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

Abstract

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 W function, is characterized by an exponential term which imposes restrictions on its use. These constraints have been expressed in terms of pipe roughness (ε/D) and the Reynolds number R that are required for friction factor calculation. These constraints were determined as 8.0666 ln(R)+(ε/D)R<721.97 and 8.0666 ln(R)+(ε/D)R<5731.83, respectively, for machines using single precision and double precision computations. Using the Lambert W function, an explicit equation relating R and ε/D 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 R versus ε/D space. Before computing friction factors using the explicit Colebrook–White equation, a quick check must be performed to see if the desired combination of R and ε/D values satisfies the applicable constraint mentioned above.