## 5 Result and Discussion  
A size analysis of coal particles transferred from the feed to the concentrate product was conducted. From the size distributions and the mass flow rate of the concentrate and feed measured, a performance curve for the column was constructed. Figure 2 shows a plot of the fraction of solids recovered in the concentrate (\(C/F\)) versus particle size. The trend is as expected in flotation cells, in that the recovery is maximal in the range 30-100 \(\upmu\)m and decreases for fine and larger particles. Typically, flotation performance is highest in the region of intermediate particle size and decreases at the finer and larger sizes. There are lots of effects, such as mineral liberation, floatable mineral, reagents, flotation time etc. However, in terms of particle-bubble interaction, finer particles may be the low collision and attachment efficiency to bubble, which have insufficient inertia to cross the water streamlines around the coursing bubble [35]. In case of coarse particles, this is due to the high detachment probability to bubble including decreased mineral liberation [35; 36]. Figure 2: Performance curve as fraction to concentrate (\(C/F\)) vs. particle size (\(d_{p}\)). Based on the performance curve in Figure 2, the mean particle residence times of particles were calculated using Equation (16) and used in Equations (14), (15) and (18) to estimate the flotation rate constants that correspond to plug flow, fully mixed, and axial mixing conditions. Figure 3 shows the variation of these rate constants against particle size. It shows that the flotation rate constants are similar below about 40 \(\upmu\)m. It is, as expected, as the residence time of finer particles equals that of the water. Furthermore, it has been also reported that, for finer particles, solid axial mixing is the same as fluid axial mixing [11]. For intermediate particle sizes, the estimated rate constants vary considerably but those evaluated using plug flow and axial dispersion are closer to each other than those due to a perfect mixing model. In particular, in the case of coarse particles, the deviation is large in the case of perfect mixing, while axial mixing is suitable. Thus, it has been shown that the mixing characteristics of a flotation column affect the flotation rate constant, particularly for coarser particles. Thus, it shows that the axial mixing is prevalent in columns under the operating conditions tested and the perfect mixing model is not suitable for modelling small column performance. Normally, the mixing characteristics in column are considered to be between that of plug-flow reactors and fully mixed tanks. In addition, the axial-dispersed model can describe the hydrodynamics of laboratory or pilot scale columns properly, but due to the limitations it cannot be used for industrial columns [8]. A consequence of mixing is its effect on the residence time distribution of particles and the liquid. Figure 4 shows the ratio of the residence times of particles and liquid against  
Figure 3: Comparison of calculated collection rate constant (\(k_{c}\)) vs. particle size (\(d_{p}\)) under various flow conditions of particles in the column. particle size. As can be seen, this ratio decreases with particle size from unity at small sizes to lower values at coarser sizes. The points corresponding to this study agrees well with [PERSON] and [PERSON] (2017) but deviates from those of [PERSON] and [PERSON] (2018), particularly for finer particles. It may be due to their assumption that the axial dispersion coefficient is equal for both liquid and particles. It also illustrates the effect of solids density as [PERSON] and [PERSON] data were obtained for particle density of 5.19 and the coal used in this study has a density of 1.86. The data of [PERSON] and [PERSON] had a density of 3.6 and are close to those of this study. As can be seen from the figure, the effect of solids density is to decrease the ratio (\(\tau_{p}/\tau_{L}\)), particularly for finer particle sizes.