Adaptation of various explicit formulas for calculating Darcy-Weisbash friction factor to optimize the modeling of pressure drops during turbulent flow of phosphate slurry through the pipeline

The Darcy-Weisbash friction factor $f$D is a very important parameter for the calculation and modeling of the head losses during a fluid flow through a closed pipe, because the hydraulic gradient $J$ strongly depends on this parameter. In this paper we will focus on the case of the turbulent flow of phosphate slurry through the pipeline linking Khouribga (Morocco) and Jorf-Lasfar (Morocco) over a distance of 187 km. In this study we have presented the ten explicit formulas of the factor $f$D most used in the literature for the slurries case, we evaluated these formulas by comparing their results with the experimental results of the hydraulic gradient $J$ measurements during the flow of the phosphate slurry. We carried out a succession of computation of $f$D through the explicit formulas, and a succession of experimental measurements of the hydraulic gradient J, we noted that certain formulas of $f$D generated results close to experimental, and others were far. For this we aimed to multiply all the explicit formulas of the $f$D to an adjustment parameter (A) to bring them closer to the experimental results, through this method, we arrived at new explicit formulas for calculating $f$D optimized and adapted for the case of the turbulent flow of the phosphate slurry, and which will be used later in the modeling of the pressure drops.