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Everything you did (not) want to know about the Darcy-Weisbach Equation

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D-W Equation

g

V

D

fLhl 2

2

f = (V, D, roughness & viscosity)

hl

D VL

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f /D

Re

Friction Factor, f

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Julius Weisbach, 1845

g

V

D

fLhl 2

2

Vf

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Antoine Chézy, ~ 1770

SAPV 2

RSCV

22

4V

D

L

Chl

g

C

f 8

1

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Poiseuille (1841) & Hagen (1839)

g

V

D

Lhl 2

642

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Osborne Reynolds, 1883

VD

Re

Re

64f laminar

flow

Laminar < 2,0002,000 > Critical > 4,000

Turbulent > 4,000

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f /D

Re

Friction Factor, f

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Henry Darcy, 1857

2

2

'' V

DV

DD

Lhl

2"" V

DD

Lhl

For cast iron at high flow

For all conditions

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John Fanning, 1877

g

V

R

mLhl 2

2

4

fm

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Ludwig Prandtl

originator of boundary layer theory and advisor to von Kármán, Blasius, Nikuradse and others

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Paul Blasius, 1913

4/1

3164.0

Ref

for smooth pipes

(using data from Nikuradse)

08.0log21

ff

Re

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f /D

Re

Friction Factor, f

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Theodor von Kármán, 1930

Df

log214.1

1

for rough pipes at high Re

(based on data from Nikuradse)

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f /D

Re

Friction Factor, f

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Johann Nikuradse, 1933

artificial sand roughness

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Cyril F. Colebrook, 1939

fDf Re

35.9log214.1

1 for commercial pipe in transition zone

Colebrook and White completed several papers on pipe friction in the 1930’s. The last under Colebrook’s name contained,

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f /D

Re

Friction Factor, f

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Hunter Rouse, 1942

integration

f

Re

f

1

fRe

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Lewis Moody, 1944

“convenient form”

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A Rose by Any Other NameThe D-W equation has had a confusing history of nomenclature and usage.•Weisbach’s Eq.•Darcy’s Eq.•Chezy’s Eq.•Fanning’s Eq.(still used in Chemical Eng.)•No Name or Pipe Flow Eq.•Darcy-Weisbach – Popularized by Rouse

and adopted by ASCE in 1962.

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Misnamed?

The f vs Re diagram is almost universally credited to Moody, and the contributions of others are seldom acknowledged.

This was a sore point for Hunter Rouse even though he had sincere respect for Moody. In 1976 Rouse wrote in the third person,

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Misnamed? (cont)

"After the conference, Lewis Moody of Princeton suggested using the latter variables (f and Re) as primary rather than supplementary, as in the past, but Rouse resisted the temptation because he felt that to do would be a step backward. So Moody himself published such a plot, and it is known around the world as the Moody diagram!"

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Closing Comments

•Little has changed in the D-W application since Moody’s 1944 publication.

The critical zone is still undefined.Pipe roughness is difficult to estimate.

•It is thus surprising that the f diagram has not been modified or replaced over the last 58 years.

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A Warning from Henry Darcy

In a letter to Henri Bazin, Darcy quoted François I who wrote, "Often women vary: he is quite mad

who has faith in them",

which Darcy amended with,

"is no less true for hydraulic coefficients than for women."