A Study of a Continuous and Countercurrent Column Flotation of

Jan 9, 1998 - Figure 9 shows transient behavior of the profiles at four column heights. .... CF− = particle concentration at slurry feed port (see F...
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Ind. Eng. Chem. Res. 1998, 37, 576-583

A Study of a Continuous and Countercurrent Column Flotation of Quartz Particles Using a Bench Scale Experimental Apparatus Katsuyuki Kubota,* Masaharu Yamada, Hironobu Imakoma, Shinya Hayashi, and Kunio Kataoka Department of Chemical Science and Engineering, The Faculty of Engineering, Kobe University, Rokko-Daicho Nada-Ku 657 Kobe, Japan

Finely crushed and sized quartz particles were flotated both continuously and countercurrently with a bench-scale column flotation cell. The particles were not flotable at pH 7 in the absence of the collector. The particles were flotated by making them hydrophobic with a typical, familiar, cationic surfactant, n-dodecyltrimethylammonium bromide, as a collector. The observed particle concentration profiles that formed along the column for varying collector concentrations were linearly correlated with the column height on a semilogarithmic scale. To simulate the aforementioned particle concentration profiles and the particle removabilities, we proposed a set of free settling and sedimentation model equations that involve a collision and attachment probability or a collection probability between a rising bubble and a particle. A traditional stagewise model was employed to solve the aforementioned equations, and their solutions were tested to determine whether or not they can simulate the observed profiles and removabilities. The results indicate that the mathematical model equations supported the observed linear profiles and the removabilities, which strongly depended on the collector concentrations and were compatible with those observed for the Denver-type flotation machine. From the observed rate constant at a high collector concentration, the collection probability value was deduced and its magnitude could be supported by a theoretical model proposed by other investigators. Introduction Column ore flotation (Rubinstein, 1995) is a simple industrial separation technique where the probability of the particles being collected by an inert gas bubble in the collection zone of a column flotation cell has been considered as Ec′Ea(1-Ed), where the symbols denote hydrodynamic, attachment and detachment probability, respectively. A number of excellent works about Ec′ (Rubintein, 1995; Nguyen-Van, 1994; Schulze, 1989; Yoon and Ruttrell, 1989; Weber and Paddock, 1983; Spielman and Fitzparick, 1973) have appeared. Recently, Luttrell and Yoon (1992) proposed model equations that enable one to couple Ec′ and Ea into a collection probability Ec. These authors have also experimentally examined the equations based on a coal particle flotation system. However, to date, versatile, convenient, and simple model equations for designing a column flotation cell have not been reported. In view of recent works on column flotation design, we propose a new series of convenient model equations that enable one to design a column flotation cell. We also verify the proposed equations, through a series of experiments in a bench-scale slender column flotation cell, and experimentally examine to what extent the aforementioned probability, Ec, can be supported by the present equations that involve a flotation rate constant. 1. Experimental Figure 1 shows the experimental apparatus. A typical cationic surfactant, n-dodecyltrimethylammonium bromide [CH3(CH2)11N(CH3)3Br; DTAB; (MW ) 308.35)], * Corresponding author. Fax: (078)-(803)-1171. Telephone: (078)-(803)-1152. E-mail: [email protected].

was employed as a collector to make the particles hydrophobic and flotable. Finely crushed and sized quartz particles, with diameters ranging from 53 to 74 µm and an arithmetic mean diameter of 64 µm, were employed as a gangue. The gangue is an invaluable component in a pulp for an ore flotation system. Highgrade quartz (supplied from Minas Gerais, Brazil) was crushed by a stamp mill, followed by a vibration mill. To remove as much organic contaminant as possible, the slurry was immersed in 1 N aqueous HNO3 solution and shaken in a water bath at 323 K for ∼5 h. The slurry was stored in a polystyrene bottle, and then the particles were rinsed with doubly distilled water several times until the pH of the slurry increased to ∼5. The particles were dried overnight at 373 K, stored in a desiccator, and used for the experiment. Nitrogen (Figure 1, no. 10) was used as an inert gas and was fed into the ball glass filter (Figure 1, no. 9) to form gas bubbles. Doubly distilled water dissolved with the collector, and the supporting electrolyte was forced to circulate the column by way of the solid-liquid separator (Figure 1, no. 8) and the slurry container (Figure 1, no. 14). After the flow rates of the internally circulating solution and of the bubble swarm stabilized, the dried-up particles with a constant flow rate for each run were introduced into the container with the glass capillary feeder (Figure 1, no. 2). This time is denoted the start of the flotation (θ ) 0). A make-up collector solution (Figure 1, no. 7) was also continuously fed at a constant flow rate of 3 × 10-7 m3/s during the course of the flotation because the particles adsorb the collector, which would lead to a decrease in the circulating liquid concentration. The flotation was carried out for ∼15-30 min (θ1 ranged from 15 to 30 min), depending on the run. The

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Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 577

was obtained by placing a rectangular chamber madeup plane-parallel transparent acrylic plate filled with water around the column (Figure 1, nos. 4 and 5). The pH of the collector solutions (the circulating and make-up solutions) was adjusted to 7 by adding a very small amount of KOH and/or HNO3 to the surfactant solutions. To depress double-layer potentials on the bubble and the particle surfaces, the collector solution was dissolved with potassium nitrate at a constant concentration of 10-3 kmol/m3. 2. Theory The following assumptions are made in developing this analysis: 1. The particles are uniformely sized and are of single flotable species type under the condition of constant reagent addition, which ensures that all particles are equally flotable. 2. A differentially continuous model in height is suitable to describe the dispersed (bubble) and the continuous (slurry/pulp) phases. 3. The concentration of solid particle in both phases are function of height, z, only, and variations of concentrations with radial or angular potions can be neglected. 4. Both phase hold-ups are constant throughout the column. Fundamental equations governing the particle concentration profile are as follows: For continuous phase (slurry phase or pulp phase):

∂C/∂θ ) Ep(∂2C/∂z2) + (VL + VT)(∂C/∂z) (3UgEc/2db-)C ) 0 (1) For dispersed phase (bubble phase):

∂Γ/∂θ ) -(∂Γ/∂z) + (4Ec/4)C

(2)

Plug flow is assumed in eq 2. The tank series model (Mecklemburg, 1975; Seinfeld, 1974) was employed to solve the aforementioned equations. With this method, the column is segmented into 30 equidistant stages (N ) 30), and the stages are numbered downwards from the feed stage to the bottom one (see Figure 2). Noticing that K is a volumetric collision and attachment rate constant, balance equations (eqs 3-5) can be derived for the continuous phase:

For stage 1: dC1/dθ ) C0VL/∆z C1(((VL + VLL + VT)/∆z) + K) + C2VLL/∆z (3) Figure 1. Experimental apparatus.

slurry was withdrawn through the sampling taps (Figure 1, no. 13) attached to the column wall at an appropriate distance interval. The tip of the sampling tap protrudes 5 mm inwards from the column inside wall. The cumulative amount of the flotated particles at the top and the cumulative amount of the unflotated particles at the bottom, trapped in the separator (Figure 1, no. 8), were gravimetrically determined. The continuous phase (slurry phase) concentration profiles, formed along the column, were also determined by weighing samples that were gathered through the sampling taps. An undistorted picture of the bubbles

For stages 2 ∼ (N - 1): dCI/dθ ) CI-1((VL + VLL + VT)/∆z) + CI(((VL + 2VLL + VT)/∆z) + K) - CI+1VLL/∆z (4) For stage N: dCN/dθ ) CN-1((VL + VLL + VT)/∆z) CN(((VL + VLL + VT)/∆z) + K) (5) When a part of the particles lifted by the bubbles up to the slurry/froth interface returns to the slurry phase to be flotated again, the mass balance of feed stage (stage 0) can be described by eq 6 (see Figure 1):

578 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998

Figure 3. Continuous phase particle concentration profile versus column height in the absence of the collector.

to travel the distance ∆z, and is given by eq 9 (see Figure 2):

∆θ ) ∆z/Ub ) ∆zg/Ug

(9)

Equations 6 and 8 were incorporated into eqs 3-5, which were solved by the RKG method to obtain both phase concentration profiles. Initial conditions are as follows:

CI(I ) 1 - N) ) 0 at θ ) 0 ΓI(I ) 1 - N) ) 0 at θ ) 0

(10)

The container is free from the particles at θ ) 0, and ΓN ) 0 for θ > 0. 3. Experimental Results and Discussion

Figure 2. Scheme of experimental apparatus.

C0 + (CF-VL + (6(1 - β)UgΓ0/db-))/(VL + VT) (6) Variation of the continuous phase concentration of the container as a function of the time, θ, elapsed from the introduction of the particles can be written as shown in eq 7 when the slurry in the container is well mixed:

CF- ) CF(1 - exp(-V/Wθ))

(7)

On the other hand, the balance of the dispersed phase of stage I can be derived as follows:

4π(UbgA/(4π/3)(db-/2)3)(db-/2)2ΓI,out ) 4π(UbgA/(4π/3)(db-/2)3)(db-/2)2ΓI,in + 4πAK∆θ(db-/2)2ΓICI/(4π/3)(db-/2)3(I ) 1 - N) (8) Here ΓI,in and ΓI,out are the surface excesses or densities entering into and leaving from stage I, respectively. The parameter ∆θ is the time interval required by a bubble

3.1. Case Where the Collector Is Absent. Figure 3 shows the experimental correlation of the continuous phase particle concentration versus the column height for the case where the collector is absent (experimental conditions are shown in the figure). The profile does not depend on the column height, which, indicates that the particles are not flotable and removable. In Figure 3, the broken line shows C/CF ) VL/(VL + VT), where VT was estimated according to Stokes equation:

VT ) (Fp - Ff)gdp2/18µf (Fp ) 2500 kg/m3) (11) The solid line is supported by the calculated broken line from which it can be concluded and deduced that each particle falls not hindered and independently in the column with the terminal falling velocity VT (the values of VL and VT are shown in the figure). It can also be deduced that the eddy dispersion for the particles would be much smaller for the present column diameter. To pursue the aforementioned result, three gas flow rates were also examined. The results, shown in Figure 4, indicate no noticeable difference among three flat profiles. This result indicates that the gas flow rates do not affect the falling velocity, even though the result may be restricted to within the varied flow rates. 3.2. Evaluation of Ep Value. To estimate the experimental Ep value, a few transient continuous-phase

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 579

Figure 7. Transient continuous phase particle concentration profile versus flotation time for three gas flow rates in the absence of the collector (keys are the same in Figures 3 and 4; VL ) 0.00549 m/s). Figure 4. Continuous phase particle concentration profile versus column height in the absence of the collector for three gas flow rates (VL ) 0.00549 m/s).

Figure 5. Transient continuous phase particle concentration profile versus flotation time in the absence of the collector (VL ) 0.00549 m/s).

Figure 8. Continuous phase particle concentration profile versus column height at a constant collector concentration and gas flow rate (VL ) 0.00455-0.00549 m/s).

mined as 1.8 × 10-4 m2/s by the following wellestablished relationship (Mecklemburg, 1975; Seinfeld, 1974), with b ) 0 and VL, VT values of:

PeL ) (VL + VT)L/Ep ) 2N/(1 + 2b) (L ) 1.24 m and N ) 30) (12)

Figure 6. Produced transient continuous phase particle concentration profile versus flotation time for different b values (VL ) 0.00549 m/s).

concentration curves were experimentally determined and are shown in Figures 5-7, where the collector is absent. On the basis of eqs 3-10, with β ) 0, the produced and simulated curves for varying VLL values and, accordingly, b values are also shown as the broken curves in these figures. The curve produced with b ) 0 seems to give the best fit to the observed one, which is evident in Figures 5-7. The Ep value can be deter-

Dobby and Finch (1985) reported that the Ep value does largely depend on the column diameter and is linearly correlated with the column diameter (Figure 4 in Dobby and Finch, 1985). The value of Ep of 1.8 × 10-4 m2/s from this work can be found near the origin in the correlation of Dobby and Finch, indicating that the value from this work would not substantially contradict the result of the previous investigators and would be reliable. In view of these experimental results, the gas flow rate (G) was set at G ) 2 × 10-4 m3/min for subsequent runs. 3.3. Case Where the Collector Is Present. Figure 8 shows the profiles of three feed concentrations in the presence of the collector. Three proflies are linearly correlated, and the scaled concentration at the feed stage is ∼0.6, which coincides with that shown in Figures 4-7.

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Figure 9. Comparison between predicted and observed profiles for transient state (VL ) 0.00534-0.00549 m/s).

Figure 11. Profiles for varying b values (VL ) 0.00455 m/s).

Figure 10. Profiles for varying make-up collector concentrations. [b(a) CDTAB ) 1.0 × 10-5 kmol/m3; O(b) ) 1.5 × 10-5 kmol/m3; 9(c) ) 2.0 × 10-5 kmol/m3; 0(d) ) 2.5 × 10-5 kmol/m3; 2(e) ) 3.0 × 10-5 kmol/m3; 4(f) ) 5.0 × 10-5 kmol/m3; [(g) ) 15.0 × 10-5 kmol/m3; CF ) 17.33-27.34 kmol/m3; db- ) 0.565-2.348 mm; pH ) 7; 293-300 K; G ) 2.0 × 10-4 m3/min; VL ) 0.00439-0.00549 m/s].

Figure 9 shows transient behavior of the profiles at four column heights. Figure 10 shows the correlation for varying collector concentrations. The slope increases successively with increasing concentration, indicating that the removal of the particles increases with increasing concentration. A remarkable increase is observed at a concentration of ∼3 × 10-5 kmol/m3. All lines converge at C/CF ) 0.6, although a discrepancy is observed for two lines of the concentrations 5 × 10-5 and 15 × 10-5 kmol/m3. The line of 3 × 10-5 kmol/m3 is the same in Figures 8 and 9. From Figure 10 it can

be concluded that the falling velocity is also not hindered, as discussed for Figures 3-7, in the presence of the collector, and β is unity. The arithmetic mean bubble diameter depends strongly on the collector concentration, as seen in Table 1, where each diameter was carefully determined. Substituting an assumed K value into eqs 3-5 and 8, these equations were solved under the restriction of eq 6 to obtain both phase profiles over the whole flotation performance time period. Table 1 shows the net K values that give the best fit to the observed lines in Figure 10, which were obtained after a few simple trial calculations. The broken curves in Figure 9 show the simulated transient concentration profiles based on K values in Table 1. A discrepancy is observed for z ) 0.6 m, but a good general trend is observed between the observed and the calculated curves. From this result it can be concluded that K can be used to simulate the profile over the whole range of flotation performance. The present simulation can also evaluate the cumulative amount of the particles removed from the top with the froth. The removability can be defined as eq 13:

Rcal )

∫0θ (6GΓ0/db ) dθ/∫0θ 1

1

-

VCF- dθ (θ1 ) 15-30 min) (13)

A good agreement between the calculated and the observed removabilities is shown in Figure 12. However, a large discrepancy is seen for the concentration 1 × 10-5 kmol/m3. The froth is very unstable at this concentration; accordingly, a significant part of the particles lifted up to the slurry-floth interface would return to the slurry phase unremoved even though this

Table 1. Supplemental Data

a

CDTAB×105, kmol/m3

K × 102, 1/s‚m3

R, %

Rcal, %

db- × 103, m

g

0 1.0 1.5 2.0 2.5 3.0 5.0 15.0

0 0.25 0.68 0.92 1.43 2.96 8.93 19.61

0.1 12.1 54.2 66.5 84.7 96.1 97.9 98.3

0.0 25.3 54.4 69.7 78.6 94.2 99.2 99.5

3.503 2.348 1.844 1.486 1.375 1.071 0.979 0.565

0.031 0.029 0.032 0.030 0.033 0.033 0.034 0.043

Methanol solution system (2.5 vol %).

G × 104, m3/min

db- × 103, m

g

1.0a 2.0a 3.0a

0.789 1.053 1.104

0.027 0.045 0.070

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 581

Figure 12. Comparison between predicted and observed removabilities. Figure 14. Comparison of the present work’s removability with that of other investigators.

Figure 14 shows a comparison of the particle removability results of the present and those of previous workers (Chen et al., 1991). A general trend noted in the curve in this work is well supported by the results of the previous study. According to Luttrell and Yoon (1992), the stream function of a liquid near the surface of a rising bubble for an intermediate Reynolds number has been given as eq 14:

ψ ) UbRb2 sin2 θ((X2/2) - (3X/4) + (1/4X) + ((Reb0.72/15)((1/X2) - (1/X) + X + 1))) (14) On the basis of eq 14, the collection efficiency or the collection probability has been derived as follows: Figure 13. Variation of dispersed phase concentration profile versus column height for Figure 10.

statement seems to contradict the result in Figure 10. The reason for this phenomenon may be that the amount that returns unflotated to the slurry phase would not largely affect the slurry concentration at the feed stage because of the amount of the bubble loading would be very small at this concentration, which in turn would not affect the slurry concentration even though a significant part of the bubbles at this stage would collapse. Figure 11 shows how the b value effects the profile at a constant K value. The linear correlation breaks down and becomes more flat with increasing dispersion, showing a pronounced effect for a lower part rather than a higher part of the column. The figure also suggests that the eddy particle dispersion is not large. There is no noticeable change in bubble size along the column for each collector concentration, which can be deduced from the good linear correlation of the experimental profile. Figure 13 shows the variation of the simulated surface excess as a function of the column height. A steep increase in the surface density is simulated at a zone of the column just above the bubbler because the bubbles near the bubbler are free from the particles, reflecting that the profile is more strongly affected by the particle dispersion than the remainder zone of the column, as can be deduced from Figure 11.

Ec ) (R0/(Rb + Rp + Hc))2 (Hc ) 150 nm) (15) On the basis of eqs 14 and 15, a theoretical Ec value can be assessed as 0.0176 for the highest concentration of 15 × 10-5 kmol/m3. Complete removal is achieved at this concentration, which indicates that the surface of the solid particle would be highly hydrophobic. From the experimental net rate constant K, the collection probability is assessed to be 0.01, according to the relationship K ) 3UgEc/2db-. As a result, a good agreement in the order of magnitude between the observed and the predicted probabilities is obtained. All calculations through this work relied on a digital computer. 4. Conclusions Finely crushed and sized quartz particles were flotated both continuously and countercurrently at pH 7 with a cationic surfactant, n-dodecyltrimethylammonium bromide, as a collector and a slender bench-scale bubble column. The following conclusions are reached on the basis of the results: 1. The particles almost freely settled down the slurry phase, unhindered, with a terminal falling velocity within the tested gas flow ranges for the case where the colector is absent (Figures 3-7). 2. Under the coexistence of this cationic surfactant, the particles were flotable. To simulate the continuous and the dispersed phase particle concentrations formed along the column and the flotability, we have proposed

582 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998

a series of mathematical equations based on a traditional free settling and sedimentation model. As a result, a good agreement was observed between the experimental and the simulated results (Figures 8-12). Recently, Itoyokumbul (1992a,b) criticized the collision model and defined a new rate constant; however, the collision model would be also useful for designing a column flotation cell, which was confirmed through this work. This results of the present work also show that the required collection zone length depends strongly on the collector concentration, even though this conclusion was derived from the experiments that relied on the bench-scale flotation cell. 3. The removability achieved in the present cell was compatible with that of Denver-type flotation machine (Figure 14). 4. The theoretical value of the collection probability at the largest collector concentration of a 15 × 10-5 kmol/m3, deduced from the net flotation rate constant, is compatible with that of previous invesigators. The magnitude of the hydrophbicity of the coal particle may be compatible with that of the quartz particles adsorbed by the present surfactant. 5. The eddy dispersion coefficient of the particles was small, which is supported by the previous investigators (Figures 4-7). Appendix Referring to Figure 2, eqs 3-5 yield eqs A1-A3. For stage 1:

dC1/dθ ) C0VL/∆z - C1((q(1 + b)/∆z) + K) + C2qb/∆z (A1) For stage 2 ∼ (N - 1):

dCI/dθ ) CI-1q(1 + b)/∆z + CI((q(1 + 2b)/∆z) + K) CI+1qb/∆z (A2) For stage N:

dCN/dθ ) CN-1q(1 + b)/∆z - CN((q(1 + b)/∆z) + K) (A3) Incorporating BCs and IC, (10) and eq 6 into eqs A1A3, these equations were solved up to 1st time interval (∆θ) by the RKG method to get the concentrations C1CN, which are substituted into CI(I ) 1 - N) in eq 8 to get ΓI,out(I ) 1 - N). Then, ΓI(I ) 1 - N) values are shifted upward a stage, and the new feed concentration, C0, is calaculated from eq 6. The new ΓN ) 0 is introduced into stage N for 2nd time interval, ∆θ. The same procedure is repeated from start-up to steady state. The time step in the RKG method is 0.1 × ∆θ. Nomenclature A ) column cross sectional area (m2) b ) dimensionless parameter b ) VLL/(VL + VT) C ) continuous phase particle concentration (kg/m3) CDTAB ) make-up collector concentration (kmol/m3) CF ) CF ) w/V feed particle concentration (kg/m3) CF- ) particle concentration at slurry feed port (see Figure 1) (kg/m3) Cmethanol ) methanol concentration (vol %)

CN ) continuous phase particle concentration at Nth stage (kg/m3) CN-1 ) continuous phase particle concentration at (N - 1)th stage (kg/m3) C0 ) continuous phase particle concentration at feed stage (kg/m3) C1 ) continuous phase particle concentration at 1st stage (kg/m3) C2 ) continuous phase particle concentration at 2nd stage (kg/m3) CI ) continuous phase particle concentration at Ith stage (kg/m3) db- ) arithmetic mean bubble diameter (m) dp ) arithmetic mean particle diameter (m) Ea ) attachment probability Ec ) collection probability Ec′ ) hydrodynamic collision probability Ed ) detachment probability Ep ) particle eddy dispersion coefficient (m2/s) G ) gas flow rate (m3/s) g ) acceleration due to gravity (m/s2) Hc ) critical separation (m) K ) net volumetric flotation rate constant (1/s m3) L ) effective collection column height (m) N ) equidistant stage number (m) PeL ) Peclet number q ) q ) VL + VT (m/s) R ) cylindrical coordinate of stream line (m) R ) experimental removability Rb ) bubble radius (m) Rcal ) calculated removability Reb ) Reynolds number based on bubble diameter Rp ) particle radius (m) R0 ) limiting radius (m) Ub ) Ub ) Ug/g ) bubble rise velocity (m/s) Ug ) superficial gas velocity (m/s) V ) circulating liquid flow rate (m3/s) VL ) superficial liquid velocity (m/s) VT ) terminal falling velocity of particle (m/s) VLL ) hypothetical internal backward superficial liquid velocity (m/s) W ) liquid volume in slurry container (m3) w ) particle feed rate (kg/s) X ) X ) R/Rb dimensionless group in eq 14 z ) distance measured from bottom of column (m) Greeks β ) fractional constant, defined in eq 6 Γ ) particle surface density or excess (kg/m2) Γ0 ) particle surface density or excess at feed stage (kg/ m2) ΓI ) particle surface density or excess at Ith stage (kg/m2) ∆θ ) time interval defined in eq 9 (s) ∆z ) stage length (m) g ) gas void fraction θ ) time (s) θ ) angle measured from stagnation point of bubble (radian) θ1 ) flotation time period (s) µf ) liquid viscosity (kg/m s) Ff ) liquid density (kg/m3) Fp ) particle density (kg/m3) ψ ) stream function

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Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 583 Itoyokumbul, M. T. A new modeling approach to flotation column design. Min. Eng. 1992a, 5, 685. Itoyokumbul, M. T. A transfer approach to flotation column design. Chem. Eng. Sci. 1992b, 47, 3605. Luttrell, G. H.; Yoon, R. H. A hydrodynamic model for bubbleparticle attachment. J. Colloid Interface Sci. 1992, 154, 129. Mecklemburg, J. C.; Hartland, S. The Theory of Backmixing; John Wiley and Sons: New York, 1975. Nguyen-Van, The collision between fine particles and single air bubbles in flotation. J. Colloid Interface Sci. 1994, 162, 123. Rubinstein, B. Column Flotation, Process, Designs and Practices; Gordon and Breach Science: New York, 1995. Schulze, H. J. Hydrodynamics of Bubble-Mineral Particle Collisions. In Frothing in Flotation; Laskowski, J. S., Ed.; Gordon and Breach Science: New York, 1989. Seinfeld, J. H.; Lapidus, L. Mathematical Methods in Chemical Engineering, Vol. 3 (Process Modelling, Estimation and Identification); Prentice-Hall: New York, 1974.

Spielman, L. A.; Fitzpatrick, J. A. Theory for particle collection under London and gravity forces. J. Colloid Interface Sci. 1973, 42, 607. Yoon, R. H.; Luttrell, G. H. The Effect of Bubble Size on Fine Particle Flotation. In Frothing in Flotation; Laskowski, J. S., Ed.; Gordon and Breach Science: New York, 1989. Weber, M. E.; Paddock, D., Interceptional and gravitational collision efficiencies for single collectors at intermediate Reynolds numbers. J. Colloid Interface Sci. 1983, 94, 328.

Received for review June 6, 1997 Revised manuscript received October 29, 1997 Accepted November 4, 1997 IE9704096