This report reconciles the numerical results of both forms of the Colebrook-White Friction Factor equations. It should
be noted that although the Colebrook equation represents the data to within +/- 10 percent, the equations are not
reconciled to within the 5 decimal place. This reconciliation is of great importance to the researches of explicit-forms
of the Colebrook equation as it now can be compared to
either form-1 (Used in the generation of the Moody Diagram) or form-2, preferred by some investigators. The author prefers form-1 as it is the most compact-form
1. Fluid Flow 2.9
For fully developed laminar-viscous flow in a pipe, the loss is
evaluated from Equation (8) as follows:
(27)
where Thus, for laminar flow, the
friction factor varies inversely with the Reynolds number.
With turbulent flow, friction loss depends not only on flow con-
ditions, as characterized by the Reynolds number, but also on the
nature of the conduit wall surface. For smooth conduit walls, empir-
ical correlations give
(28a)
(28b)
Generally, f also depends on the wall roughness ε. The variation
is complex and best expressed in chart form (Moody 1944) as
shown in Figure 13. Inspection indicates that, for high Reynolds
numbers and relative roughness, the friction factor becomes inde-
pendent of the Reynolds number in a fully-rough flow regime. Then
(29a)
Values of f between the values for smooth tubes and those for the
fully-rough regime are represented by Colebrook’s natural rough-
ness function:
(29b)
A transition region appears in Figure 13 for Reynolds numbers
between 2000 and 10 000. Below this critical condition, for smooth
walls, Equation (27) is used to determine f ; above the critical con-
dition, Equation (28b) is used. For rough walls, Figure 13 or Equa-
tion (29b) must be used to assess the friction factor in turbulent flow.
To do this, the roughness height ε, which may increase with conduit
use or aging, must be evaluated from the conduit surface (Table 2).
Fig. 13 Relation Between Friction Factor and Reynolds Number
(Moody 1944)
HL( )f
L
ρg
------
8µV
R
2
----------
32LνV
D
2
g
-----------------
64
VD ν⁄
---------------
L
D
----
V
2
2g
------
= = =
Re VD ν and f 64 Re.⁄=⁄=
f
0.3164
Re
0.25
----------------= for Re 10
5
<
f 0.0032
0.221
Re
0.237
-----------------+= for 10
5
Re 3< < 10
6
×
1
f
--------- 1.14 2 log D ε⁄( )+=
1
f
--------- 1.14 2 log D ε⁄( )+= 2 log 1
9.3
Re ε D⁄( ) f
--------------------------------+–
The Reconciliation of the Colebrook-White Friction Factor Equations
by
Julio C. Banks, MSME, PE
e-Mail: Sell-A-Vision@Outlook.com
2001_Fundamentals_Handbook.pdf
2. Colebrook-White Friction Factor Equation Derivation 1 of 3.mcdx Page 1 of 2
Reference: "ASHRAE HVAC 2001 Fundamental Handbook" Equation 29b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
D
ε
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎜
⎝
+1 ――――
9.311
⋅⋅Re ―
ε
D
‾‾f
⎞
⎟
⎟
⎠
((29 b))
The constants 1.138 and 9.311 are results of the reconciliation by the author of the
contributing equations. Transform Equation 29b to equation 2
=λ ―
ε
D
((1))
=――
1
‾f
⋅−2 log
⎛
⎜
⎜
⎝
+―
λ
a
0
―――
b
0
⋅Re ‾f
⎞
⎟
⎟
⎠
((2))
Symbolic Solution
Substitute Eq. 1 into Eq. 29 b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
1
λ
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎝
+1 ――――
9.311
⋅⋅Re λ ‾‾f
⎞
⎟
⎠
((3))
=――
1
‾f
−(( −1.138 ⋅2 log((λ)))) ⋅2 log
⎛
⎜
⎝
⋅
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
―
1
λ
⎞
⎟
⎠
=――
1
‾f
−1.138 ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((4))
Let =⋅−2 log((x)) 1.138 ((5))
Solve for x
≔x =10
⎛
⎜⎝
−――
1.138
2
⎞
⎟⎠
0.2698 ((6))
Substitute Eq. 5 into Eq. 4
=――
1
‾f
−⋅−2 log((x)) ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((7))
Julio C. Banks, PE
3. Colebrook-White Friction Factor Equation Derivation 1 of 3.mcdx Page 2 of 2
=――
1
‾f
⋅−2 log
⎛
⎜
⎝
+⋅λ x ―――
⋅9.311 x
⋅Re ‾‾f
⎞
⎟
⎠
((8))
Recall Equaton 2
=――
1
‾f
⋅−2 log
⎛
⎜
⎜
⎝
+―
λ
a
0
―――
b
0
⋅Re ‾f
⎞
⎟
⎟
⎠
((2))
Compearing equations 8 and 2 provides for the numerical balues of the constant
parameters, and .a
0
b
0
≔a
0
=―
1
x
3.707 ((9))
and
≔b
0
=⋅9.311 x 2.512 ((10))
The standard Colebrook-White equation (used in the generation of the Moody Diagram) is
=――
1
‾f
⋅−2 log
⎛
⎜
⎝
+――
λ
3.7
―――
2.51
⋅Re ‾‾f
⎞
⎟
⎠
((11))
The author discoverd that the two versions of the Colebrook-White equations can be
reconciled within 5 significant figures only if the constants given by equations 9 and 10 are
used in Eq. 2 instead of using Eq. 11.
Julio C. Banks, PE
4. Colebrook-White Friction Factor Equation Derivation 2 of 3.mcdx Page 1 of 2
Reference: "ASHRAE HVAC 2001 Fundamental Handbook" Equation 29b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
D
ε
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎜
⎝
+1 ――――
9.311
⋅⋅Re ―
ε
D
‾‾f
⎞
⎟
⎟
⎠
((29 b))
The constants 1.138 and 9.311 are results of the reconciliation by the author of the
contributing equations. Transform Equation 29b to equation 2
=λ ―
ε
D
((1))
=――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎜
⎝
+⋅2 λ ―――
b
1
⋅Re ‾‾f
⎞
⎟
⎟
⎠
((2))
Symbolic Solution
Substitute Eq. 1 into Eq. 29 b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
1
λ
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎝
+1 ――――
9.311
⋅⋅Re λ ‾‾f
⎞
⎟
⎠
((3))
=――
1
‾f
−(( −1.138 ⋅2 log((λ)))) ⋅2 log
⎛
⎜
⎝
⋅
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
―
1
λ
⎞
⎟
⎠
=――
1
‾f
−1.138 ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((4))
Let =−1.74 ⋅2 log((x)) 1.138 ((5))
Solve for x
≔x =10
⎛
⎜⎝
――
0.602
2
⎞
⎟⎠
2.000 ((6))
Substitute Eq. 5 into Eq. 4
=――
1
‾f
−−1.74 ⋅2 log((x)) ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((7))
Julio C. Banks, PE
5. Colebrook-White Friction Factor Equation Derivation 2 of 3.mcdx Page 2 of 2
=――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎝
+⋅x λ ―――
⋅9.311 x
⋅Re ‾f
⎞
⎟
⎠
((8))
Recall Equaton 2
=――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎝
+⋅2 λ ―――
b1
⋅Re ‾‾f
⎞
⎟
⎠
((2))
≔b1 =⋅9.311 x 18.62 ((9))
The standard Colebrook-White equation (used in the generation of the Moody Diagram) is
Colebrook-White Form-1 Equation: =――
1
‾f
⋅−2 log
⎛
⎜
⎜
⎝
+―
λ
a
0
―――
b
0
⋅Re ‾f
⎞
⎟
⎟
⎠
((10))
≡a
0
b
0
⎡
⎣
⎤
⎦
3.707 2.512[[ ]]
An alternate equivalent-form of the Colebrook-White equation 10 is
Colebrook-White Form-2 Equation: =――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎜
⎝
+⋅2 λ ―――
b
1
⋅Re ‾‾f
⎞
⎟
⎟
⎠
((11))
=b1 18.62
The author discoverd that the two versions of the Colebrook-White equations, form-1 (Eq. 10)
and form-2 (Eq. 11) can be reconciled within 5 significant figures only if the constants given in
this reconciliation report are used instead of the original constant parameters pubished by
Colebrook.
Julio C. Banks, PE
6. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 1 of 5
Derivation of the Colebrook transformation of the Prandtl's equation for the friction factor
in smooth pipes in turbulent flow
by Julio C. Banks, PE
Reference
"Fundamentals of Pipe Flow" by Robert P. Bennedict. ISBN 0-471-03375-8
The Prandtl's equation for the friction factor in smooth pipes in turbulent flow is
=――
1
‾‾fS
−⋅2 log⎛
⎝ ⋅Re ‾‾fS
⎞
⎠ 0.8 ((6.12))
(Page 235 of the reference)
Colebrook took the Prandtl's Equation (6.12) and transformed it into a form
having the constant term, 1.74. Derive Colebrook's transformation of the
Prandtl's equation.
Let
=−0.8 −1.74 ⋅2 log((x)) ((1))
Solve for x from Eq. 1
=⋅2 log((x)) +1.74 0.8
=⋅2 log((x)) 2.54
≔x =10
――
2.54
2
18.62 ((2))
Substitute Eq. 1 into Eq. 6.12
=――
1
‾‾fS
+⋅2 log⎛
⎝ ⋅Re ‾‾fS
⎞
⎠ (( −1.74 ⋅2 log((x))))
=――
1
‾‾fS
−−1.74 ⋅2 log((x)) ⋅2 log
⎛
⎜
⎝
―――
1
⋅Re ‾‾fS
⎞
⎟
⎠
=――
1
‾‾fS
−1.74 ⋅2 log
⎛
⎜
⎝
―――
x
⋅Re ‾‾fS
⎞
⎟
⎠
((3))
Julio C. Banks, PE
7. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 2 of 5
Substitute the numberical value of the constant parameter, x, from Eq. 3 into Eq. 4
=――
1
‾‾fS
−1.74 ⋅2 log
⎛
⎜
⎝
―――
18.62
⋅Re ‾‾fS
⎞
⎟
⎠
((4))
Equation 4 is the Colebrook transformation (first equation on page 239 of the reference) of
the Prandtl's equation 6.12. Subsequently, Colebrook took Eq. 4 and and transformed it into
the form:
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
α
⋅Re ‾‾fS
⎞
⎟
⎠
((5))
Let =⋅−2 log((y)) 1.74 ((6))
Solve for y from Eq. 6
≔y =10
−
⎛
⎜⎝
――
1.74
2
⎞
⎟⎠
0.1349 ((7))
Substitute Eq. 6 into Eq. 4
=――
1
‾‾fS
−⋅−2 log((y)) ⋅2 log
⎛
⎜
⎝
―――
18.62
⋅Re ‾‾fS
⎞
⎟
⎠
((4))
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
⋅18.62 y
⋅Re ‾‾fS
⎞
⎟
⎠
((8))
Comparing equations 5 and 8 it can be seen that
≔α =⋅18.62 y 2.512 ((9))
Substitute Eq. 9 into Eq. 5 we get the following equation
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
2.512
⋅Re ‾‾fS
⎞
⎟
⎠
((10))
Julio C. Banks, PE
8. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 3 of 5
Colbrook took the von Karman's fully rough pipe law of friction, namely Eq. 6.15 (page
257 of the reference) and transformed it into Eq. 11
=――
1
‾‾fR
+⋅2 log
⎛
⎜⎝
―
R
ε
⎞
⎟⎠
1.74 ((6.15))
=――
1
‾‾fR
⋅−2 log
⎛
⎜
⎜
⎝
―
―
ε
D
β
⎞
⎟
⎟
⎠
((11))
Let =λ ―
ε
D
Substitue Eq. 12 into Eq. 6.15 and 11
=――
1
‾‾fR
+⋅−2 log(( ⋅2 λ)) 1.74 ((12))
=――
1
‾‾fR
⋅−2 log
⎛
⎜⎝
―
λ
β
⎞
⎟⎠
((13))
Substitute Eq. 6 into Eq. 12
=――
1
‾‾fR
+⋅−2 log(( ⋅2 λ)) ⋅−2 log((y)) ((14))
=――
1
‾‾fR
⋅−2 log(( ⋅⋅2 y λ)) ((15))
Comparing equations 13 and 15 we obtain the -parameterβ
≔β =――
1
⋅2 y
3.707 ((16))
Substitute Eq. 16 into Eq. 13
=――
1
‾‾fR
⋅−2 log
⎛
⎜⎝
――
λ
3.707
⎞
⎟⎠
((17))
Julio C. Banks, PE
9. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 4 of 5
Colebrook then took equations 5 and 13 and combined them into a single expression
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
α
⋅Re ‾‾fS
⎞
⎟
⎠
((5))
≔α =⋅18.62 y 2.512 ((9))
=――
1
‾‾fR
⋅−2 log
⎛
⎜⎝
―
λ
β
⎞
⎟⎠
((13))
≔β =――
1
⋅2 y
3.707 ((16))
Solve for the arguments of the logarithms for subsequent combination
=10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾‾fR
⎞
⎟
⎠
―
λ
β
((17))
=10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fS
⎞
⎟
⎠
―――
α
⋅Re ‾‾fS
((18))
Add equations 17 and 18
=+10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾‾fR
⎞
⎟
⎠
10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fS
⎞
⎟
⎠
+―
λ
β
―――
α
⋅Re ‾‾fS
((19))
Let =10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fT
⎞
⎟
⎠
+10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fR
⎞
⎟
⎠
10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fS
⎞
⎟
⎠
((20))
Where is the Transition Friction Factor.fT
Substitute Eq. 20 into Eq. 19
=10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fT
⎞
⎟
⎠
+―
λ
β
―――
α
⋅Re ‾‾fT
((21))
Julio C. Banks, PE
10. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 5 of 5
Solve for from Eq. 21――
1
‾‾fT
=――
1
‾‾fT
⋅−2 log
⎛
⎜
⎝
+―
λ
β
―――
α
⋅Re ‾‾fT
⎞
⎟
⎠
((22))
Where
≔α =⋅18.62 y 2.512 ((9))
≔β =――
1
⋅2 y
3.707 ((16))
Julio C. Banks, PE
11. FUNDAMENTALS OF
PIPE FLOW
Julio e. :Bank:S.
Robert P. Benedict
Fellow Mechanical Engineer
Westinghouse Electric Corporation
Steam Turbine Division
Adjunct Professor ofMechanical Engineering
Drexel University
Evening College
Philadelphia, Pennsylvania
A WILEY-INTERSCIENCE PUBLICATION
JOHN WILEY & SONS
New York • Chichester • Brisbane • Toronto • Singapore
3
14. Errata
The numerical results do not correspond to
the Colebrook Equation 6.19 (page 240) but
the alternative Form-1 Equation on page 239
which I labeled 6.19b.