The pressure drop due to friction for flow in a pipe is a function of average flow velocity (V), density (rho), viscosity (μ), pipe length (L), diameter (D), and the roughness of the wall (ε). Show that the relationship for the pressure drop in the pipe can be rewritten as:
rhoV² / 2Δp = f(D/L, μ/rhoVD, D/ε)
Note that the factor of 2 is commonly introduced for mathematical convenience. Furthermore, experiments show that the pressure drop varies linearly with D/L, and so this can be simplified to:
rhoV² L / 2ΔpD = f(μ/rhoVD, D/ε)
The above functional relationship is commonly rewritten as f = (Re, D/ε), where f is the friction factor, Re is the Reynolds number for pipe flow (sometimes written Re_D), and ε/D is the roughness ratio.