1 Introduction

Flotation is one of the most valuable and versatile mineral beneficiation technique in which certain suspended particles are separated based on their surface properties. Flotation column is a kind of multiphase contacting device where the liquid is in continuous phase while the gas and solid are in dispersed phase. If a mixture of hydrophobic and hydrophilic particles are suspended in water, and the air is bubbled through the suspension, the hydrophobic particles will tend to attach to the air bubbles and float to the surface while hydrophilic particles remain suspended in water. Flotation columns are intensively utilized as multiphase reactors in chemical, petrochemical and biochemical industries. Flotation column used for wide range of application due to its number of advantages in both design and operation as compared to other processes. Flotation column requires a little maintenance and low operating cost because of compact structure and lack of moving part. In general, most of the industrial flotation columns are counter current due to its better efficiency of separation.

The knowledge of mixing characteristics in the liquid and gas phase is essential for modeling, design and optimization of a flotation column. Mixing in flotation column controls the fine bubble generation, dispersion, and particle suspension. The variation of axial or radial gas holdup causes pressure fluctuations, which result in backmixing of phases and liquid circulation. Liquid circulation governs the rate of mixing and particle recovery in flotation column. The quality of mixedness can be defined as a technique to compare the variation of the concentration distribution of any known sample in a column, which varies from complete separate state to final mixing state [1–3]. According to Ottino [4], the complete mixing can be defined as if the local concentration of a particular component is same as the concentration of that component in the whole system. Ogawa and Ito [5] explained the definition of quality of mixedness by making use of entropy in information theory. They studied the quality of mixedness using an aqueous solution of KCl in a vessel at the different rotation. They realize that the conventional technique to investigate quality mixedness is not sufficient when applying to real world problems. Considering that the mixing phenomenon can be better understood in terms of probabilities. In order to apply the probability theory, it is required to divide the column into n regions. It was assumed that the tracer would take up some of the divided regions of the column. The probability of the presence of tracer in each divided region of the column is known as probability density and it is a strong function of entropy for mixing. Nedeltchev et al. [6, 7] applied the information entropy theory to study the quality of mixedness to scale up the bubble column. The quality of mixedness varies from zero (no mixing) to unity (complete mixing).

Since the quality of mixing is time dependent parameter, in order to scale up it is necessary to study the quality of mixedness with respect to the dimensionless time. They developed empirical correlations to interpret quality of mixedness against dimensionless time. They found that at the beginning, the quality of mixedness is higher for smaller gas velocity, which results in uniform distribution of tracer concentration in the column. Camesasca et al. [8] reported the quality of mixing of fluid systems in the two-dimensional cavity using the information entropy theory. They calculated the entropy using the probability of detection of the particular group of a component at a given location in a system. Nedeltchev et al. [9] studied the flow regime identification based on the quality of mixedness in the air–water bubble column. They used information entropy theory to calculate the quality of mixedness by deriving the data from Computer-automated radioactive particle tracking (CARPT). A more detail explanation of information entropy theory has been provided in Sect. 2.1 of this article.

The residence time distribution (RTD) technique was used by Mavros and Daniilidou [10] to investigate the mixing characteristics in the flotation column. They analyzed the effect of superficial liquid and gas velocities and column diameter on the mixing characteristics. There are two experimental methods to determine the RTD: (1) transient method (introduction of disturbance and monitoring the system response); and (2) the steady state injection of tracer technique (tracer is injected into the column and wait until steady state condition reached). According to Dobby and Finch [11], the mixing feature in the recovery zone is vital, since the particle collection procedure that happens in this zone is commonly the rate-deciding step in the overall flotation process. Kouchi et al. [12] reported the expressions for the probability of bubble-particle attachment at different flow conditions, which is the function of bubble size, bubble Reynolds number, particle size and induction time. The expression signifies, as there is a decrease in the particle size and bubble size, the attachment efficiency of the particle to gas bubble increases. With further continuous reduction in the bubble size and particle size, the attachment probability reaches a critical value and then it starts to decrease. Literature survey shows that the probability of attachment of very fine particle to the gas bubble is much small due to poor selectivity.

Yoon [13] observed that the first order flotation rate is directly proportional to the superficial gas velocity and inversely proportional to the cube of bubble diameter. He analyzed that the reducing bubble diameter is more effective than increasing superficial gas velocity in order to increase the flotation rate. Recovery mainly depends on the probability of a particle collected by an air bubble in the flotation cell, flotation rate constant and Peclet number. Peclet number (zero for complete mixing and ∞ for no mixing) is the ratio of the rate of transport by convection to the rate of transport by diffusion. It is used to quantify the mixing in the flotation cell [14]. The probability of a particle collection by air bubble is the function of the probability of particle collision, the probability of attachment and probability of detachment. The probability of detachment of fine particles is negligibly small due to low inertia, which may be neglected. Tuteja et al. [15] discussed the flotation kinetics and demonstrated the kinetic and non-kinetic models of recovery. They reported that the kinetic model like Finch and Dobby model and VPI Model is based on the axial dispersion theory. VPI model uses Peclet number as a parameter to characterize the mixing condition in the flotation column. Mavros [16] investigated the mixing characteristics in the flotation column. They developed an empirical correlation to predict the performance and intensity of mixing in the flotation process.

Kouchi et al. [12] used the Pyke et al. [17] model to calculate the collision and attachment efficiency of the bubble. They studied the effect of the flow condition, the hydrophobicity of the particle and bubble size on attachment and collision efficiency. They also calculated the flotation rate constant of the particle by an analytical model, which was derived by Yoon and Luttrell [18]. Mills and O’Connor [19] examined the mixing characteristics by RTD method in flotation column. They developed empirical correlations to predict the mixing for scale-up processes. Xu and Finch [20] analyzed the solid mixing in flotation column by RTD method using axial dispersion model. Yiantos et al. [21] estimated the mean residence time by the direct measurement of solid and liquid time response of self-aerated copper flotation cell. They used the radioactive tracer technique and non-floatable minerals particles as a solid tracer to predict the mixing. Nguyen et al. [22] reported the method of the probability of bubble-particle attachment on a single bubble in flotation column. They used the experimental data to induce the dependence of induction time on particle size. Based on calculated data they showed that induction time increases as particle size decreases. Mavros et al. [23] studied the effect of a vertical baffle, its type, and various gas sparger on mixing processes in flotation column. They found complete mixing which was due to the presence of vertical baffle that accelerates the liquid flow rate. Yiantos et al. [24] analyzed the characteristics of large size industrial flotation column cells using the non-invasive technique like the use of radioactive tracer, which will measure the residence time distribution of liquid and solid. They also reported the effect of particle sizes on RTD.

From the literature, it is observed that there have been no studies regarding the quality if mixedness based on the information entropy theory in the flotation column. The purpose of the present study is, therefore, to investigate the quality of mixedness with respect to the dimensionless time and recovery based on the quality of mixedness in presence of different dynamic and geometric variables.

2 Method of Analysis

2.1 Quality of Mixedness Based on Information Entropy Theory

The performance of any multiphase device depends on the mixing characteristics of the phases inside the multiphase device. In this study, a flotation column is assumed to be operated co-currently with gas-liquid-solid where gas and solid are dispersed in a continuous liquid phase. To determine the efficiency of the flotation column based on the information entropy theory, the knowledge of the extent of quality of mixedness are required. In the following sections, the theories for the quality of mixedness and efficiency of recovery of a mineral particle in a flotation column have been formulated based on information entropy theory. In information theory, information entropy is a measure of the uncertainty associated with a random variable [6, 25]. The randomness of variables is completely defined by its cumulative distribution function (CDF) which can be used to determine the probability distribution over a particular control volume. The probability distribution function completely describes the probability distribution of a real-valued random variable of X. For every real number x, the CDF of X is given by

$$x \to F_{x} (x) = P(X \le x)$$
(1)

The information entropy of a discrete random variable X, which can take the range of possible values \(\left\{ {x_{1} , \ldots ,x_{n} } \right\}\) which is defined as

$$H(X) = - \sum\limits_{i = 1}^{n} {p(x_{i} )\log (x_{i} )}$$
(2)

where p(x i ) = Pr (X = x i ) is the probability mass function of X. The information content is a measure of the information associated with the outcome of a random variable. By definition, the amount of self-information contained in a probabilistic event depends only on the probability of that event. The smaller is its probability, the larger the self-information associated with receiving the information that the event indeed occurred. The probability theory is useful to study the most of the mixing phenomena. The entire column has to be divided axis symmetrically into layers of semi-cylindrical shells of sub-volumes in order to apply the probability theory. The concentration of tracer needs to be known at each cylindrical shell into which, the column is divided to calculate the information entropy. The flotation column was divided into 1000 parts of the cylindrical shell. The flow of tracer in the flotation column considered to analyze the quality of mixedness. The probability of tracer come out in any semi-cylindrical shell is specified as per information entropy theory as

$$P_{i} (t) = \frac{{C_{i} (t)V_{i} }}{{\sum\nolimits_{i = 1}^{n} {C_{i} (t)V_{i} } }}$$
(3)

where \(C_{i}\) is the tracer concentration and \(V_{i}\) is the volume of each shell. This can be estimated using residence time distribution (RTD) by experiment. According to theory, the total amount of information content in each shell is given by

$$I_{i} (t) = - \log (P_{i} (t))$$
(4)

Thus, the information entropy can act as

$$H(t) = \sum\limits_{i = 1}^{n} {P_{i} (t)} - (\log (P_{i} ))$$
(5)

The minimum entropy corresponds to the state when there is no mixing in the column whereas maximum entropy corresponds to the state of uniform mixing. The quality of mixedness is a measure to express the degree of asymptotical approach to the equilibrium state and it is defined as [6]

$$M(t) = \frac{{H(t) - H_{\hbox{min} } }}{{H_{\hbox{max} } - H_{\hbox{min} } }}$$
(6)

The range of variation of mixedness factor is \(0 < M(t) < 1\). \(M(t)\) can be calculated from the Eq. (6). The relation between mixedness factor and time can be developed with the experimental data as follows

$$M(t) = 1 - \frac{1}{{ 1 { + }\left( {{\text{t/(t}}_{\text{m}} x_{0} )} \right)^{\beta } }}$$
(7)

where t m is the mean residence time of the fluid element. In the Eq. (7) the value of x o and β can be predicted by fitting the M(t) from experimental data.

2.2 Efficiency of Flotation Process Based on Mixedness Factor

The recovery in flotation column depends on two main factors, one is flotation kinetics, and the other is a degree of mixing. Yoon et al. [13] conducted research work on the modeling of flotation columns for over a decade. They reported an expression to predict the recovery based on axial dispersion theory. The recovery of the flotation process in a flotation column is represented as [13]

$$R = 1 - \frac{{4ae^{Pe/2} }}{{(1 + a)^{2} e^{a/(2Pe)} - (1 - a)^{2} e^{ - a/(2Pe)} }}$$
(8)

where

$$a = \left[ {1 + \left( {\frac{4kt}{Pe}} \right)} \right]^{1/2}$$
(9)

The Peclet number in Eqs. (8) and (9) can be expressed as [26]

$$Pe = 0.7\left( {\frac{L}{{d_{c} }}} \right)^{0.63} \left( {\frac{{u_{l} }}{{u_{g} }}} \right)^{0.5}$$
(10)

The first order flotation rate constant is directly proportional to the superficial gas flow rate and inversely proportional to the size of the bubble, which can be represented as [13]

$$k = (3E_{coll} /2d_{b} )u_{g}$$
(11)

The residence time as a function of quality of mixedness can be obtained by rearranging Eq. (7), which is given as

$$t = t_{m} x_{0} \left( {\frac{M(t)}{1 - M(t)}} \right)^{1/\beta }$$
(12)

2.3 Particle Collection Efficiency

The overall collection efficiency (\(E_{Coll}\)) of a particle collected by an air bubble, can be expressed as [13]

$$E_{coll} = E_{c} E_{a} E_{s}$$
(13)

where \(E_{c}\), \(E_{a}\), and \(E_{s}\) are the efficiencies of the subprocesses of particle–bubble collision, attachment, and stability related to the detachment efficiency respectively. The efficiency of the bubble-particle collision is determined by the hydrodynamics of the system, which, is affected by the particle size, bubble size, and the turbulence of the system. Weber and Paddock [27] derived an analytical expression to predict the efficiency of collision, which can be applicable for wide range of bubble and particle size. The efficiency of collision can be expressed as [27]

$$E_{c} = 1.5\left[ {1 + \frac{{3/16\text{Re}_{l} }}{{1 + 0.249\text{Re}_{b}^{0.56} }}} \right]\left( {\frac{{d_{p} }}{{d_{b} }}} \right)^{2}$$
(14)

The bubble diameter was calculated using correlation [28]

$$d_{b} = 13.4u_{l}^{0.052} u_{g}^{0.248} \gamma^{0.008} \sigma^{0.034}$$
(15)

where \(\gamma\) is the dynamic viscosity of liquid in Pa s, \(u_{l}\) and \(u_{g}\) are in mm/s, \(\sigma\) is in dyne/cm and \(d_{b}\) is in mm. Bubble rise velocity (\(u_{b}\) in mm) is calculated as follows [28]

$$u_{b} = 18d_{b}^{0.989}$$
(16)

The probability of particle adhesion \(\left( {E_{a} } \right)\) is affected by the hydrodynamics but is largely a function of the surface chemistry. Yoon and Luttrell [18] developed an expression for probability of particle adhesion based on stream profile of intermediate Reynolds number range, which is represented by

$$E_{a} = \sin^{2} \left[ {2\tan^{ - 1} \exp \left( {\frac{{ - (45 + 8\text{Re}^{0.72} u_{b} t_{i} }}{{15d_{b} (d_{b} /d_{p} + 1)}}} \right)} \right]$$
(17)

where \(t_{i}\) is the induction time which is a function of particle hydrophobicity [13] and bubble size [29]. The probability of particle adhesion (\(E_{a}\)) increases with a decrease in bubble size [18] and particle size. The efficiency of stability in the turbulent flow field can be determined as [30]

$$E_{s} = 1 - \exp \left\{ {1 - \left( {\frac{{d_{p\hbox{max} } }}{{d_{p} }}} \right)^{2} } \right\} = 1 - \exp \left\{ {1 - \frac{{2.34\sigma^{6/5} (1 - \cos \theta )}}{{d_{p}^{2} \Delta \rho \varepsilon_{i}^{4/5} \rho_{l}^{1/5} }}} \right\}$$
(18)

This equation is valid when the hysteresis of the contact angle is \(\Delta \theta = \theta_{A} - \theta_{R}\), where \(\theta_{A}\) and \(\theta_{R}\) are the (maximum) advancing and (minimum) receding contact angle. If the contact angle hysteresis is significant (i.e., \(\Delta \theta > \theta_{R}\)), then the following equation can be used [30]

$$E_{s} = 1 - \exp \left[ {\frac{{1.17\sigma^{6/5} \sin \theta_{R} \sin \Delta \theta }}{{d^{2}_{p} \Delta \rho \varepsilon_{i}^{4} \rho_{l}^{1/5} }}} \right]$$
(19)

Equations (18) and (19) are applicable when the turbulent tensile stresses are dominant. If the turbulent shear stresses or the vibration of bubbles rising to the pulp-froth interface is dominant, then the Eqs. (18) and (19) must be modified, as given by Nguyen and Schulze [31].

Mavros and Daniilidou [10] investigated the RTD of liquid phase using KCl as a liquid tracer in three different column diameter. They used tap water throughout the experiment at 25 ± 1 °C. the experimental finding was expressed as the plot of E (called profile of residence time distribution) against dimensionless time. Dobby and Finch [11] and Xu and Finch [20] studied the RTD of liquid and solid phase, dye as a liquid tracer and manganese dioxide (MnO2) as a solid tracer. They studied RTD of solid using seven different particle size and expressed experimental finding as of the plot of outlet concentration against time.

The raw data (typical data shown in Fig. 1) has been extracted from the profile of E versus dimensionless time and then this E data is converted into the concentration data using known quantity of liquid tracer. Concentration data is used to calculate the quality of mixedness by using Eqs. (3) to (6). Recovery is calculated by using Eqs. (8) to (17). The value of x 0 and β are obtained from Eqs. (22) and (23). Since recovery is a function of parameter t (Eqs 8 and 9) which is related to M(t) (Eq. 12), recovery is compared against the quality of mixedness. The recovery expression contains term ‘a’ (Eq. 9) which is a function of rate constant k (Eq. 9). The rate constant k expressed by Eq. (11) is a function of E coll and it can be defined by Eq. (13). Further, Eqs. (14) to (17) help in calculation of the overall collection efficiency.

Fig. 1
figure 1

Residence time distribution of tracer in a flotation column. The data is extracted from the published article of Mavros and Daniilidou [10] to analyze the present work

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3 Results and Discussion

3.1 Time Independent Profile of Quality of Mixedness at Different Operating and Geometric Variable

The information entropy theory is used to analyze the quality of mixedness in the flotation column. The quality of mixedness is calculated based on the model suggested by Nedeltchev et al. [6]. The analysis was carried out based on the experimental data taken from the literature as shown in Table 1.

Table 1 List of variables covered in this section to analyze quality of mixedness
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The quality of mixedness increases with dimensionless time to ultimate value of unity as demonstrated in Fig. 2. It represents the quality of mixedness at different operating and geometric variables. The change of quality of mixedness is same at different gas velocities as shown in Fig. 2a. The initial response of quality of mixedness is greater corresponding to the curve with highest gas velocity as it can be shown in Fig. 2b, c. It may be due to the turbulence generated by higher superficial gas velocity. Quality of mixedness increases in the flotation column due to liquid recirculation as turbulence caused by tiny bubble motion. The liquid circulation caused by superficial gas velocity is responsible for the rate of mixing and recovery efficiency of flotation column.

Fig. 2
figure 2

Time independent profile of quality of mixedness

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3.2 Variation of Recovery Based on Quality of Mixedness

The recovery of particles reflects the efficiency of the flotation process and the performance of the equipment. The recovery of the particle by flotation depends on the mixing efficacy of the process equipment. An increase in quality of mixedness enhances the degree of recovery in the flotation column. The analysis is based on the experimental data obtained from literature as shown in Table 2. The recovery responses (as shown in Fig. 3a) are same corresponding to the particle diameters of 0.000044 and 0.00063 m. In beginning, the curves belonging to mean time of 90 and 92 min show better response of recovery than the other curves up to certain value of M(t); at the end all reaches to the ultimate value. Initially both recovery curve provides better response than others do due to maximum mean time and lower particle diameter. The initial response of recovery of particle diameter, 0.000088 m (due to smaller particle diameter) should be greater than the particle diameter, 0.000125 m; but, it is not true because the mean time (23 min) corresponding to smaller particle diameter is smaller than mean time (60 min) corresponding to the greater particle diameter. It is observed that the mean time has more significant effect on the initial response to recovery than the particle size. In Fig. 3b, it is observed that the curve with mean time (23 min) takes more time to reach the ultimate value (unity) of quality of mixedness. Figure 3a, b demonstrate that the same curve with mean time (23 min) possesses smaller initial response of recovery. Same argument can be applied for Fig. 3c, d.

Table 2 List of geometric and dynamic variables covered in this section
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Fig. 3
figure 3

Variation of recovery and quality of mixedness at different dynamic and geometric variable

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3.3 Development of Model for Quality of Mixedness as a Function of Time

A correlation for quality of mixedness is developed as a function of time and mean time. From the calculated data of quality of mixedness against dimensionless time a model is developed based on the logistic model. The logistic model is expressed as (as per Origin 6.0 software)

$$Y = \left( {(A_{1} - A_{2} )/(1 + (z/x_{0} )^{p} } \right) + A_{2}$$
(20)

where parameters A 1, A 2, z and x 0 are calculated based on the E versus dimensionless time data. Then, the obtained parameters are correlated as a function of geometric and dynamic variable of system using dimensional analysis. Finally, the model is developed by multiple regression analysis in Microsoft excel. Based on the available experimental data, the logistic model equation has been fitted by least square method using Origin 6.0 software. The model equation obtained can be represented as

$$M(t) = 1 - \frac{1}{{ 1 { + }\left( {{\text{t/(t}}_{\text{m}} x_{0} )} \right)^{\beta } }}$$
(21)

From the experimental data, the parameter x 0 and β were found to be a strong function of the dynamic and, geometric variables and the physical properties of the liquid. By dimensional analysis, the parameters obtained are correlated as a function of these variables. The correlations are obtained by regression analysis based on calculated data for both the parameters and represented as

$$x_{0} = 275.9\,\text{Re}_{l}^{ - 0.84}$$
(22)

(with correlation coefficient and standard error of 0.907 and 0.512, respectively), and

$$\beta = 37.31\text{Re}_{l}^{ - 0.30}$$
(23)

(with correlation coefficient of 0.919 and standard error of 0.1574, respectively).

4 Conclusions

Time independent profile of quality of mixedness was analyzed based on the residence time distribution data. From the present study, it can be concluded that quality of mixedness has the significant effect on recovery in flotation column. Recovery increases as the quality of mixedness increase. The quality of mixedness and recovery in flotation process is the function of gas and liquid velocity, column diameter, bubble size, mean time, and column length. Mean time of flotation process plays a significant role in recovery. The present study may be useful for further understanding and scaling up the flotation column based on the quality of mixedness.