Engineering & Scientific
Haaland’s Explicit Darcy Friction Factor & Laminar-Turbulent Head-Loss Gate
Calculate Darcy-Weisbach pipe friction factor f directly without Colebrook circular iteration using S. E. Haaland’s (1983) 1.11-exponent Moody equation plus a laminar 64/Re switch.
Why Haaland’s 1.11 roughness exponent tracks implicit Colebrook within ~1.4%
Implicit Colebrook-White 1 / SQRT(f) = -2.0 * LOG10((eps/D)/3.7 + 2.51/(Re * SQRT(f))) traps f on both sides of the equals sign. S. E. Haaland (1983) eliminated circularity with a single explicit log10 expression: 1 / SQRT(f) = -1.8 * LOG10(((eps / D) / 3.7)^1.11 + 6.9 / Re) for turbulent flow (Re >= 4,000, and commonly applied above the Re = 2,300 transition), while exact Hagen-Poiseuille laminar flow below Re <= 2,300 obeys f = 64 / Re independently of wall roughness.
Once f is evaluated, Darcy-Weisbach frictional head loss across pipe length L, diameter D, and bulk velocity V is h_f = f * (L / D) * (V^2 / (2 * 9.81)) in meters of fluid head.
| Hydraulic Quantity | Formula | 200 mm Commercial Steel Pipe (Re = 100,000) |
|---|---|---|
| Relative Roughness eps / D | B3 / B2 | 0.0002 (0.04 mm / 200 mm) |
| Haaland Darcy f | (1 / (-1.8 * LOG10(((eps/D)/3.7)^1.11 + 6.9/Re)))^2 | 0.0187 |
| 100 m Pipe Head Loss h_f | f * (L / D) * (V^2 / (2 * 9.81)) | 1.91 meters (at V = 2.0 m/s) |
Regime-Switching Laminar (64/Re) + Haaland Turbulent Friction Formula
Return 64 / Re when Re <= 2300 and Haaland’s explicit square-reciprocal formula above Re > 2300.
=IF(B4 <= 2300, ROUND(64 / B4, 4), ROUND((1 / (-1.8 * LOG10((((B3 / B2) / 3.7) ^ 1.11) + (6.9 / B4)))) ^ 2, 4))Darcy-Weisbach Head Loss Formula (Meters)
Multiply Darcy f by (Length / Diameter) and velocity head V^2 / (2 * 9.81).
=ROUND(E2 * (B5 / B2) * ((B6 ^ 2) / (2 * 9.81)), 2)Try the example
Use the original values or change them to test your understanding.