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 $W$ function, is characterized by an exponential term which imposes restrictions on its use. These constraints have been expressed in terms of pipe roughness $(\epsilon /D)$ and the Reynolds number R that are required for friction factor calculation. These constraints were determined as $8.0666 \ln \left(R\right)+(\epsilon /D)R<721.97$ and $8.0666 \ln \left(R\right)+(\epsilon /D)R<5731.83\text{,}$ respectively, for machines using single precision and double precision computations. Using the Lambert $W$ function, an explicit equation relating R and $\epsilon /D$ 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 $\epsilon /D$ 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 $\epsilon /D$ values satisfies the applicable constraint mentioned above.