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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

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Lecture 7 – Axial flow turbines

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  1. 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

  2. Expansion occurs in stator and in relative frame of rotor Axial 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

  3. 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

  4. How is the temperature drop related to the blade angles ? • We study change of angular momentum at mid of blade (as approximation)

  5. 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:

  6. Principle of angular momentum Stage work output w: Ca constant:

  7. 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.

  8. Dimensionless parameters Blade loading coefficient, temperature drop coefficient: Degree of reaction: Exercise: show that this expression is equal to => when C3= C1

  9.  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:

  10.  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.

  11. 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

  12. 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

  13. 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.

  14. Turbine loss coefficients: Nozzle (stator) loss coefficients: Nozzle (rotor) loss coefficients:

  15. Problem 7.1

  16. 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)

  17. 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

  18. 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:

  19. Choice of blade profile, pitch and chord • As for compressors - profile families are used for thickness distributions. For instance: • T6, C7 (British types)

  20. 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

  21. 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

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