Polder Windpump

What is modelled

A historic polder windpump is a slow, high-torque machine. The rotor torque falls as it speeds up while a piston pump asks for an almost constant torque, so the rotor settles where the two curves cross; below a certain wind speed the rotor cannot start.

Equations

• Wind power: P = ½ ρ A Cp(λ) v3, λ = ω R / v; model Cp = Cpk(2x - x2), x = λ/λ₀, C_pk = 0.32.
• Pump torque: Tp = ρ g H Vd/(2π η) + Tf; flow Q = Vd ω/2π.
• Dynamics: J dω/dt = Tw - Tp.
• Betz: Cp,max = 16/27.

Checks run in the page

• Betz limit by numerical scan.
• Steady rotor speed against the analytic root.
• Water power against the torque-balance prediction.
• Cut-in speed from starting torque and the linearised relaxation time J/|dT/dω|.

Honest simplifications

The Cp curve is a simple parabola chosen for a slow multi-blade rotor, not measured data; pump friction is a constant torque, gusts are a simple Ornstein-Uhlenbeck process, there is no yaw or furling and no losses in gearing. Real windpumps vary their stroke and use a tail vane.

• "Calm (stalls)": the rotor never starts because the wind torque is below the pump torque.
• Increase the pump displacement: more water per revolution but a higher cut-in wind.
• "Storm": the rotor speeds up but the pump torque limits the power it can use.