Minerals Engineering 24 (2011) 42–49
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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
New model of screening kinetics Milan Trumic a,⇑, Nedeljko Magdalinovic b a b
University of Belgrade, Technical Faculty, VJ 12, Bor, Serbia Megatrend University, Faculty of Management, Park suma kraljevica bb, Zajecar, Serbia
a r t i c l e
i n f o
Article history: Received 24 April 2010 Accepted 16 September 2010 Available online 14 October 2010 Keywords: Mineral processing Screening Particle size
a b s t r a c t An understanding of screening requires a knowledge of screening kinetics. The new model of screening kinetics presented in this paper is theoretically described and experimentally proven. Samples of different raw materials, different in their density, bulk density, particle size distribution and particle shape, were used. The conducted experiments confirmed that the new model of screening kinetics successfully describes the process of screening of various raw materials even when they are exposed to a range of factors that most influence the screening process (dimensions of the screen, particle size distribution of raw materials, particle shape, and thickness of the bed on the screen). The advantage of this model of screening kinetics lies in its simplicity as it is characterized by the fact that it has only one parameter, i.e., (k) – the screening rate constant, which is experimentally measured. This constant can be measured on the basis of a single screening experiment. This fact is very significant for practical applications of the model to an industrial screen. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Previous studies have used two approaches to describe screening kinetics: a stochastic approach, and a kinetic approach. Within the stochastic approach (Baldwin, 1963; Brereton and Dymott, 1974; Brüderlein, 1972; Meinel and Schubert, 1972; Schlebusch, 1969), two models were studied – in terms of theory and experiment: (1) Free screening, where the particles do not have an impact on one another on the screening surface (Kluge, 1951; Klockhaus, 1952) which was further developed and modified by more authors – (Brüderlein, 1972; Meinel and Schubert, 1972; Schranz and Bergholz, 1954; Schubert, 1968). The main theoretical explanation was based on the assumption that the screening surface depends on the number of particles that make contact with the screening surface in time (t) and probability p = [1 (d/a)2] (Gaudin, 1939) that the particle moves through an aperture, where d = particle diameter and a = aperture size. (2) Tight screening with interaction between the particles was present during this operation. This model was analyzed in detail by Meinel and Schubert, (1971, 1972) by applying and solving the differential equation of Fokker, Plank and ⇑ Corresponding author. Tel./fax: +381 30421749. E-mail address:
[email protected] (M. Trumic). 0892-6875/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2010.09.013
Kolmogorov for a very restricted condition where one of them states that the path of a particle ends in direct contact with the screen surface, in other words the screening probability is p = 1. The theoretical interpretation of this screening model was studied and perfected by a number of authors. (Baldwin, 1963; Brereton and Dymott, 1974; Brüderlein, 1972; Ferrara and Preti, 1975; Meinel and Schubert, 1971, 1972; Schlebusch, 1969; Standish and Meta, 1985; Standish, 1985; Standish et al., 1986; Subasinghe et al., 1989a,b, 1990). Application of the tight screening model under real-time conditions requires changing and defining new limiting conditions, which further lead to extremely complex calculations. The other approach to the screening process, adopted by a number of authors (Andreev et al., 1966; Baldwin, 1963; Bodziony, 1961; Nepomnjašcˇij, 1962), is a kinetic one. More than one model (Andreev et al., 1966; Lynch, 1977; Nepomnjašcˇij, 1962) was suggested for the mathematical interpretation of screening kinetics. The one given in Eq. (1) is the simplest and the most appropriate for practical use.
dm ¼ km dt
ð1Þ
where dm is the rate particles of size (a + 0) pass through the screen dt in time t, k the screening rate constant, m the mass of the particles of size (a + 0) on the screen at time t, and a is the aperture size.
M. Trumic, N. Magdalinovic / Minerals Engineering 24 (2011) 42–49
43
The integral form of Eq. (1) would be:
m ¼ ekt m0
or 1
m ¼ 1 ekt m0
or
m0 m ¼ 1 ekt m0
ð2Þ
that is:
E ¼ 1 ekt
ð3Þ
where the screen undersize recovery (E) is ratio: the mass of undersize particles passing through the screen (m0–m) in time t and the mass of undersize in the feed (m0) in time (t = t0 = 0). In order to test Eq. (3), its logarithm can be used:
ln
1 ¼ kt 1E
ð4Þ
which presents the equation of a straight line in the coordinate sys1 tem t; ln 1E . Experiments conducted under different screening conditions and using various raw materials have shown that Eq. (4) does not always describe the screening process. Therefore, Perov (Andreev et al., 1966) suggested a more complex equation:
E ¼ 1 ek1 t
n
ð5Þ
In order to test Eq. (5), its logarithm can be used:
ln ln
1 ¼ n ln t þ ln k1 1E
ð6Þ
which presents an equation of a straight line in the coordinate sys1 tem ln t; ln ln 1E . Compared to the stochastic models, the kinetic ones are simple and have one or, at the most, two parametres which are to be identified by experiment. Figs. 1 and 2 are graphical illustrations of the kinetic models given by Eqs. (3) and (5). Observing the point arrangements on Figs. 1 and 2, which describe the screening kinetics of different raw materials by using the Eqs. (3) and (5), the deviation in relation to the straight line is clearly noticed. Results of the screening kinetics being approximated by a linear relationship, we get the correlation coefficient within the limits from 0.7343 to 0.9406. Somewhat a lower correlation coefficient
Fig. 2. Graphical illustration of the screening kinetics model described by Eq. (5).
suggests that the tested screening kinetics model does not describe the screening kinetics in the best way. Unfortunately, no serious work dealing with the study of the kinetics of sieving have been published in the last 10 years. 2. Theory of the new model of screening kinetics The model tested in this study was basically the kinetic model modified by the introduction of the screening probability coefficient (kp). The starting point of the new model of screening kinetics is that the speed of screening depends not only on the composition of a particle (a + 0) in the screen at a particular time, as assumed in model (1), but also on the change of probability of screening through time:
dm ¼ kmkp dt
ð7Þ
where dm is the rate particles of size (a + 0) pass through the screen dt in time t, k the screening rate constant, m the mass of a particle of size (a + 0) on the screen at time t, a the aperture size, and kp is the change of the probability of screening coefficient. The probability of screening depends on the relation between the diameter of the particle (d) and the aperture size (a):
p¼1
Fig. 1. Graphical illustration of the screening kinetics model described by Eq. (3).
2 d a
ð8Þ
During screening, the sequence of particles going through the screen is: ‘small’, (particles with the middle, diameter (ds) smaller than 0.75a, where a is the aperture size, 0 < ds < 0.75a), ‘small-tolarge’ (particles with mean diameter (dsl) larger than 0.75a and smaller than a, 0.75a < dsl < a), and then ‘large’ (particles with the middle diameter (d1) larger than a and smaller than 1.5a,
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M. Trumic, N. Magdalinovic / Minerals Engineering 24 (2011) 42–49
a < d1 < 1.5a). In other words, the relation d/a is increased for the remaining particles (a + 0), while the probability of screening decreases. The mathematically defined change of the probability of screening enables a more accurate model of the screening kinetics to be defined, which describes the screening process from beginning to end. Searching for a mathematical definition which describes the change of probability of screening, the starting point was the assumption that the coefficient of change of probability of screening (kp) can be defined in the following way:
kp ¼
m m0
That is:
E ¼ kt 1E
ð16Þ
Eq. (16) represents the new model of screening kinetics. It represents E the equation of a straight line in the coordinate system t; 1E , (Fig. 4). Based on Fig. 4, the following can be concluded:
k ¼ tg a ¼
E2 E1 a 1E2 1E1 ¼ b t2 t1
ð9Þ
where m is the mass of particles of size (a + 0) on the screen at time t, a the aperture size, m0 the initial mass of a particle (a + 0). This assumption is the result of the experiment shown in Fig. 3. Fig. 3 clearly shows that the shape of the change of the probability of screening depends on the proportions of ‘small’ particles in the particle size range (a + 0). In other words, the speed of decreasing probability at the beginning grows as the proportions of ‘small’ particles in the particle size range (a + 0) increases. Substitution of (kp) from Eq. (9) into the differential Eq. (7), gives:
dm m2 ¼ k dt m0
ð10Þ
By integrating Eq. (10), the following equation is obtained:
Z
m
m2 dm ¼
m0
k m0
Z
t
dt
ð11Þ
Fig. 4. Graphics of the function (16).
0
That is:
m0 ¼ 1 þ kt m
ð12Þ
m 1 ¼ m0 1 þ kt
ð13Þ
The screen undersize recovery (E) may be given by the following formula:
E¼1
m m0
ð14Þ
Substitution of (m/m0) from Eq. (13) into Eq. (14) gives:
E¼1
1 1 þ kt
ð15Þ
0,8
Proportions of ‘small’ particles in the
0,7
particle sizeparticles range (-a+0) Participation of ‘light’ in lower class of l
0,6
0%
20%
50%
70%
kp
0,5 0,4 0,3 0,2 0,1 0
0
50
100
150
200
250
300
Time, sec Fig. 3. Graphics of the coefficient of change of probability of screening magnesitelarge on a screen the opening of which was 6.68 mm, with different proportions of ‘small’ particles in the particle size range (a + 0).
Fig. 5. Vibrating sieve shaker.
ð17Þ
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M. Trumic, N. Magdalinovic / Minerals Engineering 24 (2011) 42–49
The first and most important characteristic of the new model of screening kinetics is its simplicity. It has only one parameter (k), which is identified by experiment. This parameter (k), which is the screening rate constant, can be obtained based on the results of only one screening process. This fact is very important for practical application of the model to the industrial screen, where conditions for conducting experiments are directly connected with numerous difficulties and limitations.
3. Experimental work An ILM-LABOR sieve shaker was used to study sieving kinetics in the laboratory (Fig. 5). The sieving was carried out dry, using Tyler series sieves. Screening kinetics using the sieve shaker was studied on eight raw material samples: metal balls, quartz sand, mica, small and large magnesite, coal, copper ore and gravel. The particle size distribution, density and embankment density of screening are given in Table 1. The samples were in a loose condition and filled the whole area of the screen in one layer and in many characteristic layers. Each experiment of the screening kinetics in the laboratory used one sample, which was removed from the sieve shaker after a specific period in order to define the screened mass and then returned to the sieve shaker for further screening. The data concerning the content of the particle size range (a + 0) were obtained in the following way: screening was continued to the moment when no more than 0.1% of mass per minute allowed though the screen. The screening kinetics were studied under semi-industrial conditions using a continuous semi-industrial vibration-screen of the brand ‘‘DENVER” (Fig. 6), the dimensions of which were B L = 300 600 mm. The vibration-screen has circular and ellipsoid vibrational movements. The screening surface used in the experiments was square with openings of 5 mm, 8 mm and 10 mm. The inclination of the screen was 12°. The screening was carried out dry. For the purposes of examining the screening kinetics, the screening area was devided into five sections: 100, 200, 300, 400, and 600 mm. For each experiment under semi-industrial conditions, one sample was allowed through the first section (0–100 mm), while
Fig. 6. Continuous semi-industrial vibration-screen of the brand ‘‘DENVER”Fig. 7 Particle shape of the raw material samples employed in the laboratory experiments.
the others were covered. After measurement, the sample was allowed through the first two sections (0–200 mm), while the others were covered. The same procedure was performed until all five sections (0–600 mm) had been uncovered. The screening kinetics on the semi-industrial vibration-screen was studied on samples of six raw materials: calcite, andesite, limestone, coal, copper ore and gravel.
Table 1 Characteristics of the samples for the laboratory experiments. Particle size range (mm)
Copper ore
Large magnesite
Small magnesite
Quartz sand
Metal balls
Mica
W (%)
Gravel D (%)
W (%)
Coal D (%)
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
13.33 + 9.52 9.52 + 6.68 6.68 + 4.699 4.699 + 3.327 3.327 + 2.362 2.362 + 1.651 1.651 + 1.168 1.168 + 0.833 0.833 + 0.589 0.589 + 0.417 0.417 + 0.295 0.295 + 0.208 0.208 + 0.149 0.149 + 0.106 0.106 + 0
13.6 13.3 10.4 8.4 8.9 14.1
100.0 86.4 73.1 62.7 54.2 45.3
38.7 20.9 15.1 7.4 5.7 12.2
100.0 61.3 40.4 25.3 17.9 12.2
30.2 27.2 10.7 9.3 5.5 17.1
100.0 69.8 42.6 31.9 22.6 17.1
18.7 37.7 18.6 9.9 6.2 8.9
100.0 81.3 43.6 25.0 15.1 8.9
31.2
31.2
– – – – 15.8 22.3 15.6 11.9 9.4 25.0
– – – – 100.0 84.2 61.9 46.2 34.4 25.0
– – – – – – – 2.3 9.5 24.8 24.2 19.5 19.7
– – – – – – – 100.0 97.7 88.2 63.4 39.2 19.7
– – – – – – – 13.3 22.8 22.0 21.7 14.5 5.7
– – – – – – – 100.0 86.7 63.9 41.9 20.2 5.7
– – – – – – – – – 100.0 91.0 70.8 46.6 27.6 14.6
100.0
100.0
100.0
100.0
100.0
100.0
100.0
– – – – – – – – – 9.0 20.2 24.2 19.0 13.0 14.6 100.0
Density (kg/m3)
2672
1494
2710
2922
2922
2630
4986
2655
Bulk density (kg/m3)
1754
792
1531
1500
1347
1450
3234
998
Note: W – particular weight percent. D – cumulative weight percent undersize.
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M. Trumic, N. Magdalinovic / Minerals Engineering 24 (2011) 42–49
The particle size distribution, density and embankment density of the screening are given in Table 2. The experiments were conducted on raw materials the particles of which differed in shape and size. The shapes of the different particles employed in the laboratory and semi-industrial experiments are given in Figs. 7 and 8, respectively.
4. Results The testing of the new model of screening kinetics was conducted on dry samples of different raw materials, with particles of different shapes and sizes, different thicknesses of the bed on the screen, different aperture sizes and different proportions of ‘small’ and ‘small-to-large’ particles. This means that all the most important factors were varied in the screening process. The results of the screening kinetics on the laboratory discontinual screens are graphically shown in the coordinate system E t; 1E in Figs. 9–11. The results of the screening kinetics on the semi-industrial vibration-screen of continual effect are graphically shown in the E coordinate system L; 1E in Fig. 12.
The straight-line sequence of experimental points in all graphs, for all samples under various screening conditions, with very high correlation coefficients R = 0.980–0.999, shows that the new model of screening kinetics describes the screening process very well, both on the laboratory and semi-industrial vibration-screens. The assumption based on the experiments that the screening speed is in direct proportion to the mass of the particles (a + 0) that is on the screen at a given point in time and the coefficient of change of screening probability, was proven to be absolutely correct. Based on the experiment results, the screening rate constant (k) was determined using the least squares method. Table 3 shows numerical values of the screening rate constant for experiments on the laboratory vibration-screen and Table 4 for those on the semi-industrial screen. The only parameter that affects the screening speed in this new model is the screening rate constant (k). Therefore, the dependency of the constant on the most important factors derived from the particle size distribution, particle shape as well as the conditions where screening takes place will be described in more detail. The first conclusion to be drawn is that the screening rate constant varies for different raw materials although the aperture size
Table 2 Characteristics of the samples for the semi-industrial experiments. Particle size range (mm)
17.00 + 13.33 13.33 + 9.52 9.52 + 6.68 6.68 + 4.699 4.699 + 3.327 3.327 + 2.362 2.362 + 1.168 1.168 + 0
Gravel
Coal
Copper ore
Limestone
Andesite
Calcite
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
W (%)
D (%)
– 13.6 13.3 10.4 8.4 54.2
– 100.0 86.4 73.1 62.7 54.2
16.8 22.2 13.8 8.1 6.5 32.7
100.0 83.2 61.0 47.2 39.1 32.7
20.0 14.3 12.1 9.4 7.6 36.6
100.0 80.0 65.7 53.6 44.2 36.6
14.4 22.5 15.8 11.8 9.1 26.4
100.0 85.6 63.1 47.3 35.5 26.4
15.3 34.6 16.9 8.9 6.0 18.3
100.0 84.7 50.1 33.2 24.3 18.3
9.9 12.1 18.8 36.8 19.1 3.3
100.0 90.1 78.0 59.2 22.4 3.3
100.0
100.0
100.0
100.0
100.0
Density, (kg/m3)
2672
1494
2710
2600
3015
2644
Bulk density (kg/m3)
1750
790
1530
1530
1570
1490
Note: W – particular weight percent. D – cumulative weight percent undersize.
Fig. 7. Particle shape of the raw material samples employed in the laboratory experiments.
100.0
M. Trumic, N. Magdalinovic / Minerals Engineering 24 (2011) 42–49
47
Fig. 8. Particle shape of the raw material samples employed in the semi-industrial experiments.
and copper ore (mass of 600 g) which have approximately the same density, on a screen with an aperture size of 4.699 mm, the screening rate constant (k) will be 3.268 for gravel (round particles) and 0.389 for copper ore (spiky shape).
Fig. 9. Screening kinetics of quartz sand (the mass was 100 g).
and other conditions may be the same. Three factors are important here: particle shape, proportions of ‘small’ and ‘small-to-large’ particles in the particle size range (a + 0) (particle size distribution of raw materials) and density of raw materials, which affects the dynamic movement of the particles along the screen. As far as particle shape is concerned, the experiments clearly showed that the rate constant is lower when the particles are not round (as was the case with coal, mica, magnesite and copper ore), and that it is significantly higher when they are round (gravel, metal balls). This can be illustrated by the following example: if we take gravel
Fig. 10. Screening kinetics of quartz sand (the mesh size was 0.295 mm).
Fig. 11. Screening kinetics of gravel, coal, magnesite-large, copper ore (the mass was 600 g, mesh size was 3.327 mm).
Fig. 12. Screening kinetics of limestone, coal, copper ore, andesite, calcite (mesh size 8 mm).
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M. Trumic, N. Magdalinovic / Minerals Engineering 24 (2011) 42–49
Table 3 Numerical values of the screening rate constant (k) from the experiments on the laboratory vibration-screen. Mesh size (mm)
Gravel Mass, g
Coal Mass, g
600 9.52 6.68 4.699 3.327 2.362 1.651
900
3.2679 1.1058 1.0091 0.8127
3.8010 1.3484 0.8301 0.5616
1200
1.6642 0.8533 0.8130 0.5729
Magnesite small Mass, g
2.362 1.651 1.168 0.833 0.589 0.5 0.417 0.295 0.208 0.149
Copper ore Mass, g
300
450
600
0.3730 0.2519 0.2348
0.3278 0.3075 0.1210 0.0982
0.2397 0.0730 0.0690 0.0661
Quartz sand Mass, g
300
500
800
0.2686 0.1699 0.1340
0.2819 0.1582 0.1554 0.1315
0.2036 0.1683 0.1615 0.0935
Magnesite-large Mass, g
400
600
800
600
900
1200
0.6338 0.3906 0.6612
0.3656 0.6717 0.3894 0.6598
0.3053 0.4295 0.3485 0.5237
0.3749 0.1830 0.3057 0.1357
0.4929 0.1292 0.1840 0.1108
0.3257 0.0912 0.0923 0.0793
50
100
200
0.0583 0.0526 0.0419
0.0569 0.0409 0.0401
0.0546 0.0398 0.0381
Metal balls Mass, g
100
300
500
0.4881 0.3455 0.1525 0.1093 0.0806
0.2338 0.2067 0.0810 0.0800 0.0541
0.1876 0.1805 0.0764 0.0689 0.0366
Mica Mass, g
400
600
800
2.7400 0.8997 0.5024 0.3569 0.1625
1.7872 0.5406 0.4046 0.3261 0.1330
1.5239 0.8365 0.4099 0.2635 0.0916
Table 4 Numerical values of the screening rate constant (k) from the experiments on the semi-industrial vibration-screen. Mesh size (mm) 10 8 5
Gravel
Coal
Copper ore
0.0920
0.2361 0.3191
0.2639 0.5579
Fig. 13. Graphical presentation of the screening kinetics of large magnesite on a screen the opening of which was 6.68 mm, for different amounts of ‘small’ particles in the particle size range (a + 0).
The second conclusion to be drawn is that the greater is the proportions of ‘small’ particles in the raw material, the higher will be the screening rate constant. This can be illustrated by the example of the screening of large magnesite on a screen the opening of Table 5 Screening rate constant (k) of magnesite-large. Content of ‘small’ particles (%)
Screening rate constant (k)
Correlation coefficient (R)
0 20 50 70
0.085 0.089 0.102 0.345
0.997 0.998 0.996 0.996
Andesite 0.0637 0.1380
Limestone
Calcite
0.1055 0.2337
0.0900
which was 6.68 mm with a changeable content of ‘small’ particles in the particle size (a + 0) ranges from 0% to 70%. The graphical illustration of these results is shown in Fig. 13, while Table 5 contains the values the constant (k) for different proportions of ‘small’ particles. The change of the content of ‘small’ particles in the particle size (a + 0) from 0% to 50% causes a relatively small increase in the constant (k). The constant (k) increased more when the content of ‘small’ particles was increased from 50% to 70%. When dealing with raw materials of lower density with the other conditions being constant (the same aperture size, the same thickness of the bed on the screen and the same particle shape), the constant (k) is lower. This conclusion can clearly be illustrated if the constant for coal and that for copper ore on the laboratory screen the opening of which was 6.68 mm are compared (Table 3). The particles of coal and copper ore had more or less the same shape (Fig. 7). For approximately the same thickness of the bed (coal 300 g, copper ore 600 g), the screening rate constant for coal was 0.373 and for copper ore 0.672. The larger constant for copper ore is the result of the greater density of copper ore in comparison to that of coal. The greater density results in more dynamic movement of the particles, which enables the particles (a + 0) to move faster through the larger particles and finally to slip through the screen. Generally, the constant decreases when the thickness of the bed is greater. This is because particles (a + 0) take more time to come to the screen if the bed on the screen is thicker. Certain results of the conducted experiements show some exceptions to this rule. For example, in the case of large magnesite, copper ore and gravel, when aperture size were a = 9.52, 6.68, and 4.699 mm, the coefficient was here lower than when a thicker bed was screened. This can be explained by the fact that a smaller thickness means more dynamic movement of the particles and that further means lower
M. Trumic, N. Magdalinovic / Minerals Engineering 24 (2011) 42–49
screening speed. This was observed during conducting the experiments. For the same raw material when the proportions of ‘small’ particles was the same, the constant (k) decreased as the aperture size became smaller. This is understandable and expected if one takes into account that a smaller opening means a lower coefficient of the open area. Theoretically, the screening probability is proportion to the coefficient of the open area. The results of certain experiments (copper ore, large magnesite) showed exceptions to this rule. For example, for copper ore of mass m = 600 g, and an aperture size a = 9.52 mm, k was 0.366, while k was 0.672 on a screen with opening a = 6.68 mm. This may occur when the negative effect of the lowered coefficient of the open area is lower than the positive effect of the greater proportions of ‘small’ particles in the particle size range (a + 0). In the case when the opening was a = 9.52 mm, the proportions of ‘small’ particles was 61%, and for the opening a = 6.68 mm, it was 75%, i.e., much more, which in accordance with the results given in Table 3. The above drawn conclusions based on the results of experiments conducted using the laboratory vibration-screen are also valid for the results of the experiments in the semi-industrial vibration-screen. 5. Conclusion The research conducted in this paper gives the theoretical basis of a new model of screening kinetics, which was also experimentally proven on laboratory and semi-industrial vibration-screens, and its differential equation is as follows:
dm ¼ kmkp dt Its integral form is:
E¼
kt 1þkt
where dm is the rate particles of size (a + 0) pass through the screen dt in time t, k the screening rate constant, (1/s), kp the change of probability of screening coefficient, t the screening time (s), E is the screen undersize recovery (unit parts) All the conducted experiments showed a significant straightline sequence of experimental points of a high correlation coefficient (R = 0.98–0.999), which indicates that the new model of screening kinetics describes well the process of screening different types of raw materials under a range of the most influential factors (dimensions of the screen, particle size distribution of the raw material, particle shape and thickness of the bed on the screen). One of the advantages of the new model lies in its simplicity which can be seen in the fact that it has only one parameter (k) which is obtained by experiment. The screening rate constant (k) can be obtained from the results of only one screening experiment. This fact is extremely significant for practical application of the model to an industrial screen, where the conditions for conducting experiments are connected with numerous difficulties and limitations. The practical value of the model, applied to the industrial screen, may be seen in the possibility of optimization and automatization of the screening process in the sense that the capacity of a screen for a given screen undersize recovery (E) can be determined, and vice versa. In that way, it is possible to manage the screening process under given conditions with the given influential characteristics of the raw materials. The only parameter, that in this model predetermines the screening speed is the screening rate constant (k). The results show that this constant depends on: the dimensions of the aperture size, the particle size distribution of the raw material, or, more accu-
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rately, on the proportions of ‘small’ and ‘small-to-large’ particles in the particle size range (a + 0), the thickness of the bed on the screen, the particle shape and the density of the raw material). For the same raw material, the rule is that for approximately the same proportions of ‘small’ particles in the particle size range (a + 0), the screening rate constant (k) as the aperture size becomes smaller. This is expected, bearing in mind that the smaller the opening is, the lower is the open area coefficient, and that the probability of screening is in proportion to the open area coefficient. The greater the proportions of ‘small’ particles in the lower class of largeness, the higher the screening rate constant (k). Changing the content of ‘small’ particles in the lower class of largeness from 0% to 50% causes a relatively insignificant increase in the screening rate constant (k). A relatively greater increase in the constant occurs when the content of ‘small’ particles is over 50%. When dealing with raw materials of greater particle density when the other screening conditions are the same, the screening rate constant (k) is higher. The greater particle density causes more dynamic movement of the particles, which enables particles (a + 0) to move faster between the larger particles and slip through the screen. The screening rate constant (k) decreases as the mass of the initial sample, i.e., the thickness of the bed on the screen, increases. This occures because the particles (a + 0) take more time to move through larger particles and slip through the screen when the thickness on the screen is greater. References Andreev, S.E., Zverevic, V.V., Perov, V.A. 1966. Droblenie, izmel’cˇenie i grochocˇenie poleznych iskopaemych, Izdatel’stvo ‘’Nedra’’, Moskau. Baldwin, P.L., 1963. The continuous separation of solid particles by flat deck screens. Transactions of the Institution of Chemical Engineers and the Chemical Engineer 41, 255–263. Bodziony, J., 1961. Mathematische erfassung des siebvorganges. Bergakademie 2, 90–101. Brereton, T., Dymott, K.R. 1974. Some factors which influence screen performance. Proceedings of 10th International Mineral Processing Congress, London, pp. 181–194. Brüderlein, J., 1972. Bewegungsvorgänge auf siebmaschinen. Aufbereitungs Technik 7, 401–407. Ferrara, G., Preti, U.A. 1975. A contribution to screening kinetics. Eleventh International Mineral Processing Congress, Cagliari, pp. 183–217. Gaudin, A.M., 1939. Principles of Mineral Dressing. Mc-Graw-Hill, New York. Klockhaus, W., 1952. Fördergeschwindigkeit von Schwingrinnen und Schwingsieben. Erdöl und Kohle 5, 493–495. Kluge, W., 1951. Neuzeitliche Siebmaschinen für die Aufbereitungs Technik. Erdöl und Kohle 11, 705–711. Lynch, A.J., 1977. Mineral Crushing and Grinding Circuits, Their Simulation, Optimization, Design and Control. Elsevier, New York. Meinel, A., Schubert, H., 1971. Principles of fine screening. Aufbereitungs Technik 3, 128–133. Meinel, A., Schubert, H., 1972. Über einige Zusammenhänge zwichen der Einzelkorndynamik und der stochastichen Siebertheorie bei der Klassierung auf Stösselschwingsiebmaschinen. Aufbereitungs Technik 7, 408–416. Nepomnjašcˇij, E.A., 1962. Nektorye resul’taty teoreticˇeskogo analiza processa grochocˇenija. Obogašcˇenie rud 5, 29–35. Schlebusch, L., 1969. Dünnschichtsiebung und Systematik der direkterregten Siebe. Aufbereitungs Technik 7, 341–348. Schranz, H., Bergholz, W., 1954. Die bewegungsvorgänge bei wurfsieben. Bergbauwissenschaften 8, 223–234. Schubert, H. 1968. Aufbereitung fester mineralischer Rohstoffe, Band I, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig. Standish, N., Meta, I.A., 1985. Some kinetic aspects of continuous screening. Powder Technology 41, 165–171. Standish, N., 1985. The kinetic of batch sieving. Powder Technology 41, 57–67. Standish, N., Bharadwaj, A.K., Hariri-Akbari, G., 1986. A study of the effect of operating variables on the efficiency of a vibrating screen. Powder Technology 48, 161–172. Subasinghe, G.K.N.S., Schaap, W., Kelly, E.G., 1989a. Modelling the screening process-an empirical approach. Minerals Engineering 2 (2), 235–244. Subasinghe, G.K.N.S., Schaap, W., Kelly, E.G., 1989b. Modelling the screening process: a probabilistic approach. Powder Technology 59, 37–44. Subasinghe, G.K.N.S., Schaap, W., Kelly, E.G., 1990. Modelling screening as a conjugate rate process. International Journal of Mineral Processing 28 (3–4), 289–300.