Water-Powered Incline

What is modelled

Two trolleys carrying boats hang on the two ends of one rope that passes over a drum, so gravity on the descending boat nearly pays for lifting the rising one. The turbine only supplies the force for the mass difference and for the rolling resistance, and its force falls as the rope runs faster.

Equations

• Load force FL = (M1 − M2) g sinθ + c (M1 + M2) g cosθ.
• Gate flow Q = Cd A √(2 g H), shaft power at best speed Pmax = η ρ g Q H.
• Turbine force F(v) = F0 (1 − v / v0) with F0 = 4 Pmax / v0.
• Motion m dv/dt = F(v) − FL, steady speed v = v0 (1 − FL/F0).

Checks run in the page

• A Runge-Kutta run reaches the closed-form steady speed.
• The turbine work equals the load work plus the kinetic energy.
• The turbine power peaks at half the runaway speed with the hydraulic power, and a pond drains as Torricelli predicts.
• A free trolley accelerates at g (sin a − c cos a) and coasts when tan a = c.

Honest simplifications

The turbine curve is a straight line, the drum inertia is a fixed lump, the rope mass and its sag are ignored, and the boats are treated as rigid loads. The numbers are in the range of a historical canal incline but this is not a design tool.

• Press Balanced pair and see how little water the lift needs.
• Press Heavy boat, weak flow and open the gate until the trolley starts to move.
• Press Overhauling downhill: gravity now drives the rope and the turbine acts as a brake.