Lecture 1 Axial Fans: Aerodynamic Design

N09- 085/1 A

Lecture 1

Axial Fans: Aerodynamic Design

Th. Carolus

UNIVERSITÄT SIEGEN Institut für Fluid- und Thermodynamik

N09- 085/2 A

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

1.8 Performance Analysis via Computational Fluid Dynamics (CFD)

1.9 Prototyping and Scaling Up

UNIVERSITÄT SIEGEN Institut für Fluid- und Thermodynamik

1 1.1 Axial Cascade Blade Element (BE) N09- 085/3 A

Rotor diameter: Da = 2ra Chord length: l Hub diameter: Di = 2ri Circumferential spacing t Radial extension of BE: δr Blade angle at leading edge (LE) βs1 Number of blades: z Blade angle at leading edge (LE) βs2

1.2 Actual vs. Ideal Flow N09- 085/4 A

Actual: Ideal: • 3D • 1D • unsteady • steady

2 1.3 Frames of References N09- 085/5 A

G u G w stationary (absolute) rotating frame frame of reference of reference

G u = velocity of rotating system (relative system) G w = velocity of a particle in relative system

⇒ Oberserver in the absolute frame of reference sees absolute particle velocity G G G c =+u w (1-1)

Velocity triangle

1.4 Co-ordinate System and Components of Velocity N09- 085/6 A

G Velocity c and relevant components: G c • in z-direction ⇒ axial component z G • in ϕ- dircetion ⇒ circumferential (= tangential) component c G u • in r-direction ⇒ radial component cr G GGG cc= zur++ c c (1-2) G GG In addition useful: Meridional component: cccmzr= + (1-3)

3 1.5 Velocity triangles on BE (turbine) N09- 085/7 A

1 2

1.6 EULER‘s Equation of Turbomachinery (Angular Momentum Analysis) N09- 085/8 A

control volume

δ Fu

Mshaft

δ r

δ Fu

4 N09- 085/9 A

Apply to angular momentum conservation to CV „blade element“ (BE) GG (r ×∂c) ρ GGGG + ρ ()()rcc× ⋅ ndA= Mshaft ∫V ∫A ∑ ∂t (1-4) angular momentum external unsteady term torques

With incremental mass flow through BE

δ mcAc==ρδmm ρ2 πδ r (1-5)

one gets the incremental torque at shaft

δ mrc = δ M ()22ushaft change of angular momentum from entrance to exit of blade channel

Note: Incremental tangential force is

δ Fcm=−δ uu 2 (1-6) δ F change of momentum u from entrance to exit of blade channel

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With the incremental power

δ PMBE= δΩ shaft (1-7)

and

ur= Ω (1-8) SEGNER‘s waterwheel analyzed by one eventually obtains the specific work L. EULER δ P Yuc≡=BE δ m 22u (1-9a)

or, if cu1 ≠ 0

Yucuc=−22uu 11 (1-9b)

EULER equation of turbomachinery (1754) LEONHARD EULER 1707 - 1783

5 1.7 Blade Element Design N09- 085/11 A

1st way to get circumferential force on BE

Recall circumferential force on BE from angular momentum analysis

δ Fcmuu=− 2δ (1-6)

Now, CV enclosing only one blade and thus

δ mcActr==ρδmm ρ δ yields CV

δ Fcctruum=− 2ρδ (1-10)

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2nd way to get circumferential force on BE

Flow induced forces on one blade w 2 • lift δ Lc= ρδ∞ lr L 2 w 2 • drag δ Dc= ρδ∞ lr D 2

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

δ DLδεε≡≈sin (1-11)

and δ FL≈ δ

δ FFu =−sin(βεδ∞ )

w 2 ⇒ δ Fclr=−sin()βερ∞ δ (1-12) uL∞ 2

6 N09- 085/13 A

w ∞ , β ∞ ????

For cascades with small flow deflection, i. e. for blades with small camber, replace up- and downstream velocities by their vector-mean:

GGG1 www≡+() (1-13) ∞ 2 12

Hence, from velocity triangles we get

w ∞ and the vector-mean flow angle

β∞

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

lcc−2 um2 ⇒ cL = tw∞∞sin()β − ε

Inserting in this equation

• the blade spacing 2π r t = (1-14) z • the solidity

lz (1-15) σ ≡ from selected 2πr airfoil

⇒ σ = f(velocity triangle s , ccLD, ) (1-16)

aerodynamic design point

7 1.8 Performance Analysis via Computational Fluid Dynamics (CFD) N09- 085/15 A

• Computational domain and numerical grid

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• Typical CFD results (I)

Streamlines Pressure distribution

8 N09- 085/17 A

• Typical CFD results (II)

Spanwise blade loading (pressure distribution)

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• Typical CFD results (III)

Performance Curves 0.6 Design

CFD 0.5 Experiment ηfac

0.4 [-] fa

η 0.3 , Design fa ψ

0.2

Experiment ψfa 0.1

CFD 0 0 0.05 0.1 0.15 0.2 0.25 0.3 φ [-]

9 1.9 Prototyping and Scaling Up N09- 085/19 A

Prototyping

CNC-Milling Machine

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Experimental model test

Full scale

10 Short presentation of the University of Siegen Fan Design Code .... N09- 085/21 A