Hazen-Williams Equation — Forms, Constants & Units
Hazen-Williams is an empirical friction-loss relation for water in pressurized pipe. Its appeal is that roughness enters as a single coefficient C that does not depend on velocity or Reynolds number, so head loss solves in closed form. Its cost is that the equation is dimensionally inhomogeneous — the lead constant changes with every unit set, and most wrong answers come from pairing the right constant with the wrong units.
Velocity Form
| Units | Equation | Where |
|---|---|---|
| US customary | V = 1.318·C·R0.63·S0.54 | V in ft/s, R in ft |
| SI | V = 0.849·C·R0.63·S0.54 | V in m/s, R in m |
R is the hydraulic radius (D/4 for a full circular pipe) and S is the slope of the energy grade line, hf/L — dimensionless in both systems.
Head Loss Form — the One You Actually Use
| Units | Equation | Q | D | hf, L |
|---|---|---|---|---|
| US, cfs & ft | hf = 4.73·L·Q1.852 / (C1.852·D4.87) | cfs | ft | ft |
| US, gpm & in | hf = 10.44·L·Q1.852 / (C1.852·D4.8655) | gpm | in | ft |
| SI | hf = 10.67·L·Q1.852 / (C1.852·D4.87) | m³/s | m | m |
| NFPA 13 (sprinkler) | p = 4.52·Q1.852 / (C1.852·d4.87) | gpm | in | psi per ft |
Flow Form (Solved for Capacity)
| Units | Equation | Q | D | Gradient term |
|---|---|---|---|---|
| US, cfs & ft | Q = 0.432·C·D2.63·S0.54 | cfs | ft | S = hf/L, ft/ft |
| US, gpm & in | Q = 0.282·C·D2.63·S0.54 | gpm | in | S = hf/L, ft/ft |
| US, gpm & in, pressure | Q = 0.442·C·D2.63·(Δp/L)0.54 | gpm | in | Δp/L, psi per ft |
| SI | Q = 0.278·C·D2.63·S0.54 | m³/s | m | S = hf/L, m/m |
These follow from the velocity form via Q = VA with R = D/4, and are the exact algebraic inverses of the head loss constants above — 0.432 → 4.73, 0.282 → 10.44, 0.278 → 10.67. Rounding of the exponent 4.87 versus 4.8655 accounts for the small differences between published values; any of them is well within the empirical accuracy of the method itself.
The Exponents, and Why They Matter
| Term | Exponent | Consequence |
|---|---|---|
| Flow, Q | 1.852 | Head loss is not quadratic in Q — doubling flow raises loss by 21.852 ≈ 3.61×, not 4× |
| Roughness, C | −1.852 | Dropping C from 130 to 100 raises head loss by about 63 percent |
| Diameter, D | −4.87 | Going one nominal size up is by far the cheapest way to kill head loss |
| Slope, S | 0.54 | Reciprocal of 1.852; the two are the same relation rearranged |
Validity Limits
| Condition | Valid range | Outside it |
|---|---|---|
| Fluid | water only | Use Darcy-Weisbach — C carries no viscosity term |
| Temperature | ~40–75°F | Error grows at both extremes; hot-water and chilled systems need D-W |
| Flow regime | fully turbulent | Invalid for laminar or transitional flow (Re < 4000) |
| Diameter | 2 in and larger | Small-bore service tubing is outside the calibration set |
| Velocity | below ~10 ft/s | Accuracy degrades at high velocity |
| Pressure condition | full, pressurized | Partial-flow gravity pipe is Manning's, not Hazen-Williams |
Sources: Williams, G.S. & Hazen, A. (1920). Hydraulic Tables. AWWA M11 (steel pipe), M22 (sizing water service lines), M32 (distribution system modeling). NFPA 13, Standard for the Installation of Sprinkler Systems — friction loss formula. Mays, L.W. (2010). Water Distribution Systems Handbook. Hwang & Houghtalen, Fundamentals of Hydraulic Engineering Systems.
Related cheat sheets and tools
The companion to this page is the Hazen-Williams C coefficient table — material-by-material, new versus aged. For the physically-based alternative see absolute roughness ε for Darcy-Weisbach and confirm the regime with Reynolds number. Fittings and valves are handled separately — see minor loss K values and equivalent lengths. If the pipe runs partial-flow as gravity sewer, switch to Manning's n. For whole-network pressurized and gravity modeling, see HydroComplete, the SaaS sister product to PE-Calc.