Water Resources HubFree calculator
Pump Operating Point — System Curve vs Pump Curve
A pump delivers the flow where its curve meets the system curve. Enter the system (static head plus friction) and the pump curve — a sample pump is pre-loaded, adjust it freely — to solve for the operating flow, head, and shaft power, in US or SI units.
Switching converts every value into the new system (it doesn't just relabel). Your choice is remembered on this device.
System curve — Hsys = Hstatic + KQ²
Pump curve — Hpump = H0 − aQ − bQ²
Affinity laws — applied to the operating point above
For study and checking only. These calculators run entirely in your browser — nothing is sent anywhere. They are meant to verify your hand calculations while you study; on the exam you will work by hand with the NCEES reference handbook.
Quick answer: the operating point and the power
The operating point solves Hsys = Hpump, i.e. (b + K)Q² + aQ + (Hstatic − H0) = 0 — take the positive root.
Water power (power delivered to the fluid): US: Pw [hp] = Q [gpm] · H [ft] / 3960 · SI: Pw [kW] = 9.79 · Q [m³/s] · H [m]. Brake (shaft) power: Pb = Pw / η. The 3960 constant already contains the water unit weight (62.4 lb/ft³) and the hp conversion (550 ft·lbf/s).
Affinity laws (same pump, speed N1 → N2): Q2/Q1 = N2/N1 · H2/H1 = (N2/N1)² · P2/P1 = (N2/N1)³.
Worked example: operating point → shaft power (US units)
At the operating point a pump delivers Q = 793.1 gpm against H = 82.0 ft at efficiency η = 0.75. Find the water power, the brake power, and a motor size. This is the calculator's default load — it reproduces every number below.
- Water power: Pw = QH/3960 = (793.1 × 82.0)/3960 = 16.43 hp
- Brake (shaft) power: Pb = Pw/η = 16.43/0.75 = 21.90 hp
- Select the next standard motor size up — 25 hp — to leave margin.
In SI (toggle above): Q = 50.0 L/s, H = 25.0 m → water power 12.25 kW, brake power 16.33 kW, next standard motor 18.5 kW.
Exam tip: the operating point is where the pump curve crosses the system curve — raise the static lift or close a valve and the duty point moves UP the pump curve to a lower flow; it never stays at the catalogue "best efficiency" flow by default.
Free formula resources
Grab the free formula resources — every FE Civil equation in one searchable index, including pump power and the affinity laws.
Water power vs brake power — the exam's favorite mix-up
Water power is the useful power transferred to the fluid: Pw = γQH. It depends only on the duty point — flow, head, and the fluid's unit weight. Brake power is what the motor must actually deliver at the shaft: Pb = Pw/η, always larger, because the pump wastes some input as heat, leakage, and recirculation.
The exam tests this two ways. First, by giving you η and asking for motor power — divide by η, don't multiply. Second, by asking for water power directly and planting the efficiency as a distractor (or vice versa: giving no η at all and asking for brake power is a trap — you can't compute it). The calculator shows both, side by side, with the motor-size step engineers actually take next.
The constants: γ = 62.4 lb/ft³ (9.79 kN/m³) for water, 1 hp = 550 ft·lbf/s. The 3960 in P[hp] = Q[gpm]·H[ft]/3960 is just γ and the unit conversions folded together: (62.4 lb/ft³ × Q/448.831 ft³/s × H ft) / 550 = QH/3960.
Affinity laws — speed changes without a new pump curve
For the same pump at a different speed, flow scales with N, head with N², and power with N³. The affinity panel above applies them to the live operating point: at N2/N1 = 1.10, the default duty point becomes Q = 872.4 gpm, H = 99.2 ft, Pb = 29.15 hp.
Exam notes: efficiency is assumed unchanged by the affinity laws (a stated idealization — the exam uses it, practice derates it). And the cubic on power is why a 10% speedup costs 33% more power — the most-tested consequence. Affinity laws also apply, approximately, to trimming an impeller diameter (replace N with D), which the PE exam occasionally tests.
Common exam traps
- The operating point is not the BEP. The pump runs where its curve crosses your system curve — not at the catalogue best-efficiency point. Efficiency η in the power formula is the efficiency at the operating point.
- Raise the static head → less flow. Steepening the system curve (larger K, or more static lift) slides the duty point up the pump curve: more head, less flow. The calculator demonstrates this live — bump Hstatic and watch Q fall.
- Efficiency goes in the denominator. Brake power = water power / η. Multiplying by η gives a smaller number — the classic sign error, and it's always an answer choice.
- 3960 vs 550. Use QH/3960 when Q is in gpm; use γQH/550 when Q is in ft³/s. Mixing the flow unit with the wrong constant is the pump-power version of the Manning 1.486 trap.
- Affinity power is cubic. A 10% speed increase is a 33% power increase (1.1³ = 1.331). Problems that ask "what happens to power" are testing whether you cube it.
Pump operating point — FAQ
What exactly is the operating point?
The single (Q, H) point where the pump curve and the system curve intersect — the only flow at which the head the pump produces equals the head the system demands. The calculator solves (b + K)Q² + aQ + (Hstatic − H0) = 0 and takes the positive root.
Water horsepower vs brake horsepower?
Water horsepower is power delivered to the fluid (QH/3960); brake horsepower is shaft input power (water hp / η). Motor sizing uses brake horsepower, rounded up to the next standard size.
Where does 3960 come from?
It's the water unit weight and unit conversions combined: P[hp] = γ·Q·H/550 with Q in gpm gives (62.4/448.831/550)·Q·H = Q·H/3960. In SI the equivalent is P[kW] = 9.79·Q[m³/s]·H[m].
Do the affinity laws change efficiency?
On the exam, no — efficiency is assumed constant along the affinity transformation. In practice there's a small penalty, but exam problems use the ideal laws: Q ∝ N, H ∝ N², P ∝ N³.
What if the curves don't intersect?
Then the pump can't do the job — typically H0 (shutoff head) is below the static lift, so even at zero flow the pump can't push water over the top. The calculator says so explicitly instead of returning a nonsense number.
Keep building exam speed
Pumps and system curves are core PE Civil WRE material — and pump power is a perennial FE question. The PE WRE Flagship ($119) drills every pump question type — soft launch, join the waitlist from the contact page. For a full rehearsal under timed conditions, the 110-question FE Civil practice exam ($79) is a 5-hour-20-minute run with detailed solutions.
Last reviewed: 2026-10-03. Formulas follow the NCEES FE Reference Handbook conventions; always confirm against the current handbook.