Estimation of bubble size in flotation columns

Estimation of bubble size in flotation columns

Minerals Engineering, Vol. 8, Nos 1/2, pp. 77-89, 1995 Pergamon 0892--6875(94)001114-9 Copyright O 1994 EIBevief Science Ltd Printed in Great Britai...

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Minerals Engineering, Vol. 8, Nos 1/2, pp. 77-89, 1995

Pergamon 0892--6875(94)001114-9

Copyright O 1994 EIBevief Science Ltd Printed in Great Britain. All rights re~fved 0892-6875/95 $9.50+0.00

ESTIMATION OF BUBBLE SIZE IN FLOTATION COLUMNS

M.T. ITYOKUMBUL§*, A.I.A. SALAMA ~f, and A.M. AL TAWEEL:~ Mineral Processing Section, Department of Mineral Engineering, 115 Hosler Building, Pennsylvania State University, University Park, PA 16802, USA 1" Western Research Centre, CANMET, 1 Oil Patch Drive, P.O. Bag 1280, Devon, Alberta, T0C 1E0, CANADA :[: Department of Chemical Engineering, Technical University of Nova Scotia, P.O. Box 1000, Halifax, Nova Scotia, B3J 2X4, CANADA * Author to whom correspondence should be addressed. (Received 11 July 1994; accepted 2 September 1994)

ABSTRACT

A non-iterative procedure for the estimation of bubble size in flotation columns is presented. In a bubble swarm, the bubble terminal rise velocity is estimated using an appropri~Ite drift flux relation which depends on the frother type. For Dowfroth 250C, MIBC anti Triton X-IO0, these drift flux relation were found to be those of Marrucci [1], Turner [2.1, and, Richardson and Zaki [3] respectively. By using dimensionless groups in the correlation procedure, dimensionally consistent correlations are presented. The analysis .~hows that in the presence of frothers, the drag coe~cient of the air bubbles ino'eases. This is attributed to surfactant adsorption at the gas-liquM interface. Keywor~t Bubble size, drift flux, terminal rise velocity, flotation column

INTRODUCTION The recovery of hydrophobic particles in a flotation column is an industrially important separation process. The air bubbles are generated at the base of the flotation column and as they rise due to buoyancy, they contact a countercurrent flow of slurry. The hydrophobic particles collide with and adhere to the air bubbles and are transported to the froth zone. In the froth zone, loosely attached gangue particles are washed back into the pulp phase and the clean product overflows at the cell lip. Owing to the interfacial nature of flotation, the rate of flotation depends on the availability of bubble surface. The design and scale-up of flotation columns require knowledge of the bubble characteristics (size and rise velocity). The available bubble surface area may be computed from knowledge of the bubble size and gas hold-up, while the bubble residence time in the recovery zone may be calculated from its rise velocity. Both of these hydrodynamic parameters (i.e. bubble residence time and surface area) are required in computation of the bubble loading [4]. The bubble rise velocity is a function of its size and the physical properties of the liquid. The relationship between the bubble size and terminal rise velocity in a flotation column has been studied by a number of authors [5-9]. In all of these studies, the drift flux approach of Wallis [10] is used to compute the terminal rise velocity of gas bubbles from knowledge of the phase velocities and hold-ups. The bubble size is estimated from an appropriate correlation of the terminal rise velocity of a single bubble as a function of size. However, th~ use of this method is predicated on the following: HE 8-1/2-F

7"7

78

M.T. ITYOKUMBUL et al.

(i)

A correct relationship for estimating the single bubble rise velocity in the swarm is known; and

(ii)

That the computed bubble rise velocity in the swarm is equal to the single bubble rise velocity determined under the same conditions (e.g frother concentration).

Unfortunately, the validity of the assumptions has not been established to the authors' knowledge. For example, we note that the surface phenomena occurring in a single bubble and bubble swarm are unlikely to be the same. Firstly, the effective frother concentration at the bubble surface will hardly approach equilibrium conditions that may exist in single bubble systems. This is due to the rather low frother concentrations and high bubble densities typically encountered in mineral processing operations. Secondly, for volatile frothers, loss by evaporation will cause the frother concentration to decrease at a much faster rate than in single bubble systems. Thus the effect of frother concentration on the bubble rise velocity in a bubble swarm may not be a strong as in a single bubble system. This observation is consistent with the recent experimental data of Tucker et al. [11].

THEORY Estimation of terminal rise velocity of a single bubble According to the drift flux theory, the bubble slip velocity is defined as

Us

Ug e8

U1 1 - eg

(1)

where eg is the gas hold-up, and the (+) and (-) signs refer to the countercurrent and concurrent flow of liquid and gas respectively. Various expressions for relating the terminal rise velocity to the slip velocity have been proposed in the literature [1-3,12-15]. In general, the bubble slip velocity is related to the gas hold-up and terminal rise velocity of a single bubble, Ub, by an expression of the form: U,

-- U b F(es)

(2)

A summary of the relationships reported in the literature for F(Eg) are given below [1-3,12-15].

u,

-

u,

= ub

(4)

u.

= u b(l-%)

(S)

Us

=

Ub (1 - %)

U b (l

- Eg) I'$9

(3)

(6)

Estimation of bubble size

us

= Ub ( I I

-

5/3 eI

U s = U b (1 - es)!39 (1 + 2.55 e38)

79

(7)

(8)

It should be noted that Eqs. 6-8 are modifications of the Richardson and Zaki [3] drift flux expression. Eq. 3 was suggested for large bubbles [12,13] while Eq. 5 was proposed for fine bubbles [13,14]. With the exception of the relationships of Marrueei [1] and Locker and Kirkpatriek [15], Eq. 2 may be written as"

U.

= U b (I - ee)m'1

(9)

For gas hold-up to 30%, Lockett and Kirkpatriek [15] recommend the Richardson and Zaki's [3] relation for drift flux since it fits most of the available literature data. However, it is not clear at the present time if the same relationship will be applicable under flotation conditions. For example, we note that Griffith and Wallis [13] and Bhaga and Weber [14] assumed that m = 2 for small bubbles, while Yiannatos et al. [7] assumed m to be a function of the bubble Reynolds number. Since the action of frothers differs, it is also not clear if the use of a single drift flux expression is justified. Most of these precedures for estimating the bubble size from the bubble rise velocity are iterative in nature [5,6,16]. V~ile the procedure of Zhou et al [9,17,18] is not iterative, it requires knowledge of the contamination factor, C¢, which is dependent on the frotber type and concentration. It is our observation that for typical concentrations of frothers used in mineral processing, the bubble rise velocity should not be a function of frother concentration. Since different F(eg) functions have be,on proposed in the literature, one of the objectives of this study is to determine the expression that best describes the rise velocity of air bubbles under :flotation conditions. Another objective is to determine if a single drift flux relationship may be used for al]I flotation systems. The final objective is to provide a dimensionally consistent and noniterative procedure for the determination of bubble size under flotation conditions. DETERMINATION OF F(eg) FOR A BUBBLE SWARM In order to determine the F(eg) relationship that best describes flotation column data, the bubble slip velocity was determined from reported literature data [5-9] using Eq. 1. For each F(eg) relationship (i.e. Eqs. 3-8), the terminal bubble rise velocities were calculated. Since recent studies [9,17-19] have shown that the action of frothers differs, the correlation procedure was applied to each frother system. The calculated bubble llerminal rise velocities were correlated to the bubble size by the expression:

U,.

= a Db n

(10)

where a and n are constants to be determined. Tables 1-3 shows the relationships that have been determined for the variation of bubble terminal rise velocity with size. For Dowfroth 250C, high correlation coefficients were obtained with the data of Yiannatos et al [7] and Zhou et al [9] (see Table 1). In addition, the bubble diameter exponents for these data were also comparable. Thus the authors combined these two data sets and repeated the correlation procedure. When this was done the best correlation was obtained with the use of the Marrucci drift flux expression (i.e. Eq. 7) resulting in the relationship:

8o

M. T. I T Y O K U M B U L

U b

=

et al.

'f30.95 72.6 ~ b

(11)

The correlation coefficient for Eq. 11 is 0.974. Since the bubble diameter is the parameter that is required, it may be determined from a rearrangement of Eq. 11:

Db =

I I 1.05

(12)

0.011 '-'b

T A B L E 1 Regression coefficients for Eq. 10 with Dowfroth 250C Drift flux Relation

Data of Ref. [7]

Data of Ref. [8]

a

n

a

Eq.3 [12,13]

142

1.09 0.885 0.0006

~ . 4 t2]

122

~ . 5 [13,14]

105

Data of Ref. [91

Data of Ref. [7,91 combined

a

n

a

-0.69 -0.857

12.1

0 . 7 1 0.991 221

1.15 0.923

1.04 0.995 0.047

-0.09 -0.443

38.5

0.87

0.992 106

1.02 0.969

0.99

0.52

0.856

107

1.0l

0.995 54.3

0.90

0.963

165

1.07 0.994 40.9

0.85

0.933

r1

n

0.967 3.69

r

r

n

r

Eq. 6131

98.2

0.97

0.947 20.6

0.76

0.866

~ . 7 [11

113

1.01

0.970 0.39

0.20

0.673

80.6

0.97

0.993 73.1

0.95

0.974

~ . 8 [151

103

0.98

0.961 4.23

0.53

0.850

165

1.07

0.994

45.1

0.87

0.949

Correlation mefficeint T A B L E 2 Regression coefficients for Eq. 10 with MIBC. Drift flux

Data of Ref. [7,9]

Data of Ref. [9]

Data of Rcf. [7]

combined

relation a

n

r

a

n

r

a

Eq.3 [12,13]

18800

1.74

0.921

73.4

0.95

0.801

736

1 . 2 9 6,981

Eq. 4 [12]

2200

1.42

0.922

169

1.06

0.821

262

I. 13

0.982

Eq. 5113,14]

239

1.09

0.918

295

1.13

0.801

93.4

0.96

0.977

Eq. 6 [13]

107

0.97

0.922

426

1.18 0.794

60.0

0.89

0.965

Eq. 7 [1]

629

1.23 0.919

253

1.11 0.810

138

1 . 0 2 0.976

~t. 8 [15]

132

1.0

427

1.18 0.794

68.7

0.906

b

r

0.91

0.966

T A B L E 3 Regression coefficients for Eq. 10 for other frothers. Drift flux relation

TEB: Data of Ref. [7]

Triton x-100: Data of

Ref. [8] a

n

r

a

n

r

Eq. 3 [12,13]

738,000

2.29

0.956

.133

.005

0.060

~ . 4 [2]

42,7oo

1.87

0.955

0.58

.23

0.850

Eq. 5 [13,14]

2930

1.47

0.954

2.08

.42

0.921

~ . 6 [3]

947

1.30

O.95O 3.54

.50

0.933

~ . 7 [1]

1o4oo

1.65

O.955

1.33

.35

0.915

~ . 8 [15]

1700

1.38

0.955

3.32

.49

0.934

Estimation of bubble size

81

A comparison of tl~Lepredicted bubble diameters with those experimentally determined is shown in Figure 1. It may be seen that the agreement is quite good. Thus, for Dowfroth 250C, the terminal bubble rise velocity may be estimated using the Marrucci drift flux relationship and the bubble size determined using Eq. 12. 1.6

m

u

i



i



[



1.41

E E 0 °

1.2 u

e .a

1.0

o

.Q "u

o

0.8

o o

'u Q. 0.6

0.4

0.4

m

m

m

m

|

0,6

0.8

1,0

1.2

1.4

Measured

bubble eize,

! 1.6

mm

Fig. 1 Comparison of measured and predicted bubble size for Dowfroth 250C (Bubble rise velocity calculated using the Marrucci drift flux relation, i.e. Eq. 7). For MIBC, the correlation coefficients were higher with the data of Yiarmatos et al. [7]. However, the highest correlation coefficients were obtained when the data of Yiannatos et al [7] and Zhou et al. [9] were combined (see Table 2). It may be seen from Table 2 that the highest correlation coefficient was obtained with the use of the Turner relationship for the drift flux, i.e. Eq. 4. The relationship between the bubble size and rise velocity was determined to be: Ub = 262Dd a3

(13)

with a correlation coefficient of 0.982. Thus for MIBC, the bubble size may be estimated from a rearrangement of Eq. 13 to yield: Db = 0.0072 1T "b°'"5

(14)

The parity plot for Eq. 14 is shown in Figure 2. In this case as well the agreement is quite good. The bubbles generated with Triton X-100 were considerably larger than those observed with either MIBC or Dowfroth 250C. Thus, it was expected that Eq. 3 would provide the best estimate for the bubble rise velocity in the swarm. However, the present results show that the highest correlation coefficients were obtained with

82

M.T. ITYOKUMBUL et al.

the use of either the Richardson and Zaki relationship or the Lockett and Kirkpatrick drift flux relations. In addition, both relationships give bubble diameter exponents that are approximately 0.5. Consequently, we adopted the Richardson and Zaki relationship (i.e. Eq. 6) for the drift flux. It is noted that a similar conclusion was reported by Pal and Masliyah [8]. Since the Richardson and Zaki drift flux relation was derived for rigid spheres, this would suggest that the addition of Triton X-100 tends to produce bubbles that behave as solid spheres. In the presence of Triton X-100, the terminal rise velocity is related to the bubble size by the expression: Ub

=

3.54 D °'5

(15)

Similarly, the bubble size may be estimated from a rearrangement of Eq. 15 to give: D b

=

0.08

(16)

U 2

1.4

E E



!



!



|



1.2

N

_e .Q &i :S .a

1.0

"G

e U

@.

0.8

0.6

0.6



I

.

0.8

Measured

I

i

1.0

bubble size,

I

1.2

I

1.4

mm

Fig.2 Comparison of measured and predicted bubble size for MIBC (Bubble rise velocity calculated using the Turner drift flux relation, i.e. Eq. 4). A comparison of the predicted and experimentally observed bubble diameter is shown in Figure 3. It may be seen that the agreement is quite good. For triethoxy butane (TEB), the correlation coefficients were not significantly different (0.950 to 0.956). In addition, there is no terminal rise velocity data for single bubbles in water containing TEB. The nondiscriminatory nature of the correlations may be due to the rather narrow range of bubble sizes reported by Yiannatos et al. [7]. Thus, additional information is required for determining the most appropriate drift flux expression in the presence of TEB.

Estimation of bubble size

3.5

.

.

.

.

J

.

.

.

.

i

.

83

.

.

.

i

.

.

.

.

3.0

II

e A ,41

2.5

-l ,,Q "U

e

o • I:).

2.0

1.5 1.5

.

.

.

.

i

.

.

.

.

2.0

Meuurecl

t

2.5

.

.

.

.

|

3.0

.

.

.

.

3.5

bubble size, mm

Fig.3 Comparison of measured and predicted bubble size for Dowfroth 250C (Bubble rise velo,-ity calculated using the Richardson and Zaki drift flux relation, i.e. Eq. 6). The results obtained in this study do not support the use of'a single drift flux expression for the calculation of terminal bubbl4~ rise velocity. For example, the most appropriate drift flux relation for Dowfroth 250C, MIBC and Triton X-100 are those of Marrncci (Eq. 7), Turner (Eq. 4) and Richardson and Zaki (Eq. 5) respectively.

DIMENSIONAL ANALYSIS FOR BUBBLE RISE VELOCITY In order to provide dimensionally consistent correlations, the authors have used dimensional analysis. It can be readily shown that (see Appendix A): U2

=

f ( DbUbPl DbplU2)

g Db

~t1

(17)

o

Eq. 17 shows that the bubble Froude number is a function of the Reynolds and Weber numbers. Peebles and Garber [20] have shown that the critical Weber number for bubble stability occurs at: DI,plU~ We,~ttic~ -

a

-

3.65

(18)

84

M.T. ITYOKUMBULet al.

For typical flotation conditions (o > 55 mN/m, D b < 3 mm), the Weber number is consistently lower than 0.5. Thus it maybe reasonable to assume that the effect of the Weber number on the bubble rise velocity may be safely ignored. Similarly, the bubble size stability diagram developed by Berghmans [21] also confirms that bubbles generated with injector-type nozzles will be stable under these conditions. Consequently, the bubble Froude number may be taken to be a function of the Reynolds number. Fr

f(Re)

=

(19)

Indeed thesame conclusion (i.e. the Froude number is a function of the Reynolds number) may also be obtained from a force balance. By definition, the drag coefficient of a rising spherical bubble of equivalent diameter, D b, is given by:

CD

4 Db g

(Pl - Ps)

3 U~

Pl

-

(20)

Eq. 20 may be rearranged to give

U~

_

g Db

4

( P l - Pg)

3 CD

Pl

(21)

For the air water system (typical of flotation systems)

(Pl = Pg)

~

1

(22)

Pl Thus,

U~

_

g Db

4

(23)

3 CD

The left side of Eq. 3 represents the bubble Froude number, while the right is a function of the drag coefficient only. Since the drag coefficient is frequently correlated as a function of the Reynolds number, the observation that the bubble Froude number is a function of the Reynolds number appears to be justified. With the definition of the drag coefficient (i.e. Eq. 23), the drag coefficients for air bubbles rising in Dowfroth 250C, MIBC and Triton X-100 were determined to be:

CD

_

10.2 Re 0.46

(24)

Estimation of bubble size 17.7

85 (25)

C~ = R e °'s9

Cv = 1.03

(26)

respectively. The authors note that Eq. 25 agrees rather well with that given by Lapple [22] for spherical particles. The exponent for the Reynolds number in Eqs. 24-26 differs considerably. This is an indication of the differences in the nature and action of these frothers. Eq. 26 suggest that in the presence of Triton X-100, the drag coefficient is independent of Reynolds number. Even though the bubble Reynolds numbers were in the range 270-620, the constancy of the drag coefficient would suggest the existence of turbulent conditions. For large bubbles rising in pure water, Prantl [23] recommend that the terminal rise velocity may be calculated using:

U b = 4.02 D °'5

(27)

From Eq. 27, the drag coefficient under turbulent conditions is computed to be: %

-- 0 8 1

(28)

A comparison of Eqs. 26 and 28 shows that in the presence of Triton X-100, the drag coefficient on the bubble is increased by 27 %. The increase in the drag coefficient is attributed to surfactant adsorption at the air-water interface [24,25]. From the definition of the drag coefficient, the relation between the bubble size and rise velocity for Dowfroth 250C, MIBC and Triton X-100 may be written in dimensional form as:

U b = 0.267 gO.~ (__Pl)o.3o "-'br~0"gs

U b = 0.160 go.71 (P_2)o.42 Dbt.t3 I-tt

(29)

(30)

and U b

=

1.14 (g Db)1/2

(31)

respectively. The lower and upper limits for the application of Eqs. 29 and 30 is not known at the present time. However, the analysis of Peebles and Garber [20] would suggest a Reynolds number in the range 2 < Re < 700.

86

M.T. ITYOKUMBULet al. CONCLUSION

The use of the drift flux method for the estimation of bubble size in flotation vessels has been presented. It is shown that in the presence of Dowfroth 250C, MIBC, and Triton X-100 frothers, the Marrucci, Turner, and Richardson and Zaki relations respectively best describe the relationship between the terminal and bubble slip velocity. The results obtained show that the terminal rise velocity of gas bubbles is reduced in the presence of frothers. This is attributed to the interaction frother molecules at the gas-liquid interface. It is shown that a single drift-flux relationship is not applicable for all frother types.

NOTATION CD

drag coefficient, [-]

Db

bubble diameter, m

Fr

Froude number, Ub2/(gDb), [-]

g

acceleration due to gravity, m/s2

m

constant, [-]

Re

bubble Reynolds number, (UbDbPl) /1~1, [-]

Ub

terminal bubble rise velocity, m/s superficial gas velocity, ntis

Ul

superficial liquid velocity, m/s

Us

bubble slip velocity, m/s

We

Weber number, DbPlUb2/O, [-]

Greek Symbols eg

gas holdup

pg

density of gas, kg/m 3

Pl

density of liquid, kg/m 3

I.tg

viscosity of gas, Pa.s

I.q

viscosity of liquid, Pa.s

o

surface tension, N/m

Estimationof bubblesize

87

REFERENCES .

2. 3. 4. 5. 6. 7. .

9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25.

Marrucci, G., Ind. Eng. Chem. Fund., 4, 224 (1965). Turner, J.C.R., Chem. Eng. Sci. 21, 971 (1966). Richardson, J.F. & W.N. Zaki, Trans. Inst. Chem. Engrs., 32, 32 (1954). King, R.P., Hatton, T.A & Hulbert, D.G., Trans. Inst. Min. Met., 85, Cl12 (1975). Dobby, G.S. & Finch, J.A., Can. Metall. Q., 25, 9 (1986). Dobby, G.S., Yiannatos, J.B. & Finch, J.A., Can. Metall. Q. 27, 85 (1988). Yiannatos, J.B., Finch, J.A. Dobby, G.S. & Xu, m., J. Colloid and Interface Science 126, 37 (1988). Pal, R. & Masliyah, J.B., Can. J. Chem. Eng., 67, 916 (1989). Zhou, Z.A., Egiebor, N.O. & Plitt, L.R., Minerals Engineering, 6, 55 (1993). Wallis, G.B., One-Dimensional Two Phase Flow, McGraw-Hill, New York (1969). Tucker, J.]?., Deglon, D.A, Franzidis, J-P., Harris, M.C. & O'Connor, C.T., Minerals Engineering, 7, 667 (1994). Davidson, J.F. & Harrison, D., Chem. Eng. Sci. 21, 731 (1966). Griffith, P. & Wallis, G.B., ASME Trans. J. of Heat Transfer, 83, 307 (1961). Bhaga, D. & Weber, M.E., Can. J. Chem. Eng. 50, 323 (1972). Lockett, 1V[.J. & Kirkpatrick, R.D., Trans. Inst. Chem. Engrs., 53, 267 (1975). Xu, M., Radial Gas Holdup ProyTle and Mixing in the collection Zone of Flotation Columns, Ph.D. Thesis, McGill University (1991). Zhou, Z.A., Egiebor, N.O. & Plitt, L.R., Effect of frothers on bubble rise velocity in a flotation column-- I. Single bubble system, In G. E. Agar, B. J. Huls and D. B. Hymn (eds.) Column '91, CIM, Canada, 249 (1991). Zhou, Z.A., Egiebor, N.O. & Plitt, L.R., Can. Metall. Q. 31, 11 (1992). Laughlin, LD.M., Quirm, P., Robertson, R. & Agar, G.E., XVIll Int. Mineral Processing Congress, 637 (1993), Peebles, F N. & Garber, H.J., Chem. Eng. Prog., 49, 88 (1953). Berghmans, J., Chem. Eng. Sci., 28, 2005 (1973). Lapple, C.E., FIuM and Particle Mechanics, University of Delaware Publishers, Newark, Delaware, 284 (1953). Prantl, L., Hydro-aeromechanics, in V.I. Klassen, and V.A. Mokrousov, An Introduction to the Theory of Flotation, 468, Butterworths, London (1963). Davis, R.I:I. & Acrivos, A., Chem. Eng. Sci. 21, 681 (1966). Shah, Y.T., Kelkar, B.G., Godbole, S.P. & Deckwer, W.-D., AIChE J., 28, 353 (1982).

APPENDIX A Dimensional Analysis for Bubble rise Velocity

The bubble rise velocity, Ub, depends on the bubble size, Db, the liquid and gas densities, Ol and pg, the liquid and gas viscosities, i~l and I~g, the acceleration due to gravity, g, and the liquid surface tension, a. The Buckingham Pi method is used to determine the dimensionless groups into which the variables may be combined. The fundamental units or dimensions of the variables, v, are three (mass M, length L, and time T). The number of variables , q, are eight. According to the Buckingham Pi method, the number of dimensionless groups or r~'s should be q - v, or 8 - 3 = 5. Db, U b, and Pl are selected to be the core variables common to the five groups. Then the five dimensionless groups are

88

M.T. ITYOKUMBULet

al.

~1

= D : Ubb p~ o 1

(A.1)

=2

= D~ U~ p~ gl

(A.2)

i



1

(A.3)

k p!1 ~sl

(A.4)

n4

=

D~U~

ns

=

D~ U~ p~ plg

(A.5)

Consider the first dimensionless group ~1. The exponents a, b, and c are evaluated by substituting the dimensions for each variable as shown below: M°L°T ° =

1 =

L a (L/T) b (M/L3) ¢ (M/T2) 1

(A.6)

Equating the exponents o f L, M and T on both sides o f Eq. (A.6) gives: (L) 0 = a + b - 3c

(A.7)

(M) ° = c + 1

(A.8)

(T) 0 = b + 2

(A.9)

Solving these equations, a = -1, b = -2, and c = -1. Substituting these values into Eq. A. 1 gives

x1

o

1

-

-

D b U ~ Pl

(A. 10)

We

Repeating the procedure for ~2, n3, ~4 and n5 gives ~2

-

g Db

Ub2

-

1

Fr

(A.11)

Estimation of bubble size I.tl k

7g3

89

(A. 12)

Db Ub I~I

~4 -

~s DbUb Pt

(A.13)

~s

Pg

(A.14)

-

Pl On inspection, it can be seen that the denominators of n 3 and ~4 are identical. A new n 4 is obtained by dividing g4 by ~3. n,

= ~

(A.15)

It is evident from this treatment that two of these dimensionless groups are the density and viscosity ratios, P J P l and I.tg/I.tr F'rom an engineering point of view, neither of these ratios will be much greater than zero. Thus their contributions may be neglected without much loss of accuracy. This leaves three dimensionless groups which may be combined to yield Fr

= fiRe, We)

(A.16)

In flotation, the concentration of frothers or surfactants used is typically low. Thus, the change in the surface tension with frother and collector addition is often negligible. Furthermore, available data indicates that for typical flotation size air bubbles ( Db < 3 mm), the Weber number is generally small. As a result of this, the contribution of surface tension forces is unlikely to be important. Thus, the Froude number may be taken to be a function of the Reynolds number Fr

= f(Re)

At the present time, this dependency is assumed to be via the drag coefficient.

(A.17)