- Email: [email protected]
Transient thermoelastohydrodynamic gas lubrication of face seals
CHAPTER
Transient thermoelastohydrodynamic gas lubrication of face seals
6
For gas-lubricated bearings and seals, the wear resistance and lubrication reliability of the friction pair surface largely depend on the stability of the lubricating gas film thickness. Studies under isothermal conditions show that the stiffness and damping of the lubricating gas film of bearings [15] and seals [6] change significantly with the external disturbance frequency. When the influence of thermal effect is taken into account, the dynamic load characteristics of the lubricating gas film are different from those of isothermal conditions. Sim [7] analyzed the thermal effect of the tilting tile gas bearing and showed that the increase of rotor temperature and gas viscosity can improve the stiffness and damping coefficient of the bearing, but the bearing is prone to thermal instability when it is not sufficiently cooled. Paulsen [8] pointed out through theoretical analysis of gas sliding bearings that the influence of gas lubrication thermal effect on the steady and dynamic performance of bearings should be considered under heavy load conditions. Under the conditions of high pressure and high speed, the pressure distribution, load capacity, and leakage rate characteristics of the sealing gas film are significantly changed because of the face deformation, fluid rheological characteristics, and the pumping effect of the face spiral groove. The reliability design of the face seal with high parameters depends on the accurate analysis and mastery of the law of dynamic load characteristics of the gas film. In this chapter, the gas spiral groove face seal is taken as the main research object, and the dynamic load characteristics of the seal gas film under the conditions of isothermal lubrication, rigid-surface thermal lubrication, and thermoelastohydrodynamic lubrication (TEHL) are emphatically discussed.
6.1 Fundamental equations For gas face seal, the modeling analysis can be simplified to two degrees of freedom, namely, axial channeling and angular oscillation. Fig. 6.1 shows the force and freedom of motion of the friction pair on the spiral groove face seal. Assuming that the stator does not move, the motion of the friction pair is reflected Gas Thermo-hydrodynamic Lubrication and Seals. DOI: https://doi.org/10.1016/B978-0-12-816716-8.00006-1 Copyright © 2019 Tsinghua University Press Limited. Published by Elsevier Inc. All rights reserved.
121
122
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.1 Force and motion freedom of the friction pair on the spiral groove face seal.
in the description of the rotor, including the equations of motion and the equations of lubrication. Following is a brief description:
6.1.1 Dynamic equations The equations of motion of the flexibly mounted rotor along the axial and angular direction can be expressed as follows: 8 ðð d2 z > > m 1 F 2 ðp 2 p0 Þrdrdθ 5 0 > c > < dt2 ðð d2 α > > > Iα 2 1 kα α0 1 ðp 2 p0 Þr 2 cosθdrdθ 5 0 > : dt
(6.1)
where m is the mass of seal rotor, Fc is the closing force acting on seal rotor, Iα is the rotational inertia, kα is the additional tilt stiffness, and α0 is the initial tilt angle.
6.1.2 Lubrication equations 1. Reynolds equation According to the derivation in Chapter 2, Gas lubrication equations, the Reynolds equation of a polar coordinate system is @ h3 ρ @p @ rh3 ρ @p @ðρhÞ @ðρhÞ 1 5 6ω 1 12 r@θ η r@θ r@r η @r @θ @t
(6.2)
6.1 Fundamental equations
2. Energy equation According to the derivation in Chapter 2, Gas lubrication equations, the energy equation of the gas film in polar coordinates is 0
1 0 1 " # 2 2 3 3 2 2 3 h @p ωrh @T h @p @T ηω r h @p @p @ A A 2 1@ 52 1 1 @r r@θ 2 r@θ 12η r@θ 12η @r @r hρcv 12ηρcv kgs1 kgs2 Ru @Th ðTs1 2 TÞ 2 ðTs2 2 TÞ 2 2 ρcv ρcv cv @t
(6.3)
3. Heat conduction equation The heat conduction equation of the seal rings in polar coordinates is @2 Ts 1@ @Ts @2 Ts ρ cs @Ts r 1 2 5 s 1 2 2 r @r @r @z kc @t r @θ
(6.4)
where kc is the heat conduction coefficient of the seal ring material, ρs is the density of the seal ring material, and cs is the specific heat of the seal ring material. The gas state equation, film thickness equation, viscosity equation, and interface equation used in the analysis are the same as those used in Chapter 5, Gas thermoelastichydrodynamic lubrication of face seals.
6.1.3 Boundary conditions The fluid and solid boundary conditions involved are the same as those in the TEHL analysis in Chapter 5, Gas thermoelastichydrodynamic lubrication of face seals.
6.1.4 Dynamic characteristic parameter The perturbation method is used to analyze dynamic characteristics of gas film. It is assumed that the displacement induced by the ambient perturbation is defined as Δh 5 a sin(ωtt), in which a is the ambient perturbation amplitude and ωt is the ambient perturbation frequency with a dimensionless of Ω 5 ωt/ω0, and ω0 is the characteristic frequency of seal rotor ring expressed as rffiffiffiffiffi k0 ω0 5 m
(6.5)
where k0 is axial gas film stiffness when the ambient perturbation frequency is zero. Further, define dimensionless parameters as follows: Z5
z ; h0
α5
αri ; h0
ΔP 5
Δp ; pa
R5
r ; ri
τ 5 ω0 t;
A5
Δh0 a
(6.6)
123
124
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
So the dimensionless additional gas film thickness ΔH induced by the ambient perturbation can be expressed by following equation: ΔH 5 Z 2 αR 2 A sinðΩτÞ
(6.7)
Dimensionless additional gas pressure ΔP induced by the ambient perturbation can be represented as follows: ΔP 5 Gz Z 1 Gα α 2 Gz A sinðΩτÞ
(6.8)
where Gz and Cα are the complex numbers, its real and imaginary part, respectively, representing stiffness and damping coefficients of gas film obtained as follows: ÐÐ 8 ΔPRdRdθ > > G 5 > z > < Z ÐÐ ΔPR cos θdRdθ > > > > Gα 5 : α
(6.9)
Kz, Kα, and Cz, Cα are dimensionless gas film stiffness and damping induced by film thickness and tilt angle, respectively, and can be computed as follows: 8 Kz 5 ReðGz Þ > > > ImðGz Þ > > C 5 > > < z Ω Kα 5 ReðGα Þ > > > > ImðGα Þ > > C > α5 : Ω
(6.10)
6.2 Dynamic characteristics of isothermal gas lubrication In this section, the dynamic characteristics of the seal gas film under isothermal lubrication are discussed, including the stiffness and damping characteristics of the axial and angular directions, as well as the variation of the amplitude and frequency characteristics.
6.2.1 Axial stiffness and damping The gas film stiffness refers to the gas film load capacity caused by the change of unit film thickness, and it is defined that the stiffness is positive when the gas film thickness decreases and the load capacity increases. According to the analysis of gas lubrication law in the previous chapter, the reduction of gas film thickness increases the load capacity; that is, the gas film stiffness is positive, which is conducive to avoiding contact wear of the seal friction pair.
6.2 Dynamic characteristics of isothermal gas lubrication
FIGURE 6.2 Distribution of additional pressure on axial gas film stiffness generated by unit vibration amplitude.
When the external disturbance is taken into account, the vibration of seal rings produces the extrusion effect on the gas film, causing the pressure of the gas film to rise, forming the increment of additional pressure distribution and changing the original pressure distribution. This additional pressure distribution is related to the disturbance amplitude and frequency of the gas film thickness. Fig. 6.2 shows the distribution of additional pressure of gas film stiffness per unit amplitude under specific vibration frequency. As the gas film thickness decreases due to compression, the additional pressure generated is positive; that is, the axial stiffness is positive. This additional pressure distribution is related to the disturbance frequency. As shown in Fig. 6.3, when the external disturbance frequency is low, the change of gas film stiffness is not obvious. When the disturbance frequency increases to a certain value, the stiffness increases rapidly with the increase of the external disturbance frequency and then tends to be stable. According to the vibration principle of mechanical system, when the compression vibration produces additional stiffness pressure, the system damping restrains the generation of additional stiffness pressure. Fig. 6.4 shows the additional pressure distribution of gas film damping generated by unit amplitude under specific vibration frequency. When the gas film thickness decreases and the stiffness is positive, the damping additional pressure generated is positive; that is, the axial damping is positive, which is beneficial to restrain the rapid increase of the stiffness value and is related to the disturbance frequency. As shown in Fig. 6.5, when the external disturbance frequency is low, the change of gas film damping is not obvious. When the frequency of external disturbances increases to a certain value, the damping decreases rapidly with the increasing frequency of external disturbances and then tends to be stable.
125
126
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.3 Frequency-domain characteristics of axial gas film stiffness.
FIGURE 6.4 Additional pressure distribution of gas film damping generated by unit vibration amplitude.
6.2.2 Angular stiffness and damping For the angular vibration of the seal rings, the angular vibration of the rotor in the lubrication area of the seal face is symmetrical with the diameter as the boundary to form two kinds of film thickness changes. Fig. 6.6 shows the additional pressure distribution of angular gas film stiffness. It can be seen that positive pressure distribution is generated in the compression region of the gas film, and negative pressure distribution is generated in the divergent region of the gas film.
6.2 Dynamic characteristics of isothermal gas lubrication
FIGURE 6.5 Frequency-domain characteristics of axial gas film damping.
FIGURE 6.6 Distribution of additional pressure on the gas film stiffness.
Fig. 6.7 shows the response curve in the frequency domain of angular stiffness of seal gas film. It can be seen that when the external disturbance frequency is low, the angular stiffness does not change significantly. When the external disturbance frequency reaches a certain value, the angular stiffness rapidly approaches zero and remains stable. The reason for this phenomenon can be explained as follows: the formation of gas film pressure distribution in the seal clearance depends on shear flow caused
127
128
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.7 Frequency-domain characteristics of angular gas film stiffness.
FIGURE 6.8 Additional pressure distribution of angular gas film damping generated by unit amplitude.
by rotation speed or pressure flow formed by seal pressure; that is, the pressure distribution of viscous fluid has timeliness. When the film thickness changes faster than the flow of viscous fluid, the pressure distribution does not change significantly; that is, the angular stiffness tends to zero. When the angular vibration produces additional stiffness pressure, the system produces the damping simulation of additional angular damping pressure. Fig. 6.8
6.2 Dynamic characteristics of isothermal gas lubrication
FIGURE 6.9 Frequency-domain response curve of angular damping.
shows the additional pressure distribution of angular gas film damping generated by unit amplitude at a specific vibration frequency. It can be seen that in the region where the gas film thickness decreases, the additional stiffness pressure is positive, and the additional damping pressure is positive. When the additional stiffness pressure is negative in the region where the gas film thickness decreases, the additional damping pressure is negative. According to the analysis of the previous section, the load capacity of the gas film is not a linear relation with the thickness of gas film. Under the condition of big gap, because the fluid flow resistance is lower and there is less time to form stable pressure distribution, the formation of convergence gap at the side with the gap increased, which is beneficial to increase the bearing capacity of gas film. On the other hand, the divergent lubrication clearance formed at the side with the gap decreased, and the load capacity decreases. Thus, negative angular damping makes the vibration increase, which is related to the vibration frequency. This means that external disturbance will cause angular self-excited vibration, which will easily lead to contact friction on the ring edge. Fig. 6.9 shows the frequency-domain response curve of the angular damping. It can be seen that when the external disturbance frequency is low, the angular damping change is not obvious. When the frequency of external disturbances is large, the value of angular damping decreases rapidly to zero as the frequency of external disturbances increases, and then remains stable.
6.2.3 Amplitude-frequency characteristics of gas film The amplitude-frequency characteristic curve of the rotor is shown in Fig. 6.10. It can be seen that when the external disturbance frequency is low, the amplitude of
129
130
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.10 Axial amplitude-frequency characteristics of isothermal gas film.
gas film is small and does not change significantly with the increase of the disturbance frequency. After reaching a certain frequency, the amplitude increases sharply with the increase of outside disturbance frequency. According to the principle of mechanical vibration, when the maximum amplitude is reached near the characteristic frequency and beyond the characteristic frequency, the forced vibration is formed; that is, the dynamic response of the seal rings disappears, and the seal face is prone to contact wear. In addition, it can be seen from Fig. 6.10 that the large seal pressure produces high gas film stiffness while the high stiffness produces small amplitude under the condition of low disturbance frequency. Therefore, for the design of the seal face, increasing the stiffness of gas film is beneficial to reducing the vibration of seal rings and improving the stability of gas film.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces When the gas is compressed, work is done to the gas, and the temperature, density, and pressure of the gas rise simultaneously. On the other hand, the gas expansion to work, temperature, density, and pressure simultaneously decline. There is a coupling effect relationship between the three variables. Therefore the change of gas film temperature is bound to lead to a change of dynamic load characteristics.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces
In this section, the thermal dynamic characteristics of rigid-surface gas film are discussed. Without considering the influence of face distortions, the axial and angular film stiffness and damping characteristics of the gas spiral groove face seal are given, as well as the variation of film amplitude-frequency characteristics.
6.3.1 Axial stiffness and damping Fig. 6.11 shows the additional temperature and pressure distribution of axial gas film stiffness generated by unit amplitude under a certain disturbance frequency. It can be seen that under the condition of dynamic load, the gas film produces an obvious additional temperature rise of stiffness, and the change of gas film
FIGURE 6.11 Additional temperature and pressure distribution of axial gas film stiffness generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
131
132
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
temperature leads to a change of additional pressure distribution of gas film stiffness. Compared with the additional pressure under isothermal conditions, the pressure distribution value increased significantly, and the pressure distribution is also significantly different. The temperature change of gas film also has a significant influence on its damping property. Fig. 6.12 shows the additional temperature and pressure distribution of angular damping generated by unit amplitude at a certain vibration frequency. It can be seen that under the dynamic loading, the gas film produces a significant increase in the additional damping temperature, and the change of the gas film temperature leads to a change of the additional pressure distribution of the gas film damping. Compared with the distribution of additional damping pressure under isothermal conditions, the pressure distribution values in the figure were significantly increased, and the distribution was also significantly different.
FIGURE 6.12 Additional temperature and pressure distribution of angular damping generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces
Fig. 6.13 shows the frequency-domain response curves of the axial stiffness and damping of rigid-surface gas film considering thermal effects. It can be seen that, compared with the isothermal calculation results in the previous section, under the condition of low vibration frequency, the thermal effect makes the gas film stiffness significantly greater than the isothermal calculation results under the condition of low vibration frequency, and it is more significant under the condition of high seal pressure.
FIGURE 6.13 Frequency-domain response curves of the axial stiffness and damping of rigid-surface gas film considering thermal effects. (A) Axial stiffness and (B) axial damping.
133
134
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
Correspondingly, under the condition of low vibration frequency, the thermal effect also makes the gas film damping significantly greater than the isothermal calculation results, and the increase is more significant under the condition of high seal pressure.
6.3.2 Angular stiffness and damping Fig. 6.14 shows the additional temperature and pressure distribution of angular gas film stiffness generated by unit amplitude under a certain disturbance frequency. It can be seen that the angular vibration of seal ring makes the seal gas film produce obvious angular stiffness additional temperature rise, and the change of gas film temperature leads to the change of additional pressure distribution of
FIGURE 6.14 Additional temperature and pressure distribution of angular gas film stiffness generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces
gas film stiffness. Compared with the additional pressure of angular stiffness under isothermal conditions, the thermal effect of gas film significantly increases the value of additional pressure, and the distribution is also significantly different, which means that the thermal effect of gas film also has a significant impact on the angular stiffness of gas film. Fig. 6.15 shows the additional temperature and pressure distribution of angular gas film damping generated by unit amplitude at a certain vibration frequency. It can be seen that the angular vibration of the seal ring makes the seal gas film produce obvious angular damping additional temperature rise and corresponding changes in the distribution of additional pressure. Compared with the angular damping additional pressure under isothermal condition, the thermal effect of gas film increases the value of additional pressure significantly.
FIGURE 6.15 Additional temperature and pressure distribution of angular gas film damping generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
135
136
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
Fig. 6.16 shows frequency-domain response curves of angular stiffness and damping of rigid-surface gas film considering thermal effects. IIt can be seen that, the thermal effect makes angular stiffness and damping present different variation rules from the isothermal results in the previous section. Considering the thermal effect of the gas film, under the condition of low vibration frequency and high seal pressure, the angular stiffness and damping are both negative, indicating that the performance of the seal gas film to resist the angular vibration is reduced. When angular damping is negative, it means that external disturbance will cause angular self-excited vibration and easily cause contact friction at the ring edge.
FIGURE 6.16 Frequency-domain response curves of angular stiffness and damping of rigid-surface gas film considering thermal effects. (A) Angular stiffness and (B) angular damping.
6.4 Dynamic characteristics of gas thermoelastohydrodynamic
FIGURE 6.17 Amplitude-frequency characteristic curves of rigid-surface gas film considering thermal effect. (A) Axial stiffness and (B) axial damping.
6.3.3 Amplitude-frequency characteristics of gas film Fig. 6.17 shows the amplitude-frequency characteristic curves of the rigid-surface seal gas film considering thermal effect. It can be seen that the axial amplitudefrequency characteristics present the same variation pattern as the isothermal condition after taking into account the thermal effect. With the increase of disturbance frequency, the maximum amplitude is reached near the feature frequency, and the forced vibration is formed after exceeding the feature frequency.
6.4 Dynamic characteristics of gas thermoelastohydrodynamic lubrication Under the condition of high seal pressure, the elastic distortion and thermal distortion of the seal faces lead to the change of seal gap and affect the pressure distribution and temperature distribution of the seal gas film, thus influencing the dynamic load characteristics of the gas film. Based on the first two sections, this section further compares and analyzes the TEHL characteristics of high-pressure gas face seals.
6.4.1 Axial stiffness and damping Fig. 6.18 shows the frequency-domain response curves of the axial stiffness and damping of seal gas film under different lubricating conditions. The calculation
137
138
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
results of three lubrication conditions are given in the figure, in which the isothermal lubrication condition does not consider thermal effect and face distortions. The thermal lubrication condition of rigid surface takes into account the thermal effect of gas film but does not consider the face distortions. The TEHL condition considers both the thermal effect of the gas film and the thermoelastic distortions of the seal faces. As can be seen from the figure, with an increase of external disturbance frequency, the thermal effect makes the axial stiffness of the gas film increase rapidly, and the axial damping of the gas film increases significantly
FIGURE 6.18 Comparison of frequency-domain response of axial stiffness and damping of seal gas film. (A) Axial stiffness and (B) axial damping.
6.4 Dynamic characteristics of gas thermoelastohydrodynamic
correspondingly. When the face distortions are further considered, the axial stiffness and damping of the gas film are smaller than that of the rigid surface. However, the gas film stiffness and damping under TEHL conditions are still significantly higher than under isothermal conditions.
6.4.2 Angular stiffness and damping of gas film Fig. 6.19 shows the comparison of frequency-domain response of the angular stiffness and damping of the gas film under different lubricating conditions. It can be seen that when the external disturbance frequency is low, the thermal effect
FIGURE 6.19 Comparison of frequency-domain response of angular stiffness and damping of seal gas film. (A) Axial stiffness and (B) axial damping.
139
140
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
causes the angular stiffness of gas film to decrease rapidly with the increase of disturbance frequency, and the obvious negative stiffness appears, which leads to the greater negative damping of angular damping. However, the angular stiffness and damping of air film are still significantly different from the isothermal working condition.
6.4.3 Amplitude-frequency characteristics of gas film Fig. 6.20 shows the comparison of the axial amplitude-frequency characteristic of the seal gas film. It can be seen from the figure that under the three lubrication conditions, the axial gas film amplitude and frequency characteristics show the same variation rule. However, in the transition interval between lowfrequency band and high-frequency band, the thermal effect reduces the axial amplitude of gas film significantly, and the amplitude after seal end face deformation has no obvious change compared with the rigid thermal lubrication condition. This is because the pressure of the gas film mainly depends on the size of the pressure flow formed by seal pressure. Compared with the shear flow, seal gap has little influence on pressure distribution of pressure flow. When the choked flow effect and entrance pressure loss are further considered, the dependence of load capacity on the change of film thickness is further weakened, which leads to the fact that the face distortions have no obvious change in the stiffness of seal gas film.
FIGURE 6.20 Comparison of axial amplitude-frequency characteristic of the seal gas film.
References
References [1] X.L. Fu, H.Y. Zhang, The analysis of gas dynamic forces of lightly loaded magnetic head, Chin. J. Comput. 6 (1992) 401407. [2] X.L. Fu, S.B. Yang, Analysis of dynamic characteristics of air-lubricated slider bearings, Chin. J. Comput. 17 (s) (1994) 5968. [3] X.L. Fu, Z. Sun, Dynamic characteristics of light-loaded slider bearing in magnetic recording disk files, ACTA Aerodyn. Sin. 12 (2) (1994) 225230. [4] Y.J. Wang, Y.F. Chen, P. Zhuang, et al., Dynamic coefficients and dynamic stability in magnetic recording, China Mech. Eng. 13 (3) (2002) 256259. [5] P. Zhuang, Y.J. Wang, Y.F. Chen, Roughness effects on dynamic response of high density disk drivers under high Knudsen number, China Mech. Eng. 12 (10) (2001) 11911194. [6] I. Green, R.M. Barnsby, A simultaneous numerical solution for the lubrication and dynamic stability of noncontacting gas face seals, J. Tribol. 123 (2) (2001) 388394. [7] K. Sim, D. Kim, Thermohydrodynamic analysis of compliant flexure pivot tilting pad gas bearings, J. Eng. Gas Turb. Power 130 (3) (2008) 032502. [8] B.T. Paulsen, S. Morosi, I.F. Santos, Static, dynamic, and thermal properties of compressible fluid film journal bearings, Tribol. Trans. 54 (2) (2011) 282299.
141
Transient thermoelastohydrodynamic gas lubrication of face seals
6
For gas-lubricated bearings and seals, the wear resistance and lubrication reliability of the friction pair surface largely depend on the stability of the lubricating gas film thickness. Studies under isothermal conditions show that the stiffness and damping of the lubricating gas film of bearings [15] and seals [6] change significantly with the external disturbance frequency. When the influence of thermal effect is taken into account, the dynamic load characteristics of the lubricating gas film are different from those of isothermal conditions. Sim [7] analyzed the thermal effect of the tilting tile gas bearing and showed that the increase of rotor temperature and gas viscosity can improve the stiffness and damping coefficient of the bearing, but the bearing is prone to thermal instability when it is not sufficiently cooled. Paulsen [8] pointed out through theoretical analysis of gas sliding bearings that the influence of gas lubrication thermal effect on the steady and dynamic performance of bearings should be considered under heavy load conditions. Under the conditions of high pressure and high speed, the pressure distribution, load capacity, and leakage rate characteristics of the sealing gas film are significantly changed because of the face deformation, fluid rheological characteristics, and the pumping effect of the face spiral groove. The reliability design of the face seal with high parameters depends on the accurate analysis and mastery of the law of dynamic load characteristics of the gas film. In this chapter, the gas spiral groove face seal is taken as the main research object, and the dynamic load characteristics of the seal gas film under the conditions of isothermal lubrication, rigid-surface thermal lubrication, and thermoelastohydrodynamic lubrication (TEHL) are emphatically discussed.
6.1 Fundamental equations For gas face seal, the modeling analysis can be simplified to two degrees of freedom, namely, axial channeling and angular oscillation. Fig. 6.1 shows the force and freedom of motion of the friction pair on the spiral groove face seal. Assuming that the stator does not move, the motion of the friction pair is reflected Gas Thermo-hydrodynamic Lubrication and Seals. DOI: https://doi.org/10.1016/B978-0-12-816716-8.00006-1 Copyright © 2019 Tsinghua University Press Limited. Published by Elsevier Inc. All rights reserved.
121
122
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.1 Force and motion freedom of the friction pair on the spiral groove face seal.
in the description of the rotor, including the equations of motion and the equations of lubrication. Following is a brief description:
6.1.1 Dynamic equations The equations of motion of the flexibly mounted rotor along the axial and angular direction can be expressed as follows: 8 ðð d2 z > > m 1 F 2 ðp 2 p0 Þrdrdθ 5 0 > c > < dt2 ðð d2 α > > > Iα 2 1 kα α0 1 ðp 2 p0 Þr 2 cosθdrdθ 5 0 > : dt
(6.1)
where m is the mass of seal rotor, Fc is the closing force acting on seal rotor, Iα is the rotational inertia, kα is the additional tilt stiffness, and α0 is the initial tilt angle.
6.1.2 Lubrication equations 1. Reynolds equation According to the derivation in Chapter 2, Gas lubrication equations, the Reynolds equation of a polar coordinate system is @ h3 ρ @p @ rh3 ρ @p @ðρhÞ @ðρhÞ 1 5 6ω 1 12 r@θ η r@θ r@r η @r @θ @t
(6.2)
6.1 Fundamental equations
2. Energy equation According to the derivation in Chapter 2, Gas lubrication equations, the energy equation of the gas film in polar coordinates is 0
1 0 1 " # 2 2 3 3 2 2 3 h @p ωrh @T h @p @T ηω r h @p @p @ A A 2 1@ 52 1 1 @r r@θ 2 r@θ 12η r@θ 12η @r @r hρcv 12ηρcv kgs1 kgs2 Ru @Th ðTs1 2 TÞ 2 ðTs2 2 TÞ 2 2 ρcv ρcv cv @t
(6.3)
3. Heat conduction equation The heat conduction equation of the seal rings in polar coordinates is @2 Ts 1@ @Ts @2 Ts ρ cs @Ts r 1 2 5 s 1 2 2 r @r @r @z kc @t r @θ
(6.4)
where kc is the heat conduction coefficient of the seal ring material, ρs is the density of the seal ring material, and cs is the specific heat of the seal ring material. The gas state equation, film thickness equation, viscosity equation, and interface equation used in the analysis are the same as those used in Chapter 5, Gas thermoelastichydrodynamic lubrication of face seals.
6.1.3 Boundary conditions The fluid and solid boundary conditions involved are the same as those in the TEHL analysis in Chapter 5, Gas thermoelastichydrodynamic lubrication of face seals.
6.1.4 Dynamic characteristic parameter The perturbation method is used to analyze dynamic characteristics of gas film. It is assumed that the displacement induced by the ambient perturbation is defined as Δh 5 a sin(ωtt), in which a is the ambient perturbation amplitude and ωt is the ambient perturbation frequency with a dimensionless of Ω 5 ωt/ω0, and ω0 is the characteristic frequency of seal rotor ring expressed as rffiffiffiffiffi k0 ω0 5 m
(6.5)
where k0 is axial gas film stiffness when the ambient perturbation frequency is zero. Further, define dimensionless parameters as follows: Z5
z ; h0
α5
αri ; h0
ΔP 5
Δp ; pa
R5
r ; ri
τ 5 ω0 t;
A5
Δh0 a
(6.6)
123
124
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
So the dimensionless additional gas film thickness ΔH induced by the ambient perturbation can be expressed by following equation: ΔH 5 Z 2 αR 2 A sinðΩτÞ
(6.7)
Dimensionless additional gas pressure ΔP induced by the ambient perturbation can be represented as follows: ΔP 5 Gz Z 1 Gα α 2 Gz A sinðΩτÞ
(6.8)
where Gz and Cα are the complex numbers, its real and imaginary part, respectively, representing stiffness and damping coefficients of gas film obtained as follows: ÐÐ 8 ΔPRdRdθ > > G 5 > z > < Z ÐÐ ΔPR cos θdRdθ > > > > Gα 5 : α
(6.9)
Kz, Kα, and Cz, Cα are dimensionless gas film stiffness and damping induced by film thickness and tilt angle, respectively, and can be computed as follows: 8 Kz 5 ReðGz Þ > > > ImðGz Þ > > C 5 > > < z Ω Kα 5 ReðGα Þ > > > > ImðGα Þ > > C > α5 : Ω
(6.10)
6.2 Dynamic characteristics of isothermal gas lubrication In this section, the dynamic characteristics of the seal gas film under isothermal lubrication are discussed, including the stiffness and damping characteristics of the axial and angular directions, as well as the variation of the amplitude and frequency characteristics.
6.2.1 Axial stiffness and damping The gas film stiffness refers to the gas film load capacity caused by the change of unit film thickness, and it is defined that the stiffness is positive when the gas film thickness decreases and the load capacity increases. According to the analysis of gas lubrication law in the previous chapter, the reduction of gas film thickness increases the load capacity; that is, the gas film stiffness is positive, which is conducive to avoiding contact wear of the seal friction pair.
6.2 Dynamic characteristics of isothermal gas lubrication
FIGURE 6.2 Distribution of additional pressure on axial gas film stiffness generated by unit vibration amplitude.
When the external disturbance is taken into account, the vibration of seal rings produces the extrusion effect on the gas film, causing the pressure of the gas film to rise, forming the increment of additional pressure distribution and changing the original pressure distribution. This additional pressure distribution is related to the disturbance amplitude and frequency of the gas film thickness. Fig. 6.2 shows the distribution of additional pressure of gas film stiffness per unit amplitude under specific vibration frequency. As the gas film thickness decreases due to compression, the additional pressure generated is positive; that is, the axial stiffness is positive. This additional pressure distribution is related to the disturbance frequency. As shown in Fig. 6.3, when the external disturbance frequency is low, the change of gas film stiffness is not obvious. When the disturbance frequency increases to a certain value, the stiffness increases rapidly with the increase of the external disturbance frequency and then tends to be stable. According to the vibration principle of mechanical system, when the compression vibration produces additional stiffness pressure, the system damping restrains the generation of additional stiffness pressure. Fig. 6.4 shows the additional pressure distribution of gas film damping generated by unit amplitude under specific vibration frequency. When the gas film thickness decreases and the stiffness is positive, the damping additional pressure generated is positive; that is, the axial damping is positive, which is beneficial to restrain the rapid increase of the stiffness value and is related to the disturbance frequency. As shown in Fig. 6.5, when the external disturbance frequency is low, the change of gas film damping is not obvious. When the frequency of external disturbances increases to a certain value, the damping decreases rapidly with the increasing frequency of external disturbances and then tends to be stable.
125
126
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.3 Frequency-domain characteristics of axial gas film stiffness.
FIGURE 6.4 Additional pressure distribution of gas film damping generated by unit vibration amplitude.
6.2.2 Angular stiffness and damping For the angular vibration of the seal rings, the angular vibration of the rotor in the lubrication area of the seal face is symmetrical with the diameter as the boundary to form two kinds of film thickness changes. Fig. 6.6 shows the additional pressure distribution of angular gas film stiffness. It can be seen that positive pressure distribution is generated in the compression region of the gas film, and negative pressure distribution is generated in the divergent region of the gas film.
6.2 Dynamic characteristics of isothermal gas lubrication
FIGURE 6.5 Frequency-domain characteristics of axial gas film damping.
FIGURE 6.6 Distribution of additional pressure on the gas film stiffness.
Fig. 6.7 shows the response curve in the frequency domain of angular stiffness of seal gas film. It can be seen that when the external disturbance frequency is low, the angular stiffness does not change significantly. When the external disturbance frequency reaches a certain value, the angular stiffness rapidly approaches zero and remains stable. The reason for this phenomenon can be explained as follows: the formation of gas film pressure distribution in the seal clearance depends on shear flow caused
127
128
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.7 Frequency-domain characteristics of angular gas film stiffness.
FIGURE 6.8 Additional pressure distribution of angular gas film damping generated by unit amplitude.
by rotation speed or pressure flow formed by seal pressure; that is, the pressure distribution of viscous fluid has timeliness. When the film thickness changes faster than the flow of viscous fluid, the pressure distribution does not change significantly; that is, the angular stiffness tends to zero. When the angular vibration produces additional stiffness pressure, the system produces the damping simulation of additional angular damping pressure. Fig. 6.8
6.2 Dynamic characteristics of isothermal gas lubrication
FIGURE 6.9 Frequency-domain response curve of angular damping.
shows the additional pressure distribution of angular gas film damping generated by unit amplitude at a specific vibration frequency. It can be seen that in the region where the gas film thickness decreases, the additional stiffness pressure is positive, and the additional damping pressure is positive. When the additional stiffness pressure is negative in the region where the gas film thickness decreases, the additional damping pressure is negative. According to the analysis of the previous section, the load capacity of the gas film is not a linear relation with the thickness of gas film. Under the condition of big gap, because the fluid flow resistance is lower and there is less time to form stable pressure distribution, the formation of convergence gap at the side with the gap increased, which is beneficial to increase the bearing capacity of gas film. On the other hand, the divergent lubrication clearance formed at the side with the gap decreased, and the load capacity decreases. Thus, negative angular damping makes the vibration increase, which is related to the vibration frequency. This means that external disturbance will cause angular self-excited vibration, which will easily lead to contact friction on the ring edge. Fig. 6.9 shows the frequency-domain response curve of the angular damping. It can be seen that when the external disturbance frequency is low, the angular damping change is not obvious. When the frequency of external disturbances is large, the value of angular damping decreases rapidly to zero as the frequency of external disturbances increases, and then remains stable.
6.2.3 Amplitude-frequency characteristics of gas film The amplitude-frequency characteristic curve of the rotor is shown in Fig. 6.10. It can be seen that when the external disturbance frequency is low, the amplitude of
129
130
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
FIGURE 6.10 Axial amplitude-frequency characteristics of isothermal gas film.
gas film is small and does not change significantly with the increase of the disturbance frequency. After reaching a certain frequency, the amplitude increases sharply with the increase of outside disturbance frequency. According to the principle of mechanical vibration, when the maximum amplitude is reached near the characteristic frequency and beyond the characteristic frequency, the forced vibration is formed; that is, the dynamic response of the seal rings disappears, and the seal face is prone to contact wear. In addition, it can be seen from Fig. 6.10 that the large seal pressure produces high gas film stiffness while the high stiffness produces small amplitude under the condition of low disturbance frequency. Therefore, for the design of the seal face, increasing the stiffness of gas film is beneficial to reducing the vibration of seal rings and improving the stability of gas film.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces When the gas is compressed, work is done to the gas, and the temperature, density, and pressure of the gas rise simultaneously. On the other hand, the gas expansion to work, temperature, density, and pressure simultaneously decline. There is a coupling effect relationship between the three variables. Therefore the change of gas film temperature is bound to lead to a change of dynamic load characteristics.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces
In this section, the thermal dynamic characteristics of rigid-surface gas film are discussed. Without considering the influence of face distortions, the axial and angular film stiffness and damping characteristics of the gas spiral groove face seal are given, as well as the variation of film amplitude-frequency characteristics.
6.3.1 Axial stiffness and damping Fig. 6.11 shows the additional temperature and pressure distribution of axial gas film stiffness generated by unit amplitude under a certain disturbance frequency. It can be seen that under the condition of dynamic load, the gas film produces an obvious additional temperature rise of stiffness, and the change of gas film
FIGURE 6.11 Additional temperature and pressure distribution of axial gas film stiffness generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
131
132
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
temperature leads to a change of additional pressure distribution of gas film stiffness. Compared with the additional pressure under isothermal conditions, the pressure distribution value increased significantly, and the pressure distribution is also significantly different. The temperature change of gas film also has a significant influence on its damping property. Fig. 6.12 shows the additional temperature and pressure distribution of angular damping generated by unit amplitude at a certain vibration frequency. It can be seen that under the dynamic loading, the gas film produces a significant increase in the additional damping temperature, and the change of the gas film temperature leads to a change of the additional pressure distribution of the gas film damping. Compared with the distribution of additional damping pressure under isothermal conditions, the pressure distribution values in the figure were significantly increased, and the distribution was also significantly different.
FIGURE 6.12 Additional temperature and pressure distribution of angular damping generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces
Fig. 6.13 shows the frequency-domain response curves of the axial stiffness and damping of rigid-surface gas film considering thermal effects. It can be seen that, compared with the isothermal calculation results in the previous section, under the condition of low vibration frequency, the thermal effect makes the gas film stiffness significantly greater than the isothermal calculation results under the condition of low vibration frequency, and it is more significant under the condition of high seal pressure.
FIGURE 6.13 Frequency-domain response curves of the axial stiffness and damping of rigid-surface gas film considering thermal effects. (A) Axial stiffness and (B) axial damping.
133
134
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
Correspondingly, under the condition of low vibration frequency, the thermal effect also makes the gas film damping significantly greater than the isothermal calculation results, and the increase is more significant under the condition of high seal pressure.
6.3.2 Angular stiffness and damping Fig. 6.14 shows the additional temperature and pressure distribution of angular gas film stiffness generated by unit amplitude under a certain disturbance frequency. It can be seen that the angular vibration of seal ring makes the seal gas film produce obvious angular stiffness additional temperature rise, and the change of gas film temperature leads to the change of additional pressure distribution of
FIGURE 6.14 Additional temperature and pressure distribution of angular gas film stiffness generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
6.3 Dynamic characteristics of thermal gas lubrication of rigid surfaces
gas film stiffness. Compared with the additional pressure of angular stiffness under isothermal conditions, the thermal effect of gas film significantly increases the value of additional pressure, and the distribution is also significantly different, which means that the thermal effect of gas film also has a significant impact on the angular stiffness of gas film. Fig. 6.15 shows the additional temperature and pressure distribution of angular gas film damping generated by unit amplitude at a certain vibration frequency. It can be seen that the angular vibration of the seal ring makes the seal gas film produce obvious angular damping additional temperature rise and corresponding changes in the distribution of additional pressure. Compared with the angular damping additional pressure under isothermal condition, the thermal effect of gas film increases the value of additional pressure significantly.
FIGURE 6.15 Additional temperature and pressure distribution of angular gas film damping generated by unit amplitude. (A) Additional temperature and (B) additional pressure.
135
136
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
Fig. 6.16 shows frequency-domain response curves of angular stiffness and damping of rigid-surface gas film considering thermal effects. IIt can be seen that, the thermal effect makes angular stiffness and damping present different variation rules from the isothermal results in the previous section. Considering the thermal effect of the gas film, under the condition of low vibration frequency and high seal pressure, the angular stiffness and damping are both negative, indicating that the performance of the seal gas film to resist the angular vibration is reduced. When angular damping is negative, it means that external disturbance will cause angular self-excited vibration and easily cause contact friction at the ring edge.
FIGURE 6.16 Frequency-domain response curves of angular stiffness and damping of rigid-surface gas film considering thermal effects. (A) Angular stiffness and (B) angular damping.
6.4 Dynamic characteristics of gas thermoelastohydrodynamic
FIGURE 6.17 Amplitude-frequency characteristic curves of rigid-surface gas film considering thermal effect. (A) Axial stiffness and (B) axial damping.
6.3.3 Amplitude-frequency characteristics of gas film Fig. 6.17 shows the amplitude-frequency characteristic curves of the rigid-surface seal gas film considering thermal effect. It can be seen that the axial amplitudefrequency characteristics present the same variation pattern as the isothermal condition after taking into account the thermal effect. With the increase of disturbance frequency, the maximum amplitude is reached near the feature frequency, and the forced vibration is formed after exceeding the feature frequency.
6.4 Dynamic characteristics of gas thermoelastohydrodynamic lubrication Under the condition of high seal pressure, the elastic distortion and thermal distortion of the seal faces lead to the change of seal gap and affect the pressure distribution and temperature distribution of the seal gas film, thus influencing the dynamic load characteristics of the gas film. Based on the first two sections, this section further compares and analyzes the TEHL characteristics of high-pressure gas face seals.
6.4.1 Axial stiffness and damping Fig. 6.18 shows the frequency-domain response curves of the axial stiffness and damping of seal gas film under different lubricating conditions. The calculation
137
138
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
results of three lubrication conditions are given in the figure, in which the isothermal lubrication condition does not consider thermal effect and face distortions. The thermal lubrication condition of rigid surface takes into account the thermal effect of gas film but does not consider the face distortions. The TEHL condition considers both the thermal effect of the gas film and the thermoelastic distortions of the seal faces. As can be seen from the figure, with an increase of external disturbance frequency, the thermal effect makes the axial stiffness of the gas film increase rapidly, and the axial damping of the gas film increases significantly
FIGURE 6.18 Comparison of frequency-domain response of axial stiffness and damping of seal gas film. (A) Axial stiffness and (B) axial damping.
6.4 Dynamic characteristics of gas thermoelastohydrodynamic
correspondingly. When the face distortions are further considered, the axial stiffness and damping of the gas film are smaller than that of the rigid surface. However, the gas film stiffness and damping under TEHL conditions are still significantly higher than under isothermal conditions.
6.4.2 Angular stiffness and damping of gas film Fig. 6.19 shows the comparison of frequency-domain response of the angular stiffness and damping of the gas film under different lubricating conditions. It can be seen that when the external disturbance frequency is low, the thermal effect
FIGURE 6.19 Comparison of frequency-domain response of angular stiffness and damping of seal gas film. (A) Axial stiffness and (B) axial damping.
139
140
CHAPTER 6 Transient thermoelastohydrodynamic gas lubrication
causes the angular stiffness of gas film to decrease rapidly with the increase of disturbance frequency, and the obvious negative stiffness appears, which leads to the greater negative damping of angular damping. However, the angular stiffness and damping of air film are still significantly different from the isothermal working condition.
6.4.3 Amplitude-frequency characteristics of gas film Fig. 6.20 shows the comparison of the axial amplitude-frequency characteristic of the seal gas film. It can be seen from the figure that under the three lubrication conditions, the axial gas film amplitude and frequency characteristics show the same variation rule. However, in the transition interval between lowfrequency band and high-frequency band, the thermal effect reduces the axial amplitude of gas film significantly, and the amplitude after seal end face deformation has no obvious change compared with the rigid thermal lubrication condition. This is because the pressure of the gas film mainly depends on the size of the pressure flow formed by seal pressure. Compared with the shear flow, seal gap has little influence on pressure distribution of pressure flow. When the choked flow effect and entrance pressure loss are further considered, the dependence of load capacity on the change of film thickness is further weakened, which leads to the fact that the face distortions have no obvious change in the stiffness of seal gas film.
FIGURE 6.20 Comparison of axial amplitude-frequency characteristic of the seal gas film.
References
References [1] X.L. Fu, H.Y. Zhang, The analysis of gas dynamic forces of lightly loaded magnetic head, Chin. J. Comput. 6 (1992) 401407. [2] X.L. Fu, S.B. Yang, Analysis of dynamic characteristics of air-lubricated slider bearings, Chin. J. Comput. 17 (s) (1994) 5968. [3] X.L. Fu, Z. Sun, Dynamic characteristics of light-loaded slider bearing in magnetic recording disk files, ACTA Aerodyn. Sin. 12 (2) (1994) 225230. [4] Y.J. Wang, Y.F. Chen, P. Zhuang, et al., Dynamic coefficients and dynamic stability in magnetic recording, China Mech. Eng. 13 (3) (2002) 256259. [5] P. Zhuang, Y.J. Wang, Y.F. Chen, Roughness effects on dynamic response of high density disk drivers under high Knudsen number, China Mech. Eng. 12 (10) (2001) 11911194. [6] I. Green, R.M. Barnsby, A simultaneous numerical solution for the lubrication and dynamic stability of noncontacting gas face seals, J. Tribol. 123 (2) (2001) 388394. [7] K. Sim, D. Kim, Thermohydrodynamic analysis of compliant flexure pivot tilting pad gas bearings, J. Eng. Gas Turb. Power 130 (3) (2008) 032502. [8] B.T. Paulsen, S. Morosi, I.F. Santos, Static, dynamic, and thermal properties of compressible fluid film journal bearings, Tribol. Trans. 54 (2) (2011) 282299.
141