Size a piping run and find where a pump will actually operate — friction and fitting losses by Darcy–Weisbach, the system resistance curve, and its crossover with the pump curve that fixes the true duty flow and head.
Vertical rise from source to destination, plus any pressure head required at the outlet.
Count of each fitting × its K factor. These dominate in short runs with many bends.
Simple parabolic pump curve H = H₀(1 − (Q/Q_max)²). Enter two points from a real curve if you have them.
The system curve (blue) rises with Q² from the static lift; the pump curve (brass) falls. Where they cross is the duty point — the flow and head you'll actually get, not the flow you asked for. The dashed line marks your design flow.
Velocity & regime. v = Q/A, Re = vD/ν. Laminar below Re 2300 (f = 64/Re), turbulent above. Aim for 1–3 m/s in liquid lines — faster wastes head to friction, slower risks settling.
Friction factor. Turbulent f from the Swamee–Jain explicit fit to Colebrook–White: f = 0.25 / [log₁₀(ε/3.7D + 5.74/Re^0.9)]².
Head loss. Major (pipe) loss h_f = f·(L/D)·v²/2g; minor (fitting) loss h_m = ΣK·v²/2g. Total system head at flow Q is H = H_static + h_f + h_m, and since both losses scale with v² ∝ Q², the system curve is a parabola rooted at the static lift.
Operating point. The pump delivers head along its own curve; the system demands head along the parabola. The intersection — solved here numerically — is the only flow where supply equals demand, so it's the flow you actually get. Sizing a pump means placing this crossing near its best-efficiency point.
Power. Hydraulic power P = ρ·g·Q·H; shaft power divides by pump efficiency.
Single pipe, steady incompressible flow. Networks, NPSH/cavitation, water hammer, and two-phase flow are separate analyses.