Chalmers University of Technology Discussion on design task 1 Elementary axial turbine theory –Velocity triangles –Degree of reaction –Blade loading coefficient,

Chalmers University of Technology

• Discussion on design task 1

• Elementary axial turbine theory– Velocity triangles

– Degree of reaction

– Blade loading coefficient, flow coefficient

• Problem 7.1

• Some turbine design aspects– Choice of blade profile, pitch and chord

Lecture 7 – Axial flow turbines


Axial flow turbines

• Expansion occurs in stator and in relative frame of rotor

• Working fluid is accelerated by the stator and decelerated by the rotor

• Boundary layer growth and separation does not limit stage loading as in axial compressor


Elementary theory• Energy equation for control

volumes (again):

0103

0103

21

1

23

3

00103

22TTchh

Vh

Vhwq p

gasPerfect

hh

• Adiabatic expansion process (work extracted from system - sign convention for added work = +w)– Rotor => -w = cp(T03-T02) <=>

w = cp(T02-T03)– Stator => 0 = cp(T02-T01)

=> T02= T01


How is the temperature drop related to the blade angles ?

• We study change of angular momentum at mid of blade (as approximation)


Governing equations and assumptions• Relative and absolute refererence

frames are related by:

23

velocityrelativefor direction

of change Assume

23

22332233

2233

radiusconstant at Flow

wwww

wwww

ww

CCUCCU

UCUCrCrCworklTheoretica

torquelTheoreticarCrC

momentumangularofchangeofRate

UCV • We only study designs where:

– Ca2=Ca3

– C1=C3

• You should know how to extend the equations!!!• We repeat the derivation of theoretical work used

for radial and axial compressors:


Principle of angular momentum

Stage work output w:

3322

32

tantan aa

ww

CCU

CCUw

Ca constant:

32

3322

tantan

tantan

a

aa

UC

CCUw



Combine derived equations =>

32 tantan aUCw

stagep Tcw ,0

Exercise: derive the correct expression when 3 is small enough to allow 3 to be pointing in the direction of rotation.

(7.3) tantan 32,0 astagep UCTc

Energy equation

Energy equation:

We have a relation between temperature drop and blade angles!!! :


Dimensionless parametersBlade loading coefficient, temperature drop coefficient:

(7.6) tantan2

7.3Equation

21 32

2

,0

U

C

U

Tcastagep

Degree of reaction:31

32

TT

TT

Exercise: show that this expression is equal to =>when C3= C1 0301

32

TT

TT


can be related to the blade angles!

C3 = C1 =>

32,0 tantan UCTcTc stagepstagep

Relative to the rotor the flow does no work (in the relative frame the blade is fixed). Thus T0,relative is constant =>

22

3222

22

3 tantan2

1

2

1 arotorp CVVTc

Exercise: Verify this by using the definition of the relative total temperature: p

relative c

VTT

2

2

,0


can be related to the blade angles!

Plugging in results in definition of =>

(7.7) tantan2 23

31

32

U

C

TT

TT a

The parameter quantifies relative amount of ”expansion” in rotor. Thus, equation 7.7 relates blade angles to the relative amount of expansion. Aircraft turbine designs are typically 50% degree of reaction designs.


Dimensionless parameters Finally, the flow coefficient:

5.0

0.50.3

0.18.0

Current aircraft practice (according to C.R.S):

U

Ca

Aircraft practice => relatively high values on flow and stage loading coefficients limit efficiencies


Dimensionless parameters Using the flow coefficient in 7.6 and 7.7 we obtain:

(7.8) tantan2 32

The above equations and 7.1 can be used to obtain the gas and blade angles as a function of the dimensionless parameters

22

1

2

1 tan 2

22

1

2

1 tan 3

1

tan tan 22

1

tan tan 33

(7.9) tantan2

23


• Exercise: show that the velocity triangles become symmetric for = 0.5. Hint combine 7.1 and 7.9

• Exercise: use the “current aircraft practice” rules to derive bounds for what would be considered conventional aircraft turbine designs. What will be the range for 3? Assume = 0.5.

Two simple homework exercises


Turbine loss coefficients:Nozzle (stator) loss coefficients:

202

0201

22

22

2

pp

ppY

cC

TT

N

p

N

3,03

,03,02

22

33

2

pp

ppY

cV

TT

rel

relrelR

p

R

Nozzle (rotor) loss coefficients:


Problem 7.1


3D design - vortex theory

• U varies with radius

• Cw velocity component at stator exit => static pressure increases with radius => higher C2 velocity at root

• Twist blades to take changing gas angles into account– Vortex blading

3D optimized blading (design beyond free vortex design)


3D design in steam turbines• Keep blade angles from

root to tip (unless rt/rr high)

• Cut cost• Rankine cycle relatively

insensitive to component losses


Choice of blade profile, pitch and chord• We want to find a blade that will minimize loss and perform the required

deflection

• Losses are frequently separated in terms:

s losssecondary termone into Grouped

loss flowSecondaryAnnulus

Lossclearance Tip

cascadein Measured

ProfileTotal


Choice of blade profile, pitch and chord• As for compressors - profile families are used for thickness distributions.

For instance:– T6, C7 (British types)


Choice of blade profile, pitch and chord• Velocity triangles determine gas angles not blade angles.

– arccos(o/s) should approximate outflow air angle:

• Cascade testing shows a rather large range of incidence angles for which both secondary and profile losses are relatively insensitive


Choice of blade profile, pitch and chord• Selection of pitch chord:

– Blade loss must be minimized (the greater the required deflection the smaller is the optimum s/c - with respect to λProfile loss)

– Aspect ratio h/c. Not critical. Too low value => secondary flow and tip clearence effects in large proportion. Too high => vibration problems likely. 3-4 typical. h/c < 1 too low.

– Effect on root fixing• Pitch must not be too small to allow safe fixing to turbine disc rim

Documents

Chalmers University of Technology Discussion on design task 1 Elementary axial turbine theory –Velocity triangles –Degree of reaction –Blade loading coefficient,