Examining and extending the concept of machine acceleration used in modelling the bubble-particle detachment in flotation

The bubble-particle (BP) detachment is a significant factor in controlling the recovery of coarse particles in mechanical flotation cells. It has been quantified by balancing the restoring force of surface tension and the centrifugal force exerted on the particle-bubble aggregate by the turbulent flow field using a “machine acceleration”. The concept of machine acceleration is useful because it links the mean energy dissipation rate of turbulence with the condition of the BP detachment. Here we further examine the concept of machine acceleration by applying the theory of isotropic turbulence. We confirm the known results of the first approximation for the inertial subrange. We also show that the turbulence acceleration has two principal components, i.e., the longitudinal and transverse components measured relatively to the BP centre-line. Significantly, the longitudinal component corresponds to the centrifugal force of turbulence against the restoring force of surface tension. The transverse component can be significant to quantifying the BP detachment if the turbulence shear is strong. We also extend the theory to cover the full range of isotropic turbulence, from the viscous to inertial subranges. Our estimation of the transition from the viscous to inertial subrange shows that the viscous effect can critically affect the BP detachment. Finally, our assessment of the contributions of the longitudinal and transverse components to the machine acceleration reveals the importance of the transverse component which can lead to a rather poor approximation for the machine acceleration as currently used. This paper shows that the effect of turbulence on the BP detachment should be better quantified using both the longitudinal and transverse components of turbulence acceleration rather than their modulus as the first approximation being termed the machine acceleration