Darcy Weisbach Calculator

Friction head loss is the energy per unit weight of fluid that wall shear converts irreversibly into heat along a run of pipe. The Darcy-Weisbach form hf = f (L/D) (V^2 / 2g) splits the problem into a geometric term L/D, the velocity head V^2/2g that carries the units of feet, and a dimensionless friction factor f that absorbs the physics of the wall boundary layer. Because f is dimensionless, the same equation serves any Newtonian fluid once the Reynolds number Re = VD/nu and the relative roughness epsilon/D are known. Below Re of roughly 2000 the flow is laminar and f = 64/Re exactly, independent of roughness. Above about 4000 the Swamee-Jain expression used here reproduces the Colebrook-White curve to within roughly 1 percent for Re from 5000 to 10^8. Typical values anchor the output. Commercial steel is taken as epsilon = 0.00015 ft, cast iron 0.00085 ft and PVC 0.000005 ft, so turbulent f in ordinary water lines usually lands between 0.012 and 0.045, and mains are commonly sized for 3 to 8 ft/s with 1 to 5 ft of loss per 100 ft. Three caveats matter. The equation covers straight, full-flowing pipe only, so valve, elbow and entrance losses must be added separately as K V^2/2g. Kinematic viscosity is temperature sensitive, falling from 1.217e-5 ft2/s at 60 degrees F to about 7.39e-6 at 100 degrees F, which shifts Re by nearly a factor of two. And tuberculation in old unlined iron can raise effective roughness tenfold, so as-new values understate loss in aged systems.

The equation carries two names because it was assembled in stages. Julius Weisbach, a professor at the mining academy in Freiberg, Saxony, published the modern dimensionless form in his 1845 textbook Lehrbuch der Ingenieur- und Maschinen-Mechanik, recognizing that loss scales with velocity head and with length over diameter. Henry Darcy, the French engineer behind the water supply of Dijon, ran the careful pipe experiments reported in 1857 that showed the friction coefficient depends on the condition of the pipe wall and not on the fluid alone. For decades afterward there was no general way to predict f, so trade-specific empirical formulas proliferated in waterworks, gas and steam practice. The gap closed in the 1930s. Johann Nikuradse, working under Ludwig Prandtl in Gottingen, published experiments in 1933 on pipes artificially roughened with graded sand grains, mapping the laminar, smooth-turbulent, transitional and fully rough regimes. Cyril Colebrook and Cedric White at Imperial College then showed that commercial pipe passes through the transition more gradually than uniform sand grains do, and Colebrook published his implicit interpolation formula in 1939. Lewis Moody of Princeton plotted that relation as the chart presented to the ASME in 1944, which became the standard graphical design tool for a generation. Explicit algebraic substitutes followed once machine computation made repeated evaluation cheap, notably the Swamee and Jain approximation of 1976 and the Haaland form of 1983, both of which avoid iteration while tracking Colebrook to within a few percent.