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Lecture 7 – Axial flow turbines 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

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lecture 7 axial flow turbines
Lecture 7 – Axial flow turbines
  • 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
axial flow turbines
Expansion occurs in stator and in relative frame of rotorAxial flow turbines
  • 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
Elementary theory
  • Energy equation for control volumes (again):
  • 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
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
Governing equations and assumptions
  • Relative and absolute refererence frames are related by:
  • 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
Principle of angular momentum

Stage work output w:

Ca constant:

slide8
Combine derived equations =>

Energy equation

Energy equation:

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

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

dimensionless parameters
Dimensionless parameters

Blade loading coefficient, temperature drop coefficient:

Degree of reaction:

Exercise: show that this expression is equal to =>

when C3= C1

can be related to the blade angles
 can be related to the blade angles!

C3 = C1 =>

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

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

can be related to the blade angles11
 can be related to the blade angles!

Plugging in results in definition of  =>

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 parameters12
Dimensionless parameters

Finally, the flow coefficient:

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

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

dimensionless parameters13
Dimensionless parameters

Using the flow coefficient in 7.6 and 7.7 we obtain:

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

two simple homework exercises
Two simple homework exercises
  • 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.
turbine loss coefficients
Turbine loss coefficients:

Nozzle (stator) loss coefficients:

Nozzle (rotor) loss coefficients:

3d design vortex theory
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
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
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:
choice of blade profile pitch and chord20
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 chord21
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 chord22
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