The Prony equation (named after Gaspard de Prony) is a historically important equation in hydraulics, used to calculate the head loss due to friction within a given run of pipe. It is an empirical equation developed by Frenchman Gaspard de Prony in the 19th century:

${\displaystyle h_{f}={\frac {L}{D}}(aV+bV^{2})}$

where h_f is the head loss due to friction, calculated from: the ratio of the length to diameter of the pipe L/D, the velocity of the flow V, and two empirical factors a and b to account for friction.

This equation has been supplanted in modern hydraulics by the Darcy–Weisbach equation, which used it as a starting point.

References

• Simmons, Craig (2007), "Henry Darcy (1803–1858): Immortalised by his scientific legacy", in Chery, Laurence (ed.), Aquifer Systems Management: Darcy's Legacy in a World of Impending Water Shortage: Selected Papers from the International Association of Hydrogeologists (IAH) Dijon Symposium, Dijon, France, 30 May–1 June 2006, Leiden, The Netherlands: Taylor & Francis / Balkema, pp. 3–24, ISBN 9780415443555. The Prony equation and its replacement by the Darcy–Weisbach equation are on pp. 11–12.