Terraced Water Lift

What is modelled

A centrifugal pump gives less head the more water it passes. The pipe asks for the static lift plus a friction and valve loss that grows with the square of the flow. The flow settles where the two curves meet. The water in the hilltop reservoir adds to the static head, so as the reservoir fills the flow falls, until the supply matches the town demand or the reservoir is full.

Equations

• Pump curve H = N2 H0 − a Q2, a = H0 / Qm2.
• System curve H = Hz + h + (Kf + Kv) Q2, with Kf = 8 f L / (π2 g D5) and Kv = K0 (1/o2 − 1).
• Operating flow Q = √((N2 H0 − Hz − h) / (a + Kf + Kv)).
• Power P = ρ g Q H / η; reservoir A dh/dt = Q − Qdemand.

Checks run in the page

• The operating point by bisection against the closed form.
• Lift, friction and valve power adding up to the pump hydraulic power.
• The time for the reservoir to move toward its settling level, by RK4 against a Simpson integral.
• The pump affinity exponents 1, 2 and 3 with no static lift.

Honest simplifications

The pump curve is a single parabola scaled by the affinity laws, the efficiency is a parabola around 60 % of the free-delivery flow with a peak of 78 %, the pipe is one 800 m line with a constant friction factor, the three terraces are only a picture of the static lift, and the reservoir has vertical walls with a sluice at 5 m. Use it for the trends, not for sizing a pump.

• Press Throttled by the valve and watch the flow fall and the valve loss take power.
• Press Pump too weak for the hill and see the flow drop to zero.
• Press Slow pump, high demand and watch the reservoir drain, then raise the pump speed.