Darcy–Weisbach equation

In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after Henry Darcy and Julius Weisbach. Currently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the Moody diagram or Colebrook equation. The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient. In a cylindrical pipe of uniform diameter D, flowing full, the pressure loss due to viscous effects Δp is proportional to length L and can be characterized by the Darcy–Weisbach equation: where the pressure loss per unit length Δp/L (SI units: Pa/m) is a function of: the density of the fluid (kg/m3); the hydraulic diameter of the pipe (for a pipe of circular section, this equals D; otherwise DH = 4A/P for a pipe of cross-sectional area A and perimeter P) (m); the mean flow velocity, experimentally measured as the volumetric flow rate Q per unit cross-sectional wetted area (m/s); the Darcy friction factor (also called flow coefficient λ). For laminar flow in a circular pipe of diameter , the friction factor is inversely proportional to the Reynolds number alone (fD = 64/Re) which itself can be expressed in terms of easily measured or published physical quantities (see section below). Making this substitution the Darcy–Weisbach equation is rewritten as where μ is the dynamic viscosity of the fluid (Pa·s = N·s/m2 = kg/(m·s)); Q is the volumetric flow rate, used here to measure flow instead of mean velocity according to Q = π/4Dc2 (m3/s). Note that this laminar form of Darcy–Weisbach is equivalent to the Hagen–Poiseuille equation, which is analytically derived from the Navier–Stokes equations.