Pipe Flow Darcy Weisbach Calculator

This solver works in SI units and characterizes the fluid by dynamic viscosity rather than kinematic viscosity, so the Reynolds number is formed as Re = rho V D / mu with rho in kg/m3 and mu in Pa.s. Water at 20 degrees C has rho near 998 kg/m3 and mu near 0.001002 Pa.s, a kinematic viscosity of about 1.004e-6 m2/s. Below Re of 2300 the flow is laminar and f = 64/Re exactly. Above 4000 the Colebrook-White relation 1/sqrt(f) = -2 log10(eps/(3.7 D) + 2.51/(Re sqrt(f))) applies, and because f appears on both sides it is solved here by fixed-point substitution from a starting guess of 0.02, which converges to a tolerance of 1e-8 in fewer than ten passes for ordinary pipe conditions. Head loss follows as hf = f (L/D) V^2 / (2 x 9.81) in metres and pressure drop as dp = f (L/D) rho V^2 / 2 in pascals, from which pump power is dp times volumetric flow. Representative roughness is 0.0015 mm for drawn copper or plastic, 0.045 mm for commercial steel, 0.26 mm for cast iron and 0.3 to 3.0 mm for concrete. Water transmission mains typically run at 1 to 3 m/s, while viscous products such as heavy fuel oil can sit at Reynolds numbers in the hundreds, where f exceeds 0.2 and viscosity rather than roughness dominates. Limitations: the 2300 to 4000 band is indeterminate and only approximated here, fitting losses are excluded, Colebrook itself scatters a few percent against experiment, and non-Newtonian or two-phase flows are out of scope.

Julius Weisbach set out the dimensionless form of the pipe friction equation in 1845, and Henry Darcy published his experimental campaign on cast iron and other pipes in Paris in 1857 under the title Recherches experimentales relatives au mouvement de l eau dans les tuyaux, establishing that the wall condition governs the coefficient. Confusion persisted for decades between the Darcy friction factor and the Fanning friction factor used in chemical engineering, which is exactly one quarter as large. Johann Nikuradse supplied the first systematic map of the friction factor in 1933 by gluing graded sand grains inside pipes and measuring loss across the full range of Reynolds numbers. The formula implemented in this calculator comes from Cyril Colebrook and Cedric White at Imperial College, who reported in the late 1930s that commercial pipes make a gentler transition from smooth to fully rough behaviour than Nikuradse artificial roughness, and Colebrook published the interpolating expression in the Journal of the Institution of Civil Engineers in 1939. Because that expression is implicit, generations of engineers avoided it by reading the Moody chart of 1944 instead. Digital computing reversed the tradeoff: iteration on a few lines of code became cheaper than chart reading, and direct solution of Colebrook became routine in hydraulic software. Later analysis showed the equation can also be expressed in closed form using the Lambert W function, though iterative and explicit approximations remain the usual practice.