In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after Henry Darcy and Julius Weisbach. Currently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the Moody diagram or Colebrook equation.
The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.
Pressure-loss equation
In a cylindrical pipe of uniform diameter , flowing full, the pressure loss due to viscous effects is proportional to length and can be characterized by the Darcy–Weisbach equation:
where the pressure loss per unit length (SI units: Pa/m) is a function of: , the density of the fluid (kg/m3); , the hydraulic diameter of the pipe (for a pipe of circular section, this equals ; otherwise for a pipe of cross-sectional area and perimeter ) (m); , the mean flow velocity, experimentally measured as the volumetric flow rate per unit cross-sectional wetted area (m/s); , the Darcy friction factor (also called flow coefficient ). For laminar flow in a circular pipe of diameter , the friction factor is inversely proportional to the Reynolds number alone () which itself can be expressed in terms of easily measured or published physical quantities (see section below). Making this substitution the Darcy–Weisbach equation is rewritten as
where
is the dynamic viscosity of the fluid (Pa·s = N·s/m2 = kg/(m·s)); is the volumetric flow rate, used here to measure flow instead of mean velocity according to (m3/s).
Note that this laminar form of Darcy–Weisbach is equivalent to the Hagen–Poiseuille equation, which is analytically derived from the Navier–Stokes equations.
Head-loss form
The head loss (or ) expresses the pressure loss due to friction in terms of the equivalent height of a column of the working fluid, so the pressure drop is
where:
= The head loss due to pipe friction over the given length of pipe (SI units: m); = The local acceleration due to gravity (m/s2).
It is useful to present head loss per length of pipe (dimensionless):
where is the pipe length (m).
Therefore, the Darcy–Weisbach equation can also be written in terms of head loss:
In terms of volumetric flow
The relationship between mean flow velocity and volumetric flow rate is
where: = The volumetric flow (m3/s), = The cross-sectional wetted area (m2).
In a full-flowing, circular pipe of diameter ,
Then the Darcy–Weisbach equation in terms of is
Shear-stress form
The mean wall shear stress in a pipe or open channel is expressed in terms of the Darcy–Weisbach friction factor as
The wall shear stress has the SI unit of pascals (Pa).
Darcy friction factor
The friction factor is not a constant: it depends on such things as the characteristics of the pipe (diameter and roughness height ), the characteristics of the fluid (its kinematic viscosity [nu]), and the velocity of the fluid flow . It has been measured to high accuracy within certain flow regimes and may be evaluated by the use of various empirical relations, or it may be read from published charts. These charts are often referred to as Moody diagrams, after L. F. Moody, and hence the factor itself is sometimes erroneously called the Moody friction factor. It is also sometimes called the Blasius friction factor, after the approximate formula he proposed.
Figure 1 shows the value of as measured by experimenters for many different fluids, over a wide range of Reynolds numbers, and for pipes of various roughness heights. There are three broad regimes of fluid flow encountered in these data: laminar, critical, and turbulent.
Laminar regime
For laminar (smooth) flows, it is a consequence of Poiseuille's law (which stems from an exact classical solution for the fluid flow) that
where is the Reynolds number
and where is the viscosity of the fluid and
is known as the kinematic viscosity. In this expression for Reynolds number, the characteristic length is taken to be the hydraulic diameter of the pipe, which, for a cylindrical pipe flowing full, equals the inside diameter. In Figures 1 and 2 of friction factor versus Reynolds number, the regime demonstrates laminar flow; the friction factor is well represented by the above equation.