lecture 1 axial fans: aerodynamic design...for design conditions usually d
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1
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)
2
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
3
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)
4
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
5
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
6
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
7
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
8
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)
9
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
10
19N09- 085/ A
CNC-Milling Machine
Prototyping
1.9 Prototyping and Scaling Up
20N09- 085/ A
Experimental model test
Full scale
11
21N09- 085/ AShort presentation of the University of Siegen Fan Design Code ....
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