Windmill and Piston Pump Matched from Torque First Principles: Cut-In Speed, the Best Rotor–Pump Pairing, and the Calm-Month Shortfall

Matching a windmill rotor to a piston pump is a torque problem before it is a power problem: the pump's peak torque demand over a stroke sets the wind speed at which the rotor can start and keep turning. For six rotor–pump pairings against a 30 m head, this paper computes cut-in speeds from 9.13 m/s for a 3 m rotor on a 50 mm pump — a pairing that effectively never starts in a 5 m/s wind regime — down to 4.84 m/s for the best pairing, a 6 m rotor on a 75 mm pump. Weighting monthly volume by the Rayleigh fraction of time the wind exceeds cut-in, that pairing delivers 3,663.5 m³ in a normal month at a cost of 7405 dollars, the best yield per unit cost of the six. **In a calm month at 3 m/s mean wind it delivers only 213.9 m³ against a demand of 300 m³ — a shortfall of 86.1 m³, or 28.7% of demand.** SIMULATED, NOT MEASURED.