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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

Solution

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How It Works

The Colebrook-White equation estimates the Darcy friction factor for turbulent pipe flow by combining the effects of Reynolds number and relative roughness. Because the friction factor appears on both sides of the equation, the calculator solves it iteratively rather than by a single algebraic rearrangement. Once the friction factor converges, you can use it in Darcy-Weisbach head-loss and pressure-drop calculations.

Example Problem

Find the Darcy friction factor for a pipe with absolute roughness ε = 0.15 mm, diameter D = 50 mm, and Reynolds number Re = 100,000.

  1. Convert the length inputs into consistent units before forming the roughness ratio.
  2. Compute relative roughness: ε / D = 0.15 mm / 50 mm = 0.003.
  3. Substitute ε / D and Re into the Colebrook-White equation.
  4. Because f appears inside the logarithm and also under the square root, start with an initial guess for f.
  5. Iterate until the friction-factor estimate stops changing within the solver tolerance.
  6. For this case, the Darcy friction factor converges to about 0.0260.

Key Concepts

The Darcy friction factor is a dimensionless measure of wall-friction losses in pipe flow. In turbulent flow, it depends on both Reynolds number and relative roughness, which is why the Colebrook-White equation remains a standard reference in hydraulics, fluid mechanics, and piping design. A smoother pipe or a larger diameter lowers relative roughness and usually lowers the friction factor, while rougher walls tend to increase it.

Applications

  • Pipe-system design: estimating friction losses before sizing pumps, valves, and pressure margins
  • HVAC and building services: checking ductless water loops, condenser lines, and chilled-water piping
  • Process engineering: comparing pressure-drop behavior across different pipe materials and diameters
  • Hydraulics education: learning how Reynolds number and roughness interact instead of relying only on the Moody chart

Common Mistakes

  • Using the Colebrook equation for laminar flow even though laminar friction factor follows f = 64 / Re instead of the turbulent roughness relation
  • Confusing absolute roughness with relative roughness and forgetting that the Colebrook equation depends on the ratio ε / D
  • Mixing Darcy and Fanning friction factors even though the Darcy value is four times the Fanning value

Frequently Asked Questions

What is the Colebrook equation used for?

It is used to calculate the Darcy friction factor for turbulent flow in pipes. That friction factor is then used in equations such as Darcy-Weisbach to estimate head loss or pressure drop.

What is the formula for the Colebrook-White equation?

The standard implicit form is 1/√f = -2 log10(ε/(3.7D) + 2.51/(Re√f)), where f is the Darcy friction factor, ε is absolute roughness, D is pipe diameter, and Re is Reynolds number.

Why does the Colebrook equation require iteration?

Because the unknown friction factor appears on both sides of the equation, including inside the logarithm and under the square root. That makes it an implicit equation rather than a simple closed-form solve.

What is relative roughness?

Relative roughness is the ratio ε / D, where ε is the pipe's absolute roughness and D is the pipe diameter. It tells you how significant the wall roughness is compared with the pipe size.

Does the Colebrook equation work for laminar flow?

No. It is intended for turbulent flow. In laminar flow, the Darcy friction factor is simply f = 64 / Re and does not depend on roughness in the same way.

What is the difference between Darcy and Fanning friction factor?

They are related but not identical. The Darcy friction factor is four times the Fanning friction factor, so mixing them can create a 4× error in downstream calculations.

When should I use the Colebrook equation instead of the Moody chart?

Use the Moody chart for quick visual estimates and the Colebrook equation when you want a precise numerical friction factor that you can carry into later head-loss or optimization calculations.

Colebrook Equation Formula

The Colebrook-White equation is an implicit turbulent-flow formula, which means the Darcy friction factor appears on both sides of the equation. That is why the calculator uses iteration instead of a simple single-step solve.

1/√f = −2 log₁₀(ε/(3.7D) + 2.51/(Re√f))
  • f — Darcy friction factor
  • ε — absolute roughness of the pipe wall
  • D — pipe diameter
  • Re — Reynolds number of the flow

Worked Examples

Commercial Steel

How do you solve the Colebrook equation for a 100 mm steel pipe?

A commercial steel water line uses a 100 mm pipe with an absolute roughness of 0.15 mm and Reynolds number 100,000.

  • Enter ε = 0.15 mm, D = 100 mm, and Re = 100,000.
  • The calculator converts both roughness and diameter into the same base length unit.
  • It computes the relative roughness ε/D, which is the roughness term used by the Colebrook-White equation.
  • Then it iteratively solves the implicit equation because f appears on both sides.
  • For this case, ε/D = 0.0015 and the turbulent-flow friction factor converges near 0.0222.
  • That friction factor can then feed the Darcy-Weisbach equation for head-loss calculations.

A 100 mm pipe with ε = 0.15 mm at Re = 100,000 has a Darcy friction factor of about 0.0222.

This is a realistic pipe-design example because it combines both Reynolds number and wall roughness, rather than assuming a smooth pipe.

Galvanized Pipe

What is the friction factor for a rougher 50 mm pipe?

A rougher small pipe uses ε = 0.15 mm, D = 50 mm, and Reynolds number 100,000.

  • Enter ε = 0.15 mm, D = 50 mm, and Re = 100,000.
  • The calculator converts both inputs into meters before solving.
  • Relative roughness is ε/D = 0.00015 / 0.05 = 0.003.
  • That larger ε/D pushes the solution toward a higher friction factor than the smoother 100 mm case.
  • The iterative Colebrook solution converges near 0.0260.
  • Use that larger friction factor when estimating head loss in smaller, rougher piping.

A 50 mm pipe with ε = 0.15 mm at Re = 100,000 has a Darcy friction factor of about 0.0260.

This is the kind of side-by-side comparison that makes the Colebrook equation more useful than a rough chart lookup.

Smooth Pipe Check

How does a smoother pipe change the answer?

A smoother pipe uses ε = 0.01 mm, D = 100 mm, and Reynolds number 100,000.

  • Enter ε = 0.01 mm, D = 100 mm, and Re = 100,000.
  • Convert the roughness and diameter into the same base length unit.
  • Relative roughness becomes ε/D = 0.00001 / 0.1 = 0.0001.
  • That smaller roughness term reduces the right-hand side of the Colebrook equation.
  • The iterative solution converges near 0.0185, which is lower than the rougher-pipe examples.
  • Lower roughness means less wall drag and therefore a lower turbulent friction factor.

A smoother 100 mm pipe with ε = 0.01 mm at Re = 100,000 has a Darcy friction factor of about 0.0185.

This example is a nice sanity check because it shows how the solution trends downward as relative roughness decreases.

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