A slip factor calculation in centrifugal impellers based on linear cascade data

A slip factor calculation in centrifugal impellers based on

linear cascade data

Dr. Abraham FrenkBecker Turbo System Engineering (2005).

(Presented by Dr. E. Shalman)

AbstractAccurate modeling of the flow slip against direction of rotation is essential for correct prediction of the centrifugal impeller performance. The process is characterized by a slip factor. Most correlations available for calculation of the slip factor use parameters characterizing basic impeller geometry (review of Wiesner (1967) and Backstrom (2006)).

Approach presented below is based on reduction of radial cascade to equivalent linear cascade. The reduction allows to calculate characteristics of radial blade row using well established experimental data obtained for linear cascades, diffusers and axial blade rows.

2

Rotors with splitters or with high loading must be divided into few radial blade rows. In the case of multiple rotor blade rows the slip angle is calculated for each blade row. The slip angle of the rotor is a sum of the slip angles obtained for each row. Suggested reduction of radial blade rows allows also calculating of other parameters essential for impeller design.

Suggested method allows also determine additional causes influencing slip factor. The slip factor depends not only on parameters characterizing basic impeller geometry, but on difference of inlet flow angle from stall flow angle. In the rotors with the same basic geometry parameters slip factor depends on the length of the blade. Slip factor increases with blade length.

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Axial compressor Centrifugal compressor4

ZRv ks 22 sinβπθ Ω=

Zk2sin

1βπ

σ −=

2k

Ca2

Cu2

C2

Stodola (1927)Busemann (1928) called this displacement flow; other authors refer to its rotating cells as relative eddies.

(definition)2

1R

v s

Ω−= θσ

5Slip flow in Centrifugal compressor

Axial compressor Centrifugal compressor

t

( ) ( )( )kka

u

c22

2

2 tantan1 βδβσ −+−=

7.02sin

1Z

kβσ −=

( ) ( )( ) tx kf θδβδ ++= 22 002.0223.0

(Wiesner)

Deviation angle δ = β2 - β2k.

Slip factor σ or slip angle δ = β2 - β2k.

Xf

Howell (1945)

6

Z-number of blades

Ca

Cu

2k

2

1

2211

12

−=

∆w

w

w

P

ρ

General Inviscid

( )( )

2

,2

,1

2

1

2211 cos

cos11

2

−=

−=

∆

eqv

eqveqv

w

w

w

P

ββ

ρ

In Linear cascade Ca1= Ca2

General blade row

( )( )

2

2

1

2

1

2211 cos

cos11

2

−=

−=

∆DK

w

w

w

P

ββ

ρ

Equivalent linear cascade∆P = ∆Peqvβ2 =β2,eqv

( ) ( )1,1 coscos ββ Deqv K=

2,1 ββε −= eqveqv

eqveqv i θδε =+−

( ) ( )( ) tx kf θδβδ ++= 22 002.0223.0

δββ += k22

ik += 11 ββ

W2

Ca2

β2,eqvW1

Ca1

β1,eqv

7

b

Calculation of equivalent pitch (solidity)

( ) ( )( ) ( )1002.0223.0 22

2,2 −+++= eqveqvkfkeqvk tx θδβββ

( ) ( )( ) eqveqvkfeqv tx θδβδ ++= 22 002.0223.0

( ) ( )21

22

121

22 22

tt

t

b

t

btt

tteqv +

=+

=

8

( ) ( )( )eqvkeqveqvka

u

c,2,2

2

2 tantan1 βδβσ −+−=

( )( )

1.0

212

12 2

/

/

=

= h

b

ww

ww

bh

Experimental data for diffusers

1. Influence of the channel height h.

Calculation of the coefficient KD

Stratford (1959) obtained the height (h) of the diffuser with given length (b) that has maximal static pressure rise coefficient. The velocity ratio w2/w1 depends on the ratio b/h. The experimental data for linear cascade were obtained for h=2b. The experimental data of Stafford may be approximated by equation

( ) ( )1,1 coscos ββ Deqv K=9

2. Influence of the ratio Ca1/Ca2.

Calculation of the coefficient KD

( )( )2

1

1

2

1

2

cos

cos

ββ

a

a

c

c

w

w=

Cascade with constant height and solidity:

1.0

2

1

1

2 2

=

h

b

t

t

c

cK

a

aT

If the solidity is changed, then velocity ratio in the equivalent cascade must be multiplied by the pitch ratio

( )( )2

1

1

2

cos

cos

ββ

TKw

w=

W2

Ca2

β2,eq

v

W1

Ca1

β1,eq

v

( ) ( )1,1 coscos ββ Deqv K=10

Calculation of the coefficient KD

( )( )

( )( )2

1

,2

,1

1

2

cos

cos

cos

cos

ββ

ββ

T

eqv

eqv Kw

w==

For axial compressors

TKTD KK 2

1

=

Hence, KT=KD

In General case 1.0

2

1

1

2 2

=

h

b

t

t

c

cK

a

aTwhere

W2

Ca2

β2,eq

v

W1

Ca1

β1,eq

v

( ) ( )1,1 coscos ββ Deqv K=11

(effect of boundary layer separation and blockage)

1.0

2

1

1

2 2

=

h

b

t

t

c

cK

a

aT

where

The rotor with splitters is divided into sequence of rotors. Each rotor has the same number of blades at inlet and exit.

For high loaded rotors the loading of each equivalent linear cascade must be less then maximal allowed.

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Impellers with high blade loading and splitters

Calculate slip anglefor each rotor.

The slip angle of the rotor is a sum of the slip angles obtained for each row

Rotor 1

Rotor 2

13

Impellers with high blade loading and splitters

14

( ) ( )1,1 coscos ββ Deqv K=

High blade loading

Stall conditions

TKTD KK 2

1

=1.0

2

1

1

2 2

=

h

b

t

t

c

cK

a

aT

Maximal pressure gradient

( )( ) tmw

w

w

P

ststeqv

steqv

st

stst

+=

−=

−=

∆1

124.1

cos

cos11

2

2

,2

,1

2

,1

,2211 β

βρ

Analysis of experimental data of Howell (1945), Emery (1957), Bunimovich (1967)allows obtaining following equation for coefficient of static pressure rise at stall conditions (maximalpressure rise)

Parameter mst characterizes type of the cascade. E.g. For cascades studied by Howell mst =1.

btt /= is relative pitch of the linear cascade

( ) ( )∫∫ −−=b

essureSide

b

eSuctionSidst dbwwdbwwm0

Pr1

0

1

15

( ) ( )1,1 coscos ββ Deqv K= TKTD KK 2

1

=1.0

2

1

1

2 2

=

h

b

t

t

c

cK

a

aT

1. Stage of turbojet Olympus 2. Stage tested by Stechkin3. Stage designed by Beker Engineering and tested at

Concept. 4. Stage designed by Beker Engineering , not tested5. Stage C1 (tandem blades) (data from USSR) 6. Monig et al. Trans ASME. Journal of turbomachinery

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 turbine conference presentation 87-GT-20.

Compressor Stages used to test the model

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Results

510.616560.050.6121.01.7819.92.896.3612/248

52016146.250.1750.562.186.252.756.7918/367

42516050.470.3160.742.08.482.03.8615/306

455185.346.650.3460.252.129.02.74.5427/275

550182.956.330.6411.502.148.281.03.877/144

500189.357.480.3971.271.8912.00.485.2311/223

48317032.7-0.502.3033.583.03.95122

472.318057.80.6951.01.8316.990.414.356/121

u2

m/s(tip

velocity)

ca2

m/s

β1

(inlet

flow

angle )

t2,mean(outlet relative

pitch)

t1,mean(inlet relative

pitch)

β2k(outlet blade

angle)

G

Kg/s

Rotorpressure

ratio

Znumber

of blades

Rotor

No.

Table 1. Rotor parameters.

1

2

r

r

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ResultsTable 2. Slip factor and exit angle.

0.8770.8610.8870.8950.8900.89335.035.98

0.9260.9310.9100.9190.9340.93317.817.97

0.8960.9140.8900.9080.8960.89923.023.326

0.9030.9100.8760.9020.9010.91520.620.175

0.8700.8340.7620.8740.875-56.3-4

0.8650.8800.8480.8880.8880.88727.027.03

0.8400.8230.720.8400.896-43.8-2

0.7840.7180.8600.8320.8780.88132.031.81

σEck

σStechkin

σStodola

σWiesner

σPresent

work

σmeasured

β2Present

work

β2

measured

Rotor

No.

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Measured and calculated slip factor for 6 impellers

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Calculated slip factor for 8 impellers

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Conclusions• Suggested method allowing reduction of

arbitrary rotor blade passage to equivalent linear cascade

• Method allows establishing equivalence between axial and centrifugal impellers

• Method allows calculating slip factor based on well established linear cascade correlations for deviation angle

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