Simple Performance Prediction Methods Module 2 Momentum Theory

© L. Sankar Wind Engineering, 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, 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, Continuity Area A 1 Density  1 Velocity V 1 Area A 2 Density  2 Velocity V 2 Rate at which mass enters=  1 A 1 V 1 Rate at which mass leaves=  2 A 2 V 2

© L. Sankar Wind Engineering, Continuity In compressible flow through a “tube”  AV= constant In incompressible flow  does not change. Thus, AV = constant

© L. Sankar Wind Engineering, 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, Continuity (Continued..) 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. High Velocity Low Velocity

© L. Sankar Wind Engineering, Venturi Tube is a Device for Measuring Flow Rate we will study later. Low velocity High velocity

© L. Sankar Wind Engineering, Continuity Station 1 Density  1 Velocity V 1 Area A 1 Station 2 Density  2 Velocity V 2 Area A 2 Mass Flow Rate In = Mass Flow Rate Out  1 V 1 A 1 =  2 V 2 A 2

© L. Sankar Wind Engineering, Momentum Equation (Contd..) Density  velocity V Area =A Density  d  velocity V+dV Area =A+dA Momentum rate in= Mass flow rate times velocity =  V 2 A Momentum Rate out= Mass flow rate times velocity =  VA (V+dV) Rate of change of momentum within this element = Momentum rate out - Momentum rate in =  VA (V+dV) -  V 2 A =  VA dV

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

© L. Sankar Wind Engineering, 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, Forces acting on the Stream tube Pressure times area=pA (p+dp)(A+dA) Horizontal Force = Pressure times area of the ring=(p+dp/2)dA 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, Momentum Equation From the previous slides, Rate of change of momentum when fluid particles flow through the stream tube =  AVdV 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:  VdV+dp=0 This equation is called the Euler’s Equation

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

© L. Sankar Wind Engineering, 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, 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, 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, Control Volume V Disk area is A Total area S Station1 Station 2 Station 3 Station 4 V- v 2 V-v 3 Stream tube area is A 4 Velocity is V-v 4

© L. Sankar Wind Engineering, Conservation of Mass

© L. Sankar Wind Engineering, Conservation of Mass through the Rotor Disk Thus v 2 =v 3 =v There is no velocity jump across the rotor disk The quantity v is called velocity deficit at the rotor disk V-v 2 V-v 3

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

© L. Sankar Wind Engineering, Conservation of Momentum at the Rotor Disk V-v p2p2 p3p3 Due to conservation of mass across the Rotor disk, there is no velocity jump. Momentum inflow rate = Momentum outflow rate Thus, drag D = A(p 2 -p 3 )

© L. Sankar Wind Engineering, 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 V-v V-v 4

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

© L. Sankar Wind Engineering, Induced Velocities V V-v V-2v 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.

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

© L. Sankar Wind Engineering, 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.