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Magnetogasdynamic inclined thrust bearings
209
Wear, 29 (1974) 209-217 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
~GNETOGASDYNA~IC
INCLINED
THRUST
BEARINGS
A. C. MAHANTI and G. RAMANAIAH Department of Applied Mathematics, Indian Institute of Technology, Kharagpur (indin) (Received December 10, 1973; in final form February 25, 1974)
SUMMARY
A study of magnetoga~ynamic inclined thrust bearings with an axial magnetic field showed that, for a given flow rate, load capacities of bearings with converging and diverging film thickness are greater under adiabatic than under isothermal conditions. At constant pump pressure, load capacities of bearings with converging and diverging film thicknesses are lower under adiabatic conditions than under isothermal conditions.
NOMENCLATURE
inner radius of the bearing outer radius of the bearing uniform applied axial magnetic field lihn thickness inner film thickness outer film thickness ratio of outer film thickness to inner film thickness dimensionle~ film thickness = B, ho (CT/~)*Hartmann Number pressure of fluid dimensionless pressure dimensionless pump pressure external pressure mass flow rate dimensionless mass flow rate radial distance dimensionless radial distance radial velocity load capacity d~ensionl~ load capacity axial distance ratio of inner radius to outer radius of the bearing viscosity coefficient
210 CJ
P
Y
A. C. MAHANTI,
G. RAMANAIAH
electrical conductivity density polytropic constant
1. INTRODUCTION
Hughes and Elco’ studied thrust bearings lubricated with an incompressible and electrically conducting fluid, with an axial magnetic field and a radial current. For a given flow rate, load capacity and pump pressure increase with increase of magnetic field. For a constant pump pressure load can be sustained with lower pump pressures with increase of magnetic field. With a diverging inclination of the bearing surface Shukla2 showed that the magnetic field increases the load capacity. With a thrust bearing with an electrically conducting gas lubricant and a uniform axial magnetic field Agrawa13 showed that the load capacity is greater under adiabatic than iso-thermal conditions. This paper considers the effects of converging and diverging inclination of the bearing surfaces. The results are compared with those of Agrawa13. 2. ANALYSIS
Figures 1 and 2 show bearings with converging (hi < 1) and diverging (hi > 1) film thicknesses respectively; inertia terms are neglected. The equation of motion of the lubricating substance is given by
In eqn. (1) the Hartmann Number M =B, h,(a/p)* isused. The resulting equation is solved by using the boundary condition u=O,
z=o, to get
h=
rh,+~(r-a( a
h
__
__b---
Fig. 1. Bearing
with a converging
film.
MAGNETOGASDYNAMIC
211
INCLINED THRUST BEARINGS
Fig. 2. Bearing with a diverging film.
Equation (3) is substituted in the equation of continuity h
! 0
udz = - Q 2rcpr
(4
to get
pQM3
dp
z=3
(5)
2zprh,
The lubricating fluid satisfies the relation P PY Fe= 0 Z . Combining eqns. (2) (5) and (6) with the following dimensionless quantities
(6)
(7) gives @’
+ d/Y
-Z-=
(1 +y)Q(l -I)M3 r{l(r;l-l)+(li-l)(l-A)}(2
tanh$
(8) - Mh)*
Equation (8) is diffkult to integrate in the closed form so it was solved for small and high values of the Hartmann Number. Solution for small Hartmann
Number
The power series expansion of the hyperbolic function is truncated after the term containing M6 and substituted in eqn. (8). The resulting equation is d-(‘+Y)/Y
P
dIi ____
120(1+y)Q(1-4 = ~{~(~,-l)+(t;-1)(1-I)}ti3(MZ~2-10)’
212
A. C. MAHANTI, G. RAMANAIAH
Equation (9) was expressed in partial fraction form and integrated using the boundary condition p=l
at /i=t5 1 *
(10)
The pressure distribution is given by
when A = (~~~-1)+~(~~-10)(1-~)-~~(~~-1)(~~~+~-2~ n(ti, - 1)&P- lO)(& +n-2)
_ c Pa)
B = ~-1)-F(1R,-1)(M~-10)-1E(h,-1)(ilt11+1-2) A(!%,-1)(A42- lo)(nf;, +A-2)
(f2b)
1 c = - lO(Lh,-1)
(12c)
IV&?@,- I) D = loo((/z& -l)%lO(l-4)‘)
02d)
kP(l-1) E = - lo~(nr;,-1)~M~-10(1-E.)~~
(12e)
and (/I- 1)5 F = (n~,-1)“(M2(;Ujl-1)2-10(A-1)2)
*
On using I;= 1, in eqn. (II), the pump pressure is given by -(l+Y)/y_1
Pi
_
When Ti, = 1 (bearing with parallel surfaces), eqns. (11) and (13) reduce to
and &l+Y)/Y_
1 =
(120
MAGNETOGASDYNAMIC
INCLINED
THRUST
213
BEARINGS
Combining eqns. (14) and (15) gives :
-(l+YYY_l
=
P
in the
($+Y)lY_l)_
(16)
Equation (16) which agrees with Agrawa13 represents the pressure distribution bearing with a uniform film under a given pressure.
Solution for high Hartmann
Number
For high values of the Hartmann Number, tanh M/i/2 is nearly equal to unity. The reduced form of eqn. (8) is djj”
+ Y)/Y
(l+y)Q(l-1)M3 y{&-l)+(ti-l)(l-;1)}(2-Mt;)
Yz-=
(17,
Equation (17) was solved using the condition (10) to get -cl+
P
Y)/Y _ 1 =
_
-
2
log,
(g$)]
(18)
and 8~+r)/r_l
=
(l+y)(Z(l-A)M3
- &llog&)]
1
Y
(19)
when
(l-4
(204
A’ = 2(1-I)+M(&-1) and M B1 = 2(1-L)+M(lt;,-1)
The bearing with parallel surfaces corresponds reduce to -(l+Y)lY_l
P
=
Q;;@+y;3
log,
to h, = 1. Thus eqns. (18) and (19)
A
(21)
0r
and jjy+Y)iY_ 1 =
Eliminating Q from the eqns. (20) and (21) gives eqn. (16). Combining (18) and (19) gives @I-1)+(1-W-l) }-(l+Y)/Y_l=($+Y)/Y_q
P
h-
Bllog.(~)
1)
(23) I
214
A. C. MAHANTI, G. RAMANAIAH
When pi is constant, the limiting value of eqn. (23) as M tends to infmity is the same as the eqn. (16). The load capacity of.the bearing is given as ‘b
W= 71U2(pi-p,)+271
1a
r(p-p,)dr
which, on using the eqn. (2) and the dimensionless quantities (7), is written in dimensionless form as WC
&
= P(p,-
1)+2;1(&)
e
w 6
!‘(jj1
_________‘(.,+________________--------
--______________________-
t
l)dh+
2 (&+)’
j:;(r;l)(i-
l)dt;.
1
_______--
__________+,____-----1
Fig. 3. Load capacity vs. Hartmann number at constant flow rate. (a) Small Hartmann number; (b) large Hartmann number.
MAGNETOGASDYNAMIC
215
INCLINED THRUST BEARINGS
3. DISCUSSION AND CONCLUSIONS
To evaluate the load capacities (eqn. (24)) of bearings with converging and diverging film thicknesses, the specific values of ti, are considered as 0.5 and 1.5. In each case, a fifth degree polynomial passing through six equidistant points at a spacing 0.1 is considered. It was found that, for a given flow rate, the load capacity of a bearing with a converging film is greater and that of a bearing with a diverging film is less than that of a bearing with a uniform film. The load capacities of bearings with converging,and diverging film thicknesses are greater under adiabatic than under isothermal conditions. Load capacity increases as the magnetic field increases (see Figs. 3(a) and (b)).
6
0
I
0 680
0.2
hm0.01 ; as.1 -- _____ _ x,.0.5 B,= 1.5 _._._ ii,= 1
I
O-4
M
0.6
0.6
1.0
b
_+__-M---J
____-;--.~-----.-._ 45
--------;,,
_z
120 ----:zz:~.40 60 4 Fig. 4. Pump pressure us. Hartmann number at constant flow rate. (a) Small Hartmarm number; (b) large Hartmann number.
216
A. C. MAHANTI,
G. RAMANAIAH
The pump pressures of bearings with converging and diverging films are greater and less respectively than that of a bearing with a uniform tilm. Pump pressure is less under isothermal than under adiabatic conditions. Pump pressure increases with increase of the magnetic field (see Figs. 4(a) and (b)). At constant pump pressure the load capacity of a bearing with a converging film is greater and that with a diverging film is lower than that of a bearing with a uniform film. The load capacity of the bearing with a converging film diminishes and that of a bearing with a diverging film increases as the Hartmann Number increases. It follows from the limiting form of the eqn. (23) that the pressure profiles and the resulting load capacities of bearings with converging, diverging and uniform film thicknesses are similar when the Hartmann Number is infinitely large. The load 5.2
cl1
x=0.01 &i-l0 +5 -----
________ ________ --___ y.,.+--i_______ ;: :1;5--
3.9
---_-__
_________
____________________
y-,.&6
_______-
2.6
1 0 ,5.2
I
I
I
J
0.6
0.8
1.0
I
0.2
0.4
M
b
3.9 I-
iii ‘------------y*,_________
2.6 ,
‘------------
-----_--________-_______
---_____
____________________y~,.4______
------------Y.1.6c----_--__________________________
___--
.--------____________
_.-._.-.
J.,
---------------------q-!jS._.-
r.1.4 -“.66._._._ _._._.-.-.-. _._._._-ys1.64-.-‘-‘-‘-’ ___.-_-.
1.3
C
i.0
d
._.-.
-.-.-y:;.:K
__._.-.-.-.-
I
45
Fig. 5. Load capacity 0s. Hartmann (b) large Hartmann number.
I
6
7=1.66-.-.-
I
I
50
number
M
55
at constant
t
pump
pressure.
(a) Small Hartmann
number;
MAGNETOGASDYNAMIC
I
0
INCLINED
I
I
0.2
0.4
M
THRUST
217
BEARINGS
I
I
0.6
0.8
0.
b
01
40
I
45
Fig. 6. Flow rate us. Hartmann Hartmann number.
I
5O M number at constant
I
55 pump pressure.
f (a) Small Hartmann
number;
(b) large
capacity of any bearing is greater under isothermal than under adiabatic conditions, (see Figs. 5(a) and (b)). Th e flow rates of bearings with converging and diverging film thicknesses diminish with increase of the magnetic field. The flow rate of a bearing with a converging film is less and that of a bearing with a diverging film greater than the flow rate of a bearing with a uniform film. The flow rate of any bearing is greater under isothermal than under adiabatic conditions (see Figs. 6(a) and (b)). REFERENCES 1 W. F. Hughes and R A. Elco, J. Fluid Me&., 2 J. B. Shukla, Appl. Sci. Rex, Al3 (1964) 432. 3 V. K. Agrawal, Wear, 15 (1970) 79-82.
13 (1962) 21.
Wear, 29 (1974) 209-217 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
~GNETOGASDYNA~IC
INCLINED
THRUST
BEARINGS
A. C. MAHANTI and G. RAMANAIAH Department of Applied Mathematics, Indian Institute of Technology, Kharagpur (indin) (Received December 10, 1973; in final form February 25, 1974)
SUMMARY
A study of magnetoga~ynamic inclined thrust bearings with an axial magnetic field showed that, for a given flow rate, load capacities of bearings with converging and diverging film thickness are greater under adiabatic than under isothermal conditions. At constant pump pressure, load capacities of bearings with converging and diverging film thicknesses are lower under adiabatic conditions than under isothermal conditions.
NOMENCLATURE
inner radius of the bearing outer radius of the bearing uniform applied axial magnetic field lihn thickness inner film thickness outer film thickness ratio of outer film thickness to inner film thickness dimensionle~ film thickness = B, ho (CT/~)*Hartmann Number pressure of fluid dimensionless pressure dimensionless pump pressure external pressure mass flow rate dimensionless mass flow rate radial distance dimensionless radial distance radial velocity load capacity d~ensionl~ load capacity axial distance ratio of inner radius to outer radius of the bearing viscosity coefficient
210 CJ
P
Y
A. C. MAHANTI,
G. RAMANAIAH
electrical conductivity density polytropic constant
1. INTRODUCTION
Hughes and Elco’ studied thrust bearings lubricated with an incompressible and electrically conducting fluid, with an axial magnetic field and a radial current. For a given flow rate, load capacity and pump pressure increase with increase of magnetic field. For a constant pump pressure load can be sustained with lower pump pressures with increase of magnetic field. With a diverging inclination of the bearing surface Shukla2 showed that the magnetic field increases the load capacity. With a thrust bearing with an electrically conducting gas lubricant and a uniform axial magnetic field Agrawa13 showed that the load capacity is greater under adiabatic than iso-thermal conditions. This paper considers the effects of converging and diverging inclination of the bearing surfaces. The results are compared with those of Agrawa13. 2. ANALYSIS
Figures 1 and 2 show bearings with converging (hi < 1) and diverging (hi > 1) film thicknesses respectively; inertia terms are neglected. The equation of motion of the lubricating substance is given by
In eqn. (1) the Hartmann Number M =B, h,(a/p)* isused. The resulting equation is solved by using the boundary condition u=O,
z=o, to get
h=
rh,+~(r-a( a
h
__
__b---
Fig. 1. Bearing
with a converging
film.
MAGNETOGASDYNAMIC
211
INCLINED THRUST BEARINGS
Fig. 2. Bearing with a diverging film.
Equation (3) is substituted in the equation of continuity h
! 0
udz = - Q 2rcpr
(4
to get
pQM3
dp
z=3
(5)
2zprh,
The lubricating fluid satisfies the relation P PY Fe= 0 Z . Combining eqns. (2) (5) and (6) with the following dimensionless quantities
(6)
(7) gives @’
+ d/Y
-Z-=
(1 +y)Q(l -I)M3 r{l(r;l-l)+(li-l)(l-A)}(2
tanh$
(8) - Mh)*
Equation (8) is diffkult to integrate in the closed form so it was solved for small and high values of the Hartmann Number. Solution for small Hartmann
Number
The power series expansion of the hyperbolic function is truncated after the term containing M6 and substituted in eqn. (8). The resulting equation is d-(‘+Y)/Y
P
dIi ____
120(1+y)Q(1-4 = ~{~(~,-l)+(t;-1)(1-I)}ti3(MZ~2-10)’
212
A. C. MAHANTI, G. RAMANAIAH
Equation (9) was expressed in partial fraction form and integrated using the boundary condition p=l
at /i=t5 1 *
(10)
The pressure distribution is given by
when A = (~~~-1)+~(~~-10)(1-~)-~~(~~-1)(~~~+~-2~ n(ti, - 1)&P- lO)(& +n-2)
_ c Pa)
B = ~-1)-F(1R,-1)(M~-10)-1E(h,-1)(ilt11+1-2) A(!%,-1)(A42- lo)(nf;, +A-2)
(f2b)
1 c = - lO(Lh,-1)
(12c)
IV&?@,- I) D = loo((/z& -l)%lO(l-4)‘)
02d)
kP(l-1) E = - lo~(nr;,-1)~M~-10(1-E.)~~
(12e)
and (/I- 1)5 F = (n~,-1)“(M2(;Ujl-1)2-10(A-1)2)
*
On using I;= 1, in eqn. (II), the pump pressure is given by -(l+Y)/y_1
Pi
_
When Ti, = 1 (bearing with parallel surfaces), eqns. (11) and (13) reduce to
and &l+Y)/Y_
1 =
(120
MAGNETOGASDYNAMIC
INCLINED
THRUST
213
BEARINGS
Combining eqns. (14) and (15) gives :
-(l+YYY_l
=
P
in the
($+Y)lY_l)_
(16)
Equation (16) which agrees with Agrawa13 represents the pressure distribution bearing with a uniform film under a given pressure.
Solution for high Hartmann
Number
For high values of the Hartmann Number, tanh M/i/2 is nearly equal to unity. The reduced form of eqn. (8) is djj”
+ Y)/Y
(l+y)Q(l-1)M3 y{&-l)+(ti-l)(l-;1)}(2-Mt;)
Yz-=
(17,
Equation (17) was solved using the condition (10) to get -cl+
P
Y)/Y _ 1 =
_
-
2
log,
(g$)]
(18)
and 8~+r)/r_l
=
(l+y)(Z(l-A)M3
- &llog&)]
1
Y
(19)
when
(l-4
(204
A’ = 2(1-I)+M(&-1) and M B1 = 2(1-L)+M(lt;,-1)
The bearing with parallel surfaces corresponds reduce to -(l+Y)lY_l
P
=
Q;;@+y;3
log,
to h, = 1. Thus eqns. (18) and (19)
A
(21)
0r
and jjy+Y)iY_ 1 =
Eliminating Q from the eqns. (20) and (21) gives eqn. (16). Combining (18) and (19) gives @I-1)+(1-W-l) }-(l+Y)/Y_l=($+Y)/Y_q
P
h-
Bllog.(~)
1)
(23) I
214
A. C. MAHANTI, G. RAMANAIAH
When pi is constant, the limiting value of eqn. (23) as M tends to infmity is the same as the eqn. (16). The load capacity of.the bearing is given as ‘b
W= 71U2(pi-p,)+271
1a
r(p-p,)dr
which, on using the eqn. (2) and the dimensionless quantities (7), is written in dimensionless form as WC
&
= P(p,-
1)+2;1(&)
e
w 6
!‘(jj1
_________‘(.,+________________--------
--______________________-
t
l)dh+
2 (&+)’
j:;(r;l)(i-
l)dt;.
1
_______--
__________+,____-----1
Fig. 3. Load capacity vs. Hartmann number at constant flow rate. (a) Small Hartmann number; (b) large Hartmann number.
MAGNETOGASDYNAMIC
215
INCLINED THRUST BEARINGS
3. DISCUSSION AND CONCLUSIONS
To evaluate the load capacities (eqn. (24)) of bearings with converging and diverging film thicknesses, the specific values of ti, are considered as 0.5 and 1.5. In each case, a fifth degree polynomial passing through six equidistant points at a spacing 0.1 is considered. It was found that, for a given flow rate, the load capacity of a bearing with a converging film is greater and that of a bearing with a diverging film is less than that of a bearing with a uniform film. The load capacities of bearings with converging,and diverging film thicknesses are greater under adiabatic than under isothermal conditions. Load capacity increases as the magnetic field increases (see Figs. 3(a) and (b)).
6
0
I
0 680
0.2
hm0.01 ; as.1 -- _____ _ x,.0.5 B,= 1.5 _._._ ii,= 1
I
O-4
M
0.6
0.6
1.0
b
_+__-M---J
____-;--.~-----.-._ 45
--------;,,
_z
120 ----:zz:~.40 60 4 Fig. 4. Pump pressure us. Hartmann number at constant flow rate. (a) Small Hartmarm number; (b) large Hartmann number.
216
A. C. MAHANTI,
G. RAMANAIAH
The pump pressures of bearings with converging and diverging films are greater and less respectively than that of a bearing with a uniform tilm. Pump pressure is less under isothermal than under adiabatic conditions. Pump pressure increases with increase of the magnetic field (see Figs. 4(a) and (b)). At constant pump pressure the load capacity of a bearing with a converging film is greater and that with a diverging film is lower than that of a bearing with a uniform film. The load capacity of the bearing with a converging film diminishes and that of a bearing with a diverging film increases as the Hartmann Number increases. It follows from the limiting form of the eqn. (23) that the pressure profiles and the resulting load capacities of bearings with converging, diverging and uniform film thicknesses are similar when the Hartmann Number is infinitely large. The load 5.2
cl1
x=0.01 &i-l0 +5 -----
________ ________ --___ y.,.+--i_______ ;: :1;5--
3.9
---_-__
_________
____________________
y-,.&6
_______-
2.6
1 0 ,5.2
I
I
I
J
0.6
0.8
1.0
I
0.2
0.4
M
b
3.9 I-
iii ‘------------y*,_________
2.6 ,
‘------------
-----_--________-_______
---_____
____________________y~,.4______
------------Y.1.6c----_--__________________________
___--
.--------____________
_.-._.-.
J.,
---------------------q-!jS._.-
r.1.4 -“.66._._._ _._._.-.-.-. _._._._-ys1.64-.-‘-‘-‘-’ ___.-_-.
1.3
C
i.0
d
._.-.
-.-.-y:;.:K
__._.-.-.-.-
I
45
Fig. 5. Load capacity 0s. Hartmann (b) large Hartmann number.
I
6
7=1.66-.-.-
I
I
50
number
M
55
at constant
t
pump
pressure.
(a) Small Hartmann
number;
MAGNETOGASDYNAMIC
I
0
INCLINED
I
I
0.2
0.4
M
THRUST
217
BEARINGS
I
I
0.6
0.8
0.
b
01
40
I
45
Fig. 6. Flow rate us. Hartmann Hartmann number.
I
5O M number at constant
I
55 pump pressure.
f (a) Small Hartmann
number;
(b) large
capacity of any bearing is greater under isothermal than under adiabatic conditions, (see Figs. 5(a) and (b)). Th e flow rates of bearings with converging and diverging film thicknesses diminish with increase of the magnetic field. The flow rate of a bearing with a converging film is less and that of a bearing with a diverging film greater than the flow rate of a bearing with a uniform film. The flow rate of any bearing is greater under isothermal than under adiabatic conditions (see Figs. 6(a) and (b)). REFERENCES 1 W. F. Hughes and R A. Elco, J. Fluid Me&., 2 J. B. Shukla, Appl. Sci. Rex, Al3 (1964) 432. 3 V. K. Agrawal, Wear, 15 (1970) 79-82.
13 (1962) 21.