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Turbulence in flotation cells, a simplified approach
International Journal of Mineral Processing, 34 ( 1992 ) 133-135 Elsevier Science Publishers B.V., Amsterdam
133
Turbulence in flotation cells, a simplified approach Kai Fallenius Harjuviita 16 A 9 SF-02110 Espoo, Finland (ReceivedJuly 6, 1990; accepted after revision May 17, 1991)
ABSTRACT Failenius, K., 1992. Turbulence in flotation cells, a simplified approach. Int. J. Miner. Process., 34: 133-135. An easy way of deriving the equations for the turbulent fluctuation of velocityin a flotation cell is presented. The equations are of the same form as derived earlier, but the constants are different.
In the article Turbulence in flotation cells (Fallenius, 1987), a n u m b e r of relationships for flotation cells are derived. The derivation is rather long. The general form of these results can however be obtained by a very simple way based on a known equation o f turbulence theory (Rotta, 1972; Hinze, 1975 ): u ~3 E----A - L The right hand side represents the work done by the energy-containing eddies per unit time and mass (m E s - 3 ) . This is under certain conditions equal to the total turbulent dissipation ~ ( m 2 s - 3 ) . u ' = x / ( - ~ ) = t u r b u l e n t fluctuation o f velocity prevalent in the cell space ( m s -1 ). u = turbulence c o m p o n e n t (m s-1 ). The integral length scale L ( m ) is determined mainly by the larger energy-containing eddies, and can be held proportional to some dimension o f the flow, which determines the size of these eddies. A = numerical constant o f the order o f unity. The equation is valid a long enough time after the outset of the turbulence and while a high enough Reynold's n u m b e r Re prevails. Re = u' L v - 1. ~ is the kinematic viscosity (m 2 s - 1). Both conditions are met in this case (Rotta, 1972; Reynolds, 1974; Hinze, 1975 ).
0301-7516/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
134
k ~A~[[NIIiS
The dissipation must be equal to the power supplied to a unit mass: P ¢:= pV P = P o w e r (W); V=cell volume (m3);p=slurry density (kg m-- ~). In the cell space there is one large toroidal eddy above the radial jet and another below it (Fallenius, 1987 ). The former can be rather high. The height of the slurry space H (m) and its diameter T(m) determine the dimensions of these eddies and are candidates for L. The height of the eddy system is tried forL. Equating the expressions and substituting L = H yields:
u'=
(l)
b = numerical constant depending on the vessel form. This equation is of the same form as equation 19 in Fallenius (1987). Substituting the expression for P:
P= Cpp N3D 5 yields: ,,1/3
/~\2/3
D--diameter of the rotor (m); N = rotational speed of the rotor (s -1); C v = numerical constant. This equation is of the same form as equation 18 in Fallenius ( 1987 ). Selection of the height of the slurry space to represent L proved to be right. In principle it should be possible to select the length parameter based on statistical analysis of empirical data on flotation cells (Fallenius, 1987 ). The data available on a given family, however, indicates a strong mutual dependence between various geometrical parameters. This is due to the high degree of geometric similarity of cells of various size. This results in that no significant difference is obtained between parameters like H and T. In the experiment treated in FaUenius ( 1987 ); Cp= 3 and D~ T= 0.486. For a cylindrical vessel with H = T, b = 7t/4. IfA = 1, we obtain: u
U2-n \bAJ
=0.31
This is in accordance with the earlier results. The dissipation e, and therefore also u', is a local function, the maximum values of ~ being 10 to 30 times higher than the mean value. They are found downstream around the impeller (Schubert, 1989; Weiss and Schubert, 1989 ).
TURBULENCEINFLOTATIONCELLS
135
Equations 1 and 2 are the crucial results obtained in Fellenius ( 1987 ). The constants in these equations are different and useful.
REFERENCES Fallenius, K., 1987. Turbulence in flotation cells. Int. J. Miner. Process., 21: 1-23. Hinze, J.O., 1975. Turbulence. McGraw-Hill, New York, NY, pp. 221-226. Reynolds, A.J., 1974. Turbulent Flows in Engineering. Wiley, London, pp. 99-102. Rotta, J.C., 1972. Turbulente Str6mungen. B.G. Teubner, Stuttgart, pp. 90-114. Schubert, H., 1989. Role of turbulence in mineral processing unit operations. In: K.V.S. Sastry and M.C. Fuerstenau (Editors), Challenges in Mineral Processing. Proc. Symp. Honoring D.W. Fuerstenau, Berkeley. SME, Littleton, CO, pp. 272-289. Weiss, Th. and Schubert, H., 1989. Der Einfluss des Feinstkornes aufdie Turbulenz der Mehrphasenstr~Smung fest/fliissig. In: K. Leschonski (Editor), 4. Europ~iisches Symposium Partikelmesstechnik Niirnberg. pp. 679-693.
133
Turbulence in flotation cells, a simplified approach Kai Fallenius Harjuviita 16 A 9 SF-02110 Espoo, Finland (ReceivedJuly 6, 1990; accepted after revision May 17, 1991)
ABSTRACT Failenius, K., 1992. Turbulence in flotation cells, a simplified approach. Int. J. Miner. Process., 34: 133-135. An easy way of deriving the equations for the turbulent fluctuation of velocityin a flotation cell is presented. The equations are of the same form as derived earlier, but the constants are different.
In the article Turbulence in flotation cells (Fallenius, 1987), a n u m b e r of relationships for flotation cells are derived. The derivation is rather long. The general form of these results can however be obtained by a very simple way based on a known equation o f turbulence theory (Rotta, 1972; Hinze, 1975 ): u ~3 E----A - L The right hand side represents the work done by the energy-containing eddies per unit time and mass (m E s - 3 ) . This is under certain conditions equal to the total turbulent dissipation ~ ( m 2 s - 3 ) . u ' = x / ( - ~ ) = t u r b u l e n t fluctuation o f velocity prevalent in the cell space ( m s -1 ). u = turbulence c o m p o n e n t (m s-1 ). The integral length scale L ( m ) is determined mainly by the larger energy-containing eddies, and can be held proportional to some dimension o f the flow, which determines the size of these eddies. A = numerical constant o f the order o f unity. The equation is valid a long enough time after the outset of the turbulence and while a high enough Reynold's n u m b e r Re prevails. Re = u' L v - 1. ~ is the kinematic viscosity (m 2 s - 1). Both conditions are met in this case (Rotta, 1972; Reynolds, 1974; Hinze, 1975 ).
0301-7516/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
134
k ~A~[[NIIiS
The dissipation must be equal to the power supplied to a unit mass: P ¢:= pV P = P o w e r (W); V=cell volume (m3);p=slurry density (kg m-- ~). In the cell space there is one large toroidal eddy above the radial jet and another below it (Fallenius, 1987 ). The former can be rather high. The height of the slurry space H (m) and its diameter T(m) determine the dimensions of these eddies and are candidates for L. The height of the eddy system is tried forL. Equating the expressions and substituting L = H yields:
u'=
(l)
b = numerical constant depending on the vessel form. This equation is of the same form as equation 19 in Fallenius (1987). Substituting the expression for P:
P= Cpp N3D 5 yields: ,,1/3
/~\2/3
D--diameter of the rotor (m); N = rotational speed of the rotor (s -1); C v = numerical constant. This equation is of the same form as equation 18 in Fallenius ( 1987 ). Selection of the height of the slurry space to represent L proved to be right. In principle it should be possible to select the length parameter based on statistical analysis of empirical data on flotation cells (Fallenius, 1987 ). The data available on a given family, however, indicates a strong mutual dependence between various geometrical parameters. This is due to the high degree of geometric similarity of cells of various size. This results in that no significant difference is obtained between parameters like H and T. In the experiment treated in FaUenius ( 1987 ); Cp= 3 and D~ T= 0.486. For a cylindrical vessel with H = T, b = 7t/4. IfA = 1, we obtain: u
U2-n \bAJ
=0.31
This is in accordance with the earlier results. The dissipation e, and therefore also u', is a local function, the maximum values of ~ being 10 to 30 times higher than the mean value. They are found downstream around the impeller (Schubert, 1989; Weiss and Schubert, 1989 ).
TURBULENCEINFLOTATIONCELLS
135
Equations 1 and 2 are the crucial results obtained in Fellenius ( 1987 ). The constants in these equations are different and useful.
REFERENCES Fallenius, K., 1987. Turbulence in flotation cells. Int. J. Miner. Process., 21: 1-23. Hinze, J.O., 1975. Turbulence. McGraw-Hill, New York, NY, pp. 221-226. Reynolds, A.J., 1974. Turbulent Flows in Engineering. Wiley, London, pp. 99-102. Rotta, J.C., 1972. Turbulente Str6mungen. B.G. Teubner, Stuttgart, pp. 90-114. Schubert, H., 1989. Role of turbulence in mineral processing unit operations. In: K.V.S. Sastry and M.C. Fuerstenau (Editors), Challenges in Mineral Processing. Proc. Symp. Honoring D.W. Fuerstenau, Berkeley. SME, Littleton, CO, pp. 272-289. Weiss, Th. and Schubert, H., 1989. Der Einfluss des Feinstkornes aufdie Turbulenz der Mehrphasenstr~Smung fest/fliissig. In: K. Leschonski (Editor), 4. Europ~iisches Symposium Partikelmesstechnik Niirnberg. pp. 679-693.