Colebrook Equation Calculator
Calculate the Darcy-Weisbach friction factor (λ) for turbulent pipe flow using the implicit Colebrook-White equation. Supports Reynolds number or velocity-based input with automatic SI/FPS conversion.
The Colebrook equation is the most widely used implicit formula for calculating the Darcy friction factor in turbulent pipe flow. It seamlessly combines the smooth-pipe and rough-pipe asymptotes into a single expression valid for the entire turbulent regime (Re > 4000). Because λ appears on both sides of the equation, this calculator uses the robust Newton-Raphson method for fast, precise convergence.
Colebrook-White Friction Factor Calculator
Colebrook-White Formulas
Implicit Colebrook Equation:
\[ \frac{1}{\sqrt{\lambda}} = -2 \log_{10} \left( \frac{2.51}{\text{Re} \sqrt{\lambda}} + \frac{\epsilon/D}{3.72} \right) \]
Where: λ = Darcy friction factor, Re = Reynolds number, ε = absolute roughness, D = hydraulic diameter.
Explicit Swamee-Jain Approximation (Initial Guess):
\[ \lambda \approx \frac{0.25}{\left[ \log_{10} \left( \frac{\epsilon/D}{3.7} + \frac{5.74}{\text{Re}^{0.9}} \right) \right]^2} \]
Used to seed the Newton-Raphson iterative solver for rapid convergence (typically < 5 iterations).
Reynolds Number (from Velocity):
\[ \text{Re} = \frac{u \cdot D}{\nu} \]
Where: u = flow velocity, ν = kinematic viscosity.
About the Colebrook Equation
Published in 1939 by Cyril Colebrook and Cedric White, this equation is an implicit correlation that fits experimental data for friction factors in both smooth and rough pipes. Because λ appears on both sides of the equation, it cannot be rearranged into a direct algebraic formula and must be solved iteratively. This calculator uses the Newton-Raphson method for fast, precise convergence.
Common Pipe Roughness Values
| Pipe Material | Roughness ε (mm) | Relative Roughness (ε/D) for 100mm pipe |
|---|---|---|
| Drawn tubing (glass, copper) | 0.0015 | 0.000015 |
| Plastic (PVC, PE) | 0.001 – 0.007 | 0.00001 – 0.00007 |
| Commercial steel | 0.045 | 0.00045 |
| Galvanized iron | 0.15 | 0.0015 |
| Cast iron | 0.26 | 0.0026 |
| Concrete | 0.3 – 3.0 | 0.003 – 0.03 |
| Riveted steel | 0.9 – 9.0 | 0.009 – 0.09 |
Darcy vs. Fanning Friction Factor
It is critical to distinguish between the two common friction factors. The Darcy friction factor (λ) (also called the Moody friction factor) is 4 times the Fanning friction factor (f). The Colebrook equation yields the Darcy factor. Always verify which factor your pressure drop formula requires:
Darcy: ΔP = λ · (L/D) · (ρu²/2)
Fanning: ΔP = 4f · (L/D) · (ρu²/2)
References & Further Reading
- • Colebrook, C.F. (1939). "Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws." Journal of the Institution of Civil Engineers, 11(4), 133-156.
- • Swamee, P.K. & Jain, A.K. (1976). "Explicit Equations for Pipe-Flow Problems." Journal of the Hydraulics Division, ASCE, 102(5), 657-664.
- • Munson, B.R., et al. (2013). Fundamentals of Fluid Mechanics (7th ed.). John Wiley & Sons. (Moody chart and friction factor derivations)
- • Crane Co. (1988). Flow of Fluids Through Valves, Fittings, and Pipe (Technical Paper No. 410).
Frequently Asked Questions
The friction factor λ appears on both sides of the equation — inside the logarithm and as the quantity being solved for. This means it cannot be rearranged into a direct algebraic formula and must be solved iteratively (or via explicit approximations like Swamee-Jain).
The Colebrook equation is valid for turbulent flow only, typically Re > 4000. For laminar flow (Re < 2100), use λ = 64/Re. The transitional zone (2100–4000) is highly unpredictable and the equation should not be used.
Yes — use the hydraulic diameter Dh = 4A/P, where A is the cross-sectional area and P is the wetted perimeter. This is a well-established and accurate approximation for turbulent flow in rectangular or annular ducts.
In the "fully rough" turbulent regime (very high Re), the friction factor becomes independent of the Reynolds number and depends solely on the relative roughness (ε/D). In the "transitionally rough" regime, it depends on both Re and ε/D.