Relative roughness is calculated by dividing the absolute roughness of a pipe’s interior surface (ε) by the pipe’s inside diameter (D). The formula is simply ε/D. Both values must be in the same units, and the result is a dimensionless number, typically a small decimal like 0.001 or 0.0004. This ratio is one of two inputs you need to find a friction factor on a Moody chart or through the Colebrook equation, the other being the Reynolds number.
The Formula
The calculation is straightforward division:
Relative roughness = ε / D
Where ε (epsilon) is the absolute roughness of the pipe material and D is the inside diameter. You can use any unit of length (millimeters, inches, feet) as long as both ε and D share the same unit. Because you’re dividing a length by a length, the units cancel out and you get a pure ratio with no units attached.
What Absolute Roughness Means
Absolute roughness represents the average height of tiny bumps and irregularities on the inside wall of a pipe. It’s formally defined as the height of tightly packed, uniformly sized sand grains that would produce the same pressure loss as the actual pipe surface. You don’t measure this yourself. Instead, you look it up in a reference table based on the pipe material. Here are common values in millimeters:
- PVC or plastic pipes: 0.0015 mm
- Commercial steel: 0.045 to 0.09 mm
- Galvanized steel: 0.15 mm
- New cast iron: 0.25 to 0.8 mm
- Worn cast iron: 0.8 to 1.5 mm
- Ordinary concrete: 0.3 to 3 mm
- Riveted steel: 0.9 to 9 mm
Notice the enormous range. A smooth plastic pipe has a roughness value thousands of times smaller than a riveted steel pipe. That difference directly affects the relative roughness and, ultimately, how much energy you lose to friction in the flow.
A Worked Example
Say you have a 150 mm (6-inch) diameter commercial steel pipe. The absolute roughness for commercial steel is about 0.045 mm. Convert the diameter to the same unit if needed (it’s already in millimeters here), then divide:
Relative roughness = 0.045 mm / 150 mm = 0.0003
Now suppose you have the same steel pipe but only 25 mm in diameter. The roughness of the material hasn’t changed, but the ratio shifts dramatically:
Relative roughness = 0.045 mm / 25 mm = 0.0018
The smaller pipe has a relative roughness six times higher, even though the wall texture is identical. This is the core insight behind the calculation: the same material creates more friction resistance in a smaller pipe because the bumps on the wall are larger relative to the flow area.
Non-Circular Conduits
For rectangular ducts, annular spaces, or other non-circular cross sections, you replace D with the hydraulic diameter. The hydraulic diameter equals four times the cross-sectional area divided by the wetted perimeter (the total length of the boundary touching the fluid). Once you have the hydraulic diameter, the relative roughness calculation works the same way: ε divided by the hydraulic diameter.
How Pipe Aging Changes the Calculation
The roughness values in reference tables represent new or typical pipe conditions. Over years of service, corrosion byproducts and mineral deposits accumulate on interior walls, increasing the effective roughness and reducing the actual inside diameter. Both changes push relative roughness higher. Rusted steel, for instance, can have a roughness of 0.15 to 4 mm, compared to 0.045 to 0.09 mm for a new commercial steel pipe. If you’re analyzing an existing system rather than designing a new one, using the “new pipe” roughness value will underestimate friction losses. Engineering references often list separate roughness values for aged or corroded conditions for this reason.
Using Relative Roughness to Find Friction Factor
The whole reason you calculate relative roughness is to determine the Darcy friction factor, which tells you how much pressure a fluid loses as it moves through the pipe. There are two main ways to get there.
The Moody Chart
A Moody chart plots friction factor on the vertical axis against Reynolds number on the horizontal axis, with a family of curves representing different relative roughness values (ranging from about 0.000001 up to 0.05). You find your Reynolds number along the bottom, follow it up to the curve matching your relative roughness, and read the friction factor off the left side. At high Reynolds numbers, the curves flatten out. This means the friction factor depends almost entirely on relative roughness and stops changing with flow speed. At lower Reynolds numbers, both the roughness and the viscosity of the fluid matter.
The Colebrook Equation
The Colebrook equation is the mathematical basis behind the Moody chart. It looks like this:
1/√f = −2.0 × log₁₀[(ε/D)/3.7 + 2.51/(Re × √f)]
The term (ε/D)/3.7 is your relative roughness contribution, and 2.51/(Re × √f) accounts for viscous effects. This equation is valid for Reynolds numbers above about 4,000, covering the full turbulent range up past 10 million. It’s implicit, meaning f appears on both sides, so you either solve it iteratively or use one of several explicit approximations available in engineering references. For fully rough flow at high Reynolds numbers, the viscous term becomes negligible and the equation simplifies to 1/√f = −2.0 × log₁₀(ε/3.7D), which you can solve directly.
Typical Relative Roughness Ranges
In practice, relative roughness values for engineered piping systems fall between about 0.000001 and 0.05. Smooth plastic pipe in a large diameter might produce a relative roughness near 0.00001, while a small-diameter concrete pipe could land near 0.01 or higher. Values below roughly 0.00001 behave essentially as smooth pipes, where the wall bumps are so small relative to the diameter that they don’t influence the friction factor in any measurable way. Roughness values at the high end (0.03 to 0.05) represent severely rough or very small-diameter conduits where friction losses are substantial.
If your calculated relative roughness seems unreasonably high or low, double-check that your units match. Mixing millimeters for roughness with inches for diameter is one of the most common mistakes in this calculation.