Blade Nomenclature

Blade Nomenclature

Axial and Radial Flow TurbinesDifferences between turbine and compressor:

Long Short

Blade 1 Last blade

Compressor Turbine

►Work as diffuser►Work as nozzle

►Direction of rotation is opposite

to lift direction

►Direction of rotation is same as

Life

►Number of stages are many►Number of stages is small <3

►Temperatures are relative low

►Temperature is high, sometimes

blade cooling is required

Axial and Radial Flow TurbinesDifferences between Radial and Axial Types.

Radial(Centrifugal)

Axial

►Used for small engines►Used for large engines

►Small mass flow rates►Large mass flow rates

►Lower efficiencies►Better efficiencies

►Cheap►Expensive

►Easy to manufacture►Difficult to manufacture

Axial Flow Turbines

Most of the gas turbines employ the axial flow turbines.

The chapter is concerned with axial flow turbines.

The radial turbine can handle low mass flows more efficiently than the axial flow machines.

Axial Flow TurbineElementary Theory of Axial Flow Turbine

► Velocity Triangles.

■ The velocity triangles for one axial flow turbine stage and the nomenclature employed are shown. The gas enters the row of nozzle blades with a static pressure and temperature P1, T1, and a velocity C1, is expanded to P2, T2, with an increased velocity C2 at an angle α2.

■ The rotor blade angle will be chosen to suit the direction β2 of the gas velocity V2 relative to the blade at inlet.

■ V2 and β2 are obtained from the velocity diagram of known C2, α2, and U.

Axial Flow Turbine• Elementary Theory The gas leaves the rotor at β3, T3, with relative velocity

V3 at an angle β3. C3 and α3 can be obtained from the velocity diagram.

Axial Flow Turbine► Single Stage Turbine ■ C1 is axial → α1 = 0, and C1 = Cα1. For similar

stages (same black shapes) C1 = C3, and α1 = α3, called repeating stage.

■ Due to change of U with radius, velocity triangles vary from root to tip of the blade.

Axial Flow Turbine► Assumptions

■ Consider conditions at the mean diameter of the annulus will represent the average picture of what happen to total mass flow.

■ This is valid for low ratio of tip radius to root radius.

■ For high radii ratio, 3-D effects have to be considered.

■ The change of tangential (whirl) mass is . This amount produces useful torque.

■ The change in axial component produces the axial thrust on the rotor.

■ Also there is an axial thrust due to P2 – P3. ■ These forces (net thrust on turbine rotor) are

normally balanced by the thrust on the compressor rotor.

Axial Flow Turbine

Axial Flow Turbine► Calculation of Work

Assume Ca= constant

2 3Ca Ca

Ca

2 2

3 3

tan tan

tan tan

U

Ca

2 2 3tan tan tan tane

2 2tan tanU Ca Ca

) 1(

Axial Flow TurbineApplying principle of angular momentum

2 3

2 3

(

( )(tan tan )sW U C C

U Ca

From Equation (1)

2 3(tan tan )sW UCa

Steady-state energy equation:sp oW C T

Thus:

2 3(tan tan ) /

1.148, 1.333 and 41

so p

p

T U Ca C

C

Axial Flow TurbineElementary theory of axial flow turbine

1 3,

3,

1

1

1

1 3

1

1

11

/

s isent

isent

isent

o s o

s o o

o

s oo

s o

o o

T T

T T

TT

T

TP P

Axial Flow Turbineηs is the isentropic stage efficiency based on

stagnation (total) temperature.

1 3

1 3

o os

o o

T T

T T

1 3

1 3

( ) o os

o

T Ttotal to static

T T

(used for land-based gas turbines).

Definingψ = blade loading coefficient (temperature drop

coefficient)

2

2sp oC T

U

Axial Flow TurbineThus,

2 32 (tan tan ) /aC U Degree of reaction: 0 ≤ Λ ≤ 1

2,3 2,3 2 3

1,3 1 31,3

rotor

total

h T T T

T T Th

For, Ca = const. and C3 = C1

1 31 3

2 3

( )

(tan tan )

p p o o

a

C T T C T T

U C

and relative to rotor blades no work, thus

(a)

Axial Flow Turbine

2 22 3 3 2

2 2 23 2

2 2 23 2

1( )

21

sec sec21

tan tan2

p

a

a

C T T V V

C

C

2 213 222 3

1 3 2 3

tan tan

(tan tana

a

CT T

T T U C

13 2(tan tan )

2

C

U

Substitute in (a):

Axial Flow Turbine

3 2

3 2

2 22 2 2 23 2

2 2

2 2 2 23 2tan tan

a w a w

w w

a a

V V C C u C C u

C u C u

C C

2 3 3 2and

3 2 3 2andV V

3 2 3 2,C C aC

U

Λ = 0.5 → Symm. velocity triangles

● Λ = 0 : Impulse turbine

● Λ = 1 :

Defining flow coefficient:

Axial Flow Turbine

2 1

3 2

2 (tan tan )

(tan tan )2

Adding:3

1 1tan 2

2 2

2

1 1tan 2

2 2

From:2 2

3 3

3 3

2 2

(tan tan )

(tan tan )

1tan tan

1tan tan

a

a

U C

U C

Axial Flow TurbineIf , Λ, and are assumed, blade angles can be determined. ● For aircraft applications:

3 < ψ < s, 0.8 < < 1 ● For industrial applications:

is less (more stages) is less (larger engine size)α3 < 20 (to min. losses in nozzle)

● Loss coefficient:

1 2

1

2 22( )2

2

/ 2n nozzle statorp

o oN

o

T T

C C

P PY

P P

Λ and Y: The proportion of the leaving energy which is degraded by friction.

Axial Flow TurbineExample (Mean diameter design)Given:

1

1 3

1 3

1

Single-stage turbine

= 20 kg/s

= 0.9

= 1100 K

Temperature drop, = 145 K

Pressure ratio, / = 1.873

Inlet pressure, = 4 bar

t

o

o o

o o

o

m

T

T T

P P

P

Assumptions:Rotational speed fixed by compressor: N = 250 rpsMean blade speed: 340 m/sNozzle loss coefficient:

2 222 / 2N

p

T T

C C

Axial Flow Turbine

/t rr r

2 3 1 3

1

,

0

a aC C C C

Calculation:a)Λ degree of reaction at mean radiusb)Plot velocity diagramsc)Blade height h, tip/root radius,

Assume:

3

2 2

2 2 1.148 145 102.88

340sp oC T

U

flow coefficient 0.8aC

U

The temperature drop coefficient:

Assume (try):

Axial Flow Turbine

3 3

1tan tan

3tan 1.25

3

1 1tan 2

2 2

0.28

■ To get Λ use

This is low as a mean radius value because Λ will be low or negative at the root.

This introduce a value for α3.

Take α3 = 10°

* To calculate degree of reaction Λ:

■ Get β3:α3 = 0

Axial Flow Turbine

3 3 3

3

1tan tan tan 1.426

1 1tan 2

2 2

0.421 (Acceptable)

Reaction at root should be checked.Thus α3 = 10°, β3 = tan-1 1.426 =

54.96

2

2

1 1tan 2 0.374

2 2

0.421

2.88

0.8

20.49

Axial Flow Turbine2 2

2

1tan tan 1.624

58.38

3 3 2 2, , , , U

/t rr r

11 :axial aC C

With knowledge of

plot velocity diagrams.

* Determine blade height h and tip/root radius ratio, .

Assumption:

Calculation of area at Section 2 (exit of nozzle)

Axial Flow Turbine

2

2

2 1

2

2 2 2

22

2 2

340 0.8 272 m/s

cos 519 m/s

1100 K

5.9 K2

a

a

o o

op

C U

C C C

T T

CT T T

C

22

2 2

2

0.05 117.3 5.9 K2

976.8 K

Np

CT T

C

T

1 1 1

2

/ 1

4

22

2.49 baroPo o

o

P TP

P T

Axial Flow TurbineFor the nozzle:

1

1

21 2

1 1

1

1/(2 ) 1 11

2 2

1 42.16

2 1.853

o p

oc

c

MT T C CM

T T

PP

P

P2 > Pc, the nozzle is not choked. 2, 2.49throatThus P P 32

2 22

22 2 2 2

2

22 2

22 2 2 2 2 2 2

0.833 /

, , m , 0.0833

throat area of nozzles; A

, m 0.0437 , also A cos

aa

Pkg m

RT

mA or C A A m

C

mN

C

or C A N A N m A N

Axial Flow TurbineCalculate areas at section (1) inlet nozzle and (3) exit rotor.

3

1 1

1

1 1

1

1 1 3 33

21

1 1

11 1

1

311 1

1

21 1 1

, but C C , 276.4 /cos

10672

3.54

1.155 /

0.626

aa a

op

o o

a

CC C C and C m s

CT T T K

c

P TP bar

P T

Pkg m

RT

m C A A m

Axial Flow Turbine

3 1 5

3

3 3

o

23

3 3

13 3

3

333 5

5

Similarly at outlet of stage ( rotor)

T 1100 145 955 ,

9222

1.856

0.702 /

o o

op

o o

T T K given

CT T T K

c

P TP bar

P T

Pkg m

RT

3

23 5 5

23 3 3

3/ 0.702 /

0.1047

Blade height and annulus radius ratio

a

P RT kg m

m C A A m

Axial Flow TurbineMean radius

m

3402 0.216

2 (250)

for known (A); A 2 r

m m mu Nr r m

also h

t r , 2 2 2m r m

m

A h hh then r r r

r

using areas at stations 1,2,3 thus

21mAmh1

/t rr r

Location123

0.06260.08330.1047

0.040.06120.077

1.241.331.43

Axial Flow TurbineBlade with width WNormally taken as W=h/3Spacing s between axial blades

t

r

a t

space0.25, should not be less than 0.2 W

width

r* should be 1.2 1.4

r

unsatisfactory values such as 0.43 can be reduced by

changing axial velocity through .

increasing C reduce r check has to

s

w

will

v be made for mach number M .

Axial Flow TurbineVortex TheoryThe blade speed ( u=r) changes from root to tip, thus velocity triangles must vary from root to tip.

Free Vortex designaxial velocity is constant over the annulus.Whirl velocity is inversely proportional to annulus.

,C ,tan

tanC ,tan

33

22

constrtconsC

tconsrtconsC

a

a

Along the radius.

2 3 2 3

( ) tansW u C C C r C r cons t

Axial Flow TurbineFor variable density, m is given by

t

r

r

r

a

a

rdrCm

Crrm

2

2

2

2

2

)2(

2 2

2

2

a 2

2 22

3 33

tan tan

C cosntant, thus changes as

tan tan (a)

tan tan (b)

a

mm

mm

C r cons t r C

but is

r

r

similarly

r

r

Axial Flow Turbine

2 2

2

2

s 3

3

2 2 2 2

m2

2

a 3 3

3 33 3

tan tan , , tan tan

rtan (c)

r

for exit of rotor u C tan tan

tan tan (d)

a aa

m

m a

a

mm

m a

uu C C thus

C

urm

r C

C

r r uthus

r r C

Ex: Free vortexResults from mean diameter calculations

2 2m 3

3 2

3

58.38, 20.49, 10 ,

54.96, 0.0612, 0.216,

0.077,2

om m

m m

r m

h r

hh r r

Axial Flow Turbine

Tip54.9308.5258.33

Root62.1539.3212.1251.13

mean58.3820.491054.96

3 3

2 3

2

32 3

m

a

1.164, ( ) 0.877, 1.217, 0.849

u 1also 1.25, Results are

C

m m m m

t t r t

m

a

r r r r

r r r r

u

C

2 2

Axial Flow Turbine

'3o1o

3o1os

)1/(

1o

3o1os

1o

'3o

1os3o1oos

12a12a

32a32a3o1oposp

3322a

TT

TT where

))p

p(1(T)

T

T1(TTTT

)tan(tanUCm)tan(tanUCm

)tan(tanUCm)tan(tanUCm)TT(cmTcmW

tantantantanC

U

EES Design Calculations of Axial Flow Turbine

Known Information

To1 = 1100 [K]

P ratio = 1.873

DelTs = 145

Etta turbine = 0.9

Assumptions

U = 340 [m/s]

N rps = 250

= 0.8

3 = 10

Loss nozzle = 0.05

EES Design Calculations of Axial Flow Turbinecp = 1148 R = 0.287 = 1.333

DelTs = To1 – To3

P ratio = Po1

Po3

Ca = C2 · cos ( 2 )

= Ca

U

Gamr =

– 1

Epsi = 2 · cp · DelTs

U2

Epsi = 2 · · ( tan ( 2 ) + tan ( 3 ) )

Reaction =

2 · ( tan ( 3 ) – tan ( 2 ) )

U = Ca · ( tan ( 2 ) – tan ( 2 ) )

U = Ca · ( tan ( 3 ) – tan ( 3 ) )

EES Design Calculations of Axial Flow TurbineCalculate A2

Loss nozzle = T2 – T2dash

C22

2 · cp

To2 = To1

To2 – T2 = C2

2

2 · cp

Po1

P2 =

To1

T2dash

Gamr

Po1

Pc =

+ 1

2

Gamr

Pth = P2

Rho2 = Pth

R · T2

A2 = m

Rho2 · Ca

A2 · cos ( 2 ) = A2N

EES Design Calculations of Axial Flow Turbine

Calculate A1

To1 – T1 = C1

2

2 · cp

Po1

P1 =

To1

T1

Gamr

Rho1 = P1

R · T1

C1 = Ca

A1 = m

Rho1 · Ca

Calculate A3

To3 – T3 = C3

2

2 · cp

Po3

P3 =

To3

T3

Gamr

Rho3 = P3

R · T3

C3 = Ca

A3 = m

Rho3 · Ca

EES Design Calculations of Axial Flow Turbine

Blade height

U = 2 · · N rps · rm

Blade height at section 1

A1 = 2 · · rm · h1

r t1 = rm + h1

2

r r1 = rm – h1

2

rratio1 = r t1

r r1

Blade height at section 2

A2 = 2 · · rm · h2

r t2 = rm + h2

2

r r2 = rm – h2

2

rratio2 = r t2

r r2

Blade height at section 3

A3 = 2 · · rm · h3

r t3 = rm + h3

2

r r3 = rm – h3

2

rratio3 = r t3

r r3

EES Design Calculations of Axial Flow Turbine

A1 = 0.06345 A2 = 0.08336 A2N = 0.04372 A3 = 0.1046 2 = 58.37

3 = 10

2 = 20.49 3 = 54.97 C1 = 272 C2 = 518.7

C3 = 272 Ca = 272 cp = 1148 [J/kgK] DelTs = 145 Epsi = 2.88

Ettaturbine = 0.9 = 1.333 Gamr = 4.003 h1 = 0.04666 h2 = 0.06129

h3 = 0.07692 Lossnozzle = 0.05 m = 20 [kg/s] Nrps = 250 [rev per sec] P1 = 355.1

P2 = 248.8 P3 = 186.1 Pc = 215.9 = 0.8 Po1 = 400 [kPa]

Po3 = 213.6 Pth = 248.8 Pratio = 1.873 R = 0.287 [kJ/kgK] Reaction = 0.4211

Rho1 = 1.159 Rho2 = 0.8821 Rho3 = 0.7029 rratio1 = 1.242 rratio2 = 1.33

rratio3 = 1.432 rm = 0.2165 rr1 = 0.1931 rr2 = 0.1858 rr3 = 0.178

rt1 = 0.2398 rt2 = 0.2471 rt3 = 0.2549 T1 = 1068 T2 = 982.8

T2dash = 977 T3 = 922.8 To1 = 1100 [K] To2 = 1100 [K] To3 = 955

U = 340 [m/s]

Axial Flow Turbine

Axial Flow Turbine