Laminar/turbulent flow and friction factor

Laminar and turbulent flow

• Laminar flow: fluid particles move in smooth, parallel layers without mixing; viscous forces dominate; $Re < 2000$ in pipes. Example: thick oil in a thin tube.
• Turbulent flow: particles move randomly with eddies and mixing; inertia forces dominate; $Re > 4000$ in pipes. Example: water in a penstock.

Numerical

Given: $D = 0.7$ m, $Q = 1200$ L/s $= 1.2$ m³/s, $L = 50$ m, $h_f = 0.6$ m.

Velocity:

$$\begin{aligned} A &= \frac{\pi}{4}(0.7)^2 = 0.3848\ \text{m}^2 \\ V &= \frac{Q}{A} = \frac{1.2}{0.3848} = 3.118\ \text{m/s} \\ \frac{V^2}{2g} &= \frac{3.118^2}{2 \times 9.81} = 0.4956\ \text{m} \end{aligned}$$

Darcy–Weisbach equation:

$$h_f = f\frac{L}{D}\frac{V^2}{2g}$$ $$\begin{aligned} f &= \frac{h_f\, D}{L}\cdot\frac{2g}{V^2} \\ &= \frac{0.6 \times 0.7}{50 \times 0.4956} = 0.01695 \end{aligned}$$

Answer: Darcy friction factor $f \approx 0.017$. (If the form $h_f = \frac{4fLV^2}{2gD}$ is used, the coefficient of friction is $f/4 \approx 0.0042$.) At $V = 3.12$ m/s in a 0.7 m pipe, $Re \approx 2.2\times10^6$, so the flow is turbulent.