Estimation of flotation rate constant and particle-bubble interactions considering key hydrodynamic parameters and their interrelations

Minerals Engineering 141 (2019) 105836

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Estimation of flotation rate constant and particle-bubble interactions considering key hydrodynamic parameters and their interrelations

T

Ahmad Hassanzadeha,b, , Asghar Azizic, Sabri Kouachid, Mohsen Karimie, Mehmet S. Celika ⁎

a

Mineral Processing Engineering Department, Faculty of Mines, Istanbul Technical University, Maslak 34469, Istanbul, Turkey Department of Processing, Helmholtz-Institute Freiberg for Resource Technology, Helmholtz-Zentrum Dresden-Rossendorf, Chemnitzer Straße 40, Freiberg 09599, Germany c Faculty of Mining, Petroleum and Geophysics, Shahrood University of Technology, Shahrood, Iran d Applied Chemistry and Materials Technology Laboratory, Larbi Ben M'hidi University, OEB 04000, Algeria e Department of Chemistry and Chemical Engineering, Chalmers University of Technology, 41296 Gothenburg, Sweden b

ARTICLE INFO

ABSTRACT

Keywords: Flotation rate constant Particle–bubble interactions Particle density Turbulence dissipation rate Response surface modeling

Particle-bubble sub-processes cannot be directly and physically obtained in froth flotation due to the complexity of the process as well as numerous and dynamic interactions of particles and bubbles in an extremely intensive turbulent condition. Therefore, over the last three decades, two fundamental model configurations have been used as an only solution for prediction of particle-bubble collection efficiencies (Ecoll). Additionally, the relative intensity of the main flotation parameters on flotation rate constant, particle–bubble interactions together with their interrelations is not adequately addressed in the literature. The present study attempts in two separate phases to overcome these difficulties. In the first stage, prediction and evaluation of particle-bubble sub-processes are critically discussed by categorizing them in two configurations. The analytical models (approach I) commonly applied generalized Sutherland equation (EcGSE ), modified Dobby–Finch (EaDF ) and modified Schulze stability (EsSC ) models. The second approach, numerical models, utilized Yoon–Luttrell (EcYL ), Yoon–Luttrell (intermediate) (EaYL ) and modified Schulze stability (EsSC ) models. In the second stage, relative intensity and interrelation of key effective hydrodynamic parameters on the probability of particle–bubble encounter (Ec) and flotation rate constant (k) are obtained and optimized by means of the response surface modeling (RSM) based on central composite design (CCD). Five key factors including particle size (1–100 µm), particle density (1.3–4.1 kg/m3), bubble size (0.05–0.10 cm) and bubble velocity (10–30 cm/s) together with turbulence dissipation rate (18–30 m2/s3) are considered in order to maximize the responses including the k and Ec. The results obtained show that the Ecoll calculated by numerical techniques (configuration (II)) is greater than that of analytical approaches (configuration (I)) due to assumptions involved in using Yoon–Luttrell collision and attachment models. It is also found that under the conditions studied, particle size and bubble velocity are the most effective factors on Ec and k, respectively. Furthermore, not only the relative significance of factors on Ec and k but also the interrelation of cell turbulence and bubble size as well as bubble velocity and turbulence are shown to be inconsistent in the literature and thus require further studies. We briefly reported the main longstanding challenges in flotation kinetic modeling and emphasized on a serious need for fulfilling lack of physical observations. Finally, the presented analyses with respect to three-zone model offer a new concept for the extension of common flotation modeling approach using analytical and numerical techniques.

1. Introduction Flotation is a powerful separation technique with its extensive applications in different industries such as solid-solid separation, paper (Venditti, 2004) and plastic industries (Wang et al., 2015),

petrochemical industry (Zheng and Zhao, 1993) electrolyte cleaning, remediation (Vanthuyne et al., 2003), oil sands processing (Kasongo et al., 2000) and wastewater treatment (Rubio et al., 2002). Flotation takes place in the presence of one continuous phase (liquid (water)) and two disperse phases (i.e., solid (particles), and gas (air bubbles)). The

⁎ Corresponding author at: Department of Processing, Helmholtz-Institute Freiberg for Resource Technology, Helmholtz-Zentrum Dresden-Rossendorf, Chemnitzer Straße 40, Freiberg 09599, Germany. E-mail addresses: [email protected], [email protected] (A. Hassanzadeh).

https://doi.org/10.1016/j.mineng.2019.105836 Received 29 August 2018; Received in revised form 16 May 2019; Accepted 27 May 2019 Available online 19 June 2019 0892-6875/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature k Np Nb Zpb Ec Ea Es dp db Vp Vb ε θ f

tind Gfr Vcell Jg CD ν f Fatt Fdett t p

d32 a

Ib NB

K3 and β dimensionless numbers EcGSE generalized Sutherland equation EcYL Yoon–Luttrell collision model EcSU Sutherland equation EaDF modified Dobby–Finch model EaYL Yoon–Luttrell attachment model EsSC modified Schulze stability model Kst Stokes number Reb bubble Reynolds number Sb bubble surface area flux λ micro–turbulence EcAK Afruns-Kitchner collision model Cp particle circularity BO Bond number g gravitational constant angle of tangency t σ surface tension MRTb mean bubble residence time gas hold-up g RTD residence time distribution CCD central composite design ANOVA analysis of variance MLR multi linear regression ANN artificial neural network RSM response surface modeling Ecoll collection efficiency As empirical stabilization constant = 0.50 TKE total kinematic energy of turbulence DF degree of freedom DOE design of experimental MB microbubble A and B constants in attachment efficiency model

flotation rate constant number of particles per unit volume number of bubbles per unit volume collision frequency particle-bubble encounter efficiency particle-bubble attachment efficiency particle-bubble stability efficiency particle size bubble size particle relative velocity bubble relative velocity turbulence dissipation rate contact angle fluid density induction time gas flow rate volume of vessel superficial gas velocity drag coefficient liquid viscosity kinematic viscosity constant value = 2.03 attachment force detachment force time particle density Sauter mean diameter adhesion angle interfacial area of bubbles nanobubble

properties of each phase play a vital role in the efficiency of the flotation process. Flotation process encompasses three important factors involving ore features (e.g. mineralogy and degree of liberation), cell hydrodynamics (e.g. air flowrate, cell geometry and agitation type) and chemical reagents (e.g. collector, frother, modifier, activator and depressant) to obtain desirable metallurgical outcomes (i.e. grade, recovery and selectivity index (SI)) (Hassanzadeh and Hasanzadeh, 2016). From the microscopic point of view, the particle–bubble encounter efficiency (Ec) which is predominantly controlled by hydrodynamic forces can be considered as the most effective initial sub–process in flotation rate constant. Estimation of the Ec and its modeling are generally performed using three different techniques viz. analytical, numerical and experimental approaches. Each method has some restrictions leading to deviation of the measured and/or estimated values from the actual amounts. Detailed insights in this regard are reported elsewhere (Min et al., 2008; Hassanzadeh, 2018; Wang et al., 2018). In the previous work, we found that direct experimental visualization of the particle-bubble sub-processes is very complicated owing to the difficulties in terms of isolating these microprocesses from each other in an actual flotation separation system (Hassanzadeh et al., 2018a). However, the fundamental deviations between analytical and numerical approaches based on the applied models were not discussed for specific case studies which are covered in this research work. Apart from the particle-bubble interactions, flotation rate constant (k) is a key factor for optimization and improvement of circuit flowsheets and scale-up principles (Reuter and van Deventer, 1992; Mesa and Brito-Parada, 2018). Macroscopic modeling of rate constant was extensively studied over the past decades which is now basically categorized in two empirical and phenomenological (population balance, mathematical and probabilistic) models (Gharai and Venugopal, 2016;

Prakash et al., 2018). From the 1930′s to early 1990, various mathematical and practical flotation kinetic models were proposed and examined under different flotation conditions (Zuniga, 1935; Kelsall, 1961; Arbiter and Harris, 1962; Imauzimi and Inoue, 1963; Woodburn, 1970; Jowett and Safvi, 1960; Trahar and Warren, 1976; Harris, 1978; Klimpel, 1980; Agar et al., 1989; Mehrotra and Padmanabhan, 1990; Kelly and Carlson, 1991; Ek, 1992; Yuan et al., 1996). Nevertheless, it is now accepted that there is no unique flotation kinetic model applicable to the froth flotation processes (Dowling et al., 1985; Nguyen and Schulze, 2004). The majority of the empirical models have fitted first–order kinetic models to the experimental data which are valid only at steady–state conditions (Azizi et al., 2015; Albijanic et al., 2015; Ni, et al., 2016; Hassanzadeh and Karakaş, 2017a). The inability of these empirical and phenomenological models to describe the flotation rate constant under various flotation conditions has made them unsuitable for assessment, control and monitoring purposes. Additionally, these models are suffered from overfitting as it has been reported in several research studies (Bu et al., 2017; Sahoo et al., 2016; Hassanzadeh et al., 2018b) and information criteria (IC) was recommended as a reliable approach rather than using common regression methods due to the consideration of number of model parameters and model complexities (Hassanzadeh, 2017a; Sahoo et al., 2016). Therefore, it appears that the first principle and fundamental analytical modeling of flotation rate constant can provide a universal way of evaluating the flotation kinetics with a distribution of particle floatabilities (Nguyen and Schulze, 2004). Several factors such as particle size (Trahar, 1981; Agheli et al., 2018), bubble diameter (Tao, 2004), percent solids (Azizi et al., 2015), particle residence time distribution (Lelinski et al., 2002; Hassanzadeh, 2017b), angle of tangency (Firouzi et al., 2011), water chemistry (Michaux et al., 2018), particle roughness (Guven and Celik, 2016), 2

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morphology (Verrelli et al., 2014), density (Shi and Fornasiero, 2009) and particle terminal settling velocity (Nguyen et al., 1997), contact angle (Chaua et al., 2009), bubble surface contamination (Malysa et al., 2005), bubble velocity (Hassanzadeh et al., 2016), superficial gas velocity and gas hold-up (Newcombe et al., 2013), rotor speed (Newcombe et al., 2018) and power input (Safari et al., 2016), cell aspect ratio (Tabosa et al., 2016), fluid flow regime (Kouachi et al., 2015) and turbulent dispersion rate (Fallenius, 1987) play significant roles in maximizing both Ec and k in the froth flotation processes. A brief explanation describing the main effect of each parameter on Ec and k is given below and schematically represented in Table 1. Turbulence dissipation rate ( ): An optimum flotation rate constant is essential not only to maximize the particle–bubble encounter efficiency but also to minimize the particle–bubble detachment (Schubert, 2008). More specifically, the cell turbulence leads to a slight increase in the flotation rate constant of fine and intermediate sizes of dense particles and in turn to an increase in particle–bubble collision frequency (Zpb), production of smaller bubbles by breakup and volume of pulp that sweeps through the bubble (Vpath). On the contrary, a substantial decrease in the flotation rate constant of coarse particles occurs due to a decrease in the particle–bubble aggregate stability, particularly for denser minerals. Contact angle ( ): Particle contact angle leads to an increase in the collection efficiency and flotation rate constant due to an increase in the attachment efficiency (Najafi et al., 2008). Both static and dynamic contact angles vary substantially as a function of mineral surface roughness, heterogeneity together with particle shape and size properties (Chau et al., 2009). For a given particle size, there is a critical contact angle below which particles do not attach to bubble interfaces; contact angle increases with a decrease in particle size (Chipfunhu et al., 2012). Meanwhile, the flotation rate constant increases rapidly and slightly with an increase in contact angle of intermediate and coarse size fractions, respectively (Muganda et al., 2011). Percent solids (%S): There is an optimal pulp density where maximum k and Ec can be achieved in froth flotation. Increasing % solid results in greater collision frequency, however, it negatively affects the pulp rheology inducing an increase in entrainment of gangue minerals in the froth zone. Furthermore, a high percent solids and/or attractive interaction between fine particles can also increase the pulp viscosity leading to a decrease in turbulent energy dissipation rate, which in turn induces a negative effect on the Zpb (Fornasiero and Filippov, 2017). Particle size (dp): There is a large amount of evidence indicating that Ec increases with increasing the particle size. Fine particles have low relative velocity and thus a lower collision probability with bubbles compared with coarse particles. However, the behavior of k versus dp follows different trends. As a general rule, k decreases gradually when the range of particle size becomes finer due to an increase in the number of particles per unit weight, the deteriorating conditions for particle–bubble collision in connection with low mass and inertia, insufficient kinetic energy for rupturing the thin water film in particlebubble interface and formation of three–phase line of contact (TPLC), possessing lower critical contact angle than needed for flotation and such effects as increasing surface oxidation of the particles. The k also decreases suddenly above an optimum particle size either due to a low degree of mineral liberation or reduced ability of bubbles to lift the coarse particles (Trahar, 1981; Hassanzadeh and Karakas, 2017b). Particle density ( p ): It is believed that increasing particle density

enhances the Ec, however, numerical results reveal that there are two distinct zones where the particle density plays completely different roles. In the first zone, increasing particle density reduces the Ec, whereas, in the second zone, increasing particle density leads to an increase in the Ec (Liu and Schwarz, 2009). Increasing the particle density at constant particle size, in general, reduces the flotation rate constant (Shi and Fornasiero, 2009). There is a critical set of particle diameter and its density where the collision angle is minimal (Nguyen et al., 2006; Firouzi et al., 2011). Angle of tangency ( t ): The analytical results predict a continuous decrease in the angle of tangency with increasing the particle size (Kouachi et al., 2017). However, numerical outcomes indicate that there is a critical particle size where the angle of tangency is minimal (Firouzi et al., 2011). Bubble size (db): A decrease in the bubble size increases the particle–bubble encounter and attachment efficiencies and in turn causes an increase in the flotation rate and recovery of fine particles (Reay and Ratcliff, 1973; Hassanzadeh et al., 2016). Therefore, enhancement of flotation efficiency particularly in the range of fine (< 20 µm) and ultrafine (< 10 µm) particles using combination of conventional macrobubbles (1 mm < CMBs < 100 µm) with microbubbles (1 µm < MBs < 100 µm) and nanobubbles (NBs < 1 µm) is one of the key developments in this field (Temesgen et al., 2017). Small bubbles have low relative velocities leading to a substantial increase in their numbers at the same airflow rate, which brings an increase in the particle collision probability in interaction with fine particles. However, if the bubbles are too small, they cannot lift the particles to the quiescent zone owing to insufficient buoyancy force. Increasing the bubble size at a constant bubble velocity results in an increase in the flotation rate constant of coarse particles but leads to a decrease in the flotation rate constant of fine particles (Pyke, 2003). Bubble velocity (vb): Increasing bubble velocity above a threshold decreases the Ec due to a decrease in the maximum possible collision angle and likewise, k is reduced owing to a reduction in the attachment efficiency (Pyke, 2003). Addition of frother reduces bubble rise velocity which depends on the type and dosage of the frother. It is known that the flotation process is affected by many interlinked factors. For instance, changing the db impacts significantly on the vb (Ghatage et al., 2013) and varying surface contamination of the bubble is highly effective on bubble diameter and bubble velocity (Laskowski et al., 2008; Kulkarani and Joshi, 2005; Rafiei Mehrabadi, 2009). Also, turbulence has a major effect on db (Amini et al., 2013) and vb (Nesset et al., 2006) in a flotation cell. Bubble size and shape can also be influenced by solids concentration (Rocha et al., 2008; Finch et al., 2008; Vazirizadeh, 2015). Bubbles become smaller in the presence of fine particles (air-water-solid system) than that in the absence of them (airwater system) (Hoang et al., 2019) due to inhibiting bubble coalescence and forming larger bubbles. Further, bubbles become more rounded as the %S increases and the dp decreases (Rocha et al., 2008). Moreover, the interaction of air flow rate and froth thickness (pulp level) induces a significant effect on flotation rate constant (Cilek, 2004). Particle residence time distribution (RTD) and its measurement are impacted by slurry density and particle density (Newcombe et al., 2013). The interrelated relation of gas hold–up, the interfacial area of bubbles and bubble surface area flux significantly affect k (Vazirizadeh et al., 2015). Therefore, it is not preferred to use one–factor at a time (OFAT) method due to neglecting the interaction effects of the factors. Despite a wide

Table 1 A schematic summary on the impact of key hydrodynamic factors on the Ec and k. Parameter Ec (%) k (1/min)

(m2/s3) ↑ ↑↓

(˚) ↑ ↑

%S (%)

dp(µm)

↑ ↑↓

↑ ↑↓

p (g/cm

↑↓ ↓

3

3

)

t (˚)

↑↓ ↑↓

db(cm)

vb(cm/s)

↓ ↓

↓ ↓

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series of scientific researches on kinetic behavior of flotation and the particle-bubble collision interactions, the relative intensity of main flotation parameters on the particle–bubble interactions together with their interrelations has not been adequately investigated. The effects of key parameters on Ec and k are less explored in the absence and presence of cell turbulence. Less attention has been given to evaluate the simultaneous effects of important variables. On the other hand, the complication of the flotation mechanism and the interdependence of the effective factors usually make the quantitative and predictive modeling much more difficult. Over the last three decades, in terms of inability in physical observations of microscopic phenomena, the two most common techniques in flotation modeling were numerical and analytical approaches to study the flotation system in actual conditions. Therefore, the initial aim of the present study is to evaluate and analyze two most common model configurations (i.e. (EcGSE , EaDF and EsSC ) and (EcYL , EaYL and EsSC )) used for estimating the particle–bubble collection efficiencies in analytical and numerical studies while five crucial parameters in flotation are varied. Following the first aim, the second purpose is to identify the main and interactive significant factors as well as their order on estimation and optimization of the Ec and k. The central composite design (CCD) method is introduced as an efficient approach to cope with this difficulty in connection with finding a suitable modeling method to predict the particle–bubble encounter probability and flotation rate constant.

2. Theory and methodology Several thermodynamic and kinetic-based approaches were presented in the last century of flotation modeling with the aim of predicting the rate constant which was summarized in detail by Massey (2011). In the present study, the most common flotation kinetic equation (Eq. (1)) was used (Pyke et al., 2003; Duan et al., 2003; Chipfunhu et al., 2012; Govender et al., 2013; Karimi et al., 2014 a, 2014b; Popli et al., 2015). It was initially proposed by Ahmed and Jameson (1989) based on the assumptions that the reaction is first–order, bubble concentration in the pulp is constant, and the volume of particles removed is negligible. According to Derjaguin and Dukhin (1993), the collection efficiency (Ecoll, so–called capture efficiency, Ecap) was calculated as the product of three probability functions (Ec.Ea.Es), well-known as threezone model, quantifying the collision, attachment, and detachment (stability) efficiencies presuming that sub-processes are independent of each other, particle diameters are smaller than bubble sizes and particles and bubbles are spherical.

dNp dt

=

kNp =

Zpb Ec Ea Es

(1)

where t (s) is time, Np the number of particles and k (1/s) the flotation rate constant. Ec, Ea and Es were consecutive sub–processes comprising the particle–bubble collision, attachment and stability. Zpb (m3/s) is the collision frequency per unit volume between particles and bubbles of diameters dp and db (m), respectively.

Table 2 Classification of previous works in two groups incorporated two fundamental modeling configurations for prediction of Ecoll.

Configuration I (EcGSE , EaDF and EsSC )

Configuration II (EcYL, EaYL and EsSC )

Author(s)

Year

Notes

Dai et al.,

1998

Pyke et al.,

2003 and 2004

Duan et al.,

2003

Newell

2006

Ralston et al.,

2007

Kouachi et al.,

2010

Karimi et al.,

2014a

Karimi et al.,

2014b

Kouachi et al.,

2017

Koh and Schwarz

2003

Koh and Schwarz

2006

Koh and Schwarz

2007

Evans et al.,

2008

Govender et al.,

2012

Schwarz et al.,

2016

Zhou et al.,

2019

A good agreement between experimental and calculated collection efficiencies of angular and smooth quartz particles was reported using single particle-bubble interaction by assuming unity for Es. Agreement between the GFK model and experimental data given by floating methylated quartz, chalcopyrite and galena particles in Smith-Partidge and Rushton flotation cells was satisfactory, depending upon the mineral and particle size involved. k-values of chalcopyrite in a complex sulfide ore floated in a Rushton flotation cell were found in a good agreement with calculated k-values Computed-k for different minerals validated the analytical computations using the experimental measurements of the flotation rate constants. Industrial data taken from rougher flotation stage of an operating plant using a property-based model approach was shown. Analytical approach was applied for calculating k-values of quartz, chalcopyrite and galena using GFKM while , Reb, db, and vb varied in specific intervals demonstrating the impact of particle inertial forces. The numerical (CFD) predictions were validated against experimental data on quartz particles and analytical computations using the fundamental flotation model of Pyke et al. (2003) under different ranges of hydrophobicity, agitation speed and gas flow rates. Qualitative and quantitative agreements between k-values obtained by developed CFD-kinetic model and experimental data (Pyke et al., 2003) for chalcopyrite and galena in the same physical setup were reported. Three minerals i.e. quartz, chalcopyrite and galena were used to study the role of particle density and particularly particle’s inertial effect on particle-bubble interactions and flotation rate constant. CFD of a CSIRO flotation cell was performed using an Eulerian–Eulerian approach for particle sizes 7.5, 15, 30 and 60 µm at 800, 1000 and 1200 rpm stirring speeds. Collection rates were found greater than observed flotation rates. The flotation kinetic rates were obtained using CFD technique in a semi-batch process (3.78L) via a multi-phase flow equations and an Eulerian–Eulerian approach. Flotation kinetics in a Denver laboratory cell was studied using CFD modelling at various impeller speeds. Correlating the results with the literature data for quartz (Ahmed and Jameson, 1985) showed reasonable agreements in the presence of buoyancy reduction factor. It was reported that simulation results obtained for a Rushton turbine were fairly in good agreement with the practical results presented by Ahmed and Jameson, 1985. The multiphase CFD model developed by Ragab and Fayed (2012) was utilizes and applied to two industrial case studies of porphyry copper rougher-scavenger flowsheets comprised of Wemco and Dorr-Oliver machines operating in high and low rpms. Surprisingly, extremely large numbers for pseud k-values (0 < k < 300 min−1) were reported. A sequential multi-scale modelling approach using multi-phase CFD models were applied to largescale flotation cells. Pure pyrite samples were floated by means of a micro-flotation column focusing on induction time and flotation rate constant.

4

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To calculate Zpb, several formulas were presented by Smoluchowski (1917), Camp and Stein (1943), Saffman and Turner (1956), and Abrahamson (1975) which were discussed in detail elsewhere (Hassanzadeh et al., 2018a). Abrahamson (1975)’s model (Eq. (2)), well–accepted for the flotation processes (Schubert, 1999; Koh et al., 2000; Liu and Schwarz, 2009), was used in this study. According to assumptions of Abrahamson’s model (i.e. highly turbulent condition and infinite Stokes number), the Ec is approximately 1. However, in an actual flotation condition, as the turbulent is not infinitive, the use of collision efficiency model provides much less value in serving as a correction factor for Zpb (Sherrell, 2004).

Zpb = 5Np Nb (

dp + db 2

2

attachment and detachment, the two most common model configurations were selected as follows: (I) Dukhin so–called generalized Sutherland equation (EcGSE ), modified Dobby–Finch attachment (EaDF ) and modified Schulze stability (EsSC ) models (i.e. (EcGSE , EaDF , EsSC ) and, (II) Yoon–Luttrell (EcYL ), Yoon–Luttrell (intermediate) (EaYL ) and modified Schulze stability (EsSC ) models (i.e. (EcYL , EaYL , EsSC )). For more than two decades, these two model configurations have been utilized for estimating the Ecoll and k. One school of mind has used configuration I (mostly in analytical works) and the other made use of configuration II (numerical studies) for prediction of Ecoll and k values as shown in Table 2. However, to the best of the authors’ knowledge, the discrepancy of these configurations has not been discussed at all in the literature which is addressed in the first phase of this present work. The following briefly expresses concepts and assumptions in connection with EcGSE vs. EcYL as for collision well as EaDF vs. EaYL for attachment. Dukhin (1983) collision model (so–called EcGSE ) was experimentally verified and used by Dai (1998), Pyke (2003), Sherrell (2004), Duan et al. (2003), Newell (2006) and Miettinen (2007). However, recent numerical studies demonstrated its inaccuracy due to poor estimation of the collision angle and disregarding the microhydrodynamics and bubble wall effects (Phan et al., 2003; Liu and Schwarz, 2009; Firouzi et al., 2011). On the other hand, the collision model developed by Yoon and Luttrell (1989) (EcYL ) was accepted as an accurate model and used by several researchers (Koh and Schwarz, 2006; Evans et al., 2008; Shahbazi et al., 2009; Jamson 2010; Govender et al., 2013; Yoon et al., 2016; Hoang et al., 2018; Zhou et al., 2019). They applied the interception mechanism and an empirical stream function valid for intermediate flow conditions where the bubble Reynolds number is between 1 and 100. However, it was only applicable to particles finer than 100 µm and bubbles smaller than 1 mm with immobile surfaces which did not cover particle and bubble ranges in flotation conditions. The EcYL (Eq. (6)) and EcGSE (Eq. (7)) were selected to estimate the Ecs.

2

) 2 (V p + Vb )

(2)

where Np and Nb were the number of particles and bubbles per unit

volume, Vp and Vb (m/s) were turbulent root–mean square (RMS) velocities of particle and bubble relative to the turbulent fluid velocity which could be approximated by the following expression (Eq. (3)) (Schubert and Bischofberger, 1978; Schubert,1999): 2

(Vi )1/2

0.33

4 7 9 di9 1

v3

(

p

f 2

)3

(3)

f

where the subscript i refers to the particle or bubble, (W/kg) denotes the dispersion rate of the turbulent kinetic energy per unit mass, v (m2/ s) is the kinematic viscosity of the fluid, p and f (kg/m3) are particle and fluid densities, respectively. By combining Eqs. (1) and (2), it could be shown that; 4

k = 5Nb

2 3

7

9d 9 db2 [0.33 1b 4 v3

p

f

] Ec Ea Es

(4)

f 3

where Nb represented by gas flow rate Gfr (cm /min) and bubble residence time, MRTb (min), per unit volume of a vessel (Vcell). By assuming that db > > dp (which is broadly verified) (Chipfunhu et al., 2012), Eq. (5) can be eventually shown as:

k = 2.39

Gfr db Vcell

[0.33

4 7 9 db9 1 v3

p

f f

2 3

1 ] Ec Ea Es vb

EcYL = (

(5)

The capability of Eq. (5) was experimentally examined for the flotation of quartz, chalcopyrite and galena in several studies (Pyke et al., 2003; Pyke, 2004; Duan et al., 2003; Newell, 2006; Newell and Garno, 2006). Additionally, Karimi et al. (2014a, 2014b) presented a good agreement between the experimental and numerical data on the flotation of quartz, chalcopyrite and galena using computational fluid dynamics (CFD) under various contact angles, agitation and gas flow rates. More importantly, it was validated by industrial data taken from the rougher flotation stage of an operating plant using a property-based model approach (Ralston et al., 2007). Therefore, the agreement of experimental, numerical and industrial results with the theoretical model of Eq. (5) was led us to confidentially calculate the flotation rate constants. Two most widely used model configurations to obtain the Ecoll are examined in the following section.

4Reb0.72 Rp 2 3 + )( ) 2 15 Rb

(6)

where Reb is bubble Reynolds number, Rp and Rb indicate particle and bubble radii, respectively.

EcGSE = EcSU sin2 2

t

× exp 3K3cos

cos3 t 3 2EcSU sin2 t

9K3 ( 3 +

t

ln

3 EcSU

1.8

cos t ) (7) (EcSU

3dP ), dB

= represented Sutherland collision model where t in degree referred to the angle of tangency (Eq. (8)) (maximum collision angle) which was described as the angle above which no encounter was possible and K3 was given by Eq. (9).

EcSU

t

= arcsin (2 ( 1 +

2.1. Selection of two model configurations

K3 = Kst

As noted in Eq. (5), the most crucial term is estimation of Ecoll = Ec × Ea × Es by presuming that (i) sub-processes are independent of each other (ii) particles and bubbles are spherical (iii) particles are finer than bubbles (iv) flow around the bubble is modeled as if the bubbles are stationary in a flow field giving the equivalent bubble rise velocity and (v) only one particle interacts with each bubble. Considering the existing analytical models of the collision,

(

p

f p

)

=

2

(8)

))1/2

4vb (

p

f

)

9 db

(9)

where the Kst term represents the particle Stokes number as Kst =

and being the liquid viscosity. Dimensionless number lated by Eq. (10).

= 5

2fEcSU 9K3

2 p vb dp

9 db

was calcu-

(10)

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where f is a constant value equal to 2.034 (Dukhin and Rulyov, 1977). Basically, Ea depends upon the surface characteristics of the mineral and the degree of collector adsorption on the mineral surface (Albijanic et al., 2010; Xing et al., 2017). Attachment efficiencies measured in several studies (Dai, 1998; Duan et al., 2003; Pyke et al., 2003) aimed to test the particle–bubble attachment model showed good agreement with those predicted using the modified Dobby and Finch attachment efficiency model (EaDF ) (Dobby and Finch, 1987). Likewise, other researchers used EaDF in their studies (Dai et al., 2000; Pyke, 2004; Newell, 2006; Karimi et al., 2014a; Hassanzadeh et al., 2016; Kouachi et al., 2017; Popli, 2017). Similarly, Yoon–Luttrell attachment model (EaYL ) was utilized for the estimation of Ea in several studies (Koh and Schwarz, 2003, 2006; 2007; Evans et al., 2008; Govender et al., 2013; Schwarz et al., 2016; Yoon et al., 2016; Zhou et al., 2019). More detailed information concerning both EaYL and EaDF can be found elsewhere (Kouachi et al., 2010). Thus, EaYL (Eq. (11)) Yoon–Luttrell (1989) and EaDF (Dobby and Finch, 1987) (Eq. (12)) were selected to calculate the Ea;

EaYL

=

sin2 (2 × arctan(exp[

vb tind (45 + 8Reb0.72 15db (db / (dp + 1))

where B was constant and independent of the particle size ( 0.6 ± 0.1) (Duan et al., 2003) and A was inversely proportional to the particle contact angle (Dai et al., 1999). Modified Schulze (1992) stability model was chosen for determining detachment efficiency of particle–bubble as below:

Estab = 1

dp2 ( BO =

sin2 sin2

(12)

t

where a the adhesion angle in (°), was the specific collision angle where its sliding time being equal to the induction time. The a was the collision angle where sliding time equaled its induction time (tind):

a

= 2arctanexp

tind

2(vP + vb) + (vb )

(

db dp + db

)

3

d p + db

(13)

The induction time calculated using Eq. (14).

tind = AdpB

1 BO

exp As 1

(15)

p

f )g

+ 1.9

2 p 3

(

dp 2

+

6 sin

(

db 2

1 3

)

4

+ 1.5dp ( d

b

2

) sin(

db pf g ) sin2 (

2

)

+ 2)

where (°) was the contact angle, (N/m) surface tension, g (m/s2) gravitational constant and (W/kg) was energy input. The power value of each parameter in Eq. (16) was experimentally examined by Safari and Deglon (2018) with the aim of proposing a new attachment–detachment flotation kinetic model. As a general rule, particle size fractions in the range of 1–100 µm were selected in this study to cover fine (< 20 µm), middling or intermediate (20–75 µm) and coarse (75–100 µm) fraction sizes (Trahar, 1981; Feng and Aldrich, 1999). Depending on the reagent type and dosages, impeller speed and other relevant operating factors, bubble size and its velocity were considered in the typical range of 0.05–0.10 cm (Tao, 2004; Grau and Heiskanen, 2005; Laskowski et al., 2008) and 10–30 cm/s (Kowalczuk et al., 2017), respectively. Particle density was considered < 4.1 g/cm3 since the large discrepancies between models and experimental data were found mostly for the flotation of dense minerals, e.g. galena (Pyke, 2004; Safari and Deglon, 2018; Kouachi et al., 2017; Pyke, 2004). Contact angle of the materials were considered 75° for p = 2.7 g/cm3 (e.g. quartz) (Miettinen, 2007; Guven et al., 2015), 62° for p = 1.3 g/cm3 (e.g. coal) (He et al., 2018), 56° for p = 4.1 g/cm3 (e.g. chalcopyrite) (Pyke et al., 2003; Abreu

(11)

a

=1

(16)

where tind (s) is induction time described in detail elsewhere (Dai et al., 1999)

Ea =

Fatt Fdet

where As was a constant value of 0.5 (Bloom and Heindel, 2002; Govender et al., 2013). The ratio of the detachment force (Fdet) to the attachment force (Fatt) was a dimensionless term indicated as Bond number (BO ) by Eq. (16) (Pyke et al., 2003; Koh and Schwarz, 2007).

]))

sin2 90

exp As 1

(14)

Fig. 1. A comparison between the particle-bubble interactions using two model configurations: (a) encounter (Dukhin (GSE) vs. Yoon-Luttrell (YL)), (b) attachment (Dobby-Finch (DF) vs. Yoon-Luttrell (YL)) and (c) configuration I vs. configuration II. 6

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et al., 2010), 68° for p = 3.4 g/cm3 (e.g. apatite) (Zhou et al., 2015), 62° for p = 2.0 g/cm3 (e.g. Trona) (Ozdemir et al., 2009) to cover the values greater than the critical contact angles. Turbulence dissipation rate was given in the range of 18–30 m2/s3 in order to provide intensive turbulence system in a Rushton turbine cell as described by Pyke (2004) and Duan et al. (2003). The rest of the parameters were kept constant as gas flow rate of 3500 cm3/min, liquid viscosity of 0.00891 g/cms, water surface tension of 72dyne/cm, the fluid density of 0.997 g/cm3 and kinematic viscosity of 0.0089 cm2/s as detailed by Pyke (2004).

dependence of theoretical and experimental collision efficiencies on different bubble (0–1.2 mm) and particle (12, 18, 27, 31 and 41 µm) sizes. For theoretical estimations, EcYL and Anfruns and Kitchener (1977) (EcAK ) collision models were used for coal and quartz, respectively. It was concluded that theoretical predictions overestimate Ec by a factor of about two for bubbles approaching 1 mm in diameter. Most recently, Darabi et al. (2019) applied Nguyen and Schulz, EcYL (intermediate I and II), and Schulze collision models to estimate overall Ec values in an aerated mechanical flotation cell (10.5L) as a function of particle size and superficial gas velocity (Jg). It was reported that for coarse particles, EcYL increased more sharply than the other models which further confirms the presented results (e.g. runs 13, 21 and 25). As shown by Fig. 1b, except for a few runs (3, 6, 13, 18, 21 and 25) EaYL varies very slightly in the vicinity of one. In other words, the estimated value of the Ea using EaYL does not change upon varying the effective factors. However, EaDF shows different values as a function of 30 runs. In this regard, Shahbazi et al. (2009) used EaYL for predicting particle–bubble attachment probability of quartz in a mechanical flotation cell and reported that the results were approximately zero for all experiments. Kouachi et al. (2010) analyzed the EcYL and EaYL under various flotation conditions for quartz and galena minerals. It was reported that a large maximum collision angle (90°) used in the Yoon–Luttrell model resulted in small attachment efficiencies. Newell (2006) and Miettinen (2007) proposed a new attachment model (1/W attachment model) based on the ratio between the rate of interaction force controlled interparticle collision with and without electrostatic double layer repulsion. Experimental attachment efficiencies compared with corresponding values of 1/W attachment model, Yoon and Mao model (EaYM ) and EaDF . It was disclosed that EaDF and 1/W attachment model agreed with the experimental findings contrary to EaYM which showed an entirely opposite behavior. Finally, it is found in Fig. 1c that the Ecoll estimated by applying the configuration II gives greater values than that of using configuration I. It implies that the numerical approach overestimates the Ecoll. In this regard, Koh and Schwarz (2003) admitted that the collection rates obtained by CFD approach were fast in comparison to the flotation rates generally observed in either batch or plant-scale cells. Surprisingly, the results of Govender et al. (2013) extremely overestimated k-values for two case studies which are in complete disagreement with reported results by Pyke et al. (2003), Karimi et al. (2014b), Jameson (2012), Kouachi et al. (2017), Safari and Deglon (2018) and Jameson (2010) who considered k-maximum < 10 min−1. Therefore, according to the data given in the literature (Table 2) together with the results obtained from Fig. 1, the configuration I was chosen for the calculation of Ecap in Eq. (5) to optimize k-values by evaluating five effective factors.

3. Optimization technique–response surface method (RSM) Design of experimental (DOE) methodology was used to incorporate the interaction effects into the modeling technique. The response surface methodology (RSM), a well–known statistical technique (Box and Wilson, 1951; Box and Hunter, 1957; Martinez–L et al., 2003; Kalyani et al., 2005; Dashti and Eskandari Nasab, 2013), was utilized for modeling and optimization of the flotation process. In this regard, second–order models were applied as they provide a high degree of flexibility and applicability in a wide variety of functional forms. These advantages lead to a good approximation of the true response surface. Also, it is very easy to estimate the parameters in a second–order model using the method of least squares. To determine a critical point (maximum, minimum, or saddle), it is necessary for the polynomial function to contain quadratic terms according to the following equation:

Y=

0

+

k i=1

i xi

k
ii x i

2

+

k 1
ij x i x j

+

(17)

where k, 0 , i , x i , ii , ij and represent number of variables, constant term, coefficients of the linear parameters, variables, coefficients of the quadratic parameters, coefficient of the interaction parameters and residual associated to the experiments, respectively (Bezera et al., 2008; Mehrabani et al., 2010). Finally, the codes were calculated as functions of the range of interest of each factor. A detailed discretion in this regard can be found elsewhere (Azizi et al., 2012). 4. Results and discussions 4.1. Estimation of Ec, Ea and Ecoll Fig. 1 displays the particle–bubble interaction probabilities for 30 runs. Fig. 1a shows the differences between the particle–bubble encounter efficiencies using EcGSE and EcYL models. Fig. 1b presents the values given by modified EaDF and EaYL (intermediate) attachment models. Also, Fig. 1c indicates the particle–bubble collection efficiencies using two types of model configurations (i.e. (EcGSE , EaDF and EsSC ) and (EcYL , EaYL and EsSC )). It can be seen in Fig. 1a that EcYL and EcGSE have relatively similar trends. However, it can be found that EcYL is much more dependent on particle diameter thanEcGSE . The minimum and maximum Ecs for EcYL occur when the particle size is respectively coarser than 78 µm (runs 13, 21, and 25) and finer than 33 µm (runs 8, 12, 14, 20, and 23) regardless of the values of other factors. However, it appears that the EcGSE takes the combination of the factors into account. In this regard, Schulze (1989) and Dai et al. (1998) noted that the assumptions involved in predicting EcYL are unrealistic. Following this, Dai et al. (2000) compared experimental and predicted encounter efficiencies for several models in a specific flotation condition (dp < 70 µm, db = 0.08 and 0.15 cm, vb = 31.60 and 19.60 cm/s and p = 2.65 g/cm3) assuming that the particle–bubble stability is equal to unity. It was pointed out that EcYL underestimates Ec in comparison with EcGSE due to ignoring the particle inertial forces and assuming a uniform distribution of collision over the entire upper half surface of the bubble; this is in a relatively good agreement with our findings. Also, King (2001) indicated the

4.2. Estimation, order and interrelation of effective parameters on Ec A series of 30 tests with appropriate combinations of particle size (A), particle density (B), bubble size (C) and bubble velocity (D) were conducted using the central composite design method. The design matrix of these variables in coded units is given in Table 3 along with the predicted values of the response (Ec). The factors were studied with their codified values to simplify the calculations and for uniform comparison. The factors are coded according to the following equation:

xi =

Xi

X0 X

(18)

where xi is the dimensionless coded value of the ith factor, Xi is the actual value of the factor, X0 is the value of Xi at the center point and ΔX is the step change value. In order to describe the effect of factors on Ec, it is first necessary to choose a suitable empirical model. Therefore, experimental data in Table 3 were fitted to a full quadratic second order model equation by applying multiple regression analysis for Ec using Design Expert software (Demo v. 7.0.0, Stat–Ease, Inc.). Consequently, the final equation 7

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surrounded streamlines around bubbles and eventually obtain poor Ecvalues. One possible solution is to aggregate fine particles to act as if coarse particles which indeed opens up new avenues to research studies on the floc-flotation for floating fine and ultrafine particles in future. In this context, Safari et al. (2016) estimated k-values of three sulfide and oxide minerals using oscillating grid flotation cell (OGC). It was reported that the energy/power input as a key factor in flotation led to an increase in the flotation rate for fine particles, an optimum k for more moderate particles and a reduction in the flotation rate for coarse particles. Nevertheless, Tabosa et al. (2016) indicated that the size of turbulence zone is the main factor affects flotation recovery than the energy input. As reported in several studies (Reay and Ratcliff, 1973; Ahmed and Jameson, 1985; Dobby and Finch, 1987; Tao, 2004), small bubbles (i.e. MBs or NBs) enhance the Ec specifically in the case of fine particles. A minimum improvement of ca. 10% in recovering coal, quartz (Nazari et al., 2019), phosphate and rare earth elements (REEs) were reported in the presence of an appropriate combination of macro-, micro- and nano-bubbles. For instance, Pan et al. (2012) reported that the use of MBs was more effective for increasing the kinetics of film thinning and hence the flotation rate. The reason is attributed to long residence and interaction time, high specific surface area and efficient mass transfer. A detailed study on the effect of db and vb is given elsewhere (Hassanzadeh et al., 2016) by exampling chalcopyrite’s flotation. In this regard, Eskanlou et al., 2018) conducted experimental trials and estimated the Ecs for pyrite and chalcopyrite using multiple linear regression (MLR) and artificial neural network (ANN) techniques with the aim of studying the impact of bubble surface area flux (Sb ), microturbulence ( ), particle circularity (Cp), bubble Reynolds number (Reb), particle size (dp) and particle density ( p ). The most influential factors on Ec were introduced as Sb, Reb and dp. However, it could be found that CP had a stronger impact on Ec than , dp and p . Other than that, negatively influenced Ec which is in a complete disagreement with fundamental concepts of the particle–bubble interaction (Duan et al., 2003; Pyke et al., 2003; Govender et al., 2013; Cheng et al., 2017). Generally, the influence of p and bubble rising velocity can be

Table 3 Central composite design with actual/coded values for the parameters and results of the Ec. Run

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

(g/cm3) (B)

Bubble size (db)(cm) (C)

Bubble velocity (vb) (cm/s) (D)

Collision efficiency (Ec)

4.10 3.40 3.40 2.70 3.40 3.40 1.30 2.00 2.70 2.70 3.40 3.40 2.00 2.00 2.70 2.70 2.00 2.00 2.00 2.70 2.70 2.70 3.40 2.70 3.40 2.00 2.70 2.00 2.70 2.70

0.08 0.09 0.09 0.08 0.06 0.06 0.08 0.09 0.08 0.10 0.06 0.09 0.06 0.09 0.08 0.08 0.06 0.09 0.09 0.08 0.08 0.08 0.09 0.08 0.06 0.06 0.08 0.06 0.08 0.05

20 15 25 20 15 15 20 15 20 20 25 25 25 25 20 10 15 25 15 20 20 20 15 20 25 25 20 15 30 20

0.07 0.08 0.08 0.08 0.07 0.11 0.15 0.07 0.08 0.06 0.06 0.04 0.13 0.06 0.08 0.10 0.17 0.10 0.12 0.03 0.10 0.08 0.05 0.08 0.09 0.09 0.08 0.10 0.07 0.12

Particle size (dp) (µm) (A)

Particle density (

55 (0) 78 (1) 78 (1) 55 (0) 33 (−1) 78 (1) 55 (0) 33 (−1) 55 (0) 55 (0) 33 (−1) 33 (−1) 78 (1) 33 (−1) 55 (0) 55 (0) 78 (1) 78 (1) 78 (1) 10 (−2) 100 (2) 55 (0) 33 (−1) 55 (0) 78 (1) 33 (−1) 55 (0) 33 (−1) 55 (0) 55 (0)

p)

(2) (1) (1) (0) (1) (1) (−2) (−1) (0) (0) (1) (1) (−1) (−1) (0) (0) (−1) (−1) (−1) (0) (0) (0) (1) (0) (1) (−1) (0) (−1) (0) (0)

(0) (1) (1) (0) (−1) (−1) (0) (1) (0) (2) (−1) (1) (−1) (1) (0) (0) (−1) (1) (1) (0) (0) (0) (1) (0) (−1) (−1) (0) (−1) (0) (−2)

(0) (−1) (1) (0) (−1) (−1) (0) (−1) (0) (0) (1) (1) (1) (1) (0) (−2) (−1) (1) (−1) (0) (0) (0) (−1) (0) (1) (1) (0) (−1) (2) (0)

(Eq. A(1), Table A.1) representing the Ec in terms of coded factors after removing insignificant terms was obtained as shown in Appendix A. Evidently, the regression coefficients of all four factors together with interactions of the AB (dp and p ) and BC ( p and db) terms were found to be significant. The analysis of variance (ANOVA) was also applied to estimate the adequacy of the model and its significance at the 95% confidence level. The coefficient of determination (R2) was found to be 0.97, which means that the model could explain 97% of the total variations in the system. The p–value was found < 0.0001 indicating the significance of the model due to being smaller than 0.05. Fig. 2 shows the perturbation plot of the effects of the main factors on the Ec, which is simulated from model fitting (Eq. A(1), Appendix A). This plot helps to compare the effect of all the factors at a particular point in the design space. A steep slope or curvature in a factor displays whether the efficiency is sensitive to that particular factor. In addition, 3D response surface plots were applied to gain a better understanding of the influence of factors and their interactive effects. Fig. 3 displays the response surface plots of the effect of four factors on the particle–bubble collision efficiency. Figs. 2 and 3 show that the order of significance of the studied factors is as particle size, particle density, bubble diameter, and bubble velocity (dp > p > db > vb). In other words, particle size positively affects Ec the most whereas p , db and vb influence negatively under the studied flotation conditions. The following contains a brief explanation concerning the impact of each parameter along with a discussion with the reported works in the literature. The effect of dp has been extensively illustrated in the literature; the Ec increases with increasing the particle size particularly in quiescent conditions due to interceptional, gravitational and inertial forces (Schulze, 1989; Dai et al., 2000; Ralston et al., 2002; Hassanzadeh et al., 2018a). However, fine particles have extremely low affinity to collide with bubbles because of low mass ratios leading to following the

discussed by two dimensionless numbers as Stokes (Kst = vb f db

2 p vb dp

9 db

) and

bubble Reynolds (Reb = ) numbers. The KSt is used to characterize the behavior of particles suspended in a fluid flow. It is defined as the ratio of the characteristic time of a particle to a characteristic time of the flow. A particle with a low Stokes number (KSt < < 1) follows fluid

Fig. 2. Perturbation plot showing the relative significance of factors on the Ec. 8

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Fig. 3. 3D response surface plots showing the effect of two factors on the Ec; (a) particle size and particle density and (b) bubble size and particle density.

streamlines and inertial forces have practically no effect on the motion of the particle. However, a particle with a large Stokes number (KSt > > 1) is dominated by its inertia and continues along its initial trajectory. The Reb is generally used to assess the flow conditions of fluid in proximity to particles and described as the inertial forces to the viscous forces of the fluid. Stokes flow (so–called creeping flow) conditions apply when the Reb is very much less than unity (Reb < < 1) and potential flow conditions apply at 80 < Reb < 500. An increase in p and vb increases the KSt. Thus, the particle–bubble encounter interaction changes from interception to inertial. This facilitates collisions between particles and bubbles. An increase in vb affects also the fluid flow conditions around the bubble surface. At a greater vb, potential flow conditions apply, which are more advantageous for particle–bubble encounter than Stokes flow conditions (King, 2001). In addition to that, it is evidenced that ε can affect terminal rise velocity of bubbles and settling velocity of particles (Evans et al., 2008). In fact, pulp turbulence reduces the particle and bubble velocities by inducing an increase in drag coefficient (CD) which is in conjunction with the inverse of Re (Eq. (19)) (Nguyen and Schulze, 2004). 2

CD = (24/ Re ) × (1 + 0.169Re 3 ), 0 < Re

700

coefficient are related. The results have been reported on Appendix B, Table B.1. Furthermore, it can be observed that the role of p on Ec is greater than db and vb. It is worth noting that unlike other features of the particle, the role of the p on the Ec has not been practically investigated due to some difficulties involved in the process. It is recently shown by analytical and numerical methods that there are two distinct zones where particle densities play completely different roles. In the first zone, increasing the particle density leads to reducing the Ec due to the negative effect of inertia of water flow. However, in the second zone, increasing the particle density follows by an increase in the Ec as it overcomes the negative effect of water flow inertia by the particle inertia (Nguyen et al., 2006; Liu and Schwarz, 2009; Firouzi et al., 2011; Hassanzadeh et al., 2017; Kouachi et al., 2017). A physical observation is highly needed to overcome this discrepancy. Fig. 4 shows the significant interaction effects of AB and BC on the Ec. As can be seen, at high and low levels of particle density, increasing the dp and reducing the db results in an increase in the Ec. Also, it is observed from Fig. 4 that at high and low levels of dp and db, encounter efficiency is reduced with increasing the p . Further experimental studies should be implemented to verify these statements. The Ec was optimized to obtain the maximum efficiency using Design Expert software. Fig. 5 demonstrates the trend of factors for

(19)

where CD is dimensionless drag coefficient. A parametric study was performed to investigate further how turbulence, Re number and drag

Fig. 4. Interaction effect plot between (a) particle size and particle density and (b) particle density and bubble size on the Ec. 9

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movement towards the optimal point. Evidently, changing the parameters toward the optimal point decreases the Ec. The optimum dp, p , db and vb were found to be 78 µm, 2 g/cm3, 0.06 cm and 15 cm/s, respectively. Under these conditions, the maximum Ec was determined about 0.1575 in the absence of cell turbulence. 4.3. Estimation, order and interrelation of effective parameters on k Table 3 represents the coded and values of five factors and the response (i.e. the flotation rate constant) for 50 runs using CCD method. Appropriate combinations of five factors as particle size (A), particle density (B), bubble size (C), bubble velocity (D) and turbulence dissipation rate (E) were used in this study. The obtained results from 2 ANOVA analysis showed that R2 and Radj. were 0.95 and 0.93 together with the p–value of < 0.0001 showing the significance of the model. As can be seen in Table 4, the minimum and maximum k-values are 0.13 min−1 (Run 10) and 10.24 min−1 (Run 39), respectively. In this regard, Safari and Deglon (2018) used a large pile of data to derive empirical correlations for describing the relationship between the attachment/detachment rate constants and the particle size, particle density, bubble size, collector dosage and energy input. The obtained kinetic rates were in the range of 0.1–10 min−1 which is in good agreement with the presented results. Shi and Fornasiero (2009)

Fig. 5. Perturbation plot showing the optimal conditions of factors to obtain the maximum Ec.

Table 4 Central composite design with actual/coded values for the parameters and results of the k values. Run

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43*

Particle size (dp)(µm) (A)

Particle density ( p )(g/cm3)

33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 33 (−1) 78 (1) 10 (−2) 100 (2) 55 (0) 55 (0) 55 (0) 55 (0) 55 (0) 55 (0) 55 (0) 55 (0) 55 (0)

2.00 2.00 3.40 3.40 2.00 2.00 3.40 3.40 2.00 2.00 3.40 3.40 2.00 2.00 3.40 3.40 2.00 2.00 3.40 3.40 2.00 2.00 3.40 3.40 2.00 2.00 3.40 3.40 2.00 2.00 3.40 3.40 2.70 2.70 1.30 4.10 2.70 2.70 2.70 2.70 2.70 2.70 2.70

(B) (−1) (−1) (1) (1) (−1) (−1) (1) (1) (−1) (−1) (1) (1) (−1) (−1) (1) (1) (−1) (−1) (1) (1) (−1) (−1) (1) (1) (−1) (−1) (1) (1) (−1) (−1) (1) (1) (0) (0) (−2) (2) (0) (0) (0) (0) (0) (0) (0)

Bubble size (db)(cm) (C)

Bubble velocity (vb)(cm/s) (D)

Turbulence ( ) (m2/s3) (E)

Kinetic rate (k) (1/min)

0.07 0.07 0.07 0.07 0.09 0.09 0.09 0.09 0.07 0.07 0.07 0.07 0.09 0.09 0.09 0.09 0.07 0.07 0.07 0.07 0.09 0.09 0.09 0.09 0.07 0.07 0.07 0.07 0.09 0.09 0.09 0.09 0.08 0.08 0.08 0.08 0.05 0.10 0.08 0.08 0.08 0.08 0.08

15 15 15 15 15 15 15 15 25 25 25 25 25 25 25 25 15 15 15 15 15 15 15 15 25 25 25 25 25 25 25 25 20 20 20 20 20 20 10 30 20 20 20

21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 24 24 24 24 24 24 24 24 18 30 24

4.50 2.61 7.33 4.50 5.27 4.73 5.29 4.65 0.48 0.13 0.93 0.27 1.01 0.47 1.87 1.33 5.61 3.76 8.93 6.65 5.91 5.23 9.23 10.01 0.77 0.24 1.47 0.59 1.17 0.56 2.17 1.53 3.13 1.33 0.97 3.19 0.95 3.66 10.24 0.43 2.47 3.07 2.79

(−1) (−1) (−1) (−1) (1) (1) (1) (1) (−1) (−1) (−1) (−1) (1) (1) (1) (1) (−1) (−1) (−1) (−1) (1) (1) (1) (1) (−1) (−1) (−1) (−1) (1) (1) (1) (1) (0) (0) (0) (0) (−2) (2) (0) (0) (0) (0) (0)

* Run 43 is the average for 8 central tests. 10

(−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (1) (1) (1) (1) (1) (1) (1) (1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (1) (1) (1) (1) (1) (1) (1) (1) (0) (0) (0) (0) (0) (0) (−2) (2) (0) (0) (0)

(−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (−1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (0) (0) (0) (0) (0) (0) (0) (0) (−2) (2) (0)

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studied the effects of dp, p and on flotation rate and recovery of pure quartz, chalcopyrite and galena. The rate of flotation increased with increasing dp up to a maximum value before decreasing for coarser particles. The particle size for maximum flotation rate was found to decrease with increasing mineral density. The flotation rate also increased with an agitation rate up to a maximum value before decreasing at high agitation rates. It can be found by ANOVA (Table A.2, Appendix A) that the influential degree of the important factors on k is bubble velocity, particle density, cell turbulence, particle size, and bubble diameter (i.e. vb > p > > dp > db). An attachment–detachment kinetic model developed by Safari and Deglon (2018) presented their order of factors as p > > db > dp which relatively overlaps our represented order of factors. Yoon (1993) also reported that decreasing the db yields a more effective mean than increasing the gas rate to reach faster kinetics. Vazirizadeh et al. (2015) also studied the relative effect of gas hold–up ( g ), the interfacial area of bubbles (Ib) and bubble surface area flux (Sb). It was reported that for fine (53–75 µm) and intermediate particles (75–106 µm), Sb had the strongest effect than g and Ib on the flotation kinetic constant (i.e. Sb > g > Ib) which is in good agreement with other studies (Gorain et al., 1997; Massinaei et al., 2009). As the particle size increased (106–150 µm), the contribution of the Sb decreased and importance of g and Ib increased (Ib > g > Sb) which is in agreement with the results given by Kracht et al. (2005). In the same context, Deglon (1998) investigated the effect of dp, db and level of agitation on the flotation of quartz in a laboratory batch flotation cell. It was reported that k-values were weakly dependent on particle size and strongly to bubble size and energy input. Following this, Eskanlou et al. (2018) were identified Sb and λ as the most effective factors on k.

Fig. 7. Perturbation plot showing the relative significance of factors on the k (min−1).

Furthermore, Massey (2011) obtained experimental k-values for quartz particles (< 100 µm) using an oscillating grid flotation cell. It was reported that the effect of on flotation kinetics was strongly dependent on both dp and db. The results obtained above and outcomes presented

Fig. 6. 3D response surface plots showing the effect of two factors on the k (min−1): (a) particle size and bubble size; (b) particle density and bubble velocity; (c) particle density and turbulence; (d) bubble velocity and turbulence. 11

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in the literature leads to the conclusion that the relative influence of flotation variables on flotation kinetic rates in both batch and industrial cells requires further studies. Most importantly, we found that there is a substantial lack of analyzing methods available for identifying the relative intensity of influential factors and their interconnection in flotation systems. Therefore, in addition to statistical approaches, new constructive and practical techniques should be introduced to tackle this difficulty. Fig. 6 displays response surface plots focusing on the interactive effects and respective flotation rate constants. The shapes of the contour and response surface plots show the nature and extent of the interactions of the different components. It is obvious from the results presented in Fig. 7 that the flotation rate constant depends strongly on the interaction among factors. It can be found that the ranking of the significant interaction on the flotation kinetics is as follows: interactive effects of bubble velocity and turbulence > particle density and bubble velocity > particle density and turbulence > particle size and bubble size. It can be pointed out that the interaction of .vb negatively impacts on k. Regarding the .vb effect, Amini et al. (2013) examined the influence of turbulence kinetic energy (TKE) on db in two mechanical

flotation cells with the same geometry but different sizes as laboratory (5L) and full–scale (60L) cells. It was reported that increasing TKE reduced bubble Sauter mean diameter (d32) in the 5L cell until a critical value (0.18 m2/s2) after which there was no significant further influence. The TKE had no effect on the d32 in the 60L cell. As seen, the interconnected role of .db on k is insignificant in the conditions studied. In this context, we found contradictory reports in the literature indicating inconsistent results. Some demonstrating a reduction in the d32 upon increasing impeller speed (Grau et al., 2005; Shahbazi et al., 2009; Sovechles et al., 2016; Hoang et al., 2019), some an increase (Girgin et al., 2006) and others showing little to no effect (Finch et al., 2008; Nesset, 2011; Sovechles et al., 2016) which is in agreement with our findings. Nevertheless, none of these studies examined the interaction effect of .vb on k. We recently reported that the possible reason for reducing d32 with increasing is that by increasing the agitation rate, the initially generated bubbles are split-up in the shear flow and trapped below the impeller zone. It leads to the formation of a bubble vortex in this region of the flotation cell that decreases buoyancy of the small bubbles (Hoang et al., 2019). Other than investigating the interactive effects of .vb and .db on k,

Fig. 8. Interaction effect plot between (a) particle size and bubble size; (b) particle density and bubble velocity; (c) particle density and turbulence; and (d) bubble velocity and turbulence on the flotation kinetic rates.

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most of these studies assumed isotropic turbulence in terms of Ec and k modeling in flotation, however, it is not exactly true in most flotation vessels. Interaction graphs were constructed to provide better understanding and insight of interactive effects among factors on flotation kinetics, which are displayed in Fig. 8. As can be seen, at a high level of particle sizes (78 µm), increasing the db resulted in an increase at the flotation rate constant (Fig. 8a). The increment in the bubble size leads to a slight increase of flotation kinetics at the low level of dp (33 µm) (Fig. 8a). It is also obvious from the results that the bubble size has less significant on the kinetic rate, which is probably attributed to the fact that, the range chosen for this parameter (33–78 µm) is too small to have any confidence in the outcome of the tests. Meanwhile, it is observed that at high and low levels of p and , kinetic rate strongly decreases with increased bubble velocity. It can be also seen that the vb has maximum influence on the flotation kinetic (Fig. 8b, d, Eq. (A.1) and Table A.2). This variable has the largest F-value among the selected parameters. Thus, the variation of this term leads to a large difference in the rate constant. Moreover, Pyke’s equation (Eq. (5)) relates k to the inverse of vb; this also shows the importance of vb and interestingly enough the ANOVA table further confirms this effect. Basically, increasing vb decreases the Ec (Table 1) due to a decrease in the maximum possible collision angle and likewise, k is reduced owing to the substantial reduction in the attachment efficiency. We reported additional k-values previously for three different bubbles velocities (18, 24 and 31 cm/s) under experimental conditions for quartz and chalcopyrite minerals while other parameters kept constant (Kouachi et al., 2017). In addition, it is found from Fig. 8c that the has a positive effect on the flotation kinetics of dense particles in the range studied; turbulence has a negligible effect on the kinetics at low particle densities. The impact of turbulence dissipation rate (varies at 18, 21, 24, 27 and 30 m2/s3) in a constant gas flow rate (3500 cm3/min) is shown in Fig. 9. The following states in Fig. 9(a–e) take place within the flotation cell while the factor slowly increases. In the first graph (Fig. 9A), dispersion of particles and bubbles is very poor and they only disperse in the center of the vessel. By increasing the up to 24 m2/s3 (Fig. 9b and c), particles and bubbles distribute over the entire cross–section of the cell above the agitator. Under these conditions, only small mass transfer intensities are feasible at low kinetic rates. However, once the turbulence dissipation rate rises to 30 m2/s3 (Fig. 9d and e), particles and bubbles distribute in the lower part of the cell leading to improved particle–bubble interactions and greater mass transfer rates. Moreover, greater impeller speed breaks up the large bubbles, leading to small bubbles and consequently resulting in higher Ec. In order to find an optimum condition with the highest flotation rate constant, Eq. (A.2) in Appendix A was optimized using quadratic programming of the mathematical software package within the experimental range studied (Fig. 10). The optimal conditions suggested by Design Expert software are particle size of 33 µm, the particle density of 3.4 g/cm3, bubble size of 0.09 cm, bubble velocity of 15 cm/s and turbulence dissipation rate of 27 m2/s3. Under these optimal conditions, the maximum k of 8.61 min−1 was approximately obtained.

Fig. 10. Perturbation plot showing the optimal conditions of factors to obtain the maximum k.

4.4. Main challenges and opportunities in flotation kinetic modeling Despite the efforts of researchers in the modeling of froth flotation processes by means of theoretical, experimental and numerical techniques, the following challenges still require further investigations:

• developing accurate visualization techniques to observe local par• • •

ticle-bubble interactions for verification of microscopic aspects of flotation modeling conducting dynamic contact angle measurements of minerals at bubble interfaces considering materials’ heterogeneities applying local particle and bubble turbulent velocities under an intensive turbulent environment incorporating local turbulent kinetic energies in eddies of various sizes rather than its mean value into first principle flotation modeling

5. Conclusions Prediction of the particle-bubble microscopic sub-processes along with Ecoll is clearly identified as the most initiative and crucial aspect in froth flotation which directly influences the estimation of k. After three decades, it is now well-known that in-situ and direct visualization and measurement of particle-bubble interactions in an actual flotation system is a very sophisticated process. Therefore, analytical and numerical approaches have been widely used for this purpose by

Fig. 9. A schematic view of increasing turbulence dissipation rate from 18 to 30 m2/s3, (a) 18 m2/s3, (b) 21 m2/s3, (c) 24 m2/s3, (d) 27 m2/s3, (e) 30 m2/s3 on dispersion of particles and bubbles inside the flotation cell.

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considering a large number of simplifications and assumptions. These techniques have utilized two different model configurations for which the discrepancies have not been addressed in the literature so far. Additionally, flotation is a complex process affected by many interlinked hydrodynamic parameters which their interactions might have a substantial impact on the particle-bubble sub-processes and in turn flotation rate constant. However, there is not a promising method to handle this issue and reports in the literature are surprisingly inconsistent. The present research study tackled these difficulties by undertaking in two phases of work. In the first stage, two commonly used model configurations were applied to estimate the particle–bubble collision, attachment and detachment efficiencies as well as Ecoll. We found that configuration I was mainly favored by analytical methods while configuration II was used solely in numerical approaches. The second configuration provided greater Ecoll-values than the first one due to incorporating EcYL and EaYL which respectively overlooked the particle inertial forces, assumed a uniform distribution of collision over the entire upper half surface of the bubble and presumed a collision angle of 90°. Moreover, it was shown that using model configuration II was only sensitive to particle and bubble sizes disregarding the impact of other influential parameters such as energy dissipation rate and bubble velocity. The presented analyses with respect to the three-zone model offered a new concept for the extension of common flotation modeling approach using analytical and numerical techniques. In the second stage, the configuration I was applied to a first principle flotation kinetic model (GFKM) to estimate and optimize flotation

rate constants by central composite design methodology (CCD) varying five key hydrodynamic parameters. The variables were considered as particle size (10, 33, 55, 78 and 100 µm), particle density (1.3, 1.0, 2.7, 3.4 and 4.1 g/cm3), bubble size (0.05, 0.06, 0.08, 0.09 and 0.1 cm), bubble velocity (10, 15, 20, 25 and 30 cm/s) turbulence dissipation rate (18, 21, 24, 27, 30 m2/s3). The aim of the second stage was to identify the relative intensity of the parameters studied with a main focus on the interrelation of the factors on Ec and k. The important factors on k was found as vb, p , , dp, and db. It was clarified that there is not a clear statement in the literature and good agreement with studies concerning the relative importance of effective parameters on flotation kinetic constant. Additionally, the interlinked relationship between and d32 as well as and vb were found inconsistence in the literature. Further specific studies are required to address these issues from practical and modeling points of view. Further, a critical and conceptual overview is essentially needed to be oriented to the size of turbulent zones rather than merely energy input in hydrodynamic aspects of flotation cells. Acknowledgment Authors would like to sincerely appreciate Istanbul Technical University (ITU) (Turkey) and Helmholtz-Institute Freiberg for Resource Technology (Germany) for supporting this research. Also, the authors thank the anonymous reviewers for their detailed and constructive comments.

Appendix A The equation represents the Ecs in terms of coded factors after removing insignificant terms given by Design Expert software.

Ec = + 0.08 + 0.02 × A 0.017 × B 0.014 × C 0.009167 × D 0.00375 × A × B + 0.00375 × B × C 0.003958 × A2 + 0.007292 × B2 + 0.002292 × C 2 + 0.003542

(A.1)

Final equation given by the software in terms of coded factors for calculation of the flotation rate constants (see Table A.1).

k = + 2.77

0.44 × A + 0.68 × B + 0.42 × C 2.37 × D + 0.46 × E + 0.25 × A × C 0.43 × B × D + 0.32 × B × E 0.45 × D × E + 0.51 × D2

(A.2)

Table A1 ANOVA results of quadratic model to predict the EC. Source

Sum of squares

DF

Mean square

F value

p-value Prob > F

Model A-particle size B-particle density C-bubble size D-velocity AB BC A2 B2 C2 D2 Residual Pure Error Cor Total Std. Dev. R2 Adj R2 Adeq Precision

0.027 9.60E-03 7.35E-03 4.82E-03 2.02E-03 2.25E-04 2.25E-04 4.30E-04 1.46E-03 1.44E-04 3.44E-04 8.50E-04 0 0.028 6.69E-03 0.9692 0.953 32.921

10 1 1 1 1 1 1 1 1 1 1 19 5 29

2.67E-03 9.60E-03 7.35E-03 4.82E-03 2.02E-03 2.25E-04 2.25E-04 4.30E-04 1.46E-03 1.44E-04 3.44E-04 4.47E-05 0

59.76 214.59 164.29 107.67 45.08 5.03 5.03 9.61 32.6 3.22 7.69

< 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001 0.037 0.037 0.0059 < 0.0001 0.0887 0.0121

14

Significant

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Table A2 ANOVA results of quadratic model to predict the flotation rate constants. Source

Sum of squares

DF

Mean square

F value

p-value Prob > F

Model A-particle size B-particle density C-bubble size D-bubble velocity E-turbulence AC BD BE DE D2 Residual Pure Error Cor Total Std. Dev. R2 Adj R2 Adeq Precision

321.28 8.31 20.22 7.57 242.93 9.15 1.94 5.81 3.35 6.5 15.5 18.16 0 339.44 0.68 0.9465 0.9328 37.719

10 1 1 1 1 1 1 1 1 1 1 39 7 49

32.13 8.31 20.22 7.57 242.93 9.15 1.94 5.81 3.35 6.5 15.5 0.47 0

69 17.85 43.43 16.27 521.73 19.65 4.16 12.47 7.2 13.97 33.28

< 0.0001 0.0001 < 0.0001 0.0002 < 0.0001 < 0.0001 0.0483 0.0011 0.0107 0.0006 < 0.0001

significant

Appendix B See Table B.1. Table B1 Estimation of drag coefficients in terms of db, Reb and vb in the studied range. vb (cm/s)

db (cm)

Re

CD

vb (cm/s)

db (cm)

Re

CD

10

0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.04 0.05 0.06 0.07 0.08 0.09 0.10

40 50 60 70 80 90 98 80 100 120 140 160 180 196 120 150 180 210 240 270 294

1.79 1.58 1.44 1.33 1.24 1.17 1.12 1.24 1.11 1.02 0.95 0.90 0.85 0.82 1.02 0.92 0.85 0.80 0.75 0.72 0.69

15

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.04 0.05 0.06 0.07 0.08 0.09 0.10

60 75 90 105 120 135 147 100 125 150 175 200 225 245

1.44 1.28 1.17 1.09 1.02 0.97 0.93 1.11 1.00 0.92 0.86 0.81 0.77 0.75

20

30

25

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