The Hazen-Williams Equation
Friction loss for water flowing in a pipe
What we are computing
When water flows through a pipe, energy is lost to friction against the pipe wall. The Hazen-Williams equation is an empirical correlation that gives the pressure drop per unit length as a function of flow rate, pipe diameter, and a roughness coefficient \(C\) that captures the smoothness of the pipe wall.
It is the standard tool in fire-protection hydraulics — NFPA 13 explicitly specifies it for sprinkler system calculations — and is widely used in water-distribution design as well.
1. The equation, US units
In the form used in NFPA 13, friction loss per foot of pipe is
\[ \boxed{\; p_f \;=\; \dfrac{4.52\, Q^{1.852}}{C^{1.852}\, d^{4.87}} \qquad [\text{psi/ft}] \;} \]where
- \(Q\) is the volumetric flow rate in gallons per minute,
- \(d\) is the actual inside diameter of the pipe in inches, and
- \(C\) is the dimensionless Hazen-Williams roughness coefficient (a smoother pipe has a higher \(C\)).
Total friction loss over a length \(L\) of pipe (in feet) is then
\[ P_f \;=\; p_f \cdot L \qquad [\text{psi}]. \]The constant \(4.52\) is what falls out when you write the original Hazen-Williams correlation in these particular US engineering units. Different unit systems give different numerical constants but the same underlying physics.
2. The equation, SI units
The same equation in coherent SI units is
\[ h_f \;=\; \dfrac{10.67\, L\, Q^{1.852}}{C^{1.852}\, D^{4.87}} \qquad [\text{m of water}] \]with \(L\) and \(D\) in meters and \(Q\) in m\(^3\)/s. Convert head to pressure with
\[ \Delta P \;=\; \rho\, g\, h_f \;\approx\; 9.806\, h_f \qquad [\text{kPa}] \]where \(\rho = 1000\) kg/m\(^3\) and \(g = 9.806\) m/s\(^2\) for water at ordinary temperatures.
The SI form makes it explicit that the only unit conversions involved are linear scalings of flow, diameter, and length; the exponents \(1.852\) on \(Q\) and \(4.87\) on \(D\) are universal.
3. Flow velocity
The mean velocity in the pipe follows directly from continuity:
\[ V \;=\; \dfrac{Q}{A} \;=\; \dfrac{4\,Q}{\pi\, d^{2}}. \]In US units, with \(Q\) in gpm and \(d\) in inches, this simplifies to
\[ V \;\approx\; \dfrac{0.4085\,Q}{d^{2}} \qquad [\text{ft/s}]. \]Fire-protection design practice limits velocity to roughly 20 ft/s (6 m/s) in supply mains; higher velocities cause water-hammer concerns and excessive friction loss. This calculator flags velocities above 10 ft/s as a cautionary note.
4. C-factors
The roughness coefficient \(C\) is the largest source of uncertainty in any Hazen-Williams calculation. NFPA 13 Table 23.4.4.7.1 specifies the following design values:
| Pipe / tubing material | C |
|---|---|
| Unlined cast or ductile iron | 100 |
| Black steel (dry & pre-action systems) | 100 |
| Black steel (wet & deluge systems) | 120 |
| Galvanized (all) | 120 |
| Plastic (listed) | 150 |
| Cement-lined cast or ductile iron | 140 |
| Copper tube or stainless steel | 150 |
| Asbestos cement | 140 |
| Concrete | 140 |
For older, in-service pipe the effective \(C\) can be substantially lower than these values, especially for unlined ferrous pipe carrying water with significant hardness or oxygen content. If a more representative value is available from field measurement, use it.
5. Actual vs. nominal diameter
Friction loss is extremely sensitive to diameter (it scales as \(d^{-4.87}\)). For nominal Schedule 40 carbon steel, the actual inside diameter differs from the nominal size:
| Nominal size | Actual ID (in) | Nominal size | Actual ID (in) |
|---|---|---|---|
| 1/2″ | 0.622 | 3″ | 3.068 |
| 3/4″ | 0.824 | 4″ | 4.026 |
| 1″ | 1.049 | 6″ | 6.065 |
| 1-1/4″ | 1.380 | 8″ | 7.981 |
| 1-1/2″ | 1.610 | 10″ | 10.020 |
| 2″ | 2.067 | 12″ | 11.938 |
| 2-1/2″ | 2.469 | — | |
For copper, CPVC, and other materials, use the manufacturer’s ID. Selecting a nominal size in the calculator auto-fills the Sch. 40 ID; for other materials, enter the ID directly.
6. Limitations
- The Hazen-Williams equation is empirical and valid for water at ordinary temperatures (40–75 °F / 4–24 °C) and turbulent flow. It is not valid for other fluids, for compressible flow, or at very low Reynolds numbers.
- It systematically under-predicts friction loss at very high velocities (> ~10 ft/s) and over-predicts at very low velocities relative to the Darcy-Weisbach equation. For most fire-protection design flows the agreement is within a few percent.
- This tool computes loss in a single straight pipe of one material and diameter. To handle multi-segment systems, branches, or networks, use a dedicated hydraulic calculation program.
References
- NFPA 13, Standard for the Installation of Sprinkler Systems, §23.4.4 (Hazen-Williams formula) and Table 23.4.4.7.1 (C-factors).
- Williams, G.S., and Hazen, A., Hydraulic Tables, 3rd ed., John Wiley & Sons, 1933.
- ASME B36.10M, Welded and Seamless Wrought Steel Pipe (Schedule 40 dimensions).