## Henry Darcy and His Law: The History of the Darcy-Weisbach Equation

December 21, 2008*Henry Darcy and His Law*

**The History of the Darcy-Weisbach Equation**

Glenn Brown

Oklahoma State University

6/7/00

revised 2/21/02

What we call the Darcy-Weisbach equation has had a long history of development. It is named after two of the great hydraulic engineers of the middle 19th century, but others have also played a major role. Julies Weisbach (1806-1871) a native of Saxony, proposed in 1845 the equation we now use,

*h _{l} = fL/D * V^{2}/2g*

where hl is the head loss, *L* is the pipe length, *D* is the pipe diameter, *V *is the average velocity, *g* is the acceleration of gravity and *f* is a friction factor. However, he did not provide adequate data for the variation in *f* with velocity. Thus, his equation performed poorly compared to the empirical Prony equation (Gaspard Clair Francois Marie Riche de Prony, 1755-1839) in wide use at the time;

*h _{l} = L/D * (aV + bV^{2})*

where *a* and *b* are empirical friction factors for the velocity and velocity squared.

While Weisbach was ahead of most other engineers, his was not the first work in the area. In about 1770 Antoine Chézy (1718-1798), an early graduate of l’Ecole des Ponts et Chaussées, published an equation for flow in open channels that can be reduced to the same form. Unfortunately, Chézy’s work was lost until 1800 when his former student, Prony published an account describing it. Surprisingly, Prony developed his own equation, but it is believed that Weisbach was aware of Chézy’s work from Prony’s publication. Darcy, (Prony’s student) in 1857 published new relations for the Prony coefficients based on a large number of experiments. His new equation was,

*h _{l} = L/D * [(c + d/D^{2}*)

*V + (d + e/D)V*

^{2}]where *c, d *and *e* are empirical coefficients for a given type of pipe. Darcy thus introduced the concept of the pipe roughness scaled by the diameter; what we now state as the relative roughness when applying the Moody diagram. Therefore, it is traditional to call *f,* the “Darcy f factor”, even though Darcy never proposed it in that form. Fanning apparently was the first effectively put together the two concepts in (1877). He published a large compilation of *f* values as a function of pipe material, diameter and velocity. His data came from French, American, English and German publications, with Darcy being the single biggest source. However, it should be noted that Fanning used hydraulic radius, instead of *D* in the friction equation, thus “Fanning *f*” values are only 1/4th of “Darcy *f*” values.

Parallel to the development in hydraulics, viscosity and laminar flow were defined by Jean Poisseuille (1799-1869) and Gotthilf Hagen (1797-1884), while Osborne Reynolds (1842-1912) described the transition from laminar to turbulent flow in 1883. During the early 20th century, Ludwig Prandtl (1875-1953) and his students Th. von Kármán (1881-1963) Paul Blasius (1883-?) and Johnann Nikuradse (1894-1979) attempted to provide an analytical prediction of the friction factor using both theoretical considerations and data from smooth and uniform sand lined pipes. Their work was complimented by Colebrook and White’s analysis of pipes with non-uniform roughness in 1939. The Darcy-Weisbach equation was not made universally useful until the development of the Moody diagram (Moody, 1944) which built on the work of Hunter Rouse. Rouse (1946) gives a good feel for the development of the f factor, but he doesn’t reference Moody. Rouse felt that Moody was given too much credit for what Rouse himself and others did (Rouse, 1976).

The name of the equation through time is also curious and may be tracked in hydraulic and fluid mechanics textbooks. Early texts generally do not name the equation. Starting in the mid 20th century some authors including at least one German named it “Darcy’s Equation”, an obvious confusion point with “Darcy’s Law”. Rouse in 1946 appears to be the first to call it “Darcy-Weisbach”, but that naming did not become universal until the late 1980’s. It is a good enough name, but as pointed out previously, it leaves out many important contributions. So if you wanted give full credit and confuse people, call it the “Chézy-Weisbach-Darcy-Poiseuille-Hagen-Reynolds-Fanning-Prandtl-Blasius-Kármaán-Nikuradse-Colebrook-White-Rouse-Moody equation”.

From a practical standpoint, the Darcy-Weisbach equation has only become popular since the advent of the electronic calculator. It requires a lot of number crunching compared to empirical relationships, such as the Hazen-Williams equation, which are valid over narrow ranges. However, because of its general accuracy and complete range of application, the Darcy-Weisbach Equation should be considered the standard and the others should be left for the historians. A recent interesting discussion on the topic is presented by Liou (1998), Christensen (2000), Locher (2000) and Swamee (2000).

**References**

Christensen, B.A., 2000. Discussion of “Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering”, ASCE.

Darcy, H. 1857. Recherches Experimentales Relatives au Mouvement de L’Eau dans les Tuyaux, 2 volumes, Mallet-Bachelier, Paris. 268 pages and atlas. (“Experimental Research Relating to the Movement of Water in Pipes”)

Fanning, 1877. Treatise on Water Supply.

Liou, C.P., 1998. Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering, ASCE. Vol. 124.

Locher, F. A., 2000. Discussion of “Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering”, ASCE.

Moody, L. F., 1944. Friction factors for pipe flow. Transactions of the ASME, Vol. 66.

Poiseuille, J. L., 1841. Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres, Comptes Rendus, Académie des Sciences, Paris, 1841.

Reynolds, O., 1883. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channel, Philo. Trans. of the Royal Soc., 174:935-982.

Rouse, H., 1946. Elementary Mechanics of Fluids. John Wiley and Sons, New York.

Rouse, H., 1976. History of Hydraulics in America, 1776-1976.

Swamee, P. K., 2000. Discussion of “Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering”, ASCE.

Weisbach, J., 1845. Lehrbuch der Ingenieur- und Maschinen-Mechanik, Braunschwieg.

**Source:**

** http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm**

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