Pipe Flow

Pipe flow analysis deals with the transport of liquids or gases through closed conduits under pressure. The governing equation for steady, incompressible flow is the Bernoulli equation extended with a head-loss term: P1/(ρg) + V1²/(2g) + z1 = P2/(ρg) + V2²/(2g) + z2 + h_L, where h_L is the total head loss between sections 1 and 2. Head loss has two components: major loss due to wall friction along the pipe and minor loss due to fittings, valves, bends, entrances, and exits. The Darcy-Weisbach equation gives the major head loss: h_f = f (L/D) (V²/2g), where f is the Darcy friction factor, L the pipe length, D the internal diameter, and V the mean velocity. The friction factor depends on the flow regime. For laminar flow (Re < 2000), f = 64/Re, which is independent of roughness. For turbulent flow (Re > 4000), f is obtained from the Colebrook-White equation or the Moody chart, which relates f to the Reynolds number Re = V D / ν and the relative roughness ε/D. The Colebrook equation is implicit: 1/√f = -2 log10(ε/(3.7D) + 2.51/(Re√f)). The Haaland equation is an explicit approximation. In the fully rough turbulent regime, f depends only on ε/D. The transition region (2000 < Re < 4000) is uncertain and should be avoided in design. Minor losses are expressed as h_m = K V²/(2g), where K is the loss coefficient for the fitting. Alternatively, the equivalent length method replaces a fitting with a length of straight pipe that produces the same loss: L_eq = K D / f. The total head loss is h_L = h_f + Σh_m. The flow rate is Q = V A = V (πD²/4). The hydraulic gradient is i = h_f / L. For water distribution networks, the empirical Hazen-Williams equation is often used: V = k C R_h^0.63 S^0.54, where C is the Hazen-Williams roughness coefficient, R_h the hydraulic radius, and S the hydraulic gradient. Pipe roughness ε depends on material and age: drawn tubing (0.0015 mm), commercial steel (0.045 mm), cast iron (0.26 mm), and concrete (0.3–3 mm). Energy losses increase with velocity squared, so doubling the velocity quadruples the friction head loss, which is why pipe sizing has a strong effect on pumping cost. In design, the pipe diameter is selected so that velocity remains within economic limits (typically 1–3 m/s for water) to balance capital cost against pumping energy.