Abstract:

A one-dimensional steady-state model is developed for the
prediction of axial variation of the gas holdup in flotation column
froths. Froth is considered as an inverse fluidized bed of bubbles and
hence the frictional pressure gradient is obtained based on the energy
balance. Pressure gradient can also be obtained from the Ergun equation
with adjustable constants. The model correctly captures the effect of
superficial gas velocity, superficial liquid velocity and bubble
diameter on the variation of the gas holdup along the froth height. The
predictions of the model are in agreement with the experimental data
from the literature.

On a etabli un modele en regime permanent unidimensionnel pour la prediction de la variation axiale de la retention de gaz dans l'ecume de colonnes de flottation. L'ecume est consideree comme un lit fluidise de bulles inverse, et de ce fait le gradient de pression frictionnelle est obtenu d'apres un bilan energetique. Le gradient de pression peut egalement etre obtenu a partir de l'equation d'Ergun avec des constantes ajustables. Ce modele capture correctement l'effet de la vitesse de gaz superficielle, la vitesse de liquide superficielle et le diametre des bulles sur la variation de la retention de gaz sur toute la hauteur de la mousse. Les predictions du modele montrent un bon accord avec les donnees experimentales de la litterature scientifique.

Keywords: froth, gas holdup, flotation column, frictional pressure gradient

On a etabli un modele en regime permanent unidimensionnel pour la prediction de la variation axiale de la retention de gaz dans l'ecume de colonnes de flottation. L'ecume est consideree comme un lit fluidise de bulles inverse, et de ce fait le gradient de pression frictionnelle est obtenu d'apres un bilan energetique. Le gradient de pression peut egalement etre obtenu a partir de l'equation d'Ergun avec des constantes ajustables. Ce modele capture correctement l'effet de la vitesse de gaz superficielle, la vitesse de liquide superficielle et le diametre des bulles sur la variation de la retention de gaz sur toute la hauteur de la mousse. Les predictions du modele montrent un bon accord avec les donnees experimentales de la litterature scientifique.

Keywords: froth, gas holdup, flotation column, frictional pressure gradient

Authors:

Bhole, Manish R.

Joshi, Jyeshtharaj B.

Joshi, Jyeshtharaj B.

Pub Date:

06/01/2007

Publication:

Name: Canadian Journal of Chemical Engineering Publisher: Chemical Institute of Canada Audience: Academic Format: Magazine/Journal Subject: Engineering and manufacturing industries Copyright: COPYRIGHT 2007 Chemical Institute of Canada ISSN: 0008-4034

Issue:

Date: June, 2007 Source Volume: 85 Source Issue: 3

Accession Number:

192099552

Full Text:

INTRODUCTION

Column flotation in mineral processing industries for the concentration of ore is essentially a counter-current bubble column. For the maximum recovery from the ore, the hydrodynamics of the flotation column must be clearly understood. In a typical flotation column, a slurry feed is introduced against a plume of rising gas bubbles. The portion of the column below the slurry feed point operates in the bubbly flow (homogeneous) regime and is referred to as collection zone (Finch and Dobby, 1990). The froth which is formed in the column above the slurry feed point serves the important purpose of cleaning and transporting the concentrate mineral to overflow. The design and operating parameters must be maintained such that the bubbly flow regime and froth regime co-exist in a flotation column.

The height of the froth zone can vary considerably with the gas and liquid flow rates and the nature and the amount of surface-active components (frothers) present in the liquid phase. A small amount of water is sprinkled on the top of the froth and it is referred to as wash water. A part of wash water overflows with the froth bubbles and the remaining part flows down the froth counter-current to the gas phase and is referred to as bias water. A net downward flow of water through the froth is referred to as "positive bias" in the flotation literature (Finch and Dobby, 1990). The height of the froth zone increases considerably due to the practice of addition of wash water and maintenance of a positive bias. The column froths are relatively wet (higher liquid content) compared to conventional foams. They contain spherical or deformed gas bubbles as against the cellular structure of polyhedral bubbles observed in conventional foams (Bhakta and Ruckenstein, 1996).

The schematic of a column froth is shown in Figure 1. The two distinct zones, namely the homogeneous bubbly zone and the froth zone, are separated by an interface. Typically, the gas holdup in the collection zone can be 10-20% depending upon the superficial gas velocity and bubble size. At the interface, there is an abrupt increase in the gas holdup to about 60% (Yianatos et al., 1986). The gas holdup can increase along the froth height from about 60% (at the interface) to about 80% (near the wash water addition point). Mean bubble size at any axial location in the froth can be measured by photographic analysis. Figure 2 shows the axial profile of mean bubble size along the froth height (Finch and Dobby, 1990). Typically, the sauter mean diameter increases from 1 mm to 3 mm along the froth height due to bubble coalescence (Yianatos et al., 1986; Ata et al., 2003).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Various investigators have modelled the gas holdup in the froth. Finch and Dobby (1990) have carried out drift flux analysis. This methodology relies on the drag law used to describe the hindered motion of the bubbles in the swarm. Yianatos et al. (1986) obtained the gas holdup as a function of bubble diameter in the froth. They have considered the froth to be made up of two different sections viz: expanded bubble bed and packed bubble bed. Separate models were developed for the two sections. While the model for the packed bubble bed was based on a foam drainage equation, the model for the expanded bed was based on the analysis of frictional pressure drop. However, their expression for the frictional pressure gradient needs careful attention. Further, the assumption of laminar flow is restrictive for typical bubble Reynolds numbers encountered in flotation columns. In this work, we present the analysis of frictional pressure gradient based on energy balance. Both laminar and turbulent contributions to the frictional pressure gradient based on Ergun equation are considered. Most importantly, the effect of superficial gas and liquid velocities on the gas holdup is clearly brought out.

MATHEMATICAL MODEL

One-dimensional steady-state axial flow of the gas and liquid is considered. The hydrodynamic variables like gas holdup, bubble diameter and flow rates, etc., can be considered to be averaged over the column cross-section. We do not consider the presence of a solid phase in the froth, although it is known to affect bubble coalescence and froth stability (Ata et al., 2003). The bubble coalescence is not addressed directly. Instead, the variation of mean bubble diameter along the froth height is assumed or taken from the literature. The axial profile of gas holdup is obtained accordingly. Thus, the total froth height is already assumed. The model is based on the mass and energy balances for the gas and liquid flowing in the froth. Since a positive bias is assumed in the column, the net liquid flow is counter-current to the rising bed of bubbles.

Consider a cylindrical column of diameter D in which the froth is present. The axial location z = 0 corresponds to the froth-liquid interface (shown in Figure 1). The bubbles continuously enter the froth at this location and they exit the froth bed from the top (z = H) either due to coalescence with air or the movement into the concentrate. Steady state is assumed in the analysis and hence the total height of the froth is constant. Imagine a small volume element of diameter D and thickness [DELTA]z at any location z in the froth. The gas and liquid flow countercurrent to each other in this volume element. The thickness of the volume element, [DELTA]z is small enough to consider uniformity of gas holdup over it and let this gas holdup be represented as [[member of].sub.G] (z). Typically [DELTA]z can be of the order of ten times the bubble diameter. An energy balance is considered over this volume element. The kinetic energies associated with the incoming and outgoing gas have been neglected. The same is the case with the kinetic energy of the liquid. The gas entering the volume element must overcome the pressure head acting on it and hence, it possesses higher-pressure energy than the gas leaving the control volume. However, the gas leaving the control volume has higher potential energy due to an extra elevation, [DELTA]z. Hence, the energy dissipated per unit time by the gas in the control volume is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

Similarly, the energy dissipated per unit time by the liquid in the control volume is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

The net energy dissipated per unit time in the control volume due to counter-current flow of gas and liquid is obtained from Equations (1) and (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Here, we have utilized the fact that the slip velocity for the counter-current gas-liquid flow is given by:

[V.sub.S] (z) = [V.sub.G]/[[member of].sub.G](z) + [V.sub.L]/ [[member of].sub.L](z) (4)

The frictional pressure drop and the net energy dissipated per unit time in the control volume are related to each other through the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

From Equations (3) and (5), the frictional pressure gradient is given by:

[[DELTA]P.sub.F]/[DELTA]z = ([[rho].sub.L] - [[rho].sub.G])[[member of].sub.G] (z)g (6)

The pressure gradient given by Equation (6) is analogous to the one in the case of fluidized beds. The friction at the column wall and the friction at the gas-liquid interface both contribute to the pressure gradient. Since the total gas-liquid interfacial area is much higher than the surface area of the wall, the friction at the wall can be neglected. The frictional pressure drop in the packed bed of bubbles can be described by Ergun equation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

The constants A and B have the values of 150 and 1.75, respectively, for the packed bed of solids (Ergun, 1952). The two terms on the RHS of Equation (7) correspond to the viscous and inertial contributions to the frictional pressure gradient. When the Reynolds number is less than 10, the viscous contribution to the pressure drop dominates and the second term on RHS on Equation (7) can be neglected. On the other hand, when the Reynolds number is greater than 1000, the inertial contribution to the pressure drop dominates and the first term on the RHS of Equation (7) can be neglected. Typically in the column froth with bubble diameter about 2 mm and bubble slip velocity about 30 mm/s, the Reynolds Number is about 60 (Yianatos et al., 1986). Thus, the froth flow is in the transition regime and the viscous as well as inertial contributions are significant.

The values of constants A and B, as they appear in the Ergun equation, are for packed bed of solids. The solid-fluid interface is rigid. With this condition and the energy balance, Pandit and Joshi (1998) have provided the justification for the values of the constants (A = 150 and B = 1.75). Although the froth can be treated analogous to a packed bed, there is an important difference between the solid particles and bubbles. The gas-liquid interface is generally partially mobile. The extent of mobility of the interface depends upon the bubble Reynolds number, [Re.sub.B] and the presence of surface-active agents at the interface. Hence, the values of constants A and B, in case of a froth, need a modification. The greater the mobility of the interface, the smaller will be the velocity gradients in the liquid surrounding the bubble. This would lead to relatively smaller values of pressure drop and hence the constants A and B are expected to be smaller in case of packed bed of bubbles. In this work, we have found that A = 65 and B = 0.8 best describe the available experimental data of the gas holdup in the froth (as shown in Figure 3). These constants are obtained from the regression analysis and have been maintained throughout the work. Langberg and Jameson (1992) have also modified the values the constants appearing in Ergun equation while analyzing the conditions for the coexistence of froth and liquid phases in a flotation column. It is instructive to note that these constant can vary with the nature and the concentration of the frother added in the flotation column.

Equations (6) and (7) can be combined to obtain the following equation for the gas holdup:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

The slip velocity in the above equation is to be obtained from Equation (4). Equation (8) represents a non-linear equation in the gas holdup. Knowing the superficial gas and liquid velocities and the bubble size at any axial location, the gas holdup at that location can be obtained. The z-coordinate does not appear explicitly in Equation (8). In other words, we assume that the froth height is already given and then predict the gas holdup at various axial locations. Obviously, Equation (8) predicts a constant gas holdup for the entire froth bed if the bubble size is the same for all z's. Whether or not the bubble size remains the same over the entire froth height depends upon the bubble coalescence, which in turn, is dependent upon the nature and amount of surface-active components present in the froth. In this analysis, we have assumed that the bubbles are perfectly spherical. However, the bubbles in the froth are deformed. They can be oblate ellipsoidal as observed by Pal and Masliyah (1989). If the aspect ratio of the ellipsoidal bubbles is known, its shape factor can be calculated and appropriately included in Equation (8). The aspect ratio of a single bubble in an infinite liquid medium can be obtained as a function of bubble diameter (Clift et al., 1978). However, with the present status of knowledge, it is very difficult to obtain the aspect ratio of a bubble in the presence of other bubbles in the froth. Hence, we have analyzed the problem with the assumption of spherical bubbles only.

[FIGURE 3 OMITTED]

RESULTS AND DISCUSSION

A case of constant bubble diameter for the entire froth bed is considered first. In this case, Equation (8) predicts the average gas holdup in the froth when the values of constants A and B are known. An extensive experimental database of the mean bubble size and gas holdup for various combinations of superficial gas and liquid velocities in a flotation column froth has been obtained by Pal and Masliyah (1989). They have conducted the two sets of experiments employing two different surfactants to generate the froth. In set I, 30 mg/L of Dowfroth 250-C was used whereas in set II, 15 mg/L of Triton x-100 was used. Utilizing their experimental data in Equation (8), the constants A and B were evaluated. It was found that A = 65 and B = 0.8 allows a reasonable fit for the gas holdup data in the froth as shown in Figure 3. As noted earlier, these constants are smaller than the standard constants for a packed bed of spherical solids (i.e., A = 150 and B = 1.75). A frictional pressure loss which occurs at the gas-liquid interface in a packed bed of bubbles is smaller due to partial mobility of the interface. Hence, the constants A and B for a froth are naturally expected to be smaller. Although a surfactant is present at the gas-liquid interface in a typical froth, it is not sufficient to impart the complete rigidity to the interface.

We now consider the case of axially varying bubble size. In this case, the gas holdup also varies along the froth height. Employing A = 65 and B = 0.8 in Equation (8), the gas holdup at various axial locations (z) can be obtained when the bubble diameter as a function of z is known from Figure 2. Typical values of the superficial gas and liquid velocities prevalent in flotation columns are considered for the predictions. At any axial location in the froth, the gas holdup is seen to decrease with an increase in the superficial gas velocity as well as superficial liquid velocity. This result is in agreement with the experimental findings of Yianatos et al. (1986). It is observed that the effect of superficial gas velocity on the gas holdup is different in the homogeneous bubbly flow and froth flow. In case of homogeneous bubbly flow, the gas holdup increases almost linearly with an increase in the superficial gas velocity due to concomitant increase in the number of bubbles per unit volume (Joshi et al., 1998). However, in case of froth, the number density of bubbles cannot increase in this manner. Froth bed, being a packed bed of bubbles, the number density depends upon the arrangement of bubbles and the gas holdup can change only when the nature of packing of bubbles changes. In fact, with the increase in the superficial gas velocity, more amount of liquid is entrained in the froth zone through the interface and hence the gas holdup decreases (Finch and Dobby, 1990). This effect is captured in Figure 4.

Similarly, the effect of superficial liquid velocity is also different in the case of column froths and homogeneous bubbly regime. The froth becomes wetter and hence the gas holdup decreases with the increase in the superficial liquid velocity as shown in Figure 5. However, in case of counter-current bubble column operating in the homogeneous regime the gas holdup increases with the increase in superficial liquid velocity as observed by Finch and Dobby (1990) due to increase in the residence time of bubbles in the column. Thus, the hydrodynamics of the column froths is significantly different from the homogeneous bubbly regime.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Yianatos et al. (1986) have measured the bubble size distribution as well as gas holdup at various axial locations in a froth in a continuous flotation column. Bubble sizes were obtained by analysis of photographic images of the froth whereas the gas holdup was obtained by pressure sensors or conductivity measurements. Utilizing their experimental data for sauter mean bubble diameter, the gas holdup at various axial locations in the froth is obtained and it is compared with the experimental gas holdup values in Figure 6. A good agreement between the two essentially highlights the utility of the model.

The model developed in this work is applicable for expanded bubble bed. Considering an arrangement of mono-sized spherical gas bubbles in hexagonal close packing, the upper limit on the gas holdup for an expanded bubble bed is 74%. However, this upper limit on the gas holdup can increase when a bubble size distribution exists and the bubbles are deformed. It is in this spirit, we have utilized the model even for the gas holdup higher than 74%. However, we note that the model becomes increasingly inapplicable at very high gas holdup (above 85%) where the polyhedral bubbles structure exists. In such a case, the analysis of liquid drainage through the plateau borders under the influence of gravity and capillary forces becomes imperative (Bhakta and Ruckenstein, 1996; Neethling et al., 2000; Stevenson et al., 2003).

[FIGURE 6 OMITTED]

CONCLUSION

A one-dimensional steady-state model developed in this work is based on the idea that the froth can be treated as the inverse fluidized bed of bubbles. It can be taken as a first approximation to the analysis of two-phase froths and its predictions are reliable as compared with the laboratory data ([+ or -] 5% from the experimental results). Further, it correctly captures the effect of superficial gas and liquid velocities and bubble diameter on the axial variation of the gas holdup in the froth. A mathematical model that describes accurately the froth behaviour in flotation columns is important because it gives the possibility to understand the relationship between the hydrodynamics of the froth and the hydrodynamics of collection zone, both as a unit. In this regard, the model developed here serves as a point of reference for the development of elaborate representations that may include the effect of bubble shape and even the presence of a solid phase.

Manuscript received October 10, 2006; revised manuscript received February 10, 2007; accepted for publication March 16, 2007

REFERENCES

Ata, S., N. Ahmed and G. J. Jameson, "A Study of Bubble Coalescence in Flotation Froths," Int. J. Miner. Process. 72, 255-266 (2003).

Bhakta, A. and E. Ruckenstein, "Modeling of the Generation and Collapse of Aqueous Foams," Langmuir 12, 3089-3099 (1996).

Clift, R., J. R. Grace and M. E. Weber, "Bubbles, Drops and Particles ," Academic Press, NY (1978). Ergun, S., "Fluid Flow through Packed Columns," Chem. Eng. Prog. 48, 89-94 (1952).

Finch, J. A. and G. S. Dobby, "Column Flotation," Pergamon Press, Oxford (1990).

Joshi, J. B., U. Parasu Veera, Ch. V. Prasad, D. V. Phanikumar, N. S. Deshpande, S. S. Thakare and B. N. Thorat, "Gas Holdup Structure in Bubble Column Reactors," Proc. Indian Natl. Sci. Acad. 64A, 441-567 (1998).

Langberg, D. E. and G. J. Jameson, "The Coexistence of the Froth and Liquid Phases in a Flotation Column," Chem. Eng. Sci. 47, 4345-4355 (1992).

Neethling, S. J., J. J. Cilliers and E. T. Woodburn, "Prediction of the Water Distribution in a Flowing Foam," Chem. Eng. Sci. 55, 4021-4028 (2000).

Pal, R. and J. Masliyah, "Flow Characterization of a Flotation Column," Can. J. Chem. Eng. 67, 916-923 (1989).

Pandit, A. B. and J. B. Joshi, "Pressure Drop in Fixed, Expanded and Fluidized Beds, Packed Columns and Static Mixers--A Unified Approach," Rev. Chem. Eng. 14, 321-371 (1998).

Stevenson, P., C. Stevanov and G. J. Jameson, "Liquid Overflow from a Column of Rising Aqueous Froth," Miner. Eng. 16, 1045-1053 (2003).

Yianatos, J. B., J. A. Finch and A. R. Laplante, "Holdup Profile and Bubble Size Distribution in Flotation Column Froths," Can. Metall. Q. 25, 23-29 (1986).

Manish R. Bhole and Jyeshtharaj B. Joshi * Institute of Chemical Technology, University of Mumbai, N. P. Marg, Matunga, Mumbai, Maharashtra, India 400019

* Author to whom correspondence may be addressed. E-mail address: jbj@udct.org

Column flotation in mineral processing industries for the concentration of ore is essentially a counter-current bubble column. For the maximum recovery from the ore, the hydrodynamics of the flotation column must be clearly understood. In a typical flotation column, a slurry feed is introduced against a plume of rising gas bubbles. The portion of the column below the slurry feed point operates in the bubbly flow (homogeneous) regime and is referred to as collection zone (Finch and Dobby, 1990). The froth which is formed in the column above the slurry feed point serves the important purpose of cleaning and transporting the concentrate mineral to overflow. The design and operating parameters must be maintained such that the bubbly flow regime and froth regime co-exist in a flotation column.

The height of the froth zone can vary considerably with the gas and liquid flow rates and the nature and the amount of surface-active components (frothers) present in the liquid phase. A small amount of water is sprinkled on the top of the froth and it is referred to as wash water. A part of wash water overflows with the froth bubbles and the remaining part flows down the froth counter-current to the gas phase and is referred to as bias water. A net downward flow of water through the froth is referred to as "positive bias" in the flotation literature (Finch and Dobby, 1990). The height of the froth zone increases considerably due to the practice of addition of wash water and maintenance of a positive bias. The column froths are relatively wet (higher liquid content) compared to conventional foams. They contain spherical or deformed gas bubbles as against the cellular structure of polyhedral bubbles observed in conventional foams (Bhakta and Ruckenstein, 1996).

The schematic of a column froth is shown in Figure 1. The two distinct zones, namely the homogeneous bubbly zone and the froth zone, are separated by an interface. Typically, the gas holdup in the collection zone can be 10-20% depending upon the superficial gas velocity and bubble size. At the interface, there is an abrupt increase in the gas holdup to about 60% (Yianatos et al., 1986). The gas holdup can increase along the froth height from about 60% (at the interface) to about 80% (near the wash water addition point). Mean bubble size at any axial location in the froth can be measured by photographic analysis. Figure 2 shows the axial profile of mean bubble size along the froth height (Finch and Dobby, 1990). Typically, the sauter mean diameter increases from 1 mm to 3 mm along the froth height due to bubble coalescence (Yianatos et al., 1986; Ata et al., 2003).

[FIGURE 1 OMITTED]

Various investigators have modelled the gas holdup in the froth. Finch and Dobby (1990) have carried out drift flux analysis. This methodology relies on the drag law used to describe the hindered motion of the bubbles in the swarm. Yianatos et al. (1986) obtained the gas holdup as a function of bubble diameter in the froth. They have considered the froth to be made up of two different sections viz: expanded bubble bed and packed bubble bed. Separate models were developed for the two sections. While the model for the packed bubble bed was based on a foam drainage equation, the model for the expanded bed was based on the analysis of frictional pressure drop. However, their expression for the frictional pressure gradient needs careful attention. Further, the assumption of laminar flow is restrictive for typical bubble Reynolds numbers encountered in flotation columns. In this work, we present the analysis of frictional pressure gradient based on energy balance. Both laminar and turbulent contributions to the frictional pressure gradient based on Ergun equation are considered. Most importantly, the effect of superficial gas and liquid velocities on the gas holdup is clearly brought out.

MATHEMATICAL MODEL

One-dimensional steady-state axial flow of the gas and liquid is considered. The hydrodynamic variables like gas holdup, bubble diameter and flow rates, etc., can be considered to be averaged over the column cross-section. We do not consider the presence of a solid phase in the froth, although it is known to affect bubble coalescence and froth stability (Ata et al., 2003). The bubble coalescence is not addressed directly. Instead, the variation of mean bubble diameter along the froth height is assumed or taken from the literature. The axial profile of gas holdup is obtained accordingly. Thus, the total froth height is already assumed. The model is based on the mass and energy balances for the gas and liquid flowing in the froth. Since a positive bias is assumed in the column, the net liquid flow is counter-current to the rising bed of bubbles.

Consider a cylindrical column of diameter D in which the froth is present. The axial location z = 0 corresponds to the froth-liquid interface (shown in Figure 1). The bubbles continuously enter the froth at this location and they exit the froth bed from the top (z = H) either due to coalescence with air or the movement into the concentrate. Steady state is assumed in the analysis and hence the total height of the froth is constant. Imagine a small volume element of diameter D and thickness [DELTA]z at any location z in the froth. The gas and liquid flow countercurrent to each other in this volume element. The thickness of the volume element, [DELTA]z is small enough to consider uniformity of gas holdup over it and let this gas holdup be represented as [[member of].sub.G] (z). Typically [DELTA]z can be of the order of ten times the bubble diameter. An energy balance is considered over this volume element. The kinetic energies associated with the incoming and outgoing gas have been neglected. The same is the case with the kinetic energy of the liquid. The gas entering the volume element must overcome the pressure head acting on it and hence, it possesses higher-pressure energy than the gas leaving the control volume. However, the gas leaving the control volume has higher potential energy due to an extra elevation, [DELTA]z. Hence, the energy dissipated per unit time by the gas in the control volume is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

Similarly, the energy dissipated per unit time by the liquid in the control volume is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

The net energy dissipated per unit time in the control volume due to counter-current flow of gas and liquid is obtained from Equations (1) and (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Here, we have utilized the fact that the slip velocity for the counter-current gas-liquid flow is given by:

[V.sub.S] (z) = [V.sub.G]/[[member of].sub.G](z) + [V.sub.L]/ [[member of].sub.L](z) (4)

The frictional pressure drop and the net energy dissipated per unit time in the control volume are related to each other through the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

From Equations (3) and (5), the frictional pressure gradient is given by:

[[DELTA]P.sub.F]/[DELTA]z = ([[rho].sub.L] - [[rho].sub.G])[[member of].sub.G] (z)g (6)

The pressure gradient given by Equation (6) is analogous to the one in the case of fluidized beds. The friction at the column wall and the friction at the gas-liquid interface both contribute to the pressure gradient. Since the total gas-liquid interfacial area is much higher than the surface area of the wall, the friction at the wall can be neglected. The frictional pressure drop in the packed bed of bubbles can be described by Ergun equation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

The constants A and B have the values of 150 and 1.75, respectively, for the packed bed of solids (Ergun, 1952). The two terms on the RHS of Equation (7) correspond to the viscous and inertial contributions to the frictional pressure gradient. When the Reynolds number is less than 10, the viscous contribution to the pressure drop dominates and the second term on RHS on Equation (7) can be neglected. On the other hand, when the Reynolds number is greater than 1000, the inertial contribution to the pressure drop dominates and the first term on the RHS of Equation (7) can be neglected. Typically in the column froth with bubble diameter about 2 mm and bubble slip velocity about 30 mm/s, the Reynolds Number is about 60 (Yianatos et al., 1986). Thus, the froth flow is in the transition regime and the viscous as well as inertial contributions are significant.

The values of constants A and B, as they appear in the Ergun equation, are for packed bed of solids. The solid-fluid interface is rigid. With this condition and the energy balance, Pandit and Joshi (1998) have provided the justification for the values of the constants (A = 150 and B = 1.75). Although the froth can be treated analogous to a packed bed, there is an important difference between the solid particles and bubbles. The gas-liquid interface is generally partially mobile. The extent of mobility of the interface depends upon the bubble Reynolds number, [Re.sub.B] and the presence of surface-active agents at the interface. Hence, the values of constants A and B, in case of a froth, need a modification. The greater the mobility of the interface, the smaller will be the velocity gradients in the liquid surrounding the bubble. This would lead to relatively smaller values of pressure drop and hence the constants A and B are expected to be smaller in case of packed bed of bubbles. In this work, we have found that A = 65 and B = 0.8 best describe the available experimental data of the gas holdup in the froth (as shown in Figure 3). These constants are obtained from the regression analysis and have been maintained throughout the work. Langberg and Jameson (1992) have also modified the values the constants appearing in Ergun equation while analyzing the conditions for the coexistence of froth and liquid phases in a flotation column. It is instructive to note that these constant can vary with the nature and the concentration of the frother added in the flotation column.

Equations (6) and (7) can be combined to obtain the following equation for the gas holdup:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

The slip velocity in the above equation is to be obtained from Equation (4). Equation (8) represents a non-linear equation in the gas holdup. Knowing the superficial gas and liquid velocities and the bubble size at any axial location, the gas holdup at that location can be obtained. The z-coordinate does not appear explicitly in Equation (8). In other words, we assume that the froth height is already given and then predict the gas holdup at various axial locations. Obviously, Equation (8) predicts a constant gas holdup for the entire froth bed if the bubble size is the same for all z's. Whether or not the bubble size remains the same over the entire froth height depends upon the bubble coalescence, which in turn, is dependent upon the nature and amount of surface-active components present in the froth. In this analysis, we have assumed that the bubbles are perfectly spherical. However, the bubbles in the froth are deformed. They can be oblate ellipsoidal as observed by Pal and Masliyah (1989). If the aspect ratio of the ellipsoidal bubbles is known, its shape factor can be calculated and appropriately included in Equation (8). The aspect ratio of a single bubble in an infinite liquid medium can be obtained as a function of bubble diameter (Clift et al., 1978). However, with the present status of knowledge, it is very difficult to obtain the aspect ratio of a bubble in the presence of other bubbles in the froth. Hence, we have analyzed the problem with the assumption of spherical bubbles only.

[FIGURE 3 OMITTED]

RESULTS AND DISCUSSION

A case of constant bubble diameter for the entire froth bed is considered first. In this case, Equation (8) predicts the average gas holdup in the froth when the values of constants A and B are known. An extensive experimental database of the mean bubble size and gas holdup for various combinations of superficial gas and liquid velocities in a flotation column froth has been obtained by Pal and Masliyah (1989). They have conducted the two sets of experiments employing two different surfactants to generate the froth. In set I, 30 mg/L of Dowfroth 250-C was used whereas in set II, 15 mg/L of Triton x-100 was used. Utilizing their experimental data in Equation (8), the constants A and B were evaluated. It was found that A = 65 and B = 0.8 allows a reasonable fit for the gas holdup data in the froth as shown in Figure 3. As noted earlier, these constants are smaller than the standard constants for a packed bed of spherical solids (i.e., A = 150 and B = 1.75). A frictional pressure loss which occurs at the gas-liquid interface in a packed bed of bubbles is smaller due to partial mobility of the interface. Hence, the constants A and B for a froth are naturally expected to be smaller. Although a surfactant is present at the gas-liquid interface in a typical froth, it is not sufficient to impart the complete rigidity to the interface.

We now consider the case of axially varying bubble size. In this case, the gas holdup also varies along the froth height. Employing A = 65 and B = 0.8 in Equation (8), the gas holdup at various axial locations (z) can be obtained when the bubble diameter as a function of z is known from Figure 2. Typical values of the superficial gas and liquid velocities prevalent in flotation columns are considered for the predictions. At any axial location in the froth, the gas holdup is seen to decrease with an increase in the superficial gas velocity as well as superficial liquid velocity. This result is in agreement with the experimental findings of Yianatos et al. (1986). It is observed that the effect of superficial gas velocity on the gas holdup is different in the homogeneous bubbly flow and froth flow. In case of homogeneous bubbly flow, the gas holdup increases almost linearly with an increase in the superficial gas velocity due to concomitant increase in the number of bubbles per unit volume (Joshi et al., 1998). However, in case of froth, the number density of bubbles cannot increase in this manner. Froth bed, being a packed bed of bubbles, the number density depends upon the arrangement of bubbles and the gas holdup can change only when the nature of packing of bubbles changes. In fact, with the increase in the superficial gas velocity, more amount of liquid is entrained in the froth zone through the interface and hence the gas holdup decreases (Finch and Dobby, 1990). This effect is captured in Figure 4.

Similarly, the effect of superficial liquid velocity is also different in the case of column froths and homogeneous bubbly regime. The froth becomes wetter and hence the gas holdup decreases with the increase in the superficial liquid velocity as shown in Figure 5. However, in case of counter-current bubble column operating in the homogeneous regime the gas holdup increases with the increase in superficial liquid velocity as observed by Finch and Dobby (1990) due to increase in the residence time of bubbles in the column. Thus, the hydrodynamics of the column froths is significantly different from the homogeneous bubbly regime.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Yianatos et al. (1986) have measured the bubble size distribution as well as gas holdup at various axial locations in a froth in a continuous flotation column. Bubble sizes were obtained by analysis of photographic images of the froth whereas the gas holdup was obtained by pressure sensors or conductivity measurements. Utilizing their experimental data for sauter mean bubble diameter, the gas holdup at various axial locations in the froth is obtained and it is compared with the experimental gas holdup values in Figure 6. A good agreement between the two essentially highlights the utility of the model.

The model developed in this work is applicable for expanded bubble bed. Considering an arrangement of mono-sized spherical gas bubbles in hexagonal close packing, the upper limit on the gas holdup for an expanded bubble bed is 74%. However, this upper limit on the gas holdup can increase when a bubble size distribution exists and the bubbles are deformed. It is in this spirit, we have utilized the model even for the gas holdup higher than 74%. However, we note that the model becomes increasingly inapplicable at very high gas holdup (above 85%) where the polyhedral bubbles structure exists. In such a case, the analysis of liquid drainage through the plateau borders under the influence of gravity and capillary forces becomes imperative (Bhakta and Ruckenstein, 1996; Neethling et al., 2000; Stevenson et al., 2003).

[FIGURE 6 OMITTED]

CONCLUSION

A one-dimensional steady-state model developed in this work is based on the idea that the froth can be treated as the inverse fluidized bed of bubbles. It can be taken as a first approximation to the analysis of two-phase froths and its predictions are reliable as compared with the laboratory data ([+ or -] 5% from the experimental results). Further, it correctly captures the effect of superficial gas and liquid velocities and bubble diameter on the axial variation of the gas holdup in the froth. A mathematical model that describes accurately the froth behaviour in flotation columns is important because it gives the possibility to understand the relationship between the hydrodynamics of the froth and the hydrodynamics of collection zone, both as a unit. In this regard, the model developed here serves as a point of reference for the development of elaborate representations that may include the effect of bubble shape and even the presence of a solid phase.

Manuscript received October 10, 2006; revised manuscript received February 10, 2007; accepted for publication March 16, 2007

REFERENCES

Ata, S., N. Ahmed and G. J. Jameson, "A Study of Bubble Coalescence in Flotation Froths," Int. J. Miner. Process. 72, 255-266 (2003).

Bhakta, A. and E. Ruckenstein, "Modeling of the Generation and Collapse of Aqueous Foams," Langmuir 12, 3089-3099 (1996).

Clift, R., J. R. Grace and M. E. Weber, "Bubbles, Drops and Particles ," Academic Press, NY (1978). Ergun, S., "Fluid Flow through Packed Columns," Chem. Eng. Prog. 48, 89-94 (1952).

Finch, J. A. and G. S. Dobby, "Column Flotation," Pergamon Press, Oxford (1990).

Joshi, J. B., U. Parasu Veera, Ch. V. Prasad, D. V. Phanikumar, N. S. Deshpande, S. S. Thakare and B. N. Thorat, "Gas Holdup Structure in Bubble Column Reactors," Proc. Indian Natl. Sci. Acad. 64A, 441-567 (1998).

Langberg, D. E. and G. J. Jameson, "The Coexistence of the Froth and Liquid Phases in a Flotation Column," Chem. Eng. Sci. 47, 4345-4355 (1992).

Neethling, S. J., J. J. Cilliers and E. T. Woodburn, "Prediction of the Water Distribution in a Flowing Foam," Chem. Eng. Sci. 55, 4021-4028 (2000).

Pal, R. and J. Masliyah, "Flow Characterization of a Flotation Column," Can. J. Chem. Eng. 67, 916-923 (1989).

Pandit, A. B. and J. B. Joshi, "Pressure Drop in Fixed, Expanded and Fluidized Beds, Packed Columns and Static Mixers--A Unified Approach," Rev. Chem. Eng. 14, 321-371 (1998).

Stevenson, P., C. Stevanov and G. J. Jameson, "Liquid Overflow from a Column of Rising Aqueous Froth," Miner. Eng. 16, 1045-1053 (2003).

Yianatos, J. B., J. A. Finch and A. R. Laplante, "Holdup Profile and Bubble Size Distribution in Flotation Column Froths," Can. Metall. Q. 25, 23-29 (1986).

Manish R. Bhole and Jyeshtharaj B. Joshi * Institute of Chemical Technology, University of Mumbai, N. P. Marg, Matunga, Mumbai, Maharashtra, India 400019

* Author to whom correspondence may be addressed. E-mail address: jbj@udct.org

NOMENCLATURE A proportionality constant for viscous term in Ergun equation B proportionality constant for inertial term in Ergun equation [d.sub.B] bubble diameter (m) D column diameter (m) g acceleration due to gravity (m/[s.sup.2]) [[DELTA]P.sub.F] frictional pressure drop (N/[m.sup.2]) [Re.sub.B] bubble Reynolds number, [d.sub.B][[rho].sub.L] [V.sub.S]/[[micro].sub.L] [V.sub.G] superficial gas velocity (m/s) [V.sub.L] superficial liquid velocity (m/s) [V.sub.S] slip velocity, i.e., relative velocity between the bubbles and liquid (m/s) z axial coordinate in the froth column (m) Greek Symbols [[member of].sub.G] fractional gas holdup at an axial location z in the froth [[member of].sub.L] fractional liquid holdup at an axial location z in the froth [[member of].sub.G] density of gas (kg/[m.sup.3]) [[rho].sub.L] density of liquid (kg/[m.sup.3]) [[micro].sub.L] viscosity of liquid (kg/m.s)

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