What the Hazen-Williams equation does
Hazen-Williams predicts how much pressure water loses to friction as it travels down a pipe. Allen Hazen and Gardner Williams published it in 1905 from measurements on water mains, and it survived because it does one thing extremely well: it gives an accurate friction loss for cool water in ordinary pipe using a single roughness number, with no viscosity, no Reynolds number and no iteration.
The whole character of the equation lives in its exponents. Flow enters to the power 1.852, so pushing 20% more water through a fixed pipe raises the loss by 1.21.852 − 1 = 41%. Diameter enters to the power −4.87, so going from a 4-inch to a 6-inch bore at fixed flow divides the loss by (6/4)4.87 = 7.2. That second exponent is why one pipe size up is so often the cheapest fix for a pressure complaint.
The coefficient C runs the other way from the roughness in Darcy-Weisbach: a high C means a smooth pipe. Plastic and copper are taken as 150 when new, cement-lined ductile iron as 140, new steel as 120, and old unlined cast iron anywhere from 100 down to 60 once tuberculation sets in.
Solving in three directions
The equation has four quantities — loss, flow, diameter and C — and you normally know three. This calculator inverts it for whichever one you are missing.
Head loss is the everyday case. You have a flow and a pipe size and you want to know what it costs in pressure. Enter both and read the loss per 100 feet.
Flow is the capacity question. You have an existing pipe and a known available pressure — say a hydrant residual or the difference between street pressure and the minimum a fixture needs — and you want to know how much water the pipe will actually pass. Because flow is recovered by raising the loss to the power 1/1.852 = 0.54, the answer is far less sensitive to an error in available head than the forward calculation is to an error in flow.
Diameter is the design question, and it is the one worth being careful with. The bore comes back as a continuous number — 4.63 in, say — which is not a pipe you can buy. Always round up to the next real inside diameter and then re-run the calculation in head-loss mode with that actual bore, because the exponent 4.87 means the difference between 4.63 and 6.065 in is not a rounding detail.
Whichever direction you solve in, the velocity output is the check you should read next. Hazen-Williams will happily hand back a bore that carries the flow at 25 ft/s within your pressure budget. That pipe will erode and scream. The pipe water velocity calculator lists the limits that actually govern.
Worked example: 500 gpm through 1,000 ft of 6-inch cast iron
A 6-inch unlined cast-iron main carries 500 gpm over 1,000 ft. Take C = 100 and a bore of 6.000 in.
- Raise the flow. Q1.852 = 5001.852 = 99,653.
- Raise the coefficient. C1.852 = 1001.852 = 5,058.3.
- Raise the diameter. d4.87 = 64.87 = 6,160.2.
- Assemble. p = 4.52 × 99,653 ÷ (5,058.3 × 6,160.2) = 450,434 ÷ 31,160,000 = 0.014456 psi per foot.
- Per 100 feet. 0.014456 × 100 = 1.446 psi per 100 ft.
- Over the whole run. 0.014456 × 1,000 = 14.456 psi.
- In feet of water. 14.456 × 2.309 = 33.38 ft.
- Velocity. V = 0.4085 × 500 ÷ 6² = 204.25 ÷ 36 = 5.67 ft/s.
Both numbers are healthy: 1.45 psi per 100 ft is a normal transmission loss and 5.7 ft/s sits inside the band the equation was fitted for. Now relay that main in cement-lined ductile iron with C = 140. The loss falls by the factor (100/140)1.852 = 0.536, to 0.775 psi per 100 ft — a 46% reduction from the lining alone, with no change in diameter.
Hazen-Williams C values for common pipe
| Pipe | C (new) | C (aged, typical design) | Loss vs C = 100, at new C |
|---|---|---|---|
| PVC, CPVC, PE, PEX | 150 | 150 | 0.472 |
| Copper tube | 150 | 140 | 0.472 |
| Cement-lined ductile iron | 140 | 130 | 0.536 |
| Asbestos cement | 140 | 130 | 0.536 |
| Steel, new welded | 120 | 100 | 0.716 |
| Cast iron, unlined | 130 | 100 | 0.616 |
| Galvanised steel | 120 | 100 | 0.716 |
| Cast iron, 20+ years unlined | — | 80 | — |
| Severely tuberculated iron | — | 60 | — |
The multiplier column is (100/C)^1.852 evaluated for each new-pipe C, so C = 150 gives (0.6667)^1.852 = 0.472 — a pipe that loses less than half what C = 100 loses at the same flow and bore.
The C you design with is not the C you will operate with
The C factor you design with is a snapshot, and Hazen-Williams gives no hint that it will not hold. Every material in the table above has two C values for a reason: what it measures new, and what a designer should assume once it has been in service. The equation itself does not distinguish; it takes whatever C you give it and returns a loss, and a pipe sized exactly to a pressure budget at the new-pipe C has no margin left for the drift that follows.
The direction of that drift is one-way. Loss varies as 1/C1.852, so a lower C always means a higher loss for the same flow and bore — there is no regime in this equation where an aging pipe's friction loss goes down. Using the table's own figures for unlined cast iron, a new C of 130 loses 61.6% of what C = 100 loses at the same flow and diameter; by the time the pipe has aged to that design C of 100, the same flow costs about 1.62 times the friction it cost when new (1 ÷ 0.616). A branch sized tight against its pressure budget at C = 130 can run short of pressure years before anything else about the installation changes.
This is why a design C and a verified C are different questions. Sizing new work from a table value is normal practice, but a system already showing pressure complaints should have its C measured rather than assumed — the two-hydrant flow test described elsewhere on this page turns the same equation around to solve for C directly from a real pressure drop, and that measured value is the one to trust over any table entry.
Why fire protection uses this equation and not Darcy-Weisbach
NFPA 13 specifies Hazen-Williams for sprinkler system hydraulic calculations, and every sprinkler design program implements the 4.52 form used here. If you compute a branch line with Darcy-Weisbach your number will be close but it will not match the plan reviewer's, and a hydraulic calculation sheet that cannot be reproduced is a rejected submittal. Use the C values the standard assigns to each pipe type — the standard fixes them, so this is not a place to exercise judgement.
The opposite is true in process and mechanical work, where Crane Technical Paper 410 and the ASHRAE handbooks use Darcy-Weisbach. Follow the governing document rather than personal preference.
Where Hazen-Williams goes wrong
- Anything but water. The equation contains no viscosity term, so it has no way to represent glycol, oil or even hot water correctly. For those, use Darcy-Weisbach.
- Very low velocities. Below roughly 3 ft/s the fit drifts, and in laminar flow the equation is simply wrong — it predicts loss going as Q^1.852 where laminar loss goes as Q^1.0.
- Small bores. The original data was taken on water mains. The equation is used routinely on half-inch branches because the tables exist, but the answer there is only as good as your choice of C.
- Nominal instead of actual diameter. With diameter to the power 4.87, using 1.000 in for a nominal 1-inch pipe whose bore is 1.049 in overstates the loss by (1.049/1.000)^4.87 = 27%.
- Straight pipe only. Fittings and valves are not in the equation. Add their equivalent length to the length input, or your calculated loss will be optimistic on a fitting-heavy run.
- Confusing pressure loss with pressure available. Friction is only one term. Elevation costs 0.433 psi per foot of lift, which the psi to feet of head converter handles, and the meter and any treatment devices take their own share.
Where this fits in a sizing job
Friction loss is never the whole calculation. A real pipe-sizing problem starts with a pressure budget: static pressure at the source, minus the pressure the fixture or sprinkler needs, minus elevation, minus meter and device losses. Whatever is left is what you may spend on friction, and Hazen-Williams tells you which bore fits inside it. The water service pipe size calculator runs that whole budget and applies this equation segment by segment.
On the demand side, you need a flow before you can size anything. In a building that flow comes from a fixture-unit count converted through the Hunter curve — the water supply fixture unit calculator does that conversion. In a fire sprinkler system it comes from the density-area curve in NFPA 13.
For gravity drainage none of this applies, because there is no pressure to spend: flow is driven by slope and the pipe runs partly full. That is Manning's territory, handled by the Manning's equation pipe flow calculator. And if you simply need to know how much water a length of pipe holds — for flushing, chlorination or a hydrostatic test — the pipe volume calculator is the faster tool.