Theory of turbo machinery / Turbomaskinernas teori Chapter 3 · 2009-09-09 · LTH /...

Theory of turbo machinery / Turbomaskinernas teori

Chapter 3

LTH / Kraftverksteknik / JK

2D cascades

Let us first understand the facts and then we may seek the causes. (Aristotle)


2D cascades

High hub-tip ratio (of radii)

• negligible radial velocities

• 2D cascades directly applicable

Low hub-tip ratio

• Blade speed varying

• Blades twisted from hub to tip


2D cascades

FIG. 3.1. Compressor cascadewind tunnels. (a) Conventional low-speed, continuousrunning cascade tunnel (adapted from Carter et al. 1950).(b) Transonic/supersoniccascade tunnel (adapted from Sieverding 1985).


2D cascades

How long must the “infinite” direction be to make derivatives negligible?


2D cascades

FIG. 3.2. Compressor cascade and blade notation.

( )y x• Camber line

• Profile thickness ( )t x

a

x

t y

( )b y a=• Max camber


2D cascades


s• Spacing

• Stagger angle

• Camber angleChange in angle of the camber line

• Blade entry angle

• Blade exit angle

• Inlet flow angle

• Incidence

ξ

θ

1 'α

2 'α

1α

i


2D cascades (incompressible)

FIG. 3.3. Forces and velocities in a blade cascade.

• Continuity:

1 1 2 2cos cos xc c cα α= =

• Momentum (x and y):

( )2 1X p p s= −

( )1 2x y yY sc c cρ= −

or

( )21 2tan tanxY scρ α α= −

Forces per unit depth!


2D cascades Energy losses

( )2 1X p p s= −

• Loss in total pressure from skin friction 0Δp

( )2 20 1 21 2

Δ 12

p p p c cρ ρ

−= + −

( ) ( ) ( )( )2 2 2 21 2 1 2 1

21 2 2

2x xy y y y y yc cc c c c c c c c− = + − + = + −

( )1 2x y yY sc c cρ= −

0 1 2Δ tan tan tan2 m

p X Y X Ys s s s

α α αρ ρ ρ ρ ρ

+− −= + = + Def of αm


2D cascades Energy losses

• Dimensionless forms are obtained normalizing with axial or absolute velocity :

02

Δ2x

pc

ζρ

=

021

Δ2

pc

ωρ

=

• Pressure rise coefficient and tangential force coefficientare

2 12 22 2px x

p p XCc scρ ρ−

= =

2 2fx

YCscρ

=

tanp f mC C α ζ= −


2D cascades Lift and drag

FIG. 3.4. Lift and drag forces exerted by a cascade blade (of unit span) upon the fluid.

cosm x mc c α=

sin cossin cos

m m

m m

L X YD Y X

α αα α

= += −

Lift and drag forces are same as Y and X, but in the coordinates of the blades

FIG. 3.5. Axial and tangential forces exertedby unit span of a bladeupon the fluid.



( )21 2 0

0

tan tan sec Δ sinΔ cos

x m m

m

L sc s pD s p

ρ α α α α

α

= − −

=

Rearranging previous equations:

( )21 2 0

2 2

02 2

tan tan sec Δ sin2 2

Δ cos2 2

x m mL

m m

mD

m m

sc s pLCc l c l

s pDCc l c l

ρ α α α αρ ρ

αρ ρ

− −= =

= =

Dimension less forms are

sec 1 cosα α=

( ) 21 2

2sec tan tan secfmLm

D

CCLD C

α α α αζ ζ

= = − = (3.20)


2D cascades Circulation and lift

http://www.sagabelladesign.se/res/Monster/dodskalleny.jpg


2D cascades Efficiency of a compressor cascade

Compressor blade cascade efficiency defined as diffuser efficiency:

( )2 1

2 21 2 2Dp pc c

ηρ

−=

−so that 0Δ 0p = when 1Dη =

( )( )

( ) ( )

2 20 1 2 02 1

22 2 2 21 21 2 1 2

-Δ 2 Δ1tan tan tan2 2D

x m

p c c pp pcc c c c

ρη

ρ α α αρ ρ

+ −−= = = −

−− −

Using equations 3.7, 3.9 and 3.25

21sin 2

DD

L m

CC

ηα

= −



,2

4 cos 2 0 45degsin 2

D mDm opt

m L m

CC

αη αα α∂

= = ⇒ =∂

Assuming constant ratio between lift and drag const.D LC C =

may be found by differentiation:21sin 2

DD

L m

CC

ηα

= −An optimum of

,max21 D

DL

CC

η = −

And the corresponding efficiency becomes



FIG. 3.6. Efficiency variation with average flow angle (adapted from Howell 1945).


2D cascades

FIG. 3.7. Streamline flow through cascades (adapted from Carter et al. 1950).


2D cascades

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


2D cascades

• Experimental Techniques in separate lecture

• Experiments should help determining

Blade shape (thickness, max camber, position…)

Space chord ratio

Deviation

…..

http://www.pagendarm.de/trapp/programming/java/profiles/NACA4.html

• Generalized experiments

http://www.pagendarm.de/trapp/programming/java/profiles/NACA4.html


2D cascades Fluid deviation


Incidence is chosen by designer

With limited number of blades:

2 2'α α≠So that the deviation may be defined as

2 2 'δ α α= −


2D cascades

FIG. 3.12. Compressor cascade characteristics (Howell 1942). (By courtesy of the Controller of H.M.S.O., Crown copyright reserved).

1 1 'i α α= −

1 2ε α α= −

Incidence:

Deflection:


2D cascades Generalizing experimental results

2 2 'δ α α= −

Deviation by Howell: Nominal deviation a function of camber and space chord ratio:

( )* nm s lδ θ=

( ) *2

0.50.23 2 500

nm a l a=

= +

with the following constants for compressor cascades


2D cascades Generalizing experimental results

FIG. 3.18. Variation of nominal deflection with nominal outletangle for several space/chordratios (adapted from Howell 1945).

Example 3.1, Howell: ( )* *2, , Ref s lε α=


2D cascades Optimum space chord ratio of turbineblades (Zweifel)

FIG. 3.27. Pressure distribution around a turbine cascade blade (after Zweifel 1945).


2D cascades Optimum space chord ratio of turbineblades (Zweifel)

22idY c bρ=

Maximum tangential load (force per unit span)

b is passage width, fig 3.27

Ratio of real to ideal load for minimum losses is around 0.8

( ) ( )22 1 22 cos tan tan 0.8T

id

Y s bY

Ψ α α α= = + ≈

For specified inlet and outlet angles s b or s l may be determined

Theory of turbo machinery / Turbomaskinernas teori Chapter 3 · 2009-09-09 · LTH /...

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Transcript of Theory of turbo machinery / Turbomaskinernas teori Chapter 3 · 2009-09-09 · LTH /...