Topic 4 of 10 — free theory
Pipe Flow
Darcy-Weisbach, Hazen-Williams, minor losses, and series/parallel pipe systems.
Take the free 5-question mini-quiz ↓
Head loss — the Darcy-Weisbach equation
Friction in a pipe eats head. The Darcy-Weisbach equation is the fundamental (and exam-favourite) way to compute it:
hf = f LD V²2g
| hf | friction head loss |
| f | Darcy friction factor (dimensionless — not the Fanning factor, which is f/4) |
| L, D | pipe length and diameter |
| V | mean velocity |
Re = VDν laminar: f = 64Re
| Re | Reynolds number; laminar below ≈ 2,300, turbulent above ≈ 4,000 |
| ν | kinematic viscosity of the fluid |
Turbulent f: Haaland approximation 1/√f ≈ −1.8 log[(ε/3.7D)1.11 + 6.9/Re]
| ε | absolute roughness of the pipe wall (commercial steel ≈ 0.00015 ft) |
| Use the Moody chart or Colebrook equation equivalently; Haaland is calculator-friendly. |
Fully rough turbulent flow: f depends only on ε/D. Smooth-pipe turbulent flow: f depends only on Re.
Hazen-Williams — the water-industry shortcut
An empirical alternative, widely used for water distribution design. Simpler than Darcy-Weisbach, but restricted: water only, turbulent flow, ordinary temperatures.
V = 1.318 · C · R0.63 · S0.54 (English units, V in ft/s)
| C | Hazen-Williams roughness coefficient (≈ 150 new PVC, 130 new steel, 100 old cast iron) |
| R | hydraulic radius = A/P (D/4 for a full circular pipe) |
| S | slope of the energy grade line = hf/L |
SI form: V = 0.849·C·R0.63·S0.54 with V in m/s. C is not the Chézy C and is not the Darcy f.
Minor losses, series and parallel pipes
hm = K V²2g
| K | minor loss coefficient (entrance ≈ 0.5, exit = 1.0, valves/fittings from tables) |
| Equivalently, fittings can be converted to an equivalent length of straight pipe. |
Series pipes: Q is the same everywhere; head losses add. Parallel pipes: head loss is the same in each branch; discharges add.
For parallel pipes, write hf1 = hf2 and solve for the flow split — with Darcy-Weisbach this usually needs iteration.
PE depth: networks, force mains, and wet wells
A distribution network must satisfy two conditions simultaneously: continuity at every junction and energy conservation around every closed loop.
Node: ΣQin − ΣQout = qdemand Loop: ΣhL = 0
Write each pipe loss as hL = rQn with a sign that follows the assumed flow direction. For Darcy-Weisbach, n is approximately 2 when f is treated as fixed; for Hazen-Williams, n = 1.852. A negative corrected Q simply means the actual flow opposes the original arrow.
Wet-well cycle check
If inflow Qin is steady and one pump discharges Qp, a usable wet-well volume Vu fills in Vu/Qin while the pump is off and drains in Vu/(Qp−Qin) while it runs. The total cycle time is the sum. If Qp ≤ Qin, the level cannot recover.
PE trap: a force main uses pressure-pipe equations even though it carries wastewater. Do not use gravity-sewer Manning geometry once the conduit is flowing full under pressure.
Worked example Friction loss in a steel water main
Given:
- L = 1,000 ft of 12-in. commercial steel pipe (ε = 0.00015 ft).
- Q = 3 cfs of water at 60°F (ν = 1.22×10−5 ft²/s).
- Neglect minor losses.
Solution:
- V = Q/A = 3 / 0.785 = 3.82 ft/s. Re = VD/ν = 3.82×1.0/1.22×10−5 = 313,000 (turbulent).
- Relative roughness ε/D = 0.00015. Haaland: 1/√f = −1.8 log[(0.00015/3.7)1.11 + 6.9/313000] = 8.01, so f = 0.0156.
- hf = 0.0156 × (1000/1.0) × (3.82²/64.4) = 0.0156 × 1000 × 0.2266 = 3.53 ft.
Answer: Friction head loss ≈ 3.53 ft over the 1,000-ft run.