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Colebrook Equation Calculator

One over the square root of f equals negative two times the log base ten of epsilon over 3.7 D plus 2.51 over Reynolds number times the square root of f

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Colebrook-White Equation

The Colebrook-White equation is an implicit formula that relates the Darcy friction factor to the pipe’s relative roughness and Reynolds number. The calculator uses iterative methods to converge on a solution.

1/√f = −2 log₁₀(ε/(3.7D) + 2.51/(Re√f))

How It Works

The Colebrook-White equation is an implicit formula that calculates the Darcy friction factor for turbulent pipe flow. Because the friction factor appears on both sides, the calculator uses fixed-point iteration (up to 100 steps, tolerance 10⁻¹⁰) to converge on a solution. The result feeds directly into the Darcy-Weisbach equation for head loss calculations.

Example Problem

Find the Darcy friction factor for water flowing through a 50 mm galvanized iron pipe (ε = 0.00015 m) at Re = 100,000.

  1. Relative roughness: ε/D = 0.00015 / 0.05 = 0.003
  2. Iterate the Colebrook equation until convergence
  3. Result: f ≈ 0.0269

This matches the Moody diagram value for ε/D = 0.003 at Re = 10⁵.

Key Concepts

The Colebrook-White equation is an implicit formula — the friction factor f appears on both sides — so it must be solved iteratively. It bridges the gap between smooth-pipe (Blasius) and fully-rough (von Kármán) flow regimes by incorporating both the Reynolds number and relative roughness. The resulting friction factor feeds directly into the Darcy-Weisbach equation for head loss calculations.

Applications

  • Piping design: determining friction factors for pressure-drop calculations in water supply and HVAC systems
  • Oil and gas: sizing pipelines and predicting pumping power for crude oil and natural gas transport
  • Chemical engineering: calculating friction losses in process piping with various fluids and pipe materials
  • Fire protection: verifying flow capacity and pressure requirements in sprinkler system piping

Common Mistakes

  • Using the Colebrook equation for laminar flow (Re < 2,300) — laminar friction factor is simply f = 64/Re and does not depend on roughness
  • Confusing Darcy friction factor with Fanning friction factor — Darcy f is four times Fanning f; mixing them gives head losses off by 4×
  • Using absolute roughness without dividing by diameter — the equation uses relative roughness ε/D, not ε alone
  • Assuming a single iteration is sufficient — convergence typically requires 5-10 iterations for engineering accuracy

Frequently Asked Questions

What is the Colebrook equation used for?

It calculates the Darcy friction factor for turbulent flow in pipes (Re > 4,000). The friction factor is then used in the Darcy-Weisbach equation to determine pressure loss due to pipe friction.

What is the difference between absolute and relative roughness?

Absolute roughness (ε) is the physical height of surface irregularities in meters. Relative roughness is ε/D, the ratio of roughness to pipe diameter. Commercial steel has ε ≈ 0.045 mm; drawn copper is about 0.0015 mm.

Does the Colebrook equation work for laminar flow?

No. For laminar flow (Re < 2,300), the friction factor is simply f = 64/Re and does not depend on roughness. The Colebrook equation only applies to the turbulent regime.

What is the Moody diagram?

The Moody diagram is a graphical plot of the Colebrook equation, showing friction factor versus Reynolds number for various relative roughness values. This calculator replaces manual chart reading with precise iterative computation.

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