job number/bt/ serial number/ page number generalised energy extraction limits peter jamieson april...

Job number/BT/ serial number/ page number

Generalised Energy Extraction Limits

Peter JamiesonApril 2008


Background


Open uniform flow Constrained flow

Definitions


V0V0

V0

V0 1 a0

SYSTEM

No energy extraction


V0 1

3V0CP

16

27

CP16

271 a0

CT8

9

SYSTEM

Optimum energy extraction


Rotor Plane Induction

Far wake

V (1-2b)

Extraction planeV (1-a)

Reference planeV (1-b)

SystemFreestream

V

Conservation of mass flow )a1(VA = )b1(VAref

When no energy extraction, 0b,aa 0 )a1(VA 0 = VAref

refA = )a1(A 0

b =

0

0

a1

aa


Generalised Equations

Far wake

V (1-2b)

Extraction planeV (1-a)

Reference planeV (1-b)

SystemFreestream

V

Thrust T = Ap

and T = T2

0 CAV2

1

Hence p = T2

0 CV2

1

Everywhere TC = )b1(b4

= 2

0

0

)a1(

)a1()aa(4

b =

0

0

a1

aa


System Equations

GENERAL OPERATION OPTIMUM OPERATION

Open Flow Constrained

Flow Open Flow Constrained

Flow Axial induction at rotor plane

a a 3

1

3

a21 0

Axial induction in far wake

a2 0

0

a1

aa21

3

2

3

2

Performance coefficient, pC

2a1a4

2

0

20

a1

a1aa4

27

16 0a127

16

Thrust coefficient, TC

a1a4 2

0

0

a1

a1aa4

9

8 9

8

Pressure difference across rotor

20V)a1(a2

202

0

20 Va1

a1aa2

2

0Vp9

4 202

0 a1Vp9

4


Primary purpose of a wind turbine?

• To extract energy from natural air flow

• To extract kinetic energy from natural air flow

• To extract potential energy from natural air flow


Optimum operation with same upstream

wind speed

9 m/s 6 m/s 3 m/s

P = 1.000

P = 1.125 P = 2.000

9 m/s 6.75 m/s 3 m/s9 m/s 18 m/s 3 m/s


Optimum operation with same

speed local to turbine

9 m/s 6 m/s 3 m/s

3 m/s 6 m/s 1 m/s8 m/s 6 m/s 2.67 m/s

P = 1.00

P = 0.79 P = 0.11


Ideal Rotors and Ideal Systems

A rotor can operate with fixed geometry in variable speed to preserve optimal flow geometry.

A diffuser or a hill cannot change geometry to suit rotor loading!


Validation of the Generalised Limit TheoryIn open flow, Cp may be expressed in terms of Ct as;

ttp C11C2

1C

5.0

t2

02

S0S0tp Ca1a1a1C2

1C

In generalised flow, considering system losses, Cp may be expressed in terms of Ct as;


Comparison of New Limit Theory and CFD

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

thrust coefficient

po

wer

per

form

ance

co

effi

cien

t

CFDEquation (22)


Optimum State

Torque tracking curve

23

pm5 CRp

2

1Q

Optimum chord distribution

mL

mp21

2m

222m

tL

a1xC

a1C1a1xa1B

C2

R

cC

Optimum twist distribution

0m

m1

a1x

a1tanx


Optimum Twist Distribution

θ(x) = tan-1(1-am/ λ x (1+a`m)) - α0 a0 am

am = (1+2ao)/3 0.0 0.333-0.1 0.267

α0 6 deg -0.2 0.200

a`m 0 -0.5 0.000λ 9 -2.0 -1.000

x 0 -0.1 -0.2 -0.5 -20.2 14.3 16.2 18.0 23.1 42.00.4 4.5 5.5 6.5 9.5 23.10.6 1.0 1.7 2.4 4.5 14.30.8 -0.7 -0.2 0.3 1.9 9.51.0 -1.8 -1.3 -0.9 0.3 6.5

a0

0m

m1

a1(x

a1tanx

where 3

a21a 0

m

, 0am , 9 and 60


Optimum State with a0 ≠ 0

• Torque tracking curve (optimal mode gain) changes

• Optimum chord changes but very little

• Optimum twist changes significantly – pitch change can approximate


System Issues

• Pressure recovery

• Optimal loading

• Turbine control

V0 = 9 m/s

49V94p 2 N/m2

5.1p%3 N/m2

5

0 10p N/m2


Optimal loading

• Only an ideal system realises Ct = 8/9

• Ideal systems are variable geometry and don’t exist in reality

• If there is not full pressure recovery and wind turbines are coupled in series then they have to share proportions of 8/9

• There may be compensation in entrainment of wakes by flows that do not pass through the wind turbines (Opt Ct > 8/9 possible in a well designed diffuser)


Turbine Control

• Considering wind direction, atmospheric characteristics and terrain characteristics, it will probably be impossible to modify control deterministically.

• Can a slow controller vary below rated pitch and optimal mode gain to tune the turbine for specific operational conditions?


Conclusions

•Lanchester-Betz limit is special case for open flow

•It appears that whilst 16/27 is not a universal number in actuator ffffdisc theory, 8/9 may be

•The theory has obvious applicability to wind turbines in diffuser ffffand ducted turbines in all types of energy systems and may also ffffhave significant implications for wind farm operation

•The theory is derived analytically without any requirement to use ffffempirical information. A very useful validation is obtained ffffcomparing with CFD results.

job number/bt/ serial number/ page number generalised energy extraction limits peter jamieson april...

Documents

universal number

cfd slide

natural air flow slide

designed diffuser slide

generalised flow

optimal flow geometry

open flow

turbine p