Friction factor of flow in circular pipe
\( \displaystyle \textbf{Laminar flow (Re}<2100):\; f=\frac{64}{Re}. \)
\( \displaystyle \textbf{Turbulent flow (Re}\gtrsim 4000,\;\text{Haaland, explicit}):\;
\frac{1}{\sqrt f} = -1.8\,\log_{10}\!\left[\left(\frac{\varepsilon/D}{3.7}\right)^{1.11}+\frac{6.9}{Re}\right]. \)
Symbols
- Re — Reynolds number \(Re=\dfrac{\rho V D}{\mu}\)
- \(\rho\) — fluid density (kg·m\(^{-3}\))
- V — average pipe velocity (m·s\(^{-1}\))
- \(\mu\) — dynamic viscosity (Pa·s)
- D — pipe diameter (m)
- f — friction factor (Darcy \(f\))
- \(\varepsilon\) — equivalent roughness (m)
- \(\varepsilon/D\) — relative roughness (dimensionless)
Transition (Re ≈ 2100–4000): onset of turbulence depends on disturbances and surface condition. There is no formula for the transition region. The Moody Chart below is created using the above formulas.
ε/D: 2.00e-4
\( \text{Colebrook: }\;\frac{1}{\sqrt f}=-2\log_{10}\!\Big(\frac{\varepsilon/D}{3.7}+\frac{2.51}{Re\sqrt f}\Big) \)