Theory of turbo achinery / Turboaskinernas teori Chapter 3
D cascades Let us first understand the facts and then we ay seek the causes. (Aristotle)
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
D cascades FIG. 3.1. 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).
D cascades How long ust the infinite direction be to ake derivatives negligible?
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.
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.
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. 3.3. Forces and velocities in a blade cascade. Forces per unit depth!
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 1 1 1 ( 1 ) Y = ρsc c c x y y Δp0 X Y tanα1+ tanα X Y = + = + tanα Def of α ρ sρ sρ ρs ρs
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
D cascades Lift and drag FIG. 3.4. 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. 3.5. Axial and tangential forces exerted by unit span of a blade upon the fluid. L= X sinα + Ycosα D= Ysinα X cosα
D cascades Lift and drag Rearranging previous equations: ( ) L= ρsc tanα tanα secα sδp sinα D = sδp x 1 0 0 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
D cascades Lift and drag
D cascades Circulation and lift
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 1 0 1 0 1 ρ 1 Using equations 3.7, 3.9 and 3.5 η D = 1 C L CD sin α
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
D cascades Efficiency of a copressor cascade FIG. 3.6. Efficiency variation with average flow angle (adapted fro Howell 1945).
D cascades FIG. 3.7. Strealine flow through cascades (adapted fro Carter et al. 1950).
D cascades FIG. 3.8. Contraction of strealines due to boundary layer thickening (adapted fro Carter et al. 1950).
D cascades Experiental Techniques in separate lecture Experients should help deterining Blade shape (thickness, ax caber, position ) Space chord ratio Deviation.. http://www.pagendar.de/trapp/prograing/java/profiles/naca4.htl Generalized experients
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.
D cascades Incidence: i = α α ' 1 1 Deflection: ε = α1 α FIG. 3.1. Copressor cascade characteristics (Howell 194). (By courtesy of the Controller of H.M.S.O., Crown copyright reserved).
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
D cascades Generalizing experiental results FIG. 3.18. 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)
D cascades Optiu space chord ratio of turbine blades (Zweifel) FIG. 3.7. Pressure distribution around a turbine cascade blade (after Zweifel 1945).
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