Colebrook-White Formula for Pipe Flows

Flow-resistance laws-as used, for example, in water-supply pipe networks-are formulas relating the volume flow rate, q, along a pipe to the pressure-head difference, t, between its ends, such that q= (t), in which is monotonic. The simple Hazen-Williams power law is often used, but in appropriate circumstances the more complicated Colebrook-White law (CW) may better represent aspects of the experimental data. Result 1, the first and easiest-to-state result in the paper, is that φCW, the inverse of CW, can be expressed in terms of the Lambert W-function (Corless et al. 1993). Result 2 summarizes one use of this, and of related results, in convex optimization problems describing equilibrium flows in pipe networks.