Flow Through Pipes Notes

Flow Through Pipes

Major loss — Darcy–Weisbach, Chezy, Manning and Hazen–Williams formulas; minor losses at sudden enlargement, contraction, entrance, exit, bends, fittings and obstructions; HGL and TEL; pipes in series (equivalent pipe) and parallel; branched pipes and three-reservoir problems; siphons; power transmission through pipes and maximum power condition; nozzle diameter for maximum power; pipe networks and the Hardy Cross method — with solved numericals.

Last reviewed 16 Sept 2026 · 6 min read

Losses in pipes

Energy losses are of two kinds:

Major loss — due to friction along the pipe length.
Minor losses — due to changes in velocity or direction: enlargements, contractions, entrance, exit, bends, valves and fittings. (In long pipelines these are small; in short pipes they can dominate.)

Major (friction) loss

FormulaDarcy–Weisbach equation

h_{f} = \frac{f L V ^{2}}{2 g D} = \frac{f L Q ^{2}}{12.1 D ^{5}}

$f$ = Darcy friction factor (dimensionless). Some texts use $h_{f} = \frac{4 f ^{'} L V ^{2}}{2 g D}$ with Fanning coefficient $f^{'} = f /4$ .

Wall shear stress: $τ_{0} = \frac{f ρ V ^{2}}{8}$

$h_{f} \propto Q^{2} / D^{5}$ for a given $f$ : halving the diameter increases friction loss about 32 times at the same discharge.
Laminar: $f = 64/ R e$ ; turbulent: from Blasius, Colebrook or the Moody diagram (see Laminar & Turbulent Flow).

Other empirical formulas

Formula	Expression	Notes
Chezy	$V = C mi$	$m = A / P = D /4$ (hydraulic mean depth); $i = h_{f} / L$ ; $C = 8 g / f$
Manning	$V = \frac{1}{n} R^{2/3} S^{1/2}$	Common for open channels and sewers
Hazen–Williams	$V = 0.849 C_{H W} R^{0.63} S^{0.54}$ (SI)	Water supply mains; $C_{H W}$ ≈ 130–150 for new smooth pipes, lower for old pipes

Minor losses

FormulaMinor losses

Situation	Head loss
Sudden enlargement	$h_{e} = \frac{( V _{1} - V _{2} ) ^{2}}{2 g}$
Sudden contraction	$h_{c} = (\frac{1}{C _{c}} - 1)^{2} \frac{V _{2}^{2}}{2 g}$ ≈ $0.5 \frac{V _{2}^{2}}{2 g}$ if $C_{c}$ not given (0.375 with $C_{c}$ = 0.62)
Entrance to pipe (sharp)	$0.5 \frac{V ^{2}}{2 g}$
Exit from pipe into a reservoir	$\frac{V ^{2}}{2 g}$
Obstruction (area $a$ in pipe area $A$ )	$(\frac{A}{C _{c} ( A - a )} - 1)^{2} \frac{V ^{2}}{2 g}$
Bends, valves, fittings	$k \frac{V ^{2}}{2 g}$ ( $k$ from tables)

Sudden enlargement loss follows from the momentum equation (Borda–Carnot).
Gradual expansions (cone angle about 6–8°) greatly reduce the loss.
Equivalent length of a fitting: $L_{e} = k D / f$ — the length of straight pipe with the same loss.

Hydraulic gradient and total energy lines

TEL drops by friction along the pipe and suddenly at each minor loss; just inside the entry the TEL is $0.5 V^{2} /2 g$ below the reservoir level and the HGL is $1.5 V^{2} /2 g$ below it.
At the exit into a reservoir the HGL meets the reservoir surface (all velocity head lost).
HGL above the pipe → positive pressure; below → negative pressure.

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