axial flow turbines

1Lect- 23

Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay 2

Lect-23

In this lecture...

Tutorial on axial flow turbines


Lect-23

Problem # 1 A single stage gas turbine operates at its

design condition with an axial absolute flow at entry and exit from the stage. The absolute flow angle at the nozzle exit is 70 deg. At stage entry, the total pressure and temperature are 311 kPa and 850oCrespectively. The exhaust static pressure is 100 kPa, the total to static efficiency is 0.87 and mean blade speed is 500 m/s. Assuming constant axial velocity through the stage, determine (a) the specific work done (b) the Mach number leaving the nozzle (c) the axial velocity (d) total to total efficiency (e) stage reaction.


Lect-23

Problem # 1

U

C1

V3

V2C2

Rotor

Stator/Nozzle

1

2

33

2

1

3

2

C3


Lect-23

Solution: Problem # 1

[ ][ ]

[ ]

s/m/U/wCUCw,C,trianglevelocitytheFrom

RT/CM

isnumberMachthe,exitnozzletheAt)b(

kg/kJ)./(.

)P/P(Tcw,isworkSpecific

)P/P(Tcw

,efficiencystatictototalthatknowWe

tw

wtw

.

/)(ptst

/)(p

tts

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0

2761131111231148870

1

1

32

23

222

2480

101301

101301

===

==

=

==

=

=


Lect-23


930

07751102762

2008701

21

200

970973287331588

973

588

3

223

22

2

222

1012

222

.

..w

C1: as efficiency static-to-total

the to related is efficiency total-to-total The)d(

s/mcosCC,velocityaxialThe)c(

../M,Hence

Kc/CTTthatknowWes/msin/CC

tt

ttstt

a

p

w

=

=

==

==

==

==

==


Lect-23


s

T01

1

203

2s

3s

303s

P01P1

P2

P03P3

pcC2

23

Expansion process in a turbine stage


Lect-23


45101000274502001

1

1

221

223

2321

./.

tan)U/C(RC/UtantanandC/Utan

,trianglevelocitytheFrom)tan)(tanU/C(R,reactionofDegree)e(

ax

aa

ax

=

=

=

==

=


Lect-23

Problem # 2Combustion gases enter the first stage of a gas turbine at a stagnation temperature and pressure of 1200 K and 4.0 bar. The rotor blade tip diameter is 0.75m, the blade height is 0.12 m and the shaft speed is 10,500 rpm. At the mean radius the stage operates with a reaction of 50%, a flow coefficient of 0.7 and a stage loading coefficient of 2.5. Determine (a) the relative and absolute flow angles for the stage; (b) the velocity at nozzle exit; (c) the static temperature and pressure at nozzle exit assuming a nozzle efficiency of 0.96 and the mass flow.


Lect-23


RotorStator/Nozzle

1 2 3

U

V2

V3

V3

C2

3

C3 C2

2

V2

2

U


Lect-23


oo

ooxa

aa

ax

aww0

.and.,stagereaction%aFor.and.

Rand)U/C(,ofvaluesngSubstituti)U/C/()R/(tanand)U/C/()R/(tan

,equationsabovethegSimplifyin/)tan)(tanU/C(R,Also

)tan)(tanU/C(U/)VV(U/hby given is loading stage The)a(

984626850

9846268

22

2

2332

23

23

23

23232

====

==

=

=+=

=

+=+==


Lect-23


K.)/(.c/CTT exit, nozzle the at etemperatur staticThe)c(

s/m..cos/.cos/CC,exitnozzletheatvelocity,Therefore

ands/m.UC,velocityaxialThes/m..)/(U,speedbladethe

m./)..(r radius, mean the At)b(

p2

a

ma

m

m

31016116028665212002

8665226845242

452423634631503010500

31502120750

22202

22

===

===

==

==

==


Lect-23


s/kg...)./10(1.986mCA)RT/P(CAm is rate flow mass The

bar..P

.T/T)P/P(

)P/P(T/T

hhhh,efficiencynozzleThe

5aa2

.n

/)(

/)(s

n

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98618405204

8405201

1

11

2222

030342

0121012

1012

012

201

201

==

==

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=

=

=

=


Lect-23

Problem # 3A single stage axial flow turbine operates with an inlet temperature of 1100 K and total pressure of 3.4 bar. The total temperature drop across the stage is 144 K and the isentropic efficiency of the turbine is 0.9. The mean blade speed is 298 m/s and the mass flow rate is 18.75 kg/s. The turbine operates with a rotational speed of 12000 rpm. If the convergent nozzle is operating under choked condition determine (a) blade-loading coefficient (b) pressure ratio of the stage and (c) flow angles.


Lect-23

Problem # 3

U

C1

V3

V2C2

Rotor

Stator/Nozzle

1

2

33

2

1

3

2

U

C3


Lect-23

Problem # 3

[ ]

bar.Pand.PP

is turbine the of ratio pressure The

.TT

PPor

)P/P(TT

TTTT,turbineaofefficiencyisentropicThe

KTTTKTT)b(

.U

Tcas defined is loading blade The)a(

)/(

t

/)(

st

p

81318751

53301

1

95614411001100

86151298

1441148

0303

01

1

01

0

01

03

1010301

0

0301

0301

00103

0102

220

==

=

=

=

=

===

==

=

==


Lect-23

Problem # 3

oa

a

2

2

2

and./C/Ccos,trianglesvelocitytheFrom

.s/m.UC velocity axial The.s/m.C therefore,

is nozzle choked the leaving gases the of velocity absolute The.K. Tis exit nozzle the at etemperatur static The

.TTand

RTC Therefore,

unity. isnumber Mach exit the choked, is nozzle the Since)c(

6247160600283

2839502983600

2944

16512

1

222

2

02

2

====

===

=

=

=+

=

=


Lect-23

Problem # 3

oa

oa

op

aopt

o

a

.,thereforeand

.tantanortantanCU

.or

..tanUC

Tctanor

)tan(tanUCTcw,workspecificturbineThe

.or.tantan

tantanCU

5448

13211544

07930880712832981441148

63982801

1

3

3333

3

23

32

222

22

=

=+==

=

=

==

+==

===

==


Lect-23

Problem # 4A multi-stage axial turbine is to be designed with impulse stages and is to operate with an inlet pressure and temperature of 6 bar and 900 K and outlet pressure of 1 bar. The isentropic efficiency of the turbine is 85 %. All the stages are to have a nozzle outlet angle of 75o and equal inlet and outlet rotor blade angles. Mean blade speed is 250 m/s and the axial velocity is 150 m/s and is a constant across the turbine. Estimate the number for stages required for this turbine.


Lect-23

Problem # 4

U

V2

V3

V3

C2

RotorStator/Nozzle

1 2 3

3

22

3

C3 C2

2

V2

2

U

3

Ca

Cw2

Cw3

Vw3

Vw2


Lect-23


K.TThhorhhhhR

R reaction, of degree the turbine, impulse an is this Since

K./.c/CTTnozzle, the in etemperatur stagnation in change no is there Since,

c/CTTc/CTTs/m.cos/cos/CC,trianglesvelocitytheFrom

K.).(.)TT(T,HenceK.T

PP

TT

known, is ratio pressure overall the Since

x

x

p

pp

a

es01toverall

es

././)(

e0es

01

7753

0

77531148255799002

22

55797515062749576900850

9576

6

32320301

32

222012

22022

22202

22

00

0

3313301

0

01

===

=

=

===

=+=

===

===

=

=

=


Lect-23


.stages../.)TT/(TisturbinetheforrequiredstagesofnumberThe

K..TTisstageperdropetemperaturThe

K./..c/CTT,Therefore

.velocityaxialttanconsforCVVthatseecanWes/m.cos/CV

../)sin.(C/)UsinC(tan

entry,rotor at triangles velocity the From

overall

p

2a

o2

a2

38927946274

79428805900

28805

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153441664

0652150250755579

03010

0301

223203

332

2

22

==

==

=

+=+=

==

==

=

===


Lect-23

Exercise Problem # 1An axial flow turbine operating with an overall stagnation pressure of 8 to 1 has a polytropic efficiency of 0.85. Determine the total-to-total efficiency of the turbine. If the exhaust Mach number of the turbine is 0.3, determine the total-to-static efficiency. If, in addition, the exhaust velocity of the turbine is 160 m/s, determine the inlet totaltemperature.

Ans: 88%, 86.17%, 1170.6 K


Lect-23

Exercise Problem # 2The mean blade radii of the rotor of a mixed flow turbine are 0.3 m at inlet and 0.1 m at outlet. The rotor rotates at 20,000 rev/min and the turbine is required to produce 430kW. The flow velocity at nozzle exit is 700 m/s and the flow direction is at 70 to the meridional plane. Determine the absolute and relative flow angles and the absolute exit velocity if the gas flow is 1 kg/s and the velocity of the through-flow is constant through the rotor.

Ans: 2=70 deg, 2=7.02 deg, 3=18.4 deg, 3=50.37 deg


Lect-23

Exercise Problem # 3An axial flow gas turbine stage develops 3.36MW at a mass flow rate of 27.2 kg/s. At the stage entry the stagnation pressure and temperature are 772 kPa and 727C, respectively. The static pressure at exit from the nozzle is 482 kPa and the corresponding absolute flow direction is 72 to the axial direction. Assuming the axial velocity is constant across the stage and the gas enters and leaves the stage without any absolute swirl velocity, determine (a) the nozzle exit velocity; (b) the blade speed; (c) the total-to-static efficiency; (d) the stage reaction.

Ans: 488m/s, 266.1 m/s, 0.83, 0.128


Lect-23

Exercise Problem # 4A single stage axial turbine has a mean radius of 30 cm and a blade height at the stator inlet of 6 cm. The gases enter the turbine stage at 1900 kPa and 1200 K and the absolute velocity leaving the stator is 600 m/s and inclined at an angle of 65 deg to the axial direction. The relative angles at the inlet and outlet of the rotor are 25 deg and 60 deg respectively. If the stage efficiency is 0.88, calculate (a) the rotor rotational speed, (b) stage pressure ratio (c) flow coefficient (d) degree of reaction and (e) the power delivered by the turbine.

Ans: 13550 rpm, 2.346, 0.6, 0.41, 34.6 MW

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axial flow turbines

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