A slip factor calculation in centrifugal impellers based on linear ...

A <strong>slip</strong> <strong>factor</strong> <strong>calculation</strong> <strong>in</strong>
<strong>centrifugal</strong> <strong>impellers</strong> <strong>based</strong> on
l<strong>in</strong>ear cascade data
Dr. Abraham Frenk
Becker Turbo System Eng<strong>in</strong>eer<strong>in</strong>g (2005).
(Presented by Dr. E. Shalman)

Abstract
Accurate model<strong>in</strong>g of the flow <strong>slip</strong> aga<strong>in</strong>st direction
of rotation is essential for correct prediction of the
<strong>centrifugal</strong> impeller performance. The process is
characterized by a <strong>slip</strong> <strong>factor</strong>. Most correlations
available for <strong>calculation</strong> of the <strong>slip</strong> <strong>factor</strong> use
parameters characteriz<strong>in</strong>g basic impeller geometry
(review of Wiesner (1967) and Backstrom (2006)).
Approach presented below is <strong>based</strong> on reduction of
radial cascade to equivalent l<strong>in</strong>ear cascade. The
reduction allows to calculate characteristics of
radial blade row us<strong>in</strong>g well established
experimental data obta<strong>in</strong>ed for l<strong>in</strong>ear cascades,
diffusers and axial blade rows.
2

Rotors with splitters or with high load<strong>in</strong>g must be
divided <strong>in</strong>to few radial blade rows. In the case of
multiple rotor blade rows the <strong>slip</strong> angle is calculated
for each blade row. The <strong>slip</strong> angle of the rotor is a
sum of the <strong>slip</strong> angles obta<strong>in</strong>ed for each row.
Suggested reduction of radial blade rows allows also
calculat<strong>in</strong>g of other parameters essential for impeller
design.
Suggested method allows also determ<strong>in</strong>e additional
causes <strong>in</strong>fluenc<strong>in</strong>g <strong>slip</strong> <strong>factor</strong>. The <strong>slip</strong> <strong>factor</strong>
depends not only on parameters characteriz<strong>in</strong>g
basic impeller geometry, but on difference of <strong>in</strong>let
flow angle from stall flow angle. In the rotors with the
same basic geometry parameters <strong>slip</strong> <strong>factor</strong>
depends on the length of the blade. Slip <strong>factor</strong>
<strong>in</strong>creases with blade length.
3

Axial compressor Centrifugal compressor
4

Slip flow <strong>in</strong> Centrifugal compressor
Busemann (1928) called this
displacement flow;
other authors refer to its
rotat<strong>in</strong>g cells as
relative eddies.
C 2
C u2
C a2
σ = 1−
v R s<strong>in</strong> β
s 2 2k
π = Ω
θ
π s<strong>in</strong> β
σ = 1−
Z
2k
vθ s
ΩR
2
Stodola (1927)
(def<strong>in</strong>ition)
2k
Z
5

Axial compressor Centrifugal compressor
X f
Howell (1945)
Deviation angle δ = β 2 - β 2k .
2 ( 2 ) + 0.
002(
β δ )
( 0.
23
) θ t
δ +
= x f
2k
t
c
σ = 1− − β
u
σ = 1−
C u
Z-number of blades
Slip <strong>factor</strong> σ or <strong>slip</strong> angle δ = β 2 - β 2k .
( tan(
β + δ ) ( ) )
a2
2
2k
tan
s<strong>in</strong> β
Z
0.
7
2k
C a
2k
(Wiesner)
2k
6

∆P
ρ w
1
General Inviscid
∆P
ρ w
eqv
2
1
1
2
2 ( 2 ) + 0.
002(
β δ )
( 0.
23
) θ t
δ +
= x f
2k
W 1
C a1
2
1
β 1,eqv
2
⎛ w
= 1−
⎜
⎝ w
⎛ w
= 1−
⎜
⎝ w
2
1
⎞
⎟
⎠
2
W 2
2
1
C a2
⎟ ⎞
⎠
In L<strong>in</strong>ear cascade C a1 = C a2
⎛ cos
= 1−
⎜
⎝ cos
2
β 2,eqv
( β1,
eqv )
( β )
2,
eqv
⎟ ⎞
⎠
2
General blade row
∆P
ρ w
1
2
1
⎛ w
= 1−
2 ⎜
⎝ w
2
1
⎞
⎟
⎠
2
⎛ cos
= 1−
⎜
⎝ cos
( β1)
( β )
2
K D
Equivalent l<strong>in</strong>ear cascade
∆P = ∆P eqv
β 2 =β 2,eqv
( ) = K cos(
)
cos β1
ε − β
β1, eqv D
eqv = β1,
eqv 2
eqv − i + δ θ eqv
2 = β 2 k δ
k i + = 1 β1
ε =
β +
β
7
⎞
⎟ ⎟
⎠
2

Calculation of equivalent pitch (solidity)
β
eqv
2k
, eqv
( ( ) ( ) 2
0.
23 2x
f + 0.
002 β k δ θeqv
teqv
δ +
= 2
b
t eqv
( t + t ) b b ( t ) 1 + t2
( . 23(
2 ) 0.
002(
) ( 1)
2
x + β + δ θ t −
= β2k
+ 0 f
2k
eqv eqv
=
1
2t
c
σ =
1− − β
u
( tan(
β + δ ) ( )
a2
2
2k
, eqv eqv tan
2
2
2
1
=
t
2
2k
, eqv
2t
2
8

( ) w / w
( w / w )
2
2
( ) = K cos(
)
cos β
β1, eqv D
Calculation of the coefficient K D
1. Influence of the channel height h.
Stratford (1959) obta<strong>in</strong>ed the height (h) of the diffuser
with given length (b) that has maximal static pressure
rise coefficient. The velocity ratio w 2 /w 1 depends on the
ratio b/h. The experimental data for l<strong>in</strong>ear cascade
were obta<strong>in</strong>ed for h=2b. The experimental data of
Stafford may be approximated by equation
Experimental data for diffusers
1
1
h=
2b
=
⎛
⎜
⎝
2b
h
1
⎞
⎟
⎠
0.
1
9

W 1
C a1
Calculation of the coefficient K D
β 1,eq
v
2. Influence of the ratio C a1/C a2.
Cascade with constant height and solidity:
w 2 =
w
1
K
c
c
T
a2
a1
=
cos
cos
c
c
a2
a1
( β ) 1
( β )
If the solidity is changed, then velocity ratio <strong>in</strong> the
equivalent cascade must be multiplied by the pitch ratio
w 2 =
w
1
W 2
KT
C a2
β 2,eq
v
cos
cos
( ) = K cos(
)
cos β
β1, eqv D
( β ) 1
( β ) 2
1
t
t
1
2
2
⎛
⎜
⎝
2b ⎞
⎟⎠
h
0.
1
10

W 1
K
Calculation of the coefficient K D
C a1
In General case
D
β 1,eq
v
=
( ) = K cos(
)
cos β
K
β1, eqv D
W 2
1
2
T
C a2
KT
β 2,eq
v
where
For axial compressors
w
w
2
1
where
1
=
cos
cos
K
( β ) 1,
eqv
( β )
T
2,
eqv
=
c
c
a2
a1
=
t
t
1
2
K
T
cos
cos
( β ) 1
( β )
2
Hence, K T =K D
⎛
⎜
⎝
2b ⎞
⎟⎠
h
(effect of boundary layer separation and blockage)
K
T
=
c
c
a2
a1
t
t
1
2
⎛
⎜
⎝
2b ⎞
⎟⎠
h
0.
1
0.
1
11

Impellers with high blade load<strong>in</strong>g
and splitters
The rotor with splitters
is divided <strong>in</strong>to
sequence of rotors.
Each rotor has the
same number of blades
at <strong>in</strong>let and exit.
For high loaded rotors
the load<strong>in</strong>g of each
equivalent l<strong>in</strong>ear
cascade must be less
then maximal allowed.
12

Impellers with high blade load<strong>in</strong>g
Rotor 1
and splitters
Rotor 2
Calculate <strong>slip</strong> angle
for each rotor.
The <strong>slip</strong> angle of
the rotor is a sum
of the <strong>slip</strong> angles
obta<strong>in</strong>ed for each
row
13

( ) = K cos(
)
cos β
β1, eqv D
High blade load<strong>in</strong>g
1
K
D
=
Stall conditions
K
1
2K
T
T
K
T
=
c
c
a2
a1
t
t
1
2
⎛
⎜
⎝
2b ⎞
⎟⎠
h
0.
1
14

Maximal pressure gradient
Analysis of experimental data of Howell (1945), Emery (1957),
Bunimovich (1967)allows obta<strong>in</strong><strong>in</strong>g follow<strong>in</strong>g equation for
coefficient of static pressure rise at stall conditions (maximal
pressure rise)
m
( ) = K cos(
)
cos β
st
β1, eqv D
∆P
2
ρ w
1
=
b
∫
0
1
st
2
⎛
= 1−
⎜
⎝
w
w
2,
st
1,
st
⎞
⎟
⎠
2
1
⎛ cos
= 1−
⎜
⎝ cos
( )
2
β ⎞ 1eqv,
st ⎟
1.
124
=
( β ) ⎟ 1+
m t
2eqv,
st
( w ) ∫
SuctionSide
− w1
db ( w1
− wPr
essureSide)
Parameter mst characterizes type of the cascade. E.g. For
cascades studied by Howell mst =1.
t = t / b is relative pitch of the l<strong>in</strong>ear cascade
b
0
K
D
=
K
K
1
2
T
T
⎠
K
T
db
=
c
c
a2
a1
st
t
t
1
2
⎛
⎜
⎝
2b ⎞
⎟⎠
h
0.
1
15

Compressor Stages used to test the model
1. Stage of turbojet Olympus
2. Stage tested by Stechk<strong>in</strong>
3. Stage designed by Beker Eng<strong>in</strong>eer<strong>in</strong>g and tested at
Concept.
4. Stage designed by Beker Eng<strong>in</strong>eer<strong>in</strong>g , not tested
5. Stage C1 (tandem blades) (data from USSR)
6. Monig et al. Trans ASME. Journal of turbomach<strong>in</strong>ery
1993,v115, p.565-571.
7. Stage C9 (data from USSR)
8. Musgrave D., Plehn N.J. Mixed flow stage design and test
results with a pressure ratio of 3:1. ASME Gas turb<strong>in</strong>e
conference presentation 87-GT-20.
16

Results
510.6
165
60.05
0.612
1.0
1.78
19.9
2.89
6.36
12/24
8
520
161
46.25
0.175
0.56
2.18
6.25
2.75
6.79
18/36
7
425
160
50.47
0.316
0.74
2.0
8.48
2.0
3.86
15/30
6
455
185.3
46.65
0.346
0.25
2.12
9.0
2.7
4.54
27/27
5
550
182.9
56.33
0.641
1.50
2.1
48.28
1.0
3.87
7/14
4
500
189.3
57.48
0.397
1.27
1.89
12.0
0.48
5.23
11/22
3
483
170
32.7
-
0.50
2.30
33.58
3.0
3.95
12
2
472.3
180
57.8
0.695
1.0
1.83
16.99
0.41
4.35
6/12
1
u 2
m/s
(tip
velocity)
c a2
m/s
β 1
(<strong>in</strong>let
flow
angle )
t 2,mean
(outlet
relative
pitch)
t 1,mean
(<strong>in</strong>let
relative
pitch)
β 2k
(outlet
blade
angle)
G
Kg/s
Rotor
pressure
ratio
Z
number
of blades
Rotor
No.
Table 1. Rotor parameters.
1
2
r
r
17

Results
Table 2. Slip <strong>factor</strong> and exit angle.
0.877
0.861
0.887
0.895
0.890
0.893
35.0
35.9
8
0.926
0.931
0.910
0.919
0.934
0.933
17.8
17.9
7
0.896
0.914
0.890
0.908
0.896
0.899
23.0
23.32
6
0.903
0.910
0.876
0.902
0.901
0.915
20.6
20.17
5
0.870
0.834
0.762
0.874
0.875
-
56.3
-
4
0.865
0.880
0.848
0.888
0.888
0.887
27.0
27.0
3
0.840
0.823
0.72
0.840
0.896
-
43.8
-
2
0.784
0.718
0.860
0.832
0.878
0.881
32.0
31.8
1
σ
Eck
σ
Stechk<strong>in</strong>
σ
Stodola
σ
Wiesner
σ
Present
work
σ
measured
β 2
Present
work
β 2
measured
Rotor
No.
18

Measured and calculated <strong>slip</strong> <strong>factor</strong> for 6 <strong>impellers</strong>
19

Calculated <strong>slip</strong> <strong>factor</strong> for 8 <strong>impellers</strong>
20

Conclusions
• Suggested method allow<strong>in</strong>g reduction of
arbitrary rotor blade passage to
equivalent l<strong>in</strong>ear cascade
• Method allows establish<strong>in</strong>g equivalence
between axial and <strong>centrifugal</strong> <strong>impellers</strong>
• Method allows calculat<strong>in</strong>g <strong>slip</strong> <strong>factor</strong>
<strong>based</strong> on well established l<strong>in</strong>ear cascade
correlations for deviation angle
21