How it works
A pumping system and a pump meet at one point. The system demands head that grows with the square of flow (friction), while the pump delivers head that falls with flow. Their intersection is the only flow rate the pump can actually deliver through that system.
1 · System curve
where \(H_{st}\) gathers static elevation difference and any pressure-head difference between receiving and source levels, and \(K\) is a single friction coefficient calibrated from one known duty: \(K = H_{f,D}/Q_D^{\,2}\). Because pipe friction follows the Darcy–Weisbach law \(h_f = f(L/D)(V^2/2g)\), every friction term scales with \(Q^2\) at fixed resistance.
2 · Pump curve and affinity scaling
A two-parameter parabola through shutoff \((0,H_0)\) and BEP \((Q_{bep},H_{bep})\) — the same normalized curve-shape logic as the classic ns-based characteristic family. Changing speed \(r=N/N_{rated}\) follows the affinity laws, and the parabola stays a parabola:
$$H_{p}(Q,r) \;=\; H_{0}\,r^{2} \;-\; a\,Q^{2}$$3 · Operating point (closed form)
Valid when \(H_0r^2 > H_{st}\); otherwise the pump cannot overcome static head and flow is zero.
4 · Efficiency and power
a parabola peaking exactly at BEP (zero at zero flow, falling off at runout — the classic normalized shape). Shaft power then follows the standard hydraulic identity:
$$P \;=\; \frac{\rho\,g\,Q H}{\eta}\;\;\text{SI},\qquad P(\text{hp}) \;=\; \frac{Q(\text{gpm})\,H(\text{ft})\cdot SG}{3960\,\eta}\;\;\text{US}$$5 · Throttling vs variable speed
To reduce flow to \(Q_t\): a throttle valve forces the pump to keep riding the full-speed curve — you pay for head \(H_p(Q_t)\) but the system only needs \(H_{sys}(Q_t)\); the difference is burned across the valve. A VFD lowers speed so the reduced curve meets the system curve exactly at \(Q_t\):
$$r_{VFD} \;=\; \sqrt{\frac{H_{st}+(a+K)\,Q_t^{2}}{H_{0}}}$$and power follows the cubic law. The savings figure uses your operating hours and electricity price. Note the static head fraction: the higher \(H_{st}\) relative to friction, the smaller the savings — the classic VFD screening rule.
Worked example
A classic handbook case: 1,000 gpm of specific gravity 0.8 pumped through 8-in suction / 6-in discharge piping with 3 ft + 25 ft of friction, against 50 ft of static lift plus a 100 psi receiving pressure — total head TH = 372 ft (both the gauge-difference method and the arbitrary-point method agree). Enter Hst=60 ft (50 lift + 10 head from pressure difference... equivalently use pressure head), friction 28 ft at 1,000 gpm to reproduce the same intersection logic on this canvas.
Engineering criteria applied
- 70–110% of BEP flow — preferred operating window (green); 50–120% cautionary (amber), beyond that red (recirculation, radial load, cavitation risk).
- Rise to shutoff ≥ 15–20% — the "continuously rising to shutoff" stability rule; flat or drooping curves risk hunting and unstable parallel operation.
- Affinity-law savings — throttling losses scale roughly with the head difference; VFD conversion saves according to the cubic law, damped by static head share.