## 2 Mixing in Flotation Columns  
In evaluating the performance, it is important to assess the residence time distribution of the particles. Due to the upward flow of air and the downward flow of wash water and feed slurry, some degree of mixing occurs in column in an axial direction. The degree of mixing depends on the flowrates of these streams and the dimensions of the column.  
The axial mixing in a column may be described by:  
\[\frac{\partial\mathbb{C}}{\partial t}=E\left(\frac{\partial^{2}C}{C\partial^{2 }}\right)-u\frac{\partial\mathbb{C}}{\partial x}\, \tag{1}\]  
where \(E\) is the axial dispersion coefficient and quantify the degree of mixing and \(C\) is the concentration of a given phase (solids or liquid).  
As the residence time distribution (RTD) of the given phase is also dependent on the column height (\(H_{c}\)) and its superficial velocity, a vessel dispersion number (\(N_{d}\)) is defined to characterize the mixing process.  
\[N_{d}=\frac{E}{u\;H_{c}}, \tag{2}\]  
It is recognized that the dispersion number for solids differs from that of the liquid and may be defined as:  
\[\text{for liquid }N_{dL}=\frac{E_{L}}{u_{i}\;H_{c}}, \tag{3}\]  
\[\text{for solid particles }N_{dp}=\frac{E_{p}}{u_{p}\;H_{c}} \tag{4}\]  
The interstitial liquid velocity, \(u_{i}\), is given by:  
\[u_{i}=\frac{u_{L}}{1-\varepsilon_{g}}, \tag{5}\]  
where \(u_{L}\) is the superficial liquid velocity. The particle velocity \(u_{p}\) may be calculated in terms of its slip velocity \(U_{sp}\) as;  
\[u_{p}=u_{i}+U_{sp}, \tag{6}\]  
The slip velocity of particles is estimated by [PERSON] [21] as:  
\[U_{sp}=\frac{g\;d_{p}^{2}\;(\rho_{p}-\rho_{sl})\;(1-\mathcal{O}_{s})^{2.7}}{18 \;\mu_{f}\big{(}1+0.15\;Re_{p}^{0.687}\big{)}}, \tag{7}\]  
where the particle Reynold's no is given by: \(Re_{p}=\frac{d_{p}\;U_{sp}\;\rho_{l}\;(1-\mathcal{O}_{s})}{\mu_{f}}\).  
[PERSON] and [PERSON] [26] argued that the axial dispersion coefficient for the liquid and solid phases may be taken as equal and estimated by:  
\[E_{p}=E_{l}=0.063\;d_{c}\;(\frac{J_{g}}{1.6})^{0.3}. \tag{8}\]  
[PERSON] [25] proposed a relationship that is applicable to a wide range of column diameters, albeit, for a limited superficial gas flowrates.  
\[E=(9.3\;d_{c}-30.1)\;J_{g}^{0.3}, \tag{9}\]  
Substituting Equations (5)-(9) in (3),  
\[\text{For liquid }:\;N_{dL}=\frac{0.063\;d_{c}\;(\frac{J_{g}}{1.6})^{0.3}}{ \big{[}\big{(}J_{sl}\big{)}/\big{(}1-\varepsilon_{g}\big{)}\big{]}\;H_{c}} \tag{10}\]\[N_{dL}=\frac{\left(9.3\,d_{c}-30.1\right)\,J_{g}^{0.3}}{\left[\left(J_{sl}/\left(1- \varepsilon_{g}\right)\right)\right]\,H_{c}} \tag{11}\]  
\[\text{For particles : }N_{dp}=\frac{0.063\,d_{c}\,\left(\frac{I_{g}}{1_{b}} \right)^{0.3}}{\left[\left(J_{sl}\right)/\left(1-\varepsilon_{g}\right)+U_{ sp}\right]\,H_{c}}, \tag{12}\]  
Dobby and Finch [26],  
\[N_{dp}=\frac{\left(9.3\,d_{c}-30.1\right)\,J_{g}^{0.3}}{\left[\left(J_{sl}/ \left(1-\varepsilon_{g}\right)\right)+U_{sp}\right]\,H_{c}}, \tag{13}\]  
[PERSON] [25],  
Note: \(J_{sl}\) includes the superficial rate of wash water and that of feed water.  
These relationships are used with the data from the coal processing application reported in this work to evaluate the extent of mixing in the column.