Lecture 7 – Axial flow turbines

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# Lecture 7 – Axial flow turbines - PowerPoint PPT Presentation

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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Presentation Transcript
Lecture 7 – Axial flow turbines
• Discussion on design task 1
• Elementary axial turbine theory
• Velocity triangles
• Degree of reaction
• Problem 7.1
• Some turbine design aspects
• Choice of blade profile, pitch and chord
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
• 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 ?
• We study change of angular momentum at mid of blade (as approximation)
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

Stage work output w:

Ca constant:

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

Degree of reaction:

Exercise: show that this expression is equal to =>

when C3= C1

 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 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 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 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
• 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:

Nozzle (stator) loss coefficients:

Nozzle (rotor) loss coefficients:

3D design - vortex theory
• 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

3D optimized blading (design beyond free vortex design)

• 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:
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