lecture 8 – axial turbines 2 + radial compressors 2

Chalmers University of Technology

Lecture 8 – Axial turbines 2 + radial compressors 2

• Axial turbines– Turbine stress considerations– The cooled turbine– Simplified 3D axisymmetric inviscid flow

• Free vortex design method

• Radial compressors 2– Diffuser and vaneless space– Compressor maps

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Choice of blade profile, pitch and chordRotor blade stresses:

1 centrifugal stress:

2 gas bending stresses reduce as cube of chord:

3 centrifugal bending stress

Annulus area

ns)interactio estator wak toduen fluctuatio to

subject(MN/m 93 ... exampleour

1

22

3

velocityin whirl Change

32max

zc

h

n

CCm mwmwgb

bar 2000MN/m 200]example fromgeometry [

3

4 taperNormal

2

22

max

alloy

CoCrNi

b

t

rr

bct ANardr

a

Steady stress/Creep Combination steady/fluctuating

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The cooled turbine• Cooled turbine

– application of coolant to the nozzle and rotor blades (disc and blade roots have always been cooled). This may reduce blade temperatures with 200-300 K.

– blades are either: • cast - conventional, directionally solidified, single crystal

blade• forged

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The cooled turbineTypical cooling

distribution for stage:

Distribution required for operation at 1500 K

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The cooled turbine - methods• Air cooling is divided into the following

methods– external cooling

• Film cooling

• Transpiration cooling

– internal cooling

Techniques to cool rotor blade

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The cooled turbine - methods

Techniques to cool stator blade

• Stator cooling– Jet impingement cools the hot leading

edge surface of the blade.

– Spent air leave through slots in the blade surface or in the trailing edge

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3D axi-symmetric flow (inviscid)• Allow radial velocity components.

– Derive relation in radial direction– Balance inertia, FI, and pressure forces

(viscous forces are neglected)

• Derived results can be used to interpret results from CFD andmeasurements

onaccelerati

SS

SS

Sw

massdirectionblade

inwidthunit

streamlinealongonacceleratiRadial

iii

streamlinecurvedtodueforceRadial

ii

forcelCentripeta

iI

dt

dC

r

C

r

Cdrrd

FFFF

sincos22

)(

)()(

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3D flow (inviscid)• Pressure forces FP balancing the

inertia forces in the radial direction are:

rdpdpdrdprdrdpdpdrdprd

termsorderhighneglect

drddp

pprdddrrdppF

dd

P

222)(

directionradialin actingelement of

sideson force Pressure

22sin

• Equating pressure forces and inertia forces yields:

SS

SS

Sw

dt

dC

r

C

r

C

dr

dp

sincos1 22

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3D flow

• The above equation will be usedto derive an energy relation.

r

C

dr

dp w21

• For many design situations rs can beassumed to be large and thus αs small.These approximations give the radial equilibrium equation:

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3D flow

• The radial variation is therefore:

222

0 2

1

2 wa CChC

hh

• The stagnation enthalpy at any radius is (neglecting radial components):

dr

dCC

dr

dCC

dr

dh

dr

dh ww

aa 0

• We have the thermodynamic relation:

which produces:

dp

dhTds

2

1

11

dr

dp

dr

dsTtermsorderhigherneglect

dpdr

d

dr

dp

dr

dTds

dr

dsT

dpTds

dr

d

dr

dh

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3D flow• We now have:

dr

dCC

dr

dCC

r

C

dr

dsT

dr

dCC

dr

dCC

dr

dp

dr

dsT

dr

dh

ww

aa

w

ww

aa

termmequilibriuradialThe

2

0 1

• If we neglect the radial variation of entropy we get the vortex energy equation:

dr

dCC

dr

dCC

r

C

dr

dh ww

aa

w 2

0

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Theory 8.1 – The free vortex design methodUse:

and design for:– constant specific work at all radii– maintain Ca constant across the annulus

11222

1

1

1

221

1

212

2

1

nIntegratio2

lnln)ln( ln)ln()ln(

)ln( 0

rCrCr

r

r

rr

C

CCC

Cr

dr

C

dC

dr

dCC

r

C

www

www

ww

www

w

Thus Cwr must be kept constant to fulfill our design assumption.This condition is called the free vortex condition

– Designs based on free vortex principle sometimes yields a marked variation of degree of reaction with radius

dr

dCC

dr

dCC

r

C

dr

dh ww

aa

w 2

0

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Design methods (Λ m = 0.50)

Free vortex blading (n = -1)gives the lowest degree of reaction in the root region!

• For low root tip ratios a high degreeof reaction is required in the mid to ensure positive reaction in the root

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Free vortex design - turbines• We have shown that if we assume

– constant specific work at all radii, i.e. h0 constant over annulus (dh0/dr=0)

– maintain Ca constant across the annulus (dCa/dr=0)

• We get– Cwr must then be kept constant to satisfy

the radial equilibrium equation

• Thus we have Cw r = Ca tanα r r = constant. But Ca constant => tanα r r = constant, which leads to the radial variations:

mm

mm

r

r

r

r

33

3

22

2

tantan

tantan

33

33

3

22

222

22

tantan

tantantan

a

m

mm

m

a

m

mm

m

a

C

U

r

r

r

r

C

U

r

r

r

r

C

U

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Radial compressor 2 - General characteristics

• Suitable for handling small volume flows– Engines with mass flows in this range will have very small geometrical

areas at the back of an axial compressor when operating at a pressure ratio of around 20.

– Typical for turboshaft or turboprop engines with output power below 10MW

• Axial compressor cross section area may only be one half or a third of the radial machine

• Better at resisting FOD (for instance bird strikes)

• Less susceptible to fouling (dirt deposits on blade causing performance degradation)

• Operate over wider range of mass flow at a particular speed

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Development trends

3

2

• Pressure ratios over 8 possible for one stage (in production – titanium alloys)

• Efficiency has increased around % per year the last 20 years

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Axial centrifugal combination - T700

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The vaneless space - diffuser

!!!Constant

0

rC

space

vanelessinTorque

w

)(γ

)(γ

r Mγ

MAP

RTm 12

1

2

0

0

2

11

Use Cw and guessed

Cr => C => T => M, Mr

Perform check on area (stagnation properties

constant):

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The diffuser• Boundary layer growth

and risk of separation makes stagnation process difficult

• Diffuser design will be a compromise between minimizing length and retaining attached flow

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Shrouds• Removes losses in

clearance.

• Not used in gas turbines– Add additional mass– Unacceptable for high

rotational speed where high stresses are produced

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Non-dimensional numbers - mapsWe state that:

design of scalelinear

viscositykinematic

speed rotational

),,,,,,,,(

),,,,,,,,(

01012

0101102

D

n

DdesignRnTPmf

DdesignRnTPmfP

c

based on the observation that we can not think of any more variables on which P02 and ηc depends.

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Non-dimensional parameters

• Nine independent parameters

• Four primary variables– mass, length, time and temperature

• 9 - 4 = 5 independent non-dimensional parameters– According to pi teorem.

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Non-dimensional numbers• Several ways to form non-dimensional

numbers exist. The following is the most frequently used formulation:

),,,,(

),,,,(

2

012

01

012

2

012

01

011

01

02

designnD

RT

nD

DP

RTmf

designnD

RT

nD

DP

RTmf

P

P

c

Chalmers University of Technology

Non-dimensional numbersFor a given design and working fluid we obtain:

),,(

),,(

number Re

2

012

01

012

number Re

2

012

01

011

01

02

nD

RT

nD

DP

RTmf

nD

RT

nD

DP

RTmf

P

P

c

Compressors normally operate at such high Reynolds numbers that they become independent of Re!!!

Chalmers University of Technology

Non-dimensional numbersWe arrive at the following expressions:

),(),(

),(),(

0101

012

012

01

012

0101

011

012

01

011

01

02

T

n

P

Tmf

RT

nD

DP

RTmf

T

n

P

Tmf

RT

nD

DP

RTmf

P

P

c

Compressors normally operate at such high Reynolds numbers that they become independent of Re!!!

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Compressor maps• Data is usually collected

in diagrams called compressor maps– What is meant by surge– What happens at

right-hand extremities of rotational speed lines

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SurgeWhat will happen in point D if mass flow drops infinitesimally– Delivery pressure drops– If pressure of air downstream of

compressor does not drop quickly enough flow may reverse its direction

– Thus, onset of surge depends on characteristics of compressor and components downstream

Surge can lead to mechanical failure

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Choke• What happens for increasing

mass flow?– Increasing mass flow

– Decreasing density

– Eventually M = 1 in some section in impeller (frequently throat of diffuser

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Overall turbine performance

• Typical turbine map– Designed to choke in stator

– Mass flow capacity becomes independent of rotational speed in choking condition

– Variation in mass flow capacity below choking pressure ratio decreases with number of stages

– Relatively large tolerance to incidence angle variation on profile and secondary losses give rise to limited variation in efficiency with rotational speed

Chalmers University of Technology

Learning goals• Have a basic understanding of how cooling is introduced

in gas turbines• Be familiar with the underlying theory and know what

assumptions the radial equilibrium design principle is based on

• Have some knowledge about – the use and development of radial compressor

– the physics governing the diffuser and

vaneless space • Understand what are the basis for compressor and

turbine maps.– Know about limitations inherent to the maps