Darcy friction factor formulae

In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open-channel flow.

The Darcy friction factor is also known as the Darcy–Weisbach friction factor, resistance coefficient or simply friction factor; by definition it is four times larger than the Fanning friction factor.^[1]

Notation

In this article, the following conventions and definitions are to be understood:

The Reynolds number Re is taken to be Re = V D / ν, where V is the mean velocity of fluid flow, D is the pipe diameter, and where
ν is the kinematic viscosity μ / ρ, with μ the fluid's viscosity, and ρ the fluid's density.
f stands for the Darcy friction factor. Its value depends on the flow's Reynolds number Re and on
The pipe's relative roughness ε / D, where ε is the pipe's effective roughness height and D the pipe (inside) diameter.
The log function is understood to be base-10 (as is customary in engineering fields): if x = log(y), then y = 10^x.
The ln function is understood to be base-e: if x = ln(y), then y = e^x.

Flow regime

Which friction factor formula may be applicable depends upon the type of flow that exists:

Laminar flow
Transition between laminar and turbulent flow
Fully turbulent flow in smooth conduits
Fully turbulent flow in rough conduits
Free surface flow.

Laminar flow

The Darcy friction factor f for laminar flow in a circular pipe (Reynolds number less than 2320) is given by the formula

f={\frac {64}{{\mathrm {Re}}}}

Transition flow

Transition (neither fully laminar nor fully turbulent) flow occurs in the range of Reynolds numbers between 2300 and 4000. The value of the Darcy friction factor is subject to large uncertainties in this flow regime.

Turbulent flow in smooth conduits

The Blasius correlation is the simplest equation for computing the Darcy friction factor. Because the Blasius correlation has no term for pipe roughness, it is valid only to smooth pipes. However, the Blasius correlation is sometimes used in rough pipes because of its simplicity. The Blasius correlation is valid up to the Reynolds number 100000.

Turbulent flow in rough conduits

The Darcy friction factor for fully turbulent flow (Reynolds number greater than 4000) in rough conduits can be modeled by the Colebrook–White equation.

Free surface flow

The last formula in the Colebrook equation section of this article is for free surface flow. The approximations elsewhere in this article are not applicable for this type of flow.

Choosing a formula

Before choosing a formula it is worth knowing that in the paper on the Moody chart, Moody stated the accuracy is about ±5% for smooth pipes and ±10% for rough pipes. If more than one formula is applicable in the flow regime under consideration, the choice of formula may be influenced by one or more of the following:

Required precision
Speed of computation required
Available computational technology:
- calculator (minimize keystrokes)
- spreadsheet (single-cell formula)
- programming/scripting language (subroutine).

Colebrook–White equation

The phenomenological Colebrook–White equation (or Colebrook equation) expresses the Darcy friction factor f as a function of Reynolds number Re and pipe relative roughness ε / D_h, fitting the data of experimental studies of turbulent flow in smooth and rough pipes.^[2]^[3] The equation can be used to (iteratively) solve for the Darcy–Weisbach friction factor f.

For a conduit flowing completely full of fluid at Reynolds numbers greater than 4000, it is expressed as:

{\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon }{3.7D_{\mathrm {h} }}}+{\frac {2.51}{\mathrm {Re} {\sqrt {f}}}}\right)

{\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon }{14.8R_{\mathrm {h} }}}+{\frac {2.51}{\mathrm {Re} {\sqrt {f}}}}\right)

where:

Hydraulic diameter, $D_{{\mathrm {h}}}$ (m, ft) – For fluid-filled, circular conduits, $D_{{\mathrm {h}}}$ = D = inside diameter
Hydraulic radius, $R_{{\mathrm {h}}}$ (m, ft) – For fluid-filled, circular conduits, $R_{{\mathrm {h}}}$ = D/4 = (inside diameter)/4

Note: Some sources use a constant of 3.71 in the denominator for the roughness term in the first equation above.^[4]

Solving

The Colebrook equation is usually solved numerically due to its implicit nature. Recently, the Lambert W function has been employed to obtain explicit reformulation of the Colebrook equation.^[5]^[6]^[7]

$x={\frac {1}{\sqrt {f}}},b={\frac {\varepsilon }{14.8R_{h}}},a={\frac {2.51}{Re}}$

$x=-2log(ax+b)$

$10^{-{\frac {x}{2}}}=ax+b$

$p=10^{-{\frac {1}{2}}}$

will get:

p^{x}=ax+b

x=-{\frac {W(-{\frac {\ln p}{a}}\,p^{-{\frac {b}{a}}})}{\ln p}}-{\frac {b}{a}}

then:

f={\frac {1}{({\frac {2W({\frac {\ln 10}{2a}}\,10^{\frac {b}{2a}})}{\ln 10}}-{\frac {b}{a}})^{2}}}

Expanded forms

Additional, mathematically equivalent forms of the Colebrook equation are:

{\frac {1}{\sqrt {f}}}=1.7384\ldots -2\log \left({\frac {2\varepsilon }{D_{\mathrm {h} }}}+{\frac {18.574}{\mathrm {Re} {\sqrt {f}}}}\right)

where:

1.7384... = 2 log (2 × 3.7) = 2 log (7.4)

18.574 = 2.51 × 3.7 × 2

and

{\frac {1}{\sqrt {f}}}=1.1364\ldots +2\log \left(D_{\mathrm {h} }/\varepsilon \right)-2\log \left(1+{\frac {9.287}{\mathrm {Re} (\varepsilon /D_{\mathrm {h} }){\sqrt {f}}}}\right)

{\frac {1}{\sqrt {f}}}=1.1364\ldots -2\log \left({\frac {\varepsilon }{D_{\mathrm {h} }}}+{\frac {9.287}{\mathrm {Re} {\sqrt {f}}}}\right)

where:

1.1364... = 1.7384... − 2 log (2) = 2 log (7.4) − 2 log (2) = 2 log (3.7)

9.287 = 18.574 / 2 = 2.51 × 3.7.

The additional equivalent forms above assume that the constants 3.7 and 2.51 in the formula at the top of this section are exact. The constants are probably values which were rounded by Colebrook during his curve fitting; but they are effectively treated as exact when comparing (to several decimal places) results from explicit formulae (such as those found elsewhere in this article) to the friction factor computed via Colebrook's implicit equation.

Equations similar to the additional forms above (with the constants rounded to fewer decimal places, or perhaps shifted slightly to minimize overall rounding errors) may be found in various references. It may be helpful to note that they are essentially the same equation.

Free surface flow

Another form of the Colebrook-White equation exists for free surfaces. Such a condition may exist in a pipe that is flowing partially full of fluid. For free surface flow:

{\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon }{12R_{\mathrm {h} }}}+{\frac {2.51}{\mathrm {Re} {\sqrt {f}}}}\right).

Approximations of the Colebrook equation

Haaland equation

The Haaland equation was proposed in 1983 by Professor S.E. Haaland of the Norwegian Institute of Technology.^[8] It is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation, but the discrepancy from experimental data is well within the accuracy of the data.

The Haaland equation^[9] is expressed:

{\frac {1}{\sqrt {f}}}=-1.8\log \left[\left({\frac {\varepsilon /D}{3.7}}\right)^{1.11}+{\frac {6.9}{\mathrm {Re} }}\right]

Swamee–Jain equation

The Swamee–Jain equation is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation.^[10]

{\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {5.74}{\mathrm {Re} ^{0.9}}}\right)

Serghides's solution

Serghides's solution is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. It was derived using Steffensen's method.^[11]

The solution involves calculating three intermediate values and then substituting those values into a final equation.

A=-2\log \left({\frac {\varepsilon /D}{3.7}}+{12 \over \mathrm {Re} }\right)

B=-2\log \left({\frac {\varepsilon /D}{3.7}}+{2.51A \over \mathrm {Re} }\right)

C=-2\log \left({\frac {\varepsilon /D}{3.7}}+{2.51B \over \mathrm {Re} }\right)

{\frac {1}{\sqrt {f}}}=A-{\frac {(B-A)^{2}}{C-2B+A}}

The equation was found to match the Colebrook–White equation within 0.0023% for a test set with a 70-point matrix consisting of ten relative roughness values (in the range 0.00004 to 0.05) by seven Reynolds numbers (2500 to 10⁸).

Goudar–Sonnad equation

Goudar equation is the most accurate approximation to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. Equation has the following form^[12]

a={2 \over \ln(10)}

b={\frac {\varepsilon /D}{3.7}}

d={\ln(10)\mathrm {Re}  \over 5.02}

s={bd+\ln(d)}

q={{s}^{{s/(s+1)}}}

g={bd+\ln {d \over q}}

z={\ln {q \over g}}

D_{{LA}}=z{{g \over {g+1}}}

D_{{CFA}}=D_{{LA}}\left(1+{\frac {z/2}{(g+1)^{2}+(z/3)(2g-1)}}\right)

{\frac {1}{\sqrt {f}}}={a\left[\ln \left(d/q\right)+D_{CFA}\right]}

Brkić solution

Brkić shows one approximation of the Colebrook equation based on the Lambert W-function^[13]

S=\ln {\frac {\mathrm {Re} }{\mathrm {1.816\ln {\frac {1.1\mathrm {Re} }{\ln \left(1+1.1\mathrm {Re} \right)}}} }}

{\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.71}}+{2.18S \over \mathrm {Re} }\right)

The equation was found to match the Colebrook–White equation within 3.15%.

Blasius correlations

Early approximations for smooth pipes^[14] by Paul Richard Heinrich Blasius in terms of the Moody friction factor are given in one article of 1913:^[15]

f=.316{\mathrm {Re}}^{{-{1 \over 4}}}

Johann Nikuradse in 1932 proposed that this corresponds to a power law correlation for the fluid velocity profile.

Mishra and Gupta in 1979 proposed a correction for curved or helically coiled tubes, taking into account the equivalent curve radius, R_c:^[16]

f=0.316{\mathrm {Re}}^{{-{1 \over 4}}}+0.0075{\sqrt {{\frac {D}{2R_{c}}}}}

with,

R_{c}=R\left[1+\left({\frac {H}{2\pi R}}\right)^{2}\right]

where f is a function of:

Pipe diameter, D (m, ft)
Curve radius, R (m, ft)
Helicoidal pitch, H (m, ft)
Reynolds number, Re (dimensionless)

valid for:

Re_tr < Re < 10⁵
6.7 < 2R_c/D < 346.0
0 < H/D < 25.4

Table of Approximations

The following table lists historical approximations to the Colebrook–White relation^[17] Note that the Churchill equation^[18] (1977) is the only one that returns a correct value for friction factor in the laminar flow region (Reynolds number < 2300). All of the others are for transitional and turbulent flow only.

Table of Colebrook equation approximations
Equation	Author	Year	Range
$f=.0055\left[1+\left(2\times 10^{4}\cdot {\frac {\varepsilon }{D}}+{\frac {10^{6}}{\mathrm {Re} }}\right)^{\frac {1}{3}}\right]$	Moody	1947	$Re=4000-5.10^{8}$ $\varepsilon /D=0-0.01$
$f=.094\left({\frac {\varepsilon }{D}}\right)^{0.225}+0.53\left({\frac {\varepsilon }{D}}\right)+88\left({\frac {\varepsilon }{D}}\right)^{0.44}\cdot {\mathrm {Re} }^{-{\Psi }}$ where $\Psi =1.62\left({\frac {\varepsilon }{D}}\right)^{0.134}$	Wood	1966	$Re=4000-5.10^{7}$ $\varepsilon /D=0.00001-0.04$
${\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.715}}+{\frac {15}{\mathrm {Re} }}\right)$	Eck	1973
${\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {5.74}{\mathrm {Re} ^{0.9}}}\right)$	Swamee and Jain	1976	$Re=5000-10^{8}$ $\varepsilon /D=0.000001-0.05$
${\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.71}}+\left({\frac {7}{\mathrm {Re} }}\right)^{0.9}\right)$	Churchill	1973	Not specified
${\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.715}}+\left({\frac {6.943}{\mathrm {Re} }}\right)^{0.9}\right)$	Jain	1976
$f=8\left[\left({\frac {8}{\mathrm {Re} }}\right)^{12}+{\frac {1}{(\Theta _{1}+\Theta _{2})^{1.5}}}\right]^{\frac {1}{12}}$ where $\Theta _{1}=\left[-2.457\ln \left(\left({\frac {7}{\mathrm {Re} }}\right)^{0.9}+0.27{\frac {\varepsilon }{D}}\right)\right]^{16}$ $\Theta _{2}=\left({\frac {37530}{\mathrm {Re} }}\right)^{16}$	Churchill	1977
${\frac {1}{\sqrt {f}}}=-2\log \left[{\frac {\varepsilon /D}{3.7065}}-{\frac {5.0452}{\mathrm {Re} }}\log \left({\frac {1}{2.8257}}\left({\frac {\varepsilon }{D}}\right)^{1.1098}+{\frac {5.8506}{\mathrm {Re} ^{0.8981}}}\right)\right]$	Chen	1979	$Re=4000-4.10^{8}$
${\frac {1}{\sqrt {f}}}=1.8\log \left[{\frac {\mathrm {Re} }{0.135\mathrm {Re} (\varepsilon /D)+6.5}}\right]$	Round	1980
${\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {4.518\log \left({\frac {\mathrm {Re} }{7}}\right)}{\mathrm {Re} \left(1+{\frac {\mathrm {Re} ^{0.52}}{29}}(\varepsilon /D)^{0.7}\right)}}\right)$	Barr	1981
${\frac {1}{\sqrt {f}}}=-2\log \left[{\frac {\varepsilon /D}{3.7}}-{\frac {5.02}{\mathrm {Re} }}\log \left({\frac {\varepsilon /D}{3.7}}-{\frac {5.02}{\mathrm {Re} }}\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {13}{\mathrm {Re} }}\right)\right)\right]$ or ${\frac {1}{\sqrt {f}}}=-2\log \left[{\frac {\varepsilon /D}{3.7}}-{\frac {5.02}{\mathrm {Re} }}\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {13}{\mathrm {Re} }}\right)\right]$	Zigrang and Sylvester	1982
${\frac {1}{\sqrt {f}}}=-1.8\log \left[\left({\frac {\varepsilon /D}{3.7}}\right)^{1.11}+{\frac {6.9}{\mathrm {Re} }}\right]$	Haaland^[9]	1983
${\frac {1}{\sqrt {f}}}=\Psi _{1}-{\frac {(\Psi _{2}-\Psi _{1})^{2}}{\Psi _{3}-2\Psi _{2}+\Psi _{1}}}$ or ${\frac {1}{\sqrt {f}}}=4.781-{\frac {(\Psi _{1}-4.781)^{2}}{\Psi _{2}-2\Psi _{1}+4.781}}$ where $\Psi _{1}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {12}{\mathrm {Re} }}\right)$ $\Psi _{2}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {2.51\Psi _{1}}{\mathrm {Re} }}\right)$ $\Psi _{3}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {2.51\Psi _{2}}{\mathrm {Re} }}\right)$	Serghides	1984
$A=0.11\left({\frac {68}{Re}}+\varepsilon \right)^{0.25}$ if $A\geq 0.018$ then $f=A$ and if $A<0.018$ then $f=0.0028+0.85A$	Tsal	1989
${\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {95}{\mathrm {Re} ^{0.983}}}-{\frac {96.82}{\mathrm {Re} }}\right)$	Manadilli	1997	$Re=4000-10^{8}$ $\varepsilon /D=0-0.05$
${\frac {1}{\sqrt {f}}}=-2\log \left\lbrace {\frac {\varepsilon /D}{3.7065}}-{\frac {5.0272}{\mathrm {Re} }}\log \left[{\frac {\varepsilon /D}{3.827}}-{\frac {4.657}{\mathrm {Re} }}\log \left(\left({\frac {\varepsilon /D}{7.7918}}\right)^{0.9924}+\left({\frac {5.3326}{208.815+\mathrm {Re} }}\right)^{0.9345}\right)\right]\right\rbrace$	Monzon, Romeo, Royo	2002
${\frac {1}{\sqrt {f}}}=0.8686\ln \left[{\frac {0.4587\mathrm {Re} }{(S-0.31)^{\frac {S}{(S+1)}}}}\right]$ where: $S=0.124\mathrm {Re} {\frac {\varepsilon }{D}}+\ln(0.4587\mathrm {Re} )$	Goudar, Sonnad	2006
${\frac {1}{\sqrt {f}}}=0.8686\ln \left[{\frac {0.4587\mathrm {Re} }{(S-0.31)^{\frac {S}{(S+0.9633)}}}}\right]$ where: $S=0.124\mathrm {Re} {\frac {\varepsilon }{D}}+\ln(0.4587\mathrm {Re} )$	Vatankhah, Kouchakzadeh	2008
${\frac {1}{\sqrt {f}}}=\alpha -{\frac {\alpha +2\log \left({\frac {\mathrm {B} }{\mathrm {Re} }}\right)}{1+{\frac {2.18}{\mathrm {B} }}}}$ where $\alpha ={\frac {0.744\ln(\mathrm {Re} )-1.41}{1+1.32{\sqrt {\varepsilon /D}}}}$ $\mathrm {B} ={\frac {\varepsilon /D}{3.7}}\mathrm {Re} +2.51\alpha$	Buzzelli	2008
$f={\frac {6.4}{(\ln(\mathrm {Re} )-\ln(1+.01\mathrm {Re} {\frac {\varepsilon }{D}}(1+10{\sqrt {\frac {\varepsilon }{D}}})))^{2.4}}}$	Avci, Kargoz	2009
$f={\frac {0.2479-0.0000947(7-\log \mathrm {Re} )^{4}}{(\log \left({\frac {\varepsilon /D}{3.615}}+{\frac {7.366}{\mathrm {Re} ^{0.9142}}}\right))^{2}}}$	Evangelides, Papaevangelou, Tzimopoulos	2010
$f=1.613\left[\ln \left(0.234\varepsilon ^{1.1007}-{\frac {60.525}{Re^{1.1105}}}+{\frac {56.291}{Re^{1.0712}}}\right)\right]^{-2}$	Fang	2011
$f=\left[-2\log \left({\frac {2.18\beta }{Re}}+{\frac {\varepsilon }{3.71}}\right)\right]^{-2}$ , $\beta =\ln {\frac {Re}{1.816\ln \left({\frac {1.1Re}{\ln \left(1+1.1Re\right)}}\right)}}$	Brkić	2011
$f=1.325474505\log _{e}\left(A-0.8686068432B\log _{e}\left(A-0.8784893582B\log _{e}\left(A+(1.665368035B)^{0.8373492157}\right)\right)\right)^{-2}$ where $A={\frac {\varepsilon /D}{3.7065}}$ $B={\frac {2.5226}{\mathrm {Re} }}$	Alashkar	2012

References

↑ Manning, Francis S.; Thompson, Richard E. (1991). Oilfield Processing of Petroleum. Vol. 1: Natural Gas. PennWell Books. ISBN 0-87814-343-2. , 420 pages. See page 293.
↑ Colebrook, C. F.; White, C. M. (1937). "Experiments with Fluid Friction in Roughened Pipes". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 161 (906): 367–381. Bibcode:1937RSPSA.161..367C. doi:10.1098/rspa.1937.0150. Often erroneously cited as the source of the Colebrook-White equation. This is partly because Colebrook (in a footnote in his 1939 paper) acknowledges his debt to White for suggesting the mathematical method by which the smooth and rough pipe correlations could be combined.
↑ Colebrook, C F (1939). "TURBULENT FLOW IN PIPES, WITH PARTICULAR REFERENCE TO THE TRANSITION REGION BETWEEN THE SMOOTH AND ROUGH PIPE LAWS.". Journal of the ICE. 11 (4): 133–156. ISSN 0368-2455. doi:10.1680/ijoti.1939.13150.
↑ VDI Gesellschaft (2010). VDI Heat Atlas. Springer. ISBN 978-3-540-77876-9.
↑ More, A. A. (2006). "Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes". Chemical Engineering Science. 61 (16): 5515–5519. doi:10.1016/j.ces.2006.04.003.
↑ Brkić, D. (2012). "Lambert W Function in Hydraulic Problems" (PDF). Mathematica Balkanica. 26 (3–4): 285–292.
↑ Keady, G. (1998). "Colebrook-White Formula for Pipe Flows". Journal of Hydraulic Engineering. 124 (1): 96–97. doi:10.1061/(ASCE)0733-9429(1998)124:1(96).
↑ Haaland, SE (1983). "Simple and Explicit Formulas for the Friction Factor in Turbulent Flow". Journal of Fluids Engineering. 105 (1): 89–90. doi:10.1115/1.3240948.
1 2 Massey, Bernard Stanford (1989). Mechanics of fluids. Chapman & Hall. ISBN 978-0-412-34280-6.
↑ Swamee, P.K.; Jain, A.K. (1976). "Explicit equations for pipe-flow problems". Journal of the Hydraulics Division. 102 (5): 657–664.
↑ T.K, Serghides (1984). "Estimate friction factor accurately". Chemical Engineering Journal. 91 (5): 63–64. ISSN 0009-2460.
↑ Goudar, C. T; Sonnad, J. R. (2008). "Comparison of the iterative approximations of the Colebrook-White equation: Here's a review of other formulas and a mathematically exact formulation that is valid over the entire range of Re values". Hydrocarbon processing. 87 (8).
↑ Brkić, Dejan (2011). "An Explicit Approximation of Colebrook’s equation for fluid flow friction factor". Petroleum Science and Technology. 29 (15): 1596–1602. doi:10.1080/10916461003620453.
↑ Massey, B. S. (2006). Mechanics of Fluids (8th ed.). Chapter 7 eq 7.5: Taylor & Francis. p. 254. ISBN 0-415-36205-9.
↑ Trinh, Khanh Tuoc, On the Blasius correlation for friction factors (PDF), arXiv:1007.2466 
↑ Bejan, Adrian; Kraus, Allan D. (2003). Heat Transfer Handbook. John Wiley & Sons. ISBN 978-0-471-39015-2.
↑ Beograd, Dejan Brkić (March 2012). "Determining Friction Factors in Turbulent Pipe Flow". Chemical Engineering: 34–39. (subscription required)
↑ Churchill, S.W. (November 7, 1977). "Friction-factor equation spans all fluid-flow regimes". Chemical Engineering: 91–92.

External links

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Darcy friction factor formulae

Notation

Flow regime

Laminar flow

Transition flow

Turbulent flow in smooth conduits

Turbulent flow in rough conduits

Free surface flow

Choosing a formula

Colebrook–White equation

Solving

Expanded forms

Free surface flow

Approximations of the Colebrook equation

Haaland equation

Swamee–Jain equation

Serghides's solution

Goudar–Sonnad equation

Brkić solution

Blasius correlations

Table of Approximations

References

Further reading

External links