The Colebrook-White formula for pipe networks

SUMMARY Flow resistance laws, as used for example in water-supply pipe networks, are formulae relating the volume ow rate q along a pipe to the pressure-head diierence t between its ends, q = (t). is monotonic. The simple Hazen-Williams power \law" is often used: it has been claimed that the more complicated Colebrook-White law (CW) better represents aspects of the experimental data. Equilibrium ows in pipe networks are described by convex optimization problems, where the objective function in a primal form involves the indeenite integral of. This is analytically tractable for both Hazen-Williams and (CW). There is also a dual of this optimization problem, the starting point for which is , the inverse to. It turns out that for (CW) can be expressed in terms of a known transcendental function, the Lambert W-function. We explain how these facts might be useful in computer codes. Some simple numerical work, and some phenomena of independent interest are brieey reported. See Appendices. NOMENCLATURE The following symbols are used in this article and in 18]. A arc set for a nework; U dual objective function; b i consumption at node i; V primal objective function; c 1 , c 2 , c 3 constants in CW ; W Lambert W-function; D a diameter of pipe a; viscosity of water g acceleration due to gravity; inverse to ; K a pipe roughness for a; integral of ; L a length of pipe a; ow function; N node set for a nework; integral of ; p i pressure head at node i; P power consumed; q a ow along pipe a; k a (separable) conductivity factor; t a pressure head diierence along a; (separable) ow function. These are not used in 18].