Theory of turbo machinery / Turbomaskinernas teori. Chapter 4

Transcription

1 Theory of turbo machinery / Turbomaskinernas teori Chapter 4

2 Note direction of α 2 FIG Turbine stage velocity diagrams.

3 Assumptions: Hub to tip ratio high (close to 1) Negligible radial velocities No changes in circumferential direction (wakes and nonuniform outlet velocity distribution neglected)

4 Continuity equation for uniform steady flow: ρ A c = ρ A c = ρ Ac 1 1 x1 2 2 x2 3 3 x3 Assuming constant axial velocity c = c = c = c x1 x2 x3 x ρ A = ρ A = ρ A

5 Work done on rotor by unit mass of fluid ( ) ΔW = W m= h01 h03 = U c2 y + c3 y Please note: No work done in nozzle row: h = h With ( x y) h h c c c = + 2= + 2 And using above equations: 2 2 ( x y) 2 ( y y ) h h = h h + c + c = U c + c

6 Rewriting this in terms of relative velocity c U = w y2 y2 c + U = w y3 y3 c + c = w + w y2 y3 y2 y3 Combining above equations: 2 2 ( y y ) h2 h3 + w 2 + w 3 2= 0 w y2 c y2 U with wx 2 = wx3 = cx and w + w = w x y 2 2 ( ) h2 h3 + w2 w3 2= 0 Relative stagnation enthalpy,, does not change across rotor h 0,rel

7 Nozzle row (1 to 2): Static pressure: Stagnation enthalpy: Stagnation pressure: (isentropic: p = p ) p h p p 1 2 = h > p Subscript s denotes isentropic change and ss denotes both rows isentropic FIG Mollier diagram for a turbine stage.

8 Rotor row (2 to 3): Static pressure: Stagnation enthapy: Stagnation pressure: h p > h > p p p 2 3 However: Relative Stagnation enthapy, h = h + w 2 = h 2 02, rel , rel FIG Mollier diagram for a turbine stage.

9 Turbine stage total to total efficiency: η tt Actual work output = = Ideal work output when operating to same back pressure h h h h ss For a normal stage, no changes in are made in velocities from inlet to outlet: c1 = c3 and α1 = α3. Further assuming c3ss = c3 the efficiency becomes: η tt h h h h = = h h h h ss 1 3ss

10 Defining enthalpy loss coefficients for the nozzle and rotor respectively: ζ N h h and ζ h = = h 2 2s 3 3s 2 R 2 c2 2 w3 2 Neglecting rotor temperature drop, the stage efficiencies may be expressed as: η η tt ts ζ Rw = ζ c N 2 ( h h ) ζ Rw + ζ c + c = 1 + 2( h1 h3) N 2 1 1

11 Soderberg s correlation: Large set of data compiled Design assuming Zweifel s criteria for optimum space axial chord ratio Y 2 Ψ T = = 2( sb) cos α2( tanα1+ tanα2) 0.8 Y id Result: Turbine blade losses are a function of Deflection Blade aspect ratio Blade thickness-chord ratio Reynolds number ε Hb tmax l

12 Deflection Blade aspect ratio: Blade thickness-chord ratio ε = α1+ α2 Hb= 3 tmax l = h h h 2 2 ( ) 5 Re = ρ cd μ D defined at exit throat D = 2sHcosα scosα + H = 10 l b t max H is height of blade (radial direction) s

13 For turbines: Deflection, ε = α + α, is large, but 1 2 Deviation, δ = α α, is small 2 2 ' α α 2 2 ' ε = α ' + α ' 1 2

14 FIG Soderberg s correlation of turbine blade loss coefficient with fluid deflection (adapted from Horlock (1960).

15 Corrections for Reynolds number 5 Re 10 ζ 5 14 * 10 * cor ζ = Re Blade aspect ratio Nozzles: Rotors: ( ζ )( bh) ( ζ )( bh) * * 1+ ζcor = * * 1+ ζcor = Tip clearance losses and disc friction not included

16 Design considerations Rotor angular velocity (stresses, grid phasing) Weight (aircraft) Outside diameter (aircraft) Efficiency (almost always)

17 Consider a case with given Blade speed Specific work Axial velocity U ( 2 3) ΔW = U cy + cy c x The only remaining parameter to define is since Triangles may be constructed Loss coefficients determined from Soderberg Efficiencies computed from loss coefficients ΔW U cy2 cy3 = cy2

18 Stage loading factor: ΔW 2 U cx flow coefficient: U Aspect ratio: H b FIG Variation of efficiency with c y2 /U for several values of stage loading factor ΔW/U 2 (adapted from Shapiro et al. 1957).

19 Stage reaction, R Alternative description to cy2 U Several definitions available Here: ( ) ( ) R= h h h h E.g: R = 0.5 ( ) ( ) 0.5 = h h h h h h = h h R = 0.5

20 For a normal stage, c = c ( )( ) R= h h h h Using eq. 4.4: ( ) h2 h3+ w2 w3 2= 0 and Euler R R = w w ( 2 + c 3) U c 2 y y ( )( ) 2U( cy + cy ) w w w + w w w = = U

21 Relative tangential velocity w = c tan β y x R w3 w2 c x ( tan β tan β ) = = 2U 2U 3 2 Or using cy2 = wy2 + U w w w + U w R 2U 2U 1 cx = U y2 = = = ( tan β tanα )

22 Zero reaction stage R c x ( ) = tan β3 tan β2 = 0 if β3 = β2 2U FIG Velocity diagram and Mollier diagram for a zero reaction turbine stage.

23 50% reaction stage c x 1 R = + ( tan β3 tanα2) = 0.5 if β3 = α2 2 2U FIG Velocity diagram and Mollier diagram for a 50% reaction turbine stage.

24 FIG Velocity diagram for 100% reaction turbine stage.

25 FIG. 4.4 ΔW Cy R = U U 2 FIG Influence of reaction on total-to-static efficiency with fixed values of stage loading factor.

26 FIG Mollier diagram for an impulse turbine stage.

27 Alternative representation for specified reaction: η = f ( Ψ, Φ) where ΔW Ψ = is the stage loading and 2 U cx Φ = is the flow coefficient U FIG Design point total-to-total efficiency and deflection angle contours for a turbine stage of 50 percent reaction.

28 FIG Design point total-to-total efficiency and rotor flow deflection angle for a zero reaction turbine stage.

29 Centrifugal stresses 2 dfc = Ω d dm= ρ Adr r m dσ c df c Ω ρ = ρa = 2 rdr With constant cross section this may be integrated 2 σ r 2 t Utip c rh = Ω rdr 1 ρ = rh 2 rt FIG Centrifugal forces acting on rotor blade element.

30 Tapering: Reduction of cross sectional area in radial direction, in order to reduce stresses Pure fluid dynamics would recomend the opposit FIG Effect of tapering on centrifugal stress at blade root (adapted from Emmert 1950).

31 FIG Maximum allowable stress for various alloys (1000 hr rupture life) (adapted from Freeman 1955).

32 FIG Properties of Inconel 713 Cast (adapted from Balje 1981).

33 Turbine blade cooling. Why is the efficiency of the gas turbine comparable to that of a Rankine cycle? (given that we do have to pay a considerable amount of energy to the compressor, whereas compression of water in the Rankine cycle is cheap)

34 FIG Turbine thermal efficiency vs inlet gas temperature (adapted from le Grivès 1986).