Flow Rate Calculator

Volumetric flow through a closed conduit follows from conservation of mass as Q = A x V, where A is the internal area pi x D^2 / 4 and V the average velocity over it. Two subtleties matter. The velocity is a cross-sectional average, not the centreline maximum: fully developed laminar flow has a parabolic profile peaking at twice the mean, while turbulent flow is much flatter, which is why the simple product works well in the turbulent regime and less well in the laminar. And the diameter must be the actual internal bore, not the nominal pipe size, which for schedule-40 steel differs by several percent. Mass flow is Q times density, and it is preferred for gases and energy balances because it stays constant when pressure and temperature change the volume. The Reynolds number Re = rho x V x D / mu decides the regime, comparing inertial to viscous forces. Below roughly 2300 flow is laminar, above about 4000 turbulent, with an unpredictable band between; most industrial piping runs turbulent at Re in the tens or hundreds of thousands. Regime sets the friction factor in the Darcy-Weisbach equation dP = f x (L/D) x rho x V^2 / 2: exactly 64/Re in laminar flow, but roughness-dependent and implicit in turbulent flow. Design velocities balance capital against pumping cost, typically 1 to 3 m/s for water, 0.5 to 1.5 for viscous oils, 6 to 10 for compressed air. Roughness also grows with corrosion and scale, sometimes doubling pressure drop over a pipe's life.

Pipe flow resisted theory because the laminar and turbulent cases obey different laws. Gotthilf Hagen in 1839 and Jean Leonard Marie Poiseuille in 1840, the latter motivated by blood flow in capillaries, independently established that flow in narrow tubes varies with the fourth power of diameter and inversely with viscosity. Julius Weisbach published the general friction-loss form in 1845, and Henry Darcy's experiments on the water mains of Dijon, reported in 1857, supplied the roughness dependence, giving the combined Darcy-Weisbach equation still standard today. The unifying insight came from Osborne Reynolds at Owens College in Manchester, whose 1883 dye-filament experiment showed a thin coloured streak holding steady at low flow and bursting into eddies above a critical value of the dimensionless group now named after him. Johann Nikuradse's sand-grain roughness experiments in the early 1930s quantified how surface texture affects turbulent friction, and Cyril Colebrook and Cedric White combined smooth and rough behaviour into a single implicit correlation in 1937. Lewis Ferry Moody turned that into the log-log chart published in 1944 that generations read by hand, and explicit approximations such as the Swamee-Jain form of 1976 made it programmable. Crane Technical Paper No. 410, first issued in 1942, remains the working reference for valve and fitting resistances.