modelling the crushing-sizing procedure of industrial gyratory crushers

Presented at the 11th European Symposium on Comminution, October 9-12, 2006, Budapest, Hungary, www.comminution2006.mke.org.hu

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Modelling the crushing-sizing procedure of industrial gyratory crushers

K.G. Tsakalakis

School of Mining and Metallurgical Engineering, National Technical University of Athens, 15780-Zografou, Athens, Greece

e-mail: [email protected]

Keywords: Crushing process, Particle size distribution, modelling

Abstract: In the present paper we propose empirical models for the description of the particle size-distributions of industrial gyratory-crusher products. For the development of the models, typical product size-distributions of gyratory crushers were used. Various curves from two sources referred to “scalped” and “run of mine” feeds were extensively analyzed. From the whole work it was proved that the product size distribution (cumulative fraction passing through a square screen aperture of size x) can be satisfactorily described from exponential equations. The models give the cumulative fraction passing P as a function of the product x and the open-side setting (discharge opening) Oss of the gyratory crusher.The current work shows that the proposed functions can be used as alternatives to the RRSB and G-G-S models for the prediction of the product size-distributions and the improvement of crushing-sizing circuits during the design procedure.

Introduction

The known functions used for the description of the particle size distributions of primary crushing products are [1], [2], [3]:

1. The Gates-Gaudin-Schuhmann (G-G-S) power function given by:

m

k

xP

, 0≤x≤k, m>0, (1)

where P is the cumulative fraction passing from a square screen aperture size x,k is the “size modulus” of the distribution (i.e. k is the maximum particle size,[P(k) = 1] and m is a parameter called “distribution modulus” (slope) of the particle size distribution.

2. The Rosin-Rammler-Sperling-Benett (RRSB) exponential function given by:

n

k

x

eR

, (2)

where R is the cumulative fraction of the screened material retained on a square screen aperture of size x, k is the “size modulus” of the distribution [R(k) = e-1= 0.3679] and n is the “distribution modulus” (slope) of the particle size distribution.


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It has already been proved [4], [5], [6] that the graphical representation of the product size-distributions of a crushed material is actually given, in both RRSB and G-G-S graphs, from a continuous line (curve) approached by two intersected lines as shown in Figs. 1 and 2. The two lines show quite different slopes (distribution moduli or uniformity factors). They represent the coarse part (“residue”) and the fine fraction (“fragments”) of the size distribution, respectively.

Fig. 1. Typical RRSB plots of gyratory crusher products.

Fig. 2. Typical G-G-S plots of gyratory crusher products.


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Table 1. Percent of product passing a square opening equal to the open side-setting of the crusher [7]

According to Table 1, the hardness of the crushed material, expressed through its Bond work-index (wi), affects significantly the percentage of the product passing the open side-setting Oss

(discharge opening) of the crusher. This percentage ranges from 75-90% from hard to soft materials, respectively. In the present paper we consider 80% by weight of the product, as the percent passing the open side-setting of the crusher. This means, that only 80% of the product passes a square opening equal to the open side-setting of the crusher, with the remainder being “oversize” or “uncrushed” material.

Mathematical treatment of size-distributions data

The equations tested here for the description of the crushed product size-distributions were expressed by the general form:

),(

)(

1),( ssOx

xf

ss eOxPP

(3)where P is the cumulative percentage passing from a square opening x, Oss is the open side-setting of the crusher, f(x) is a function of x and φ (x, Oss) is a function of x and the open side setting Oss.Assuming that the functions f(x) = x and φ(x) = a x+ b, Eq. (3) becomes:

bax

x

ss eOxPP

1),( (4)

Data [8], obtained from Fig. 3, were used for the determination of the parameters a and b. This can be easily achieved through the linearized form of Eq. (4). For this purpose, least-squareslinear regression analysis is applied to pairs (P, x) received from Fig.1.Then, correlating the various values of the open side-setting Oss with the parameters a, b previously predicted, the equations configured are:

ssss OOdca ln83.2954.8ln11 (5)

ssss OOdcb 017.63.16322 (6)

The coefficient of determination is r2 = 0.99718 for Eq. (5) and the corresponding correlation coefficient of Eq. (6) is r = 0.9997.

Finally, substituting Eqs. (5) and (6) into Eq. (4), yields:

Product curve selection(High energy work index, wi)

Percent passing

Soft (wi ≤ 10) 90

Medium (10 ≤wi ≤ 15) 85

Hard (wi ≥ 15) 75


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Fig. 3. Typical product size-distribution curves of a gyratory crusher fed with screened or “scalped” feed, Source: Unit Operations [8].

Fig 4. Typical product size-distribution curves of a gyratory crusher fed with “run of mine” feed, after Nordberg [9]

)3.163017.6()ln83.2954.8(1),(

ssss OxO

x

ss eOxPP (7)

Applying Eq. (7) for the prediction of the cumulative percent passing P (%) as a function of Oss and x, we designed Fig. 5. From this figure, a good correlation between observed and predicted P values is obvious. Eq. (7) describes efficiently the 80-85% of the whole product size distribution, when “scalped” material is fed to a gyratory crusher. However, when P = 99.9 % or 99.99%, solving Eq. (7) for x, a very good approximation of the maximum particle size (P ≈ 100%) of the product can also be achieved.

The same procedure, as that previously described, is followed for the curves shown in Fig. 2. These curves represent the product size distributions, when pure “run of mine” feed is crushed in gyratory crushers. The typical RRSB and C-G-S plots, corresponding to these curves and indicated with “2”, are shown in Figs. 1and 2.Among the equations tested, the better correlation was given by:

cbxax

x

ss eOxPP

2

2

1),( , (8)


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y = 1,0234x + 0,0027

r 2 = 0,9987

0

1020

3040

5060

7080

90

0 10 20 30 40 50 60 70 80 90

Cumulative percent passing P (%) predicted from Eq. (5)

Cu

mu

lativ

e p

erc

en

t pa

ssin

g

P (

%)

ob

serv

ed

in F

ig. 1

Cum. passing percentages compared

Solid line corresponds to y=x

where a, b and c are parameters.

Afterwards, Eq. (8) was transformed as follows:

)/1()/1()1

1ln( 2xcxba

P

, (9)

Applying multiple linear regression analysis to Eq. (9), the parameters a, b and c are predictedIn this case, it was proved that a = -0.5378 (constant), while b, c are given from:

713.6164.1 ssss OlkOb (10)with a correlation coefficient r = 0.99958 and

58.27498.9 ssss OnmOc (11)The correlation coefficient is r = 0.9949.

Taking into account Eqs. (10) and (11), Eq. (8) becomes:

)58.27498.9()713.6164.1(5378.0 2

2

1),(

ssss OxOx

x

ss eOxPP (12)

Eq. (12) is used for the efficient description of the 80% of the product size distribution corresponding to the coarse (≥12-20 mm) material.In Figs. 5 and 6 a comparison is made between the values predicted from the proposed models and those calculated from Figs. 1 and 2. From the “distribution” of the points around the lines of comparison (y=x, angle 45o), the good agreement of the results, obtained from Eqs. (7) and (12) and those shown in Figs. 3 and 4, is obvious.

Fig. 5. Comparison of the cumulative percent passing predicted from Eq. (5) with that observed in Fig. 1, for “scalped” feed crushed in gyratory crushers.

Fig. 6. Comparison of the cumulative percent passing predicted from Eq. (9) with that observed in Fig. 2, for “run of mine” feed crushed in gyratory crushers.

y = 1,0269x - 1,6444

r 2 = 0,9993

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Cumulative percent passing P(%) predicted from Eq. (9)

Cum

ulat

ive

perc

ent p

assi

ng

obse

rved

in F

ig. 2

Cum. passing percentages compared

Solid line corresponds to y=x


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CONCLUSIONS

In the present work, exponential equations were derived, which give the cumulative percent passing P as a function of: the open side-setting Oss (mm) of the gyratory crusher and the product size x (mm). The proposed models are used for the description of the product size distributions in cases, where the crusher is fed with either “scalped” or “run of mine” material.From this work it was shown that these equations approach very well the values observed in Figs. 3and 4.The proposed models are continuous functions which describe the product very well, especially in the central “regions” of the distribution, where the known RRSB and G-G-S distributions usually fail. However, differences are encountered regarding the distribution lower end e.g. in the “fine” region of the product size distribution. Their main advantage is that they don’t require 4 parameters (2 pairs for the two lines) for the description of the whole distribution.

ACKNOWLEDGEMENTS

The author is greatly indebted to Prof. E. Mitsoulis for helpful discussions and suggestions.

REFERENCES

1. Taggart, A.F., 1954, Handbook of Mineral Dressing, John Wiley & Sons, Inc., New York p.4-18 to 4-36 & 19-145 to 19-150.

2. Beke, B., 1964, Principles of Comminution, Akadèmiai Kiadó, Budapest, p.23-31.3. Arbiter, N., Harris, C.C. and Stamboltzis G.A., 1969, Single Fracture of Brittle Spheres,

AIME Transactions, Vol. 244, pp.118-131.4. Stamboltzis, G.A., 1989, Calculation of Gates-Gaudin-Schuhmann and Rosin-Rammler

Parameters from the Size Analysis of the Coarse Part of the Distribution, Mining and Metallurgical Annals, Vol.72-73, pp.29-38, (in Greek with English abstract).

5. Stamboltzis, G.A., 1983, Modification of Rosin-Rammler and Gates-Gaudin-Schuhmann Equations in Repeated Single Fracture of Brittle Materials, Mining and Metallurgical Annals, Vol.54, pp.36-44 (in Greek with English abstract).

6. Tsakalakis, K.G., 1990, Product Size Distributions of Industrial Gyratory Crushers, Mining and Metallurgical Annals, Vol. 75, pp.41-48, (in Greek with English abstract).

7. Nordberg Superior Gyratory Crushers, Brochure No. 1253-01-02-CS/Appleton-English, www.metsominerals.com

8. Curie, M.J., 1973, Unit Operations, British Columbia, Canada.9. SME, 1985, Mineral Processing Handbook, Editor in Chief A. Weiss, SME of AIME, N.Y.,

p. 3B (43-46).

modelling the crushing-sizing procedure of industrial gyratory crushers

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