## 1 Introduction  
Froth flotation exploits the differences in surface hydrophobicity of the different constituent minerals selectively separates the valuable minerals from gangue by attaching them to air bubbles and recovering them from the mineral laden froth [1; 2; 3; 4]. Conventional flotation that uses mechanical cells are widely used in a rougher and more scavenger-like process due to good mixing of pulps but has been known to be ineffective in processing finer mineral particles because of entrainment of fine gangue [5; 6]. Column flotation has found applicability in the mineral industry owing to its effectiveness in processing fine particles, particularly in cleaning operations where gangue entrainment is reduced [7]. Its effectiveness, simplicity of implementation, and convenience in incorporating automatic control has made it very attractive to be used in processing complex ores which require fine grid sizes [6; 8]. The flotation process requires the hydrophobic particles to collide with air bubbles and subsequent bubble-particle attachment to occur for effective separation [9]. In conventional flotation cells, the degree of mixing is relatively high as they process particle sizes that are coarser than those treated in flotation columns. Continuous conventional cells are modelled as fully mixed reactors while batch operations are treated as plug-flow reactors consistent with the residence time of particles being equal [9; 10]. In columns, however, the degree of mixing varies considerably due to their size, aeration rates, size of rising bubbles, and wash water used to reduce gangue entrainment [11]. Thus, the mixing characteristics have been considered to be between that of fully mixed tanks and plug-flow reactors. Hydrodynamics play an important role in the performance of columns since it affects gas dispersion properties, residence time of particles and liquids, as well as mixing and carrying rates [1; 12; 13; 14; 15]. Gas dispersion properties, such as gas superficial velocity (\(J_{g}\)), bubble size (\(d_{b}\)), gas holdup (\(\varepsilon_{g}\)), and bubble surface area flux (\(S_{b}\)) in column flotation, are important parameters which are related to flotation efficiency [16; 17]. Especially, bubble size which affects the bubble residence time, the bubble surface area flux, and the carrying rate has long been recognized as an essential factor in evaluating the performance of columns. Many researchers have measured or estimated the bubble size using the relationships between bubble size, its terminal velocity, or its slip velocity within an operating column and gas holdups [2; 11; 18; 19; 20; 21]. Gas holdups depend on numerous factors, such as frother dosage, bubble size, solid percentage, air velocity, particle size, etc. [15; 22]. Considerable research efforts have been made to quantify the mixing in columns. Control of mixing is important because the degree of mixing in column flotation may deteriorate flotation efficiency due to selectivity and entrainment [11; 23]. The collision/attachment between particles and bubbles resulted from mixing in columns is mainly due to pulp recirculation and the turbulence due to bubble motion in liquid, solid, and gas phases [24; 25]. In the case of assuming equal dispersion coefficient for liquids and solids, a decrease in dispersion number results in perfect mixing [23; 26]. Meanwhile, [PERSON] et al. [22] reported that a highly turbulent mixture in the collection zone affects bubble-particle adhesion, which consequently affects the flotation process. Residence time distribution arises from a complex interaction between the profile velocity, diffusion, and turbulence, and it greatly influences the metallurgical performance of flotation [27]. Many studies on the mixing characteristics of industrial columns have been conducted focusing on the perfect mixing model. [PERSON] et al. [13] showed that large and small tank-in-series and N perfect mixers in series were the best models to present the flow pattern of liquid transferred to tailing and concentrate, respectively.