Minerals
08924875(01)ooooc1
Vol. 14. No. 3, pp. 349-357,200l 0 2001 Published by Elsevier Science Ud All rights reserved 0892-6875/01/$ - see. front matter
Engineering,
PREDICTION OF TI-IESETTLING VELOCITY OF IRREGULARLY SHAPED PARTICLES K.G. TSAKALAKISq and G.A. STAMBOLTZIS’ ‘j Dept. of Mining and Metallurgical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Str., 15780-Athens, Greece, E-mail:
[email protected] 8 Ex-Associate Professor, National Technical University of Athens (Received 9 October 2000; accepted 18 December 2000)
ABSTRACT This work presents a new empirical relationship between Reynolds number Re, and Archimedes number Ar. From this relationship it is possible to calculate the free settling (terminal) velocity of irregularly shaped (e.g. crushed) solids in water. Experimental terminal velocities for various size fractions from earlier works were correlated with the new equation. The data refer to crushed quartz and galena particles. The new relationship gives for a very wide range of Archimedes numbers cflow regimes) a good correlation between experimental and calculated terminal velocities. Comparisons with the Ganguly relationship also showed a good agreement for the prediction of the terminal velocities. The current work shows that the terminal velocities in water can be calculuted with a single equation for various irregularly shaped solids and for various size fractions. 0 2001 Published by Elsevier Science Ltd. All rights reserved.
Keywords Classification; dense medium separation; gravity concentration; thickening; modelling
INTRODUCTION There are many processes in various engineering fields in which the accurate calculation of the free settling velocity is necessary. In mineral processing operations (e.g. flotation, thickening, dense media separation, etc.), where solids are settled, the behavior of these solids in the pulp due to its own weight and volume is of great importance. Many attempts have been made by various researchers in the last fifty years to propose suitable equations for the calculation of free settling velocity of solids in liquids or in a gaseous medium. The equations derived were theoretical, empirical and graphical. A detailed description and analysis was made in the paper published by Ganguly (1990). In the theoretical equations the particles settled are considered to be spherical. The empirical equations are different for various flow regimes (Archimedes or Reynolds numbers) and involve in its form the voidage factor &. The modified equations used for the prediction of Ret and u, refer to the extrapolated values, where the voids fraction E = 1.
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The theoretical and empirical equations do not include the shape factor or sphericity A. In the graphical methods the settling velocity is determined from plots of voids fraction E versus Rep (Reynolds number corresponding to the velocity of a single particle), extrapolated at E = 1 (Ganguly, 1990). But in these methods considerable error is encountered as a result of personal observations. In his paper, Ganguly (1990) proposed a new empirical equation, which was derived from experimental data. It has the form:
Re, = 0,103x(Ar)0~8M~(4~)0~745
(1)
where Re, is the Reynolds number at settling velocity of the particle and Ar is the Archimedes number. In Eq. (l), Ganguly introduced the shape factor & as a parameter of the equation. The ranges of Ar vary from 53 to 3761, and six different kinds of minerals were examined. The author calculated the standard deviation given by Eq. (1) in the + 15.2% range. In our previous paper (Stamboltzis and Tsakalakis, 1994), we used the same data as in the present paper and proposed equations for the calculation of free settling velocity in water. The equations were of the power form: u, =a-Pm
(2)
where U, is the free terminal velocity of crushed particles in water, a is a coefficient, p is the average particle size fraction of the particles and m is a coefficient depending on the flow regime (laminar, transitional or turbulent). The calculated values of m were: 1. 2. 3.
m = 2 for the laminar flow regime (Stoke’s law, U, a,@) 0.5~ m c2 for the transitional flow regime (Newton-Rittinger equation, U, ccpm) and m = 0.5 for the turbulent flow regime (Newton equation, at afloV5).
The above Eq. (2) was also recast and expressed as: u, =
