Theory of turbo machinery / Turbomaskinernas teori. Chapter 3

Transcription

1 Theory of turbo achinery / Turboaskinernas teori Chapter 3

2 D cascades Let us first understand the facts and then we ay seek the causes. (Aristotle)

3 D cascades High hub-tip ratio (of radii) negligible radial velocities D cascades directly applicable Low hub-tip ratio Blade speed varying Blades twisted fro hub to tip

4 D cascades FIG Copressor cascade wind tunnels. (a) Conventional low-speed, continuous running cascade tunnel (adapted fro Carter et al. 1950). (b) Transonic/supersonic cascade tunnel (adapted fro Sieverding 1985).

5 D cascades How long ust the infinite direction be to ake derivatives negligible?

6 D cascades Caber line y( x) Max caber b = y( a) Profile thickness tx ( ) t y x a FIG. 3.. Copressor cascade and blade notation.

7 D cascades Spacing Stagger angle Caber angle Change in angle of the caber line Blade entry angle Blade exit angle Inlet flow angle Incidence s ξ θ α ' 1 α ' α1 i FIG. 3.. Copressor cascade and blade notation.

8 D cascades (incopressible) Continuity: c1cosα1 = ccosα = cx Moentu(xand y): ( ) X = p p s 1 ( 1 ) Y = ρsc c c or x y y ( tan tan ) Y = sc x ρ α1 α FIG Forces and velocities in a blade cascade. Forces per unit depth!

9 D cascades Energy losses Loss in total pressure fro skin friction Δp0 Δp0 p1 p 1 = + ρ ρ ( c ) 1 c ( ) X = p p s ( y cx) ( y cx) ( y1 y)( y y) c c = c + c + = c + c c c ( 1 ) Y = ρsc c c x y y Δp0 X Y tanα1+ tanα X Y = + = + tanα Def of α ρ sρ sρ ρs ρs

10 D cascades Energy losses Diensionless fors are obtained noralizing with axial or absolute velocity : ζ = Δp ρc 0 x Δp ω = ρc 1 0 Pressure rise coefficient and tangential force coefficient are C C p f p p X = = ρ = 1 cx ρscx Y ρsc x C = C tanα ζ p f

11 D cascades Lift and drag FIG Lift and drag forces exerted by a cascade blade (of unit span) upon the fluid. c = c cosα x Lift and drag forces are sae as Y and X, but in the coordinates of the blades FIG Axial and tangential forces exerted by unit span of a blade upon the fluid. L= X sinα + Ycosα D= Ysinα X cosα

12 D cascades Lift and drag Rearranging previous equations: ( ) L= ρsc tanα tanα secα sδp sinα D = sδp x cosα secα = 1 cosα Diension less fors are C C L D L D ( ) ρsc tanα tanα secα sδp sinα L x 1 0 ρcl = = ρcl D sδp cosα = = ρcl cl 0 ρ = CL secα C f ( tanα1 tanα) sec α C = ζ = ζ (3.0) D

13 D cascades Lift and drag

14 D cascades Circulation and lift

15 D cascades Efficiency of a copressor cascade Copressor blade cascade efficiency defined as diffuser efficiency: η D = ρ p p 1 ( c c ) 1 so that Δp 0 = 0 when η D = 1 η D ( ) ρ ( ) 1 ( ) ρcx tanα( tanα1 tanα) p p -Δp + c c Δp = = = ρ c c c c ρ 1 Using equations 3.7, 3.9 and 3.5 η D = 1 C L CD sin α

16 D cascades Efficiency of a copressor cascade Assuing constant ratio between lift and drag C C = const. D L An optiu of η D = 1 C L CD sin α ay be found by differentiation: η α 4C cos α = = 0 α = 45deg D D CLsin α, opt And the corresponding efficiency becoes η D,ax C = C 1 D L

17 D cascades Efficiency of a copressor cascade FIG Efficiency variation with average flow angle (adapted fro Howell 1945).

18 D cascades FIG Strealine flow through cascades (adapted fro Carter et al. 1950).

19 D cascades FIG Contraction of strealines due to boundary layer thickening (adapted fro Carter et al. 1950).

20 D cascades Experiental Techniques in separate lecture Experients should help deterining Blade shape (thickness, ax caber, position ) Space chord ratio Deviation.. Generalized experients

21 D cascades Fluid deviation Incidence is chosen by designer With liited nuber of blades: α ' α So that the deviation ay be defined as δ = α α ' FIG. 3.. Copressor cascade and blade notation.

22 D cascades Incidence: i = α α ' 1 1 Deflection: ε = α1 α FIG Copressor cascade characteristics (Howell 194). (By courtesy of the Controller of H.M.S.O., Crown copyright reserved).

23 D cascades Generalizing experiental results Deviation by Howell: Noinal deviation a function of caber and space chord ratio: δ = α α ' δ = θ ( s l) * n with the following constants for copressor cascades n = 0.5 ( ) * = 0.3 a l + a 500

24 D cascades Generalizing experiental results FIG Variation of noinal deflection with noinal outlet angle for several space/chord ratios (adapted fro Howell 1945). Exaple 3.1, Howell: ε * = f ( s l, α *,Re)

25 D cascades Optiu space chord ratio of turbine blades (Zweifel) FIG Pressure distribution around a turbine cascade blade (after Zweifel 1945).

26 D cascades Optiu space chord ratio of turbine blades (Zweifel) Maxiu tangential load (force per unit span) Yid = ρc b b is passage width, fig 3.7 Ratio of real to ideal load for iniu losses is around 0.8 Y Ψ T = = ( sb) cos α( tanα1+ tanα) 0.8 Y id For specified inlet and outlet angles sb or sl ay be deterined