Solve the implicit Colebrook–White equation for turbulent pipe flow at Re = 100,000 and ε/D = 0.001. Both inputs are dimensionless, so this result is independent of the length-unit system.
1. Calculator Inputs
Select the units below before entering the numbers. These are fixed benchmark inputs for checking the calculator.
| Calculator field | Value | Unit / setting |
|---|---|---|
| Reynolds number input | Enter Reynolds number | |
| Reynolds number | 100,000 | dimensionless |
| Roughness input | Relative roughness, ε/D | |
| Relative roughness | 0.001 | dimensionless |
2. Step-by-step Math
Write the Darcy-form equation
1/√f = −2 log₁₀[ (ε/D)/3.7 + 2.51/(Re√f) ]
f(next) = {−2 log₁₀[0.001/3.7 + 2.51/(100000√f)]}⁻²The equation contains f on both sides. Use base-10 logarithms and iterate, starting with f = 0.020.
Iterate until the value stops changing
Start: 0.020000000 Iteration 1: 0.022290508 Iteration 2: 0.022168795 Iteration 3: 0.022174821 Iteration 4: 0.022174522 Iteration 5: 0.022174537 Iteration 6: 0.022174536
The fixed-point hand calculation converges to the calculator’s Colebrook–White result. Retain more digits internally than shown here.
Convert Darcy to Fanning
fFanning = fDarcy / 4 fFanning = 0.0221745359445 / 4 fFanning ≈ 0.00554
Use the Darcy value in ΔP = f(L/D)ρV²/2. A Fanning-based pressure-loss equation includes a separate factor of four.
3. Compare Your Answers
Keep full precision during the calculation. The calculator may display fewer digits; compare answers rounded to three significant figures; retain unrounded intermediate values.
| Result | Expected answer | Your answer |
|---|---|---|
| Darcy friction factor | 0.0222 | __________ |
| Fanning friction factor | 0.00554 | __________ |
| Flow regime | Turbulent | __________ |
| Method | Colebrook-White | __________ |
Second check
Change Reynolds number to 1,000. The calculator switches to the laminar relation fDarcy = 64/Re = 0.064; fFanning = 0.016. Relative roughness does not enter this fully developed laminar relation.
4. Assumptions and Method
This example concerns conventional single-phase pipe flow. Roughness is specified rather than estimated for a material. Transition-region results use a different treatment and should not be checked with this turbulent iteration.
Equation definitions and implementation: Darcy Friction Factor Calculator. See also calculation methodology and other worked examples.