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


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


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


The cooled turbineTypical cooling

distribution for stage:

Distribution required for operation at 1500 K


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


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


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

)(

)()(


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


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:


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


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


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


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


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


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


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


Axial centrifugal combination - T700


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):


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


Shrouds• Removes losses in

clearance.

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

rotational speed where high stresses are produced


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.


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.


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


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!!!


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!!!


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


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


Choke• What happens for increasing

mass flow?– Increasing mass flow

– Decreasing density

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


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


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

lecture 8 – axial turbines 2 + radial compressors 2

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