Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation

We analyze the general applicability of a recent explicit expression of the Colebrook.White equation for turbulent flow friction factor calculation. This explicit expression, which is based on the Lambert \IW\N function, is characterized by an exponential term which imposes restrictions on its use. These constraints have been expressed in terms of pipe roughness (ϵ/\ID\N) and the Reynolds number R that are required for friction factor calculation. These constraints were determined as 8.0666 ln(R) + (ϵ/\ID\N)R<721.97 and 8.0666 ln(R) + (ϵ/\ID\N)R<5731.83, respectively, for machines using single precision and double precision computations. Using the Lambert W function, an explicit equation relating Rand ϵ/\ID\N was derived at the limiting case which allowed for a graphical representation of the applicability of the explicit form of the Colebrook.White equation in the Rversus ϵ/\ID\N space. Before computing friction factors using the explicit Colebrook-White equation, a quick check must be performed to see if the desired combination of Rand ϵ/\ID\N values satisfies the applicable constraint mentioned above.