1 Turbomachinery Class 12. 2 Axial vs. Radial Machines Need to determine what type of turbine is...

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Turbomachinery

Class 12

2

Axial vs. Radial Machines

Need to determine what type of turbine is most efficient for application - function of Ns for both compressors and turbine

3

2 23/ 4 3/ 4

1 /0

0201

01

/ /

1

s

ideal

p

N m N mN

h pc T

p

Need to determine what type of turbine is most efficient for application - function of Ns for both compressors and turbine

4

Radial Radial OutflowOutflow Turbine Turbine

Ljungstrom Steam Turbine: Dixon - steam turbine design - No stator blades counter-rotating blades - radial outflow - large amount of work per stage - rugged

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Radial Outflow Turbines

• Ljungstrom Turbine arrangement

Compatible with expanding steam, more area with same blade height as density drops

Vaneless - Counter rotating

Old Configuration recently re-invented for gas turbines

– axial counter-rotating

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Counter-Rotating Turbines

• Counter Rotation High Stage Work

Compare: – Conventional Axial Stage, 50% Reaction & 90 Gas Turning

vs. – Counter Rotating, Vaneless Stages with 90 Gas Turning

Cx1 = Cx2, U1 = U2 = Cx/U= 0.6

Repeating Stages

Counter Rotation U changes direction

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Radial Flow Turbine Analysis

• Remember from Class:

2

2tan 1

RE

2

22tan 1

RE

2

2tan 2

RE

2

22tan 2

RE

0

1

2

1

2

3

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Radial Flow Turbine Analysis

• In this problem, for the axial stage

=0.6, R=0.5, and 1212

- Iteration:

Guess 1

From Calculate E.

From Calculate 2

Iterate until turning (12 is correct

• For the counter-rotating stage…..match turning

2

2tan 1

RE

2

2tan 2

RE

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Radial Flow Turbine Analysis Conventional vs Counter-Rotating

Counter Rotating

Cx/U 0.600 0.600Beta 1 25.097 15.482Alpha 1 64.903 62.774Turning 90.000 90.000Beta 2 -64.903 -74.518Alpha 2 -25.097 -62.774Reaction 0.500 1.000E -1.562 -2.332

2nd StageCx/U -0.600Alpha 3 -62.774Beta 1 -15.482Error 0.000

Convergence 2.135 3.610

Conventional

Counter-rotating: high stage work (E)

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Radial Flow Turbine Analysis - Conventional Design

50% Reaction, 90 Deg Turning

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

C1 U

W1

W2

C2

50% Reaction, 90 Deg Turning

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

C1 U

W1

W2

C2

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Radial Flow Turbine Analysis - Counter-Rotating Design

01 2 90

Vaneless, Counter Rotating, 90 Deg Turning

W1

C1 U

W2U

U'

W3

C2

Vaneless, Counter Rotating, 90 Deg Turning

W1

C1 U

W2U

U'

W3

C2

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Radial Inflow [90Radial Inflow [9000 IFR] Turbines IFR] Turbines

Kinematic view Thermodynamic view

Exit part of rotor (exducer) is curved to remove most of tangential component of velocity

Advantage of IFR turbine: efficiency equal to axial turbine, greater amount of work per stage, ease of manufacture, ruggedness

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Radial Flow Turbines

• Radial Inflow Turbine with Scroll

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• Radial Inflow Turbine Stator/Rotor

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• Radial Inflow Turbine Stator/Rotor [No shroud]

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Radial Flow Turbines• Radial Inflow Turbine Scroll

Scroll or distributor - streamwise decreasing cross flow area - provide nearly uniform properties at exit

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Radial Flow Turbines• Radial Inflow Turbine Scroll - Stator

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Radial Flow Turbines• Radial Inflow Turbine Impeller

Note - direction of rotation - rotor rearward curvature

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Radial Flow Turbine Design• Nominal Stator / Rotor Design:

Station 1 – Inlet to StatorStation 2 – Exit of Stator, Inlet to Rotor

[Radially inward]Station 3 – Exit of Rotor [Absolute velocity is axial]Station 4 - Exit of Diffuser

• Rotor inlet relative velocity is radially inward

- For Zero Incidence at Rotor Inlet, W2=Cr2 and C2=U2

• Rotor exit absolute flow is axial

- For Axial Flow at Rotor Exit, C3=Cx3 and C3=0

C2

Cm2=Cr2=W2

U2

Cm3=C3=Cx3

U3

W3

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Radial Flow Turbine Design- 900 IFR

• For adiabatic irreversible [friction] processes in rotating components

• From the Alternate Euler Equation:

• and

2 2 2 2 2 22 3 2 3 2 3

0 2Rotor

U U W W C Ch

gJ

2 2 22 2 2C W U 2 2 2

3 3 3W C U

2 3

2 22

2 32 2 2orel orel orel

U UUI h h h

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Radial Flow Turbine Design

• substituting:

• Thus from Alternate Euler’s Equation :

gJ

UUWUCUWCh Rotor 2

23

22

23

22

22

23

23

22

0

22

0 01 03 02 03Rotor

Uh h h h h

gJ

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Radial Flow Turbine Design• Example: Dixon 8.1

• The rotor of an IFR turbine, designed to operate at nominal condition, – Diameter is 23.76 cm and rotates at 38,140 rev/min.

– At the design point the absolute flow angle at the rotor entry is 72 deg.

– The rotor mean exit diameter is ½ the rotor diameter

– The relative velocity at the rotor exit is twice the relative velocity at the inlet.

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2

3 2

3 2

02

22

2 2 2

2 2 2

23.76

38,140

/ 2 12.88

2

72

38,140 0.2376 / 60 474.560cot 154.17

/ sin 498.9

mean

Given

D cm

N rpm

D D cm

W W

rotor inlet design point flow angle

NDU mps

W U mps

C U mps

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2 22 2 2 2 23 3 3

2 2 2 2 22 3 2

2 2 2 2 23 2 2

2 2 2 22 3

2 2 2 2 2 22 3 2 3 2 3 2 2

0

2 154.17 0.5 474.5 38,786 /

1 0.25 168,863 /

3 71,305 /

210,115 /

225,142 /2

C W U m s

U U U m s

W W W m s

C C m s

Examing relative sources of specific work

U U W W C CW h m s

W

2 2 20 2

0.375( ) 0.158( ) 0.476( ) [% ]

225,142 /

U W C of total

Also

W h U m s

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Radial Flow Turbine Design• Example: Baskharone p. 434-8

• 0=inlet 1=stator exit 2=rotor inlet 3=rotor exit

• Stator / nozzle exit Mach number M1=0.999

00 0

4.16 / s 71,600 1.33

8.84 1205 0.038stator

m kg N rpm

p bar T K

1 1 1 1

01 00 00

010 1 1

01 1

1156.71

0.999 1034.4 628.35 590.6

[ ] 73.0cos

pc R

M T a C

p p p

m T RFP f M

p A

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Radial Flow Turbine Design• Example: Baskharone p. 434-8 cont’d

• In constant area interstage duct, apply free-vortex condition to flow from stator exit to rotor inlet

1 1 1

1 1 1

cos 172.6

sin 564.8r

u

C C mps

C C mps

2 1 1 2

2 1 1 2

2

22 02 2

2 2

2

( / ) 184.8

( / ) 605.1

632.69

/ 2 1032

627.62

1.008

r r

u u

p

C C r r mps

C C r r mps

C mps

T T C c K

a RT mps

M

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• Mollier Diagram for Radial Flow Turbine (Dixon):

s

1/2 C2spouting

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• Spouting Velocity [from hydraulic turbine practice]:

Spouting Velocity, C0: the velocity which has an associated kinetic energy equal to the isentropic enthalpy drop from the turbine inlet stagnation pressure P01 to the final exhaust pressure.

• By taking the exhaust pressure as the isentropic exit static pressure, and assuming no diffuser, then

• For an ideal radial turbine with KE recovery, this leads to

or

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01 03

1;

2 ss

Ch h ss refers to ideal work at same back pressure

gJ

Jg

C

gJ

Uh Rotor

20

22

0 2

1 707.0

0

2 C

U

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01 03

01 3

2 202 03 2 3 2 3

01 3 02 03 03 3 3 3 3 3

0

20 3 3 3 3 3

1

2

1( ) ( )

2

tsts ss

ss ss s s ss

Rotorts

Rotor s s ss

h hactual W

ideal m h h h

W h h h h C C

h h h h h h W h h h h

h

h C h h h h


• Nominal Design Point Efficiency:

Efficiency is centrifugal devices is often represented by total-to-static conditions: ts = total-static

s

22U

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• The enthalpy losses for the stator (nozzle) and rotor can be expressed as a fraction, , of the exit kinetic energy:

• For a constant pressure process, ds = dh/T so that:

23 3 3

1for rotor

2s rh h w

2 33 3 2

2

1for nozzle

2s ss n

Th h c

T

33 3 2 2

2

( )s ss s

Th h h h

T

ss

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• Substituting into the efficiency definition:

• For the Nominal Design Point velocity triangles, we see that:

0

2 2 2 33 3 2

2

11 ( )

2

Rotorts

r n

h

TC W C

T

2 2 2

3 3 3

3 3 3

csec

csec

cot

C U

W U

C U

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• Substituting, we get for the efficiency:

• The temperature ratio, T3/T2, generally only has a very minor effect on the value of ts so that it can be neglected.

2

2 2 23 32 3 3

2 2

1

11 csec ( csec cot )

2

ts

n r

T r

T r

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• In addition, the values of r3 and 3are taken to apply at the mean exit radius of the rotor, i.e. r3 = ½ (r3h+ r3t), so that:

2

32 2 22 3 3

2

1

11 csec ( csec cot )

2

ts

avgn r avg avg

r

r

Read Dixon Example 8.2, page 254

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From Centrifugal Compressor Notes

• Slip: flow does not leave impeller at metal angle [even for inviscid flow]

• If absolute flow enters impeller with no swirl, =0.• If impeller has swirl (wheel speed) , relative to the impeller the

flow has an angular velocity - called the relative eddy [from Helmholtz theorem].

• Effect of superimposing relative eddy and through flow at exit is one basis for concept of slip.

Relative eddy Relative eddy with throughflow

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• Static pressure gradient across passage causes streamline to shift flow towards suction surface

• In reality, the incidence to the rotor varies over the pitch of the rotor as:

due to – Potential and wake interaction with the vane.– Relative eddy effect seen at exit of compressor– Effect produces a LE slip factor

This variation over the pitch leads to an - optimal incidence and - optimal number of blades

where the efficiency of the rotor is a maximum.


2 2 , rCf

U

P=pressureS=suction

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• Rotor Inlet Velocity Triangle (with incidence):

– Average relative velocity W and avg. relative incidence 2

• If we define an incidence factor, [like slip factor in compressors]:


2

2

U

CU

U2

W2CR2=CM2

C2

CU2

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• From the work of Stanitz regarding slip factors:

• Then from the rotor inlet velocity triangles, the inlet flow angle to the rotor is:

0.63 21 1 where Z=Number of Rotor Blades

Z Z

2 22

2 2

2tan where = M

M

U C

Z C U

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• Criteria for the Optimal Number of Blades:

2

2

min

min

2

1 12

2

1

2

0;

r

T T T T T T

From particle physics analogy

dWF ma f r f r W

dtp W p

W r and Wr r

WGets r implying that

W at given r is not constant across passage p s

W W r

At r r if W U r and W r U U

min

min 2

2

2 2 tan

T

T

T

Z

UZ

W

Jamieson model

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• Criteria for the Optimal Number of Blades:

Optimum blade number balances loading & friction

• Rohlick model uses (quantities at the inlet to rotor):

• Jamieson model

min2

2 min

2

min 2

2

0.03 57 12

UC Z

U Z

Z

min 22 tanZ

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• Other Correlations for Optimal Number of Blades (Rohlick results similar to Jamieson):

from Dixon

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This is to clarify some of the confusing notation in Dixon regarding blade count

• Stanitz correlation

– uses blade number and flow coefficient to calculate the relative radial turbine exit flow angle.

• Other correlations – uses semi-empirical expressions for calculating the optimum

[minimum] blade count Z for an optimum efficiency design, where

– For such a design the exit flow will be radial [in the absolute frame], therefore 2=0 and the correlations are in terms of the corresponding absolute frame air angles [2], e.g.

22

2

2 2tan

M

U

ZC Z

2 2 2tanrU C

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This is to clarify some of the confusing notation in Dixon regarding blade count

• Jamieson

• Rohlik

min 22 tanZ

2

min 20.03 57 12Z

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• Specific Speed relates impeller shape & velocity diagram

• Rohlick shows:

where

4/3

2/1

H

NQN s

3/ 4 3/ 2 1/ 2 3/ 23/ 2

3 3 32/

3 3 2 3

is

ii

h C D hUN

C U D Dh

/0 sin exit

0 0in exit

ideal work, P to P

ideal work, P to Pi

i

h

h

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Efficiency [ts] of 900 IFR Turbine with Ns [Rohlick analysis]

3 2

3

2 2

3 2

3 2

2

/ 2.0

0

. .

/ 0.7

/ 0.4

,

av

U

opt

shroud

hub

Constraints of analsis

W W

C

i e zero incidence

r r

r r

For each value of shaded

area governed by geometric

scale limits as above

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Efficiency [ts] of 900 IFR Turbine with Ns [Rohlick analysis]

Dixon Fig. 8.14

Ns units: Nondimensional=radians

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Radial Flow Turbine Design• Rohlik Model shows Distribution of Losses

Clearance: pressure diff. effects between rotor and shroud

Windage: friction effects between rotor and shroud

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Radial Flow Turbine Design• Flow Path Shape is Related to Specific Speed

Maximum efficiency designs

1 Turbomachinery Class 12. 2 Axial vs. Radial Machines Need to determine what type of turbine is...

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