Simple Performance Prediction Methods

1. Simple Performance Prediction Methods Module 2 Momentum Theory

2. Overview • In this module, we will study the simplest representation of the wind turbine as a disk across which mass is conserved, momentum and energy are lost. • Towards this study, we will first develop some basic 1-D equations of motion. • Streamlines • Conservation of mass • Conservation of momentum • Conservation of energy © L. Sankar Wind Engineering, 2009

3. Continuity • Consider a stream tube, i.e. a collection of streamlines that form a tube-like shape. • Within this tube mass can not be created or destroyed. • The mass that enters the stream tube from the left (e.g. at the rate of 1 kg/sec) must leave on the right at the same rate (1 kg/sec). © L. Sankar Wind Engineering, 2009

4. Continuity Rate at which mass leaves=r2A2V2 Rate at which mass enters=r1A1V1 Area A1 Density r1 Velocity V1 Area A2 Density r2 Velocity V2 © L. Sankar Wind Engineering, 2009

5. Continuity In compressible flow through a “tube” rAV= constant In incompressible flow, r does not change. Thus, AV = constant © L. Sankar Wind Engineering, 2009

6. Continuity (Continued..) AV = constant If Area between streamlines is high, the velocity is low and vice versa. High Velocity Low Velocity © L. Sankar Wind Engineering, 2009

7. Continuity (Continued..) High Velocity AV = constant If Area between streamlines is high, the velocity is low and vice versa. In regions where the streamlines squeeze together, velocity is high. Low Velocity © L. Sankar Wind Engineering, 2009

8. Venturi Tube is a DeviceforMeasuring Flow Ratewe will study later. High velocity Low velocity © L. Sankar Wind Engineering, 2009

9. Continuity Mass Flow Rate In = Mass Flow Rate Out r1 V1 A1 =r2 V2 A2 Station 1 Density r1 Velocity V1 Area A1 Station 2 Density r2 Velocity V2 Area A2 © L. Sankar Wind Engineering, 2009

10. Momentum Equation (Contd..) Density r velocity V Area =A Density r+dr velocity V+dV Area =A+dA Momentum Rate out= Mass flow rate times velocity = r VA (V+dV) Momentum rate in= Mass flow rate times velocity = rV2A Rate of change of momentum within this element = Momentum rate out - Momentum rate in = r VA (V+dV) - rV2A = r VA dV © L. Sankar Wind Engineering, 2009

11. Momentum Equation (Contd..) Density r velocity V Area =A Density r+dr velocity V+dV Area =A+dA Rate of change of momentum as fluid particles flow through this element= r VA dV By Newton’s law, this momentum change must be caused by forces acting on this stream tube. © L. Sankar Wind Engineering, 2009

12. Forces acting on the Control Volume • Surface Forces • Pressure forces which act normal to the surface • Viscous forces which may act normal and tangential to control volume surfaces • Body forces • These affect every particle within the control volume. • E.g. gravity, electrical and magnetic forces • Body forces are neglected in our work, but these may be significant in hydraulic applications (e.g. water turbines) © L. Sankar Wind Engineering, 2009

13. Forces acting on the Stream tube Horizontal Force = Pressure times area of the ring=(p+dp/2)dA (p+dp)(A+dA) Pressure times area=pA Area of this ring = dA Net force = pA + (p+dp/2)dA-(p+dp)(A+dA)=- Adp - dp • dA/2-Adp Product of two small numbers © L. Sankar Wind Engineering, 2009

14. Momentum Equation From the previous slides, Rate of change of momentum when fluid particles flow through the stream tube = rAVdV Forces acting on the stream tube = -Adp We have neglected all other forces - viscous, gravity, electrical and magnetic forces. Equating the two factors, we get: rVdV+dp=0 This equation is called the Euler’s Equation © L. Sankar Wind Engineering, 2009

15. Bernoulli’s Equation Euler equation: rVdV + dp = 0 For incompressible flows, this equation may be integrated: Bernoulli’s Equation Kinetic Energy + Pressure Energy = Constant © L. Sankar Wind Engineering, 2009

16. Actuator Disk Theory: Background • Developed for marine propellers by Rankine (1865), Froude (1885). • Used in propellers by Betz (1920) • This theory can give a first order estimate of HAWT performance, and the maximum power that can be extracted from a given wind turbine at a given wind speed. • This theory may also be used with minor changes for helicopter rotors, propellers, etc. © L. Sankar Wind Engineering, 2009

17. Assumptions • Momentum theory concerns itself with the global balance of mass, momentum, and energy. • It does not concern itself with details of the flow around the blades. • It gives a good representation of what is happening far away from the rotor. • This theory makes a number of simplifying assumptions. © L. Sankar Wind Engineering, 2009

18. Assumptions (Continued) • Rotor is modeled as an actuator disk which adds momentum and energy to the flow. • Flow is incompressible. • Flow is steady, inviscid, irrotational. • Flow is one-dimensional, and uniform through the rotor disk, and in the far wake. • There is no swirl in the wake. © L. Sankar Wind Engineering, 2009

19. Control Volume V Station1 Disk area is A Station 2 V- v2 Station 3 V-v3 Stream tube area is A4 Velocity is V-v4 Station 4 Total area S © L. Sankar Wind Engineering, 2009

20. Conservation of Mass © L. Sankar Wind Engineering, 2009

21. Conservation of Mass through the Rotor Disk V-v2 V-v3 Thus v2=v3=v There is no velocity jump across the rotor disk The quantity v is called velocity deficit at the rotor disk © L. Sankar Wind Engineering, 2009

22. Global Conservation of Momentum Mass flow rate through the rotor disk times velocity loss between stations 1 and 4 © L. Sankar Wind Engineering, 2009

23. Conservation of Momentum at the Rotor Disk Due to conservation of mass across the Rotor disk, there is no velocity jump. Momentum inflow rate = Momentum outflow rate Thus, drag D = A(p2-p3) V-v p2 p3 V-v © L. Sankar Wind Engineering, 2009

24. Conservation of Energy Consider a particle that traverses from station 1 to station 4 We can apply Bernoulli equation between Stations 1 and 2, and between stations 3 and 4. Not between 2 and 3, since energy is being removed by body forces. Recall assumptions that the flow is steady, irrotational, inviscid. 1 V-v 2 3 4 V-v4 © L. Sankar Wind Engineering, 2009

25. From an earlier slide, drag equals mass flow rate through the rotor disk times velocity deficit between stations 1 and 4 Thus, v = v4/2 © L. Sankar Wind Engineering, 2009

26. Induced Velocities V The velocity deficit in the Far wake is twice the deficit Velocity at the rotor disk. To accommodate this excess Velocity, the stream tube has to expand. V-v V-2v © L. Sankar Wind Engineering, 2009

27. Power Produced by the Rotor © L. Sankar Wind Engineering, 2009

28. Summary • According to momentum theory, the velocity deficit in the far wake is twice the velocity deficit at the rotor disk. • Momentum theory gives an expression for velocity deficit at the rotor disk. • It also gives an expression for maximum power produced by a rotor of specified dimensions. • Actual power produced will be lower, because momentum theory neglected many sources of losses- viscous effects, tip losses, swirl, non-uniform flows, etc. • We will add these later. © L. Sankar Wind Engineering, 2009