General formulation of drag force on assemblage of spherical particles in fluids: A critical review and a new empirical formula

Establishment of a simple but accurate formula for the drag force acting on an assemblage of particles moving relative to its carrier fluid is critical to modeling the flow of a particle–fluid two-phase system. In this paper, research published in the past century is reviewed. For clarity, the general problem is distinguished by different flow regimes. Representative results valid in each flow regime are summarized. Among various empirical formulas, the Richardson–Zaki type, which is basically valid at relatively small concentrations, and the Ergun type, which is applicable to relatively large concentration problems, are paid special attention. As many as 3265 high-quality data records, covering single particle cases and the assemblage cases with various particle concentration are collected to verify the existing formulas for the drag coefficient as well as to establish a new one. The new formula is shown reducing to the Stokes' law and the Newton's law as the particle Reynolds number tends to zero and a sufficiently large value, respectively, and converging to the single particle and the porous medium cases as the particle volume concentration tends to zero and its maximum value, respectively. The accuracy of the new formula is confirmed to be evidently higher than others.