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Influence of temperature on nonlinear dynamic characteristics of spiral-grooved gas-lubricated thrust bearing-rotor systems for microengine
Tribology International 61 (2013) 138–143
Contents lists available at SciVerse ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Influence of temperature on nonlinear dynamic characteristics of spiralgrooved gas-lubricated thrust bearing-rotor systems for microengine Xiao-Qing Zhang, Xiao-Li Wang n, Ren Liu, Ben Wang School of Mechanical Engineering, Beijing Institute of Technology, Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 September 2012 Received in revised form 14 December 2012 Accepted 24 December 2012 Available online 3 January 2013
The operating temperature of the gas thrust bearings for microengine is up to 1300–1700 K. The influence of temperature on both the gas viscosity and the rarefaction is considered to study the characteristics of the bearing-rotor systems. The molecular gas-film lubrication equation, discretized by employing the finite volume method, is systematically coupled with the kinetic equations and solved simultaneously. The results show that the temperature range can be divided into two regions: the viscosity effect domain region and the rarefaction effects domain region. The system exhibits the weakest stability when operating in the junction zone of the two regions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Gas-lubricated bearings High temperature Super thin film Nonlinear response
1. Introduction With the rapid development of the micro-electro-mechanical system (MEMS) technology, Power MEMS has become the research hotspot due to its advantages of small size, light weight and high power density [1]. As a typical Power MEMS device, the micro gas turbine engine developed at MIT is designed to produce up to ten watts of power per gram. To achieve the efficiency targets, a rotor speed of order 2 106 rpm (equivalent to 500 m/s blade tip speed) is required. Therefore the bearings with low friction and low power usage are needed to support the rotor against axial and radial loads; and at the same time, they must remain stable during operation. Compared with the magnetic bearings, water-based bearings [2] and other types of bearings, gas-lubricated bearings are the best choice for the microengine based on their superior load capability and relative ease of fabrication. Hydrostatic gas thrust bearings which support the axial loads have been studied and demonstrated by Teo [3]. Compared with the hydrostatic gas thrust bearings, the hydrodynamic gas thrust bearings have more advantages, because the external air supplies and their channels are not required, which reduces the number of wafers needed [4]. To generate a pressure gradient in the gas film and thus providing the load capacity and stiffness, grooves with certain configuration, e.g., spiral groove, are shallowly etched on the thrust pad.
n
Corresponding author. Tel.: þ86 10 13021907765. E-mail address: [email protected] (X.-L. Wang).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2012.12.013
The gas thrust bearing for microengine is well outside the existing theory and empirical design practice as their fluid dynamics is far different from their conventional brethren. The lubricating film of the thrust bearing is ultra thin (only several microns) and the fluid is considered a rarified gas (0.01oKn o10). Thus the rarefaction effects should be taken into account. To achieve high power density, the required temperature in microengine is supper high [5,6], up to 1300–1700 K. The molecular mean free path (l) increases with the temperature [7], so higher temperature means stronger rarefaction effects and thus weaker load capacity. On the other hand, the increasing of the temperature enhances the gas viscosity (m) and then yields the larger load capacity [8]. As a result, the effects of temperature on viscosity and gas rarefaction have conflict and complex influence on the performance of the gas bearings. Many researchers have studied the rarefaction effects on the characteristics of micro gas bearings at room temperature [9–11]. Others [12] investigated the characteristics of micro gas bearings at different temperatures, but the influence of temperature on the gas viscosity is not considered. Unfortunately, no study on gas bearings considering both effects of temperature synthetically has been reported yet. One of the most important technological barriers encountered in the application of the gas thrust bearing is to guarantee the stable operation of the system at such a high rotating speed. Efforts have been done to provide insight to the dynamic behavior of the gas bearings in which the small perturbation method based on linearization hypothesis is usually used [12–14]. However, the gas film force is strongly nonlinear, and the forces and moments depend not only on the instantaneous kinematical state but also
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
Nomenclature D d f, F Fz fz h, H h0 hr hg iR, iy Kn k m, M mq n ! n p, P pa pin pout Qcon Qp Qp rin rout Rg R
pffiffiffiffiffiffiffiffi inverse Knudson number, D ¼ ph= m Rg T molecular diameter load capacity, [N]; non-dimensional load capacity non-dimensional external resultant force external resultant force, [N] film thickness, [m]; non-dimensional film thickness equilibrium film thickness, [m] film thickness in the ridge area, [m] film thickness in the groove area, [m] unit vectors along the R and y directions, respectively Knudsen number groove number, [pairs] rotor mass, [kg]; non-dimensional rotor mass molecular mass, [kg] rotational speed, [rpm] unit normal vector gas pressure, [Pa]; non-dimensional gas pressure ambient pressure, [Pa] pressure at inner radius, [Pa] pressure at outer radius, [Pa] flow rate coefficient of continuum flow flow rate coefficient of Poiseuille flow non-dimensional flow rate coefficient inner radius in cylindrical coordinates, [m] outer radius in cylindrical coordinates, [m] universal gas constant non-dimensional radial coordinate
upon the history of the motion, which results in coupling of the lubrication analysis with the dynamics of the rotor motion [15]. Thus in this paper, the MGL (Molecular gas-film lubrication) equation. [16, 17], valid for arbitrary Knudsen numbers, are systematically coupled with the kinetic equations of the bearing so that they could be solved simultaneously to investigate the nonlinear dynamic behavior of the spiral-grooved gas thrust bearing. The equations and the corresponding boundary conditions are discretized by using the finite volume method (FVM), as it can easily deal with the film thickness discontinuities that are present with spiral grooved face geometries. At the same time, both the viscosity effects and the gas rarefaction effects related to temperature are taken into account.
2. Analysis Fig. 1 is a schematic drawing of a silicon-based microengine with spiral-grooved gas thrust bearings. The high-pressure and high-temperature gas from the combustor flows through the turbine guide blades, and then blows on the blades located on the rotor and spins the rotor. The rotor is supported axially by a pair of micro gas hydrodynamic thrust bearings, and radially by a journal bearing. The journal bearing has been investigated by Zhang [18] and the present article focuses on the thrust bearings. The micro spiral-grooved thrust bearing and the coordinate system are captured in Fig. 2. The configuration of the grooves etched on the surface of the bearing can be expressed as r ¼ rin expðy tan aÞ. All symbols are defined in the nomenclature. The bearing is subjected to pressures pin and pout at its inner and outer boundary. The relative motion between the grooved and flat mating surfaces causes a pumping action in the gas lubricant.
r, y, z T t, t x, y, z Z
a G
g dr yg, yr L l
m O
o w
139
cylindrical coordinates temperature, [K] time, [s]; non-dimensional time, t ¼ ot rectangular coordinates non-dimensional axial displacement, Z ¼z/2h0 spiral angle, [deg] finite volume boundary width ratio of the groove and ridge, g ¼ yr/yg the ratio of the film height to the depth of the groove groove width and ridge width, respectively 2 compressibility number, A ¼ 6mor 2in =pa hg molecular mean free path fluid kinematic viscosity finite volume domain rotorangular velocity, [rad/s] radius ratio, w ¼ r in =r out
Subscripts C avg max min
node C average value maximum value minimum value
Supperscripts
t t 1
value at time t value at time t 1
2.1. Kinematic model Fig. 3 is a schematic of the dynamic motion of the rotor. The applied axial forces on the rotor include the gravity of the rotor and the gas film pressure. Assuming that when the rotor rotates at a speed of o, the equilibrium position is z¼ 0, as shown in Fig. 3. That is, the gas film force is equal to the gravity of the rotor, but has the opposite direction, and thus the gas film thickness, h, would keeps invariant, h¼h0. The film thickness at the land area is different from the groove area. That is: Land : h0 ¼ hr
ð1Þ
Groove : h0 ¼ hg þ hr
ð2Þ
When the rotor is disturbed (by displacement disturbance, by force disturbance, etc.), the clearance between the rotor and the
Level 1
Rotor
Hydrodynamic thrust bearings
Level 2
L evel 3
Fig. 1. Schematic of the bearing-rotor system for micro gas turbine engine.
140
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
y
rotor center, @z_ =@t, fulfils the following equation:
r
Ridge
@z_ f ¼ z @t m
g
The non-dimensional forms of Eqs. (4) and (5) are
Groove pin
rin rc pout
x
B
z y
Rotor Ridge x hg
Groove
hr
Stator Section B-B
Fig. 2. Schematic of the micro spiral-grooved gas bearing system (not to scale).
h O'
z Downward Force
z r
O
O
ð6Þ
@Z_ FZ ¼ @t M
ð7Þ
2.2. Lubrication model The gas film thickness of the micro spiral-grooved bearing varies from about 0.01 mm to 1.5 mm, causing the most frequent Knudsen number to be in the range of 0.04–7, approximately, which means that the continuous flow model is no longer valid. As a result, the MGL model [16,17] is adopted. With the assumptions of laminar and isothermal flow and the ignorance of the inertia item, the non-dimensional governing compressible form of Reynolds equation based on cylindrical coordinate is @ @P @ @P @ @ þ 2 Q p PH3 R Q p PH3 ¼ K ðPHÞ þ 2K ðPHÞ R@R @R @y @y @t R @y ð9Þ
Upper thrust bearing
h0
@H ¼ Z_ @t
All variables in the above equations are defined in the nomenclature. The kinematical equations and the lubrication model are coupled systematically and calculated together to investigate the nonlinear dynamic behavior of the system precisely. The lubrication model is given in the following section. Note that, the axial displacement z is restricted in the range of ( h0, h0), and going beyond this range means the contact of the rotor to the thrust pad and thus the crash of the system. Therefore, in the calculation program, the computation will be stopped if jzt j 4 h0 . Another terminal condition of the system is when the system achieves stable operation, expressed as: zt ztDt o104 mm ð8Þ
B
rout
Rotor Equilibrium Position (z = 0)
where the symbols and variables are defined in the nomenclature. In Eq. (9), Q p is the non-dimensional flow rate coefficient and equal to the ratio of Qp to Qcon:
h0
Lower thrust bearing Qp ¼ Fig. 3. Schematic of the rotor motion.
bearing would change. Take Fig. 3 for example, the rotor moves up, causing the gas film of the upper thrust bearing to become thinner and the gas film of the lower one would then become thicker. According to Ref. [13], thinner gas film yields higher load capacity, so the rotor would be exerted by a downward force, which impelled it toward the original equilibrium position. Therefore the original balance is upset and the gas film thickness would change as time goes by. The transient gas film thickness of the lower thrust bearing is h ¼ h0 þz
ð3Þ
The axial instantaneous velocity of the rotor can be expressed as @h ¼ z_ @t
ð5Þ
ð10Þ
where Qp and Qcon are the flow rate coefficients for Poiseuille flow and continuum flow, respectively. Qp and Qcon are related to the inverse Knudson number, D, and the details for their calculation can be found in Eq. (A.1). The boundary and initial conditions are 8 P > < R ¼ Rin ¼ Pin P R ¼ Rout ¼ P out ð11Þ > : P ðR, 0, tÞ ¼ P ðR, 2p, tÞ According to Fukui [16] and Zhou [7], both the molecular mean free path l and the gas viscosity m are related to the temperature, that is 16 m Rg T 1=2 l¼ ð12Þ 5 p 2p
ð4Þ
According to the Newton’s Second Law, the external resultant force exerted on the rotor, fz, and the axial acceleration of the
Qp Q con
m¼
5mq 16d
2
Rg T
1=2
p
Therefore Q p and m in Eq. (9) vary with the temperature.
ð13Þ
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
At steady state, the time dependent item in Eq. (9) is omitted, that is 2K
@ ðPHÞ ¼ 0 @t
ð14Þ
2.3. Numerical method The finite volume method is employed in the spatial discretization of the lubrication equation. The dynamic gas-lubricated equation for the micro spiral groove thrust bearing, Eq. (9), can be rewritten as @ r Q ¼ 2K ðPHÞ ð15Þ @t where Q ¼ Q p PH3 R @P=@R iR þ KRPHQ p PH3 @P=R@y iy ; r is the gradient operator and r Q ¼ @=R@R Q p PH3 R @P= @RÞÞ þ @=R@y KRPHQ p PH3 @P=R@y . Integrating Eq. (15) over finite volume surface O surrounded by the dotted line in Fig. 4 yields ZZ ZZ @ r Q dO ¼ 2K ðPHÞdO ð16Þ @t O O The Green’s theorem is applied to Eq. (16) and the derived equation is Z Z @ ! r Q! n dG ¼ 2K ðPHÞ n dG ð17Þ @ t G G ! where n is a unit normal vector along the finite volume boundary ! surrounding O (see Fig. 4). The above equation can be discretized as h i 3 @P @P KRPHQ p PH3 R@ y e KRPHQ p PH R@y w DR
@P @P Q p PH3 R Dy Q p PH3 R @R n @R s @ þ 2K ðPHÞC DyDR ¼ 0 ð18Þ @t The time derivative in Eq. (18) can be calculated by 2K
@ P t P t1 @H ðPHÞC DyDR 2KDyDRHavg C C þ2KP C DyDR @t @t Dt
ð19Þ
141
Discretizing Eq. (19) by following the power-law approximation [15] yields
PtC P tC1 1 @H ð20Þ ¼ aDy Dy þ aDR DRat 2KDyDRHavg @t Dt where 8 3 @P > aDy ¼ Q p PH3 R @P > @R n Q p PH R @R s > < @P @P aDR ¼ Q p PH3 R@ KRPH Q p PH3 R@ -KRPH y y > > e w > : at ¼ 2KPC DyDR
ð21Þ
For the estimation of steady-state characteristics, including the calculation of the pressure distribution and the load capacity, the time dependent item in the left hand side of Eq. (20) is equal to zero and the derived equation would be solved by the Newton– Raphson method. The non-dimensional gas film force F can then be obtained by the integration of the pressure distribution over the lubrication domain: Z 2p Z Rout ðP1:0ÞRdRdy F¼ ð22Þ 0
Rin
For dynamic performance, the direct numerical simulation method [14] is used to couple the lubrication model, Eq. (20), and the kinetic equations, Eqs. (6) and (7). The resulting equations are @ x ¼ z @t
ð23Þ
where oT n x ¼ P, Z_ , H
z ¼
ð24Þ
n o9 8 1 > a Dy þaDR DRat Z_ > > > = < 2KDyDRHavg Dy > > :
FZ M
Z_
> > ;
ð25Þ
Eq. (25) can be solved by using the Runge–Kutta method, and the system transient responses captured above will be then obtained. Note that, the item Fz in Eq. (25) is the resultant force of the rotor weight and the non-dimensional gas film force F, which can be calculated according to Eq. (22).
3. Results and discussions
Δ
Land Groove
N
The detailed parameters of the bearing in the present analysis are as follows: k¼12, a ¼201, rin ¼6.35 mm, hr ¼0.5 mm, w ¼0.3848, g ¼0.5, dr ¼0.17, pin ¼pout ¼pa. The influence of temperature is considered to analyze the steady-state and nonlinear dynamic behavior of the micro gas thrust bearing-rotor system.
n W ΔR
C
w
s Δ
R
3.1. Influence of temperature on steady-state characteristics
e E
R
S θ Γ
Ω
Finite volume
Boundary of FVM Mesh line
Fig. 4. Finite volume discretization in polar coordinates.
Results of non-dimensional load capacity versus temperature at different rotational speeds have been obtained and shown in Fig. 5. As can be seen, in the range from 273 K to 300 K, the change in the value of load capacity is not significant. When the temperature goes higher, the load capacity increase at the beginning and then decrease after reaching the maximum values at about 750 K. When the temperature goes beyond 1350 K, the load capacity begins to be weaker than the value at room temperature (T E300 K). Moreover, the higher the speed is, the more important the effects become. As discussed above, the increase of the temperature could cause the increase of molecular mean free path and thus cause the
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
rarefaction effects, which might decrease the load capacity, to become more important. However, on the other hand, the gas viscosity may increase with the increase of the temperature, and higher gas viscosity yields larger load capacity. In a word, when temperatures varies, the influence of the above two aspect on the bearing characteristics are opposite. The simulation results show that the temperature variation range from 300 K to 1600 K could be divided into two parts, as shown in Fig. 5. The left part is the viscosity effect domain region (300–753 K) and the other one is the rarefaction effects domain region (753–1600 K). In the first part, the influence of temperature on gas viscosity is dominant and the load capacity becomes larger as temperature increases. In the other part, the influence of temperature on rarefaction effects is more important and the inverse trend can be found as temperature goes high.
0.5 Non-dimensional axial displacement, Z
142
2.0
0.4
1.8
0.3
1.6
0.2
1.4 1.2
0.1
Upper
0.0
x
τ300 = 2676
-0.1
τ800 = 3493
x
τ1600 = 1738 x
-0.2
Lower
-0.3 -0.4
T = 300K T = 800K T = 1600K
n=2×104rpm
-0.5 0
1000
2000 3000 4000 Non-Dimensional Time τ
3.2. Influence of temperature on dynamic behavior
Viscous Effect Domain Region
6.0 5.5 5.0 4.5 4.0 3.5
753 K
3.0 2.5
Rarefaction Effects Domain Region
300 K 273.15 K
2.0 200
400
600
800 1000 Temperature T/K
1200
1400
1600
Fig. 5. Curves of non-dimensional load capacity versus temperature at different rotational speeds.
2.8 2.5 2.2 1.9 1.6 1.3
0.4 0.3 0.2
Upper
Lower x τ300 = 4950
0.1 0.0 x
-0.1
τ1600 = 2776
-0.2
T = 300K T = 800K T = 1600K
-0.3 -0.4
n = 1×106rpm
-0.5 1000
0
4000 2000 3000 Non-Dimensional Time τ
5000
6000
Fig. 7. Axial displacement of thrust bearing system at different operational temperature at n¼1 106 rpm.
a
b
0.5 Non-dimensional maximum axial displacement, Zmax
Non-dimensional load capacity, F
6.5
0.5
n = 2×104rpm n = 6×104rpm n = 8×105rpm n = 9×105rpm n = 1×106rpm
7.0
6000
Fig. 6. Axial displacement of thrust bearing system at different operational temperature at n¼2 104 rpm.
Non-dimensional axial displacement, Z
Effects of temperature on the nonlinear dynamic transient response of the micro bearing system to velocity perturbation has been studied and three cases of operating temperature (T¼ 300 K, 800 K, 1600 K) at different rotational speed (n ¼2 104 rpm, 1 106 rpm) are displayed in Fig. 6 and Fig. 7, respectively. When n ¼2 104 rpm, as shown in Fig. 6, for all the three cases of temperature, after a period of fluctuation, the amplitude of vibration decays to zero, and the state of motion then goes back to stable. The fluctuation period is defined as stability recovery time which is marked with a symbol, X. At T¼800 K, the stability recovery time is the longest and the vibration frequency is the highest of all and the reversal phenomenon can be found for the case of T¼1600 K. When the rotational speed is higher, n ¼1 106 rpm (Fig. 7), it would take longer recovery time to return to the stable operation state for cases of T¼300 K and T¼1600 K. However, when T¼ 800 K, as time goes by, the amplitude of vibration goes exceed the upper limit (Z ¼0.5) eventually, which means the failure of the gas thrust bearing system. At this time, the rotor contacts the upper thrust bearing. Its transient pressure distribution is shown in the top of Fig. 7, which is different from the pressure distribution at n ¼2 104 rpm (see the top of Fig.6). The maximum axial displacement in the history of the transient response of the rotor, Zmax, is an important parameter to characterize the dynamic behavior of the micro bearing system
5000
2.0 1.8 1.6 1.4 1.2
c
0.3 0.2 0.1
c
0.0
b 753 K
-0.1 -0.2 -0.3 -0.4 -0.5 200
d
n = 2×104rpm n = 6×105rpm n = 8×105rpm n = 9×105rpm n = 1×106rpm
d
0.4
2.8 2.5 2.2 1.9 1.6 1.3
a
Viscosity Effect Domain Region 400
600
Rarefaction Effects Domain Region 800 1000 1200 Temperature T/K
1400
1600
Fig. 8. Maximum axial displacement at different values of temperature.
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
since Hmin ¼ 0:5Z max represents the minimum distance between the rotor and the upper bearing pad. Fig. 8 shows the curves of Zmax versus temperature at different rotational speed. The contour graphs of pressure distribution corresponding to points of a, b, c and d are also displayed at the top of Fig. 8. When the rotational speed go beyond the value of 6 105 rpm, as can be seen, the stability of the thrust bearing system becomes weaker when the operating temperature lies in the range from 700 K to 1000 K, which is the junction zone between the viscosity domain region and the rarefaction domain region. In this case, the excessive deviation of the rotor position from the original equilibrium position (Z ¼0) will causes the system to crash.
4. Conclusions The steady-state characteristics and nonlinear dynamic behavior of micro gas thrust bearing system for micro gas turbine engine are studied considering the influence of temperature on both viscosity and gas rarefaction. The results show that: a) As temperature increases, the load capacity becomes larger in the viscosity effect domain region and weaker in the rarefaction effects domain region. b) At different operating temperature, the nonlinear dynamic transient response of the micro bearing system is different in stability recovery time and the ultimate equilibrium position. When operating in the junction zone between the viscosity domain region and the rarefaction domain region, the stability of the system is the weakest of all. In designing or establishing operation protocols, the influence of temperature on both the gas viscosity and the gas rarefaction should be considered simultaneously. The model developed in this paper can serve as a useful tool to provide the insight to micro gas thrust bearing-rotor systems.
Acknowledgements This work is supported by the Natural Science Foundation of China (No. 51275046), the Research Fund for the Doctoral Program of Higher Education of China (No. 20121101110015) and Beijing Natural Science Foundation (No. 3102025).
Appendix 8 Q con ¼ D=6 > > > > D 1:0653 2:1354 > Q < p ¼ 6 þ 1:0162 þ D D2
ð5r DÞ
Q p ¼ 0:13852D þ 1:25087þ ð0:15 rD o 5Þ > > > > > 0:0000694 : Q p ¼ 2:22919D þ 2:10673 þ 0:01653 ð0:01 r D o 0:15Þ D D2 0:15653 0:00969 D D2
ðA:1Þ
143
References [1] Epstein AH. Millimeter-scale, micro-electro-mechanical systems gas turbine engines. ASME Journal of Engineering for Gas Turbines and Power 2004;126:205–26. [2] Xie GX, Luo JB, Liu SH, Zhang CH, Lu XC. Micro-bubble phenomenon in nanoscale water-based lubricating film induced by external electric field. Tribology Letters 2008;29:169–76. [3] Teo CJ, Spakovszky ZS. Modeling and experimental investigation of microhydrostatic gas thrust bearings for micro-turbomachines. ASME - Journal of Turbomachinery 2006;128:597–605. [4] Wong CW, Zhang X, Jacobson SA, Epstein AH. A self-acting gas thrust bearing for high-speed micro-rotors. Journal of Microelectromechanical Systems 2004;13(2):158–64. [5] Epstein AH, Senturia SD, Ayon A, Breuer K, Chen KS, Ehrich F, et al. Power MEMS and microengines. IEEE conference on solid state sensors and actuators, Chicago; 1997. p. 753–6. [6] Epstein AH. Micro-engines-scholar lecture. ASME - Journal of Turbomachinery 2004;5(2):204–11. [7] Zhou WD, Liu B, Yu SK, et al. A generalized heat transfer model for thin film bearings at head-disk interface. Applied Physics Letters 2008;92(4):0431091–043109-3. [8] Zhang HJ, Zhu CS, Yang Q. Characteristics of micro gas journal bearings based on effective viscosity. ASME Journal of Turbomachinery 2009;131(4):041707-1– 5. [9] Murthy AN, Etsion L, Talke FE. Analysis of surface textured air bearing sliders with rarefaction effects. Tribology Letters 2007;28:251–61. [10] Zhang WM, Meng G, Huang H, Zhou JB, Chen JY, Chen D. Characteristics analysis and dynamic responses of micro-gas-lubricated journal bearings with a new slip model. Journal of Physics D: Applied Physics 2008;411553051–16. [11] Zhang WM, Zhou JB, Meng G. Performance and stability analysis of gaslubricated journal bearings in MEMS. Tribology International 2011;44:887–97. [12] Lee YB, et al. Numerical prediction of slip flow effect on gas-lubricated journal bearings for MEMS/MST-based micro-rotating machinery. Tribology International 2004;38:89–96. [13] Liu R, Wang XL, Zhang XQ. Effects of gas rarefaction on dynamic characteristics of micro spiral-grooved thrust bearing. ASME Journal of Tribology 2012;134:022201-1– 7. [14] Zhang QD, Shan XC. Dynamic characteristics of micro air bearings for Microsystems. Microsystem Technologies 2008;14(2):229–34. [15] Miller BA, Green I. Numerical formulation for the dynamic analysis of spiralgrooved gas face seals. ASME Journal of Tribology 2001;123(2):395–403. [16] Fukui S, Kaneko R. Analysis of ultra-thin gas film lubrication based on linearized boltzman equation: first report-derivation of generalized lubrication equation including thermal creep flow. ASME Journal of Tribology 1988;10:253–62. [17] Fukui S, Kaneko R. A database for interpolation of Poiseuille flows rates for high Knudsen number lubrication problems. ASME Journal of Tribology 1990;112:78–83. [18] Zhang XQ, Wang XL, Liu R. Effects of microfabrication defects on the performance of gas bearings with high aspect ratio in microengine. Tribology International 2012;48:207–15.
Contents lists available at SciVerse ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Influence of temperature on nonlinear dynamic characteristics of spiralgrooved gas-lubricated thrust bearing-rotor systems for microengine Xiao-Qing Zhang, Xiao-Li Wang n, Ren Liu, Ben Wang School of Mechanical Engineering, Beijing Institute of Technology, Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 September 2012 Received in revised form 14 December 2012 Accepted 24 December 2012 Available online 3 January 2013
The operating temperature of the gas thrust bearings for microengine is up to 1300–1700 K. The influence of temperature on both the gas viscosity and the rarefaction is considered to study the characteristics of the bearing-rotor systems. The molecular gas-film lubrication equation, discretized by employing the finite volume method, is systematically coupled with the kinetic equations and solved simultaneously. The results show that the temperature range can be divided into two regions: the viscosity effect domain region and the rarefaction effects domain region. The system exhibits the weakest stability when operating in the junction zone of the two regions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Gas-lubricated bearings High temperature Super thin film Nonlinear response
1. Introduction With the rapid development of the micro-electro-mechanical system (MEMS) technology, Power MEMS has become the research hotspot due to its advantages of small size, light weight and high power density [1]. As a typical Power MEMS device, the micro gas turbine engine developed at MIT is designed to produce up to ten watts of power per gram. To achieve the efficiency targets, a rotor speed of order 2 106 rpm (equivalent to 500 m/s blade tip speed) is required. Therefore the bearings with low friction and low power usage are needed to support the rotor against axial and radial loads; and at the same time, they must remain stable during operation. Compared with the magnetic bearings, water-based bearings [2] and other types of bearings, gas-lubricated bearings are the best choice for the microengine based on their superior load capability and relative ease of fabrication. Hydrostatic gas thrust bearings which support the axial loads have been studied and demonstrated by Teo [3]. Compared with the hydrostatic gas thrust bearings, the hydrodynamic gas thrust bearings have more advantages, because the external air supplies and their channels are not required, which reduces the number of wafers needed [4]. To generate a pressure gradient in the gas film and thus providing the load capacity and stiffness, grooves with certain configuration, e.g., spiral groove, are shallowly etched on the thrust pad.
n
Corresponding author. Tel.: þ86 10 13021907765. E-mail address: [email protected] (X.-L. Wang).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2012.12.013
The gas thrust bearing for microengine is well outside the existing theory and empirical design practice as their fluid dynamics is far different from their conventional brethren. The lubricating film of the thrust bearing is ultra thin (only several microns) and the fluid is considered a rarified gas (0.01oKn o10). Thus the rarefaction effects should be taken into account. To achieve high power density, the required temperature in microengine is supper high [5,6], up to 1300–1700 K. The molecular mean free path (l) increases with the temperature [7], so higher temperature means stronger rarefaction effects and thus weaker load capacity. On the other hand, the increasing of the temperature enhances the gas viscosity (m) and then yields the larger load capacity [8]. As a result, the effects of temperature on viscosity and gas rarefaction have conflict and complex influence on the performance of the gas bearings. Many researchers have studied the rarefaction effects on the characteristics of micro gas bearings at room temperature [9–11]. Others [12] investigated the characteristics of micro gas bearings at different temperatures, but the influence of temperature on the gas viscosity is not considered. Unfortunately, no study on gas bearings considering both effects of temperature synthetically has been reported yet. One of the most important technological barriers encountered in the application of the gas thrust bearing is to guarantee the stable operation of the system at such a high rotating speed. Efforts have been done to provide insight to the dynamic behavior of the gas bearings in which the small perturbation method based on linearization hypothesis is usually used [12–14]. However, the gas film force is strongly nonlinear, and the forces and moments depend not only on the instantaneous kinematical state but also
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
Nomenclature D d f, F Fz fz h, H h0 hr hg iR, iy Kn k m, M mq n ! n p, P pa pin pout Qcon Qp Qp rin rout Rg R
pffiffiffiffiffiffiffiffi inverse Knudson number, D ¼ ph= m Rg T molecular diameter load capacity, [N]; non-dimensional load capacity non-dimensional external resultant force external resultant force, [N] film thickness, [m]; non-dimensional film thickness equilibrium film thickness, [m] film thickness in the ridge area, [m] film thickness in the groove area, [m] unit vectors along the R and y directions, respectively Knudsen number groove number, [pairs] rotor mass, [kg]; non-dimensional rotor mass molecular mass, [kg] rotational speed, [rpm] unit normal vector gas pressure, [Pa]; non-dimensional gas pressure ambient pressure, [Pa] pressure at inner radius, [Pa] pressure at outer radius, [Pa] flow rate coefficient of continuum flow flow rate coefficient of Poiseuille flow non-dimensional flow rate coefficient inner radius in cylindrical coordinates, [m] outer radius in cylindrical coordinates, [m] universal gas constant non-dimensional radial coordinate
upon the history of the motion, which results in coupling of the lubrication analysis with the dynamics of the rotor motion [15]. Thus in this paper, the MGL (Molecular gas-film lubrication) equation. [16, 17], valid for arbitrary Knudsen numbers, are systematically coupled with the kinetic equations of the bearing so that they could be solved simultaneously to investigate the nonlinear dynamic behavior of the spiral-grooved gas thrust bearing. The equations and the corresponding boundary conditions are discretized by using the finite volume method (FVM), as it can easily deal with the film thickness discontinuities that are present with spiral grooved face geometries. At the same time, both the viscosity effects and the gas rarefaction effects related to temperature are taken into account.
2. Analysis Fig. 1 is a schematic drawing of a silicon-based microengine with spiral-grooved gas thrust bearings. The high-pressure and high-temperature gas from the combustor flows through the turbine guide blades, and then blows on the blades located on the rotor and spins the rotor. The rotor is supported axially by a pair of micro gas hydrodynamic thrust bearings, and radially by a journal bearing. The journal bearing has been investigated by Zhang [18] and the present article focuses on the thrust bearings. The micro spiral-grooved thrust bearing and the coordinate system are captured in Fig. 2. The configuration of the grooves etched on the surface of the bearing can be expressed as r ¼ rin expðy tan aÞ. All symbols are defined in the nomenclature. The bearing is subjected to pressures pin and pout at its inner and outer boundary. The relative motion between the grooved and flat mating surfaces causes a pumping action in the gas lubricant.
r, y, z T t, t x, y, z Z
a G
g dr yg, yr L l
m O
o w
139
cylindrical coordinates temperature, [K] time, [s]; non-dimensional time, t ¼ ot rectangular coordinates non-dimensional axial displacement, Z ¼z/2h0 spiral angle, [deg] finite volume boundary width ratio of the groove and ridge, g ¼ yr/yg the ratio of the film height to the depth of the groove groove width and ridge width, respectively 2 compressibility number, A ¼ 6mor 2in =pa hg molecular mean free path fluid kinematic viscosity finite volume domain rotorangular velocity, [rad/s] radius ratio, w ¼ r in =r out
Subscripts C avg max min
node C average value maximum value minimum value
Supperscripts
t t 1
value at time t value at time t 1
2.1. Kinematic model Fig. 3 is a schematic of the dynamic motion of the rotor. The applied axial forces on the rotor include the gravity of the rotor and the gas film pressure. Assuming that when the rotor rotates at a speed of o, the equilibrium position is z¼ 0, as shown in Fig. 3. That is, the gas film force is equal to the gravity of the rotor, but has the opposite direction, and thus the gas film thickness, h, would keeps invariant, h¼h0. The film thickness at the land area is different from the groove area. That is: Land : h0 ¼ hr
ð1Þ
Groove : h0 ¼ hg þ hr
ð2Þ
When the rotor is disturbed (by displacement disturbance, by force disturbance, etc.), the clearance between the rotor and the
Level 1
Rotor
Hydrodynamic thrust bearings
Level 2
L evel 3
Fig. 1. Schematic of the bearing-rotor system for micro gas turbine engine.
140
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
y
rotor center, @z_ =@t, fulfils the following equation:
r
Ridge
@z_ f ¼ z @t m
g
The non-dimensional forms of Eqs. (4) and (5) are
Groove pin
rin rc pout
x
B
z y
Rotor Ridge x hg
Groove
hr
Stator Section B-B
Fig. 2. Schematic of the micro spiral-grooved gas bearing system (not to scale).
h O'
z Downward Force
z r
O
O
ð6Þ
@Z_ FZ ¼ @t M
ð7Þ
2.2. Lubrication model The gas film thickness of the micro spiral-grooved bearing varies from about 0.01 mm to 1.5 mm, causing the most frequent Knudsen number to be in the range of 0.04–7, approximately, which means that the continuous flow model is no longer valid. As a result, the MGL model [16,17] is adopted. With the assumptions of laminar and isothermal flow and the ignorance of the inertia item, the non-dimensional governing compressible form of Reynolds equation based on cylindrical coordinate is @ @P @ @P @ @ þ 2 Q p PH3 R Q p PH3 ¼ K ðPHÞ þ 2K ðPHÞ R@R @R @y @y @t R @y ð9Þ
Upper thrust bearing
h0
@H ¼ Z_ @t
All variables in the above equations are defined in the nomenclature. The kinematical equations and the lubrication model are coupled systematically and calculated together to investigate the nonlinear dynamic behavior of the system precisely. The lubrication model is given in the following section. Note that, the axial displacement z is restricted in the range of ( h0, h0), and going beyond this range means the contact of the rotor to the thrust pad and thus the crash of the system. Therefore, in the calculation program, the computation will be stopped if jzt j 4 h0 . Another terminal condition of the system is when the system achieves stable operation, expressed as: zt ztDt o104 mm ð8Þ
B
rout
Rotor Equilibrium Position (z = 0)
where the symbols and variables are defined in the nomenclature. In Eq. (9), Q p is the non-dimensional flow rate coefficient and equal to the ratio of Qp to Qcon:
h0
Lower thrust bearing Qp ¼ Fig. 3. Schematic of the rotor motion.
bearing would change. Take Fig. 3 for example, the rotor moves up, causing the gas film of the upper thrust bearing to become thinner and the gas film of the lower one would then become thicker. According to Ref. [13], thinner gas film yields higher load capacity, so the rotor would be exerted by a downward force, which impelled it toward the original equilibrium position. Therefore the original balance is upset and the gas film thickness would change as time goes by. The transient gas film thickness of the lower thrust bearing is h ¼ h0 þz
ð3Þ
The axial instantaneous velocity of the rotor can be expressed as @h ¼ z_ @t
ð5Þ
ð10Þ
where Qp and Qcon are the flow rate coefficients for Poiseuille flow and continuum flow, respectively. Qp and Qcon are related to the inverse Knudson number, D, and the details for their calculation can be found in Eq. (A.1). The boundary and initial conditions are 8 P > < R ¼ Rin ¼ Pin P R ¼ Rout ¼ P out ð11Þ > : P ðR, 0, tÞ ¼ P ðR, 2p, tÞ According to Fukui [16] and Zhou [7], both the molecular mean free path l and the gas viscosity m are related to the temperature, that is 16 m Rg T 1=2 l¼ ð12Þ 5 p 2p
ð4Þ
According to the Newton’s Second Law, the external resultant force exerted on the rotor, fz, and the axial acceleration of the
Qp Q con
m¼
5mq 16d
2
Rg T
1=2
p
Therefore Q p and m in Eq. (9) vary with the temperature.
ð13Þ
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
At steady state, the time dependent item in Eq. (9) is omitted, that is 2K
@ ðPHÞ ¼ 0 @t
ð14Þ
2.3. Numerical method The finite volume method is employed in the spatial discretization of the lubrication equation. The dynamic gas-lubricated equation for the micro spiral groove thrust bearing, Eq. (9), can be rewritten as @ r Q ¼ 2K ðPHÞ ð15Þ @t where Q ¼ Q p PH3 R @P=@R iR þ KRPHQ p PH3 @P=R@y iy ; r is the gradient operator and r Q ¼ @=R@R Q p PH3 R @P= @RÞÞ þ @=R@y KRPHQ p PH3 @P=R@y . Integrating Eq. (15) over finite volume surface O surrounded by the dotted line in Fig. 4 yields ZZ ZZ @ r Q dO ¼ 2K ðPHÞdO ð16Þ @t O O The Green’s theorem is applied to Eq. (16) and the derived equation is Z Z @ ! r Q! n dG ¼ 2K ðPHÞ n dG ð17Þ @ t G G ! where n is a unit normal vector along the finite volume boundary ! surrounding O (see Fig. 4). The above equation can be discretized as h i 3 @P @P KRPHQ p PH3 R@ y e KRPHQ p PH R@y w DR
@P @P Q p PH3 R Dy Q p PH3 R @R n @R s @ þ 2K ðPHÞC DyDR ¼ 0 ð18Þ @t The time derivative in Eq. (18) can be calculated by 2K
@ P t P t1 @H ðPHÞC DyDR 2KDyDRHavg C C þ2KP C DyDR @t @t Dt
ð19Þ
141
Discretizing Eq. (19) by following the power-law approximation [15] yields
PtC P tC1 1 @H ð20Þ ¼ aDy Dy þ aDR DRat 2KDyDRHavg @t Dt where 8 3 @P > aDy ¼ Q p PH3 R @P > @R n Q p PH R @R s > < @P @P aDR ¼ Q p PH3 R@ KRPH Q p PH3 R@ -KRPH y y > > e w > : at ¼ 2KPC DyDR
ð21Þ
For the estimation of steady-state characteristics, including the calculation of the pressure distribution and the load capacity, the time dependent item in the left hand side of Eq. (20) is equal to zero and the derived equation would be solved by the Newton– Raphson method. The non-dimensional gas film force F can then be obtained by the integration of the pressure distribution over the lubrication domain: Z 2p Z Rout ðP1:0ÞRdRdy F¼ ð22Þ 0
Rin
For dynamic performance, the direct numerical simulation method [14] is used to couple the lubrication model, Eq. (20), and the kinetic equations, Eqs. (6) and (7). The resulting equations are @ x ¼ z @t
ð23Þ
where oT n x ¼ P, Z_ , H
z ¼
ð24Þ
n o9 8 1 > a Dy þaDR DRat Z_ > > > = < 2KDyDRHavg Dy > > :
FZ M
Z_
> > ;
ð25Þ
Eq. (25) can be solved by using the Runge–Kutta method, and the system transient responses captured above will be then obtained. Note that, the item Fz in Eq. (25) is the resultant force of the rotor weight and the non-dimensional gas film force F, which can be calculated according to Eq. (22).
3. Results and discussions
Δ
Land Groove
N
The detailed parameters of the bearing in the present analysis are as follows: k¼12, a ¼201, rin ¼6.35 mm, hr ¼0.5 mm, w ¼0.3848, g ¼0.5, dr ¼0.17, pin ¼pout ¼pa. The influence of temperature is considered to analyze the steady-state and nonlinear dynamic behavior of the micro gas thrust bearing-rotor system.
n W ΔR
C
w
s Δ
R
3.1. Influence of temperature on steady-state characteristics
e E
R
S θ Γ
Ω
Finite volume
Boundary of FVM Mesh line
Fig. 4. Finite volume discretization in polar coordinates.
Results of non-dimensional load capacity versus temperature at different rotational speeds have been obtained and shown in Fig. 5. As can be seen, in the range from 273 K to 300 K, the change in the value of load capacity is not significant. When the temperature goes higher, the load capacity increase at the beginning and then decrease after reaching the maximum values at about 750 K. When the temperature goes beyond 1350 K, the load capacity begins to be weaker than the value at room temperature (T E300 K). Moreover, the higher the speed is, the more important the effects become. As discussed above, the increase of the temperature could cause the increase of molecular mean free path and thus cause the
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
rarefaction effects, which might decrease the load capacity, to become more important. However, on the other hand, the gas viscosity may increase with the increase of the temperature, and higher gas viscosity yields larger load capacity. In a word, when temperatures varies, the influence of the above two aspect on the bearing characteristics are opposite. The simulation results show that the temperature variation range from 300 K to 1600 K could be divided into two parts, as shown in Fig. 5. The left part is the viscosity effect domain region (300–753 K) and the other one is the rarefaction effects domain region (753–1600 K). In the first part, the influence of temperature on gas viscosity is dominant and the load capacity becomes larger as temperature increases. In the other part, the influence of temperature on rarefaction effects is more important and the inverse trend can be found as temperature goes high.
0.5 Non-dimensional axial displacement, Z
142
2.0
0.4
1.8
0.3
1.6
0.2
1.4 1.2
0.1
Upper
0.0
x
τ300 = 2676
-0.1
τ800 = 3493
x
τ1600 = 1738 x
-0.2
Lower
-0.3 -0.4
T = 300K T = 800K T = 1600K
n=2×104rpm
-0.5 0
1000
2000 3000 4000 Non-Dimensional Time τ
3.2. Influence of temperature on dynamic behavior
Viscous Effect Domain Region
6.0 5.5 5.0 4.5 4.0 3.5
753 K
3.0 2.5
Rarefaction Effects Domain Region
300 K 273.15 K
2.0 200
400
600
800 1000 Temperature T/K
1200
1400
1600
Fig. 5. Curves of non-dimensional load capacity versus temperature at different rotational speeds.
2.8 2.5 2.2 1.9 1.6 1.3
0.4 0.3 0.2
Upper
Lower x τ300 = 4950
0.1 0.0 x
-0.1
τ1600 = 2776
-0.2
T = 300K T = 800K T = 1600K
-0.3 -0.4
n = 1×106rpm
-0.5 1000
0
4000 2000 3000 Non-Dimensional Time τ
5000
6000
Fig. 7. Axial displacement of thrust bearing system at different operational temperature at n¼1 106 rpm.
a
b
0.5 Non-dimensional maximum axial displacement, Zmax
Non-dimensional load capacity, F
6.5
0.5
n = 2×104rpm n = 6×104rpm n = 8×105rpm n = 9×105rpm n = 1×106rpm
7.0
6000
Fig. 6. Axial displacement of thrust bearing system at different operational temperature at n¼2 104 rpm.
Non-dimensional axial displacement, Z
Effects of temperature on the nonlinear dynamic transient response of the micro bearing system to velocity perturbation has been studied and three cases of operating temperature (T¼ 300 K, 800 K, 1600 K) at different rotational speed (n ¼2 104 rpm, 1 106 rpm) are displayed in Fig. 6 and Fig. 7, respectively. When n ¼2 104 rpm, as shown in Fig. 6, for all the three cases of temperature, after a period of fluctuation, the amplitude of vibration decays to zero, and the state of motion then goes back to stable. The fluctuation period is defined as stability recovery time which is marked with a symbol, X. At T¼800 K, the stability recovery time is the longest and the vibration frequency is the highest of all and the reversal phenomenon can be found for the case of T¼1600 K. When the rotational speed is higher, n ¼1 106 rpm (Fig. 7), it would take longer recovery time to return to the stable operation state for cases of T¼300 K and T¼1600 K. However, when T¼ 800 K, as time goes by, the amplitude of vibration goes exceed the upper limit (Z ¼0.5) eventually, which means the failure of the gas thrust bearing system. At this time, the rotor contacts the upper thrust bearing. Its transient pressure distribution is shown in the top of Fig. 7, which is different from the pressure distribution at n ¼2 104 rpm (see the top of Fig.6). The maximum axial displacement in the history of the transient response of the rotor, Zmax, is an important parameter to characterize the dynamic behavior of the micro bearing system
5000
2.0 1.8 1.6 1.4 1.2
c
0.3 0.2 0.1
c
0.0
b 753 K
-0.1 -0.2 -0.3 -0.4 -0.5 200
d
n = 2×104rpm n = 6×105rpm n = 8×105rpm n = 9×105rpm n = 1×106rpm
d
0.4
2.8 2.5 2.2 1.9 1.6 1.3
a
Viscosity Effect Domain Region 400
600
Rarefaction Effects Domain Region 800 1000 1200 Temperature T/K
1400
1600
Fig. 8. Maximum axial displacement at different values of temperature.
X.-Q. Zhang et al. / Tribology International 61 (2013) 138–143
since Hmin ¼ 0:5Z max represents the minimum distance between the rotor and the upper bearing pad. Fig. 8 shows the curves of Zmax versus temperature at different rotational speed. The contour graphs of pressure distribution corresponding to points of a, b, c and d are also displayed at the top of Fig. 8. When the rotational speed go beyond the value of 6 105 rpm, as can be seen, the stability of the thrust bearing system becomes weaker when the operating temperature lies in the range from 700 K to 1000 K, which is the junction zone between the viscosity domain region and the rarefaction domain region. In this case, the excessive deviation of the rotor position from the original equilibrium position (Z ¼0) will causes the system to crash.
4. Conclusions The steady-state characteristics and nonlinear dynamic behavior of micro gas thrust bearing system for micro gas turbine engine are studied considering the influence of temperature on both viscosity and gas rarefaction. The results show that: a) As temperature increases, the load capacity becomes larger in the viscosity effect domain region and weaker in the rarefaction effects domain region. b) At different operating temperature, the nonlinear dynamic transient response of the micro bearing system is different in stability recovery time and the ultimate equilibrium position. When operating in the junction zone between the viscosity domain region and the rarefaction domain region, the stability of the system is the weakest of all. In designing or establishing operation protocols, the influence of temperature on both the gas viscosity and the gas rarefaction should be considered simultaneously. The model developed in this paper can serve as a useful tool to provide the insight to micro gas thrust bearing-rotor systems.
Acknowledgements This work is supported by the Natural Science Foundation of China (No. 51275046), the Research Fund for the Doctoral Program of Higher Education of China (No. 20121101110015) and Beijing Natural Science Foundation (No. 3102025).
Appendix 8 Q con ¼ D=6 > > > > D 1:0653 2:1354 > Q < p ¼ 6 þ 1:0162 þ D D2
ð5r DÞ
Q p ¼ 0:13852D þ 1:25087þ ð0:15 rD o 5Þ > > > > > 0:0000694 : Q p ¼ 2:22919D þ 2:10673 þ 0:01653 ð0:01 r D o 0:15Þ D D2 0:15653 0:00969 D D2
ðA:1Þ
143
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