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 Dynamic viscosity, and ρ the fluid's density.
• The pipe's relative roughness ε / D, where ε is the pipe's effective roughness height and D the pipe (inside) diameter.
• 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.
• 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.

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 accuracy
• 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:

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

or

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

where:

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

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

${\displaystyle x=-2\log(ax+b)}$

or

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

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

will get:

${\displaystyle p^{x}=ax+b}$

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

then:

${\displaystyle f={\frac {1}{\left({\dfrac {2W\left({\frac {\ln 10}{2a}}\,10^{\frac {b}{2a}}\right)}{\ln 10}}-{\dfrac {b}{a}}\right)^{2}}}}$

Expanded forms

Additional, mathematically equivalent forms of the Colebrook equation are:

${\displaystyle \frac{1}{\sqrt{f}}}$Zdroj:https://en.wikipedia.org?pojem=Colebrook_equation
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