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Introduction To Wind Energy S21 exam
Introduktion til vindenergi (46000)
Danmarks Tekniske Universitet
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Table of contents
- TABLE OF CONTENTS Course
- 1 Wind Profiles
- 1 Velocity profile for mean wind speed and power law
- 1 Stable, neutral and unstable conditions
- 2 Wind resources
- 2 Wind distribution: Weibull
- 2 Annual Energy Production
- 2 Turbulence
- 3 Technology of wind turbines
- 3 Rotors and efficiency
- 3 Gear ratios
- 3 The planetary gear
- 3 Speed and torque relations
- 4 Aerodynamics
- 4 1D momentum theory
- 4 Theory on rotating wake
- 4 Forces acting on the blade
- 5 Economy
- 5 Table of running expenses
- 6 Structural mechanics
- 6 Second moment of inertia
- 6.1 Square box
- 6.1 Cross section for a circle
- 6.1 Approximated circular cross section
- 6.1 Sandwich beam blade
- 6.1 Parallel axis theorem for second moment of inertia
- 6 The effective modulus of elasticity, Ec
- 6 Strain in the upper most part of a beam
- 6 Formula for deflection along the blade and max deflection
- 6 Generic beam case: Cantilever beam with point load
- 6 Generic beam case: Cantilever beam with distributed load
- 6 Euler-Bernolli Theory and Natural frequency
- 6 Second moment of inertia
- 7 Composites and mixtures
- 7 Stiffness of anisotropic materials
- 7 Fiber and lay up mixture
1 WIND PROFILES Course 34322
1 Stable, neutral and unstable conditions
Why is an unstable profile more uniform than a stable one? In unstable conditions there is much more mixing in the vertical direction. Hereby momentum is transferred across the vertical direction. This induces an effective shear stress that acts to reduce the vertical velocity gradient.
Figure 2: Stable, neutral and unstable conditions
2 WIND RESOURCES Course 34322
2 Wind resources
Learning objectives:
- To calculate turbulence intensity and describe turbulent time scales
- Apply a Weibull distribution for mean wind speed
- To describe the principles of the WaS Pmethod
- To calculate the annual energy production (AEP) for a turbine
2 Wind distribution: Weibull
The parameters of the Weibull distribution is A and k. The value A is NOT the area of the wind turbine.
Figure 3: Theory on Weibull
Mean Weibull value Gamma value can be found with se.mathworks/help/matlab/ref/gamma.html
E(v)=AΓ
(
1 + 1 k
)
(3)
3 TECHNOLOGY OF WIND TURBINES Course 34322
3 Technology of wind turbines
3 Rotors and efficiency
Figure 5: Rotor and efficiency
3 Gear ratios
Figure 6: Gear ratios
3 TECHNOLOGY OF WIND TURBINES Course 34322
3 The planetary gear
Figure 7: Planetary gear
3 Speed and torque relations
Figure 8: Speed and torque relation
4 AERODYNAMICS Course 34322
4 Forces acting on the blade
Lift force: L′=dL= 12 ρcClVrel 2 [N/m]
Drag force: D′=dD= 12 ρcCdVrel 2 [N/m]
Thrust contribution: T′=dT=dTcos
(
φ
)
+dDsin
(
φ
)
Normal force: pn=dLcos
(
φ
)
+dDsin
(
φ
)
Tangential force: pt=dLsin
(
φ
)
+dDcos
(
φ
)
5 ECONOMY Course 34322
5 Economy
5 Table of running expenses
6 STRUCTURAL MECHANICS Course 34322
6 Formula for deflection along the blade and max deflection
Along the blade: u(x)= kx
2 120 EI
(
x 3 − 10 L 2 x+ 20 L 3
)
, k= 2 LQ 2 (15)
Maximum deflection umax= 1160 EIQL 3 , Q=F 3 (16)
6 Generic beam case: Cantilever beam with point load
The maximum deflection is given by u= 13 PloadL
3 EI (17)
With a momentum of M=−PL (18)
6 Generic beam case: Cantilever beam with distributed load
The maximum deflection is given by u= 18 ploadL
4 EI (19)
with a momentum of M=− 12 pL 2 (20)
6 Euler-Bernolli Theory and Natural frequency
Beam equation: pU ̈+(EIuxx)xx=p(x)
Natural frequency: ω=
√ 3 EI
mL 3
7 COMPOSITES AND MIXTURES Course 34322
7 Composites and mixtures
7 Stiffness of anisotropic materials
7 Fiber and lay up mixture
Introduction To Wind Energy S21 exam
Kursus: Introduktion til vindenergi (46000)
Universitet: Danmarks Tekniske Universitet
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