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Mathematical model for the entrainment of hydrophilic particles in froth flotation
International Journal of Mineral Processing, 35 ( 1992 ) l - 11
1
Elsevier Science Publishers B.V., Amsterdam
Mathematical model for the entrainment of hydrophilic particles in froth flotation V.M. Kirjavainen Department of Materials Science and Rock Engineering, Helsinki University of Technology, tlelsinkL Finland (Received 10 May 1991 ; accepted after revision 18 October 1991 )
ABSTRACT Kirjavainen, V.M., 1992. Mathematical model for the entrainment of hydrophilic particles in froth flotation. Int. J. Miner. Process., 35:1-11. The entrainment of hydrophilic particles was studied using a circulating flotation system. Flotation tests were carried out with granular and flaky minerals at different slurry densities using only frother. The results showed that particle mass and shape, water recovery rate and slurry viscosity control particle entrainment. A mathematical model for the interrelation of the variables is presented.
INTRODUCTION
Low selectivity is often a major problem in flotation processes with finely ground materials. It is known that the mechanical recovery of suspended particles strongly impairs separation selectivity when the proportion of fines ( < 10/tm) is significant in the system. The pioneering work in this field was done by Jowett (1966) and Johnson et al. (1974). The characteristics of entrainment of gangue minerals were described in detail both in batch and continuous systems. Further detailed studies on the entrainment mechanism were carried out by Englebrecht and Woodburn ( 1975 ) as well as by Bisshop and White (1976). More recent experimental studies include those of Warren (1985). Subrahmanyam and Forssberg (1988a), Kirjavainen and Laapas ( 1988 ) and Kirjavainen ( 1989 ). The subject was also reviewed by Thorne et al. (1976), Lynch et al. ( 1981 ), Trahar ( 1981 ), Subrahmanyam and Forssberg (1988b) and Smith and Warren (1989). In batch flotation tests the entrainment of gangue was found to be related to the recovery of water (Warren, 1985 ): Correspondence to: V.M. Kirjavainen, Helsinki University of Technology, Dept. of Materials Science and Rock Engineering, Lab of Mineral and Particle Technology, Vuorimiehentie 2A, 02150 Espoo, Finland.
0301-7516/92/$05.00
© 1992 Elsevier Science Publishers B.V. All rights reserved.
2
V.M. KIRJAVAINEN
R e = e~ Rwate r
( 1)
where Rg is the recovery of fine gangue of a given size in a given time, egi i s a constant for a given particle size and specific gravity and Rwateris the recovery of water at the same time. A similar type of relationship was described by Kirjavainen and Laapas (1988). The relationship was further modified by Kirjavainen (1989) and the concentration of a hydrophilic mineral in the cell was described using the following equation: tl
1
C=Co |
exp ( - P R w ) f(P) dP
(2)
0
where C is the concentration of a hydrophilic mineral expressed by the initial concentration (Co). Rw is the recovery of water, P is the entrainment factor and f(P) is a distribution function of the entrainment factor. It is evident that the entrainment factor is analogous to the term e~ in eq. (1). It has been found that the value of the entrainment factor varies between 0 and 1 which indicates that it is actually a probability factor. In general the models presented to account for the entrainment mechanism are simplified and do not consider other variables than particle size. This makes it difficult to compare the results from different origins. In this paper the effect of particle mass and shape as well as water recovery rate and slurry viscosity on particle entrainment is described and a mathematical model for their interrelation is presented. EXPERIMENTAL
Materials Because the aim was to study the behaviour of nonfloatable particles, monomineral materials were seen to be suitable for the purpose. Two mineral products, ( 1 ) quartz and (2) phlogopite, having different particle characteristics were used as test materials. The former forms granular gangue in various types of ores and the latter is a m e m b e r of the mica group minerals with platy cleavage. Both test materials were industrial products. The densities of the samples, determined using a helium-air pycnometer, were 2.6 for quartz and 2.9 for phlogopite. An advantage of using monomineral materials was that chemical analyses were not needed and recoveries could be calculated on the basis of sample weights.
ENTRAINMENT OF HYDROPH1LIC PARTICLES IN FROTH FLOTATION: A MODEL
3
Methods Flotation procedure Flotation tests were carried out using a Denver laboratory flotation machine fitted with a 3 1 cell. Since it was not possible to sample the froth product in a conventional batch flotation test, a circulating test procedure was used. This was arranged by returning the froth product immediately back to the cell with a small peristaltic pump. The aeration rate was controlled with a rotameter. The frother used was a mixture of polypropylene glycol ethers (Teefroth G). Rotor speed was kept constant at 1600 rpm to obtain adequate mixing in the cell. Experiments were carried out at the predetermined slurry densities at constant slurry volume (2.3 1). Tests with quartz were carried out at slurry densities of 5, 10, 20 and 40% solids by weight and with phlogopite the solid content was 5, 10 and 20%. Samples were collected at definite time intervals from the froth stream with 100 ml beakers. The sample volume was replaced with tap water. The samples were weighed and dried so that the recovery of solids and water could be calculated.
Particle size analysis Size distributions were determined by the electrical sensing zone method using a Coulter Multisizer. This method was chosen because it was essential to determine the actual volume distributions of the samples for the detailed analysis of the data.
Data reduction An IBM 3090 computer was used in data reduction. Linear regression was used for the calculation of size distribution models and nonlinear models were fitted to the data using the Levenberg-Marquardt algorithm. RESULTS AND DISCUSSION
Particle size distributions The three parametric log-normal distribution function was used to describe the size distributions. The fitting procedure can be done using linear regression if the log-normal function is first linearized. The principle of the method applied is described elsewhere (Laapas and Kirjavainen, 1988). The successful fitting is demonstrated in Fig. 1 where the size distributions of the test materials and typical froth products are presented. The degree of correlation was measured with the correlation coefficient. The value of the coefficient in all cases was equal to or higher than 0.9995.
4
V.M. KIRJAVAINEN
100 ~5
"q '3
Quartz
80
Floated
quartz
Phlogopite
6O
Floated
phlogopite
20 0
10 Particle
100
s i z e . lam
Fig. 1. Size distributions of test materials and typical froth products. Solid lines represent threeparametric log-normal functions fitted to the data. i .0
¢5 0.8 o
,,.o Ol~3 0
0.6 ti2 t~oeqj 0.4
o
Slurry
milli
density
• •
5% 10% 20%
o
0.2
40 %
[]
0.0 o.O
o
•
wN~ ~ m
•
0~
o
I
I
I
0.2
0.4
0.6
I
0.8
1.0
2 Water
recovery rate, kg/m
s
Fig. 2. Effect of recovery rate of water and slurry density on the entrainment of quartz.
Dependency of particle entrainment on related variables To study the effect of slurry volume on particle entrainment flotation tests were carried out using quartz at 2.0 and 2.3 1 slurry volumes. In both cases identical results were obtained and the relationship between the recovery of quartz and the recovery rate o f water followed a particular curve regardless of whether frother dosage or aeration rate was used as the control variable. The relationship remained unchanged in spite of the difference in slurry volume or in the thickness o f the froth bed. The volume of the further tests was fixed at 2.3 1, The results obtained showed that water recovery rate is an important variable that controls the entrainment o f particles. This is illustrated in Figs. 2 and 3 where the recovery o f solids per water recovery, which expresses the
ENTRAINMENT OF HYDROPHILIC PARTICLES 1N FROTH FLOTATION: A MODEL
5
1.0 o
o°o
o
o o
o
o~ o
o
oo
o
m~a
0.8
•
•
II
lli
.
oa 0.6
• nnnu
rain O
o~
.
i
II
.e,o
•
•
•
o an
LO
O0
ee
" .
Slurry
•
density
4
0.4 ¢m © ~0
•
oo
• into
•
0.2 0.0 o,O
5%
•
10 %
o
20 %
I
I
I
I
0.2
0.4
0.6
0.8
10
2
Water
recovery
rate. kg/m
s
Fig. 3. Effect of recovery rate of water and slurry density on the entrainment of phlogopite. average entrainment factor for the solids, is plotted versus water recovery rate. The water recovery rate is calculated per free slurry surface to make the quantity independent of the slurry volume. In general the degree of entrainment increased with increasing water recovery rate and slurry density but there were also some apparent differences between the materials. As Figs. 2 and 3 show, the changes in slurry density had m u c h less effect on the entrainment of granular quartz particles than on flaky phlogopite particles. Furthermore, the effect of water recovery rate was negligible with phlogopite at 20% slurry density. These facts indicate that particle entrainment is dependent on particle and slurry characteristics. Effect o f water recovery rate and particle mass In a more detailed analysis, the recoveries of quartz and phlogopite were determined in narrow 1-am size intervals in the size range 5.5-25.5 and 12.529.5 ~tm, respectively. Each of these values was then used as an independent observation for the recovery of the corresponding m e a n particle size. It was concluded that the water recovery rate and particle mass are both important variables, which are to be included in a mathematical model describing the entrainment mechanism. It is obvious that if the feed is very fine and the mass ( m ) of particles is very small the probability of entrainment (P) should approach unity. In such a case the recovery of solids should be equal to the recovery of water or, in other words, solids recovery per water recovery should be equal to one. It seems also clear that if the recovery rate of water (w) becomes very slow the probability of entrainment should approach zero. These requirements were met with the following equation:
P=
W a
wa + bm C
(3)
0
V.M. KIRJAVAINEN
where a, b and c are constants. Equation (3) was fitted to the data obtained with the test materials at each slurry density level. Particle mass was expressed in pikograms in all computations. As the results in Tables 1 and 2 show, eq. (3) accurately described the relationship between the entrainment factor and the variables resulting in high correlation in all other cases except with phlogopite at 20% slurry density. This exception can be explained by a difference in the rheological properties of quartz and phlogopite suspensions. It was shown by Laapas (1983) that quartz suspensions are Newtonian in the size range and at the slurry densities used in this work. Also phlogopite suspensions will follow Newtonian flow behaviour at low concentrations but they tend to become Bingham plastic at high slurry densities and with increasing fineness. This is consistent with the results shown in Fig. 3. The entrainment of phlogopite was only slightly affected by the changes in the recovery rate of water at 20% slurry density showing that flaky particles were not able to drain from the froth due to the existing yield stress.
Effect of viscosity As will appear from the results above slurry viscosity also controls particle entrainment. If the results in Tables 1 and 2 are considered it will prove obvious that exponent a is constant at slurry densities where the Newtonian flow behaviour is expected, being close to 0.7 for both test materials. Coefficient b TABLE 1 Results of fitting the data obtained with quartz to eq. (3) Slurry density
Number of obs.
a
b
c
Sum of squares
Coefficient of determination
5 10 20 40
500 760 700 400
0.7252 0.7581 0.6473 0.9185
0.00607 0.00577 0.00450 0.00147
0.4999 0.4975 0.5297 0.5833
0.14364 0.52574 0.48419 0.29542
0.99026 0.97909 0.97683 0.97513
(%)
TABLE2 Resultsoffittingthedataobtained withphlogopitetoeq.(3) Slurry density
Number of obs.
a
b
c
Sum of squares
Coefficient of determination
5 10 20
510 510 510
0.7440 0.7147 0.5081
0.1185 0.0531 0.2022
0.0960 0.1278 -0.1105
0.22016 0.27268 0.32844
0.96726 0.95976 0.32077
(%)
ENTRAINMENT OF HYDROPH1LIC PARTICLESIN FROTH FLOTATION: A MODEL
7
is dependent on the solids content in the feed. This was thought to indicate that b is dependent on slurry viscosity. Exponent c is also constant at different slurry densities but it was dependent on the characteristics of the test materials. Consequently, b was assumed to be dependent on slurry viscosity and eq. (3) was modified to the following form: P-
W a
(4)
wa+f(~/)m c
Laapas (1983) described the viscosity of quartz and phlogopite suspensions by the following empirical equation:
FCv ~=~0~ (1_cv)48
(5)
where Cv is the volume fraction of solids, ~/o denotes the viscosity of the suspending m e d i u m ( 1.0 mPa.s in this work) and F i s a constant, whose value is 1.83 and 3.80 for quartz and phlogopite, respectively. Equation (5) was used to estimate the viscosities of the quartz and phlogopite suspensions. When eq. (4) was fitted to the experimental data it was found that the relationship between the entrainment factor and the three variables could be described accurately with both of the test materials in the Newtonian region using the following function for f(q): f(t/) = b i t/b2
(6)
where b~ and b2 are constants that are characteristic for each of the test materials as shown by the results in Table 3.
Effect of particle shape The results presented above show that b~, b2 and c are dependent on particle shape and eq. (4) takes the form: W a
(7)
P-wa+f~ (~-'])~ f2(t/'t) m f 3 ( ~ )
where f~ ( ~ ) , f2 (~u) and t"3( ~ ) represent functions which are dependent on particle shape. There are several shape factors that may be used in particle TABLE3 Results of fitting the data to eqns. (4) and (6) Material
Number of obs.
a
bl
b2
c
Sum of squares
Coefficient of determination
Quartz Phlogopite
2360 1020
0.7332 0.7285
0.00521 0.16953
-0.6813 -6.3597
0.5196 0.1090
1.82196 0.50341
0.97600 0.96889
8
V.M. KIRJAVAINEN
characterization but only two are considered here: (1) the dynamic shape factor (x), defined by Tsubaki and Jimbo (1979) and (2) Wadell's sphericity factor (~Uw) (Wadell, 1934). The dynamic shape factor is defined as the ratio of the resistance to motion of a given particle divided by the resistance of a spherical particle of the same volume. When the particle settles under laminar flow conditions, as may be expected in the quiescent zone near the froth bed in a floatation cell, the terminal velocity obeys Stokes' law and the dynamic shape factor can be expressed using the volume diameter (d v) and the Stokes' diameter (dst) (Allen, 1991 ): (8)
x=(dv/dst) 2
Wadell's sphericity factor is defined as the ratio of the surface area of a sphere having the same volume as the particle divided by the surface area of the particle. Under laminar flow conditions Wadell's sphericity factor can be expressed as: (9)
~w = ( d s t / d v ) 4
It is easy to see that x = ~w 1/2. Equations (8) and (9) are of practical importance because they make it possible to determine the shape factors for a mineral using two different techniques. The volume diameter is obtained by Coulter analysis and the Stokes' diameter is obtained by gravitational sedimentation, e.g. with a Sedigraph. A sieved fraction from both test materials was analysed using these methods. On the basis of the results (Tables 4 and 5 ) the modes of the distributions were estimated to be 32.0 and 35.8 #m for the quartz fraction and 14.5 and 44.5 #m for the phlogopite fraction. Using these values the dynamic shape factors were calculated to be 1.25 for quartz and 9.42 for phlogopite. Similarly, the values of Wadell's sphericity factor were 0.64 for quartz and 0.011 for phlogopite. TABLE4 Coulter and Sedigraph analyses of a quartz fraction Sedigraph analysis
Coulter analyses #m
Differential volume, %
lzm
Differential volume, %
32.26-33.27 33.27-34.28 34.28-35.29 35.29-36.29 36.29-37.30 37.30-38.31 38.31-39.32 39.32-40.33
5.62 6.83 7.69 7.78 7.63 7.26 6.93 6.77
29-30 30-31 31-32 32-33 33-34 34-35 35-36 36-37
6.5 7.0 8.0 8.0 7.5 5.5 4.5 4.5
ENTRAINMENT OF HYDROPHILIC PARTICLES 1N FROTH FLOTATION: A MODEL TABLE 5 Coulter and Sedigraph analyses of a phlogopite fraction Coulter analysis
Sedigraph analysis
/tm
Differential volume, %
/zm
Differential volume, %
39.48-40.89 40.89-42.31 42.31-43.73 43.73-45.15 45.15-46.56 46.56-47.98 47.98-49.40 49.40-50.82
4.78 5.94 7.14 7.38 7.32 6.76 6.00 4.54
12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20
5.6 7.8 9.2 8.7 7.8 7.5 6.5 6.0
TABLE 6 Results of fitting the data obtained with quartz and phlogopite to eq. ( 7 ) Shape ~ctor
Number ofobs,
b3
Sum of squares
Coefficientof determination
(x) (~w)
3380 3380
0.00694 0.00694
3.95714 3.96931
0.95981 0.95969
Most shape factors, including Wadell's sphericity factor, have a maximum value of 1.0 representing spherical particles or other regular shapes. However, the dynamic shape factor is an exception having a minimum value of 1.0. It appeared that either of the shape factors may be used in the model and that f~ (7~), f2 ( ~ ) and f3 ( ~ ) are relatively simple functions. The best solution for eq. (7) was obtained by equation: W 0-7 P - - w °'7 + b3 ~ t / - 0 . 5 ~ m 0.5 l/if--0.4
( 10 )
where 5 u = x = 5Uw1/2 and the term b 3 is a constant, which is, however, dependent on the units in which the variables are expressed. As shown by the results in Table 6 the coefficient of determination was close to unity in both cases. It is also easy to see that in the case of spherical particles, when K= ~gw= 1, eq. (10) is reduced to wO.7
P-wo.7 +b3(m/rl)o.5
( 11)
which is not dependent on particle shape. In fact this was an important criterion when the functions for f~ ( ~ ) , f2 ( ~ ) and f3 ( ~ ) were chosen.
10
V.M. KIRJAVAINEN
CONCLUSIONS It was shown that the entrainment o f hydrophilic minerals is dependent on particle characteristics, i.e. particle mass and shape, and two adjustable variables, water recovery rate and slurry viscosity. Particle size is the most important factor that controls mechanical entrainment and to minimize the entrainment of particles flotation should be carried out using as coarse grinding as possible. It is obvious that entrainment can cause the enrichment o f fine gangue in recycled flotation circuits and that fine fractions need more cleaning stages for a particular grade compared to coarse fractions. The detrimental effects o f particle entrainment can be reduced using separate circuits for coarse and fine fractions. The entrainment mechanism is clearly a factor that should be taken into account already in process planning especially with fine materials. On the basis of the experimental data it was possible to construct a probability model that accurately described the relationship between the entrainment factor and the variables in the Newtonian region. Particle mass and shape were characterized by methods generally used in particle technology. The recovery rate o f water is a quantity that takes into account the effects of other related variables, such as frother dosage and aeration rate. Slurry viscosity is a factor that controls the drainage o f particles from the froth. The results also indicate that mineral slurries tend to b e c o m e Bingham plastic in the presence of mica minerals causing a high degree of entrainment due to the existing yield stress. Viscosity is a variable that is to be taken into account especially in the flotation of very fine materials. The model makes it possible to estimate the degree o f entrainment using easily measurable variables. The purpose is to test the model also for process simulation and control. ACKNOWLEDGEMENTS Financial support from O u t o k u m p u Oy is gratefully acknowledged.
REFERENCES Allen, T., 1991. Particle Size Measurement. Chapman and Hall, London, 806 pp. Bisshop, J.P. and White, M.E., 1976. Study of particle entrainment in flotation froths. Proc. I.M.M., 85: 191-194. Engelbrecht, J.A. and Woodburn, E.T., 1975. The effects of froth height, aeration rate an gas precipitation on flotation. J. S. Afr. Inst. Min. Metall., October: 125-132. Johnson, N.W., McKee, D.J. and Lynch, A.J., 1974. Flotation rates of non-sulphide minerals in chalcopyrite flotation processes. Trans. Am. Inst. Min. Pet. Eng., 256: 204-226. Jowett, A., 1966. Gangue mineral contamination of froth. Brit. Chem. Eng., 2: 330-333. Kirjavainen, V.M., 1989. Application of a probability model for the entrainment ofhydrophilic particles in froth flotation. Int. J. Miner. Process., 27: 63-74.
ENTRAINMENT OF HYDROPHILIC PARTICLES IN FROTH FLOTATION: A MODEL
11
Kirjavainen, V.M. and Laapas, H.R,, 1988. A study of entrainment mechanism in flotation. In: K.S.E. Forssberg (Editor), Proc. & t h e XVIth IMPC, Stockholm. Elsevier, Amsterdam, Part A, pp. 665-677. Laapas, H.R., 1983. Measurement and estimation of the theological parameters of some mineral slurries, Dissertation, Helsinki University of Technology, 71 pp. Laapas, H.R. and Kirjavainen, V.M., 1988. Fitting of size analysis into a three-parametric lognormal function with a micro-computer. In: P.J. Lloyd (Editor), Particle Size Analysis 1988. Wiley, New York~ NY, pp. 277-286. Lynch, A.J., Johnson, N.W., Manlapig, E.V. and Thorne, C.G., 1981. Mineral and coal Flotation Circuits. Their Simulation and Control. Elsevier, Amsterdam, 292 pp. Smith, P.G. and Warren, L.J., 1989. Entrainment of particles into flotation froths. In: J.S. Laskowski (Editor), Frothing in flotation. Gordon and Breach, New York, NY, pp. 123-145. Subrahmanyam, T.V. and Forssberg, E., 1988a. Froth stability, particle entrainment and drainage in flotation - - a review. Int. J. Miner. Process., 23: 33-53. Subrahmanyam, T.V. and Forssberg, E., 1988b. A study of particle entrainment in flotation with different f r o t h e r s - the case of copper ore. In: K.S.E. Forssberg (Editor). Proc. of the XVIth IMPC, Stockholm. Elsevier, Amsterdam, Part A, pp. 785-796. Thorne, C.G., Manlapig, E.V., Hall, J.S. and Lynch, A.J., 1976. Modelling of industrial sulfide flotation circuits. In: M.C. Fuerstenau (Editor), Flotation, A.M. Gaudin Memorial Volume. Am. Inst. Min. Metal. Pet. Eng., New York, NY, Vol. 2, pp. 725-752. Trahar, W.J., 1981. A rational interpretation of the role of particle size in flotation. Int. J. Miner. Process., 8: 289-327. Tsubaki, J. and Jimbo, G., 1979. The identification of particles using diagrams and distributions of shape indices. Powder Technol., 22:171-178. Wadell, H., 1934. Some new sedimentation formulas. Physics, 5:281-291. Warren, L.J., 1985, Determination of the contributions of true flotation and entrainment in batch flotation tests. Int. J. Miner. Process., 14: 33-44.
1
Elsevier Science Publishers B.V., Amsterdam
Mathematical model for the entrainment of hydrophilic particles in froth flotation V.M. Kirjavainen Department of Materials Science and Rock Engineering, Helsinki University of Technology, tlelsinkL Finland (Received 10 May 1991 ; accepted after revision 18 October 1991 )
ABSTRACT Kirjavainen, V.M., 1992. Mathematical model for the entrainment of hydrophilic particles in froth flotation. Int. J. Miner. Process., 35:1-11. The entrainment of hydrophilic particles was studied using a circulating flotation system. Flotation tests were carried out with granular and flaky minerals at different slurry densities using only frother. The results showed that particle mass and shape, water recovery rate and slurry viscosity control particle entrainment. A mathematical model for the interrelation of the variables is presented.
INTRODUCTION
Low selectivity is often a major problem in flotation processes with finely ground materials. It is known that the mechanical recovery of suspended particles strongly impairs separation selectivity when the proportion of fines ( < 10/tm) is significant in the system. The pioneering work in this field was done by Jowett (1966) and Johnson et al. (1974). The characteristics of entrainment of gangue minerals were described in detail both in batch and continuous systems. Further detailed studies on the entrainment mechanism were carried out by Englebrecht and Woodburn ( 1975 ) as well as by Bisshop and White (1976). More recent experimental studies include those of Warren (1985). Subrahmanyam and Forssberg (1988a), Kirjavainen and Laapas ( 1988 ) and Kirjavainen ( 1989 ). The subject was also reviewed by Thorne et al. (1976), Lynch et al. ( 1981 ), Trahar ( 1981 ), Subrahmanyam and Forssberg (1988b) and Smith and Warren (1989). In batch flotation tests the entrainment of gangue was found to be related to the recovery of water (Warren, 1985 ): Correspondence to: V.M. Kirjavainen, Helsinki University of Technology, Dept. of Materials Science and Rock Engineering, Lab of Mineral and Particle Technology, Vuorimiehentie 2A, 02150 Espoo, Finland.
0301-7516/92/$05.00
© 1992 Elsevier Science Publishers B.V. All rights reserved.
2
V.M. KIRJAVAINEN
R e = e~ Rwate r
( 1)
where Rg is the recovery of fine gangue of a given size in a given time, egi i s a constant for a given particle size and specific gravity and Rwateris the recovery of water at the same time. A similar type of relationship was described by Kirjavainen and Laapas (1988). The relationship was further modified by Kirjavainen (1989) and the concentration of a hydrophilic mineral in the cell was described using the following equation: tl
1
C=Co |
exp ( - P R w ) f(P) dP
(2)
0
where C is the concentration of a hydrophilic mineral expressed by the initial concentration (Co). Rw is the recovery of water, P is the entrainment factor and f(P) is a distribution function of the entrainment factor. It is evident that the entrainment factor is analogous to the term e~ in eq. (1). It has been found that the value of the entrainment factor varies between 0 and 1 which indicates that it is actually a probability factor. In general the models presented to account for the entrainment mechanism are simplified and do not consider other variables than particle size. This makes it difficult to compare the results from different origins. In this paper the effect of particle mass and shape as well as water recovery rate and slurry viscosity on particle entrainment is described and a mathematical model for their interrelation is presented. EXPERIMENTAL
Materials Because the aim was to study the behaviour of nonfloatable particles, monomineral materials were seen to be suitable for the purpose. Two mineral products, ( 1 ) quartz and (2) phlogopite, having different particle characteristics were used as test materials. The former forms granular gangue in various types of ores and the latter is a m e m b e r of the mica group minerals with platy cleavage. Both test materials were industrial products. The densities of the samples, determined using a helium-air pycnometer, were 2.6 for quartz and 2.9 for phlogopite. An advantage of using monomineral materials was that chemical analyses were not needed and recoveries could be calculated on the basis of sample weights.
ENTRAINMENT OF HYDROPH1LIC PARTICLES IN FROTH FLOTATION: A MODEL
3
Methods Flotation procedure Flotation tests were carried out using a Denver laboratory flotation machine fitted with a 3 1 cell. Since it was not possible to sample the froth product in a conventional batch flotation test, a circulating test procedure was used. This was arranged by returning the froth product immediately back to the cell with a small peristaltic pump. The aeration rate was controlled with a rotameter. The frother used was a mixture of polypropylene glycol ethers (Teefroth G). Rotor speed was kept constant at 1600 rpm to obtain adequate mixing in the cell. Experiments were carried out at the predetermined slurry densities at constant slurry volume (2.3 1). Tests with quartz were carried out at slurry densities of 5, 10, 20 and 40% solids by weight and with phlogopite the solid content was 5, 10 and 20%. Samples were collected at definite time intervals from the froth stream with 100 ml beakers. The sample volume was replaced with tap water. The samples were weighed and dried so that the recovery of solids and water could be calculated.
Particle size analysis Size distributions were determined by the electrical sensing zone method using a Coulter Multisizer. This method was chosen because it was essential to determine the actual volume distributions of the samples for the detailed analysis of the data.
Data reduction An IBM 3090 computer was used in data reduction. Linear regression was used for the calculation of size distribution models and nonlinear models were fitted to the data using the Levenberg-Marquardt algorithm. RESULTS AND DISCUSSION
Particle size distributions The three parametric log-normal distribution function was used to describe the size distributions. The fitting procedure can be done using linear regression if the log-normal function is first linearized. The principle of the method applied is described elsewhere (Laapas and Kirjavainen, 1988). The successful fitting is demonstrated in Fig. 1 where the size distributions of the test materials and typical froth products are presented. The degree of correlation was measured with the correlation coefficient. The value of the coefficient in all cases was equal to or higher than 0.9995.
4
V.M. KIRJAVAINEN
100 ~5
"q '3
Quartz
80
Floated
quartz
Phlogopite
6O
Floated
phlogopite
20 0
10 Particle
100
s i z e . lam
Fig. 1. Size distributions of test materials and typical froth products. Solid lines represent threeparametric log-normal functions fitted to the data. i .0
¢5 0.8 o
,,.o Ol~3 0
0.6 ti2 t~oeqj 0.4
o
Slurry
milli
density
• •
5% 10% 20%
o
0.2
40 %
[]
0.0 o.O
o
•
wN~ ~ m
•
0~
o
I
I
I
0.2
0.4
0.6
I
0.8
1.0
2 Water
recovery rate, kg/m
s
Fig. 2. Effect of recovery rate of water and slurry density on the entrainment of quartz.
Dependency of particle entrainment on related variables To study the effect of slurry volume on particle entrainment flotation tests were carried out using quartz at 2.0 and 2.3 1 slurry volumes. In both cases identical results were obtained and the relationship between the recovery of quartz and the recovery rate o f water followed a particular curve regardless of whether frother dosage or aeration rate was used as the control variable. The relationship remained unchanged in spite of the difference in slurry volume or in the thickness o f the froth bed. The volume of the further tests was fixed at 2.3 1, The results obtained showed that water recovery rate is an important variable that controls the entrainment o f particles. This is illustrated in Figs. 2 and 3 where the recovery o f solids per water recovery, which expresses the
ENTRAINMENT OF HYDROPHILIC PARTICLES 1N FROTH FLOTATION: A MODEL
5
1.0 o
o°o
o
o o
o
o~ o
o
oo
o
m~a
0.8
•
•
II
lli
.
oa 0.6
• nnnu
rain O
o~
.
i
II
.e,o
•
•
•
o an
LO
O0
ee
" .
Slurry
•
density
4
0.4 ¢m © ~0
•
oo
• into
•
0.2 0.0 o,O
5%
•
10 %
o
20 %
I
I
I
I
0.2
0.4
0.6
0.8
10
2
Water
recovery
rate. kg/m
s
Fig. 3. Effect of recovery rate of water and slurry density on the entrainment of phlogopite. average entrainment factor for the solids, is plotted versus water recovery rate. The water recovery rate is calculated per free slurry surface to make the quantity independent of the slurry volume. In general the degree of entrainment increased with increasing water recovery rate and slurry density but there were also some apparent differences between the materials. As Figs. 2 and 3 show, the changes in slurry density had m u c h less effect on the entrainment of granular quartz particles than on flaky phlogopite particles. Furthermore, the effect of water recovery rate was negligible with phlogopite at 20% slurry density. These facts indicate that particle entrainment is dependent on particle and slurry characteristics. Effect o f water recovery rate and particle mass In a more detailed analysis, the recoveries of quartz and phlogopite were determined in narrow 1-am size intervals in the size range 5.5-25.5 and 12.529.5 ~tm, respectively. Each of these values was then used as an independent observation for the recovery of the corresponding m e a n particle size. It was concluded that the water recovery rate and particle mass are both important variables, which are to be included in a mathematical model describing the entrainment mechanism. It is obvious that if the feed is very fine and the mass ( m ) of particles is very small the probability of entrainment (P) should approach unity. In such a case the recovery of solids should be equal to the recovery of water or, in other words, solids recovery per water recovery should be equal to one. It seems also clear that if the recovery rate of water (w) becomes very slow the probability of entrainment should approach zero. These requirements were met with the following equation:
P=
W a
wa + bm C
(3)
0
V.M. KIRJAVAINEN
where a, b and c are constants. Equation (3) was fitted to the data obtained with the test materials at each slurry density level. Particle mass was expressed in pikograms in all computations. As the results in Tables 1 and 2 show, eq. (3) accurately described the relationship between the entrainment factor and the variables resulting in high correlation in all other cases except with phlogopite at 20% slurry density. This exception can be explained by a difference in the rheological properties of quartz and phlogopite suspensions. It was shown by Laapas (1983) that quartz suspensions are Newtonian in the size range and at the slurry densities used in this work. Also phlogopite suspensions will follow Newtonian flow behaviour at low concentrations but they tend to become Bingham plastic at high slurry densities and with increasing fineness. This is consistent with the results shown in Fig. 3. The entrainment of phlogopite was only slightly affected by the changes in the recovery rate of water at 20% slurry density showing that flaky particles were not able to drain from the froth due to the existing yield stress.
Effect of viscosity As will appear from the results above slurry viscosity also controls particle entrainment. If the results in Tables 1 and 2 are considered it will prove obvious that exponent a is constant at slurry densities where the Newtonian flow behaviour is expected, being close to 0.7 for both test materials. Coefficient b TABLE 1 Results of fitting the data obtained with quartz to eq. (3) Slurry density
Number of obs.
a
b
c
Sum of squares
Coefficient of determination
5 10 20 40
500 760 700 400
0.7252 0.7581 0.6473 0.9185
0.00607 0.00577 0.00450 0.00147
0.4999 0.4975 0.5297 0.5833
0.14364 0.52574 0.48419 0.29542
0.99026 0.97909 0.97683 0.97513
(%)
TABLE2 Resultsoffittingthedataobtained withphlogopitetoeq.(3) Slurry density
Number of obs.
a
b
c
Sum of squares
Coefficient of determination
5 10 20
510 510 510
0.7440 0.7147 0.5081
0.1185 0.0531 0.2022
0.0960 0.1278 -0.1105
0.22016 0.27268 0.32844
0.96726 0.95976 0.32077
(%)
ENTRAINMENT OF HYDROPH1LIC PARTICLESIN FROTH FLOTATION: A MODEL
7
is dependent on the solids content in the feed. This was thought to indicate that b is dependent on slurry viscosity. Exponent c is also constant at different slurry densities but it was dependent on the characteristics of the test materials. Consequently, b was assumed to be dependent on slurry viscosity and eq. (3) was modified to the following form: P-
W a
(4)
wa+f(~/)m c
Laapas (1983) described the viscosity of quartz and phlogopite suspensions by the following empirical equation:
FCv ~=~0~ (1_cv)48
(5)
where Cv is the volume fraction of solids, ~/o denotes the viscosity of the suspending m e d i u m ( 1.0 mPa.s in this work) and F i s a constant, whose value is 1.83 and 3.80 for quartz and phlogopite, respectively. Equation (5) was used to estimate the viscosities of the quartz and phlogopite suspensions. When eq. (4) was fitted to the experimental data it was found that the relationship between the entrainment factor and the three variables could be described accurately with both of the test materials in the Newtonian region using the following function for f(q): f(t/) = b i t/b2
(6)
where b~ and b2 are constants that are characteristic for each of the test materials as shown by the results in Table 3.
Effect of particle shape The results presented above show that b~, b2 and c are dependent on particle shape and eq. (4) takes the form: W a
(7)
P-wa+f~ (~-'])~ f2(t/'t) m f 3 ( ~ )
where f~ ( ~ ) , f2 (~u) and t"3( ~ ) represent functions which are dependent on particle shape. There are several shape factors that may be used in particle TABLE3 Results of fitting the data to eqns. (4) and (6) Material
Number of obs.
a
bl
b2
c
Sum of squares
Coefficient of determination
Quartz Phlogopite
2360 1020
0.7332 0.7285
0.00521 0.16953
-0.6813 -6.3597
0.5196 0.1090
1.82196 0.50341
0.97600 0.96889
8
V.M. KIRJAVAINEN
characterization but only two are considered here: (1) the dynamic shape factor (x), defined by Tsubaki and Jimbo (1979) and (2) Wadell's sphericity factor (~Uw) (Wadell, 1934). The dynamic shape factor is defined as the ratio of the resistance to motion of a given particle divided by the resistance of a spherical particle of the same volume. When the particle settles under laminar flow conditions, as may be expected in the quiescent zone near the froth bed in a floatation cell, the terminal velocity obeys Stokes' law and the dynamic shape factor can be expressed using the volume diameter (d v) and the Stokes' diameter (dst) (Allen, 1991 ): (8)
x=(dv/dst) 2
Wadell's sphericity factor is defined as the ratio of the surface area of a sphere having the same volume as the particle divided by the surface area of the particle. Under laminar flow conditions Wadell's sphericity factor can be expressed as: (9)
~w = ( d s t / d v ) 4
It is easy to see that x = ~w 1/2. Equations (8) and (9) are of practical importance because they make it possible to determine the shape factors for a mineral using two different techniques. The volume diameter is obtained by Coulter analysis and the Stokes' diameter is obtained by gravitational sedimentation, e.g. with a Sedigraph. A sieved fraction from both test materials was analysed using these methods. On the basis of the results (Tables 4 and 5 ) the modes of the distributions were estimated to be 32.0 and 35.8 #m for the quartz fraction and 14.5 and 44.5 #m for the phlogopite fraction. Using these values the dynamic shape factors were calculated to be 1.25 for quartz and 9.42 for phlogopite. Similarly, the values of Wadell's sphericity factor were 0.64 for quartz and 0.011 for phlogopite. TABLE4 Coulter and Sedigraph analyses of a quartz fraction Sedigraph analysis
Coulter analyses #m
Differential volume, %
lzm
Differential volume, %
32.26-33.27 33.27-34.28 34.28-35.29 35.29-36.29 36.29-37.30 37.30-38.31 38.31-39.32 39.32-40.33
5.62 6.83 7.69 7.78 7.63 7.26 6.93 6.77
29-30 30-31 31-32 32-33 33-34 34-35 35-36 36-37
6.5 7.0 8.0 8.0 7.5 5.5 4.5 4.5
ENTRAINMENT OF HYDROPHILIC PARTICLES 1N FROTH FLOTATION: A MODEL TABLE 5 Coulter and Sedigraph analyses of a phlogopite fraction Coulter analysis
Sedigraph analysis
/tm
Differential volume, %
/zm
Differential volume, %
39.48-40.89 40.89-42.31 42.31-43.73 43.73-45.15 45.15-46.56 46.56-47.98 47.98-49.40 49.40-50.82
4.78 5.94 7.14 7.38 7.32 6.76 6.00 4.54
12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20
5.6 7.8 9.2 8.7 7.8 7.5 6.5 6.0
TABLE 6 Results of fitting the data obtained with quartz and phlogopite to eq. ( 7 ) Shape ~ctor
Number ofobs,
b3
Sum of squares
Coefficientof determination
(x) (~w)
3380 3380
0.00694 0.00694
3.95714 3.96931
0.95981 0.95969
Most shape factors, including Wadell's sphericity factor, have a maximum value of 1.0 representing spherical particles or other regular shapes. However, the dynamic shape factor is an exception having a minimum value of 1.0. It appeared that either of the shape factors may be used in the model and that f~ (7~), f2 ( ~ ) and f3 ( ~ ) are relatively simple functions. The best solution for eq. (7) was obtained by equation: W 0-7 P - - w °'7 + b3 ~ t / - 0 . 5 ~ m 0.5 l/if--0.4
( 10 )
where 5 u = x = 5Uw1/2 and the term b 3 is a constant, which is, however, dependent on the units in which the variables are expressed. As shown by the results in Table 6 the coefficient of determination was close to unity in both cases. It is also easy to see that in the case of spherical particles, when K= ~gw= 1, eq. (10) is reduced to wO.7
P-wo.7 +b3(m/rl)o.5
( 11)
which is not dependent on particle shape. In fact this was an important criterion when the functions for f~ ( ~ ) , f2 ( ~ ) and f3 ( ~ ) were chosen.
10
V.M. KIRJAVAINEN
CONCLUSIONS It was shown that the entrainment o f hydrophilic minerals is dependent on particle characteristics, i.e. particle mass and shape, and two adjustable variables, water recovery rate and slurry viscosity. Particle size is the most important factor that controls mechanical entrainment and to minimize the entrainment of particles flotation should be carried out using as coarse grinding as possible. It is obvious that entrainment can cause the enrichment o f fine gangue in recycled flotation circuits and that fine fractions need more cleaning stages for a particular grade compared to coarse fractions. The detrimental effects o f particle entrainment can be reduced using separate circuits for coarse and fine fractions. The entrainment mechanism is clearly a factor that should be taken into account already in process planning especially with fine materials. On the basis of the experimental data it was possible to construct a probability model that accurately described the relationship between the entrainment factor and the variables in the Newtonian region. Particle mass and shape were characterized by methods generally used in particle technology. The recovery rate o f water is a quantity that takes into account the effects of other related variables, such as frother dosage and aeration rate. Slurry viscosity is a factor that controls the drainage o f particles from the froth. The results also indicate that mineral slurries tend to b e c o m e Bingham plastic in the presence of mica minerals causing a high degree of entrainment due to the existing yield stress. Viscosity is a variable that is to be taken into account especially in the flotation of very fine materials. The model makes it possible to estimate the degree o f entrainment using easily measurable variables. The purpose is to test the model also for process simulation and control. ACKNOWLEDGEMENTS Financial support from O u t o k u m p u Oy is gratefully acknowledged.
REFERENCES Allen, T., 1991. Particle Size Measurement. Chapman and Hall, London, 806 pp. Bisshop, J.P. and White, M.E., 1976. Study of particle entrainment in flotation froths. Proc. I.M.M., 85: 191-194. Engelbrecht, J.A. and Woodburn, E.T., 1975. The effects of froth height, aeration rate an gas precipitation on flotation. J. S. Afr. Inst. Min. Metall., October: 125-132. Johnson, N.W., McKee, D.J. and Lynch, A.J., 1974. Flotation rates of non-sulphide minerals in chalcopyrite flotation processes. Trans. Am. Inst. Min. Pet. Eng., 256: 204-226. Jowett, A., 1966. Gangue mineral contamination of froth. Brit. Chem. Eng., 2: 330-333. Kirjavainen, V.M., 1989. Application of a probability model for the entrainment ofhydrophilic particles in froth flotation. Int. J. Miner. Process., 27: 63-74.
ENTRAINMENT OF HYDROPHILIC PARTICLES IN FROTH FLOTATION: A MODEL
11
Kirjavainen, V.M. and Laapas, H.R,, 1988. A study of entrainment mechanism in flotation. In: K.S.E. Forssberg (Editor), Proc. & t h e XVIth IMPC, Stockholm. Elsevier, Amsterdam, Part A, pp. 665-677. Laapas, H.R., 1983. Measurement and estimation of the theological parameters of some mineral slurries, Dissertation, Helsinki University of Technology, 71 pp. Laapas, H.R. and Kirjavainen, V.M., 1988. Fitting of size analysis into a three-parametric lognormal function with a micro-computer. In: P.J. Lloyd (Editor), Particle Size Analysis 1988. Wiley, New York~ NY, pp. 277-286. Lynch, A.J., Johnson, N.W., Manlapig, E.V. and Thorne, C.G., 1981. Mineral and coal Flotation Circuits. Their Simulation and Control. Elsevier, Amsterdam, 292 pp. Smith, P.G. and Warren, L.J., 1989. Entrainment of particles into flotation froths. In: J.S. Laskowski (Editor), Frothing in flotation. Gordon and Breach, New York, NY, pp. 123-145. Subrahmanyam, T.V. and Forssberg, E., 1988a. Froth stability, particle entrainment and drainage in flotation - - a review. Int. J. Miner. Process., 23: 33-53. Subrahmanyam, T.V. and Forssberg, E., 1988b. A study of particle entrainment in flotation with different f r o t h e r s - the case of copper ore. In: K.S.E. Forssberg (Editor). Proc. of the XVIth IMPC, Stockholm. Elsevier, Amsterdam, Part A, pp. 785-796. Thorne, C.G., Manlapig, E.V., Hall, J.S. and Lynch, A.J., 1976. Modelling of industrial sulfide flotation circuits. In: M.C. Fuerstenau (Editor), Flotation, A.M. Gaudin Memorial Volume. Am. Inst. Min. Metal. Pet. Eng., New York, NY, Vol. 2, pp. 725-752. Trahar, W.J., 1981. A rational interpretation of the role of particle size in flotation. Int. J. Miner. Process., 8: 289-327. Tsubaki, J. and Jimbo, G., 1979. The identification of particles using diagrams and distributions of shape indices. Powder Technol., 22:171-178. Wadell, H., 1934. Some new sedimentation formulas. Physics, 5:281-291. Warren, L.J., 1985, Determination of the contributions of true flotation and entrainment in batch flotation tests. Int. J. Miner. Process., 14: 33-44.