Theory of turbo machinery / Turbomaskinernas teori. Chapter 3

Theory of turbo machinery / Turbomaskinernas teori Chapter 3

D cascades Let us first understand the facts and then we may 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 from hub to tip

Twist

Generic airfoils NACA airfoil Double circular arcs (DCA) C4 prescribed pressure distribution

D cascades FIG. 3.1. Compressor cascade wind tunnels. (a) Conventional low-speed, continuous running cascade tunnel (adapted from Carter et al. 1950). (b) Transonic/supersonic cascade tunnel (adapted from Sieverding 1985).

D cascades FIG. 3.. Streamline flow through cascades (adapted from Carter et al. 1950).

D cascades How long must the infinite direction be to make derivatives negligible?

D cascades Camber line yx ( ) Max camber b = ya ( ) Profile thickness tx ( ) t y x a FIG. 3.4. Compressor cascade and blade notation.

Profile families Notice how the maximum thickness point differs C4 at 30%, NACA65 at 40% and DCA at 50% Maximum thickness close to leading edge => wider operating range but poorer high speed performance

Maximum t max /l for a modern high speed compressor is below 0.05 Wind turbine blade section (close to hub) requiring a very wide operating range. Low speed with extremely large t max /l

D cascades Spacing Stagger angle Camber angle Change in angle of the camber line Blade entry angle Blade exit angle Inlet flow angle Incidence s ξ θ α ' 1 α ' α1 i FIG. 3.4. Compressor cascade and blade notation.

D cascades Fluid deviation Incidence is chosen by designer With limited number of blades: α ' α So that the deviation may be defined as δ = α α ' FIG. 3.4. Compressor cascade and blade notation.

Aspect ratio Two aspect ratios exist: H/l, where H = blade height = r tip - r hub and l is the corda H/b, where b is the axial corda

D cascades Cascades are used to measure losses generated through blade boundary layers flow separation shock waves And to estimate deviation As stated, the flow does not exactly follow the blade, but it is underturned δ = α α ' FIG. 3.7. The flow through a blade cascade and the formation of wakes (Johnsson and Bullock, 1965).

D cascades performance parameters Y ζ p, compressor p, turbine = = c, is p p c 01 01 c, is p p 0 1, Y p, turbine = p p 01 01 p p 0 Pressure loss coefficient Energy loss coefficient

D cascades performance parameters The purpose is to establish functional relationships where α = f ( M, α,re) (Deviation) Y p = ζ = f f 3 1 ( M ( M 1 1 1, α,re), α,re) 1 1 1 Pressure loss coeff. Energy loss coeff.

D cascades Experimental Techniques in separate lecture Experiments should help determining Blade shape (thickness, max camber, position ) Space chord ratio Deviation.. http://www.ppart.de/aerodynamics/profiles/naca4.html Generalized experiments

Cascade flow characteristics Cascade measurements are D => properties such as deviation and loss coefficients have to be determined by integration: Incoming flow Slots for traversing instruments m = Y s 0 tanα ρc = x s 0 s p, compressor Hdy 0 c = y x ρc s x c ρc x p p Hdy Hdy 01 = p p 0 01 1 s ρc 0 s 0 s dy x 0 dm ρc 0 d m x x ρc c x ρc dy y dy dy,is Turntable As probe is traversed downstream of cascade a periodic flow is observed Stagnation pressure change as probe is traversed through one wake

D cascades (incompressible) Continuity: c cosα = c cosα = cx 1 1 Momentum (x and y): ( ) X = p p s 1 ( 1 ) Y = ρsc c c or x y y ( tan tan ) Y = sc x ρ α1 α FIG. 3.10. Forces and velocities in a blade cascade. Forces per unit depth!

D cascades Energy losses Loss in total pressure from 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 = + = + tanαm Def of α m ρ sρ sρ ρs ρs

D cascades Energy losses Dimensionless forms are obtained normalizing with axial or absolute velocity : Stagnation pressure loss coefficients 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 m

D cascades Lift and drag Corrected lift and drag coefficients versus angle of attack, α A, from wind tunnel measurements for the profiles NACA 441 to 444. Adapted from Paulsen, U.S.: Aerodynamics of a full-scale, non rotating wind turbine blade under natural wind conditions, Risø National Laboratory, Roskilde Denmark 1989 Note: many different definitions of angle of attack exist! C C L D L = ρc l = m D ρc l m

D cascades Lift and drag FIG. 3.11. Lift and drag forces exerted by a cascade blade (of unit span) upon the fluid. c = c cosα m x m Lift and drag forces are same as Y and X, but in the coordinates of the blades FIG. 3.1. Axial and tangential forces exerted by unit span of a blade upon the fluid. L= X sinα + Ycosα D= Ysinα X cosα m m m m

D cascades Lift and drag Rearranging previous equations: ( ) L = ρsc tanα tanα secα sδp sinα D= sδp x 1 m 0 m 0 cosα m secα = 1 cosα Dimension less forms are C C L D L D ( ) ρsc tanα tanα secα sδp sinα L x 1 m 0 m m ρcl m = = ρcl D sδp cosαm = = ρcl cl C 0 m ρ m secαm ζ f ( tan tan ) sec = L α1 α αm C = = D ζ (3.6 b) C

D cascades Circulation and lift Bases in potential theory Kutta-Joukowski Theorem: L = Γρc

D cascades Velocity on suction and pressure surfaces Low pressure (suction side) is obtained through high velocity At the trailing edge the velocities must match This requires a large velocity decrease (diffusion) with potential for flow separation A diffusion factor may be defined (Lieblein): DF loc = ( c c ) max, s c max, s

D cascades Incidence changes velocity, and thereby, pressure distribution FIG. 3.17. Effect of incidence on surface Mach number distribution around a compressor blade cascade

D cascades Velocity on suction and pressure surfaces What does the velocity distribution look like? (Inviscid computation) Please note: Fluid elements passing pressure and suction side do NOT pass the foil in the same amount of time

D cascades Efficiency of a compressor cascade Compressor 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 tanαm( tanα1 tanα) p p -Δp + c c Δp = = = ρ c c c c 1 0 1 0 1 ρ 1 In terms of drag and lift coefficients, this can be shown to become: η D = 1 C L CD sin α m

D cascades Efficiency of a compressor cascade Assuming constant ratio between lift and drag C C = const. D L An optimum of η D = 1 C L CD sin α m may be found by differentiation: η α 4C cos α = = 0 α = 45deg D D m m CLsin αm m, opt And the corresponding efficiency becomes η D,max C = C 1 D L

D cascades FIG. 3.. Streamline flow through cascades (adapted from Carter et al. 1950).

D cascades FIG. 3.8. Contraction of streamlines due to boundary layer thickening (adapted from Carter et al. 1950).

Profile loss (Ainley & Mathieson correlation) Axial flow turbine Determine Y p at zero incidence (Y p,i=0 ). Y p at any other incidence is predicted as a function of the ratio i/i s where i s is the stalling incidence. The stalling incidence i s is defined as the incidence where the profile loss has doubled from the zero incidence loss. Y p = p p 01 01 p p 0 (3.6) So, what about Y p,i=0?

Profile loss (Ainley & Mathieson correlation) Y p,i=0 is determined by an interpolation procedure between two extreme turbine design concepts: a) Nozzle blades: large amount of acceleration in blade row b) Impulse blades: no acceleration in relative frame From what do you know about diffusion why is the profile loss coefficient Y p much higher for the impulse blades than for the nozzle blades?!! The expression used for Y p,i=0 is: Y 1 1 α 1 tmax / l α + ( ) [ Yp( α ] ( ) 1= α ) Yp( α1 0) α 0. p( i= 0) = Yp( α = 0) = α (3.49) Use β for α 1 and β 3 for α if rotor blades are being considered

Reynolds number correction (Ainley & Mathieson correlation) Correct around the nominal Re-number = 10 5 according to: 1 η 1 η tt,corrected tt,nominal Re = 5 10 0.0 Correction is valid down to Re-number 10 4. Renumber is computed on chord at blade mean and exit flow conditions from the turbine.

Simplified correlation (Soderberg) Turbines If turbines are designed to operate at optimal space chord ratio (defined through the Zweifel criterion), and at zero incidence turbine blade losses can be correlated according to: ζ = tmax f ( ε,, AR,Re) l where ε = fluid deflection, t max /l= maximum thickness to chord ratio, AR = is blade aspect ratio and Re is Reynolds number.

Simplified correlation (Soderberg) For AR = 3.0 and Re=10 5 a nominal loss coefficient is predicted from: * ζ = + 0.04 0.06( ε ) 100 where the deflection ε is in degrees. If the aspect ratio deviates from the nominal aspect ratio AR=3.0, the following correction is used (b=axial chord, H, blade height): * 0.01b 1+ ζ 1 = (1 + ζ )(0.993 + ) ( nozzle H * 0.075b 1+ ζ 1 = (1 + ζ )(0.975 + ) ( rotor H If the Re-number deviates from Re=10 5, a correction must be predicted from: row) row)

Simplified correlation (Soderberg) If the Re-number deviates from Re=10 5, a correction must be predicted from: ζ 5 10 = Re 1/ 4 ζ 1 Where the Re-number is based on the hydraulic diameter according to: D sh cosα = h s cosα + H

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

D cascades Generalizing experimental results Deviation by Howell: Nominal deviation a function of camber and space chord ratio: δ = α α ' δ = mθ ( sl) * n with the following constants for compressor cascades n = 0.5 ( ) * m= 0.3 al + a 500

D cascades Generalizing experimental results FIG. 3.16. Variation of nominal deflection with nominal outlet angle for several space/chord ratios (adapted from Howell 1945). Example 3. recommended

D cascades Optimum space chord ratio of turbine blades (Zweifel) Note angle def! FIG. 3.8. Pressure distribution around a turbine cascade blade (after Zweifel 1945).

D cascades Optimum space chord ratio of turbine blades (Zweifel) Tangential load (force) Maximum tangential load (force) ( ) Y = m c y + c 1 y = ( p p )bh Y id 01 ( b is passage width ) Incompressible loss free flow p p = ρ 01 c Ratio of real to ideal load for minimum losses is around 0.8, after some manipulations: Y ΨT = = ( sb) cos α( tanα1+ tanα) 0.8 Y id For specified inlet and outlet angles sb or sl may be determined

Turbine limit load FIG. 3.3. Schlieren photograph of flow in a highly loaded transonic turbine cascade with an exit Mach number of 1.15 (from Xu, 1985).