In fluid dynamics, the Darcy–Weisbach equation is a phenomenological equation, which relates the head loss — or pressure loss — due to friction along a given length of pipe to the average velocity of the fluid flow. The equation is named after Henry Darcy and Julius Weisbach.
The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also called the Darcy–Weisbach friction factor or Moody friction factor. The Darcy friction factor is four times the Fanning friction factor, with which it should not be confused.^{[1]}
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Head loss can be calculated with
where
Given that the head loss h_{f} expresses the pressure loss Δp as the height of a column of fluid,
where ρ is the density of the fluid, the Darcy–Weisbach equation can also be written in terms of pressure loss:
where the pressure loss due to friction Δp is a function of:
Since the pressure loss equation can be derived from the head loss equation by multiplying each side by ρ and g.
The friction factor f or flow coefficient λ is not a constant and depends on the parameters of the pipe and the velocity of the fluid flow, but it is known to high accuracy within certain flow regimes. It may be evaluated for given conditions by the use of various empirical^{[2]} or theoretical relations, or it may be obtained from published charts. These charts are often referred to as Moody diagrams, after L. F. Moody, and hence the factor itself is sometimes called the Moody friction factor. It is also sometimes called the Blasius friction factor, after the approximate formula he proposed.
For laminar (slow) flows, it is a consequence of Poiseuille's Law that λ=64/Re, where Re is the Reynolds Number calculated substituting for the characteristic length the hydraulic diameter of the pipe, which equals the inside diameter for circular pipe geometries.
For turbulent flow, methods for finding the friction factor f include using a diagram such as the Moody chart; or solving equations such as the ColebrookWhite equation, or the SwameeJain equation. While the diagram and ColebrookWhite equation are iterative methods, the SwameeJain equation allows f to be found directly for full flow in a circular pipe.
The Darcy–Weisbach friction factor is 4 times larger than the Fanning friction factor, so attention must be paid to note which one of these is meant in any "friction factor" chart or equation being used. Of the two, the Darcy–Weisbach factor is more commonly used by civil and mechanical engineers, and the Fanning factor by chemical engineers, but care should be taken to identify the correct factor regardless of the source of the chart or formula.
Most charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow. If the formula for laminar flow is f = 16/Re, it's the Fanning factor, and if the formula for laminar flow is f = 64/Re, it's the Darcy–Weisbach factor.
Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:
The procedure above is similar for any available Reynolds number that is an integral power of ten. It is not necessary to remember the value 1000 for this procedure – only that an integral power of ten is of interest for this purpose.
Historically this equation arose as a variant on the Prony equation; this variant was developed by Henry Darcy of France, and further refined into the form used today by Julius Weisbach of Saxony in 1845. Initially, data on the variation of f with velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases. In later years it was eschewed in many specialcase situations in favor of a variety of empirical equations valid only for certain flow regimes, notably the HazenWilliams equation or the Manning equation, most of which were significantly easier to use in calculations. However, since the advent of the calculator, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.
The Darcy–Weisbach equation is a phenomenological formula obtainable by dimensional analysis.
Away from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe. The key quantities are then the pressure drop along the pipe per unit length, Δp/L, and the volumetric flow rate. The flow rate can be converted to an average velocity V by dividing by the wetted area of the flow (which equals the crosssectional area of the pipe if the pipe is full of fluid).
Pressure has dimensions of energy per unit volume and, so, the pressure drop between two points must be proportional to (1/2)ρV^{2}, which has the same dimensions as it resembles (see below) the expression for the kinetic energy per unit volume. We also know that it must be proportional to the length of the pipe between the two points L as the pressure drop per unit length is a constant. To turn that into a dimensionless quantity we can divide by the hydraulic diameter of the pipe, D, which is also constant along the pipe. Therefore,
The proportionality coefficient is the dimensionless "Darcy friction factor" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as π, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the roughness height to the hydraulic diameter).
Note that (1/2)ρV^{2} is not the kinetic energy of the fluid per unit volume, for the following reasons. Even in the case of laminar flow, where all the flow lines are parallel to the length of the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area. The average kinetic energy then involves the meansquare velocity, which always exceeds the square of the mean velocity. In the case of turbulent flow, the fluid acquires random velocity components in all directions, including perpendicular to the length of the pipe, and thus turbulence contributes to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid.
