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Sliding effect of gas-lubricated porous rectangular thrust bearings
Wear,
I41 (1991)
235
235-248
Sliding effect of ,gas-lubricated porous rectangular thrust bearings S. P. Liaw Department (Taiwan)
and D. G. Lin of Mechanical
Engineering,
Tatung
Institute
of Technology,
Taipei
(Received August 15, 1989; revised May 22, 1990; accepted June 27, 1990)
Abstract A theoretical model has been developed to investigate the dynamic characteristics of porous rectangular thrust bearings lubricated with gas in which pressurized gas is pumped through a porous pad to lubricate the sliding elements. In the model a uniform temperature distribution in the gas lubricating film is assumed and a modified form of the BeaverJoseph slip velocity boundary condition is applied at the interface between the porous pad and the lubricating film. The pressure in the porous pad is obtained in a closed form when the pad thickness is very small compared with its other two dimensions. A modified Reynold’s equation was solved numerically using a finite difference method. The load capacity, mass flow rate of gas and stiffness are obtained in dimensionless forms and calculated numerically for different operating parameters. A higher load capacity is observed for a thinner film with and without relative motion. The effect of sliding is to reduce the load capacity and the stiifness, whereas it increases the mass flow rate of gas. It is also noted that when the pad shape ratio is very small, the effect introduced by sliding becomes less significant.
1. Introduction
Externally pressurized gas-lubricated thrust bearings have received a great deal of attention in recent years, primarily because the lubricant gas is cheap, stable and produces no pollutants during the working processes. As a result of these superior features they are nowadays widely used in the fields of textile machinery, food engineering, cryogenics and applications requiring very high temperatures. Depending on the method of feeding in the gas, they can be classified into two types: hole admission bearings and porous bearings. Conventional hole admission bearings are often difficult to justify in otherwise advantageous applications because of their low load capacity and stiffness relative to the gas pressure and flow required. Externally pressurized porous gas bearings are preferable to hole admission bearings owing to their simplicity and better performance characteristics. A detailed discussion has been given by Sneck [ 11. An early theoretical study of non-rotating circular thrust bearings was made by Sheinberg and Shuster [2]. Later on, Gargiulo and Gilmour [3] derived a theoretical solution to the problem of a circular thrust bearing 0043-1648/91/$3.50
0 Elsevier Sequoia/Printed in The Netherlands
with a three-dimensional flow of gas through the porous pad. The numerical results compared favourably with the available experimental data. Tayler and Lewis [4] considered the elasticity of the porous pad and the effect of pad flexure on the performance of a bearing. This effect was shown to decrease the load capacity. Using the simplification of considering the gas in the porous pad to flow only in the axial direction, Sun [5] was able to express the pressure distribution in terms of Bessel functions. The load capacity predicted by Sun’s model agrees well with that obtained by Gargiulo and Gilmour [ 31. The central assumption in the axial flow model was eventually proven to be valid particularly for thin bearings. Rectangular thrust bearings are very important in the design of machine slideways. The characteristics of an aerostatic porous thrust bearing in a static mode were investigated by applying slip [ 61 and non-slip [ 71 boundary conditions at the interface between the porous pad and the lubricating film. Making the assumption of uniform film thickness, the static characteristics were found to vary with the thickness of the lubricating film for differing bearing geometries. However, in any realistic study, the dynamic characteristics need to be considered as well as the static ones. In this study, a runner pad is assumed to move at a uniform velocity relative to a porous pad. Assuming the Beavers-Joseph slip condition [8, 91 to apply at the permeable surface, the steady state pressure distribution is obtained numerically by solving a modified Reynold’s equation. The variations in load capacity, flow rate of gas and stiffness with velocity are presented graphically. 2. Formulation
and analysis
A schematic diagram of a porous rectangular thrust bearing lubricated by pressurized gas is shown in Fig. 1. The compressed gas is fed in at a constant pressure P, from the bottom of the porous pad. The pressure distributions in the porous pad and the gas film are denoted as P ' and P respectively. The velocity components of the gas in the x, y and z directions
Fig. 1. Schematic
diagram of a porous rectangular thrust bearing.
237
are u’, V’ and W’ respectively in the porous pad, and u, o and w respectively in the film. Usually the gas film is very thin, and so the corresponding Reynold’s number is quite small and the gas flow is expected to be laminar. In cases where the temperature variation is small and can be ignored, the flow is barotropic and the density is a function of pressure alone. The thermodynamic behaviour of the lubricating gas is assumed to follow the ideal gas law. For simplicity, the film thickness h is assumed to be uniform during the analysis, even though the inflation effect may not be avoided when the movement of the runner pad is very fast. However, the involvement of the force balance between the applied load and the pressure distribution is more complicated and is ignored in the present study. In addition, the thickness of the porous pad is very small when compared with its other two dimensions. This permits the assumption that the gas will flow only in the z direction in the porous pad. The porous material is homogeneous and isotropic. Darcy’s law [ 10 J can be applied to simulate the flow for slow motion in the pad. Neglecting gravitational force in the z direction, the Darcy equations are written as
V’=.-
K
__
aP'
fii)y
where K is the permeability coefficient of the porous pad and p is the viscosity of the gas. Both K and p are constants in the analysis. With these assumptions, we can now investigate the flow in the porous pad and the lubricating film.
The flow in the porous pad is governed by the equation of cont~ui~
Substitution of the equation of state and eqn. (1) into eqn. (2) yields
Li2P ‘2 -+ ax2
a2p I2 a2p’” +yg-= rty2
0
Assuming a very small thickness for the porous pad, the above equation can be written in an abbreviated form as -
=o Ck?
subject to the boundary conditions
238
PI=P,
at
*::-H
P' =P
at
z=o
Hence the solution of eqn. (4) is readily obtained as p$n ‘p2
_ (ps” _P2)
;
(5)
where P, and P are the suppIy and film pressures respectively. In a very thin film of pressurized gas, the pressure dist~bution does not vary in the z direction. Thus, on the basis of scale analysis, the steady momentum equations are expressed as
and @b)
No-slip conditions are assumed for the impermeable runner surface and the BeaversJoseph slip boundary conditions [9] are applied to the surface of the porous bearing pad. They are z/J=0
u= -u
at
z=h
f3W
and
where U is the relative velocity between the bearing surfaces and cy is a dimensionless constant which depends on the characteristics of the porous medium. The value of CKis independent of the film thickness and the fluid properties [ 111 and it can be determined experimentally [ 121. The solutions of eqns. (6a) and (6b) subject to the boundary conditions are
and u=
_!-i?
(+_h2)
2F ?!d
- s$-&--(z-h)
In the above equations S, defined as K?&, is the slip parameter. Since s is equal to zero, the condition described in eqn. (?b) reduces to the no-slip boundary condition approp~ate to a solid wall. Such a si~ation is referred
239
to as general Couette flow. However, as S approaches infinity, a complete slip condition is prescribed and no shear stress is applied on the fluid. The steady state continuity equation in the gas film is written as -Kpu)
+ -Kpu)
ax
I
ay
Kpul)
-0
(9)
az
Since the pressure and temperature do not vary in the z direction, integration
of the above equation gives
Substituting the velocities obtained in eqns. @a) and (8b) into eqn. (10) yields
= - 24@wo
(111
where w. is the velocity crossing the interface of the porous pad and gas film. It can be obtained by applying eqns. (1) and (5): wg= f&
(P,2-P2)
(12)
Introducing the following dimensionless quantities: 2=xlB
B=yIB P, = P,lP,
Is = PIPa
eqn. (11) takes the form
-a2P + -a2P2 +v --2s+1 aP =p g_ 4s+1 a* a2 372
(P-F,“)
(13)
In the above equation, V and /3 denote velocity and feed rate parameters respectively, and they are defined as v=
12mJ P,h”
and
12KB2 p= x
Since no relative motion is specified in the y direction, the pressure distribution in the system is symmetric relative to the x axis. When the outer boundaries of the lubricating film are assumed to be adjacent to the atmosphere, the boundary conditions are P=l
at
x=+1-
and
--r
P=l
at
y= f7”
and
-1<*<1
at
jj=o
and
-1<2<1
als -= ~
0
in which r, defined as L/B, is the shape ratio of the porous pad.
(14)
240
3. Numerical
approach
The solution of eqn. (13) satisfying the boundary conditions in eqn. (14) gives the pressure distribution in the lubricating film. Because eqn. (13) is not linear, it is not an easy task to obtain the analytical solution. Thus it is solved numerically by using a finite difference method. Setting & =p2 and choosing Ax=Ay= A, eqn. (13) when written as a five-point Gauss-Seidel finite difference equation is 6L + I(i? j) =
V(2S+l) A I + *(4S* 1) (&n(i, j))‘”
+7.
V(ZS+l)
l,io 1cAdi+
A
[
4(4S+ 1) (Q&j>)*n
+&,(i,j+
l)+Q,+,(Gj-11+-
I
QTa+,(i- L%?T p(S + 1)A2ps2 4s+1 (15)
It can be further extended using the SOR method [ 131 by ~trodu~~g relaxation factor w:
Q,+l(i,j)=Qn(i,j)x(1-,>+oxQ,,l(i,j)
a
(16)
where o ranges between 1 and 2 depending on the chosen mesh space. As proposed by Patankar [ 141, if all the coefficients in the discretizatio~ equation are positive, then the numerical scheme is stable. Following this criterion, in the present study the mesh space is chosen to be 1130. Thus, in the range of V less than 50, all the coefficients in eqn, (15) are always positive, and it is stable during the numerical calculations. The value of the pressure is iterated until it converges at an absolute iteration error of less than lo-” and, because of symmetry, only half of the bearing is considered.
4. Characteristics 4.1. Load
of the bearing
capacity
The load capacity of the bearing is determined by integrating the pressure over the pad:
(17) 0 -B
and the dimensionless form is defined as
241
1
ss )‘
(I’- 1) &d?J
Ij7=
01 0
W 4BL(PS -P,)
=
r
-1
2(p,-
1)
(18)
which represents the ratio of the actual load to the maximum possible load. It varies from zero to unity. 4.2. Centre
of pressure
The location of the centre of pressure can be calculated by taking the moment of pressure force LB
2
ss
(P - P,)x
dxdy
0 -B
xc=
(1%
W
or in a dimensionless expression 1
ss I’
(I’- l)z? tid@ 0 -1 L 2(P, - l)W 0r
,&=!5= B
(20)
4.3. Mass flow rate of gas The mass flow rate can be calculated by integrating the velocity w. over the pad surface LB
G=2
ss
wo
(21)
bdiv
0 -B
Making use of eqn. (12) and the equation of state, eqn. (21) becomes LB
G=
--$$sso _a(P.2-P2)
tidy
(221
Hence the dimensionless form gives (23)
4.4. st@?less The stiffness of the bearing is defined as the negative change in load capacity caused by a change in gas film thickness:
(24) Since the sliding parameter V, feed rate parameter p and slip parameter S are all functions of h> the stiffness K can be expressed in a pfaffian form: (25) or in a ~ensio~ess EC
Kh rnL(P, -P,)
form
=fi,+IQ+K,
(26)
where
$? can be calculated from the curves of the relations V-m,
/3-w and 5’4%
5. Results and discussion Supplied from the bottom of the porous pad as shown in Fig. 1, the compressed air flows through the porous medium and film clearance, and is finally exhausted to the atmosphere. One may imagine this process as analogous to Ohm’s Law with two resistances (porous rne~u~ub~cat~g film) in a series circuit. The pressure difference is equivalent to the applied voltage and the flow velocity to the current. If the supply pressure is kept constant, a greater pressure difference in the gas film will be obtained when the resistance is higher in it and/or lower in the porous medium. This will result in a higher load capacity. 5.1. Load capacity In order to ensure the correctness of the present analysis, the limiting solutions for no relative motion were compared with those of other investigators [6, 71. In this case, the load capacity was found to increase with the feed rate parameter for a square porous pad. In other words, a higher load capacity is obtained with a thinner film. This is consistent with the results reported by Rao [ 61 and is shown in Fig. 2.
243
In the absence of relative motion, the pressure distribution is symmetric with a maximum located at the origin. The maximum pressure is observed to decrease with the velocity parameter V and the location of the maximum is shifted in the same direction as the velocity. This is illustrated in Fig. 3 for the pressure distribution in the mid-plane. If the total load is supplied at the geometrical centre, a moment will be exerted by the unsymmetrical pressure distribution, and this may cause the runner-pad to incline slightly as it achieves a new floating position. However, the effect of the applied load is excluded in this study. The distance of the centre of pressure from the origin increases with the relative motion, whereas it decreases with an increase in the feed rate parameter. Hence the velocity effect is smaller for a thinner film. This is plotted in Fig. 4. In Fig. 3, it can also be seen that the area under the pressure curve becomes smaller for a larger velocity parameter. This shows that the decrease in the load capacity is due to the presence of relative motion. The dependence of the load capacity on the velocity parameter for differing slip parameters is indicated in Fig. 5. It can be seen that the load capacity decreases with an increase in the velocity parameter and/or slip 0.6
0.4 w 02
op
I
I
I
0
2
4
6
1 8
1
1 10
-1
-0.8
-0.6
-0.4
-0.2
P
0
0.2
0:J
0.6
0 8
1
SE
Fig. 2. Comparison of results for load capacity obtained in the present study (-) obtained by Rao [S] (0); V=O; r=l; S=O; p,=3.
with those
Fig. 3. Pressure distributions of gas film in the mid-plane; S= 0.1; p= 2.0; p.= 3; T-= 1.
Y
V
Fig. 4. Dependence of centre of pressure on velocity for differing values of the feed rate parameter; S=O.l; p*=3; r=l. Fig. 5. Influence of slip parameter on load capacity; r= 1; Ij.= 3; p= 2.
244
parameter. The rate at which the load capacity decreases becomes less at higher values of the velocity parameter and/or slip parameter. For a higher slip parameter, the shear force exerted at the interface between the porous pad and the gas film (z = 0) is smaller, and hence a higher velocity distribution can be obtained. The decrease in the resistance of the gas film flow will eliminate the pressure inside it and this leads to a lower load capability. Figure 6 shows the variation in the load capacity as a function of the velocity parameter for several values of the feed rate parameter. For a fixed value of p, the load capacity decreases as the velocity parameter increases. The rate at which the load capacity decreases is larger for a smaller value of p. This is indicative of the fact that the effect of relative motion is more important for a thicker film. Examining the definition of the feed rate parameter shows that a higher value of B or a lower value of h tends to increase the resistance in the gas film, whereas H/K is propo~ional to the resistance in the porous medium. Therefore the feed rate parameter 0 represents the ratio of the resistance in the gas film to that in the porous medium. Thus the load capacity increases with the feed rate parameter at a specified velocity as illustrated in Fig. 6. The above analysis is based on the configuration of a square pad. It is interesting to note the influence of geometry. Let us fix the width of the pad which is in the same direction as the velocity and then study the effect of varying the pad length L. From F’ig. 7, it can be seen that a higher load capacity is obtained for a larger value of r or a longer L at a given velocity. The relative motion reduces the load capacity in high values of r, whereas it plays no role in very low values of Y (r < 0.2). As r decreases, the length of the pad normal to the velocity direction decreases. Hence the velocity effect becomes less significant. It may be argued that the load capacity differs in the cases where r = 0.2 and r= 5 or T-= 0.5 and r= 2 in the static mode (V= 0).Here it should be pointed out that this arises from the definition of the d~ensionless parameters /3and V based on the chara~te~stie length B regardless of L. For example, when the corresponding feed rate parameters are 50 for r= 0.2 and 2 for r= 5, the values of i/trare found to be the same. Details can be found in ref. 15.
v Fig. 6. Influence of feed rate parameter on load capacity; S=O.l; Fig. 7. Influence of pad shape ratio on load capacity; S=O.l;
v p,=3;
1’,=3;
r=l.
p=2.
245
5.2. Mass flow rate of gas Figures 8-10 show the dependence of C? on
V for different values of S, p and P-respectively. In general, the mass flow rate of gas increases with the velocity parameter and the slip parameter. However, it decreases with an increase in the feed rate parameter and pad shape ratio. The relationships in these figures are somewhat related to those in Figs. 5-7. It is interesting to investigate the relationship between e and w. For a given pressure supply, a greater pressure distribution in the film always leads to a higher load capacity, whereas it lowers the mass flow rate of the gas. It can be easily derived from a comparison of eqns. (18) and (23). Hence the mass flow rate shows an inverse trend with load capacity.
5.3. stiJpl.f?ss The stiffness denotes the ratio of variations in the load capacity and the film thickness. One may consider it as an index of the stability of the system subject to a sudden change in the applied load. Since a change in the film thickness h results in changes in all the chosen parameters p, V and S, the stiffness can be obtained by summing these three components as in eqn. (26). For example, a square pad with chosen parameters S=O.4, p= 2, p, = 3 and V in the range O-50 is analysed, and the stiffness as well as its components are plotted in Fig. 11. In this figure, the recorded values of & are positive, whereas those of r?; and &. are negative. Moreover, the contribution from E-, is found to be the most important one, especially for low values of V. As V increases, the stiffness decreases. This implies that relative motion reduces the stability. It is also expected that at a much higher value of V, the stiffness R will fall to zero. This point is one of the operating limits of the bearing. To study their effects on the stiffness, each parameter represented in Fig. 11 was varied while the other parameters were kept constant. Figures 12-14 show the respective effects of the slip parameter, pad shape ratio and supply pressure on the stiffness. Basically, the observed behaviour is similar to that in the W-V profiles in Figs. 5-7. Higher friction or pressure inside the gas film raises both the load capacity and the stability of the i.5
4 0
10
20
30 V
Fig. 8.
40
50
0
10
20
30
50
V
Influence of slip parameter on mass flow rate of gas; T= 1; P.=3;
8=2.
Fig. 9. Influence of feed rate parameter on mass flow rate of gas; S= 0.1; p. = 3; r = 1.
50
Fig. 10. Influence of pad shape ratio on mass
flow
rate of gas; S=O.l;
P,=3;
p=2.
Fig. 11. Variation of stiffness and its components with velocity parameter; S = 0.4; p, = 3; r = 1; p=2. 06
1
Of-A,
0
1 :‘:
cc
c:
3
/
1
:c
!
si:
:c
0
Fig. 12. Effect of slip parameter S on stiffness I?; r=l;
2C
22
V
v P,=3;
p=2.
Fig. 13. Effect of pad shape ratio r on stiffness k; S=O.4; ps=3;
p=2.
Fig. 14. Effect of supply pressure ps on stifbess I?; Fig. 15. Effect of feed rate parameter p on stiRness I?; S=O.4; p%=3; r=l.
bearings. For a fixed value of V, greater stiffness is observed with higher values of r and p,, and with smaller values of S. However, the stiffness decreases with the velocity parameter in all cases. In Fig. 15, the effect of the feed rate parameter is noted. For a given value of V, there is a maximum value of the stiffness at some point in the variation of the feed rate parameter. This is consistent with reports in the literature [6, 71 referring to a static
247
mode. Since the load capacity and the stiffness of the bearings should be as large as possible, the results from this work offer a reference guide for the selection of good operating conditions. 6. Conclusions (1) The effects of sliding, on a porous bearing, have been investigated analytically in this study. (2) Sliding reduces the load capacity. (3) Sliding increases the mass flow rate of gas. (4) Sliding reduces the stiffness. (5) When the pad shape ratio is very small, the effect introduced by sliding becomes less significant. (6) The load capacity increases with the feed rate parameter and shape ratio, whereas it decreases with the slip parameter. (7) The mass flow rate shows an inverse trend with load capacity. (8) In most cases, greater stiffness is obtained with a higher load capacity. References 1 H. J. Sneck, A survey of gas-lubricated porous bearings, J. Lubr. Technd, 90 (4) (1968) 804-809. 2 S. A. Sheinberg and V. G. Shuster, Analysis of non-rotating circular thrust bearings, Mach. Tool. (7J.S.S.R.), 31 (11) (1960) 24-29. 3 E. P. Gargiuio, Jr. and P. W. Gilmour, A numerical solution for the design of externally pressurized porous gas bearings: thrust bearings, J. Lubr. Technd., 90 (4) (1968) 810-817. 4 R. Tayler and G. K. Lewis, Effect of pad flexure on characteristics of aerostatic thrust bearings, 6th Int. Gas Bearing Symp., University of Southampton, U.K., March 1974, Paper C5. 5 D. C. Sun, Stability analysis of an externally pressurized gas-lubricated porous thrust bearing, J. Lube. Technol., 95 (4) (1973) 457-468. 6 N. S. Rao, Effect of slip flow in aerostatic porous rectangular thrust bearings, Wear, 61 (1980) 77-86. 7 K. C. Singh and N. S. Rao, Static characteristics of aerostatic porous rectangular thrust bearings, Wear, 77 (1982) 229-236. 8 G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. fluid Me&., 30 (1) (1967) 197-207. 9 E. M. Sparrow, G. S. Beavers and I. T. Hwang, Effect of velocity slip on porous-walled squeeze films, J. L&r. Technol., 94 (3) (1972) 260-265. 10 C. S. Yih, Dynamics of Nonhomogeneous F’luids, Macmillan, New York, 1965, pp. 196-199. 11 G. I. Taylor, A model for the boundary condition of a porous material, Part I, J. Fluid Mech., 49 (2) (1971) 319336. 12 G. S. Beavers, E. M. Sparrow and R. A. Magnuson, Experiments on coupled parallel flows in a channel and a bounding porous medium, J. Basic Eng., 92 (4) (1970) 843-848. 13 W. F. Ames, Numerical Methods for Partial D~b?ntiul Equations, Academic Press, New York, 2nd edn., 1977. 14 S. V. Patankar, Numerical Heat Transfer and FZuid Flow, Hemisphere, New York, 1980, pp. 36-39. 15 D. G. Lin, Dynamic characteristics of porous rectangular thrust bearings lubricated with gas, Master Thesis, Tatung Institute of Technology, Taipei, 1989.
Appendix
A: Nomenclature
halfwidth of porous medium mass flow rate of gas thickness of uniform film thickness of porous pad stiffness of bearing components of stiffness defined in eqn. (25) half-length of porous pad absolute pressure in bearing clearance absolute ambient pressure gas supply pressure equal to P2 LIB, pad shape ratio gas constant P/c& slip parameter absoluie temperature velocity components in bearing clearance relative velocity between bearing surfaces 12pBl_JIP,h’, velocity parameter velocity of gas crossing interface load capacity of bearing coordinates location of centre of pressure Greek
symbols
; K
P‘ P Pa
w
constant defined in eqn. (7b) 12 KB2/Hh”, feed rate parameter permeability coefficient of porous media viscosity of gas density of gas in bearing clearance density of gas under ambient conditions relaxation factor in SOR scheme
s~p~s~p~s
porous medium dimensionless quantities
I41 (1991)
235
235-248
Sliding effect of ,gas-lubricated porous rectangular thrust bearings S. P. Liaw Department (Taiwan)
and D. G. Lin of Mechanical
Engineering,
Tatung
Institute
of Technology,
Taipei
(Received August 15, 1989; revised May 22, 1990; accepted June 27, 1990)
Abstract A theoretical model has been developed to investigate the dynamic characteristics of porous rectangular thrust bearings lubricated with gas in which pressurized gas is pumped through a porous pad to lubricate the sliding elements. In the model a uniform temperature distribution in the gas lubricating film is assumed and a modified form of the BeaverJoseph slip velocity boundary condition is applied at the interface between the porous pad and the lubricating film. The pressure in the porous pad is obtained in a closed form when the pad thickness is very small compared with its other two dimensions. A modified Reynold’s equation was solved numerically using a finite difference method. The load capacity, mass flow rate of gas and stiffness are obtained in dimensionless forms and calculated numerically for different operating parameters. A higher load capacity is observed for a thinner film with and without relative motion. The effect of sliding is to reduce the load capacity and the stiifness, whereas it increases the mass flow rate of gas. It is also noted that when the pad shape ratio is very small, the effect introduced by sliding becomes less significant.
1. Introduction
Externally pressurized gas-lubricated thrust bearings have received a great deal of attention in recent years, primarily because the lubricant gas is cheap, stable and produces no pollutants during the working processes. As a result of these superior features they are nowadays widely used in the fields of textile machinery, food engineering, cryogenics and applications requiring very high temperatures. Depending on the method of feeding in the gas, they can be classified into two types: hole admission bearings and porous bearings. Conventional hole admission bearings are often difficult to justify in otherwise advantageous applications because of their low load capacity and stiffness relative to the gas pressure and flow required. Externally pressurized porous gas bearings are preferable to hole admission bearings owing to their simplicity and better performance characteristics. A detailed discussion has been given by Sneck [ 11. An early theoretical study of non-rotating circular thrust bearings was made by Sheinberg and Shuster [2]. Later on, Gargiulo and Gilmour [3] derived a theoretical solution to the problem of a circular thrust bearing 0043-1648/91/$3.50
0 Elsevier Sequoia/Printed in The Netherlands
with a three-dimensional flow of gas through the porous pad. The numerical results compared favourably with the available experimental data. Tayler and Lewis [4] considered the elasticity of the porous pad and the effect of pad flexure on the performance of a bearing. This effect was shown to decrease the load capacity. Using the simplification of considering the gas in the porous pad to flow only in the axial direction, Sun [5] was able to express the pressure distribution in terms of Bessel functions. The load capacity predicted by Sun’s model agrees well with that obtained by Gargiulo and Gilmour [ 31. The central assumption in the axial flow model was eventually proven to be valid particularly for thin bearings. Rectangular thrust bearings are very important in the design of machine slideways. The characteristics of an aerostatic porous thrust bearing in a static mode were investigated by applying slip [ 61 and non-slip [ 71 boundary conditions at the interface between the porous pad and the lubricating film. Making the assumption of uniform film thickness, the static characteristics were found to vary with the thickness of the lubricating film for differing bearing geometries. However, in any realistic study, the dynamic characteristics need to be considered as well as the static ones. In this study, a runner pad is assumed to move at a uniform velocity relative to a porous pad. Assuming the Beavers-Joseph slip condition [8, 91 to apply at the permeable surface, the steady state pressure distribution is obtained numerically by solving a modified Reynold’s equation. The variations in load capacity, flow rate of gas and stiffness with velocity are presented graphically. 2. Formulation
and analysis
A schematic diagram of a porous rectangular thrust bearing lubricated by pressurized gas is shown in Fig. 1. The compressed gas is fed in at a constant pressure P, from the bottom of the porous pad. The pressure distributions in the porous pad and the gas film are denoted as P ' and P respectively. The velocity components of the gas in the x, y and z directions
Fig. 1. Schematic
diagram of a porous rectangular thrust bearing.
237
are u’, V’ and W’ respectively in the porous pad, and u, o and w respectively in the film. Usually the gas film is very thin, and so the corresponding Reynold’s number is quite small and the gas flow is expected to be laminar. In cases where the temperature variation is small and can be ignored, the flow is barotropic and the density is a function of pressure alone. The thermodynamic behaviour of the lubricating gas is assumed to follow the ideal gas law. For simplicity, the film thickness h is assumed to be uniform during the analysis, even though the inflation effect may not be avoided when the movement of the runner pad is very fast. However, the involvement of the force balance between the applied load and the pressure distribution is more complicated and is ignored in the present study. In addition, the thickness of the porous pad is very small when compared with its other two dimensions. This permits the assumption that the gas will flow only in the z direction in the porous pad. The porous material is homogeneous and isotropic. Darcy’s law [ 10 J can be applied to simulate the flow for slow motion in the pad. Neglecting gravitational force in the z direction, the Darcy equations are written as
V’=.-
K
__
aP'
fii)y
where K is the permeability coefficient of the porous pad and p is the viscosity of the gas. Both K and p are constants in the analysis. With these assumptions, we can now investigate the flow in the porous pad and the lubricating film.
The flow in the porous pad is governed by the equation of cont~ui~
Substitution of the equation of state and eqn. (1) into eqn. (2) yields
Li2P ‘2 -+ ax2
a2p I2 a2p’” +yg-= rty2
0
Assuming a very small thickness for the porous pad, the above equation can be written in an abbreviated form as -
=o Ck?
subject to the boundary conditions
238
PI=P,
at
*::-H
P' =P
at
z=o
Hence the solution of eqn. (4) is readily obtained as p$n ‘p2
_ (ps” _P2)
;
(5)
where P, and P are the suppIy and film pressures respectively. In a very thin film of pressurized gas, the pressure dist~bution does not vary in the z direction. Thus, on the basis of scale analysis, the steady momentum equations are expressed as
and @b)
No-slip conditions are assumed for the impermeable runner surface and the BeaversJoseph slip boundary conditions [9] are applied to the surface of the porous bearing pad. They are z/J=0
u= -u
at
z=h
f3W
and
where U is the relative velocity between the bearing surfaces and cy is a dimensionless constant which depends on the characteristics of the porous medium. The value of CKis independent of the film thickness and the fluid properties [ 111 and it can be determined experimentally [ 121. The solutions of eqns. (6a) and (6b) subject to the boundary conditions are
and u=
_!-i?
(+_h2)
2F ?!d
- s$-&--(z-h)
In the above equations S, defined as K?&, is the slip parameter. Since s is equal to zero, the condition described in eqn. (?b) reduces to the no-slip boundary condition approp~ate to a solid wall. Such a si~ation is referred
239
to as general Couette flow. However, as S approaches infinity, a complete slip condition is prescribed and no shear stress is applied on the fluid. The steady state continuity equation in the gas film is written as -Kpu)
+ -Kpu)
ax
I
ay
Kpul)
-0
(9)
az
Since the pressure and temperature do not vary in the z direction, integration
of the above equation gives
Substituting the velocities obtained in eqns. @a) and (8b) into eqn. (10) yields
= - 24@wo
(111
where w. is the velocity crossing the interface of the porous pad and gas film. It can be obtained by applying eqns. (1) and (5): wg= f&
(P,2-P2)
(12)
Introducing the following dimensionless quantities: 2=xlB
B=yIB P, = P,lP,
Is = PIPa
eqn. (11) takes the form
-a2P + -a2P2 +v --2s+1 aP =p g_ 4s+1 a* a2 372
(P-F,“)
(13)
In the above equation, V and /3 denote velocity and feed rate parameters respectively, and they are defined as v=
12mJ P,h”
and
12KB2 p= x
Since no relative motion is specified in the y direction, the pressure distribution in the system is symmetric relative to the x axis. When the outer boundaries of the lubricating film are assumed to be adjacent to the atmosphere, the boundary conditions are P=l
at
x=+1-
and
--r
P=l
at
y= f7”
and
-1<*<1
at
jj=o
and
-1<2<1
als -= ~
0
in which r, defined as L/B, is the shape ratio of the porous pad.
(14)
240
3. Numerical
approach
The solution of eqn. (13) satisfying the boundary conditions in eqn. (14) gives the pressure distribution in the lubricating film. Because eqn. (13) is not linear, it is not an easy task to obtain the analytical solution. Thus it is solved numerically by using a finite difference method. Setting & =p2 and choosing Ax=Ay= A, eqn. (13) when written as a five-point Gauss-Seidel finite difference equation is 6L + I(i? j) =
V(2S+l) A I + *(4S* 1) (&n(i, j))‘”
+7.
V(ZS+l)
l,io 1cAdi+
A
[
4(4S+ 1) (Q&j>)*n
+&,(i,j+
l)+Q,+,(Gj-11+-
I
QTa+,(i- L%?T p(S + 1)A2ps2 4s+1 (15)
It can be further extended using the SOR method [ 131 by ~trodu~~g relaxation factor w:
Q,+l(i,j)=Qn(i,j)x(1-,>+oxQ,,l(i,j)
a
(16)
where o ranges between 1 and 2 depending on the chosen mesh space. As proposed by Patankar [ 141, if all the coefficients in the discretizatio~ equation are positive, then the numerical scheme is stable. Following this criterion, in the present study the mesh space is chosen to be 1130. Thus, in the range of V less than 50, all the coefficients in eqn, (15) are always positive, and it is stable during the numerical calculations. The value of the pressure is iterated until it converges at an absolute iteration error of less than lo-” and, because of symmetry, only half of the bearing is considered.
4. Characteristics 4.1. Load
of the bearing
capacity
The load capacity of the bearing is determined by integrating the pressure over the pad:
(17) 0 -B
and the dimensionless form is defined as
241
1
ss )‘
(I’- 1) &d?J
Ij7=
01 0
W 4BL(PS -P,)
=
r
-1
2(p,-
1)
(18)
which represents the ratio of the actual load to the maximum possible load. It varies from zero to unity. 4.2. Centre
of pressure
The location of the centre of pressure can be calculated by taking the moment of pressure force LB
2
ss
(P - P,)x
dxdy
0 -B
xc=
(1%
W
or in a dimensionless expression 1
ss I’
(I’- l)z? tid@ 0 -1 L 2(P, - l)W 0r
,&=!5= B
(20)
4.3. Mass flow rate of gas The mass flow rate can be calculated by integrating the velocity w. over the pad surface LB
G=2
ss
wo
(21)
bdiv
0 -B
Making use of eqn. (12) and the equation of state, eqn. (21) becomes LB
G=
--$$sso _a(P.2-P2)
tidy
(221
Hence the dimensionless form gives (23)
4.4. st@?less The stiffness of the bearing is defined as the negative change in load capacity caused by a change in gas film thickness:
(24) Since the sliding parameter V, feed rate parameter p and slip parameter S are all functions of h> the stiffness K can be expressed in a pfaffian form: (25) or in a ~ensio~ess EC
Kh rnL(P, -P,)
form
=fi,+IQ+K,
(26)
where
$? can be calculated from the curves of the relations V-m,
/3-w and 5’4%
5. Results and discussion Supplied from the bottom of the porous pad as shown in Fig. 1, the compressed air flows through the porous medium and film clearance, and is finally exhausted to the atmosphere. One may imagine this process as analogous to Ohm’s Law with two resistances (porous rne~u~ub~cat~g film) in a series circuit. The pressure difference is equivalent to the applied voltage and the flow velocity to the current. If the supply pressure is kept constant, a greater pressure difference in the gas film will be obtained when the resistance is higher in it and/or lower in the porous medium. This will result in a higher load capacity. 5.1. Load capacity In order to ensure the correctness of the present analysis, the limiting solutions for no relative motion were compared with those of other investigators [6, 71. In this case, the load capacity was found to increase with the feed rate parameter for a square porous pad. In other words, a higher load capacity is obtained with a thinner film. This is consistent with the results reported by Rao [ 61 and is shown in Fig. 2.
243
In the absence of relative motion, the pressure distribution is symmetric with a maximum located at the origin. The maximum pressure is observed to decrease with the velocity parameter V and the location of the maximum is shifted in the same direction as the velocity. This is illustrated in Fig. 3 for the pressure distribution in the mid-plane. If the total load is supplied at the geometrical centre, a moment will be exerted by the unsymmetrical pressure distribution, and this may cause the runner-pad to incline slightly as it achieves a new floating position. However, the effect of the applied load is excluded in this study. The distance of the centre of pressure from the origin increases with the relative motion, whereas it decreases with an increase in the feed rate parameter. Hence the velocity effect is smaller for a thinner film. This is plotted in Fig. 4. In Fig. 3, it can also be seen that the area under the pressure curve becomes smaller for a larger velocity parameter. This shows that the decrease in the load capacity is due to the presence of relative motion. The dependence of the load capacity on the velocity parameter for differing slip parameters is indicated in Fig. 5. It can be seen that the load capacity decreases with an increase in the velocity parameter and/or slip 0.6
0.4 w 02
op
I
I
I
0
2
4
6
1 8
1
1 10
-1
-0.8
-0.6
-0.4
-0.2
P
0
0.2
0:J
0.6
0 8
1
SE
Fig. 2. Comparison of results for load capacity obtained in the present study (-) obtained by Rao [S] (0); V=O; r=l; S=O; p,=3.
with those
Fig. 3. Pressure distributions of gas film in the mid-plane; S= 0.1; p= 2.0; p.= 3; T-= 1.
Y
V
Fig. 4. Dependence of centre of pressure on velocity for differing values of the feed rate parameter; S=O.l; p*=3; r=l. Fig. 5. Influence of slip parameter on load capacity; r= 1; Ij.= 3; p= 2.
244
parameter. The rate at which the load capacity decreases becomes less at higher values of the velocity parameter and/or slip parameter. For a higher slip parameter, the shear force exerted at the interface between the porous pad and the gas film (z = 0) is smaller, and hence a higher velocity distribution can be obtained. The decrease in the resistance of the gas film flow will eliminate the pressure inside it and this leads to a lower load capability. Figure 6 shows the variation in the load capacity as a function of the velocity parameter for several values of the feed rate parameter. For a fixed value of p, the load capacity decreases as the velocity parameter increases. The rate at which the load capacity decreases is larger for a smaller value of p. This is indicative of the fact that the effect of relative motion is more important for a thicker film. Examining the definition of the feed rate parameter shows that a higher value of B or a lower value of h tends to increase the resistance in the gas film, whereas H/K is propo~ional to the resistance in the porous medium. Therefore the feed rate parameter 0 represents the ratio of the resistance in the gas film to that in the porous medium. Thus the load capacity increases with the feed rate parameter at a specified velocity as illustrated in Fig. 6. The above analysis is based on the configuration of a square pad. It is interesting to note the influence of geometry. Let us fix the width of the pad which is in the same direction as the velocity and then study the effect of varying the pad length L. From F’ig. 7, it can be seen that a higher load capacity is obtained for a larger value of r or a longer L at a given velocity. The relative motion reduces the load capacity in high values of r, whereas it plays no role in very low values of Y (r < 0.2). As r decreases, the length of the pad normal to the velocity direction decreases. Hence the velocity effect becomes less significant. It may be argued that the load capacity differs in the cases where r = 0.2 and r= 5 or T-= 0.5 and r= 2 in the static mode (V= 0).Here it should be pointed out that this arises from the definition of the d~ensionless parameters /3and V based on the chara~te~stie length B regardless of L. For example, when the corresponding feed rate parameters are 50 for r= 0.2 and 2 for r= 5, the values of i/trare found to be the same. Details can be found in ref. 15.
v Fig. 6. Influence of feed rate parameter on load capacity; S=O.l; Fig. 7. Influence of pad shape ratio on load capacity; S=O.l;
v p,=3;
1’,=3;
r=l.
p=2.
245
5.2. Mass flow rate of gas Figures 8-10 show the dependence of C? on
V for different values of S, p and P-respectively. In general, the mass flow rate of gas increases with the velocity parameter and the slip parameter. However, it decreases with an increase in the feed rate parameter and pad shape ratio. The relationships in these figures are somewhat related to those in Figs. 5-7. It is interesting to investigate the relationship between e and w. For a given pressure supply, a greater pressure distribution in the film always leads to a higher load capacity, whereas it lowers the mass flow rate of the gas. It can be easily derived from a comparison of eqns. (18) and (23). Hence the mass flow rate shows an inverse trend with load capacity.
5.3. stiJpl.f?ss The stiffness denotes the ratio of variations in the load capacity and the film thickness. One may consider it as an index of the stability of the system subject to a sudden change in the applied load. Since a change in the film thickness h results in changes in all the chosen parameters p, V and S, the stiffness can be obtained by summing these three components as in eqn. (26). For example, a square pad with chosen parameters S=O.4, p= 2, p, = 3 and V in the range O-50 is analysed, and the stiffness as well as its components are plotted in Fig. 11. In this figure, the recorded values of & are positive, whereas those of r?; and &. are negative. Moreover, the contribution from E-, is found to be the most important one, especially for low values of V. As V increases, the stiffness decreases. This implies that relative motion reduces the stability. It is also expected that at a much higher value of V, the stiffness R will fall to zero. This point is one of the operating limits of the bearing. To study their effects on the stiffness, each parameter represented in Fig. 11 was varied while the other parameters were kept constant. Figures 12-14 show the respective effects of the slip parameter, pad shape ratio and supply pressure on the stiffness. Basically, the observed behaviour is similar to that in the W-V profiles in Figs. 5-7. Higher friction or pressure inside the gas film raises both the load capacity and the stability of the i.5
4 0
10
20
30 V
Fig. 8.
40
50
0
10
20
30
50
V
Influence of slip parameter on mass flow rate of gas; T= 1; P.=3;
8=2.
Fig. 9. Influence of feed rate parameter on mass flow rate of gas; S= 0.1; p. = 3; r = 1.
50
Fig. 10. Influence of pad shape ratio on mass
flow
rate of gas; S=O.l;
P,=3;
p=2.
Fig. 11. Variation of stiffness and its components with velocity parameter; S = 0.4; p, = 3; r = 1; p=2. 06
1
Of-A,
0
1 :‘:
cc
c:
3
/
1
:c
!
si:
:c
0
Fig. 12. Effect of slip parameter S on stiffness I?; r=l;
2C
22
V
v P,=3;
p=2.
Fig. 13. Effect of pad shape ratio r on stiffness k; S=O.4; ps=3;
p=2.
Fig. 14. Effect of supply pressure ps on stifbess I?; Fig. 15. Effect of feed rate parameter p on stiRness I?; S=O.4; p%=3; r=l.
bearings. For a fixed value of V, greater stiffness is observed with higher values of r and p,, and with smaller values of S. However, the stiffness decreases with the velocity parameter in all cases. In Fig. 15, the effect of the feed rate parameter is noted. For a given value of V, there is a maximum value of the stiffness at some point in the variation of the feed rate parameter. This is consistent with reports in the literature [6, 71 referring to a static
247
mode. Since the load capacity and the stiffness of the bearings should be as large as possible, the results from this work offer a reference guide for the selection of good operating conditions. 6. Conclusions (1) The effects of sliding, on a porous bearing, have been investigated analytically in this study. (2) Sliding reduces the load capacity. (3) Sliding increases the mass flow rate of gas. (4) Sliding reduces the stiffness. (5) When the pad shape ratio is very small, the effect introduced by sliding becomes less significant. (6) The load capacity increases with the feed rate parameter and shape ratio, whereas it decreases with the slip parameter. (7) The mass flow rate shows an inverse trend with load capacity. (8) In most cases, greater stiffness is obtained with a higher load capacity. References 1 H. J. Sneck, A survey of gas-lubricated porous bearings, J. Lubr. Technd, 90 (4) (1968) 804-809. 2 S. A. Sheinberg and V. G. Shuster, Analysis of non-rotating circular thrust bearings, Mach. Tool. (7J.S.S.R.), 31 (11) (1960) 24-29. 3 E. P. Gargiuio, Jr. and P. W. Gilmour, A numerical solution for the design of externally pressurized porous gas bearings: thrust bearings, J. Lubr. Technd., 90 (4) (1968) 810-817. 4 R. Tayler and G. K. Lewis, Effect of pad flexure on characteristics of aerostatic thrust bearings, 6th Int. Gas Bearing Symp., University of Southampton, U.K., March 1974, Paper C5. 5 D. C. Sun, Stability analysis of an externally pressurized gas-lubricated porous thrust bearing, J. Lube. Technol., 95 (4) (1973) 457-468. 6 N. S. Rao, Effect of slip flow in aerostatic porous rectangular thrust bearings, Wear, 61 (1980) 77-86. 7 K. C. Singh and N. S. Rao, Static characteristics of aerostatic porous rectangular thrust bearings, Wear, 77 (1982) 229-236. 8 G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. fluid Me&., 30 (1) (1967) 197-207. 9 E. M. Sparrow, G. S. Beavers and I. T. Hwang, Effect of velocity slip on porous-walled squeeze films, J. L&r. Technol., 94 (3) (1972) 260-265. 10 C. S. Yih, Dynamics of Nonhomogeneous F’luids, Macmillan, New York, 1965, pp. 196-199. 11 G. I. Taylor, A model for the boundary condition of a porous material, Part I, J. Fluid Mech., 49 (2) (1971) 319336. 12 G. S. Beavers, E. M. Sparrow and R. A. Magnuson, Experiments on coupled parallel flows in a channel and a bounding porous medium, J. Basic Eng., 92 (4) (1970) 843-848. 13 W. F. Ames, Numerical Methods for Partial D~b?ntiul Equations, Academic Press, New York, 2nd edn., 1977. 14 S. V. Patankar, Numerical Heat Transfer and FZuid Flow, Hemisphere, New York, 1980, pp. 36-39. 15 D. G. Lin, Dynamic characteristics of porous rectangular thrust bearings lubricated with gas, Master Thesis, Tatung Institute of Technology, Taipei, 1989.
Appendix
A: Nomenclature
halfwidth of porous medium mass flow rate of gas thickness of uniform film thickness of porous pad stiffness of bearing components of stiffness defined in eqn. (25) half-length of porous pad absolute pressure in bearing clearance absolute ambient pressure gas supply pressure equal to P2 LIB, pad shape ratio gas constant P/c& slip parameter absoluie temperature velocity components in bearing clearance relative velocity between bearing surfaces 12pBl_JIP,h’, velocity parameter velocity of gas crossing interface load capacity of bearing coordinates location of centre of pressure Greek
symbols
; K
P‘ P Pa
w
constant defined in eqn. (7b) 12 KB2/Hh”, feed rate parameter permeability coefficient of porous media viscosity of gas density of gas in bearing clearance density of gas under ambient conditions relaxation factor in SOR scheme
s~p~s~p~s
porous medium dimensionless quantities