Estimation of flotation rate constant and particle-bubble interactions considering key hydrodynamic parameters and their interrelations

Abstract Particle-bubble sub-processes cannot be directly and physically obtained in froth flotation due to the complexity of the process as well as numerous and dynamic interactions of particles and bubbles in an extremely intensive turbulent condition. Therefore, over the last three decades, two fundamental model configurations have been used as an only solution for prediction of particle-bubble collection efficiencies (Ecoll). Additionally, the relative intensity of the main flotation parameters on flotation rate constant, particle–bubble interactions together with their interrelations is not adequately addressed in the literature. The present study attempts in two separate phases to overcome these difficulties. In the first stage, prediction and evaluation of particle-bubble sub-processes are critically discussed by categorizing them in two configurations. The analytical models (approach I) commonly applied generalized Sutherland equation ( E c G S E ), modified Dobby–Finch ( E a DF ) and modified Schulze stability ( E s SC ) models. The second approach, numerical models, utilized Yoon–Luttrell ( E c YL ), Yoon–Luttrell (intermediate) ( E a YL ) and modified Schulze stability ( E s SC ) models. In the second stage, relative intensity and interrelation of key effective hydrodynamic parameters on the probability of particle–bubble encounter (Ec) and flotation rate constant (k) are obtained and optimized by means of the response surface modeling (RSM) based on central composite design (CCD). Five key factors including particle size (1–100 µm), particle density (1.3–4.1 kg/m3), bubble size (0.05–0.10 cm) and bubble velocity (10–30 cm/s) together with turbulence dissipation rate (18–30 m2/s3) are considered in order to maximize the responses including the k and Ec. The results obtained show that the Ecoll calculated by numerical techniques (configuration (II)) is greater than that of analytical approaches (configuration (I)) due to assumptions involved in using Yoon–Luttrell collision and attachment models. It is also found that under the conditions studied, particle size and bubble velocity are the most effective factors on Ec and k, respectively. Furthermore, not only the relative significance of factors on Ec and k but also the interrelation of cell turbulence and bubble size as well as bubble velocity and turbulence are shown to be inconsistent in the literature and thus require further studies. We briefly reported the main long-standing challenges in flotation kinetic modeling and emphasized on a serious need for fulfilling lack of physical observations. Finally, the presented analyses with respect to three-zone model offer a new concept for the extension of common flotation modeling approach using analytical and numerical techniques.