with

Estimating f Graphically

f declines with increasing Re, e.g., increasing V at fixed D.

In laminar region, f = 64/Re

In turbulent region, for given e/D, f declines more slowly than in laminar region; eventually, the decline stops altogether.

Mathematical Expressions for f

Colebrook and Haaland eqns yield good estimates of f in turbulent flow

Useful for calculations in spreadsheets or special software for pipe flow analysis

1 2.712log3.7 ReD

f fe

1.111 6.91.8log3.7 ReD

fe

Dependence of hL on D and V

In laminar region:

In turbulent region, when f becomes constant:

Under typical water distribution conditions, hL in a given pipe can be expressed as kQn with n slightly <2.

2

2

64 322

'lamL

l V lh VDV D g g

kD

Q

For a given pipe

22

2L fullturb

fullturb

l Vh k QfD g

For a given pipe

Energy Losses in Bends, Valves, and Other Transitions (‘Minor

Losses’)

Minor head losses generally significant when pipe sections are short (e.g., household, not pipeline)Caused by turbulence associated with flow transition; therefore, mitigated by modifications that ‘smooth’ flow patternsGenerally much greater for expansions than for contractionsOften expressed as multiple of velocity head:K is the ratio of energy lost via friction in the device of interest to the kinetic energy of the water (upstream or downstream, depending on geometric details)

2

2L minorVh Kg

Energy Losses in Contractions22

2c cVh kg

Energy Losses in Expansions

2

2c

x

V Vh

g

22

2 2c

x,dischargeVVh

g g

22

2 2c

x,dischargeVVh

g g

Energy Losses in Pipe Fittings and Bends

2

2b bVh kg

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