RICHARDSON-ZAKI EXPONENTS FOR PARTICLES, DROPS AND BUBBLES

A broad range of experimental data and resolved-surface simulations were surveyed in terms of their Richardson-Zaki exponents for solid particles (spherical, ellipsoidal and irregular in shape) along with spherical drops and bubbles (both clean and contaminated). For these data sets, the Richardson-Zaki exponent often reasonably represents the volume fraction effect on group particle drag increase for moderate volumetric concentrations, ranging from 3% to 30%. A separable effects assumption was used to determine the influence of Reynolds number on the Richardson-Zaki exponent for spheres (including solid spherical particles as well as clean and contaminated spherical bubbles) based on a hydro-static pressure gradient for a homogeneous mixture of particles in a liquid, and this method compared well with experiments. A similar influence was found for non-spherical fluid particles with moderate deformations. However, highly deformed bubbles can cause a drag reduction at high concentrations, owing to deformations aligned with the flow, so a Richardson-Zaki exponent is no longer valid. In contrast, non-spherical solid particles introduce an additional complexity due to changes in particle orientation as volume fraction increases, especially at low Reynolds numbers, which instead accentuates the drag increase due to volume fraction.