4.3 Hydraulics: Pipe Flow, Open Channel & Pumps

Pipe Friction — Darcy-Weisbach & Hazen-Williams

Friction loss in a full pipe is found with the Darcy-Weisbach equation:

h_f = f·(L/D)·(V²/2g)

where f is the dimensionless Darcy friction factor, L is pipe length, D is diameter, and V is velocity. For laminar flow f = 64/Re; for turbulent flow read f from the Moody diagram using the Reynolds number Re and the relative roughness ε/D. The Moody chart is in the Handbook — practice reading it. A trap is using the Fanning friction factor (¼ of Darcy) by mistake.

The Hazen-Williams equation is an empirical alternative for water in pressurized pipes near room temperature (it bakes in viscosity, so no Reynolds number is needed — convenient but only valid for water). 54, where S is the slope of the energy grade line (h_f/L) and C is the Hazen-Williams coefficient (≈ 130–150 for new smooth pipe, dropping toward 100 or below as a pipe corrodes and tuberculates with age). A higher C means a smoother pipe and less head loss. Darcy-Weisbach is more fundamental and works for any fluid; Hazen-Williams is faster for routine water-distribution design.

Know both and which inputs each needs: Darcy-Weisbach needs f (from Re and ε/D), while Hazen-Williams needs only C. When a problem gives you a roughness height ε, you are on the Darcy-Weisbach/Moody path.

Minor Losses & Grade Lines

Fittings, bends, valves, entrances, and exits add minor losses:

h_m = K·(V²/2g)

where K is a loss coefficient (e.g., sharp entrance ≈ 0.5, exit ≈ 1.0, 90° elbow ≈ 0.9, fully open gate valve ≈ 0.2). Total head loss h_L = h_f + Σh_m.

Two lines visualize energy along a pipe:

• Energy Grade Line (EGL) = p/γ + V²/2g + z (total head); it always slopes downward in the flow direction (losses), except across a pump where it jumps up.
• Hydraulic Grade Line (HGL) = p/γ + z; it sits exactly one velocity head (V²/2g) below the EGL.

The HGL represents the height water would rise in a piezometer. If the HGL drops below the pipe, pressure is negative (sub-atmospheric) — a cavitation/air-entry risk.

Open-Channel Flow — Manning's Equation

For steady uniform flow in an open channel, Manning's equation gives discharge:

Q = (1.49/n)·A·R^(2/3)·S^(1/2) (USC; use 1.0 instead of 1.49 in SI)

where n is Manning's roughness (concrete ≈ 0.013, earth ≈ 0.025), A is flow area, R = A/P is the hydraulic radius (P = wetted perimeter), and S is the channel/energy slope (ft/ft). The velocity form is V = (1.49/n)R^(2/3)S^(1/2). The 1.49 factor (= 3.281^(1/3)) is the unit-conversion constant for USC and is a classic trap if you forget it.

Normal depth is the uniform-flow depth satisfying Manning's equation; solving for it is usually iterative because A and R both depend on depth. Critical depth occurs at the minimum specific energy for a given discharge, where the Froude number Fr = V/√(g·D_h) = 1 (D_h is the hydraulic depth = A/top width). Fr < 1 is subcritical (tranquil, deep/slow) flow; Fr > 1 is supercritical (rapid, shallow/fast) flow; Fr = 1 is critical.

Specific energy E = y + V²/2g (depth plus velocity head, measured from the channel bottom) is the key concept here — for a fixed Q there are two depths (one sub-, one supercritical) that carry the same flow at the same specific energy, called alternate depths. A hydraulic jump is the abrupt transition from supercritical to subcritical flow; it dissipates energy and is used deliberately in stilling basins below spillways. Channel slope classification (mild vs steep) depends on whether normal depth exceeds critical depth.

The exam often asks you to identify the regime from Fr or to compute critical depth y_c = (q²/g)^(1/3) for a rectangular channel, where q = Q/width is the unit discharge.

Pumps — Head, Power & NPSH

A pump adds head h_pump to the flow. The water (hydraulic) power delivered is:

P_water = γ·Q·H

and the brake (input shaft) power is P_brake = γQH/η, where η is pump efficiency and H is total dynamic head. In USC, horsepower = γQH/(550·η) with γ in lb/ft³, Q in cfs, H in ft.

Net Positive Suction Head (NPSH) guards against cavitation: NPSH_available = (p_atm − p_vapor)/γ + (suction-side terms). The pump is safe only when NPSH_available > NPSH_required; otherwise the fluid vaporizes at the impeller and the pump cavitates.

Worked Manning's Example

A rectangular concrete channel (n = 0.013) is 3 m wide, flows 1 m deep, on slope S = 0.001. Find Q (SI, so use 1.0).

• A = 3 × 1 = 3 m²; P = 3 + 2(1) = 5 m; R = A/P = 3/5 = 0.6 m.
• Q = (1.0/0.013)·(3)·(0.6)^(2/3)·(0.001)^(1/2)
• (0.6)^(2/3) = 0.711; (0.001)^0.5 = 0.0316.
• Q = 76.9 × 3 × 0.711 × 0.0316 = 5.18 m³/s.

Worked Example — Pump Brake Power

A pump must deliver Q = 0.10 m³/s of water against a total dynamic head H = 25 m at an efficiency η = 0.70. Find the water power and the brake (shaft input) power. Use the specific weight of water γ = 9,810 N/m³.

Step 1 — water power: P_water = γ·Q·H = (9,810)(0.10)(25) = 24,525 W ≈ 24.5 kW.

Step 2 — brake power: P_brake = P_water/η = 24,525/0.70 = 35,036 W ≈ 35.0 kW.

Step 3 — convert to horsepower: 35,036 W ÷ 746 W/hp ≈ 47 hp.

The brake power always exceeds the water power because real pumps lose energy to friction, leakage, and turbulence; the gap widens as efficiency drops. A common error is forgetting to divide by η, which understates the motor size the design actually needs.

Worked Example — Minor Loss & Critical Depth

Minor loss at a fitting. Water flows at V = 3 m/s through a fully open gate valve with loss coefficient K = 0.2. The head lost is h_m = K·V²/2g = 0.2 × (3²)/(2 × 9.81) = 0.2 × 9/19.62 = 0.2 × 0.459 = 0.092 m. A sharp pipe entrance (K ≈ 0.5) at the same velocity would lose 0.5 × 0.459 = 0.23 m — more than double, which is why rounded entrances are specified where head is scarce.

Critical depth. For a rectangular channel carrying Q = 6 m³/s over a width b = 3 m, the unit discharge is q = Q/b = 2 m²/s. Critical depth is y_c = (q²/g)^(1/3) = (2²/9.81)^(1/3) = (0.4077)^(1/3) ≈ 0.74 m.

At critical depth the Froude number equals 1 and the specific energy E = y + V²/2g is at its minimum for that discharge. If the actual flow depth exceeds 0.74 m the flow is subcritical (Fr < 1); if shallower, it is supercritical (Fr > 1). Comparing actual depth to y_c is the quickest way to classify the flow regime.