- Darcy-Weisbach equation
The

**Darcy-Weisbach equation**is an important and widely used phenomenological equation inhydraulics . It relates thehead loss orpressure loss due tofriction along a given length of pipe to the average velocity of the fluid flow.**Head loss form**Head loss can be calculated with:$h\_f\; =\; f\; cdot\; frac\{L\}\{D\}\; cdot\; frac\{V^2\}\{2g\}$where

* "h_{f}" is the head loss due to friction;

* L is the length of the pipe;

* D is thehydraulic diameter of the pipe (for a pipe of circular section, this equals the internal diameter of the pipe);

* "V" is the average velocity of the fluid flow, equal to thevolumetric flow rate per unit cross-sectional wetted area;

* "g" is the local acceleration due to gravity;

* "f" is a dimensionless coefficient called the Darcy friction factor. It can be found from aMoody diagram .**Pressure loss form**Given that the

head loss "h_{f}" expresses thepressure loss "Δp" as the height of a column of fluid,:$Delta\; p\; =\; ho\; cdot\; g\; cdot\; h\_f$

where ρ is the density of the fluid, the Darcy-Weisbach equation can also be written in terms of pressure loss:

:$Delta\; p\; =\; lambda\; cdot\; frac\{L\}\{D\}\; cdot\; frac\{\; ho\; V^2\}\{2\}$

where the pressure loss due to friction "Δp" is a function of:

* the ratio of the length to diameter of the pipe, "L/D";

* the density of the fluid, "ρ";

* the average velocity of the flow, "V", as defined above;

* a (dimensionless) coefficient oflaminar , orturbulent flow , "λ"Since the pressure loss equation can be derived from the head loss equation by multiplying each side by "ρ" and "g", clearly "λ" equals "f". The use of different symbols for the same numerical coefficient depending on whether head loss or pressure is considered is a historical accident due to different conventions being used by different communities of scientists and engineers.

**Darcy friction factor**:"See also

Darcy friction factor formulae "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 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 theMoody friction factor . It is also sometimes called theBlasius friction factor, after the approximate formula he proposed.For laminar (slow) flows, it is a consequence of

Poiseuille's Law that λ equals "64/Re," where "Re" is theReynolds 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 theColebrook-White equation , or theSwamee-Jain equation . While the diagram andColebrook-White equation are iterative methods, theSwamee-Jain equation allows "f" to be found directly for full flow in a circular pipe.**Confusion with the Fanning friction factor**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 engineers, and the Fanning factor by chemical and mechanical 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 is f=16/Re, it's the Fanning factor, and if the formula 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:

#Observe the value of the friction factor for laminar flow at a Reynolds number of 1000.

#If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the formula for the laminar Darcy friction factor: f = 64/Re.

#If the value of the friction factor is 0.016, then the Fanning friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.016 are the numerator in the formula for the laminar Fanning friction factor: f = 16/Re.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.

**History**Historically this equation arose as a variant on the

Prony equation ; this variant was developed byHenry Darcy ofFrance , and further refined into the form used today byJulius Weisbach ofSaxony 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 special-case situations in favor of a variety ofempirical equation s valid only for certain flow regimes, notably theHazen-Williams equation or theManning equation , most of which were significantly easier to use in calculations. However, since the advent of thecalculator , ease of calculation is no longer a major issue, and so the Darcy-Weisbach equation's generality has made it the preferred one.**Derivation**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 cross-sectionalarea 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,:$Delta\; p\; propto\; frac\{L\}\{D\}\; imes\; frac\{1\}\{2\}\; ho\; V^2$

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 thehydraulic 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 oflaminar 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 mean-square velocity, which always exceeds the square of the mean velocity. In the case ofturbulent 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.**References*** [

*http://biosystems.okstate.edu/darcy/DarcyWeisbach/MoodyDiagram.htm Moody diagram*] (GIF; ~25k)

* [*http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm The History of the Darcy-Weisbach Equation*]

* De Nevers (1970) Fluid Mechanics. Addison-Wesley ISBN 0-201-01497-1

* Shah R.K. and A.L. London, 1978, 'Laminar Flow Forced Convection in Ducts', Supplement 1 in Advances in Heat Transfer, Academic, NY, 1978.

* Rohsenhow, W.M., J.P. Hartnett and E.N. Ganic, 1985, 'Handbook of Heat Transfer Fundamentals, 2nd Ed., McGraw-Hill Book Company.

*Wikimedia Foundation.
2010.*