Friction Factor: The Colebrook-White Equation & Moody Diagram

The Challenge of Turbulent Friction

In laminar flow ($Re < 2000$), calculating the friction factor is trivial ($f_D = 64/Re$). However, in turbulent flow, the friction factor depends heavily on the interactions between the fluid's chaotic eddies and the microscopic roughness of the pipe wall. To calculate pressure drop accurately, engineers must determine this turbulent friction factor.

The Colebrook-White Equation

In 1939, Cyril Colebrook and C.M. White formulated an equation that accurately models the friction factor across the entire turbulent regime (smooth pipes, transition zone, and fully rough pipes). It remains the industry standard today:

Where:

• $f_D$ = Darcy friction factor
• $\varepsilon$ = Absolute pipe roughness (m or mm)
• $D$ = Internal pipe diameter (m or mm, same units as $\varepsilon$)
• $Re$ = Reynolds number

The Problem: The Colebrook-White equation is implicit. The term $f_D$ appears on both sides of the equation. It cannot be solved algebraically; it requires numerical iteration (like the Newton-Raphson method) or the use of solver software.

The Moody Diagram

Before computers were ubiquitous, Lewis Moody (1944) plotted the Colebrook-White equation into a massive, highly useful chart known as the Moody Diagram. For decades, this was the only practical way for engineers to find $f_D$.

To use the Moody Diagram:

1. Calculate your Reynolds number ($Re$) and locate it on the x-axis.
2. Calculate your relative roughness ($\varepsilon/D$) and find the corresponding curved line on the right y-axis.
3. Follow the $Re$ line vertically until it intersects the $\varepsilon/D$ curve.
4. Read horizontally to the left y-axis to find $f_D$.

Explicit Approximations: Swamee-Jain

To avoid iterative calculations in spreadsheets or simple computer programs, several engineers have developed explicit approximations of the Colebrook equation. The most famous and widely used is the Swamee-Jain equation (1976):

This equation calculates $f_D$ directly. For most engineering applications ($Re$ between $5,000$ and $10^8$, and $\varepsilon/D$ between $10^{-6}$ and $0.05$), the Swamee-Jain approximation is accurate to within 1% to 2% of the true Colebrook-White value. This is well within the uncertainty of the pipe roughness itself, making it perfectly acceptable for standard design work.

Fully Rough vs. Smooth Pipes

Looking at the Colebrook equation, we can observe two extremes:

• Smooth Pipe ($Re \to \infty$ but $\varepsilon \to 0$): The $\varepsilon/D$ term vanishes. The friction factor is entirely dependent on the Reynolds number.
• Fully Rough Pipe ($Re \to \infty$ with significant $\varepsilon$): The viscous sublayer is completely destroyed by the pipe roughness. The $2.51 / (Re \sqrt{f_D})$ term vanishes. The friction factor becomes completely independent of the Reynolds number and depends only on the relative roughness. This is the flat, horizontal section on the right side of the Moody Diagram.

The equation for fully rough flow (Von Kármán equation) simplifies to:

Try the Calculator

Skip the Moody chart and the iterative math. Use the Friction Factor Calculator on this page. It uses the highly accurate Swamee-Jain explicit equation to instantly compute the Darcy and Fanning friction factors based on your pipe material and Reynolds number.