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Isothermal gas lubrication
CHAPTER
Isothermal gas lubrication
3
Gas lubrication theory developed rapidly in the 1950s. In 1952 Katto and Soda [1] gave the approximate analytical solution of the bearing capacity of a sliding bearing, based on the assumption of an infinite length bearing and an isothermal gas film. A comparison with the experimental results of Ford [2] illustrated that the value of the theoretical calculation was less than that of the experimental results. In order to test the correctness of isothermal hypothesis, the bearing capacity of a gas-lubricated bearing with different specific heat capacities was further studied. The reason for the deviation between theory and experiment is attributed to leakage loss of the bearing end. Later, Ausman offered a theoretical solution for end leakage problems of sliding bearings [3] and steel-ladder thrust bearings [4]. In 1960 Elrod and Burgdorfer [5] further improved the infinite length gas bearing theory, which explained that the temperature rise effect inside the lubrication gas film under normal working conditions can be ignored. Since then, isothermal assumption has been widely accepted in the numerical analysis of gas bearings, and isothermal gas lubrication theory has been widely used in the study of gas bearings and seals. This chapter studies the typical form of a gas lubrication friction pair such as a slider, radial bearing, thrust bearing and face seal, and introduces the modeling method of isothermal gas lubrication and basic lubrication characteristics.
3.1 Sliders The slider is a fluid dynamic bearing structure. The slider bearing with length L and width B, as shown in Fig. 3.1, presents a certain pitch angle in the velocity direction and forms hydrodynamic loading capacity depending on the velocity shear, which is also the basic lubrication structure of the read-write head/memory disk of the computer hard disk. As the magnetic storage density increases, the thickness of the lubricating air film h0 decreases to the nanometer magnitude minus the free travel of gas molecules; the shear strain rate of the gas film is as high as 1010/s; and surface slip and other surface interface effects have a significant impact on the bearing capacity of the lubricating air film, all of which demonstrate the gas film lubrication theory.
Gas Thermo-hydrodynamic Lubrication and Seals. DOI: https://doi.org/10.1016/B978-0-12-816716-8.00003-6 Copyright © 2019 Tsinghua University Press Limited. Published by Elsevier Inc. All rights reserved.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.1 Finite width slider.
3.1.1 Lubrication equation Gas lubrication is roughly divided into several regions with Knudsen number Kn, which is the slip flow field when 0.01 , Kn , 0.1. When 0.1 , Kn , 10, the slip flow field is the transition area; when Kn . 10, it allows the free flow of molecules. When the gas film is thicker, the Knudsen number is smaller, and airflow can be considered as a continuous medium in the slip flow field. When the film thickness reaches nanometer level, especially when it is less than that which would allow the free travel of molecules, the rarefaction effect of lubricating gas becomes increasingly significant. The modified gas lubrication Reynolds equation is as follows: @ @p @ @p @ðphÞ Qp h3 p 1 Qp h3 p 5 6ηu @x @x @y @y @x
(3.1)
where Qp is the flow coefficient. The flow correction coefficients of different models are as follows: The flow coefficient for the classical Reynolds equation is Qp;con 5
D 6
(3.2)
where D 5 ðπ0:5 =2KnÞ 5 ðphð2Ru T0 Þ20:5 =ηÞ, λ 5 ðη=2pÞð2πRu T0 Þ20:5 , and the Knudsen number is Kn 5 ðλ=hÞ. The flow coefficient for the first-order modified Reynolds equation is Qp;1 5
pffiffiffi D c π 1 6 2
(3.3)
3.1 Sliders
where c is the adjustment factor when air molecules collide with the wall surface c 5 ð2 2 a=aÞ. A total diffuse reflection exists when a 5 0, and a mirror reflection when a 5 1. The flow coefficient for the second-order modified Reynolds equation is Qp;2 5
pffiffiffi π D π 1 1 6 4D 2
(3.4)
The flow coefficient for the Fukui model [6] is
8 D > > Qc 5 > > > 6 > > > > > D 1:0653 2:1354 > > 2 D$5 > Qpr 5 1 1:0162 1 > > 6 D D2 > > > > < 0:15653 0:00969 Qpr 5 0:13852D 1 1:25087 1 0:15 # D , 5 2 D D2 > > > > > 0:01653 0:0000694 > > 2 0:01 # D , 0:15 Qpr 5 2 2:22919D 1 2:10673 1 > > > D D2 > > > > > > > Qp 5 Qpr > > : Qc
(3.5)
Fig. 3.2 shows the change curve of the flow coefficient with the inverse Knudsen number D. As can be seen from the figure, when D is larger, the flow coefficient calculated by each correction model is close to the curve of the continuum theory, that is, the classical correction curve. However, as D decreases, the flow coefficient begins to increase, and there is a significant difference in the
FIGURE 3.2 Flow coefficient.
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CHAPTER 3 Isothermal gas lubrication
flow correction coefficient calculated by each correction model. In the above models of flow coefficient modification, the Fukui model is more accurate under the condition of nanofilm thickness (,100 nm). In numerical solution, the equation is generally treated as being dimensionless. The dimensionless form of Reynolds equation is 2 @ @P2 L @ @P2 @ðPHÞ Qp H 3 Qp H 3 1 5 2Λ @X B @Y @X @X @Y
(3.6)
where the dimensionless parameters are defined as
8 p h x y > > P5 ; H 5 ; X5 ; Y 5 > > < pa h0 L B (3.7)
6ηuL > > Λ5 2 > > : h0 pa
where pa is ambient pressure, and Λ is named as the bearing number or seal number. For the smooth surface slider, the film thickness equation is h 5 h0 1 ðL 2 xÞsinα
(3.8)
3.1.2 Pressure boundary condition After the Reynolds equation of gas bearing is solved, the forced pressure boundary condition is generally adopted. For the slider shown in Fig. 3.1, the pressure boundary condition is: p 5 pa ; when x 5 0; L; y 5 6 0:5B
(3.9)
The dimensionless form of the pressure boundary condition is: P 5 1; when X 5 0; 1; Y 5 6 0:5
(3.10)
3.1.3 Lubrication performance parameters For slider bearings, the load capacity of gas film is the most important performance parameter for characterizing the bearing performance. For the finite width slider bearing, the load capacity can be expressed as follows: w5
ð L ð B=2 0
2B=2
pdxdy
(3.11)
3.1 Sliders
The expression of dimensionless load is W5
ð 1 ð 0:5 0
20:5
PdXdY
(3.12)
The conversion coefficient is cw 5 pa LB
(3.13)
w 5 cw W
(3.14)
That is
3.1.4 Hydrodynamic lubrication characteristics of sliders As a typical hydrodynamic bearing, the load capacity of the inclined slider is mainly dependent on the dynamic pressure effect of the lubricating fluid in the convergence gap under the action of velocity shear. For the gas lubrication Reynolds equation, when the shear flow term on the right of the equation is considered, which makes it difficult to give an analytical solution of pressure distribution, the numerical method is generally adopted to solve the equation. Fig. 3.3 shows the pressure distribution of the slider gas film. It can be seen from the figure that during the flow from the inlet to the outlet, as the gas film thickness gradually decreases, the gas is continuously squeezed under shear action, and the gas film pressure gradually increases, forming a dynamic pressure effect. The seal gas film pressure rapidly decreases near the outlet, giving the lubricating gas film a certain bearing capacity.
FIGURE 3.3 Pressure distribution of the slider gas film.
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For the slider bearings, the load capacity is mainly related to the pitch angle, film thickness, and shear velocity, and the change law of the slider load capacity is analyzed below. 1. Pitch angle Fig. 3.4 shows the influence of the pitch angle on the load capacity. It can be seen that the load capacity of gas film changes nonmonotonically with the increase of pitch angle. The load capacity increases first with the increase of the pitch angle. When the maximum value is reached, it will take effect quickly until the load capacity is lost. This means that one load capacity value corresponds to two pitch angle values, which will easily lead to unstable conditions for the design of the slider bearing. The reason is that with the increase of pitch angle, the shear rate of gas film decreases, which is not conducive to the formation of the hydrodynamic effect. On the other hand, the increase of pitch angle leads to an increase of gradient of gas film thickness, and the enhancement of convergence is conducive to the enhancement of extrusion effect and the increase of gas film pressure. The combination of the two factors leads to the extreme value on the bearing capacity curve. 2. Film thickness Fig. 3.5 shows the influence of the film thickness on the load capacity. As shown in the figure, the load capacity shows a monotonically decreasing trend with the increase of film thickness. As the viscosity of gas is low, when the film thickness increases gradually from nanometer to micron, the load capacity
FIGURE 3.4 Influence of the pitch angle on the load capacity.
3.1 Sliders
FIGURE 3.5 Influence of the film thickness on the load capacity.
decreases rapidly. In other words, under low shear rate, the hydrodynamic load capacity is significantly reduced, and the hydrodynamic load capacity with micron thickness is much lower than that of nanometer ultra-thin film lubrication. Conversely, the smaller the film thickness, the greater the hydrodynamic load capacity of the gas film. 3. Velocity Shear speed is one of the main factors affecting load capacity of slider bearings. In general, the higher the velocity, the greater the hydrodynamic load capacity. Fig. 3.6 shows the influence curve of velocity on the load capacity. It can be seen that the load capacity increases monotonically with the increase of velocity. However, when the velocity increases to a certain value, the load capacity increases slowly and approaches the limit value, which means that there is a load limit for the slider bearings.
3.1.5 Hydrodynamic lubrication characteristics of divergent sliders Conventional lubrication analysis of slider bearings mainly studies the convergence gap that can form a hydrodynamic effect, and often neglects the divergent gap lubrication law. With more and more geometrical structures applied to control the lubrication performance of the bearing surface, the influence of divergent clearances on fluid motion and lubrication characteristics is indispensable.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.6 Influence curve of velocity on the load capacity.
FIGURE 3.7 Diagram of divergent slider.
As shown in Fig. 3.7, the thickness of the lubricating film increases gradually along the direction of shear velocity, which is called the divergent slider. Theoretically, for divergent clearances, shear flow tends to increase along the direction of shear velocity because of the increase in the thickness of the smooth film, and there is bound to be a codirectional pressure flow to meet the conservation of flow in the lubrication area, so the pressure decreases rather than increases.
3.1 Sliders
FIGURE 3.8 Pressure distribution of the divergent slider.
Fig. 3.8 shows the pressure distribution of the divergent slider. It can be seen from the figure that the gas film pressure in the lubrication area is lower than the ambient pressure, that is, the lubrication film presents negative pressure distribution, and the gas film pressure value is the lowest near the outlet, which also means that the bearing capacity of the divergent slider is negative. To further analyze the load characteristics of the divergent slider, the variation curve of the load with the film thickness under different pitch angles is given in Fig. 3.9. It can be seen from the figure that, similar to the variation trend of the convergent slider, the load capacity of the divergent slider shows a monotonous decreasing trend with the increase of thickness. Theoretically, as the minimum value of gas pressure is a vacuum state, the minimum limit value of the lubricating gas film pressure of the divergent slider is 0, that is, the vacuum state. Therefore the load capacity of the divergent slider also has a limit value in the numerical value.
3.1.6 Lubrication characteristics of the magnetic head slider With the development of precision machinery and equipment, load capacity and accurate control of the bearings have become increasingly demanding but cannot be realized by controlling the bearing inclination angle alone. Surface groove technology has been widely used in the surface design of gas-lubricated friction pairs.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.9 Effect of film thickness on load capacity of the divergent slider.
The magnetic head/hard disk system is a typical application of gas-inclined slider bearing. In order to ensure the reliability of the hard disk, it is necessary to precisely control the attitude and gas film thickness of the slider bearings, and its realization mainly depends on the geometric structure of the bearing surface [710]. From the previous discussion, it can be seen that under the action of velocity shear, the convergent gap produces hydrodynamic load capacity, whereas the divergent gap produces negative load capacity. Through the geometrical structure, the magnetic head slider can maintain the stability of flight attitude in the state of nanofilm thickness. This magnetic head is called a negative pressure-type magnetic head. Here the influence law of the surface geometry structure on the gas lubrication characteristics is discussed by taking two typical slider structures of the magnetic head, as shown in Fig. 3.10, that is: Slider 1 is a dual-track magnetic head with relatively simple surface geometry, including only two parallel symmetric rectangular convex platform structures; Slider 2 is a negative pressure magnetic head with a relatively complex structure. The front end is a C-shaped convex platform, and three convex platform structures are close to the end. A concave region is formed in the middle of the sliding block; that is, the film thickness forms a divergent region. Fig. 3.11 shows the gas film pressure distribution of the two magnetic head sliders. As can be seen from the figure, because of the simple structure of Slider 1, which is equivalent to two parallel oblique slider structures, two pressure peaks
3.1 Sliders
FIGURE 3.10 Geometrical structure of magnetic head slider surface [8]. (A) Slider 1 and (B) Slider 2.
FIGURE 3.11 Gas film pressure distribution of magnetic head sliders. (A) Slider 1 and (B) Slider 2.
appear. Moreover, because of the convergence of film thickness in the lubricating area along the shear direction of velocity, only shear squeeze phenomenon exists in the process of gas flow, so the gas film pressure is higher than the ambient pressure.
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CHAPTER 3 Isothermal gas lubrication
For Slider 2, when the shear flow crosses the front end of the C-shaped convex platform, the film thickness in the lubrication area increases suddenly and presents local divergent clearances. According to the analysis in the previous section, negative pressure distribution can be easily formed for divergent clearances, especially under the condition of high shear rate. Therefore a large area of negative pressure distribution is smaller than the ambient pressure in the central region of the slider, and the convex platform forms multiple pressure peak distributions. Slider 2 is a kind of negative pressure gas bearing, which is mainly discussed in the following section. 1. Pitch angle The surface pressure distribution of the magnetic head slider is related to the geometric structure of the magnetic head surface, and the change of the pitch angle of the slider further changes the distribution of the film thickness, leading to further changes in the gas film pressure distribution and load capacity. As shown in Fig. 3.12, with an increase of pitch angle, the influence of the surface structure of the negative pressure magnetic head slider on the pressure distribution on the magnetic head slider’s surface becomes weaker and weaker. As a result, the magnetic head slider gradually degenerates into a slanted slider with limited width, and the negative pressure area disappears.
FIGURE 3.12 Influence of pitch angle on pressure distribution of negative head slider. (A) α 5 0 μrad, w 5 5.26 g, (B) α 5 30 μrad, w 5 1.68 g, (C) α 5 100 μrad, w 5 0.12 g, and (D) α 5 800 μrad, w 5 0.30 g.
3.1 Sliders
FIGURE 3.13 Influence of the pitch angle on load capacity of negative pressure sliders.
Fig. 3.13 shows the influence curve of the pitch angle on the load capacity of the negative pressure sliders. It can be seen from the figure that the bearing capacity decreases first with the increase of pitch angle, and then the second peak value appears. By comparing the influence curve of the pitch angle on the load capacity of the smooth slider in Fig. 3.4, it can be seen that the surface geometry makes the load variation rule of the slider more complicated. In addition, the geometrical structure of the surface enables the slider to generate large load capacity under small pitch angle. With the increase of pitch angle, the load capacity appears the second peak value, but the value of the second peak value is far less than that of the small pitch angle. 2. Film thickness Fig. 3.14 gives the influence curve of gas film thickness on the load capacity. It can be seen from the figure that the negative pressure slider presents a change rule similar to the smooth slide, that is, the load capacity decreases monotonously with the increase of film thickness. When the film thickness gradually increases from nanometer to micron, the load capacity decreases rapidly. It should be pointed out that the negative load of the magnetic head slider occurs within a certain range of film thickness variation, which is fundamentally different from the smooth surface slider. For the head slider, negative value means that the two surfaces of the friction pair present trend of movement toward each other. In other words, the geometric structure design of the negative pressure surface is conducive to the stability of the head/disk motion gap, thus ensuring the reliability of the head data reading and writing.
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FIGURE 3.14 Effect of film thickness on load capacity.
FIGURE 3.15 Variation of load capacity of the negative pressure slider with speed.
3. Velocity Fig. 3.15 shows the variation curve of the load capacity of the negative pressure slider with the speed. As can be seen from the figure, the load capacity of the negative pressure slider is not only small but also presents a complex law of nonmonotonic variation with the increase of velocity, compared with the
3.2 Journal bearing and radial seal
monotonic increase of the load of the smooth slider. Because of the existence of the negative pressure zone, when the velocity reaches a certain value, the load capacity of the gas film decreases, and under the condition of a large pitch angle, the phenomenon of negative load appears.
3.2 Journal bearing and radial seal The geometric structure of the radial bearing, which is mainly composed of stator and rotor, is shown in Fig. 3.16. When the rotor is eccentric, the sliding bearing structure can also be formed, but because of the low viscosity of the gas, the load capacity of hydrodynamic pressure is low. Therefore gas-lubricated radial bearings are widely used in engineering practice in the form of static bearing. As the pressure po at the inlet is greater than the pressure pi at the outlet, the gas enters the gap between the stator and rotor through the inlet and leaks out from the outlet. The rotor and stator are separated by the load capacity of the gas film in the clearance to prevent contact wear of the bearing. When the rotor has a certain eccentricity re, the gas film forms a certain hydrodynamic pressure distribution during rotation, which has an impact on the motion of the rotor. On the other hand, the geometrical structure is also the typical structure form of the radial gas seal. What is different is that load capacity of the gas film is mainly considered in design of the bearing, and the leakage rate of gas in clearance is an important index parameter to be considered in seal design. This section mainly focuses on the structure of radial bearings and discusses their gas lubrication analysis model and lubrication characteristics. Table 3.1 gives the main geometric structure parameters of radial bearing and gives the values of calculation examples.
FIGURE 3.16 Structure diagram of the radial bearing (seal).
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Table 3.1 Geometric structure parameters of radial bearings and calculation values. Parameter
Symbol
Data
Outside radius of stator Radius of the rotor Eccentric radius Width Film thickness Rotation speed Outlet pressure
ro ri re L h0 Ω pi
55 mm 30 mm 0 30 mm 550 μm 030,000 rpm 0.101 MPa
3.2.1 Lubrication equations For the radial bearing structure shown in Fig. 3.16, the gas lubrication Reynolds equation in the cylindrical coordinate system is used in the analysis. 3 @ ρh @p @ ρh3 @p @ρh 5 6ω 1 @z η @z @θ ri2 @θ η ri @θ
(3.15)
The dimensionless Reynolds equation is @ @P2 ri 2 @ @P2 @ðPHÞ 1 5 2Λ H3η H3η @θ L @Z @Z @θ @θ
(3.16)
where the dimensionless parameters are defined as
8 p h z η > > P5 ; H 5 ; Z 5 ; η5 > > pa h0 L η0 < 6η r 2 ω > > Λ 5 02 i > > : h0 pa
(3.17)
where η0 is initial viscosity of fluid. The equation of film thickness is hðθ; zÞ 5 h0 1 hg ðθ; zÞ 1 re sinðθ 2 θe Þ
(3.18)
where hg is the depth distribution of the geometric groove on the bearing cylinder. In engineering practice, to enhance the load capacity of the lubricating gas film, different dynamic pressure groove structures such as spiral grooves can be designed and processed on the working face of rotor or stator.
3.2.2 Boundary conditions Imposed pressure boundary conditions are generally applied as follows: p 5 pi ; when z 5 0
(3.19a)
p 5 po ; when z 5 L
(3.19b)
3.2 Journal bearing and radial seal
3.2.3 Lubrication parameters 1. Load capacity of gas film When the radial bearing presents as a sliding bearing and the rotor appears eccentric, the horizontal load component of the gas film is Fh 5
ð L ð 2π 0
ðp 2 pa Þri cosθdθdz
(3.20a)
0
The vertical load component of the gas film is ð L ð 2π
Fv 5
0
ðp 2 pa Þri sinθdθdz
(3.20b)
0
So, the load capacity of the gas film is F5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fh2 1 Fv2
(3.20c)
The dimensionless form of the load capacity is ð 1 ð 2π
Fh 5 Fv 5
0
0
0
0
ð 1 ð 2π
F5
ðP 2 1ÞcosθdθdZ
(3.21a)
ðP 2 1ÞsinθdθdZ
(3.21b)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Fh 1 Fv
(3.21c)
The conversion coefficient is cF 5 pa ri L
(3.22)
F 5 cF F
(3.23)
That is,
2. Leakage rate According to the derivation of Chapter 2, Gas lubrication equations, for radial bearings, the mass leakage rate is Q5
ð 2π 0
h3 @p mmol ρ 2 ri dθ 12η @z
(3.24)
The dimensionless form of the leakage rated is Q5
ð 2π
2H 3 η
0
@P2 dθ @Z
(3.25)
Define the conversion coefficient: cQ 5
mmol ρ0 pa h30 ri 24η0 L
(3.26)
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The following expression can be obtained: Q 5 cQ Q
(3.27)
3. Frictional force The horizontal component of the frictional force is Ffric;h 5
ð L ð 2π 0
2
0
@p h ωri 1η ri cosθdθdz ri @θ 2 h
(3.28a)
The vertical component of the frictional force is Ffric;v 5
ð L ð 2π 0
2
0
@p h ωri 1η ri sinθdθdz ri @θ 2 h
(3.28b)
So, we can obtain the expression of the frictional force as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Ffric;h 1 Ffric;v
Ffric 5
(3.28c)
The dimensionless horizontal component of the frictional force is F fric;h 5
ð 1 ð 2π 0
2
0
@P H η ωr 2 η 1 0 2i cosθdθdZ @θ 2 pa h0 H
(3.29a)
The dimensionless vertical component of the frictional force is F fric;v 5
ð L ð 2π 0
2
0
@P H η ωr2 η 1 0 2i sinθdθdZ @θ 2 pa h0 H
(3.29b)
So, we can obtain the dimensionless expression of the frictional force as follows: F fric 5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 F fric;h 1 F fric;v
(3.29c)
Define the conversion coefficient: cFricF 5 pa h0 L
(3.30)
The following expression can be obtained: Ffric 5 cFricF F fric
(3.31)
4. Friction torque The friction torque can be expressed as Mfric 5
ð L ð 2π 0
0
2
@p h ωri 2 ri dθdz 1η h ri @θ 2
(3.32)
The dimensionless form of the friction torque is M fric 5
ð 1 ð 2π 0
0
@P H η0 ωri2 η 1 dθdZ 2 @θ 2 pa h20 H
(3.33)
3.2 Journal bearing and radial seal
Define the conversion coefficient: cM 5 pa h0 ri L
(3.34)
The following expression can be obtained: Mfric 5 cM M fric
(3.35)
5. Average pressure For hydrostatic radial bearings or seals, because the rotor is not eccentric in theory, the load of the gas film is 0. In this case, the performance of the gas film can be characterized by the average pressure, which is defined as ðð
prdθdr pav 5 ðð rdθdr
(3.36)
The dimensionless average pressure is Pav 5
pav pa
(3.37)
3.2.4 Lubrication characteristics Fig. 3.17 shows the pressure distribution of radial bearings. In hydrostatic bearing or radial seals, the pressure of gas film is mainly caused by pressure flow,
FIGURE 3.17 Pressure distribution of radial bearing.
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CHAPTER 3 Isothermal gas lubrication
and the pressure distribution presents a convex profile because of the compressibility of gas. Under the condition of imposed pressure boundary, the influence of gas film thickness on pressure distribution is not obvious. As the film thickness increases, the pressure distribution of the gas film increases slightly. As the load capacity of the gas film depends on the pressure distribution, when the film thickness increases, the flow resistance of the fluid decreases, and the highpressure distribution is shifted toward the outlet, forming a higher load capacity of the gas film. The parameters affecting lubrication performance for hydrostatic radial bearings mainly include gas film thickness and inlet pressure, which are discussed and analyzed below. 1. Film thickness Fig. 3.18 shows the influence of film thickness on lubrication parameters of radial bearings. As can be seen from the figure, as the film thickness increases, the average pressure of the gas film increases, which means that the rigidity of the gas film increases, which is conducive to the stability of the rotor of static bearing. The increase of leakage rate means an increase of gas consumption. As shear rate decreases, friction torque decreases and power consumption decreases. 2. Inlet pressure Fig. 3.19 shows the influence curves of inlet pressure on performance of radial bearings. As can be seen from the figure, with the increase of inlet pressure, the average pressure of the gas film shows a linear increase trend, which means that the gas film has higher rigidity, which is conducive to the stability of the rotor of hydrostatic bearing. The increase of leakage rate means the increase of gas consumption. Meanwhile, both friction torque and power consumption increase.
3.3 Spiral groove thrust bearing The structure of spiral groove thrust bearing is shown in Fig. 3.20, which is composed of thrust bearing and the rotor. The face of thrust bearing is designed with a spiral groove structure. When working under the shear action of rotating speed, the spiral groove of thrust bearing surface produces the fluid hydrodynamic load capacity, which plays a supporting role on the rotor. This section mainly focuses on the spiral groove thrust bearing structure and discusses its gas lubrication analysis model and lubrication characteristics law. Table 3.2 gives the main geometric structure parameters of the bearing and gives the value of the calculation example.
3.3 Spiral groove thrust bearing
FIGURE 3.18 Influence of film thickness on lubrication parameters of radial bearings.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.19 Influence curves of inlet pressure on performance of radial bearings.
3.3 Spiral groove thrust bearing
FIGURE 3.20 Structure diagram of spiral groove thrust bearing.
Table 3.2 Geometric structure parameters of spiral groove thrust bearing and data. Parameter
Symbol
Data
Outer radius Inner radius Groove radius Groove number Spiral angle Groove depth
ro ri rg N β hg
75 mm 60 mm 65 mm 12 15 degrees 5 μm
3.3.1 Gas lubrication equations According to the derivation presented in Chapter 2, gas lubrication equations, the Reynolds equation in polar coordinates can be used for gas lubrication of thrust bearings. @ ρh3 @p @ rρh3 @p @ρh 1 5 6ω r@θ η r@θ r@r @θ η @r
(3.38)
Its dimensionless form is @ @P2 @ @P2 @ðPHÞ 1 5 2Λ H3η RH 3 η R@θ @R R@θ R@R @θ
where
(3.39)
8 p h r η > > >P5 ; H 5 ; R5 ; η5 > pa h0 ri η0 < > > > > :
Λ5
6η0 ri2 ω h20 pa
(3.40)
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CHAPTER 3 Isothermal gas lubrication
The equation of film thickness is hðr; θÞ 5 h0 1 hg ðr; θÞ
(3.41)
where hg represents the depth distribution of the spiral groove on the end face of the thrust bearing, hg(r, θ) 5 hd in the grooved area and hg(r, θ) 5 0 in the smooth area.
3.3.2 Pressure boundary conditions Imposed pressure boundary conditions can be used when the Reynolds equation is solved. pðr 5 0:5do ; θÞ 5 pa ; pðr 5 0:5di ; θÞ 5 pa
(3.42)
In order to reduce the calculation, a spiral groove periodic structure is usually adopted for analysis, and the following periodic pressure boundary conditions need to be applied. π 2 π 5 p r; θ 5 p r; θ 5 N N
(3.43)
3.3.3 Lubrication parameters 1. Load capacity The load capacity can be calculated by the following expression: F5
ð ro ð 2π ri
ðp 2 pa Þrdθdr
(3.44)
0
The dimensionless form of the load is F5
ð Ro ð 2π 1
ðP 2 1ÞRdθdR
(3.45)
0
Define the conversion coefficient cF 5 pa ri2
(3.46)
The following expression is obtained: F 5 cF F
(3.47)
2. Friction torque The friction torque can be calculated by the following expression: Mfric 5
ð ro ð 2π ri
0
@p h ωr 2 1η r dθdr r@θ 2 h
(3.48)
3.3 Spiral groove thrust bearing
The dimensionless form of the friction torque is M fric 5
ð 1 ð 2π 0
0
@P 2η ωr 2 η H1 0 2i dθdR R@θ pa h0 H
(3.49)
Define the conversion coefficient: cM 5
pa h0 ri2 2
(3.50)
The following expression is obtained: Mfric 5 cM M fric
(3.51)
3.3.4 Lubrication characteristics On the spiral groove thrust bearing face, because fluid motion resistance is small along the channel direction film thickness, gas flows along the spiral groove under the effect of circumferential shear velocity, accumulating extrusion, and the film pressure rises, leading to a high-pressure peak at the spiral groove root and forming the hydrodynamic load capacity, as shown in Fig. 3.21. For specific spiral groove structure, the main factors affecting thrust bearing load capacity include rotating speed and film thickness. 1. Rotation speed When the film thickness is given, the bearing number directly reflects the change of speed. The larger the number of bearings, the greater the speed.
FIGURE 3.21 Gas film thickness and pressure distribution of spiral groove thrust bearing.
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Fig. 3.22 shows the variation curve of the load and friction torque of the spiral groove thrust bearing with the number of bearings. As can be seen from the figure, the load capacity of the gas film increased rapidly with an increase of the bearing number. That is, with an increase of the bearing speed, the load capacity of the gas film increases rapidly, and the spiral groove shows a remarkable hydrodynamic effect. On the other hand, the dimensionless friction torque increases rapidly with an increase of the rotating speed.
FIGURE 3.22 Influence of rotation speed on lubrication performance of spiral groove thrust bearing.
3.3 Spiral groove thrust bearing
2. Film thickness Fig. 3.23 shows the influence of film thickness on the lubrication parameters of the spiral groove thrust bearing. As can be seen from the figure, as the film thickness increases, the load capacity of the gas film peaks in the value of specific small film thickness, and then decreases rapidly. On the whole, under the condition of large film thickness, the hydrodynamic effect of the fluid decreases as the shear rate of the fluid decreases, and the load capacity of the gas film is low.
FIGURE 3.23 Influence of film thickness on lubrication performance of spiral groove thrust bearing.
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CHAPTER 3 Isothermal gas lubrication
On the other hand, as film thickness increases, the shear rate of the gas film decreases and the dimensionless friction torque decreases. Because the friction torque of thrust bearing is mainly formed by velocity shear, it can be seen from Eq. (3.48) that the friction torque component formed by velocity shear is inversely related to film thickness. However, because the dimensionless friction torque is directly proportional to the film thickness, the dimensionless friction torque shows the opposite change trend, and the change trend increases with an increase of film thickness.
3.3.5 Spiral groove face seal The spiral groove face seal is the most typical and representative gas face seal. As shown in Fig. 3.24, its friction pair is mainly composed of a rotor ring and a static ring, with hydrodynamic spiral grooves on the face of the rotor ring or the static ring. In terms of working principle, the main difference from the spiral groove thrust bearing is that there is a gas pressure difference between the inner and outer diameter of face seal; that is, the seal pressure forms a radial pressure flow, whereas the inner and outer diameter of thrust bearing generally does not have a radial pressure difference. When the ambient pressure po at the outer diameter is greater than the environmental pressure pi at the inner diameter, the inner flow end face seal is formed; otherwise, it is called the outer flow seal. Because radial pressure flow and circumferential shear flow exist in the lubrication region of the end face seal, the end face seal presents different lubrication characteristics from those of the bearing.
FIGURE 3.24 Diagram of gas face seal.
3.3 Spiral groove thrust bearing
Table 3.3 Geometric parameters of face seal and data. Parameter
Symbol
Data
Outer radius Inner radius Balance radius Grooved radius Ring thickness Groove number Spiral angle Groove depth
ro ri rb rp h1, h2 N β hd
75 mm 55 mm 63.5 mm 65 mm 10 mm 12 575 degrees 6 μm
Geometrically, there are also differences between the face seal and the thrust bearing, mainly because the face seal friction pair has a balanced radius structure, as shown in Fig. 3.24. Gabriel [11] studied and discussed the fundamental characteristics of the gas spiral groove face seal. The main purpose of this section is to introduce the gas lubrication analysis model and lubrication characteristics of spiral groove face seal under isothermal conditions. Table 3.3 gives the main geometric structure parameters of the seal and the value of the calculation example.
3.3.6 Lubrication equations As with the thrust bearings, see Eqs. (3.38) and (3.39).
3.3.7 Pressure boundary conditions For the low-pressure seal condition, the pressure boundary condition generally adopts the imposed boundary condition, that is, pðr 5 ri Þ 5 pi
(3.52a)
pðr 5 ro Þ 5 po
(3.52b)
Because of the periodic distribution of seal rings in the circumference direction, in order to reduce the calculation amount, one cycle is generally taken in the numerical calculation, so the following periodic pressure boundary conditions exist in the calculation area: π 2π p r; θ 5 5 p r; θ 5 N N
(3.53)
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CHAPTER 3 Isothermal gas lubrication
3.3.8 Seal performance parameters The opening force produced by the seal gas film is the load capacity, which can be calculated by the following formula: Fo 5
ð 2π ð ro prdrdθ 0
(3.54)
ri
In subsequent analyses in this book, the load capacity of the seal gas film is expressed in terms of the opening force, consistent with the terminology used in mechanical seals. Under steady-state conditions, the radial flow rate (leakage rate) of the clearance between stator and rotor rings meets the mass conservation; that is, the radial flow rate integral at any radius is the same, so the formula for calculating the mass leakage rate can be written as Q5
ð 2π 0
h3 @p mmol ρ 2 rdθ 12η @r
(3.55)
The calculation formula of volume leakage rate is Q5
ð 2π h3 @p rdθ P 2 12η @r 0
(3.56)
The formula of friction torque is the same as that of thrust bearing.
3.3.9 Lubrication regularity As the gas viscosity is low, the hydrodynamic pressure formed by circumferential shear is less than the seal pressure. Generally, the sealing gas film pressure is dominated by the pressure flow formed by seal pressure. As shown in Fig. 3.25, the pressure of the gas film decreases gradually from the outer diameter to the inner diameter, and the pressure distribution presents generally a convex profile because of the compressibility of the gas. However, under the conditions of low seal pressure, high speed, and small film thickness, the spiral groove still presents an obvious hydrodynamic effect. At the same time, as the gas flows along the spiral groove, the pumping effect on the gas is formed, which increases the radial flow rate. For the spiral groove face seal, the factors that affect the seal gas film lubrication characteristics mainly include film thickness, rotation speed, and seal pressure. 1. Film thickness Fig. 3.26 shows the influence of film thickness on the performance of the spiral groove face seal. As can be seen from the figure, as the film thickness
3.3 Spiral groove thrust bearing
FIGURE 3.25 Distributions of film thickness and pressure of the spiral groove face seal.
increases, the opening force, that is, the load capacity of the gas film, decreases rapidly. Under the condition of small film thickness, the higher the rotation speed, the greater the opening force. On the other hand, the leakage rate increases rapidly with the increase of film thickness. At the same time, because the radial pumping effect of spiral groove, the shear rate increases the leakage rate. Dimensionless friction torque decreases rapidly with increasing film thickness. 2. Rotation speed Fig. 3.27 shows the influence of rotation speed on the performance of the spiral groove face seal. As can be seen from the figure, as the rotation speed increases, the opening force, leakage rate, and friction torque all show a linear increase trend. The higher the rotation speed, the higher the shear rate of the fluid, and the stronger the hydrodynamic effect and pumping effect of the spiral groove, leading to an increase of the opening force, leakage rate, and friction torque. 3. Seal pressure Fig. 3.28 shows the influence of seal pressure on the performance of the spiral groove face seal. As can be seen from the figure, the opening force and leakage rate increase rapidly with the increase of seal pressure. However, it should be noted that the friction torque does not change significantly with the increase of seal pressure. As can be seen from Eq. (3.48), friction torque is mainly formed by circumferential velocity shear and circumferential pressure flow. As radial pressure flow dominates the seal, especially as the seal pressure increases, the proportion of shear flow decreases, resulting in an insignificant change in friction torque.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.26 Influence of film thickness on performance of spiral groove face seal.
3.3 Spiral groove thrust bearing
FIGURE 3.27 Influence of rotation speed on performance of spiral groove face seal.
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FIGURE 3.28 Influence of seal pressure on the performance of the spiral groove face seal.
References
References [1] Y. Katto, N. Soda, Theory of lubrication by compressible fluid with special reference to air bearing, in: Proceedings of the Second Japan National Congress for Applied Mechanics, Tokyo, 1952, pp. 267270. [2] G.W.K. Ford, D.M. Harris, D. Pantall, Gas-lubricated bearings, Mach. Mark. 182 (2924) (1956) 2630. [3] J. Ausman, Finite gas-lubricated journal bearing, in: Proceedings of the Conference on Lubrication and Wear.The Institute of Mechanical Engineers, London, 1957, pp. 3945. [4] J.S. Ausman, An approximate analytical solution for self-acting gas lubrication of stepped sector thrust bearings, Tribol. Trans. 4 (2) (1961) 304313. [5] H.G. Elrod, A. Burgdorfer, Refinements of the theory of the infinitely-long, selfacting, gas-lubricated journal bearing, in: Proceedings of the International Symposium on Gas-Lubricated Bearing, 1960, pp. 93118. [6] S. Fukui, R. Kaneko, Analysis of ultra thin gas film lubrication based on linearized Boltzmann equation: first report-derivation of a generalized lubrication equation including thermal creep flow, J. Tribol. 110 (2) (1988) 253262. [7] P. Huang, L.G. Xu, Y.G. Meng, et al., Calculation and analysis on dynamic performances for ultra-thin gas film lubrication of magnetic head/disk, Lubr. Eng. (2) (2006). 1-2, 33. [8] S.X. Bai, X.D. Peng, Y.G. Meng, et al., Pseudo-nine-point finite difference method for numerical analysis of lubrication, J. Tribol. 131 (4) (2009) 044502. [9] S.X. Bai, Y.G. Meng, S.Z. Wen, et al., Influence of fixing error on flying attitude of nano-spacing head/disk, Lubr. Eng. 32 (11) (2007) 810. [10] S.X. Bai, Y.G. Meng, S.Z. Wen, Influence of Van Der Waals force on magnetic head/disk of gas film lubrication, Tribology 27 (3) (2007) 264268. [11] R.P. Gabriel, Fundamentals of spiral grooved noncontacting face seals, Lubr. Eng. 3 (1994) 215224.
71
Isothermal gas lubrication
3
Gas lubrication theory developed rapidly in the 1950s. In 1952 Katto and Soda [1] gave the approximate analytical solution of the bearing capacity of a sliding bearing, based on the assumption of an infinite length bearing and an isothermal gas film. A comparison with the experimental results of Ford [2] illustrated that the value of the theoretical calculation was less than that of the experimental results. In order to test the correctness of isothermal hypothesis, the bearing capacity of a gas-lubricated bearing with different specific heat capacities was further studied. The reason for the deviation between theory and experiment is attributed to leakage loss of the bearing end. Later, Ausman offered a theoretical solution for end leakage problems of sliding bearings [3] and steel-ladder thrust bearings [4]. In 1960 Elrod and Burgdorfer [5] further improved the infinite length gas bearing theory, which explained that the temperature rise effect inside the lubrication gas film under normal working conditions can be ignored. Since then, isothermal assumption has been widely accepted in the numerical analysis of gas bearings, and isothermal gas lubrication theory has been widely used in the study of gas bearings and seals. This chapter studies the typical form of a gas lubrication friction pair such as a slider, radial bearing, thrust bearing and face seal, and introduces the modeling method of isothermal gas lubrication and basic lubrication characteristics.
3.1 Sliders The slider is a fluid dynamic bearing structure. The slider bearing with length L and width B, as shown in Fig. 3.1, presents a certain pitch angle in the velocity direction and forms hydrodynamic loading capacity depending on the velocity shear, which is also the basic lubrication structure of the read-write head/memory disk of the computer hard disk. As the magnetic storage density increases, the thickness of the lubricating air film h0 decreases to the nanometer magnitude minus the free travel of gas molecules; the shear strain rate of the gas film is as high as 1010/s; and surface slip and other surface interface effects have a significant impact on the bearing capacity of the lubricating air film, all of which demonstrate the gas film lubrication theory.
Gas Thermo-hydrodynamic Lubrication and Seals. DOI: https://doi.org/10.1016/B978-0-12-816716-8.00003-6 Copyright © 2019 Tsinghua University Press Limited. Published by Elsevier Inc. All rights reserved.
37
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.1 Finite width slider.
3.1.1 Lubrication equation Gas lubrication is roughly divided into several regions with Knudsen number Kn, which is the slip flow field when 0.01 , Kn , 0.1. When 0.1 , Kn , 10, the slip flow field is the transition area; when Kn . 10, it allows the free flow of molecules. When the gas film is thicker, the Knudsen number is smaller, and airflow can be considered as a continuous medium in the slip flow field. When the film thickness reaches nanometer level, especially when it is less than that which would allow the free travel of molecules, the rarefaction effect of lubricating gas becomes increasingly significant. The modified gas lubrication Reynolds equation is as follows: @ @p @ @p @ðphÞ Qp h3 p 1 Qp h3 p 5 6ηu @x @x @y @y @x
(3.1)
where Qp is the flow coefficient. The flow correction coefficients of different models are as follows: The flow coefficient for the classical Reynolds equation is Qp;con 5
D 6
(3.2)
where D 5 ðπ0:5 =2KnÞ 5 ðphð2Ru T0 Þ20:5 =ηÞ, λ 5 ðη=2pÞð2πRu T0 Þ20:5 , and the Knudsen number is Kn 5 ðλ=hÞ. The flow coefficient for the first-order modified Reynolds equation is Qp;1 5
pffiffiffi D c π 1 6 2
(3.3)
3.1 Sliders
where c is the adjustment factor when air molecules collide with the wall surface c 5 ð2 2 a=aÞ. A total diffuse reflection exists when a 5 0, and a mirror reflection when a 5 1. The flow coefficient for the second-order modified Reynolds equation is Qp;2 5
pffiffiffi π D π 1 1 6 4D 2
(3.4)
The flow coefficient for the Fukui model [6] is
8 D > > Qc 5 > > > 6 > > > > > D 1:0653 2:1354 > > 2 D$5 > Qpr 5 1 1:0162 1 > > 6 D D2 > > > > < 0:15653 0:00969 Qpr 5 0:13852D 1 1:25087 1 0:15 # D , 5 2 D D2 > > > > > 0:01653 0:0000694 > > 2 0:01 # D , 0:15 Qpr 5 2 2:22919D 1 2:10673 1 > > > D D2 > > > > > > > Qp 5 Qpr > > : Qc
(3.5)
Fig. 3.2 shows the change curve of the flow coefficient with the inverse Knudsen number D. As can be seen from the figure, when D is larger, the flow coefficient calculated by each correction model is close to the curve of the continuum theory, that is, the classical correction curve. However, as D decreases, the flow coefficient begins to increase, and there is a significant difference in the
FIGURE 3.2 Flow coefficient.
39
40
CHAPTER 3 Isothermal gas lubrication
flow correction coefficient calculated by each correction model. In the above models of flow coefficient modification, the Fukui model is more accurate under the condition of nanofilm thickness (,100 nm). In numerical solution, the equation is generally treated as being dimensionless. The dimensionless form of Reynolds equation is 2 @ @P2 L @ @P2 @ðPHÞ Qp H 3 Qp H 3 1 5 2Λ @X B @Y @X @X @Y
(3.6)
where the dimensionless parameters are defined as
8 p h x y > > P5 ; H 5 ; X5 ; Y 5 > > < pa h0 L B (3.7)
6ηuL > > Λ5 2 > > : h0 pa
where pa is ambient pressure, and Λ is named as the bearing number or seal number. For the smooth surface slider, the film thickness equation is h 5 h0 1 ðL 2 xÞsinα
(3.8)
3.1.2 Pressure boundary condition After the Reynolds equation of gas bearing is solved, the forced pressure boundary condition is generally adopted. For the slider shown in Fig. 3.1, the pressure boundary condition is: p 5 pa ; when x 5 0; L; y 5 6 0:5B
(3.9)
The dimensionless form of the pressure boundary condition is: P 5 1; when X 5 0; 1; Y 5 6 0:5
(3.10)
3.1.3 Lubrication performance parameters For slider bearings, the load capacity of gas film is the most important performance parameter for characterizing the bearing performance. For the finite width slider bearing, the load capacity can be expressed as follows: w5
ð L ð B=2 0
2B=2
pdxdy
(3.11)
3.1 Sliders
The expression of dimensionless load is W5
ð 1 ð 0:5 0
20:5
PdXdY
(3.12)
The conversion coefficient is cw 5 pa LB
(3.13)
w 5 cw W
(3.14)
That is
3.1.4 Hydrodynamic lubrication characteristics of sliders As a typical hydrodynamic bearing, the load capacity of the inclined slider is mainly dependent on the dynamic pressure effect of the lubricating fluid in the convergence gap under the action of velocity shear. For the gas lubrication Reynolds equation, when the shear flow term on the right of the equation is considered, which makes it difficult to give an analytical solution of pressure distribution, the numerical method is generally adopted to solve the equation. Fig. 3.3 shows the pressure distribution of the slider gas film. It can be seen from the figure that during the flow from the inlet to the outlet, as the gas film thickness gradually decreases, the gas is continuously squeezed under shear action, and the gas film pressure gradually increases, forming a dynamic pressure effect. The seal gas film pressure rapidly decreases near the outlet, giving the lubricating gas film a certain bearing capacity.
FIGURE 3.3 Pressure distribution of the slider gas film.
41
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CHAPTER 3 Isothermal gas lubrication
For the slider bearings, the load capacity is mainly related to the pitch angle, film thickness, and shear velocity, and the change law of the slider load capacity is analyzed below. 1. Pitch angle Fig. 3.4 shows the influence of the pitch angle on the load capacity. It can be seen that the load capacity of gas film changes nonmonotonically with the increase of pitch angle. The load capacity increases first with the increase of the pitch angle. When the maximum value is reached, it will take effect quickly until the load capacity is lost. This means that one load capacity value corresponds to two pitch angle values, which will easily lead to unstable conditions for the design of the slider bearing. The reason is that with the increase of pitch angle, the shear rate of gas film decreases, which is not conducive to the formation of the hydrodynamic effect. On the other hand, the increase of pitch angle leads to an increase of gradient of gas film thickness, and the enhancement of convergence is conducive to the enhancement of extrusion effect and the increase of gas film pressure. The combination of the two factors leads to the extreme value on the bearing capacity curve. 2. Film thickness Fig. 3.5 shows the influence of the film thickness on the load capacity. As shown in the figure, the load capacity shows a monotonically decreasing trend with the increase of film thickness. As the viscosity of gas is low, when the film thickness increases gradually from nanometer to micron, the load capacity
FIGURE 3.4 Influence of the pitch angle on the load capacity.
3.1 Sliders
FIGURE 3.5 Influence of the film thickness on the load capacity.
decreases rapidly. In other words, under low shear rate, the hydrodynamic load capacity is significantly reduced, and the hydrodynamic load capacity with micron thickness is much lower than that of nanometer ultra-thin film lubrication. Conversely, the smaller the film thickness, the greater the hydrodynamic load capacity of the gas film. 3. Velocity Shear speed is one of the main factors affecting load capacity of slider bearings. In general, the higher the velocity, the greater the hydrodynamic load capacity. Fig. 3.6 shows the influence curve of velocity on the load capacity. It can be seen that the load capacity increases monotonically with the increase of velocity. However, when the velocity increases to a certain value, the load capacity increases slowly and approaches the limit value, which means that there is a load limit for the slider bearings.
3.1.5 Hydrodynamic lubrication characteristics of divergent sliders Conventional lubrication analysis of slider bearings mainly studies the convergence gap that can form a hydrodynamic effect, and often neglects the divergent gap lubrication law. With more and more geometrical structures applied to control the lubrication performance of the bearing surface, the influence of divergent clearances on fluid motion and lubrication characteristics is indispensable.
43
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.6 Influence curve of velocity on the load capacity.
FIGURE 3.7 Diagram of divergent slider.
As shown in Fig. 3.7, the thickness of the lubricating film increases gradually along the direction of shear velocity, which is called the divergent slider. Theoretically, for divergent clearances, shear flow tends to increase along the direction of shear velocity because of the increase in the thickness of the smooth film, and there is bound to be a codirectional pressure flow to meet the conservation of flow in the lubrication area, so the pressure decreases rather than increases.
3.1 Sliders
FIGURE 3.8 Pressure distribution of the divergent slider.
Fig. 3.8 shows the pressure distribution of the divergent slider. It can be seen from the figure that the gas film pressure in the lubrication area is lower than the ambient pressure, that is, the lubrication film presents negative pressure distribution, and the gas film pressure value is the lowest near the outlet, which also means that the bearing capacity of the divergent slider is negative. To further analyze the load characteristics of the divergent slider, the variation curve of the load with the film thickness under different pitch angles is given in Fig. 3.9. It can be seen from the figure that, similar to the variation trend of the convergent slider, the load capacity of the divergent slider shows a monotonous decreasing trend with the increase of thickness. Theoretically, as the minimum value of gas pressure is a vacuum state, the minimum limit value of the lubricating gas film pressure of the divergent slider is 0, that is, the vacuum state. Therefore the load capacity of the divergent slider also has a limit value in the numerical value.
3.1.6 Lubrication characteristics of the magnetic head slider With the development of precision machinery and equipment, load capacity and accurate control of the bearings have become increasingly demanding but cannot be realized by controlling the bearing inclination angle alone. Surface groove technology has been widely used in the surface design of gas-lubricated friction pairs.
45
46
CHAPTER 3 Isothermal gas lubrication
FIGURE 3.9 Effect of film thickness on load capacity of the divergent slider.
The magnetic head/hard disk system is a typical application of gas-inclined slider bearing. In order to ensure the reliability of the hard disk, it is necessary to precisely control the attitude and gas film thickness of the slider bearings, and its realization mainly depends on the geometric structure of the bearing surface [710]. From the previous discussion, it can be seen that under the action of velocity shear, the convergent gap produces hydrodynamic load capacity, whereas the divergent gap produces negative load capacity. Through the geometrical structure, the magnetic head slider can maintain the stability of flight attitude in the state of nanofilm thickness. This magnetic head is called a negative pressure-type magnetic head. Here the influence law of the surface geometry structure on the gas lubrication characteristics is discussed by taking two typical slider structures of the magnetic head, as shown in Fig. 3.10, that is: Slider 1 is a dual-track magnetic head with relatively simple surface geometry, including only two parallel symmetric rectangular convex platform structures; Slider 2 is a negative pressure magnetic head with a relatively complex structure. The front end is a C-shaped convex platform, and three convex platform structures are close to the end. A concave region is formed in the middle of the sliding block; that is, the film thickness forms a divergent region. Fig. 3.11 shows the gas film pressure distribution of the two magnetic head sliders. As can be seen from the figure, because of the simple structure of Slider 1, which is equivalent to two parallel oblique slider structures, two pressure peaks
3.1 Sliders
FIGURE 3.10 Geometrical structure of magnetic head slider surface [8]. (A) Slider 1 and (B) Slider 2.
FIGURE 3.11 Gas film pressure distribution of magnetic head sliders. (A) Slider 1 and (B) Slider 2.
appear. Moreover, because of the convergence of film thickness in the lubricating area along the shear direction of velocity, only shear squeeze phenomenon exists in the process of gas flow, so the gas film pressure is higher than the ambient pressure.
47
48
CHAPTER 3 Isothermal gas lubrication
For Slider 2, when the shear flow crosses the front end of the C-shaped convex platform, the film thickness in the lubrication area increases suddenly and presents local divergent clearances. According to the analysis in the previous section, negative pressure distribution can be easily formed for divergent clearances, especially under the condition of high shear rate. Therefore a large area of negative pressure distribution is smaller than the ambient pressure in the central region of the slider, and the convex platform forms multiple pressure peak distributions. Slider 2 is a kind of negative pressure gas bearing, which is mainly discussed in the following section. 1. Pitch angle The surface pressure distribution of the magnetic head slider is related to the geometric structure of the magnetic head surface, and the change of the pitch angle of the slider further changes the distribution of the film thickness, leading to further changes in the gas film pressure distribution and load capacity. As shown in Fig. 3.12, with an increase of pitch angle, the influence of the surface structure of the negative pressure magnetic head slider on the pressure distribution on the magnetic head slider’s surface becomes weaker and weaker. As a result, the magnetic head slider gradually degenerates into a slanted slider with limited width, and the negative pressure area disappears.
FIGURE 3.12 Influence of pitch angle on pressure distribution of negative head slider. (A) α 5 0 μrad, w 5 5.26 g, (B) α 5 30 μrad, w 5 1.68 g, (C) α 5 100 μrad, w 5 0.12 g, and (D) α 5 800 μrad, w 5 0.30 g.
3.1 Sliders
FIGURE 3.13 Influence of the pitch angle on load capacity of negative pressure sliders.
Fig. 3.13 shows the influence curve of the pitch angle on the load capacity of the negative pressure sliders. It can be seen from the figure that the bearing capacity decreases first with the increase of pitch angle, and then the second peak value appears. By comparing the influence curve of the pitch angle on the load capacity of the smooth slider in Fig. 3.4, it can be seen that the surface geometry makes the load variation rule of the slider more complicated. In addition, the geometrical structure of the surface enables the slider to generate large load capacity under small pitch angle. With the increase of pitch angle, the load capacity appears the second peak value, but the value of the second peak value is far less than that of the small pitch angle. 2. Film thickness Fig. 3.14 gives the influence curve of gas film thickness on the load capacity. It can be seen from the figure that the negative pressure slider presents a change rule similar to the smooth slide, that is, the load capacity decreases monotonously with the increase of film thickness. When the film thickness gradually increases from nanometer to micron, the load capacity decreases rapidly. It should be pointed out that the negative load of the magnetic head slider occurs within a certain range of film thickness variation, which is fundamentally different from the smooth surface slider. For the head slider, negative value means that the two surfaces of the friction pair present trend of movement toward each other. In other words, the geometric structure design of the negative pressure surface is conducive to the stability of the head/disk motion gap, thus ensuring the reliability of the head data reading and writing.
49
50
CHAPTER 3 Isothermal gas lubrication
FIGURE 3.14 Effect of film thickness on load capacity.
FIGURE 3.15 Variation of load capacity of the negative pressure slider with speed.
3. Velocity Fig. 3.15 shows the variation curve of the load capacity of the negative pressure slider with the speed. As can be seen from the figure, the load capacity of the negative pressure slider is not only small but also presents a complex law of nonmonotonic variation with the increase of velocity, compared with the
3.2 Journal bearing and radial seal
monotonic increase of the load of the smooth slider. Because of the existence of the negative pressure zone, when the velocity reaches a certain value, the load capacity of the gas film decreases, and under the condition of a large pitch angle, the phenomenon of negative load appears.
3.2 Journal bearing and radial seal The geometric structure of the radial bearing, which is mainly composed of stator and rotor, is shown in Fig. 3.16. When the rotor is eccentric, the sliding bearing structure can also be formed, but because of the low viscosity of the gas, the load capacity of hydrodynamic pressure is low. Therefore gas-lubricated radial bearings are widely used in engineering practice in the form of static bearing. As the pressure po at the inlet is greater than the pressure pi at the outlet, the gas enters the gap between the stator and rotor through the inlet and leaks out from the outlet. The rotor and stator are separated by the load capacity of the gas film in the clearance to prevent contact wear of the bearing. When the rotor has a certain eccentricity re, the gas film forms a certain hydrodynamic pressure distribution during rotation, which has an impact on the motion of the rotor. On the other hand, the geometrical structure is also the typical structure form of the radial gas seal. What is different is that load capacity of the gas film is mainly considered in design of the bearing, and the leakage rate of gas in clearance is an important index parameter to be considered in seal design. This section mainly focuses on the structure of radial bearings and discusses their gas lubrication analysis model and lubrication characteristics. Table 3.1 gives the main geometric structure parameters of radial bearing and gives the values of calculation examples.
FIGURE 3.16 Structure diagram of the radial bearing (seal).
51
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CHAPTER 3 Isothermal gas lubrication
Table 3.1 Geometric structure parameters of radial bearings and calculation values. Parameter
Symbol
Data
Outside radius of stator Radius of the rotor Eccentric radius Width Film thickness Rotation speed Outlet pressure
ro ri re L h0 Ω pi
55 mm 30 mm 0 30 mm 550 μm 030,000 rpm 0.101 MPa
3.2.1 Lubrication equations For the radial bearing structure shown in Fig. 3.16, the gas lubrication Reynolds equation in the cylindrical coordinate system is used in the analysis. 3 @ ρh @p @ ρh3 @p @ρh 5 6ω 1 @z η @z @θ ri2 @θ η ri @θ
(3.15)
The dimensionless Reynolds equation is @ @P2 ri 2 @ @P2 @ðPHÞ 1 5 2Λ H3η H3η @θ L @Z @Z @θ @θ
(3.16)
where the dimensionless parameters are defined as
8 p h z η > > P5 ; H 5 ; Z 5 ; η5 > > pa h0 L η0 < 6η r 2 ω > > Λ 5 02 i > > : h0 pa
(3.17)
where η0 is initial viscosity of fluid. The equation of film thickness is hðθ; zÞ 5 h0 1 hg ðθ; zÞ 1 re sinðθ 2 θe Þ
(3.18)
where hg is the depth distribution of the geometric groove on the bearing cylinder. In engineering practice, to enhance the load capacity of the lubricating gas film, different dynamic pressure groove structures such as spiral grooves can be designed and processed on the working face of rotor or stator.
3.2.2 Boundary conditions Imposed pressure boundary conditions are generally applied as follows: p 5 pi ; when z 5 0
(3.19a)
p 5 po ; when z 5 L
(3.19b)
3.2 Journal bearing and radial seal
3.2.3 Lubrication parameters 1. Load capacity of gas film When the radial bearing presents as a sliding bearing and the rotor appears eccentric, the horizontal load component of the gas film is Fh 5
ð L ð 2π 0
ðp 2 pa Þri cosθdθdz
(3.20a)
0
The vertical load component of the gas film is ð L ð 2π
Fv 5
0
ðp 2 pa Þri sinθdθdz
(3.20b)
0
So, the load capacity of the gas film is F5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fh2 1 Fv2
(3.20c)
The dimensionless form of the load capacity is ð 1 ð 2π
Fh 5 Fv 5
0
0
0
0
ð 1 ð 2π
F5
ðP 2 1ÞcosθdθdZ
(3.21a)
ðP 2 1ÞsinθdθdZ
(3.21b)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Fh 1 Fv
(3.21c)
The conversion coefficient is cF 5 pa ri L
(3.22)
F 5 cF F
(3.23)
That is,
2. Leakage rate According to the derivation of Chapter 2, Gas lubrication equations, for radial bearings, the mass leakage rate is Q5
ð 2π 0
h3 @p mmol ρ 2 ri dθ 12η @z
(3.24)
The dimensionless form of the leakage rated is Q5
ð 2π
2H 3 η
0
@P2 dθ @Z
(3.25)
Define the conversion coefficient: cQ 5
mmol ρ0 pa h30 ri 24η0 L
(3.26)
53
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CHAPTER 3 Isothermal gas lubrication
The following expression can be obtained: Q 5 cQ Q
(3.27)
3. Frictional force The horizontal component of the frictional force is Ffric;h 5
ð L ð 2π 0
2
0
@p h ωri 1η ri cosθdθdz ri @θ 2 h
(3.28a)
The vertical component of the frictional force is Ffric;v 5
ð L ð 2π 0
2
0
@p h ωri 1η ri sinθdθdz ri @θ 2 h
(3.28b)
So, we can obtain the expression of the frictional force as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Ffric;h 1 Ffric;v
Ffric 5
(3.28c)
The dimensionless horizontal component of the frictional force is F fric;h 5
ð 1 ð 2π 0
2
0
@P H η ωr 2 η 1 0 2i cosθdθdZ @θ 2 pa h0 H
(3.29a)
The dimensionless vertical component of the frictional force is F fric;v 5
ð L ð 2π 0
2
0
@P H η ωr2 η 1 0 2i sinθdθdZ @θ 2 pa h0 H
(3.29b)
So, we can obtain the dimensionless expression of the frictional force as follows: F fric 5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 F fric;h 1 F fric;v
(3.29c)
Define the conversion coefficient: cFricF 5 pa h0 L
(3.30)
The following expression can be obtained: Ffric 5 cFricF F fric
(3.31)
4. Friction torque The friction torque can be expressed as Mfric 5
ð L ð 2π 0
0
2
@p h ωri 2 ri dθdz 1η h ri @θ 2
(3.32)
The dimensionless form of the friction torque is M fric 5
ð 1 ð 2π 0
0
@P H η0 ωri2 η 1 dθdZ 2 @θ 2 pa h20 H
(3.33)
3.2 Journal bearing and radial seal
Define the conversion coefficient: cM 5 pa h0 ri L
(3.34)
The following expression can be obtained: Mfric 5 cM M fric
(3.35)
5. Average pressure For hydrostatic radial bearings or seals, because the rotor is not eccentric in theory, the load of the gas film is 0. In this case, the performance of the gas film can be characterized by the average pressure, which is defined as ðð
prdθdr pav 5 ðð rdθdr
(3.36)
The dimensionless average pressure is Pav 5
pav pa
(3.37)
3.2.4 Lubrication characteristics Fig. 3.17 shows the pressure distribution of radial bearings. In hydrostatic bearing or radial seals, the pressure of gas film is mainly caused by pressure flow,
FIGURE 3.17 Pressure distribution of radial bearing.
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CHAPTER 3 Isothermal gas lubrication
and the pressure distribution presents a convex profile because of the compressibility of gas. Under the condition of imposed pressure boundary, the influence of gas film thickness on pressure distribution is not obvious. As the film thickness increases, the pressure distribution of the gas film increases slightly. As the load capacity of the gas film depends on the pressure distribution, when the film thickness increases, the flow resistance of the fluid decreases, and the highpressure distribution is shifted toward the outlet, forming a higher load capacity of the gas film. The parameters affecting lubrication performance for hydrostatic radial bearings mainly include gas film thickness and inlet pressure, which are discussed and analyzed below. 1. Film thickness Fig. 3.18 shows the influence of film thickness on lubrication parameters of radial bearings. As can be seen from the figure, as the film thickness increases, the average pressure of the gas film increases, which means that the rigidity of the gas film increases, which is conducive to the stability of the rotor of static bearing. The increase of leakage rate means an increase of gas consumption. As shear rate decreases, friction torque decreases and power consumption decreases. 2. Inlet pressure Fig. 3.19 shows the influence curves of inlet pressure on performance of radial bearings. As can be seen from the figure, with the increase of inlet pressure, the average pressure of the gas film shows a linear increase trend, which means that the gas film has higher rigidity, which is conducive to the stability of the rotor of hydrostatic bearing. The increase of leakage rate means the increase of gas consumption. Meanwhile, both friction torque and power consumption increase.
3.3 Spiral groove thrust bearing The structure of spiral groove thrust bearing is shown in Fig. 3.20, which is composed of thrust bearing and the rotor. The face of thrust bearing is designed with a spiral groove structure. When working under the shear action of rotating speed, the spiral groove of thrust bearing surface produces the fluid hydrodynamic load capacity, which plays a supporting role on the rotor. This section mainly focuses on the spiral groove thrust bearing structure and discusses its gas lubrication analysis model and lubrication characteristics law. Table 3.2 gives the main geometric structure parameters of the bearing and gives the value of the calculation example.
3.3 Spiral groove thrust bearing
FIGURE 3.18 Influence of film thickness on lubrication parameters of radial bearings.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.19 Influence curves of inlet pressure on performance of radial bearings.
3.3 Spiral groove thrust bearing
FIGURE 3.20 Structure diagram of spiral groove thrust bearing.
Table 3.2 Geometric structure parameters of spiral groove thrust bearing and data. Parameter
Symbol
Data
Outer radius Inner radius Groove radius Groove number Spiral angle Groove depth
ro ri rg N β hg
75 mm 60 mm 65 mm 12 15 degrees 5 μm
3.3.1 Gas lubrication equations According to the derivation presented in Chapter 2, gas lubrication equations, the Reynolds equation in polar coordinates can be used for gas lubrication of thrust bearings. @ ρh3 @p @ rρh3 @p @ρh 1 5 6ω r@θ η r@θ r@r @θ η @r
(3.38)
Its dimensionless form is @ @P2 @ @P2 @ðPHÞ 1 5 2Λ H3η RH 3 η R@θ @R R@θ R@R @θ
where
(3.39)
8 p h r η > > >P5 ; H 5 ; R5 ; η5 > pa h0 ri η0 < > > > > :
Λ5
6η0 ri2 ω h20 pa
(3.40)
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CHAPTER 3 Isothermal gas lubrication
The equation of film thickness is hðr; θÞ 5 h0 1 hg ðr; θÞ
(3.41)
where hg represents the depth distribution of the spiral groove on the end face of the thrust bearing, hg(r, θ) 5 hd in the grooved area and hg(r, θ) 5 0 in the smooth area.
3.3.2 Pressure boundary conditions Imposed pressure boundary conditions can be used when the Reynolds equation is solved. pðr 5 0:5do ; θÞ 5 pa ; pðr 5 0:5di ; θÞ 5 pa
(3.42)
In order to reduce the calculation, a spiral groove periodic structure is usually adopted for analysis, and the following periodic pressure boundary conditions need to be applied. π 2 π 5 p r; θ 5 p r; θ 5 N N
(3.43)
3.3.3 Lubrication parameters 1. Load capacity The load capacity can be calculated by the following expression: F5
ð ro ð 2π ri
ðp 2 pa Þrdθdr
(3.44)
0
The dimensionless form of the load is F5
ð Ro ð 2π 1
ðP 2 1ÞRdθdR
(3.45)
0
Define the conversion coefficient cF 5 pa ri2
(3.46)
The following expression is obtained: F 5 cF F
(3.47)
2. Friction torque The friction torque can be calculated by the following expression: Mfric 5
ð ro ð 2π ri
0
@p h ωr 2 1η r dθdr r@θ 2 h
(3.48)
3.3 Spiral groove thrust bearing
The dimensionless form of the friction torque is M fric 5
ð 1 ð 2π 0
0
@P 2η ωr 2 η H1 0 2i dθdR R@θ pa h0 H
(3.49)
Define the conversion coefficient: cM 5
pa h0 ri2 2
(3.50)
The following expression is obtained: Mfric 5 cM M fric
(3.51)
3.3.4 Lubrication characteristics On the spiral groove thrust bearing face, because fluid motion resistance is small along the channel direction film thickness, gas flows along the spiral groove under the effect of circumferential shear velocity, accumulating extrusion, and the film pressure rises, leading to a high-pressure peak at the spiral groove root and forming the hydrodynamic load capacity, as shown in Fig. 3.21. For specific spiral groove structure, the main factors affecting thrust bearing load capacity include rotating speed and film thickness. 1. Rotation speed When the film thickness is given, the bearing number directly reflects the change of speed. The larger the number of bearings, the greater the speed.
FIGURE 3.21 Gas film thickness and pressure distribution of spiral groove thrust bearing.
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CHAPTER 3 Isothermal gas lubrication
Fig. 3.22 shows the variation curve of the load and friction torque of the spiral groove thrust bearing with the number of bearings. As can be seen from the figure, the load capacity of the gas film increased rapidly with an increase of the bearing number. That is, with an increase of the bearing speed, the load capacity of the gas film increases rapidly, and the spiral groove shows a remarkable hydrodynamic effect. On the other hand, the dimensionless friction torque increases rapidly with an increase of the rotating speed.
FIGURE 3.22 Influence of rotation speed on lubrication performance of spiral groove thrust bearing.
3.3 Spiral groove thrust bearing
2. Film thickness Fig. 3.23 shows the influence of film thickness on the lubrication parameters of the spiral groove thrust bearing. As can be seen from the figure, as the film thickness increases, the load capacity of the gas film peaks in the value of specific small film thickness, and then decreases rapidly. On the whole, under the condition of large film thickness, the hydrodynamic effect of the fluid decreases as the shear rate of the fluid decreases, and the load capacity of the gas film is low.
FIGURE 3.23 Influence of film thickness on lubrication performance of spiral groove thrust bearing.
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CHAPTER 3 Isothermal gas lubrication
On the other hand, as film thickness increases, the shear rate of the gas film decreases and the dimensionless friction torque decreases. Because the friction torque of thrust bearing is mainly formed by velocity shear, it can be seen from Eq. (3.48) that the friction torque component formed by velocity shear is inversely related to film thickness. However, because the dimensionless friction torque is directly proportional to the film thickness, the dimensionless friction torque shows the opposite change trend, and the change trend increases with an increase of film thickness.
3.3.5 Spiral groove face seal The spiral groove face seal is the most typical and representative gas face seal. As shown in Fig. 3.24, its friction pair is mainly composed of a rotor ring and a static ring, with hydrodynamic spiral grooves on the face of the rotor ring or the static ring. In terms of working principle, the main difference from the spiral groove thrust bearing is that there is a gas pressure difference between the inner and outer diameter of face seal; that is, the seal pressure forms a radial pressure flow, whereas the inner and outer diameter of thrust bearing generally does not have a radial pressure difference. When the ambient pressure po at the outer diameter is greater than the environmental pressure pi at the inner diameter, the inner flow end face seal is formed; otherwise, it is called the outer flow seal. Because radial pressure flow and circumferential shear flow exist in the lubrication region of the end face seal, the end face seal presents different lubrication characteristics from those of the bearing.
FIGURE 3.24 Diagram of gas face seal.
3.3 Spiral groove thrust bearing
Table 3.3 Geometric parameters of face seal and data. Parameter
Symbol
Data
Outer radius Inner radius Balance radius Grooved radius Ring thickness Groove number Spiral angle Groove depth
ro ri rb rp h1, h2 N β hd
75 mm 55 mm 63.5 mm 65 mm 10 mm 12 575 degrees 6 μm
Geometrically, there are also differences between the face seal and the thrust bearing, mainly because the face seal friction pair has a balanced radius structure, as shown in Fig. 3.24. Gabriel [11] studied and discussed the fundamental characteristics of the gas spiral groove face seal. The main purpose of this section is to introduce the gas lubrication analysis model and lubrication characteristics of spiral groove face seal under isothermal conditions. Table 3.3 gives the main geometric structure parameters of the seal and the value of the calculation example.
3.3.6 Lubrication equations As with the thrust bearings, see Eqs. (3.38) and (3.39).
3.3.7 Pressure boundary conditions For the low-pressure seal condition, the pressure boundary condition generally adopts the imposed boundary condition, that is, pðr 5 ri Þ 5 pi
(3.52a)
pðr 5 ro Þ 5 po
(3.52b)
Because of the periodic distribution of seal rings in the circumference direction, in order to reduce the calculation amount, one cycle is generally taken in the numerical calculation, so the following periodic pressure boundary conditions exist in the calculation area: π 2π p r; θ 5 5 p r; θ 5 N N
(3.53)
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CHAPTER 3 Isothermal gas lubrication
3.3.8 Seal performance parameters The opening force produced by the seal gas film is the load capacity, which can be calculated by the following formula: Fo 5
ð 2π ð ro prdrdθ 0
(3.54)
ri
In subsequent analyses in this book, the load capacity of the seal gas film is expressed in terms of the opening force, consistent with the terminology used in mechanical seals. Under steady-state conditions, the radial flow rate (leakage rate) of the clearance between stator and rotor rings meets the mass conservation; that is, the radial flow rate integral at any radius is the same, so the formula for calculating the mass leakage rate can be written as Q5
ð 2π 0
h3 @p mmol ρ 2 rdθ 12η @r
(3.55)
The calculation formula of volume leakage rate is Q5
ð 2π h3 @p rdθ P 2 12η @r 0
(3.56)
The formula of friction torque is the same as that of thrust bearing.
3.3.9 Lubrication regularity As the gas viscosity is low, the hydrodynamic pressure formed by circumferential shear is less than the seal pressure. Generally, the sealing gas film pressure is dominated by the pressure flow formed by seal pressure. As shown in Fig. 3.25, the pressure of the gas film decreases gradually from the outer diameter to the inner diameter, and the pressure distribution presents generally a convex profile because of the compressibility of the gas. However, under the conditions of low seal pressure, high speed, and small film thickness, the spiral groove still presents an obvious hydrodynamic effect. At the same time, as the gas flows along the spiral groove, the pumping effect on the gas is formed, which increases the radial flow rate. For the spiral groove face seal, the factors that affect the seal gas film lubrication characteristics mainly include film thickness, rotation speed, and seal pressure. 1. Film thickness Fig. 3.26 shows the influence of film thickness on the performance of the spiral groove face seal. As can be seen from the figure, as the film thickness
3.3 Spiral groove thrust bearing
FIGURE 3.25 Distributions of film thickness and pressure of the spiral groove face seal.
increases, the opening force, that is, the load capacity of the gas film, decreases rapidly. Under the condition of small film thickness, the higher the rotation speed, the greater the opening force. On the other hand, the leakage rate increases rapidly with the increase of film thickness. At the same time, because the radial pumping effect of spiral groove, the shear rate increases the leakage rate. Dimensionless friction torque decreases rapidly with increasing film thickness. 2. Rotation speed Fig. 3.27 shows the influence of rotation speed on the performance of the spiral groove face seal. As can be seen from the figure, as the rotation speed increases, the opening force, leakage rate, and friction torque all show a linear increase trend. The higher the rotation speed, the higher the shear rate of the fluid, and the stronger the hydrodynamic effect and pumping effect of the spiral groove, leading to an increase of the opening force, leakage rate, and friction torque. 3. Seal pressure Fig. 3.28 shows the influence of seal pressure on the performance of the spiral groove face seal. As can be seen from the figure, the opening force and leakage rate increase rapidly with the increase of seal pressure. However, it should be noted that the friction torque does not change significantly with the increase of seal pressure. As can be seen from Eq. (3.48), friction torque is mainly formed by circumferential velocity shear and circumferential pressure flow. As radial pressure flow dominates the seal, especially as the seal pressure increases, the proportion of shear flow decreases, resulting in an insignificant change in friction torque.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.26 Influence of film thickness on performance of spiral groove face seal.
3.3 Spiral groove thrust bearing
FIGURE 3.27 Influence of rotation speed on performance of spiral groove face seal.
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CHAPTER 3 Isothermal gas lubrication
FIGURE 3.28 Influence of seal pressure on the performance of the spiral groove face seal.
References
References [1] Y. Katto, N. Soda, Theory of lubrication by compressible fluid with special reference to air bearing, in: Proceedings of the Second Japan National Congress for Applied Mechanics, Tokyo, 1952, pp. 267270. [2] G.W.K. Ford, D.M. Harris, D. Pantall, Gas-lubricated bearings, Mach. Mark. 182 (2924) (1956) 2630. [3] J. Ausman, Finite gas-lubricated journal bearing, in: Proceedings of the Conference on Lubrication and Wear.The Institute of Mechanical Engineers, London, 1957, pp. 3945. [4] J.S. Ausman, An approximate analytical solution for self-acting gas lubrication of stepped sector thrust bearings, Tribol. Trans. 4 (2) (1961) 304313. [5] H.G. Elrod, A. Burgdorfer, Refinements of the theory of the infinitely-long, selfacting, gas-lubricated journal bearing, in: Proceedings of the International Symposium on Gas-Lubricated Bearing, 1960, pp. 93118. [6] S. Fukui, R. Kaneko, Analysis of ultra thin gas film lubrication based on linearized Boltzmann equation: first report-derivation of a generalized lubrication equation including thermal creep flow, J. Tribol. 110 (2) (1988) 253262. [7] P. Huang, L.G. Xu, Y.G. Meng, et al., Calculation and analysis on dynamic performances for ultra-thin gas film lubrication of magnetic head/disk, Lubr. Eng. (2) (2006). 1-2, 33. [8] S.X. Bai, X.D. Peng, Y.G. Meng, et al., Pseudo-nine-point finite difference method for numerical analysis of lubrication, J. Tribol. 131 (4) (2009) 044502. [9] S.X. Bai, Y.G. Meng, S.Z. Wen, et al., Influence of fixing error on flying attitude of nano-spacing head/disk, Lubr. Eng. 32 (11) (2007) 810. [10] S.X. Bai, Y.G. Meng, S.Z. Wen, Influence of Van Der Waals force on magnetic head/disk of gas film lubrication, Tribology 27 (3) (2007) 264268. [11] R.P. Gabriel, Fundamentals of spiral grooved noncontacting face seals, Lubr. Eng. 3 (1994) 215224.
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