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.

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

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

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

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.

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.

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

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.

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:

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. The equation can be used to (iteratively) solve for the Darcy–Weisbach friction factor f.

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