Transcript The History of the Darcy-Weisbach Equation for Pipe Flow Resistance

Everything you did (not) want
to know about the DarcyWeisbach Equation
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D-W Equation
hl
2
fL V
hl 
D 2g
V
D
L
f = f (V, D, roughness & viscosity)
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Friction Factor, f
e/D
f
Re
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Julius Weisbach, 1845
2
fL V
hl 
D 2g
f  

V
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Antoine Chézy, ~ 1770
V P AS
2
V  C RS
4 L 2
hl  2 V
C D
1
f

C
8
g
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Poiseuille
(1841)
& Hagen
(1839)
L V
hl  64 2
D 2g
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Osborne Reynolds, 1883
Re 
VD

Laminar < 2,000
2,000 > Critical > 4,000
Turbulent > 4,000
64
f 
Re
laminar flow
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Friction Factor, f
e/D
f
Re
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Henry Darcy, 1857
For cast iron at high flow
L
" 2
hl   "  V
D
D
For all conditions
L 
 
' 2 

hl    2  V   '  V 
D 
D 
D 

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John Fanning, 1877
2
m LV
hl 
R 2g
f
m
4
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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
for smooth pipes
0.3164
f 
1/ 4
Re
(using data from Nikuradse)
1
f


 2 log Re f  0.08
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Friction Factor, f
e/D
f
Re
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Theodor von Kármán, 1930
for rough pipes at high Re
e 
 1.14  2 log 
f
 D
1
(based on data from Nikuradse)
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Friction Factor, f
e/D
f
Re
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Johann Nikuradse, 1933
artificial sand roughness
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Cyril F. Colebrook, 1939
Colebrook and White
completed several papers on
pipe friction in the 1930’s.
The last under Colebrook’s
name contained,
for commercial pipe in transition zone
e
9
.
35
 1.14  2 log 
 D Re f
f

1
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



Friction Factor, f
e/D
f
Re
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Hunter Rouse, 1942
integration
Re
1
f
f
Re
f
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Lewis Moody, 1944
“convenient form”
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A Rose by Any Other Name
The 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."
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