Proceedings of the Institution of Mechanical
Engineers
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Orifice Restrictor Losses in Journal Bearings
E. G. Pink and K. J. Stout
Proceedings of the Institution of Mechanical Engineers 1979 193: 47
DOI: 10.1243/PIME_PROC_1979_193_005_02
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TR IBOLOGY GROUP
ORIFICE RESTRICTOR LOSSES
IN JOURNAL BEARINGS
E. G. PINK, MPhil,
K. J. STOUT, MSc, PhD, CEng, MlProdE
School of Mechanical and Production Engineering, Leicester Polytechnic
Details of a study concerning orifice restrictor losses in externally pressurized gas lubricated journal bearings are
given. The type of compensationconsidered is the pocketed orifice design.
The analysis presented includes the effects of pressure recovery in the pocket and entrance loss effects at
inlet to the bearing film. Also, by treating the flow in the bearing film local to the pocket as radial flow, the
effect of dispersion is accounted for.
It is shown that good agreement exists between computed and experimental results in pressure profides and
also for load capacity up to touchdown conditions.
From the analysis, the effect of the inherent compensationfactor, number of orifices and pocket diameter on
load capacity is discussed.
Grind,* d ( RT )
NOMENCLATURE
a
b
c
c4
d
d0
dR
D
e
h
110
K
L
m
n
Pa
pc
Pi
PO
PP
P*
R
Re
T
W
i5
Y
distance between orifice to edge of bearing
pocket depth
coefficient of discharge
coefficient of discharge (choked conditions)
o f i c e diameter
pocket diameter
bearing diameter
bearing/journal centre distance
film thickness
radial clearance
loss coeflicient
bearing length
mass flow rate through restrictor
number of orifices/row
ambient pressure
theoretical static pressure at throat of entrance to
bearing film assuming flow completely fills bearing
clearance
pressure at edge of pocket assuming purely viscous
profde
supply pressure
pocket pressure
critical static pressure at throat of orifice (i.e. sonic
conditions)
gas constant
Reynolds number at entrance to bearing film
absolute temperature
bearing load
W
(p, - Pa)LD dimensionlessload capacity
isentropic expansion index
2 2
8L
80
E
%local
MRh
do
MRho
inherent compensation factor
concentric inherent compensation factor
e
eccentricity ratio
h0
L
- feeding parameter
As‘ 4P0hO3d1+aO2) D
q
absolute viscosity
1 INTRODUCTION
Many design methods exist which predict the static, nonrotating bearing performance characteristics for externally
pressurized gas journal bearings with compensation
provided by pocketed orifices. Three typical design
methods can be found in references (1,2,3).
A recent study by Pink (4) demonstrated that the design
methods gave conflicting predictions both for the optimum
bearing geometries suggested and maximum load carrying
capacity obtainable. Also, experimental data obtained at
high eccentricities indicated the phenomenon of ‘lock-up’
or negative stiffness which undermines the validity of the
design methods, particularly for applications involving high
shock loadings.
Previous design methods also give limited and conflicting
information on the effect of varying the number of orifices
and pocket diameter. Powell (1) suggests a corrective factor
to account for the number of orifices, whilst Mechanical
Technology Incorporated (2) suggest a minimum number
dependent on the pocket diameter and other parameters.
The design methods previously mentioned all use a
simplified analysis to account for restrictor losses. These
analyses were the only ones available at the time the design
methods were formulated. However, knowledge of
restrictor losses has increased considerably in recent years
by extensive work, reported in references (5,6,7). Also,
the increased computer power now available enables more
rigorous treatment of restrictor losses.
From experimental pressure profiles produced by Pink
(8) more information has been obtained to confirm trends
observed in previous studies as well as obtaining further
correlations particularly relevant to pocketed orifice
designs.
The aim of the study reported here is to:
1. Obtain a more accurate model of restrictor losses in the
light of experimental data.
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E. G. PINK AND K. J. STOUT
48
I
Po
I
I !
!I '"
Pressure loss
through orifice
area
&
4
I
Secondary pressure loss
at edge of ,pocket
area = ndnh
IIIi
I
the low clearance side and can exceed unity at modest
eccentricity ratios. In such instances the dominant flow
restrictor becomes the flow area ndRh and not ndo2/4.
The bearing film pressure subsequently recovers dynamic
pressure as the gas slows down and eventually viscous losses
prevail in the bearing clearance. If the pressure profile given
by purely viscous losses is extrapolated back to the curtain
flow area, a pressure Pican be determined and used in the
isothermal Reynolds' equation to give correct mass flow.
Although inertia effects may exist, these are only very local
to the pocket and will have negligible effect on load
capacity.
To obtain appropriate correlations, the flow through
each portion of the gas flow path is now examined for
subsonic flow only.
Theoretical viscous
Supersonic flow is dealt with in section 2.4.
PP
--
Pi
2.1 Flow through orifice and pressure recowry
I'
Pressure recovery
in bearing film
Fig. 1. Experimental pressure profile bcal to orifice
and associated
p r a w n , brrer
2. Calculate load capacity up to high eccentricity ratios.
3. Determine the effect of varying the number of orifices,
pocket diameter and 6 on load capacity.
2 ANALYSIS
The gas flow path through the restrictor is shown in Fig. 1
with the resulting experimental pressure profile. The
pressure profie, recorded continuously in the bearing clearance whilst the pressure transducer is traversed across the
orifice, enables pressure variations over small distances to
be identified. This is particularly significant in the region of
the orifice.
The gas accelerates through the orifice area nd, / 4 from
stagnation pressure Po. The gas further undergoes a pressure
change as it enters the pocket. If the flow is unchoked at
the throat area ndO2/4,recovery can be expected, whilst if
the flow is choked .the nature of the pressure change is
dependent upon downstream conditions. In the case of
subsonic flow the gas attains a steady state pocket pressure
which is essentially at stagnation conditions. That is to say,
away from the jet stream the kinetic head in the pocket is
small compared to the static pressure head. At the edge of
the pocket the gas further expands through a secondary
restrictor given by a curtain area ndRh, where a venacontracta effect is observed as the gas enters the bearing
fdm. Bearings are normally designed such that the secondary restrictor area WdRh, is more than double the orifice
area ndo2/ 4 , i.e. the concentric inherent compensation
factor
The compressible flow through an orifice is usually assumed
to be isentropic, a reversible adiabatic process. Previous
experience has shown this to be a good first approximation
in describing the flow.
Figure 2 shows the experimental spread of coefficient of
discharge cd* for free-jetting choked flow conditions
assuming isentropic flow. The orifices considered were
watch-makers' 'ruby' jewels, and the diagram illustrates the
dependence of cd* upon the diameter of the orifices. The
variation of c d * ranges between 3 per cent and 15 per cent
and may be related primarily to the deviations in orifice
dimensions fom the nominal value stated by the manufacturers. The effect of this variation in cd* on load capacity and stiffness can be expected t o be negligible (1 1).
The flow equation can be expressed for subsonic flow
as:
The advantage of expressing riZ in terms of Pp rather than
throat pressure is that pressure recovery is accounted for.
Proc Instn Mech Engrs Vol
1
cd'
0.9
-
0.8
-
OJ
t A
I,
0.6
As the eccentricity increases, however, the local inherent
compensation factor 6~ (defined as do2/4dRh)increases in
,
11
1
Q
I
0.1
01
0.3
Fig.2. Experimental Cd* at choked amlitions against do
jewels
- ruby
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193
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49
ORIFICE RESTRICTOR LOSSES IN JOURNAL BEARINGS
confirmed that pressure recovery can occur. However,
0
1o
'
I
0
0
I
I
I
0
0
Oa8
V. I
t
because of the departure from symmetrical flow around the
feeding region in journal bearings, it is more difficult to
obtain loss coefficients than in thrust bearings.
Previously obtained correlations in thrust bearings (5)
can be used with the assumption that the pocket pressure
Pp is a stagnation pressure as experimental pressure profiles
typically shown in Fig. 1 indicate to be the case. The
correlations are thus adapted:
Po/Pa = 3 +7.8
do = 0.09 +0.30 mm
b = 0.15 +0.41 mm
pP/pO
0.6
0.6
0.7
0.8
0.9
1
Fa.3. Correlation of Cd with pressure ratio bawd on fecoved
pp - pi = V p - P c )
(4)
where K = FN(R~).
Expressing the flow through the restrictors in terms of Pi
we have:
conditions in pocket
Figure 3 shows the dependence of c d with PplPo compared with choked conditions. This is similar in form to
other studies made relating to flow discharging directly into
atmosphere where no pressure recovery is assumed.
This approachminimizes effects due to coefficient of area
(i.e. orifice vena-contracta) and to my leaks due to fitting
orifices incorrectly during the experiments in determining
c d * for choked conditions.
2.2 Restrictor in series
For our purposes we assume that the flow completely fds
the bearing clearance at the entrance to the bearing film. In
doing so, entrance loss data already obtained (5,6,7) can
be correctly applied to the flow model.
The isentropic flow through the edge of the pocket is:
By manipulating equations (1 2,3,4,5)the approximate
relationship can be found:
2.4 Choked conditions
If choked conditions exist in the orifice then the effects of
restrictor losses, apart from c d * , is of little consequence in
the overall bearing analysis. The mass flow rate is fEed and
can be used directly in the isothermal Reynolds' equation
to calculate the pressure profile within the bearing film.
Inlet pressures (Pi) and grid pressures are determined by
Were pc = theoretical static pressure at the throat of the
secondary restrictor.
By equating equations (1) and (2),an effective flow area
can be approximately expressed in terms of the overall
pressure loss as:
2.3 Entrance loss effects
Entrank loss effects have been the subject of extensive
investigations in recent years (5,6,7). Although the empirical coefficients obtained relate specifically to thrust bearings¶this data has been applied t o inherently compensated
journal bearings (9).
Vohr (6) conducted experiments on pocketed thrust
bearings. The work indicated the need to take into account
pressure recovery in the bearing clearance. Further, it has
been suggested (2) that by neglecting pressure recovery,
significant error may occur in the subsequent performance
calculations.
More recent work by Pink (8) on journal bearings has
Axial Iy
Circumferentially
Fig. 4. Grid network used in finite difference program to calculate
bearinu film pressures
Proc Instn Mech Engrs Vol 193
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50
E. G. PINK AND K. J. STOUT
employing finite difference equations and relaxation techniques in the bearing film
The choked flow is given by:
. ,*,do2
m=4
[
27
(y - 1)RT
((); *
217
7 / ( 7 - 1)
where-=
P*
-
($);I
'I2
(T$--)
PO
3 CALCULATION OF BEARING FILM PRESSURES
Orifice flow and inlet pressure Pi can be accounted for
using the previously mentioned analysis. Inlet pressures Pf
are used as the boundary conditions, and bearing pressures
are calculated for the grid points shown in the network
presented in Fig. 4. It can be seen that the flow from the
packet is represented as radial flow t o surrounding grid
points and subsequently as axial and circumferential flow.
In doing so, the effect of dispersion is accounted for.
(a) o = 0 Axial
In orifice plane
\\
-Experiment
+.SThmW
pe
.+-4
(b) E
-
-
Experiment
Theory
0 Circumferential
-
(c) c 0.5 Circumferential
Cig.5. Cornparkon between theoretical and experimental prerun profib
pa
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51
ORIFICE RESTRICTOR LOSSES IN JOURNAL BEARINGS
-
I
1 Double, a/L = 0.25
&[ -0.55
Po/Pa = 7.8
LID
6, =0.21
dR/D = 0.031
\.
/'-.
/
\
0.3
W
0.2
'
Figs. 7 and 8 respectively. It can be seen that if low
numbers of orifices are employed or small pocket diameters
are used, the load capacity reduces, and tendency to lockup increases.
Figure 9 illustrates the combined effect of these two
design variables on load capacity at E = 0.5. The load capacity has been normalized enabling dispersion effects predicted by Dudgeon and Lowe (14) to be compared directly
with our theoretical results. It can be seen that in the more
realistic case where n is in the range 8-16, differences
between Dudgeon and Lowe predictions and our predictions are within 4 per cent. In most cases the discrepancy is
less than this.
The diagram indicates that increasing the number of
orifices or pocket diameter offers diminishing returns in
0.5
0.1
Experiment
.-. -. -. Theory, no 'burr'
-
--- -
Theory 'burr'
-= 0.05
0.4
IT
0.3
ov
0
I
I
0.2
0.4
I
0.6
0.8
6
Fig.6. Comparison betwren theoretical and experimental load
capacity including effects of 'burn'
The computation method used is an iterative process. A
Newton-Raphson iteration is applied for mass flow continuity for the restrictors and finite-difference methods
used to solve grid pressures employing relaxation techniques previously developed by Stout (10).
0.2
0.1
0
0.5
0
4 COMPARISON BETWEEN THEORY AND
EXPERIMENT
Figure 5 shows a comparison between pressure profiles
obtained by the theoretical analysis and experimental results (8). The slight difference between theory and experiment in Fig. Sa can be accounted for by inertia effects,
which become more apparent local to the pocket. Rgure Sb
shows that the pressure drop between orifices and hence
dispersion effects, are accurately predicted as the difference
between theory and experiment are small. Figure 5c illustrates a loaded bearing at E = 0.5 and shows the difference
between experimented and predicted bearing film pressures
around the circumference of the bearing.
Load deflection characteristics are compared in Fig. 6.
The experimental bearings were manufactured by pressing
jewels in the bearing bore after the bore was lapped. Local
rising around the pocket was observed (measured to be of
the order of 0.05 of bearing clearance) and has been noted
previously (11,12,13). This effect has been included in the
theoretical analysis and the results illustrated in Fig. 6. It is
shown that the experimental results are within S per cent of
theoretical predictions.
5 EFFECT OF DESIGN VARIABLES
The variation of load deflection characteristicsdue to varying the number of orifices and pocket diameter is shown in
Q
E
1
Fig. 7. Effect of number of orifices on lord deflection characteristics theory
-
I
0.5
dR ~ 0 . 1 0
-
L/D = 1 Double
A& = 0.5
PJP, = 5
n-8
0.4
m
0.3
0.2
0.1
0
0
Fe.8. Effect of pockot
characteristics -theory
IMechE 1979
0.5
E
diamtrr
on
bad
deflection
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E. G. PINK AND K. J. STOUT
52
network used in computation accounts largely for the
dispersion effects often ignored in previous theoretical
work. These techniques may be confidently employed for
the preparation of comprehensive design guides and has
been used to verify design procedures recently presented
(15).
L/D = 1 Double
A& = 0.5 c; = oa
PdP, = 5
SO= 0.25
a = 0.5
-Theory
-.
ho = 25 m
Dudgeon 81Lowe
0
dR/D
0.05
-
ACKNOWLEDGEMENTS
0.10
Fii.0. Effect of number of orifices and pocket diameter on load
capacity, E
0.5
increased load capacity. For graphical convenience, the load
parameter for a bearing having n = 8 orifices around the
circumference with dR/D = 0.015 has been made unity.
The chang, in load ratios may be related to dispersion
losses in the bearing and it is for this reason that Mechanical
Technology Incorporated suggest a minimum number of
orifices and pocket diameters to comply with their analytical treatment.
Figure 10 shows the effect of the inherent compensation
factor 6 on load capacity. It can be seen that for the range
of values normally associated with these bearings, the effect
on predicted load/deflection characteristics is relatively
small.
0.E
I
60' 0.25.
0.1
liv
0.:
0.2
L/D = 1 Double
A$[ = 0.5
Po/Pa = 5
17-8
dR /D = 0.03
h, = 25 pm
0.1
CJ = 0.8
C
0.5
0
€
Flg. 10. Effect of a0 on load deflection characterirtier
1
- theory
6 CONCLUSIONS
It has been shown that bearing film pressures for concentric
and eccentric conditions as well as load deflection characteristics up to touchdown conditions are accurately predicted. Variations between theory and experiment are
typically 5 per cent. This has been possible due to a refined
analysis of the restrictor losses. Also the type of grid
The authors wish to acknowledge the co-operation received
from Southampton University, particularly MI R. W.
Woolley who assisted in the design of the test rig used in
obtaining the experimental results. In addition the authors
wish to acknowledge the co-operation of Mr A. J. Munday
for making available pressure profiles obtained from the
experimental programme.
The authors are most grateful to the Science Research
Council for funding three separate research programmes,
one at Southampton University and two at Leicester Polytechnic which has enabled this work t o be undertaken.
REFERENCES
(1) POWELL, J. W. Des& of aerostatic bearings. Machinery Pub-
lishing, 1971.
(2) WILCOCK (Ed). Design of gas bearings. Mechanical Techno-
logy Inc., Latham, New York, 1967.
(3) CONSTANTINESCU, V. N. An approximate method for the
analysis of externally pressurised gas journal bearings. Gas
Bearing Symposium, Paper 1, University of Southampton,
1967.
(4) PINK, E. G. An experimental investigation of externally
presswised gas journal bearings and comparison with design
method predictions. Gas Bearing Symposium, Paper G3,
Cambridge University, 1976.
(5) McCABE, J. T., ELROD, H. G., CARFAGNO, S., and
COLSHER, R. Entrance effects in gas bearings. Franklin Inst.
Lab. Report IC2429-1, Nov. 1969.
(6) VOHR, J. H. An experimental study of flow phenomena in the
feeding region of an externally pressurised gas bearing.
Mechanical Technology hic., Report MTI - 65TR47, Latham
New York, 1966.
(7) VOHR, J. H. A study of inherent restrictor characteristics for
hydrostatic gas bearings. Gas Bearing Symposium, Paper 30,
Southampton University, 1969.
(8) PINK, E. G. Unpublished work carried out at Southampton
University, SRC Grant No. B/RG/1740/9, 1974-76.
(9) ELROD, H. G. and GLANFIELD, G. A. Computer procedures
for the design of flexibility mounted, externally pressurised,
gas lubricated joumal bearings. Gas Bearing Symposium, Paper
22, Southampton University, 1971.
(10) STOUT, K. J. and ROWE, W. B. Ex3ernally pressurised
bearings - design for manufacture including a tolerance procedure. Tribology International, Aug. 1974.
(11) PINK, E. G. and TAWFIK, M. The effect of errors in
manufacturing on aerostatic bearing performance. 1st Joint
Polytechnic Symposium on Manufacturing Engineering, Paper
F2, Leicester Polytechnic, June 1977.
(12) MARSH, H., BENNETT, J., and HUDSON, B. C. The flow
characteristics of small orifices used in externally pressurised
gas bearings. Gas Bearing Symposium, Paper E3, Cambridge
University, 1976.
(13) Discussion of (12).
(14) DUDGEON, E. H. and LOWE, I. R. G. The prediction of
hydrostatic gas bearing journal bearing performance. ASME
Paper No. 66-LUBS-16.
(15) PINK, E. G. and STOUT, K. J. Design procedures for orifice
compensated gas journal bearings based on experimental data.
W l o g y International, Feb. 197%.
Thfs paper is published for written discusston. The MS was received
on 12th June 1978 and was accepted for publication on 18th
September 1978. 23.
0 IMechE 1979
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