Theory of turbo machinery / Turbomaskinernas teori. Chapter 4

Theory of turbo machinery / Turbomaskinernas teori Chapter 4

Axial-Flow Turbines: Mean-Line Analyses and Design Power is more certainly retained by wary measures than by daring counsels. (Tacitius, Annals)

Axial-flow turbines FIG. 4.1. Large low pressure steam turbine (Siemens)

Axial-flow turbines FIG. 4.. Turbine module of a modern turbofan jet engine (RR)

Axial-flow Turbines: -D theory Note direction of α FIG. 4.3. Turbine stage velocity diagrams.

Axial-flow Turbines: -D theory Assumptions: Hub to tip ratio high (close to 1) Negligible radial velocities No changes in circumferential direction (wakes and nonuniform outlet velocity distribution neglected)

Axial-flow Turbines: -D theory Continuity equation for uniform steady flow: ρ Ac = ρ Ac = ρ Ac 1 1 x1 x 3 3 x3 Assuming constant axial velocity c = c = c = c x1 x x3 x ρ A = ρ A = ρ A 1 1 3 3

Axial-flow Turbines: Design parameters Flow coefficient φφ = cc mm UU or φφ = cc xx UU Stage loading coefficient ψψ = Δh 0 UU Or using Euler: Δh 0 = UUΔcc θθ ψψ = Δcc θθ UU (4.) change of tangential velocity in rotor!

Axial-flow Turbines: Stage reaction Stage reaction = Static enthalpy drop across rotor Static enthalpy drop across stage R = h h 1 h h 3 3 Second law, assuming isotropic process and ignoring density changes R p p 1 p p 3 3 Tds = dh dp dh dp ρ

Axial-flow Turbines: -D theory Work done on rotor by unit mass of fluid Please note: ( c θ c ) W = W m = h h = U + 01 03 θ 3 No work done in nozzle row: h = h 01 0 Angle defined in opposite direction at ( ) ΔW = W m= h01 h03 = U c y + c3 y c cx With h0 = h + = h + + And using above equations: c θ h 0 h 03 = h h 3 + c θ c θ 3 + c c ( c c ) x x3 = U θ + θ 3

Axial-flow Turbines: -D theory Rewriting this in terms of relative velocity c c c θ θ 3 θ U = w + U = w + c θ 3 θ θ 3 = w θ + w θ 3 Combining above equations: h w w c x c θ θ 3 x3 h3 + + = with wx = wx3 = cx and 0 w + w = w x y w θ c θ U ( ) h h3 + w w3 = 0 (4.6 approximately) Relative stagnation enthalpy, h 0,rel, does not change across rotor

Axial-flow Turbines: -D theory Nozzle row (1 to ): Static pressure: Stagnation enthalpy: Stagnation pressure: (isentropic: p = p ) 01 0 p 1 < p h p = h 01 0 > p 01 0 Subscript s denotes isentropic change and ss denotes both rows isentropic FIG. 4.4. Mollier diagram for a turbine stage.

Axial-flow Turbines: -D theory Rotor row ( to 3): Static pressure: Stagnation enthapy: Stagnation pressure: h p > h 0 03 > p 0 03 p p 3 However: Relative Stagnation enthapy, h = h + w = h 0, rel 0 03, rel FIG. 4.4. Mollier diagram for a turbine stage.

Axial-flow Turbines: Repeating Stage Multi-stage turbines: Stages often similar to each other c x = const. r = const. α 1 = α 3 However, channel must be widened FIG. 4.5. General arrangement of a 6- stage repeating turbine.

Axial-flow Turbines: Repeating Stage Reaction: R = h h 3 1 1 1 1 h h 3 = h h 1 h h 3 = h h 1 01 h h 03 Relative enthalpy drop across rotor = 1- relative enthalpy drop across stator Same velocities in and out h h = h => 1 3 01 h03 h 1 h = h 01 h 0 + c c 1 = 0 + c x tan α tan α 1 φ tan α tan α1 R = 1 (4.1) ψ Stage loading ΨU = h 01 h 03

Axial-flow Turbines: Repeating Stage with c Ψ = U θ = c x ( tanα tanα ) ( tanα α ) 1 = φ tan U 1 the reaction becomes R = tan φ α tanα 1 1 (4.13 a) eq 4.13 a + eq 4.1: ( 1 tan ) ψ +φ α = R (4.14) 1

Axial-flow Turbines: Stage losses and efficiency Turbine stage total to total efficiency: η tt Actual work output = = Ideal work output when operating to same back pressure h h h h 01 03 01 03ss For a repeating 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 01 03 1 3 01 03ss 1 3ss

Axial-flow Turbines: Stage losses and efficiency Defining enthalpy loss coefficients for the nozzle and rotor respectively: ζ N h h and ζ h = = h s 3 3s R c w3 h h s

Axial-flow Turbines: Stage losses and efficiency Assuming c 1 = c 3 and neglecting rotor temperature drop, the stage efficiencies may be expressed as: η tt ζ Rw = 1 + + ζ c 3 N ( h h ) 1 3 1 (4.0 b) η ts ζ Rw + ζ c + c = 1 + ( h1 h3) 3 N 1 1 (4.1 b) (if rotor temperature drop is not negligible, these expressions are a bit more complicated)

Axial-flow Turbines: number of stages From the definition of stage loading: The number of stages may be expressed as Ψ = h01 h U 03 = W U n stage W > m ΨU (4.)

Axial-flow Turbines: Soderberg (3.6) Soderberg s correlation: Large set of data compiled Design assuming Zweifel s criteria for optimum space axial chord ratio Y ΨT = = ( sb) cos α( tanα1+ tanα) 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

Axial-flow Turbines: -D theory Deflection Blade aspect ratio: Blade thickness-chord ratio ε = α1+ α Hb= 3 tmax l = 0.15 0.3 h h h ( ) 5 Re = ρ c D µ D defined at exit throat D = sh cosα s cosα + H = 10 l b t max H is height of blade (radial direction) s

Axial-flow Turbines: -D theory For turbines: Deflection, ε = α + α, is large, but 1 Deviation, δ = α α, is small ' α α ' ε = α ' + α ' 1

Axial-flow Turbines: -D theory FIG. 3.6. Soderberg s correlation of turbine blade loss coefficient with fluid deflection (adapted from Horlock, 1960).

Axial-flow Turbines: -D theory 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+ 0.993 + 0.01 * * 1+ ζcor = 1+ 0.975 + 0.075 Tip clearance losses and disc friction not included

Axial-flow Turbines: -D theory Design considerations Rotor angular velocity (stresses, grid phasing) Weight (aircraft) Outside diameter (aircraft) Efficiency (almost always)

Axial-flow Turbines: Example Consider a case with given Blade speed U Specific work W = U c θ + c (or stage loading) c x ( ) θ 3 Axial velocity (or flow coefficent) The only remaining parameter to define is Triangles may be constructed c θ Loss coefficients determined from Soderberg Efficiencies computed from loss coefficients since c W = U c θ 3 θ

Axial-flow Turbines: -D theory Stage loading factor: ΔW U cx flow coefficient: U Aspect ratio: H b Variation of efficiency with c θ /U (Reaction) for several values of stage loading factor W/U (adapted from Shapiro et al. 1957). (Computed using Soderberg, from previous edition of Dixon)

Axial-flow Turbines: -D theory Stage reaction, R Alternative description to cy U Several definitions available Here: ( ) ( ) R= h h h h 3 1 3 E.g: R = 0.5 ( ) ( ) 0.5 = h h h h h h = h h 3 1 3 3 1 R = 0.5

Axial-flow Turbines: -D theory For a repeating stage, c1 = c3 ( ) ( ) R= h h h h Using 3 01 03 ( ) h h3 + w w3 = 0 and Euler R R = w w 3 U cy y ( + c 3) ( )( ) U( cy + cy ) w w w + w w w = = 3 3 3 3 U

Axial-flow Turbines: -D theory Relative tangential velocity w = c tan β y x R w3 w c x ( tan β tan β ) = = U U 3 Or using cy = wy + U w w w + U w R U U 1 cx = + 3 U 3 3 y = = = ( tan β tanα )

Axial-flow Turbines: -D theory Noting that both triangles have the same blade speed, U

Axial-flow Turbines: -D theory Zero reaction stage R c x ( ) = tan β3 tan β = 0 if β3 = β U FIG. 4.5. Velocity diagram and Mollier diagram for a zero reaction turbine stage.

Axial-flow Turbines: -D theory 50% reaction stage c x 1 R = + ( tan β3 tanα) = 0.5 if β3 = α U FIG. 4.7. Velocity diagram and Mollier diagram for a 50% reaction turbine stage.

Axial-flow Turbines: -D theory FIG. 4.8. Velocity diagram for 100% reaction turbine stage.

Axial-flow Turbines: -D theory From Soderberg ΔW Cy R = 1+ U U FIG. 4.9. Influence of reaction on total-to-static efficiency with fixed values of stage loading factor.

Axial-flow Turbines: -D theory FIG. 4.6. Mollier diagram for an impulse turbine stage.

Axial-flow Turbines: Smith charts Alternative representation for specified reaction Based on measurements at Rolls-Royce η = f ( Ψ, Φ) where FIG. 4.14. Design point total-to-total efficiency and deflection angle contours for a turbine stage of 50 percent reaction. ΔW Ψ = is the stage loading and U cx Φ = is the flow coefficient U

Axial-flow Turbines: -D theory FIG. 4.11. Design point total-to-total efficiency and rotor flow deflection angle for a zero reaction turbine stage.

Centrifugal stresses dfc = Ω d dm= ρ Ar d r m dσ c df c Ω ρ = ρa = rdr With constant cross section this may be integrated σ r t Utip c rh = Ω rdr 1 ρ = rh rt FIG. 4.15. Centrifugal forces acting on rotor blade element.

Axial-flow Turbines: -D theory Tapering: Reduction of cross sectional area in radial direction, in order to reduce stresses Pure fluid dynamics would recomend the opposit FIG. 4.16. Effect of tapering on centrifugal stress at blade root (adapted from Emmert 1950).

Axial-flow Turbines: -D theory FIG. 4.17. Maximum allowable stress for various alloys (1000 hr rupture life) (adapted from Freeman 1955).

Axial-flow Turbines: -D theory FIG. 4.18. Properties of Inconel 713 Cast (adapted from Balje 1981).

Axial-flow Turbines: -D theory 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)

Axial-flow Turbines: -D theory FIG. 4.0. Turbine thermal efficiency vs inlet gas temperature (adapted from le Grivès 1986).