How it works
Two related questions, one geometry. Affinity scaling predicts how a pump's Q–H–BHP characteristics move when speed or impeller diameter changes. VFD energy savings applies the same laws to the classic retrofit decision: instead of throttling a valve to reduce flow, slow the pump down so its reduced curve meets the system curve exactly at the flow you need.
1 · The three affinity laws
For the same pump running at two speeds with constant specific gravity, points that are "homologous" (on the same similarity parabola through shutoff) scale with speed \(r=N_2/N_1\) as \(Q\propto r\), \(H\propto r^{2}\), \(P\propto r^{3}\). Efficiency is assumed unchanged between homologous points — a good approximation over the normal operating band, degrading only at very low speeds.
2 · Impeller trim (cut) law — and its limits
Trimming the impeller changes the tip speed in the same proportion, so the small-trim approximation takes the same form as the speed laws. Unlike a speed change, however, trimming also alters the blade geometry relative to the casing — efficiency is not preserved. Field and handbook experience: keep the cut below 10%; beyond it efficiency falls measurably, and above a 15% trim the approximation is unreliable — re-select a smaller pump or slow it with a VFD instead.
3 · VFD savings by the head method
Model the pump as a parabola through shutoff \((0,H_0)\) and BEP \((Q_{bep},H_{bep})\), with the usual 15–20% rise to shutoff assumed stable (\(H_0=1.2\,H_{bep}\) here), and the system as static lift plus quadratic friction calibrated so the full-speed duty sits at BEP:
$$H_{p}(Q,r)=H_{0}r^{2}-aQ^{2},\quad a=\tfrac{H_{0}-H_{bep}}{Q_{bep}^{2}} \qquad H_{sys}(Q)=H_{st}+KQ^{2},\quad K=\tfrac{H_{bep}-H_{st}}{Q_{bep}^{2}}$$To reach target flow \(Q_t\) a valve forces the pump to stay on the full-speed curve \(H_{thr}=H_p(Q_t,1)\) and burns the surplus across the valve; a VFD lowers speed so the reduced curve passes through the system point \(H_{sys}(Q_t)\):
$$r_{VFD}=\sqrt{\frac{H_{st}+(a+K)\,Q_{t}^{2}}{H_{0}}} \qquad\text{then}\qquad P=\frac{\rho gQ_{t}H}{\eta} \;\;\Bigl[\text{US: } P(\text{hp})=\tfrac{Q\,H\cdot SG}{3960\,\eta}\Bigr]$$Each power is evaluated with the pump efficiency at its own duty, \(\eta(q)=\eta_{bep}(2q-q^{2})\) with \(q=Q/(r\,Q_{bep})\) — throttling runs the pump back on its own curve (off BEP), the VFD stays near BEP. The annual saving is \((P_{thr}-P_{vfd})\times\) hours \(\times\) price. This head method is the main result; the simple cubic estimate below is shown only for comparison.
4 · Static head throttles the savings
In the friction-only idealization (\(H_{st}=0\)) the duty point follows the similarity parabola, so \(r\approx Q_t/Q_{bep}\) and \(\Delta P\propto(r^{3}-r)\) — the celebrated cubic saving. Real systems have static lift: the VFD must still make \(H_{st}\) at every speed, so \(r_{VFD}>Q_t/Q_{bep}\) and the saving shrinks. The higher the static share of BEP head, the more the cubic law over-promises — the standard VFD screening rule treats a static share above roughly 60% as "savings limited", above ~70–80% as usually uneconomic. This tool always shows both numbers so you can see the gap.
5 · Slip heat — why a VFD beats slip-based speed control
Speed control by slipping (hydraulic couplings, wound-rotor resistor control, belt sheaves) dissipates the slip as heat inside the drive train. For a cubic torque load (\(P\propto x^{3}\)) the slip loss \(p_{loss}=x^{2}-x^{3}\) peaks at \(x=2/3\) speed at \(4/27\approx\) 14.8% of rated power — burned continuously at that duty. A VFD creates no slip loss: its own conversion losses are only a few percent of rated power, which is why the affinity-law saving above survives essentially intact at the meter.
Worked example
Scaling. A pump delivers 2,000 gpm at 120 ft with BHP₁ = 60 hp. Dropping to 75% speed: \(Q_2=2000\times0.75=1500\) gpm, \(H_2=120\times0.75^{2}=67.5\) ft, \(P_2=60\times0.75^{3}=25.3\) hp — power falls to 42% of reference at unchanged efficiency. (A 0.90 trim gives the same Q–H point by the cut law, but expect some efficiency loss at a 10% cut.)
VFD vs throttling. Same pump at BEP = 2,000 gpm / 120 ft / 82%, static share 30% (Hst = 36 ft, friction 84 ft at BEP), target flow 75% (1,500 gpm), 6,000 h/yr at $0.12/kWh. Throttling: the pump rides the full-speed curve to Hthr = 130.5 ft at 77% efficiency → 64.3 hp. VFD: 82% speed meets the system curve at 83.3 ft with 81% efficiency → 38.7 hp. Saving 25.6 hp ≈ 19.1 kW → ~114,400 kWh/yr → ~$13,700/yr. The friction-only cubic estimate would claim 37.2 hp — the head method correctly trims that promise because of the 36 ft of static lift.
Engineering criteria applied
- Trim limits — ≤10% cut: efficiency approximately retained; 10–15%: expect a measurable efficiency drop (amber); >15%: approximation unreliable, re-select instead.
- Static-head screening — static share >60% of BEP head: VFD savings flagged "limited" (amber); the head-method result always governs over the cubic estimate.
- Overspeed / deep turn-down — speed ratios above ~1.0–1.05 must be checked against motor and pump mechanical ratings; below ~50% speed most centrifugal pumps approach minimum-flow, recirculation and vibration limits — verify with the vendor curve.
- Slip-drive warning — retrofitting a slip coupling instead of a VFD burns up to 14.8% of rated power as heat at 2/3 speed (\(p_{loss}=x^{2}-x^{3}\)); the VFD figure here assumes direct VFD drive.