Introduction
HP = (GPM × TDH) ÷ 3960 ÷ efficiency looks suspiciously simple, but every term stands on real physics — and misunderstanding any one of them is how pumps end up wrongly sized. This guide takes the formula apart: what total dynamic head contains, where 3960 comes from, why efficiency sits in the denominator, and how the same calculation works in metric units.
If you just want the answer, the Pump Size Calculator runs all of this for you — enter flow and head, get horsepower. What follows is for everyone who wants to understand the number before trusting it, and to sanity-check results by hand.
Part 1: Total Dynamic Head (TDH)
Head is the pump industry’s universal currency for resistance: instead of juggling pressure, elevation, and friction in different units, everything gets expressed as an equivalent height of a water column. TDH is the sum of three such heights:
Static head
The vertical distance from the source water surface to the highest point the water must reach. For a submersible, measure from the pumping water level (drawdown level, not resting) to the pressure tank; for a surface pump, it’s suction lift plus discharge elevation. Horizontal distance adds zero static head — but it does add friction, which is next.
Friction loss
Water dragging against pipe walls and fittings consumes energy, expressed as feet (or meters) of head lost. Friction loss depends on five things: flow rate, pipe length, pipe diameter, pipe roughness, and fittings. The standard engineering relation is the Hazen-Williams equation:
hf = 4.73 × L × Q1.85 ÷ (C1.85 × d4.87)
where L is pipe length in feet, Q is flow in GPM, d is inside diameter in inches, and C is the roughness coefficient (150 for smooth PVC, ~140 for copper and new steel, 100–120 for aged steel — see the material guide for a full table).
Worked example: 100 ft of 1-in Sch 40 PVC at 10 GPM, C = 150. hf = 4.73 × 100 × 70.8 ÷ 10,608 ≈ 3.2 ft per 100 ft of straight pipe (fittings add equivalent length — a 1-in 90° elbow ≈ 2.7 extra feet). In practice, most people use published tables rather than hand-computing. Typical Sch 40 PVC values:
| Pipe size | Flow | Friction loss per 100 ft | Per 100 m |
|---|---|---|---|
| 3/4 in (20 mm) | 8 GPM (30 LPM) | ~8.6 ft | ~8.6 m |
| 1 in (25 mm) | 10 GPM (38 LPM) | ~3.1 ft | ~3.1 m |
| 1-1/4 in (32 mm) | 15 GPM (57 LPM) | ~2.5 ft | ~2.5 m |
| 1-1/2 in (40 mm) | 25 GPM (95 LPM) | ~3.4 ft | ~3.4 m |
| 2 in (50 mm) | 40 GPM (150 LPM) | ~2.9 ft | ~2.9 m |
Note how brutally diameter matters: the d4.87 term means upsizing one pipe size can cut friction loss by more than half — often the cheapest “head” you’ll ever buy. Undersized pipe is the silent budget-killer in pump systems, and it’s mistake #1 in the common mistakes guide.
Pressure head
System pressure converted to head: head (ft) = pressure (psi) × 2.31, or head (m) = pressure (bar) × 10.2. A household pressure switch set at 40/60 psi means the pump works against 40 psi (92 ft / 28 m) at cut-in and 60 psi (139 ft / 42 m) at cut-out — size for the cut-out pressure, the harder case. For irrigation, convert the sprinkler head’s required operating pressure the same way.
Assembling TDH
TDH = Static head + Friction loss + Pressure head, all in consistent units. Example: 60 ft static + 25 ft friction (pipe, fittings, valves) + 139 ft pressure (60 psi cut-out) = 224 ft TDH.
Part 2: From TDH to Horsepower
Where 3960 comes from
Water weighs 62.4 lb/ft³, 1 HP = 550 ft-lb/s, and 1 ft³/s = 448.831 GPM. So the power actually imparted to the water is:
Water HP = 62.4 × (Q ÷ 448.831) × H ÷ 550 = (Q × H) ÷ 3956.5
Rounded to the industry-standard 3960. That gives the hydraulic power; the motor must supply more, because no pump is 100% efficient:
Shaft HP = (GPM × TDH in feet) ÷ 3960 ÷ Efficiency
The metric version
Same physics in SI units: power = mass flow × gravity × head ÷ efficiency. With flow in L/min and head in meters:
Power (watts) = (L/min ÷ 60) × H × 9.81 ÷ Efficiency
(L/min ÷ 60 gives liters per second, i.e., kg/s of water; × 9.81 m/s² × meters of head gives watts.) A handy cross-check: 1 HP = 746 W.
Part 3: Worked Examples, Both Unit Systems
Example A — small system
10 GPM, 100 ft TDH, 60% efficiency. HP = 10 × 100 ÷ 3960 ÷ 0.60 = 1000 ÷ 3960 = 0.2525 ÷ 0.60 = 0.42 HP → round up to a 1/2 HP pump.
Metric: 38 L/min, 30 m TDH, 60% efficiency. (38 ÷ 60) × 30 × 9.81 ÷ 0.60 = 0.6333 × 30 × 9.81 = 186.4 ÷ 0.60 = 310 W. Check: 0.42 × 746 = 313 W. The two systems agree within rounding — always a good sign.
Example B — household well
12 GPM, 224 ft TDH (the 60 + 25 + 139 ft assembly above), 60% efficiency. HP = 12 × 224 ÷ 3960 ÷ 0.60 = 2688 ÷ 3960 = 0.679 ÷ 0.60 = 1.13 HP → round up to a 1.5 HP pump.
Metric: 45 L/min × 68 m: (45 ÷ 60) × 68 × 9.81 ÷ 0.60 = 834 W ≈ 1.13 HP. Consistent.
Example C — why efficiency assumptions matter
Take Example A at 80% efficiency instead of 60%: 0.2525 ÷ 0.80 = 0.32 HP — still a 1/2 HP pump, but thinner margin. On a bigger system (20 GPM × 200 ft), 60% → 1.68 HP (2 HP pump) while 80% → 1.26 HP (1.5 HP pump): the efficiency guess changed the pump size. Default to 60% for small residential pumps unless the published curve says otherwise — guessing high undersizes the system.
Part 4: What the Formula Doesn’t Cover
The formula sizes the duty point; these realities sit outside it:
- Suction limits. Surface pumps can’t practically lift more than ~25 ft (7.6 m) — deeper suction needs a submersible, regardless of the HP math.
- Variable demand. The formula sizes for peak flow. If demand swings widely, a variable-speed drive or a properly sized pressure tank matters more than extra horsepower.
- Water quality. Sand, iron, or corrosive water changes pump and material selection, not horsepower (see the material guide).
- Electrical supply. Large pumps may need three-phase power; size wire for voltage drop, especially on deep wells.
For the field procedure that produces the inputs — measuring static head, tallying friction, estimating peak flow — see the estimating guide. For a fully worked household scenario, the case study applies every formula here end to end.
Related Guides
- Pump Size Complete Guide — the full overview: terms, pump types, curves, and sizing workflow.
- Estimating Guide — measuring flow demand and each TDH component on site.
- Common Mistakes — the errors that most often corrupt the inputs to these formulas.
- Pump Size FAQ — quick answers on head, flow, and horsepower.
- Case Study — a hypothetical household well sized start to finish.
- Sizing Checklist — verify every input before you buy.
Frequently Asked Questions
What does TDH stand for?
Total Dynamic Head — the sum of static head (vertical lift), friction loss (pipes and fittings), and pressure head (system pressure as water column). It’s the “resistance” number in the horsepower formula.
Where does the 3960 in the pump formula come from?
A unit-conversion constant: 448.831 (GPM per ft³/s) × 550 (ft-lb/s per HP) ÷ 62.4 (lb/ft³) ≈ 3956.5, rounded to 3960.
How do I convert psi to feet of head?
Multiply psi by 2.31. So 40 psi = 92.4 ft, 60 psi = 138.6 ft. In metric, multiply bar by 10.2 to get meters of head.
Why do metric and imperial give slightly different answers?
They shouldn’t differ beyond rounding — 0.42 HP vs 310 W in Example A is the same answer (0.42 × 746 = 313 W). If your two calculations disagree by more than a percent or two, check unit conversions, especially GPM↔L/min (×3.785) and feet↔meters (×0.3048).
Get Your Number
Assemble TDH, multiply by flow, divide by 3960 and efficiency, round up to a standard size. Every term is hand-checkable, and metric must agree with imperial — use that cross-check. Run your numbers through the Pump Size Calculator and verify the duty point on the manufacturer’s curve.
Disclaimer: These formulas produce planning estimates for pump sizing. Real systems involve drawdown, water quality, voltage drop, and local code requirements that formulas alone don’t capture. Verify your inputs on site and consult a licensed well contractor or pump professional before purchasing or installing equipment.
