Closed Conduit Hydraulics: Energy, Friction, and Pumps
Core ideas
Apply the energy equation between sections: p1/γ + z1 + V1²/2g = p2/γ + z2 + V2²/2g + hL. The energy grade line includes velocity head; the hydraulic grade line is the EGL minus velocity head. HGL sits at the piezometric level, can never exceed EGL, and drops in the direction of flow.
Head loss has major (friction) and minor (fittings, valves, bends) components. Darcy-Weisbach: hf = f(L/D)(V²/2g), with f from the Moody chart or Colebrook using Reynolds number Re = VD/ν and relative roughness ε/D. For laminar flow (Re < 2000), f = 64/Re. Minor losses: h = K(V²/2g). Hazen-Williams is empirical: V = 1.318 C R^0.63 S^0.54 in US units, or hf = 4.73 L Q^1.852/(C^1.852 D^4.87) with Q in cfs and D in ft.
Pipes in series carry the same flow and add head losses; parallel pipes share the same head loss and split flow. Pump operating point is the intersection of the pump curve and system curve. NPSHa must exceed NPSHr to avoid cavitation.
Traps
Forgetting velocity head when computing pressure at a discharge; treating Hazen-Williams C as a roughness height; omitting minor losses near pumps and valves; and assuming parallel pipes split flow equally regardless of diameter and length.
Worked example
An 8-inch (D = 0.667 ft) pipe, L = 500 ft, Q = 1.5 cfs, f = 0.022. A = 0.349 ft², so V = 4.30 ft/s and V²/2g = 0.287 ft. Then hf = 0.022 × (500/0.667) × 0.287 = 0.022 × 750 × 0.287 ≈ 4.7 ft.