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Job number/BT/ serial number/ page number Generalised Energy Extraction Limits Peter Jamieson April 2008

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Job number/BT/ serial number/ page number Background

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Job number/BT/ serial number/ page number Open uniform flowConstrained flow Definitions

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Job number/BT/ serial number/ page number No energy extraction

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Job number/BT/ serial number/ page number Optimum energy extraction

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Job number/BT/ serial number/ page number Rotor Plane Induction

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Job number/BT/ serial number/ page number Generalised Equations

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Job number/BT/ serial number/ page number System Equations

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Job number/BT/ serial number/ page number 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

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Job number/BT/ serial number/ page number Optimum operation with same upstream wind speed P = P = 1.125P = 2.000

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Job number/BT/ serial number/ page number Optimum operation with same speed local to turbine P = 1.00 P = 0.79P = 0.11

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Job number/BT/ serial number/ page number 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!

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Job number/BT/ serial number/ page number Validation of the Generalised Limit Theory In open flow, Cp may be expressed in terms of Ct as; In generalised flow, considering system losses, Cp may be expressed in terms of Ct as;

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Job number/BT/ serial number/ page number Comparison of New Limit Theory and CFD

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Job number/BT/ serial number/ page number Optimum State

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Job number/BT/ serial number/ page number Optimum Twist Distribution

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Job number/BT/ serial number/ page number Optimum State with a 0 ≠ 0 Torque tracking curve (optimal mode gain) changes Optimum chord changes but very little Optimum twist changes significantly – pitch change can approximate

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Job number/BT/ serial number/ page number System Issues Pressure recovery Optimal loading Turbine control

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Job number/BT/ serial number/ page number 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)

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Job number/BT/ serial number/ page number 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?

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Job number/BT/ serial number/ page number 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.

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