lecture 1 axial fans: aerodynamic designn09...lecture 1 axial fans: aerodynamic design ... 1.6...

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1N09- 085/ A

Lecture 1

Axial Fans: Aerodynamic Design

Th. Carolus

UNIVERSITÄT SIEGEN Institut für Fluid- und Thermodynamik

2N09- 085/ A

UNIVERSITÄT SIEGEN Institut für Fluid- und Thermodynamik

1.1 Axial Cascade Blade Element (BE)

1.2 Actual vs. Ideal Flow

1.3 Frames of References

1.4 Co-ordinate System and Components of Velocity

1.5 Velocity triangles on BE (turbine)

1.6 EULER‘s Equation of Turbomachinery (Angular Momentum Analysis)

1.7 Blade Element Design

1.9 Prototyping and Scaling Up

1.8 Performance Analysis via Computational Fluid Dynamics (CFD)

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3N09- 085/ A1.1 Axial Cascade Blade Element (BE)

Rotor diameter: Da = 2raHub diameter: Di = 2riRadial extension of BE: δrNumber of blades: z

Chord length: lCircumferential spacing tBlade angle at leading edge (LE) βs1Blade angle at leading edge (LE) βs2

4N09- 085/ A1.2 Actual vs. Ideal Flow

Actual:• 3D• unsteady

Ideal:• 1D• steady

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5N09- 085/ A1.3 Frames of References

uc w= +

w

u = velocity of rotating system (relative system)

= velocity of a particle in relative system

⇒ Oberserver in the absolute frame of reference sees absoluteparticle velocity

stationary (absolute) frame of referencerotating frame

of reference

Velocity triangle

u

w

(1-1)

6N09- 085/ A1.4 Co-ordinate System and Components of Velocity

Velocity and relevant components:

• in z- direction ⇒ axial component• in ϕ- dircetion ⇒ circumferential (= tangential) component• in r- direction ⇒ radial component

c

zcuc

rc

= + +z u rc c c c

In addition useful: Meridional component: m z rc c c= +

(1-2)

(1-3)

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7N09- 085/ A1.5 Velocity triangles on BE (turbine)

u

1 2

8N09- 085/ A1.6 EULER‘s Equation of Turbomachinery (Angular Momentum Analysis)

controlvolume

Mshaft

δ Fu

δ Fu

r

δ r

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9N09- 085/ A

Apply to angular momentum conservation to CV „blade element“ (BE)

With incremental mass flow through BE

one gets the incremental torque at shaft

( ) ( )( )× ∂⋅+

∂× =∫∫ ∑

anguunsteady termexternaltorqu

lar momentumes

AV shaftr c c Mc

n dAr

tρ

ρ

( ) =2 2

change of angular momentum from entrance to exit

of blade channel

u shaftm r c Mδ δ

= = 2m mm c A c rδ ρ δ ρ πδ

(1-4)

(1-5)

= −change of momentum from entrance to exit

of blade channel

u uF c mδ δ2

Note:Incremental tangential force is

(1-6) δ Fu

10N09- 085/ A

With the incremental power

and

one eventually obtains the specific work

or, if

=u rΩ

≡ = 2 2BE

uPY u cm

δδ

=BE shaftP Mδ δ Ω

EULER equation of turbomachinery (1754) LEONHARD EULER

1707 - 1783

(1-7)

(1-8)

(1-9b)

≠1 0uc

= −2 2 1 1u uY u c u c

(1-9a)

SEGNER‘swaterwheelanalyzed byL. EULER

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11N09- 085/ A

Now, CV enclosing only one blade and thus

yields

1.7 Blade Element Design

= −u uF c mδ δ2

Recall circumferential force on BE from angular momentum analysis

(1-6)

2= −u u mF c c t rδ ρ δ (1-10)CV

= =m mm c A c t rδ ρ δ ρ δ

1st way to get circumferential force on BE

12N09- 085/ A

Flow induced forces on one blade

• lift

• drag

≈ F Lδ δ

∞= LwL c l rδ ρ δ

2

2

( )2

2∞

∞= −sinu LwF c l rδ β ε ρ δ⇒

and

∞= DwD c l rδ ρ δ

2

2

( )∞= −sinuF Fδ β ε δ

For design conditions usually D << L,i. e. the drag-lift ratio is

≡ ≈sinD Lδ δ ε ε (1-11)

(1-12)

2nd way to get circumferential force on BE

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13N09- 085/ A

Hence, from velocity triangles we get

and the vector-mean flow angle

∞β

( )w w w∞ ≡ +1 212

∞w

For cascades with small flow deflection,i. e. for blades with small camber, replaceup- and downstream velocities by theirvector-mean:

(1-13)

∞β∞w , ????

14N09- 085/ A

Inserting in this equation

• the blade spacing

• the solidity

Combining Eqs. (1-10) and (1-12, -13) yields

( )22

∞ ∞

−=

−sinu m

Ll c cct w β ε

= velocity trif( , )ang sle ,L Dc cσ⇒

⇒

2=

rtzπ

σπ

≡2lzr

(1-16)

(1-14)

(1-15)

aerodynamicdesign point

from selectedairfoil

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15N09- 085/ A1.8 Performance Analysis via Computational Fluid Dynamics (CFD)

• Computational domain and numerical grid

16N09- 085/ A

Pressure distributionStreamlines

• Typical CFD results (I)

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17N09- 085/ A

Spanwise blade loading (pressure distribution)

• Typical CFD results (II)

18N09- 085/ A

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

φ [-]

ψfa

, ηfa

[-]

Experiment ψfa

Experiment ηfac

CFD

CFD

Design

Design

• Typical CFD results (III)

Performance Curves

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19N09- 085/ A

CNC-Milling Machine

Prototyping

1.9 Prototyping and Scaling Up

20N09- 085/ A

Experimental model test

Full scale

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21N09- 085/ AShort presentation of the University of Siegen Fan Design Code ....