Empirical equations for meniscus depression by particle attachment.

In this paper the problem of calculating the depression of the gas-liquid meniscus by the particle attachment was solved. The analytical approximate equations obtained for small and large radii, r(tpc), of the three-phase contact were analyzed and compared to the available numerical results. The Derjaguin equation for small r(tpc) and the analytical results for large r(tpc) are accurate for r(tpc)/L< or =0.2 and r(tpc)/L> or =2, respectively, where L is the capillary length. For the meniscus depression with r(tpc)/L from 0.2 to 2, the empirical equations were obtained based on the asymptotic analysis of the analytical approximate solutions. The empirical numerical constants were obtained by fitting to the exact numerical results. The empirical equations together with the analytical approximate equations provide the accurate predictions for the meniscus depression for the whole range of the radius of the three-phase contact and are expected to be useful for modeling the detachment interaction in the flotation separation processes.