•
•
87-GT-186
THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS
345 E. 47 St., New York, N.Y. 10017
The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications.
Discussion is printed only if the paper is published in an ASME Journal. Papers are available
from ASME for fifteen months after the meeting.
Printed in USA.
Copyright © 1987 by ASME
Analysis of Windage Losses and Velocity Distribution for
a Shrouded Rotating Disk
GARTNER, W.
MTU MOnchen
Munich, Federal Rep. of Germany
ABSTRACT
•
An analysis of experimental results
was carried out, establishing a direct correlation between measured velocity profiles
and windage losses for a rotating disk inside
a shrouded stator as typically found in
turbomachinery.
The analysis gives information on the
regions of the rotating and static parts,
which have a dominant effect upon disk windage.
= kinematic viscosity of fluid
m 3 /sec
angular velocity of disk
1/sec
NOMENCLATURE
b
= disk thickness
c ,c o , c
r
C MS
z
radial-, circumferential-,
axial velocity of fluid m/sec
= dimensionless friction coefficient,
CMS
MS
C
MS
/2•
C = r • 7T • 6'
v
K
m z=
dP.,
P
tot
r,z
• R
1
INTRODUCTION
The quantity of air being pumped in
the boundary layer of a rotating disk and
the distribution of friction losses in the
cavity between the rotating disk and the
static enclosure are of interest in order
to estimate the heat pick up due to friction
losses of the air circulating in the enclosure.
In the present paper an analysis of friction
losses and measured velocity profiles in
the radial and circumferential direction
has been carried out.
5
= dimensionless pumped massflow
v
r,
2
C.)
density of fluid kg/m 3
•o
fie
r
E • c.) • R 3
r/R
I dz
1.125
r/R = 0.985
= core rotation factor,
K = c 0 /(4).r)
mass flow in radial and axial
direction kg/sec
r/R
0.75
r/R
0.456
local power fraction W
= total of positive dP. W
radial-, axial coordinate
m
R
Re
= disk outer radius m
rotational Re-number
Re = (.0•R 2 /.1.2
axial gap width m
No Superimposed
Throuohflow
s = 5 mm
= 25mm
Fig. 1 Sketch of Tested Configuration
Presented at the Gas Turbine Conference and Exhibition, Anaheim, California — May 31-June 4, 1987
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The test results, being presented in
Ref. (1), had been determined for a configuration being sketched in Fig. 1.
The measured velocity profiles and
friction power losses are correlated after
an integration of all power fractions which
were determined by use of Euler's equation
for a change in angular momentum.
Therefore no attempt has been made
to solve the equations of motion numerically, but the equations, being used, need
the empirical input of velocity profiles
to yield the distribution of friction losses
in the cavity between the rotating disk
and the static enclosure.
Due to the available test results,
the analysis has only been carried out for
cases without superimposed throughflow.
2
-2
10
8
6
4
CMS
'Ow
••■••11.
2
.••■•••••._
-3
10
8
EXPERIMENTS
6
2.1 Experimental Apparatus and Test Procedure
Tests were carried out to determine
friction losses and velocity profiles of
the circumferential and radial components
in the fluid at a rotating disk inside a
shrouded stator.
The test results are presented in Ref. (1).
The tested configuration is shown schematically
in Fig. 1. The axial gap between disk and
stator was varied from s = 5 mm to s = 25 mm.
Initial test runs were carried out
to determine friction losses at the disk's
rear side and tip, foE a range of rot ational
Re-numbers (Re = 3.10' to Re = 3.10'). In
these tests the disk's thickness was varied
(b = 50 mm and b = 100 mm) in combination
with equal axial gaps (s = 5 mm) and no
superimposed throughflow on the front and
rear side of the disk. It was assumed that
changes of geometry and superimposed throughflow only at the disk's front did not have
an effect upon the friction losses at the
disk's tip and rear side, these being unchanged
for all tests. At a given rotational Renumber these friction losses could therefore
be subtracted from the measured total friction
to obtain the losses only at the disk's
front side.
The windage power presented here only
refers to the front side of the disk.
In the tests being analysed in the
present paper, the static wall and shroud
were considered to be sealed, thus no superimposed throughflow was brought into the
enclosure. For all velocity measurements,
pressure and temperature were constant and
equal to 1 bar static pressure and 300 K
air temperature at the disk's rim.
The velocities were determined by using
calibrated 3-hole pressure probes having
a directional accuracy of 0.2 at 60 m/sec
air speed, 1 bar static pressure and 300 K
temperature.
A further, more detailed description
of the teststand is presented in Ref. (2)
and Ref. (3).
°
4
2
—
10
0.0125
0 06 , 5
I 1 1 1 1 111
10
6
s/R
s/R
2
4
6 8 10
7
2
4
Fig. 2 Disk Friction Losses at one Side
2.2 Test Results
In Fig. 2, the dimensionless parameter
C
representing the disk friction power
MS
loss at one side of the disk is plotted
versus the rotational Re-number for two
axial gaps. There is a tendency of a rising
CMS with an increasing axial gap.
The determined velocity profiles are
shown in Fig. 3 to Fig. 6 at three radii:
r/R = 0.456, r/R = 0.75 and r/R = 0.985.
The axial gap s/R = 0.0125 (Fig. 3)
had only been traversed at one rotational
speed (CO = 418.9 1/sec), the gap s/R = 0.0625
(Fig. 4-6) at three speeds (co = 314.2 1/sec,
= 418.9 1/sec, co = 502.7 1/sec). With
these rotational speeds Re-nu4ers range
from Re = 3.10 to Re = 4.8.10 .
In agreement with the results of Daily
and Nece (Ref. (4)), the tangential turbulent boundary layers are merged with an
axial gap s/R = 0.0125 while they are clearly
separated with s/R = 0.0625.
The plotted profiles for the radial
velocity show a slight deviation from zero
in the rotating core flow between the separated
boundary layers, thus continuity is not
fulfilled exactly. But in the later analysis
this error turned out to be small.
An integration of the absolute values
of c r over the axial gap yields the massflow
being pumped by the disk. The results of
these calculations are shown in Fig. 7.
It is obvious that the pumped massflow is
smaller at the narrover gap width s/R = 0.0125.
2
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,
s/R = 0.0125
o
+
X
10,10
ti
r /R =
0.985
r/8 = 0.75
r/R ,--. 0.456
. ',....., \\
ti
-.---'1
.••
/
405
Q 05
a.
..",l,
U
/
WeZ.
C''''
-0,05
1 '''.•
5 /sec
1/se
1/,ec
A
■-•
0
-Q05
0.062:,
-- . --w = 502.7
= 418.9
- - - w = 314.2
.
V'
- 0,10
- 0,10
O
- 0,15
-0,15
1,0
08
0,6
124
Q2
1,0
0,8
0
10
s/R = 0.0125
+
X
0.985
0.75
0.456
------w = 502,7
w - 478.9
Q8
- - - w . 354.2
4
, ,
r
0,2
0
s/R . 0.0625
K
,1R .
,- /R .
r/R .
s
z
0
0,6
1/sec
5/sec
1/sec
ti
c.
r* 06
Ill
, Z.
Mille
Q4
1''' /
.
"4".".......r.
111.)
I
0.2
02
0
10
0
10
08
z—
7
06
04
02
-6-
0
Fig. 3 Measured Velocity Profiles 08
3
06
04
122
0
—
Fig. 4 Measured Velocity Profiles
s/R = 0.0125
= 418.9 1/sec s/R = 0.0625
r/R = 0.456
As the boundary layers are merged at low
radii (compare Fig. 3) a further increase
of the pumped massflow along the radius
is restricted by the massflow directed inwards
at the static wall. Therefore the massflows
are nearly equal for both axial gaps at
r/R = 0.456, while they differ widely at
higher radii.
In the present configuration, the core
rotation factor K turned out to be independent
of rotational speed (see Fig. 4-6) while
the related radial velocities slightly scatter
with the disk speed.
With the rotating fluid being decelerated to zero at the static shroud (r/R = 1.125),
the core rotation factor K tends to decrease
versus increasing radius (see Fig. 4-6).
3
ANALYSIS OF TEST RESULTS
3.1 Equations Used For Analysis
Analysis of the test results, presented
above, was carried out using Euler's equation
for a change in angular momentum and the
Continuity equation. The latter was used
to determine the axial massflow as the axial
velocities had not been measured.
a) Continuity equation (stationary, incompressible fluid). Assuming an axisymmetric
flowfield continuity equation becomes:
c
l/r •
(r . c ) + Sz - 0
(1)
4 r
r
6 z
3
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b) Euler's equation
According to the sketch below the total
power for a change in angular momentum
at an infinitesimal volume element
is:
_L
I z (c 0 .th z
6
dP .G0 E
(r.c o
dz + ,
th r )dr]
(2)
+
.
t
1
t-r-
004.-■
f
Af4
.'
--,10
/10
,I:/
•,"
41,-+ "--ii"cb-
-Mt*
+
0.0625
__ -w = 502.7 1/sec
w - 41E1 .9 1/ c.. c
- ---w = 314.2 I /sec
GO + Err
ir
000-■• .1;tv,
ttr
C4
-
0,05
-
0,10
-0,15
1,0
08
0,6
0,4
Q2
0
s
10
s/R = 0.0625
K
cr
0,15
s/R
W,
,, z
— --w
w
- - - w
10,10
0.0625
=
=
,,
502..7
418.9
314.2
Q8
1/sec
1/sec
1/sec
---w =
- - -w =
502.7
418.9
314.2
1/sec
1/sec
5 /sec
ti
w
•
46
I
1
i
a<
/
•„.,.......■4,...,,s=4.
.
005
-
0.70
roi,-....----.r--,
7
10
-.-
0,8
0,6
44
42
s/R = 0.0625
r/R = 0.985
s/R = 0.0625
K
---w = 502.7
w - 418.9
- - -,4 = 314.2
48
1/sec
1/sec
1/sec
3.2 Computing Method
The measured velocity profiles c r (r,z)
and c a (r,z) (Ref. (1)) acted as an input
for tfie following numerical analysis.
After having calculated the non measured
axial velocities c (r,z) by using Eq. (1),
the mass flows M Tr,z) and M z (r,z) could
be determined. s6 all data are available
to solve Eq. (2) at each element leading
to an estimate of the local power fractions
being released or absorbed by the fluid.
46
02
06
Fig. 6 Measured Velocity Profiles
0,4
z
10
08
r
0
10
"-----------"---)
08
06
04
0.2
0
The following assumptions were made:
1.
Fig. 5 Measured Velocity Profiles
s/R = 0.0625
r/R = 0.75
Because of the very low Ma-numbers
in the flowfield, the air density is
assumed to be constant.
4
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3.
f1.0
0
0.3
08
0
c o =
0.7
O--0
•--•
•
• •
5 mm
25 ram
0.5
- four elements in radial direction
with the inner and outer boundaries:
OX
0.6
Q001
0.002
0.003
r/R
r/R
r/R
r/R
a004 a005
C V(r)
=
=
=
=
0.2625
0.456
0.75
9.985
-
-
r/R
r/R
r/R
r/R
=
=
=
=
0.456 i=1
0.75
=2
0.985 =3
1.125 =4
- eleven elements in axial direction,
(j = 1 - 11) thus resulting in a
total of 44 elements.
Fig. 7 Dimensionless Pumped Massflow
2.
cr = c z = 0
By reason of the simplified model,
c at the static shroud is assumed
0 for the calculation of
d be c the axial mass flow between the disk
rim and the shroud. This simplification
is determined by the geometry of the
elements chosen for the analysis in
accomodation of the velocity measurements.
0.6
0
The boundary condition for all static
parts and the radius r/R = 0.2625 is
c r = c = c = 0. The radius r/R =
0.2625 is de outer radius of the inlet
near the disk's axis. Here the circumferential and axial velocities of the
fluid had also been assumed to be zero
in order to simplify the calculation.
The later presentation of calculation
results will reveal that an error in
this assumption has a subordinate effect
on the calculated power distribution.
At the rotating disk the boundary condition is:
The rotating disk is totally enclosed,
thus no exchange of angular momentum
beyond the enclosure is assumed. As
the power being generated by the disk
must be balanced by the static parts
of the enclosure to fullfil momentum
balance at the rotating fluid, the
total of all local power fractions
in the cavity between the rotating
disk, the static wall and the shroud
must be zero. The total of all positive
power fractions (•Ptot) is equal to
the measured friction loss of the disk.
To determine the gradients dq /dr
for solving Eq. (1), d c r /dr' r was assumed
to be constant in the vicinity of a
considered point. The calculated powerloss was then within 15% of the measured
value.
For an improvement in accuracy of
the analysis the gradients dc /dr close
to the maximum c , near the disk's
rim, were calculEted in an iterative
procedure until measured and calculated powerlosses were within 5%.
All results being shown here are
derived from these calculations.
The measured powerlosses, being
taken for comparison with the calculated results, were determined at rotational Re-numbers corresponding to
the conditions at which the velocity
measurements had been taken. With an
angular seed w = 418.9 1/sec Re becomes:
Re = 4.10
4.
The massflows M and M are determined
by assuming linEar vel6city changes
between the edges of the elements.
5.
The power fraction P Pi calculated
by using Eq. (2) is an average value
for each element. In the following
graphs (Fig. 8 and 9) it is related
to one point of the element, being
at the outer radius and the higher
value of the z-coordinate.
3.3 Results
The results of the analysis are presented
in Fig. 8-11.
The plotted curves AP i /P, ot are the
junctions of the single element points to
which the local power fractions had been
related. The power fractions at r/R = 0.456
nearly vanish to zero, therefore they have
not been plotted in Fig. 8 and 9.
The radial distribution of power fractions at the disk's surface is presented
in Fig. 10. It is apparent that the friction
losses at radii below r/R = 0.5 are almost
zero.
A schematic breakdown of the power
being absorbed or released by the fluid
in different sections of the cavity is shown
in Fig. 11 for the axial gap s/R = 0.0625.
These numbers were produced by summing the
single power fractions of the elements inside
the considered sections.
5
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0,4 p p ;
tot .
cot_
0.3
0. 2Outer
al
•
AN_
Boundar
of Elements
____ _0.75
0
0.1
,-.4
\
0.6
0.8
C. 1
0985
0.2 ___
\
s/R = 0.0125
w = 418.9 1/sec.
0,3
,
\
Shroud
0,3
0.
0.4
0.3
0,2
0,4
0.4
Fig. 8 Calculated Average Power
Fractions for Element
s/R = 0.0125
0 2
/
0.2
al
...
4.....___
...../
>
0.8
s/R = 0.0625
w = 418.9 i/ser.
,
,
0.6
0,4
ciR
0.1
\
Outer Boundary
of Elements
_0.75
i=4
1 125
Fig. 10 Calculated Average Power
Fractions in Elements upon
Disk Surface s/R = 0.0625;
z/s = 0.04; 43 = 418.9 1/sec
0,3 p lot.
1=3
0
0 8
shroud
0,4 Ap.
i=2
0 6
0.4
Static: Shroud
1 - +30 7. u/u
0.985
.S. tatir
s
hr011d
0.2 _._1,125
0.3
Shroud
Well Boundary ///
I ayer. - 5
0,4
1
0.2
7/S
/
k Boundary
+63
/
/
/
Rotating
/ Core: +7
Fig. 9 Calculated Average Power
Fractions for Element
s/R = 0.0625
/
/
/
/
3.4 Discussion
For both axial gaps the largest amount
of power is brought into the fluid towards
the disk rim inside the disk boundary layer,
compare Fig. 8 and 9. With an axial gap
s/R = 0.0625 63% of the total power loss
was calculated in the disk boundary layer,
see Fig. 11.
As the pumped massflow is decelerated
from the disk's circumferential speed to
zero at the static shroud this amount of
power is directly drawn out of the rotating
fluid. Therefore the total of the positive
power in the disk's boundary layer is nearly
equalled by a negative power fraction at
the static shroud towards the disk's side
of the enclosure (z/s -i0), see Fig. 11.
The pumped mass flow, coming from the disk's
boundary layer, passes along the static
shroud towards the static wall, where it
begins to flow inward. Therefore it has
to be accelerated towards the circumferential
speed of the rotating core flow, resulting
Fig. 11 Schematic Breakdown of
Summarized Power Fractions
s/R = 0.0625,G) = 418.9 1/sec
in a positive power fraction being absorbed
by the fluid. This positive power fraction
reaches values up to 30% at s/R = 0.0625
(see Fig. 11).
As most of the positive power fractions
are equalled at the shroud there are only
small changes of angular momentum at the
static wall, see Fig. 8, 9 and 11.
In the range of the rotating core flow
the axial mass flow, coming from the static
wall and feeding the radially pumped flow
at the rotating disk, is accelerated to
a higher rotational speed when approaching
the disk. Therefore the fluid absorbes a
small amount of power in this flow regime
(z/s = 0.2 -0.8), being 7% of the total
power loss, see Fig. 11.
6
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4
CONCLUSIONS
Based upon measured velocity profiles
and windage power losses for an enclosed
rotating disk an analysis using Euler's
equation for a change in angular momentum
has been carried out. The measured power
losses could be reproduced by an analysis
of the measured velocity distribution in
the fluid between the rotating disk and
the stationary enclosure.
The calculated distribution of power
fractions being released or absorbed by
the fluid reveals the dominant effect of
the static shroud and the parts of the rotating disk near the rim concerning friction
losses. All other areas have only a subordinate
effect on disk friction.
Because of these results it is recommended to concentrate on the outer regions
of the rotating disk and the static shroud,
when measuring velocity profiles of c r and
c o in future researches.
ACKNOWLEDGEMENTS
The tests, being presented in Ref. 1,
had been carried out by the Institute of
Steam-and Gas Turbines at the Technical
University, Aachen, Federal Rep. of Germany.
They had been sponsored by the FVV, a West
German association for basic researches
in the field of combustion engines. Parts
of the test results, being presented here,
and a detailed description of the test stand
have been published in Ref. 2.
REFERENCES
1
FVV-Forschungsbericht Vorhaben Nr. 213
Experimentelle and theoretische Untersuchungen des Reibungseinflusses an
rotierenden Scheiben
Heft 331 (1983)
2
Dibelius, G., Radtke, F. and Ziemann, M.,
"Experiments on Friction, Velocity
and Pressure Distribution of Rotating
Discs", D.E. Metzger and N.H. Afghan,
eds., Heat and Mass Transfer in Rotating
Machinery, Hemisphere Washington, 1984
3
Zimmermann, M., Firsching A., Dibelius,
G.H., Ziemann, M.
"Friction Losses and Flow Distribution
for Rotating Disks with Shielded and
Protruding Bolts", Trans. ASME, Vol.
108, No. 3 (1986), P. 547-552 ASME
paper No. 86-GT-158
4
J.W. Daily/R.E. Nece
Chamber Dimension Effects on Induced
Flow and Frictional Resistance of Enclosed Rotating Disks
Journal of Basic Engineering: Vol. 82;
1960, pp. 217-232, ASME Paper No. 59Hyd.-9
7
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