Introduction To Wind Energy S21 exam PDF

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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...

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TABLE OF CONTENTS

Course 34322

Table of contents 1

2

3

4

5

Wind Profiles 1.1

Velocity profile for mean wind speed and power law . . . . . . . . . . . . . . . . . .

3

1.2

Stable, neutral and unstable conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Wind resources

5

2.1

Wind distribution: Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Annual Energy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.3

Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Technology of wind turbines

7

3.1

Rotors and efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.2

Gear ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.3

The planetary gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3.4

Speed and torque relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Aerodynamics

9

4.1

1D momentum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2

Theory on rotating wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.3

Forces acting on the blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Economy 5.1

6

3

11

Table of running expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Structural mechanics 6.1

11 12

Second moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

6.1.1

Square box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

6.1.2

Cross section for a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

6.1.3

Approximated circular cross section . . . . . . . . . . . . . . . . . . . . . . . .

12

6.1.4

Sandwich beam blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

6.1.5

Parallel axis theorem for second moment of inertia . . . . . . . . . . . . . . .

12

6.2

The effective modulus of elasticity, Ec . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

6.3

Strain in the upper most part of a beam . . . . . . . . . . . . . . . . . . . . . . . . . .

12

6.4

Formula for deflection along the blade and max deflection . . . . . . . . . . . . . . .

13

6.5

Generic beam case: Cantilever beam with point load . . . . . . . . . . . . . . . . . . .

13

1

7

6.6

Generic beam case: Cantilever beam with distributed load . . . . . . . . . . . . . . .

13

6.7

Euler-Bernolli Theory and Natural frequency . . . . . . . . . . . . . . . . . . . . . . .

13

Composites and mixtures

14

7.1

Stiffness of anisotropic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

7.2

Fiber and lay up mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1

1

WIND PROFILES

Course 34322

Wind Profiles

Learning objectives: • To apply basic Engineering models for the wind variation in space and time • To understand and apply the concept of roughness • To understand the difference between stable neutral and unstable conditions

1.1

Velocity profile for mean wind speed and power law

The mean velocity profile as a function of height z are given as in equation 1 and power law in 2, but it is normally plotted as in figure 1a with the height in y and speed in x. ! u∗ z −ψ U (z) = ln κ z0 U (z) = U0 (z∗ )

 α z z∗

(2)

The parameters are as follows: • κ = 0.4 (von Karman constant) • z0 , roughness lenth • u∗ , friction velovity (u∗ )2 = T/ρ • ψ, stability function. ψ > 0 : unstable, ψ = 0 : neutral, ψ < 0 : stable

(a) Wind speed profile (b) Roughness

3

(1)

1

1.2

WIND PROFILES

Course 34322

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

4

2

2

WIND RESOURCES

Course 34322

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

2.1

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 https://se.mathworks.com/help/matlab/ref/gamma.html ! 1 E (v) = AΓ 1 + k

5

(3)

2

2.2

WIND RESOURCES

Course 34322

Annual Energy Production

AEP in e.g. kWh where p (v) is the Weibull density function. Z Z 1 AEP = Power (v) × prob (v) dv = ρair Cp (v) Aturbine v3 × p (v) dv × 24 × 365 2 P (v) = Cp (v) =

2.3

1 ρCp Aturbine V 3 2

(4) (5)

P

(6)

1 ρAturbine V 3 2

Turbulence

Figure 4: Turbulence

6

3

3 3.1

TECHNOLOGY OF WIND TURBINES

Course 34322

Technology of wind turbines Rotors and efficiency

Figure 5: Rotor and efficiency

3.2

Gear ratios

Figure 6: Gear ratios

7

3

3.3

TECHNOLOGY OF WIND TURBINES

Course 34322

The planetary gear

Figure 7: Planetary gear

3.4

Speed and torque relations

Figure 8: Speed and torque relation

8

4

4 4.1

AERODYNAMICS

Course 34322

Aerodynamics 1D momentum theory

Axial induction: u = (1 − a ) V0 , a = 1/3 under optimal induction Thrust coefficient: CT = 4a (1 − a) Thrust force: T = 21ρAturbine CT V02 Power coefficient: Cp = 4a (1 − a )2 , with Betz limit 0.59 Power: P = 21 ρair Arotor CP V03

4.2

Theory on rotating wake

Tip speed ratio: λ =

ωR V0

Speed ratio along blade: x = Vωr 0

  Equation to find a: 16a 3 + 24a 2 + a 9 − 3x2 − 1 + x2 = 0

− 3a Rotational induction factor: a ′ = 41a− 1   1−a Flow angle: tan φ = (1+a ′ )x

Velocity triangle:

2 = (1 − a )2 V 2 + (1 + a )2 ω2 r2 Relative velocity: Vrel 0   R−r Prandtl’s tip loss factor: F = π2 arccos e− f , f = 2B rsin (φ) 2

Chords length: c =

8πrasin(φ) F i h B(1−a) Cl cos(φ)+Cd sin(φ)

9

4

4.3

AERODYNAMICS

Course 34322

Forces acting on the blade

2 [N /m] Lift force: L′ = dL = 21ρcCl Vrel 2 [N/m] Drag force: D′ = dD = 21ρcCd Vrel     Thrust contribution: T ′ = dT = dTcos φ + dDsin φ     Normal force: pn = dLcos φ + dDsin φ     Tangential force: pt = dLsin φ + dDcos φ

10

5

5 5.1

ECONOMY

Course 34322

Economy Table of running expenses

11

6

STRUCTURAL MECHANICS

6 6.1 6.1.1

Course 34322

Structural mechanics Second moment of inertia Square box

An applied example on how to calculate the moment of inertia can be seen in part 3.1 2017. 1 3 bh 12

(7)

 π 4 R2 − R14 4

(8)

π 3 tD 8

(9)

1 A f lapwise,side h2 2

(10)

I= 6.1.2

Cross section for a circle I=

6.1.3

Approximated circular cross section I=

6.1.4

Sandwich beam blade I=

6.1.5

Parallel axis theorem for second moment of inertia It = Inormal + Ar2

6.2

The effective modulus of elasticity, Ec   Ec = η0 E f V f + Em Vm = η0 E f V f + Em 1 − V f

6.3

(11)

(12)

Strain in the upper most part of a beam

Hooks law for a beam: σ = ǫEc

(13)

The strain is then: σ=

(h + t) M z·M M z·M = · →ǫ=z → ǫ · Ec = E · I Ec · I 2 I I c

12

(14)

6

6.4

STRUCTURAL MECHANICS

Course 34322

Formula for deflection along the blade and max deflection

Along the blade: u (x) =

 kx2  3 2Q x − 10L2 x + 20L3 , k = 2 120EI L

(15)

F 11 Q 3 L , Q= 3 60 EI

(16)

Maximum deflection umax =

6.5

Generic beam case: Cantilever beam with point load

The maximum deflection is given by u=

1 Pload L3 3 EI

(17)

With a momentum of M = −PL

6.6

(18)

Generic beam case: Cantilever beam with distributed load

The maximum deflection is given by 1 pload L4 8 EI

(19)

1 M = − pL2 2

(20)

u= with a momentum of

6.7

Euler-Bernolli Theory and Natural frequency

Beam equation: pU¨ + (EIuxx )xx = p (x) q 3EI Natural frequency: ω = mL 3

13

7

7

COMPOSITES AND MIXTURES

Course 34322

Composites and mixtures

7.1

Stiffness of anisotropic materials

7.2

Fiber and lay up mixture

14

7

COMPOSITES AND MIXTURES

Course 34322

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