Theory of turbo machinery / Turbomaskinernas teori. Chapter 4

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1 Teory of turbo macinery / Turbomaskinernas teori Capter 4

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

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

4 Axial-flow Turbines: -D teory 1 Nozzle row Rotor row 3 Note direction of FIG Turbine stage velocity diagrams.

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

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

7 Axial-flow Turbines: Design parameters Flow coefficient or Stage loading coefficient Δ 0 Or using Euler: Δ 0 Δ Δ (4.) cange of tangential velocity in rotor!

8 Axial-flow Turbines: Stage reaction Stage reaction = Static entalpy drop across rotor Static entalpy drop across stage R Second law, assuming isotropic process and ignoring density canges R p p 1 p p 3 3 Tds d dp d dp

9 Axial-flow Turbines: -D teory Δ y y W W m U c c 01 0 Please note: No work done in nozzle row: Wit And using above equations: Work done on rotor by unit mass of fluid c c U c c c c x x 0 c c c x c c U m W W Angle defined in opposite direction at

10 Axial-flow Turbines: -D teory Rewriting tis in terms of relative velocity c c c 3 U w U w c 3 3 w w 3 Combining above equations: w w c x c 3 x3 3 wit wx wx3 cx and 0 w w w x y w c U w w (4.6 approximately) Relative stagnation entalpy,, does not cange across rotor 0,rel

11 Axial-flow Turbines: -D teory Nozzle row (1 to ): Static pressure: Stagnation entalpy: Stagnation pressure: (isentropic: p p ) 01 0 p p p p 01 0 Subscript s denotes isentropic cange and ss denotes bot rows isentropic FIG Mollier diagram for a turbine stage.

12 Axial-flow Turbines: -D teory Rotor row ( to 3): Static pressure: Stagnation entapy: Stagnation pressure: p p 3 p 0 03 p 0 03 p p 3 However: Relative Stagnation entapy, w 0, rel 0 03, rel FIG Mollier diagram for a turbine stage.

13 Axial-flow Turbines: Repeating Stage Multi-stage turbines: Stages often similar to eac oter c x const. r const. 1 3 However, cannel must be widened FIG General arrangement of a 6- stage repeating turbine.

14 Axial-flow Turbines: Repeating Stage Reaction: R Relative entalpy drop across rotor = 1- relative entalpy drop across stator Same velocities in and out => c c 1 0 c x tan tan 1 tan tan 1 R 1 (4.1) Stage loading U 01 03

15 Axial-flow Turbines: Repeating Stage wit c U c x tan tan U 1 tan tan 1 te reaction becomes R tan tan 1 1 (4.13 a) eq 4.13 a + eq 4.1: 1 1 R tan (4.14)

16 Axial-flow Turbines: number of stages From te definition of stage loading: Te number of stages may be expressed as 01 U 03 W U n stage W m U Higer mass flow Higer stage loading Higer blade velocity All lead to reduced number of stages

17 Axial-flow Turbines: Stage losses and efficiency Turbine stage total to total efficiency: tt Actual work output Ideal work output wen operating to same back pressure ss For a repeating stage, no canges in are made in velocities from inlet to outlet: c1 c3 and 1 3. Furter assuming c3ss c3 te efficiency becomes: tt ss 1 3ss

18 Axial-flow Turbines: Stage losses and efficiency Defining entalpy loss coefficients for te nozzle and rotor respectively: N and s 3 3s R c w3 Neglecting rotor temperature drop, te stage efficiencies may be expressed as: tt w c , ss R 3 N 1 (4.18 c) ts w c c , ss R 3 N 1 1 (4.19 c)

19 Axial-flow Turbines: Soderberg Soderberg s correlation: Large set of data compiled Design assuming Zweifel s criteria for optimum space axial cord ratio Y T sbcos tan1tan0.8 Y id Result: Turbine blade losses are a function of Deflection Blade aspect ratio Blade tickness-cord ratio Reynolds number Hb tmax l

20 Axial-flow Turbines: Soderberg Deflection Blade aspect ratio: Blade tickness-cord ratio 1 Hb 3 tmax l Re cd D defined at exit troat D shcos scos H 10 l b t max H is eigt of blade (radial direction) s

21 Axial-flow Turbines: -D teory For turbines: Deflection,, is large, but 1 Deviation,, is small ' ' ' ' 1

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

23 Axial-flow Turbines: Soderberg Corrections for Reynolds number 5 Re * 10 * cor Re Blade aspect ratio Nozzles: Rotors: bh bh * * 1cor * * 1cor Tip clearance losses and disc friction not included

24 Axial-flow Turbines: -D teory Design considerations Rotor angular velocity (stresses, grid pasing) Weigt (aircraft) Outside diameter (aircraft) Efficiency (almost always)

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

26 Axial-flow Turbines: -D teory Stage loading factor: ΔW U cx flow coefficient: U Aspect ratio: H b Variation of efficiency wit c /U (Reaction) for several values of stage loading factor W/U (adapted from Sapiro et al. 1957).

27 Axial-flow Turbines: -D teory Stage reaction, R Alternative description to cy U Several definitions available Here: R E.g: R = R = 0.5

28 Axial-flow Turbines: -D teory For a repeating stage, c1 c3 R Using w w and Euler R R w 3 U cy cy3 w Ucy cy w w w w w w U 3

29 Axial-flow Turbines: -D teory Relative tangential velocity w c tan y x R w3 w c x tan 3 tan U U Or using cy wy U w w w U w R U U 1 cx tan 3 tan U 3 3 y

30 Axial-flow Turbines: -D teory Zero reaction stage R c x tan 3tan 0 if 3 U FIG Velocity diagram and Mollier diagram for a zero reaction turbine stage.

31 Axial-flow Turbines: -D teory 50% reaction stage c x 1 R tan 3 tan0.5 if 3 U FIG Velocity diagram and Mollier diagram for a 50% reaction turbine stage.

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

33 Axial-flow Turbines: -D teory FIG. 4.4 R ΔW Cy 1 U U FIG Influence of reaction on total-to-static efficiency wit fixed values of stage loading factor.

34 Axial-flow Turbines: -D teory FIG Mollier diagram for an impulse turbine stage.

35 Axial-flow Turbines: Smit carts Alternative representation for specified reaction Based on measurements at Rolls-Royce f (, ) were FIG Design point total-to-total efficiency and deflection angle contours for a turbine stage of 50 percent reaction. ΔW is te stage loading and U cx is te flow coefficient U

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

37 Centrifugal stresses dfc Ω d dm Adr d c df c A r m Ω rdr Wit constant cross section tis may be integrated r t Utip c r Ω rdr 1 r rt FIG Centrifugal forces acting on rotor blade element.

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

39 Axial-flow Turbines: -D teory FIG Maximum allowable stress for various alloys (1000 r rupture life) (adapted from Freeman 1955).

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

41 Axial-flow Turbines: Cooling Turbine blade cooling. Wy is te efficiency of te gas turbine comparable to tat of a Rankine cycle? (given tat we do ave to pay a considerable amount of energy to te compressor, wereas compression of water in te Rankine cycle is ceap)

42 Axial-flow Turbines: Cooling Te evolution of allowable gas temperature at te entry to te gas turbine and te contribution of superalloy development, film cooling tecnology, termal barrier coatings and (in te future) ceramic matrix composite (CMC) air foils and peraps novel cooling concepts

43 Axial-flow Turbines: Cooling FIG Turbine termal efficiency vs inlet gas temperature (adapted from le Grivès 1986).

Design and Test of Transonic Compressor Rotor with Tandem Cascade

Proceedings of the International Gas Turbine Congress 2003 Tokyo November 2-7, 2003 IGTC2003Tokyo TS-108 Design and Test of Transonic Compressor Rotor with Tandem Cascade Yusuke SAKAI, Akinori MATSUOKA,