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Variable viscosity and density effects in a porous hydrostatic thrust bearing
Wear, 69 (1981) 261 - 275 0 Elsevier Sequoia S.A., Lausanne
261 -Printed
in The Netherlands
VARIABLE VISCOSITY AND DENSITY HYDROSTATIC THRUST BEARING
EFFECTS
IN A POROUS
J. S. YADAV Department of Mathematics, Nehru Postgraduate College, Chhibramau, 209721 (India)
Farrukhabad
V. K. KAPUR Mathematics Department, (India) (Received
Harcourt Butler Technological
March 18, 1980; in revised
Institute, Kanpur 208002
form July 3, 1980)
Summary Interactions of inertia, the viscosity-density variation due to temperature and the bearing material porosity were investigated, Numerical results for the pressure distribution and load capacity were obtained and were compared with experimental and earlier theoretical results. The analytical results obtained are in good agreement with the experimental work.
1. Introduction The performance of thrust bearings has been studied extensively. The unconventional thrust bearing was analysed by Dowson [l] who considered only the isothermal aspect although the rotor was considered to be rotating at high speed. Comparison of the theoretical results of Dowson with the experimental results of Coombs and Dowson [2 ] showed a discrepancy. Inertia effects led to cavitation and thereby rupture because of the subatmospheric pressure. Hughes and Osterle [3] considered the exponential viscosity temperature relation but ignored inertia effects and the effect of the bearing step. Osterle and Hughes [4] considered heat transfer from the fluid to the surroundings but neglected the heat convection term completely because the viscosity was kept constant. However, in experimental work, when there is heat transfer, the viscosity is bound to alter. Ting and Mayer [ 5 ] considered this aspect by assuming only a radial temperature variation because the bearing clearance is small compared with the radial span. A justification of this aspect was given by Sneck [6] for the face seal problem. For bearing manufacture, design criteria and operating conditions must be known. Ting and Mayer [5] showed the interactions of inertia and thermal effects in thrust bearings. There are differences between the theoretical and experimental expressions for the pressure distribution between
262
the bearing surfaces. Kapur and Yadav [ 71 indicated that material porosity may explain these differences. The analysis including porosity effects demonstrates that the deviation between experimental and theoretical results is reduced and that coincidence of the results is obtained up to Ti= 0.6, although there is a marked deviation near the periphery of the bearing which has not been explained. In the present analysis an explanation is attempted by considering the temperature-density variation together with the viscosity variation in a hydrostatic porous recessed thrust bearing using the Beavers-Joseph [8] boundary condition at the permeable surface. The experimental and theoretical justifications of this have been given by Beavers et al. [9] and Saffmann
WI. 2. Analysis The system analysed is shown in Fig. 1. For steady incompressible laminar flow of a lubricant between the parallel surfaces of a hydrostatic porous recessed thrust bearing the momentum equations which take into account rotational inertia effects are pv2 -__ r =--
ap ar
a2u +p a2 -5
(1)
The boundary
conditions v=o
U=O
au
are
uu
1 a j- &P~u) +
0
v = rc2
-,,=p
and the continuity
W=
equation
atz=O
w=o
is
a;(pw)=o
From eqns. (1) and (2) the radial and tangential
prQ222
+ 12ph2
(2)
atz=h
h3-z3+-----
(3) velocity
profiles are
3h3
1 +a(3
(4)
253
2
6 W
n
t
CO
Fig. 1. Schematic diagram of recessed thrust bearing.
and v = rSLz/h
(5)
where p is the mean pressure across the film and is a function of r only. The variations in the viscosity and the density of the lubricant are assumed to be insignificant in the axial direction (because the lubricant film is very thin) but to vary radially according to the temperature distribution. The energy equation used to study the simultaneous density and viscosity variations for an incompressible lubricant is
From eqns. (4) - (6) integration of the energy equation across the film yields 27rrqw” + ,wC,Q g
I
l+
-where
prf2 2h3
2%
1+ I
7 (1 +ao)2
(7)
Integrating the continuity equation across the film thickness and using the boundary condition for w gives
shl
o
;
a jr
(pru)dz
(9)
= 0
From eqns. (4) and (9) the differential equation for the pressure distribution is obtained as d
prdj? -- dr i 1-1 dr )
(10)
where f(W-J) =
1 + 5/(1 + au)
(11)
1 + 3/(1 + au)
2.1. Temperature distribution It is assumed that the bearing surfaces are perfect insulators (i.e. q,” = 0) and that the heat generated is completely carried away by the lubricant. To consider the viscosity and density effects due to the temperature variation between the bearing surfaces we assume exponential relations: P = p. exp {--B(T - To1
(12)
P =
(13)
and PO
expf--6(T-
To)1
where po, p. and To are the viscosity, density and temperature respectively of the lubricant supply. From eqns. (12) and (13) the combined relation is s = so exp{-(0
- 6)(T - To)}
(14)
where s =iJlp
(15)
so = POIPO
By letting qw ” = 0, substituting dji/dr from eqn. (9) and ignoring higher order terms the energy equation (eqn. (6)) becomes + J_ 700
(rS22)2h3 s
HWJ)
(16)
265
where 3 (1 + 5/(1 + ao)j2{l + 3/(1 + aa)2} (1 + 3/(1 + au)}2 I 400
#(a/J) = 700
3
11 + 5/(1 + au)}{1 + 5/(1 + au)2} + 1 + 3/(1 + au)
-200 1
7
+ -112 1l+
2
(17)
(1 + au)2 II
From eqns. (14) and (16), s(r) is obtained as s(r) = so {(A - B)(r4 - r14) + l>-l
rI
(18)
rl
(19)
while the viscosity variation is p(r) = p. {A(r4 - r14) + 1)-l where @a
A= 2p
2P0
l%i12j.10
B=
ohgC,Q
(20)
2pokGQ
2.2. Pressure dktribu tion Integrating eqn. (9) after substituting the values of s(r) and p(r) from eqn. (18) and (19) gives the expressions for the pressure distribution on both sides of the step: 3 - - -poG!2r2f(u,u) P1 - 20
+ C, In r +DI
r. < r < r1
(21)
and A-B p2 = _ A
3 - PoQ2f(wJ) + 20 tan-’ {Ar4/(1 - Ar14)}l12
c2so
+
4((A -B)r14
-1)
(A(1 - Ar, 4))1/2
+
In
+
(A
-Nr4
-r14)
I
+D
1
r4
I
r,
2
(22)
where C1, C2, D1 and 02 are the constants of integration to be determined by using the following boundary conditions: &@0)
= 0
iWd
=P2W
F,(R)
= 0
Q&-l)
= Qdr1)
(23)
266 These constants
of integration
are
-$(l +Sf(a,o)[l - ;; +$( $ -1) -
Cl =
_
B
tan-‘{AR*/(l
-Ar14)}1/2
AR2
{Ar14/(l
-tan-l
I)
-Ar14)}1/2
{A(1 -Ar14))1’2
ld”Sf(a,a)
c2
_
B
tan-l {AR4/(1 - Ar14))1’2 -tan-l
-AR2
{Ar14/(1 - Ar14)}1/2
{A(1 - Ar14))1’2 (25)
Ill
=po 1 -Sf(a,o)‘02
+
1
2
B
I
-Ar14))1/2
tan-’ {AR4/(1
AT
--tan-l
{Ar14/(1 -Ar14)}l/2~
{A(1 -Ar14)fi1’2
11
X In r.
X
(26)
and D2 = -poSf(a,rr)
1I
+-
B + -
A
B
tan-’ {AR4/(l
AR2
-Ar14)}l12
{(1- Arl*)A)112
Q3P0
I ’
+gg-l)-
A _
B
tan+ {AR4/(1
-Ar14)J1’2
AR2
{A(1
x ln[ {(A -B)(R4
-rx4)
+
-tan-’
(Ar14/(1 -Ar14)j1L2
-Ar14)}1f2
I) X
11/R41
4((A -B)r14
- 13.
+ cy3 ln[r14{(A
-B)(R4
(27)
where A = In
4((A -B)r14 and
-rl*) - 1)
+ 1)/R4] (28)
267
3 20
s=
poR2a2 (29) DO
The following r
nondimensional
R
jj=
R
R -
5
js2=
‘-‘2 Is,
PO
Q=
are defined:
rl Fl = -
r0 To = -
f=-
p, =
quantities
(30)
-Go&
nh3jTo
20p02R ‘0
20p02R 26
$=
po2gC,hh-
po2&h4
and the pressure distribution
in the two regions takes the dimensionless
ji1=1+Sf(a,o)(F2-@)+ + JJ tan-l
Bi
$ {@S/Cj)/(l
{(PS/c?)(l
(
Sf(a,o)
I
F~2-1-;(F~~-1)+
- &!ZZ14/&)}1’2_ - PS~14/&)P2
- tan-l ((&W14/Q)/(1 -j7SF14/Cj))1’2 )-ll)ln( MWG)(l - DSG4/&)P2
+ i
lo i
tan-l { (&S/Q)/(l W/8)(1
- iLsF~4/9)}1’2
_ tan-l {@Sr”/&)/(l {(PlSQ)(l +
- fBF14/&)}1’2 _
-&S~14/&)}1’2
- 6*14/9)P2
iI +
-$(Sf(a,o) IFo2-l-$(F~2-l)+
+ p
ai
tan-l {(&S/@/(1 - &S714/&)}1’2 _ t(PSIQ)(l
-‘&sF~4/Q)}1’2
_ 1 x - tan-l (($SF14/Q)/(1 -&SF14/Q))1/2 NW&l - iWG4/&)P2 )I )
:I
form
(32)
cu3In(F,*($ - S)(l - Fld)S/& + 1) (33)
4~~B~~)SF~~~~~1~ 2.3. Load-carrying capacity The normalized loadcarrying capacity is given by
Sf(a,o)(F(J2 - 1) - 1 1 A
+ +
~3~S~(~,~)(~~2
2 (G2
-I)-
1)
@-F)(l
4A {(a - s)SF14/6j - 1)
--Fl*)s Q
112
- tan-l ~(~S~~*/~)/(l -&SF14/&)]1’2 X 1 NWQW - ~~~4/~~}1’2
f
Qa,o)i.(1 -r;&-P
;(I
-+)
+ (1 -i’*2)
x
I’= i
I +
269
X
tan-l
l/2
- w-1
(B - F)SF14 Jo i 1 -@-@-@;4/&
II2 i
t (35)
The volume flow rate Q of the lubricant through the bearing is given by Q = jkrudz 0
(36)
In dimensionless form it is
%,Q f+-.ⅈ,
(37)
270 Newton’s method is used to solve for ($ and a function F is defined as Ly3
.l
I
(33)
and (39) where F’ = U/da and n is a positive integer. Iterations are performed to obtain the value of @ by guessing Q,. It generally took three to four iterations to reach an accuracy of the order of 10-4. 3. discussion Figures 2 - 4 show the numerical values of the pressure distribution for various values of the rotational parameter S in order to discuss the interactions of the density-~s~osity variations with the pe~o~ance of porous recessed thrust bearings with a porous permeability # = 10.85 X lo-l1 cm2, slip constant a = 0.10 and film thickness h = lo-*cm. Kapur and Yadav [?I showed that up to a step pssition q = 0.4 the pressure distribution between the bearing surfaces approaches the experimental results of Coombs and Dowson [Z] , although the deviation is appreciable after 7 = 0.6. Inclusion in the analysis of the density variation due to temperature modified the analysis and the results for the pressure d~tributions before and after the step position are in close agreement with the ,experimental results [ 25 . This implies that the density variation plays a dominant role in porous bearing problems. The pressure variation is similar to that of the Ting and Mayer [5] and the Kapur and Yadav [ 7 ] theories. The pressure is never lower in the present theory than it is in previous theories [ 5,7] for any given conditions. Figures 3 and 4 illustrate the behaviour of the pressure distribution as a function of the step position. With a shift of the step towards the periphery of the bearing the relative increase in the pressure distribution is appreciable. From Fig. 3, beyond the step position there is a relative increase in the pressure distribution although the pressure decreases at every point between the bearing surfaces. In Fig. 4 this relative difference in pressure d~t~bution decreases cont~uously after the step position and finally attains its value at the periphery. The region 0.45 < f < 1 of negative pressure has reduced appreciably owing to the interaction of the density variation and the
271
0.0
0.4
0.6
1.2
r
0.0
0.4
0.8 i
Fig. 2. Radial pressure distribution in a thrust bearing (~1 = 2.18; s= 8.81; PO = 0.05; q = 0.40;6 = 0.001 “C-1): present theory, - - -, Ting and Mayer’s [ 51 theory; -., Kapur and Yadav’s [ 71 the&y ; 0, experimental results [ 21. Fig. 3. Radial pressure distribution in a thrust bearing (CY= 5.0; p = 0.708; Q, = 0.05; q = 0.20; 6 = 0.001 “C-1): present theory; - - -, Ting and Mayer’s [ 51 theory; -. -, Kapur and Yadav’s [ 71 the&y; 0, *, 0, experimental results.
1
Fig. 4. Radial pressure distribution in a thrust bearing (a! = 5.0; F= 0.708; iu = 0.05; Fl = 0.60; 6 = 0.001 “C-l): present theory; - --, Ting and Mayer’s [ 51 theory; -. -, Kapur and Yadav’s [ 71 thebry.
272
viscosity variation for all values of the rotational parameter S considered. The results for pressure and density-viscosity variation due to temperature are in good agreement with the experimental results [ 21 for S = 0,O .5 and 1.0. Simultaneous examination of Figs. 2 - 4 shows that performance can be improved significantly by considering the density variation due to temperature together with the viscosity variation. For the present theory, because the pressure distribution increases at each and every point between the bearing surfaces and approaches very close to the experimental results, it is expected that the load parameter w will also be more comparable with that of the porous thrust bearing. This is confirmed in Fig. 5. For each step position the load parameters are comparable with those of the porous bearing as are the parameters for the thermal analysis of the bearing. It is concluded that the relative difference in load parameter between the present theory and previous theories [ 5,7] is not appreciable up to a step position F1 < 0.6 but an appreciable difference is observed by just shifting the step further towards the periphery of the bearing. Figures 6 and 7 show the pressure distribution as a function of F with i=l = 0.55 and 0.70 respectively. The results are similar to those presented in Figs. 3 and 4. However, the region of negative pressure obtained in Figs. 2 and 3 is reduced further owing to the consideration of the density-temperature variation even for given design parameters. Hence it is concluded that under the present circumstances a comparatively high pressure is attained. The variations in the temperature distribution illustrated in Figs. 8 and 9 with respect to the radial distance T show a higher temperature generation than do previous theories [ 5,7] . The temperature further increases owing to an increase in the rotational parameter S but the temperature between the
I
0.4
I
/
O-6
O-8
I 1.0
il
Fig. 5. Load parameter __ present theory; theorb.
us. step positions
(a = 9.25;fl=
3.92;70
- - -, Ting and Mayer’s [ 51 theory;
= 0.493;6
= 0.001
- . -, Kapur and Yadav’s
“C-l): [7 ]
273
0.6-
0.4
0.6
0.8 i
Fig. 6. Radial pressure distribution in a thrust bearing ((.u= 9.25;B= 3.92; iTo= 0.493; Fl = 0.55;6 = 0.001 “c-l): present theory; - - -, Ting and Mayer’s [ 51 theory, -., Kapur and Yadav’s [ 71 theory. Fig. 7. Radial pressure distribution in a thrust bearing ((Y = 9.25; p= 3.92; 70 = 0.493; ?=I = 0.70;s = 0.001 c-l): present theory ; - - -, Ting and Mayer’s [ 5 ] theory; -. -, Kapur and Yadav’s [ 71 theory.
bearing surfaces decreases owing to a shift of the step towards the circumference of the bearing, a result similar to that of previous theories. It may be concluded that in order to reduce the difference between theoretical and experimental results the density-viscosity and porosity interactions must be incorporated in thrust bearing design considerations.
Nomenclature
1,B Cl, c2 c, &,D2
f(Q, 0) F &? h K n P P
slip constant quantities defined in eqn. (20) constants of integration defined in eqns. (24) and (25) specific heat of the incompressible lubricant at constant volume constants of integration defined in eqns. (26) and (27) porous factor defined in eqn. (11) function defined in eqn. (38) gravitational acceleration minimum film thickness thermal conductivity of the lubricant positive integer fluid pressure mean pressure across the film
214 1.6
1.4 t
-l I.3
P t-
5
Fig. 8. Radial temperature distribution in a thrust bearing (a = 9.25;p= 3.92; PO = 0.493 present theory; - - -, Ting and Mayer’s [ 51 theory; Fl = 0.55;6 = 0.001 “c-l): -. -, Kapur and Yadav’s [ 7 ] theory. Fig. 9. Radial temperature distribution in a thrust bearing (a! = 9.25;$= 3.92; Po = 0.493 present theory; - --, Ting and Mayer’s [ 51 theory; Pr = 0.70; 6 = 0.001 “c-l): -. -, Kapur and Yadav’s [7] theory.
PO
I% 62 P
PI, & ,I
4w
:1,QZ Q
r, 0, z r0
r1 P
PO ii
R s SO
S T
To u, v, w W
iv a!
mean pressure of the lubricant supply pressures on either side of the step dimensionless pressure across the film quantities defined in eqns. (30) total local heat transfer flux from the bearing to the surroundings volume flow rate volume flow rates on either side of the step volume flow rate parameter defined in eqns. (30) cylindrical coordinates radius of the supply hole recess radius r/R, radius ratio ro /R, radius ratio (dimensionless) r1 /R, radius ratio (dimensionless) outer radius of the bearing quantity defined in eqns. (15) quantity defined in eqns. (15) inertia parameter defined in eqn. (29) temperature temperature of the lubricant supply velocity components in the r, 8, z directions load normalized loadcarrying capacity film thickness ratio defined in Fig. 1
275 temperature coefficient of the viscosity of the lubricant viscosity parameter defined in eqns. (30) temperature coefficient of the density density parameter defined in eqns. (30) quantity defined in eqn. (28) viscosity of the lubricant viscosity of the lubricant supply density of the lubricant density of the lubricant supply !z/@/~, used in eqn. (4) porous permeability of the bearing material function defined in eqn. (17)
References 1 D. Dowson,
2
3 4 5 6 7 8 9 10 11
Inertia effects in hydrostatic thrust bearing, J. Basic Eng., 83 (2) (1961) 227. J. A. Coombs and D. Dowson, An experimental investigation of effects of lubricant inertia in hydrostatic thrust bearing, Proc. 3rd Conv. on Lubrication and Wear, in Proc., Inst. Mech. Eng., London, 179 (1964) 96. W. F. Hughes and J. F. Osterle, Heat transfer effects in a hydrostatic thrust bearing, J. Lubr. Technol., 79 (1957) 1225. J. F. Osterle and W. F. Hughes, Inertia induced cavitation in a hydrostatic thrust bearing, Wear, 4 (1961) 228. L. L. Ting and J. E. Mayer, Jr., The effects of temperature and inertia in hydrostatic thrust bearing, J. Lubr. Technol., 93 (2) (1971) 307. H. J. Sneck, Thermal effects in face seals, J. Lubr. Technol., 91 (1969) 434. V. K. Kapur and J. S. Yadav, Analytical studies of temperature and inertia on the performance of hydrostatic porous thrust bearing, Appl. Sci. Res., 35 (1979) 21. G. S. Beavers and D. D. Joseph, Boundary condition at a naturally permeable wall, J. Fluid Mech., 30 (1) (1967) 197. 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 (1970) 843. P. G. Saffmann, On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 50 (2) (1971) 93. 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961.
261 -Printed
in The Netherlands
VARIABLE VISCOSITY AND DENSITY HYDROSTATIC THRUST BEARING
EFFECTS
IN A POROUS
J. S. YADAV Department of Mathematics, Nehru Postgraduate College, Chhibramau, 209721 (India)
Farrukhabad
V. K. KAPUR Mathematics Department, (India) (Received
Harcourt Butler Technological
March 18, 1980; in revised
Institute, Kanpur 208002
form July 3, 1980)
Summary Interactions of inertia, the viscosity-density variation due to temperature and the bearing material porosity were investigated, Numerical results for the pressure distribution and load capacity were obtained and were compared with experimental and earlier theoretical results. The analytical results obtained are in good agreement with the experimental work.
1. Introduction The performance of thrust bearings has been studied extensively. The unconventional thrust bearing was analysed by Dowson [l] who considered only the isothermal aspect although the rotor was considered to be rotating at high speed. Comparison of the theoretical results of Dowson with the experimental results of Coombs and Dowson [2 ] showed a discrepancy. Inertia effects led to cavitation and thereby rupture because of the subatmospheric pressure. Hughes and Osterle [3] considered the exponential viscosity temperature relation but ignored inertia effects and the effect of the bearing step. Osterle and Hughes [4] considered heat transfer from the fluid to the surroundings but neglected the heat convection term completely because the viscosity was kept constant. However, in experimental work, when there is heat transfer, the viscosity is bound to alter. Ting and Mayer [ 5 ] considered this aspect by assuming only a radial temperature variation because the bearing clearance is small compared with the radial span. A justification of this aspect was given by Sneck [6] for the face seal problem. For bearing manufacture, design criteria and operating conditions must be known. Ting and Mayer [5] showed the interactions of inertia and thermal effects in thrust bearings. There are differences between the theoretical and experimental expressions for the pressure distribution between
262
the bearing surfaces. Kapur and Yadav [ 71 indicated that material porosity may explain these differences. The analysis including porosity effects demonstrates that the deviation between experimental and theoretical results is reduced and that coincidence of the results is obtained up to Ti= 0.6, although there is a marked deviation near the periphery of the bearing which has not been explained. In the present analysis an explanation is attempted by considering the temperature-density variation together with the viscosity variation in a hydrostatic porous recessed thrust bearing using the Beavers-Joseph [8] boundary condition at the permeable surface. The experimental and theoretical justifications of this have been given by Beavers et al. [9] and Saffmann
WI. 2. Analysis The system analysed is shown in Fig. 1. For steady incompressible laminar flow of a lubricant between the parallel surfaces of a hydrostatic porous recessed thrust bearing the momentum equations which take into account rotational inertia effects are pv2 -__ r =--
ap ar
a2u +p a2 -5
(1)
The boundary
conditions v=o
U=O
au
are
uu
1 a j- &P~u) +
0
v = rc2
-,,=p
and the continuity
W=
equation
atz=O
w=o
is
a;(pw)=o
From eqns. (1) and (2) the radial and tangential
prQ222
+ 12ph2
(2)
atz=h
h3-z3+-----
(3) velocity
profiles are
3h3
1 +a(3
(4)
253
2
6 W
n
t
CO
Fig. 1. Schematic diagram of recessed thrust bearing.
and v = rSLz/h
(5)
where p is the mean pressure across the film and is a function of r only. The variations in the viscosity and the density of the lubricant are assumed to be insignificant in the axial direction (because the lubricant film is very thin) but to vary radially according to the temperature distribution. The energy equation used to study the simultaneous density and viscosity variations for an incompressible lubricant is
From eqns. (4) - (6) integration of the energy equation across the film yields 27rrqw” + ,wC,Q g
I
l+
-where
prf2 2h3
2%
1+ I
7 (1 +ao)2
(7)
Integrating the continuity equation across the film thickness and using the boundary condition for w gives
shl
o
;
a jr
(pru)dz
(9)
= 0
From eqns. (4) and (9) the differential equation for the pressure distribution is obtained as d
prdj? -- dr i 1-1 dr )
(10)
where f(W-J) =
1 + 5/(1 + au)
(11)
1 + 3/(1 + au)
2.1. Temperature distribution It is assumed that the bearing surfaces are perfect insulators (i.e. q,” = 0) and that the heat generated is completely carried away by the lubricant. To consider the viscosity and density effects due to the temperature variation between the bearing surfaces we assume exponential relations: P = p. exp {--B(T - To1
(12)
P =
(13)
and PO
expf--6(T-
To)1
where po, p. and To are the viscosity, density and temperature respectively of the lubricant supply. From eqns. (12) and (13) the combined relation is s = so exp{-(0
- 6)(T - To)}
(14)
where s =iJlp
(15)
so = POIPO
By letting qw ” = 0, substituting dji/dr from eqn. (9) and ignoring higher order terms the energy equation (eqn. (6)) becomes + J_ 700
(rS22)2h3 s
HWJ)
(16)
265
where 3 (1 + 5/(1 + ao)j2{l + 3/(1 + aa)2} (1 + 3/(1 + au)}2 I 400
#(a/J) = 700
3
11 + 5/(1 + au)}{1 + 5/(1 + au)2} + 1 + 3/(1 + au)
-200 1
7
+ -112 1l+
2
(17)
(1 + au)2 II
From eqns. (14) and (16), s(r) is obtained as s(r) = so {(A - B)(r4 - r14) + l>-l
rI
(18)
rl
(19)
while the viscosity variation is p(r) = p. {A(r4 - r14) + 1)-l where @a
A= 2p
2P0
l%i12j.10
B=
ohgC,Q
(20)
2pokGQ
2.2. Pressure dktribu tion Integrating eqn. (9) after substituting the values of s(r) and p(r) from eqn. (18) and (19) gives the expressions for the pressure distribution on both sides of the step: 3 - - -poG!2r2f(u,u) P1 - 20
+ C, In r +DI
r. < r < r1
(21)
and A-B p2 = _ A
3 - PoQ2f(wJ) + 20 tan-’ {Ar4/(1 - Ar14)}l12
c2so
+
4((A -B)r14
-1)
(A(1 - Ar, 4))1/2
+
In
+
(A
-Nr4
-r14)
I
+D
1
r4
I
r,
2
(22)
where C1, C2, D1 and 02 are the constants of integration to be determined by using the following boundary conditions: &@0)
= 0
iWd
=P2W
F,(R)
= 0
Q&-l)
= Qdr1)
(23)
266 These constants
of integration
are
-$(l +Sf(a,o)[l - ;; +$( $ -1) -
Cl =
_
B
tan-‘{AR*/(l
-Ar14)}1/2
AR2
{Ar14/(l
-tan-l
I)
-Ar14)}1/2
{A(1 -Ar14))1’2
ld”Sf(a,a)
c2
_
B
tan-l {AR4/(1 - Ar14))1’2 -tan-l
-AR2
{Ar14/(1 - Ar14)}1/2
{A(1 - Ar14))1’2 (25)
Ill
=po 1 -Sf(a,o)‘02
+
1
2
B
I
-Ar14))1/2
tan-’ {AR4/(1
AT
--tan-l
{Ar14/(1 -Ar14)}l/2~
{A(1 -Ar14)fi1’2
11
X In r.
X
(26)
and D2 = -poSf(a,rr)
1I
+-
B + -
A
B
tan-’ {AR4/(l
AR2
-Ar14)}l12
{(1- Arl*)A)112
Q3P0
I ’
+gg-l)-
A _
B
tan+ {AR4/(1
-Ar14)J1’2
AR2
{A(1
x ln[ {(A -B)(R4
-rx4)
+
-tan-’
(Ar14/(1 -Ar14)j1L2
-Ar14)}1f2
I) X
11/R41
4((A -B)r14
- 13.
+ cy3 ln[r14{(A
-B)(R4
(27)
where A = In
4((A -B)r14 and
-rl*) - 1)
+ 1)/R4] (28)
267
3 20
s=
poR2a2 (29) DO
The following r
nondimensional
R
jj=
R
R -
5
js2=
‘-‘2 Is,
PO
Q=
are defined:
rl Fl = -
r0 To = -
f=-
p, =
quantities
(30)
-Go&
nh3jTo
20p02R ‘0
20p02R 26
$=
po2gC,hh-
po2&h4
and the pressure distribution
in the two regions takes the dimensionless
ji1=1+Sf(a,o)(F2-@)+ + JJ tan-l
Bi
$ {@S/Cj)/(l
{(PS/c?)(l
(
Sf(a,o)
I
F~2-1-;(F~~-1)+
- &!ZZ14/&)}1’2_ - PS~14/&)P2
- tan-l ((&W14/Q)/(1 -j7SF14/Cj))1’2 )-ll)ln( MWG)(l - DSG4/&)P2
+ i
lo i
tan-l { (&S/Q)/(l W/8)(1
- iLsF~4/9)}1’2
_ tan-l {@Sr”/&)/(l {(PlSQ)(l +
- fBF14/&)}1’2 _
-&S~14/&)}1’2
- 6*14/9)P2
iI +
-$(Sf(a,o) IFo2-l-$(F~2-l)+
+ p
ai
tan-l {(&S/@/(1 - &S714/&)}1’2 _ t(PSIQ)(l
-‘&sF~4/Q)}1’2
_ 1 x - tan-l (($SF14/Q)/(1 -&SF14/Q))1/2 NW&l - iWG4/&)P2 )I )
:I
form
(32)
cu3In(F,*($ - S)(l - Fld)S/& + 1) (33)
4~~B~~)SF~~~~~1~ 2.3. Load-carrying capacity The normalized loadcarrying capacity is given by
Sf(a,o)(F(J2 - 1) - 1 1 A
+ +
~3~S~(~,~)(~~2
2 (G2
-I)-
1)
@-F)(l
4A {(a - s)SF14/6j - 1)
--Fl*)s Q
112
- tan-l ~(~S~~*/~)/(l -&SF14/&)]1’2 X 1 NWQW - ~~~4/~~}1’2
f
Qa,o)i.(1 -r;&-P
;(I
-+)
+ (1 -i’*2)
x
I’= i
I +
269
X
tan-l
l/2
- w-1
(B - F)SF14 Jo i 1 -@-@-@;4/&
II2 i
t (35)
The volume flow rate Q of the lubricant through the bearing is given by Q = jkrudz 0
(36)
In dimensionless form it is
%,Q f+-.ⅈ,
(37)
270 Newton’s method is used to solve for ($ and a function F is defined as Ly3
.l
I
(33)
and (39) where F’ = U/da and n is a positive integer. Iterations are performed to obtain the value of @ by guessing Q,. It generally took three to four iterations to reach an accuracy of the order of 10-4. 3. discussion Figures 2 - 4 show the numerical values of the pressure distribution for various values of the rotational parameter S in order to discuss the interactions of the density-~s~osity variations with the pe~o~ance of porous recessed thrust bearings with a porous permeability # = 10.85 X lo-l1 cm2, slip constant a = 0.10 and film thickness h = lo-*cm. Kapur and Yadav [?I showed that up to a step pssition q = 0.4 the pressure distribution between the bearing surfaces approaches the experimental results of Coombs and Dowson [Z] , although the deviation is appreciable after 7 = 0.6. Inclusion in the analysis of the density variation due to temperature modified the analysis and the results for the pressure d~tributions before and after the step position are in close agreement with the ,experimental results [ 25 . This implies that the density variation plays a dominant role in porous bearing problems. The pressure variation is similar to that of the Ting and Mayer [5] and the Kapur and Yadav [ 7 ] theories. The pressure is never lower in the present theory than it is in previous theories [ 5,7] for any given conditions. Figures 3 and 4 illustrate the behaviour of the pressure distribution as a function of the step position. With a shift of the step towards the periphery of the bearing the relative increase in the pressure distribution is appreciable. From Fig. 3, beyond the step position there is a relative increase in the pressure distribution although the pressure decreases at every point between the bearing surfaces. In Fig. 4 this relative difference in pressure d~t~bution decreases cont~uously after the step position and finally attains its value at the periphery. The region 0.45 < f < 1 of negative pressure has reduced appreciably owing to the interaction of the density variation and the
271
0.0
0.4
0.6
1.2
r
0.0
0.4
0.8 i
Fig. 2. Radial pressure distribution in a thrust bearing (~1 = 2.18; s= 8.81; PO = 0.05; q = 0.40;6 = 0.001 “C-1): present theory, - - -, Ting and Mayer’s [ 51 theory; -., Kapur and Yadav’s [ 71 the&y ; 0, experimental results [ 21. Fig. 3. Radial pressure distribution in a thrust bearing (CY= 5.0; p = 0.708; Q, = 0.05; q = 0.20; 6 = 0.001 “C-1): present theory; - - -, Ting and Mayer’s [ 51 theory; -. -, Kapur and Yadav’s [ 71 the&y; 0, *, 0, experimental results.
1
Fig. 4. Radial pressure distribution in a thrust bearing (a! = 5.0; F= 0.708; iu = 0.05; Fl = 0.60; 6 = 0.001 “C-l): present theory; - --, Ting and Mayer’s [ 51 theory; -. -, Kapur and Yadav’s [ 71 thebry.
272
viscosity variation for all values of the rotational parameter S considered. The results for pressure and density-viscosity variation due to temperature are in good agreement with the experimental results [ 21 for S = 0,O .5 and 1.0. Simultaneous examination of Figs. 2 - 4 shows that performance can be improved significantly by considering the density variation due to temperature together with the viscosity variation. For the present theory, because the pressure distribution increases at each and every point between the bearing surfaces and approaches very close to the experimental results, it is expected that the load parameter w will also be more comparable with that of the porous thrust bearing. This is confirmed in Fig. 5. For each step position the load parameters are comparable with those of the porous bearing as are the parameters for the thermal analysis of the bearing. It is concluded that the relative difference in load parameter between the present theory and previous theories [ 5,7] is not appreciable up to a step position F1 < 0.6 but an appreciable difference is observed by just shifting the step further towards the periphery of the bearing. Figures 6 and 7 show the pressure distribution as a function of F with i=l = 0.55 and 0.70 respectively. The results are similar to those presented in Figs. 3 and 4. However, the region of negative pressure obtained in Figs. 2 and 3 is reduced further owing to the consideration of the density-temperature variation even for given design parameters. Hence it is concluded that under the present circumstances a comparatively high pressure is attained. The variations in the temperature distribution illustrated in Figs. 8 and 9 with respect to the radial distance T show a higher temperature generation than do previous theories [ 5,7] . The temperature further increases owing to an increase in the rotational parameter S but the temperature between the
I
0.4
I
/
O-6
O-8
I 1.0
il
Fig. 5. Load parameter __ present theory; theorb.
us. step positions
(a = 9.25;fl=
3.92;70
- - -, Ting and Mayer’s [ 51 theory;
= 0.493;6
= 0.001
- . -, Kapur and Yadav’s
“C-l): [7 ]
273
0.6-
0.4
0.6
0.8 i
Fig. 6. Radial pressure distribution in a thrust bearing ((.u= 9.25;B= 3.92; iTo= 0.493; Fl = 0.55;6 = 0.001 “c-l): present theory; - - -, Ting and Mayer’s [ 51 theory, -., Kapur and Yadav’s [ 71 theory. Fig. 7. Radial pressure distribution in a thrust bearing ((Y = 9.25; p= 3.92; 70 = 0.493; ?=I = 0.70;s = 0.001 c-l): present theory ; - - -, Ting and Mayer’s [ 5 ] theory; -. -, Kapur and Yadav’s [ 71 theory.
bearing surfaces decreases owing to a shift of the step towards the circumference of the bearing, a result similar to that of previous theories. It may be concluded that in order to reduce the difference between theoretical and experimental results the density-viscosity and porosity interactions must be incorporated in thrust bearing design considerations.
Nomenclature
1,B Cl, c2 c, &,D2
f(Q, 0) F &? h K n P P
slip constant quantities defined in eqn. (20) constants of integration defined in eqns. (24) and (25) specific heat of the incompressible lubricant at constant volume constants of integration defined in eqns. (26) and (27) porous factor defined in eqn. (11) function defined in eqn. (38) gravitational acceleration minimum film thickness thermal conductivity of the lubricant positive integer fluid pressure mean pressure across the film
214 1.6
1.4 t
-l I.3
P t-
5
Fig. 8. Radial temperature distribution in a thrust bearing (a = 9.25;p= 3.92; PO = 0.493 present theory; - - -, Ting and Mayer’s [ 51 theory; Fl = 0.55;6 = 0.001 “c-l): -. -, Kapur and Yadav’s [ 7 ] theory. Fig. 9. Radial temperature distribution in a thrust bearing (a! = 9.25;$= 3.92; Po = 0.493 present theory; - --, Ting and Mayer’s [ 51 theory; Pr = 0.70; 6 = 0.001 “c-l): -. -, Kapur and Yadav’s [7] theory.
PO
I% 62 P
PI, & ,I
4w
:1,QZ Q
r, 0, z r0
r1 P
PO ii
R s SO
S T
To u, v, w W
iv a!
mean pressure of the lubricant supply pressures on either side of the step dimensionless pressure across the film quantities defined in eqns. (30) total local heat transfer flux from the bearing to the surroundings volume flow rate volume flow rates on either side of the step volume flow rate parameter defined in eqns. (30) cylindrical coordinates radius of the supply hole recess radius r/R, radius ratio ro /R, radius ratio (dimensionless) r1 /R, radius ratio (dimensionless) outer radius of the bearing quantity defined in eqns. (15) quantity defined in eqns. (15) inertia parameter defined in eqn. (29) temperature temperature of the lubricant supply velocity components in the r, 8, z directions load normalized loadcarrying capacity film thickness ratio defined in Fig. 1
275 temperature coefficient of the viscosity of the lubricant viscosity parameter defined in eqns. (30) temperature coefficient of the density density parameter defined in eqns. (30) quantity defined in eqn. (28) viscosity of the lubricant viscosity of the lubricant supply density of the lubricant density of the lubricant supply !z/@/~, used in eqn. (4) porous permeability of the bearing material function defined in eqn. (17)
References 1 D. Dowson,
2
3 4 5 6 7 8 9 10 11
Inertia effects in hydrostatic thrust bearing, J. Basic Eng., 83 (2) (1961) 227. J. A. Coombs and D. Dowson, An experimental investigation of effects of lubricant inertia in hydrostatic thrust bearing, Proc. 3rd Conv. on Lubrication and Wear, in Proc., Inst. Mech. Eng., London, 179 (1964) 96. W. F. Hughes and J. F. Osterle, Heat transfer effects in a hydrostatic thrust bearing, J. Lubr. Technol., 79 (1957) 1225. J. F. Osterle and W. F. Hughes, Inertia induced cavitation in a hydrostatic thrust bearing, Wear, 4 (1961) 228. L. L. Ting and J. E. Mayer, Jr., The effects of temperature and inertia in hydrostatic thrust bearing, J. Lubr. Technol., 93 (2) (1971) 307. H. J. Sneck, Thermal effects in face seals, J. Lubr. Technol., 91 (1969) 434. V. K. Kapur and J. S. Yadav, Analytical studies of temperature and inertia on the performance of hydrostatic porous thrust bearing, Appl. Sci. Res., 35 (1979) 21. G. S. Beavers and D. D. Joseph, Boundary condition at a naturally permeable wall, J. Fluid Mech., 30 (1) (1967) 197. 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 (1970) 843. P. G. Saffmann, On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 50 (2) (1971) 93. 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961.