depth dimples.
2.4 Numerical analysis method
FIGURE 2.9 Infinitely wide Rayleigh step [7].
where p2s 5
h32 p22 ðB 2 bÞ 1 h31 p21 b h32 ðB 2 bÞ 1 h31 b
(2.72)
Another expression of the flow rate in the x-direction at the step is as follows: 3
q0s 5
hs p21 2 p22 2pa B
(2.73)
where hS is the equivalent clearance at the step, and the following equation is obtained: qs 5 q0s
(2.74)
Hence, 3
hs 5
h31 h32 B 3 h2 ðB 2 bÞ 1 h31 b
(2.75)
It should be noted that Eq. (2.75) is also accurate for continuous clearance. The definition of calculation grids is shown in Fig. 2.10. When a sealing surface has steps, discontinuous clearances exist. When shear velocity is considered, the Reynolds equation to solve pressure distribution is as follows: 2 @ @ph 3 @p h 5 12uη @x @x @x
(2.76)
Generally, to solve Eq. (2.76), the five-point finite difference method is often used [1]. And it may be assumed that the steps always are at the center of the two
31
32
CHAPTER 2 Gas lubrication equations
FIGURE 2.10 Grid definition of a rectangular step regime [7].
adjacent grid points, as shown in Fig. 2.10, when grid density is high enough. Hence, difference equation at the point i can be obtained as follows:
2 2
@ @p2 1 3 3 2 2 h3 5 h p 2 p h p 2 p 2 i i21 i11=2 i11 i21=2 i @x @x ðΔxÞ2 i
(2.77)
Where hi21=2 and hi11=2 are the equivalent clearance at the center of two adjacent grid points, their expressions can be obtained by simplified Eq. (2.75) when b 5 0.5B. 3
hi11=2 5 3
hi21=2 5
2h3i h3i11 h3i 1 h3i11
(2.78a)
2h3i h3i21 h3i 1 h3i21
(2.78b)
Eqs. (2.78a) and (2.78b) can keep flow continuity in numerical calculation and is the same as Huang’s difference equations in the case of the equidistant difference method. For the mechanical face seal, the leakage rate is actually the flow rate in the radial direction. According to the flow continuity principle, the leakage rate should be consistent at any radial, and its expression is as follows: Q5
ð 2π 0
p h3 @p rdθ 2 pa 12η @r
(2.79)
where pa is ambient pressure, and the standard atmospheric pressure is often used in the calculation Numerical analysis of the leakage rate of the step-dimpled face mechanical seal [7] shows that the flow conservation of Reynolds equation can be guaranteed by using the difference formula Eqs. (2.78a) and (2.78b), and the correctness of the sealing leakage rate formula Eq. (2.79) is confirmed. Fig. 2.11 presents the pressure distribution and leakage rate on the step-dimpled seal face. Correspondingly, as shown in Fig. 2.12, the leakage rate from the inside diameter
2.4 Numerical analysis method
FIGURE 2.11 Pressure distribution of the step-dimpled face mechanical seal [7].
FIGURE 2.12 Leakage rate of the step-dimpled face mechanical seal [7].
33
34
CHAPTER 2 Gas lubrication equations
to the outside diameter is calculated by using Eq. (2.79). The maximum deviation of the leakage rate from different radii is less than 3%.
2.4.3 Friction force balance As shown in Eqs. (2.38a) and (2.38b), and (2.40a) and (2.40b), different expressions of friction force or friction torque are often misleading in that the friction forces of the upper and lower surfaces are different. But, it should be pointed out that, this difference of friction expression is just a mathematical problem deduced by the coordinate system of the Reynolds equation, which does not affect the correct description of the Reynolds equation for the essential physical properties of fluid flow. In order to illustrate the problem further, the example of the stepdimpled face gas seal is still used. For the friction pairs of rotational motion, the friction balance problem is presented as the friction moment balance. The friction torque on solid surfaces is calculated by the following formula: M0 5
ð 2π ð ro 0
Mh 5 2
τ z50 r 2 drdθ
(2.80a)
ri
ð 2π ð ro 0
τ z5h r2 drdθ
(2.80b)
ri
where M0 and Mh are friction torques on surfaces of z 5 0 and z 5 h respectively. The shear stress of the fluid is τ5
1 @p ωrη ð2z 2 hÞ 1 2 r@θ h
FIGURE 2.13 Influence of the bearing number on friction torque.
(2.81)
References
Substitution of Eq. (2.81) into Eqs. (2.80a) and (2.80b) gives 1 @p ωrη 1 rdrdθ 2 r@θ h 0 ri ð 2π ð ro 1 @p ωrη h1 rdrdθ Mh 5 2 2 r@θ h 0 ri M0 5
ð 2π ð ro
2
(2.82a)
(2.82b)
The dimensionless method of friction torque is discussed in Chapter 3, Isothermal gas lubrication. Fig. 2.13 shows the calculation results of friction torque on the upper and lower surfaces of the seal friction pairs, as shown in Ref. [7]. It can be seen from the figure that the friction torque of the upper and lower surfaces is the same in value but opposite in direction. This means that friction or friction torque balance can also be used as a basis for the convergence of numerical results.
References [1] S. Wen, P. Huang, Theory of Tribology, second ed., Press of Tsinghua University, Beijing, 2002. [2] S. Wen, P. Yang, Theory of Elastohydrodynamic Lubrication, Press of Tsinghua University, Beijing, 1998. [3] A.L. San, T.H. Kim, Thermohydrodynamic analysis of bump type gas foil bearings: a model anchored to test data, J. Eng. Gas Turbines Power 132 (4) (2010) 042504. [4] S. Bai, X. Peng, Y. Meng, et al., Modeling of gas thermal effect based on energy equipartition principle, Tribol. Trans. 55 (6) (2012) 752761. [5] P. Huang, Lubrication Numerical Calculation Method, Higher Education Press, Beijing, 2012. [6] P. Huang, Y. Meng, H. Xu, Tribology Course, Higher Education Press, Beijing, 2008. [7] S. Bai, X. Peng, Y. Li, et al., Gas lubrication analysis method of step-dimpled face mechanical seals, J. Tribol. 134 (1) (2012) 011702.
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