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Fluid Mechanics
Darcy friction factor from the Haaland equation
Explicit estimate of the Darcy friction factor for fully developed turbulent flow in a circular pipe, from the relative roughness and the Reynolds number.
- Ref
- FLU-15
- Rev
- v1
- Field
- Fluid Mechanics
- Result
- fD
Haaland’s equation gives the Darcy friction factor for turbulent flow in a rough pipe without iterating: fD ≈ [−1.8 log₁₀((ε/D/3.7)1.11 + 6.9/Re)]−2. It is an explicit approximation to the implicit Colebrook equation, accurate to within a few per cent. Here ε is the absolute roughness and D the diameter in the same unit. It applies to fully turbulent flow (Re above about 4000).
Explicit estimate of the Darcy friction factor for fully developed turbulent flow in a circular pipe, from the relative roughness and the Reynolds number. It avoids the iteration needed for the Colebrook equation.
Calculator
- Relative roughness, ε/D4.5×10−4
- Friction factor for a smooth pipe (ε = 0) at the same Re0.01782
- Roughness penalty, f / fsmooth1.114
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For ε/D = 4.5×10−4 and Re = 100,000 the friction factor is 0.01986. A perfectly smooth pipe at the same Re would have 0.01782, so the roughness adds 11.4 %.
- A pressure drop along the pipe is fD(L/D)(ρv²/2): doubling f doubles the friction loss, so a few per cent in f is a few per cent in the pumping power needed to overcome friction.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ε Absolute roughness | 0.001 to 3 mm | 0.01786 | 0.0576 | 0.03974 | +10% | +0.958 % |
| Re Reynolds number (turbulent) | 4,000 to 1e7 | 0.01639 | 0.04075 | 0.02436 | +10% | −1.23 % |
| D Pipe diameter | 5 to 1,000 mm | 0.01801 | 0.03726 | 0.01925 | +10% | −0.886 % |
Sweep of Reynolds number (turbulent)
| Reynolds number (turbulent) | Darcy friction factor |
|---|---|
| 4,000 | 0.04075 |
| 10,640 | 0.03093 |
| 28,280 | 0.02464 |
| 75,210 | 0.02067 |
| 200,000 | 0.01837 |
| 531,800 | 0.0172 |
| 1,414,000 | 0.01668 |
| 3,761,000 | 0.01647 |
| 1e7 | 0.01639 |
Sweep one variable
Variables and how each one contributes
Darcy friction factor
What the equation gives you. Everything else feeds it.
Absolute roughness
The absolute roughness of the pipe wall, in mm (about 0.045 for new commercial steel, 0.0015 for drawn tubing). Raises the friction factor: a rougher wall disturbs the flow more. Raising it by 0.02 mm (to 0.065 mm) moves the result by +4.13 %, from 0.01986 to 0.02067.
Pipe diameter
The inside diameter of the pipe, in mm. Lowers the friction factor, because the relative roughness ε/D is smaller. Raising it by 10 mm (to 110 mm) moves the result by −0.886 %, from 0.01986 to 0.01968.
Reynolds number (turbulent)
The Reynolds number of the flow, above about 4000. The slider is logarithmic. Lowers the friction factor: at higher Reynolds numbers the viscous part falls, though it levels off for a rough pipe. Raising it by 20,000 (to 120,000) moves the result by −2.3 %, from 0.01986 to 0.0194.
Worked example with the values above
- Relative roughness: ε/D = 0.045 / 100 = 4.5×10−4; (ε/D / 3.7)1.11 = 4.512×10−5 and 6.9/Re = 6.9×10−5; their sum is 1.141×10−4.
- Logarithm: −1.8 log₁₀(1.141×10−4) = 7.097.
- fD = 1 / 7.097² = 0.01986.
Assumptions and limits
- Fully turbulent, steady flow in a full circular pipe, with Re above about 4000.
- Roughness uniform around the wall and ε and D in the same unit.
Common mistakes
- Using the Darcy friction factor in a formula written for the Fanning factor, which is a quarter as large.
- Using the equation for laminar flow, where f = 64/Re.
- Using ε in mm and D in m.
Statistics and uncertainty
The equation is within a few per cent of the Colebrook equation, while the roughness of a real pipe is uncertain by 20 % or more and changes with age and scale, so the roughness is the larger source of error. At very high Re the friction factor depends only on ε/D.
Where it is used
- The friction loss in piping for pump sizing, fuel lines and cooling circuits.
- Quick estimation in a spreadsheet without iterating Colebrook.
Maths you need
- Reynolds number: inertia forces over viscous forces
- Relative roughness: wall roughness divided by diameter