Question

1. Momentum Theory
   1. Use one-dimensional conservation of momentum together with conservation of mass (continuity) and energy (Bernoulli’s equation = mechanical energy) to derive the power an ideal, frictionless wind turbine with an infinite number of blades, uniform thrust over the rotor area and a non-rotating wake can extract from the wind. Formulate the derivation in terms of the fractional decrease in wind velocity between the velocity far upstream and at the turbine rotor, ? = (? − ?)/?, also called “axial induction factor”. Use a control volume that is defined by a stream-tube that encloses the turbine rotor, with an upstream inflow plane and downstream outflow plane parallel to the rotor plane.
   2. Define a power coefficient CP that represents the fraction of power in the wind that is extracted by the wind turbine rotor and determine for what ? it has its maximum!
   3. What is the maximum value of this power coefficient CP?

1. 4. We assumed an ideal, frictionless turbine – why is the maximum power coefficient less than one?

1. 5. For a wind turbine rotor with a diameter of 120m operating under maximum power coefficient as defined by the 1D momentum theory, from what diameter (area) far upstream of the turbine (A1) is energy extracted? Far downstream of the turbine rotor, what is the diameter (area, A4) of the flow that has passed through the rotor? (hint: use 1-D continuity, combined with conditions on velocity from the 1-D momentum theory)

1. 6. Beyond what value of induction factor ? is a physical solution with 1-D momentum theory not possible? (Hint: for what values of a is the curve of CP (a) physically not possible?)

Answer 1