Methods of granular and fragmented material packing density calculation

Int. J. Rock Mech. Min. Sei. Vol. 34, No. 2, pp. 263273, 1997

Pergamon PII:

S0148-9062(96)00029-0

© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/97 $17.00 + 0.00

Methods of Granular and Fragmented Material Packing Density Calculation S. V. TSIREL~"

This paper is the first in the series devoted to the problems of granular and fragmented material package density calculation. It considers the theory of the calculation of loosening for materials consisting of irregular shape particles. Two new methods for the density calculation of unlimited volume granular materials are offered: a crude method based on the empirical relation between density and the index (worked out by the author) of the uniform distribution of particles in individual fractions; the other method, a more precise one, allows for the consideration of fraction mixture quality. On this basis, problems pertaining to the calculation of granular and fragmented material package density in containers of various shapes and dimensions, to dilatancy packing or loosening of such media, blasted rock density computation and methods of overburden blasting, are discussed. © 1997 Elsevier Science Ltd. All rights reserved.

INTRODUCTION In mining, the package density of granular and fragmented rocks is expressed by means of the coefficient of loosening Kp, i.e. the ratio of the specific volume of the fragmented rock to that of primary rock (Kp = V/Vo). This is the coefficient that we shall use in the discussion. Here are the main problems for the solution of which the coefficient of loosening, Kp, is indispensable. (1) Quantity calculation of fragmented rock and other granular material of widely varying particle size distribution in a variety of containers (bins, wagons, buckets, lorries, etc.) (2) Prediction of the shape and structure of blasteci rock in large scale mining and blasting. (3) Capacity calculation for mining machines such as excavators, loaders, scrapers, etc. which depends on the coefficient of loosening. (4) Calculation of the quantity of material for the erection of earth embankments and dams. (5) Choice of the mechanical grading composition of the filling agent for concrete aggregates and packing rocks. (6) Technological designs of mining systems with complete packing for both underground and open-cast mining (when using either internal dumps or moving the waste rock into open areas), ore shrinkage or drawing.

?Saint-Petersburg Mining Institute (Technical University), Petersburg, 199026, 21 Liniya 2, Russian Federation.

st. 263

(7) Computation of geomechanical properties of fragmented or granular material systems created by nature or man (dams, earth embankments, zones of caving). (8) Percolation calculation for earth-fill or explosion created dams. These are not the only problems for which it is indispensable to establish the coefficient of loosening Kp. The Kp value is also indispensable, for example, when keeping a record of produced or stored crushed stone, when determining the particle size distribution of graphite powder for rods at atomic power stations, but, essentially, these are similar to the problems listed above. The existing methods for evaluating package density can be divided into theoretical and experimental. In mining and building, empirical estimations of Kp are usually used [1-10]. However, those estimations can be used only with a rather limited number of particle size distribution classes, but not with all the varieties met with in practice, which, moreover, differ in fraction mixture quality. The second approach is a theoretical one. It includes an analytical description of recurrent structures of regular form bodies [1,11,12] and a computer simulation of random structures of granular media [9,12-22]. The advantage of the computer simulation is that it offers the possibility of taking into account the very process of mixture formation and its random nature, as well as particle friction, etc. It should be noted that, in the literature known to the author, only the packing of balls and cubes was studied. Their sizes varied within a narrow band (Xmin/Xmax)and in the majority of cases only

264

TSIREL: LOOSEMEDIADENSITYCALCULATION

a 2-D problem was discussed, so it really concerned discs. At the same time, experiments show that even the qualitative aspect of package density dependence on the correlation of fractions is determined by the forms of individual pieces. Let us examine the mixture of two fractions with sizes xl and x2. F o r circular particles of regular form, the curve of dependence Kp on fraction mixture displays a minimum (which means m a x i m u m density) only when xl and x2 differ greatly. When the values of x~ and x2 are close to each other the dependence can vary (Fig. 1). For bodies of random form (including pieces obtained by crushing) as shown by numerous data in the literature, for example, in refs [6,23], as well as by the author's experiments, the curve possesses a minimum in all cases and it is only its depth that grows with the increase of ratio x ~ / x 2 . In consequence, so far the computer simulation method compares unfavourably with the experimental approach. Hence, the author has set up the problem of developing new approaches based on the use of experimental data making it possible to apply them to granular materials of any particle size distributions. The analysis of literature shows that the most comprehensive list of factors which determine the density of granular materials includes its particle size distribution, compactness and forms of particles, quality of the mixture and the container size. If the problem of calculating the package density concerns a loose mass of material, then the container size is, naturally, excluded;

the quality of mixture is hardly of any importance since it stays nearly constant. It remains to solve the problem of finding the dependence between the coefficient of loosening and the particle size distribution of the mass when particles form, compactness of their packing and the mixture quality remain constant. The calculation is based upon the evaluation of the possibility that small particles should fill the voids between greater ones. Two approaches to the phenomenon are possible. The crude method of Kp computation is based on the still rougher approach which includes the finding of the characteristics determining the fitting of small particles into the voids. Then, an empirical formula linking the coefficient of loosening with the above characteristics is worked out. The second, a more sophisticated, and at the same time more precise method, includes experimental research into the process of void filling in the most primitive situation. On the basis of these data, a complete method of Kp computation for any particle size distribution is worked out. The method provides f o r taking into consideration not only packing compactness, but also particle mixture quality. The author offers the above methods for solving, mainly, problems 1-6. In solving geomechanical problems the methods can be used as parts of more complete force and energy calculations. It is also possible to use them in solving percolation problems to obtain an approximate evaluation of permeability; to carry out precise calculations a more detailed examination of void .'.<

200 ~" 1.75

4

~

2.25 2.00!6

7

1.50 0

20

40

60

80

100

~~ "~°'~.°..°o° 1.50 I ~, ~o

10

I '<'

1.230

~= 1.226

1.215

1.222 1.22C

0

I

20

I

40

I

60

I

I

80 100 0 20 Percentage of the smaller fraction

I

40

I

60

I

80

100

Fig. 1. Variations of the coefficientof loosening Kpwhen mixing two fractions: 1 4 for circular particles; 7-11--for irregular particles; 1, 2--theoretical data (2-D case); 3-11--experimental data; 1--Xl =0.1, x2=0.05, x~/x~=2.0; 2--x~ =0.05, .v2= 0.00774, xdx2 = 6.45; 3--xl = 26.3 mm, x2 = 12 mm, x~/x2 = 2.2; 4---x~= 19.6 mm, x~ = 12 mm, x~/x2 = 1.63; 5--x~ = 70-80 mm, x_,= 5-20 mm, x~/x2 ,.~ 7.5; ~-x~ = 40-70 mm, x2 = 5-20 mm, x~/x2 ,.~ 5.3; 7--x~ = 20-40 mm, x2 = 520 mm, x~/x: ~ 2.8; 8--x~ = 5-10 mm, x2 = 2.5-5 mm, x~/x2 ~ 2.0; 9--xt = 5-10 mm, x2 = 2.5 mm, Xl/X,. ,~ 4.5; 10~x~ = 510 mm, x2 = 0.6-1 mm, x~/x2 ~ 9.1; 1l--x~ = 40-60 mm, x., = 0.5 mm, xl/x,. ~ 33; (1, 2, 11----dataof Rizhkov, Gogolin, Volkov and Karpenko; 3, 4--data of Rzhevski; 6-10---data of the author).

TSIREL: LOOSE MEDIA DENSITY CALCULATION

265

structures is necessary, which calls for an essentially more sophisticated approach.

uniformity of fragmentation together with the shape parameter of the R - R - W distribution, logarithmic dispersion and coefficient of variation. We believe that in this case n has some advantages when compared with AN APPROXIMATE METHOD FOR THE the other cited coefficients. First, the distribution COMPUTATION OF THE COEFFICIENT OF A-G-Sch which n is linked to in contrast to L - N and LOOSENING R - R - W approximates, however roughly, the cumulative The least complicated way of evaluating the coefficient curve in its entirety and does not call for the use of two of loosening is presented. Obviously, the possibility of or three values of the coefficients. Second, n is easy to small particles falling into the voids among larger find without complicated calculations or the use of particles is determinated by the ratio of the content in special networks. Third, as it will be shown further, n the intervals (Xm,x/a, Zmax),(Xmax/a2, Xmax/a), (Xmax/a3, Xmax/ characterizes a real property of the fractured medium, a 2) and so on, where a > 1. On that basis, after simple i.e. its degree of loosening. Hence, the value n will be transformations we obtain termed the index of the uniformity of fragmentation. In practice, it is convenient to find n in the following f ..... x ( d F ~ 2 way. The cumulative frequency function, to be more n = L.~. F \ d x J dx, (1) precise--a broken one, is plotted in double logarithmic co-ordinates. The product of the slope by the content of where the corresponding fraction is found for every region of F(x) = the size distribution function. the curve and the numbers are summed. Naturally, the open finest fraction (pieces falling through the Let us find the value of n for the main distributions closest-meshed sieve) cannot be computed. So, if no used in the approximation of fractured rock and information on the distribution of that fraction is granular media cumulative mass of fragments. available, an additional assumption is offered. Namely, (1) Distribution of Andreev-Gaudin-Schuhman it is assumed that the slope of the fraction is close to the next fraction (A-G-Sch):

(xy

F(x) = - -

Xmax

(2) Distribution (R-R-W):

of

, n = k.

n ~ k~ AFI + ~ k, AFt,

Rosin-Rammler-Weibull

X

AN = the portion of the i-th fraction;

k

(3) Lognormal distribution (L-N): 1 fo'l (ln2(u/d~)X~ fly/~ ueXp 2-~ ] d u ,

n ,~ 0.903/fl. (4) Linear fractional distribution:

F(x)

= qXmax +

x (1-q)x'

n-

q+l 2

(5) Gamma distribution:

exp/n

where m = the number of fractions;

F(x)=l-exp(-(~max)),n=(n-~6-1)k~O'645k"

F(x)_

(2)

i=2

~ 0 . 6 4 5 ¢ 0.645 when

d.,

0.6 < c < 7.

In contrast to cases 1-4, in the case of the F-distribution the equation is an approximate one. However, in the indicated interval the error does not exceed 5%. Thus, for every main distribution used in the approximation of the cumulative mass of fragments, the introduced value n is determined by the shape factor only. Consequently, n can be used to evaluate the

k; = the slope of the i-th fraction. As shown further, the value of n is strongly dependent on the content of the finest particles. So, if the finest open fraction includes a significant portion of the whole, it is advisable to have the least cumulative distribution function for every fraction, including the finest grained ones. Another difficulty appears when using nonsmoothed cumulative distribution functions which include regions with high values of ki as it can lead to a substantial overestimation of n values. When finding the coefficient of loosening, there exists a possibility of bypassing this difficulty, but, at the same time, the evaluation of uniformity of fragmentation may be distorted. To find the empirical relation of Kp to n (Fig. 2) several sets of data were processed [2, 5, 6,10]. The coefficient of Kendall serial correlation was - 0.926 (though in several experiments it was influenced by container walls), which indicates a rather close relation between Kp and the n index. When finding the empirical dependence of Kp from n, it was taken into account that the curve must pass through the point n = oo (all the particles are the same size). Our experiments have shown that when the size of the particles is the same, the coefficient of loosening (Kp0) is near to two. Victorov cites similar data [24]. The shape

266

TSIREL:

LOOSE MEDIA DENSITY CALCULATION

of K:--1.65-1.8). Their package density is only slightly inferior to that of random rigid packing of steel balls. According to refs [7,15] it ranges from 1.52 to 1.61. To obtain K~ for a loose mass of rock material the following formula is offered:

1.9

o 1 02

1.8

2 Kp = 1 + ~ (Kp0 - 1) arctan n,

• 3 +4

1.7

(3)

[]

1.6

if the cumulative distribution function includes short regions of high values of k,, then it is preferable to use the other version of the formula:

1.5

Kp = 1 + 2 (Kp0- 1) ~ AF~ arctan k,. 7¢

1.4 i

1.31

1.2

1.1 0.3

I

I

I

I

0.6

0.9

1.2

1.5

n

Fig. 2. Dependence of Kp with n: 1--data of Loukashov [2]; 2--data of Seinov [3]; 3--data of Rzhevski [6]; 4--data of Adushkin and Pernik [10]; 5--data according to equation (3).

of fragments influences Kp0 only slightly. According to our data (fragments of fine- and medium-grain granites obtained by mechanical crushing used in the experiments), the values of Kp0 for cubelike fragments are 1.88-1.92; for elongated fragments they are 1.93-1.96; for fiat fragments they are 2.0-2.03; for the whole mass obtained by mechanical crushing they are 1.92-1.96. Smoothed particles (sand and pebbles have lesser values

At the same time, the index of the uniformity of fragmentation is not a sufficient statistic of particle size distribution to compute the coefficient of loosening. However, variations of Kp values when the fraction mixture qualities do not differ and n values are equal, in the majority of cases do not exceed the errors in Kp measuring. In many experimental investigations, Kp is linked to the mean size of particles in the media, e.g. in ref. [5], though it follows from simple geometry that the scale cannot influence the package density. From our point of view the dependence is linked to the growth of the small-grained fractions portion (their logarithmic dispersion is substantially higher than that of the great-grained ones [25-27]) when the degree of fragmentation grows. However, this regularity is continually manifested for one rock under a uniform durability of crushing (in blasting this means a uniform diameter of boreholes). When comparing different kinds of rock fractured by different kinds of blasting, the dependence may appear to be not so rigid. To evaluate the correlation between the value n and the medium size

[]

2.8

2,8

0

[] []

2.4

[]

[]

[]

O

0 [] []

1.2

0.8

[]

2.0

aa aaCP

[]

17

[] O1:1

[]

CD 17

[]

[] []

o

ra

O 17

[]

[] 1.2

ra 00

17 []

o o []

i"1 []

[] []

13

otla 0.8

0.4-

(a)

o ~ o o°

" 7 1.6

n

m

0

[]%

2.4

[]ta

2.0

0.4

I

I

I

I

I

I

I

I

I

0,1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

d A (m)

(4)

i=[

(b)

dr (m)

Fig. 3. Correlation between the index of the uniformity of fragmentation and the average fragment size: (a) the arithmetic mean; (b) the geometric mean.

TSIREL:

LOOSE MEDIA DENSITY CALCULATION

of fragments, the data in refs [25,28,29] supplied by 43 experiments were used (Fig. 3). The coefficient of correlation for the mean arithmetic size of fragments is - 0 . 5 2 (level of significance p ~ 0.999)and the mean geometric is -0.83 (p > 0.999). The use of the least square method yielded the following constraint equations: n = (2.46 - 4.6d~) -~, n = (2.53 - 2.5dA)-~,

(5)

where the mean arithmetic (dA) and mean geometric (d~) sizes of fragments are measured in metres. Statistically significant constraints between n and the mean sizes of fragments show that the above tendency holds even when data obtained on different deposits with different borehole diameters (105-269 mm) are compared. At the same time, the degree of correlation is insufficient for finding the values of Kp on the basis of fragment mean sizes only. Hence, even when there are not more than two types in the fractured rock mass, to evaluate Kp it is necessary to use the index of the uniformity of fragmentation, and not the index of the degree of fragmentation. Greater values of the coefficient of correlation for the mean geometric fragment size result, mainly, from the content of the fine fractions which control loosening, whereas the mean arithmetic size depends, mostly, on the content of individual large fragments whose influence on the coefficient of loosening of an unbounded rock mass is limited. From our point of view, the uniformity of fragmentation involves another phenomenon. It is known that anyone with some experience in the practice of blasting or research in this field can visually notice a 10% or even lesser differences in the size of fragments in identical rocks, though, allowing for Weber-Fechter law, such precision of visual observation seems strange. By contrast, when comparing crushed rocks of different types, even an experienced man is often unable to register much greater differences in the degree of fragmentation. This is explained by the fact that the portion of small-grained fractions grows with the increase of the degree of fragmentation. Those not only fill the voids between greater-sized fractions, but partly sprinkle over them. Then, a 10% increase in the degree of fragmentation is perceived visually as a 20% increase or even more. In fact, what we register is, not the mean fragment size (dA), but the uniformity of fragmentation (n). When comparing different rocks, the correlation of degree and uniformity of fragmentation becomes less rigid, and our observation powers deteriorate, losing much of their precision. The above speculations refer in the same degree to the processing of fragment size distribution measurements (on the surface of blasted rock) obtained by planimetric and photoplanimetric methods. By virtue of that effect the latter method (in spite of important errors) can reveal small differences in medium-sized fragments, but, at the same time, the method overestimates the differences themselves. However, when analysing the results of blast crushing in different rocks (or even in identical rocks, but with such differences in the parameters of blasting that the

267

uniformity of fragmentation changes), comparative assessments of data obtained by photoplanimetric method do not yield reliable results. COMPUTATION OF THE COEFFICIENT OF LOOSENING WITH MIXING QUALITY The above cited method for the computation of the coefficient of loosening is fairly simple, but it is not universal. First, it does not make an allowance for all the characteristic features of particle size distribution or the quality of fraction mixing; second, its use precludes the possibility of computing Kp in containers. When fracturing rocks either by blasting or mechanically, rock fragmentation of all dimensions (but in different proportions) appear, and the equality of the mixture is rather high and stable. However, when the fractured mass is moved from one place to another, it is divided according to fragment dimensions and the quality of mixing deteriorates, which influences the coefficient of loosening. Similarly, the mass which results from mixing materials of different size distribution displays an inferior quality of mixture compared with that of each of those materials taken individually. Therefore, the method of the coefficient of loosening computation should take into account the uniformity of particle distribution in the bulk of the loose mass. The basis of the presented method is an imaginary process of filling space with fragments of ever diminishing sizes. An assessment of the quality of mixture of fragments, the sizes of which differ by several orders of magnitude is a difficult problem, as representative samples chosen for the evaluation of the distribution uniformity of different fractions differ themselves by several orders of magnitude. It follows that every part of the loose mass must be characterized by several indexes of mixture quality. To use one index, we shall assess only the mixture quality within the limits of the problem in hand. Let is introduce several new concepts. The ideal placement is an arrangement of loose fragments with smaller particles filling successively all accessible voids. The complete packing denotes an arrangement in which smaller particles fill all the accessible voids with a 100% probability, but, at the same time, the particles do not "know" whether the voids are filled with other particles of the same fraction or not. It is this "knowledge" which constitutes the difference between an ideal placement and a packing. Packing of quality p is an arrangement in which particles fall into accessible voids with probability p. An ordinary loose mass is a packing; a state which is nearing packing is met with in the near surface layer of the stowing mass when fine-grained material is hydraulically deposited into it (e.g. mud grouting) [23]. It should be noted that computational errors resulting from the use of only one averaged index of the mixture quality are of minor importance, but not in all cases. For instance, during vibro-packing with a small amplitude of vibration it is fine-sized fractions that are moved, and the indexes of different size pieces mixture quality can differ.

268

TSIREL:

LOOSE MEDIA

DENSITY

CALCULATION

For such material, the use of only one index may lead The initial region of curve Kp(v2//)l) corresponds to the to a substantial error in the computation of the situation when a part of voids xl remains unoccupied, coefficient of loosening. and in the case of an ideal placement/); = v2. When the Let us begin with the case of an ideal placement. The mixture is characterized by a minimum coefficient of particle size distribution will be approximated by a step loosening, pieces of fraction x2 fill the whole accessible function with a finite set of particles size values. If the volume of the voids. particle size distribution is continuous (when the mass Let us denote values A Vpor, and v2 corresponding to dimensions grow infinitely it includes pieces of all sizes), that situation Vmaxand v.... When the portion of fraction the choice of those values is random; if the size x2 increases, part of it fills a new volume, the volume of distribution is discrete, the set of values is specified. pieces filling voids xl, being then v; >~ Vmax.The function Let us denote the size of the largest section xl and its which describes the possibility of smaller pieces fitting portion in the over-all volume of the mixture Vl; the next into voids among bigger ones can be expressed as largest fraction is denoted, respectively, x2 and/)2, and so on. The relative volume of the first fraction is VI = Kpovl; the volume of voids inside V1 is Vpor,= V1 -/)1. (7) Vpor, 'V-~'2 ~-- 1 - - ~0 ,~'~,2 " The next fraction can either fill the voids remaining free or fill a new volume. Certainly, it does not follow that the whole of it is to be found on the border of space The analysis of experimental data shows that the occupied by the preceding fraction. Only part of the dependence q~ from v2/v~ manifests itself only in a certain voids formed by fraction Xl are accessible to fraction x2. range of x:/x~ values. When piece size of the two Their magnitude is, first of all, related to the correlation fractions are close, smaller pieces hardly even fall into between the sizes of the fraction considered and the next the voids. A barely noticeable minimum of fraction one. Kp(v2/Vl) corresponds to a small value of the ratio v2/vl. To investigate the accessible portion of the voids With the ratio increasing a little, Kp is again nearing Kp0, volume, the data from experiments on mixing two i.e. v~ ~ v.... When piece sizes differ markedly, the fractions were used (some of these are shown in Fig. 1, smaller pieces, in practice, fill all the voids and so both found in literature and obtained from experiments /)max ~ Vporl/K.o and, respectively, v~ ~ v. . . . The difference carried out by the author). When fraction x2 appears in between v( and /)max manifests itself within a rather the mixture, voids of three types appear, i.e. those narrow, but important region 0.2-0.3 > x2/xl > 0.05between fragments of size xl (type 1), those between 0.1. The smaller pieces "force apart", as it were, the fragments of size x2 (type 2) and those on the contacts bigger ones, and then the voids in the form of long of two different sizes (type 3), with voids inside every narrow slots become accessible for them. That results in type having a certain distribution in size. Voids of type the function Kp(v2/Vl) minimum shift to the right and its 3 can be divided into two subtypes according to the "smearing". However, the influence of that phenomnumber of fractions Xl fragments which are among enon, as seen from the comparison of experimental and others that bound them. Voids, only one boundary of theoretical data, reduces the values of Kp by not more which is formed by pieces of fraction xl, are termed than 3-5%, therefore, the ratio/)2//)1 can be crossed off subtype 1 (on average, these are somewhat smaller than the list of function q~ arguments. voids bounded by pieces x2 only); subtype 2 is bounded Nevertheless, even with this simplification it is rather by several pieces of x~ (those are, on average, somewhat difficult to find function ~0. The main problem is that any bigger than voids bounded by pieces of x2). Conse- artificially created mixture (with the exception of quently, averaging the voids sizes, it becomes possible to hydraulically filled fine-grained fractions) represents not consider only two types, the first corresponding to a placement but a packing, with the quality of mixture fraction xl and the second to fraction x2, respectively. p # 1. So if the piece sizes do not differ greatly, even Further, we shall use the size of the fraction to denote when the portion of fraction x2 is very small, part of the corresponding voids without stating their real those pieces do not fall into the voids among pieces xj. dimensions. On the contrary, even when the portion of fraction x2 is Hence, when fractions 1 and 2 are mixed, the very great, some part of accessible voids remains coefficient of loosening is unoccupied. We had to take that fact into account when processing experimental data. As, in practice, the quality Kp = 1 + Vpor,- - AVpor,+ Vpor2 AVpo,, = v~Kpo (6) of mixture p is not amenable to measuring, an /)1 + V2 Vl + V2' assumption was made. Namely, in every series of experiments irrespective of/)2 increases, the quality of where mixture remained stable (though the value p could AVpor~= voids xl, filled by pieces x2; change with every series). To establish the values of p, v~ -- volume of pieces x2 proper which fill voids xl. several similar experiments were carried out and the value of p was computed on the basis of the minimum It follows that of the stable values of the coefficient of loosening. When processing the experimental data, formulas pertaining to • AVpor I V2 packing were used (see below). Kp0 "

TSIREL: LOOSE MEDIA DENSITY CALCULATION The analysis of correlations obtained with stable values Kp0 shows that the correlations q)(x2/xl) are nearing power functions with the exception of sections x~ ~ x2 and x2<>2even if the difference between Xl and x2 is small, pieces of x2 can fill the voids. Because of this, q(Kp0) is a progressively growing monotone function which assumes a zero value when Kp0 = 1. Experiments with pieces of different geometrical forms whose Kp0 values occur in the range 1.7-2.1, demonstrated that the correlation between q and Kpo - 1 is much weaker than a linear one. Nevertheless, due to the narrowness of the range and experimental errors, it was impossible to define the correlation form with more accuracy. Consequently, after numerous calculations, the form of the correlation was chosen so that with every variety 9 f piece size distribution the growth of its coefficient Of loosening would be monotone with an increase of Kp0 value. The sophisticated way of processing the results of rather simple experiments yielded the following equation: :

(X2~ 0"4x/~ - l ,

(8)

On account of assumptions made in the course of developing equation (8), the model of calculation can be used only for continuous and discrete piece size distributions of a loose mass. When dealing with complex discrete-continuous distributions, e.g. when important lacunae in the range of piece sizes (on the logarithmic scale) occur, or in the case of a combination of large regions together with a distribution of both discrete and continuous types, the model of calculation can be greatly at variance with experimental data or eve~n lose its stability. Then the errors in the calculation of Kp can reach 15-20% and, in the most adverse cases when both phenomena take place the errors can reach 20-30%. Fortunately, such piece size distributions are never met with in mining. To return to the coefficient of loosening of an ideal placement calculation, due to equations (6) and (8) the RMMS 34/2--G

269

portion of the second fraction providing for the minimum coefficient of loosening is

Vmax Kpo =

= v,

[

1 --

lj

\71/

Kpo-I [-l Kpo L

(x Y'

J"

(9)

In the case of an ideal placement two situations are possible: (1) v2 > v.... If such is the case, fraction x2 fills all the accessible voids xl, with part of it filling an additional volume. According to the above assumptions the effective void size 12 used to calculate the location of the next fraction is equal to x2. (2) v2 ~ v.... In that situation the whole of the next fraction falls into the voids, and the overall volume of the medium does not change. The main difficulty lies in the calculation of the effective size of/2. The principle of the defining property offered by Boyarski [15] was used in this case. The role of the defining property belongs to the function /

f(x)

=

\0.4x/K-p0 - 1

2 (xc]

Wpo,.,

(I0)

, \x,i

where xc = size of the next fraction; Wpor,= void volume which is controlled by the pieces of the ith fraction. Examining all the fractions in the order of size decrease, it is possible to find the coefficients of loosening for an ideal placement. According to the structure of formula (8), it is natural to choose size ranges of the type (Xmax,Xmax/a), (Xmax/a,Xmax/a2), etc. assuming that the sizes of fragments in an individual fraction are constant. To be more exact, when averaging we should proceed from the defining property_ (9). However, the use of expressions Xmax/O.5(a"t- ~ ) , Xmax/O.5a(a+ x//-a), etc. (when the value a is near enough to 1) does not really induce any error in the calculations. The computation for different distributions used to approximate the cumulative mass of rock fragments has shown the following. The second, third, etc. fractions do not fit into the voids within the medium, and so the overall volume of voids increases. The more this volume of voids increases, the more the "non-fitting" part of every next fraction decreases. Beginning with a certain size, it assumes a zero value, and the volume of the media stops growing. Thus, first situation 1 and then situation 2 are realized. The return to the first situation is highly improbable. First, because as the piece size decreases the volume of every next fraction decreases too. Second, in situation 2 the difference between Xl and II increases continuously and, consequently, the filling of voids inaccessible for greater pieces becomes possible. The portion of fractions with sizes nearing those of grains and microporosities of

270

TSIREL: LOOSE MEDIA DENSITY CALCULATION

the rocks is rather bigger than in the adjoining fractions (0.1-1mm) [30-32]; hence, there exists a certain possibility of a return to situation 1. However, in the case of blast fragmentation, the overall portion of such particles does not exceed 1%, with an important part of them simply sticking to larger pieces [33]. Thus, in the majority of cases to find the coefficient of loosening of an ideal placement it suffices to calculate the volume of Va medium at the moment when situation 1 is developing into situation 2. To find Va for continuous distributions, amounts to solving the following linear equation: V'---=

V x

1--F 2.5x

- 2 F ' - - -

1.9 1.8

!.7 1.6 ~1.5 1.4 1.3

(11) 1.2

where

1.1

F ( x ) = cumulative mass of fragments according to

their sizes.

1.0 0

To illustrate this, let us consider solution (11) for the distribution A-G-Sch: V(£) =

k 2k - 0.4 I+---Z-6Af°4k k - - 0 - A ffk' k ~ 0 . 4 ; 1 - 2°4(1 + 0.4 In if), k = 0.4,

(12)

where = (X/Xmax) = relative size of pieces;

k = index of the degree of the distribution A-G-Sch. Let us denote the value of ~ under which the transition from situation 1 to situation 2 takes place as ~a; fractions < ffa will fill the inner voids. For the distribution A - G - S c h ~ is ___ (5k - 1)-k_-S~.4, k :~ 0.4; [

e -5.

(13)

k = 0.4.

Formula (13) shows that when k < 0.2 no transition (see above) occurs and kp ~ 1. When k > 0.2 the volume V(~a) determines the coefficient of loosening: Kp =

{

l

1 + (5k - 1)-:5~-2S_~, k ¢ 0.4; 1 + e -2.

0.2

0A

0.6

0.8

1.0

P Fig. 4. Dependence of Kp on p. Numbers by the curves denote values

of n. simplicity sake we have assumed that p = 1. In this case, situations one and two are indistinguishable, as in any case part of the new fraction will fill the voids between greater pieces, and part of it will occupy a new volume. Let us denote the quantity of the next fraction pieces by fl, and the quantity of holes where they can fall by a, then under p = 1 the mean quantity of occupied holes will be

Taking the limit, we obtain that, when adding a fraction of size x, it will fill the following volume inside the existing voids: A Vpoq = V p o r ( 1 - e x p ( -

pKpo AR

>< (1 _ (/)°'4 r4-g~ ~)- 1V~) ) )

(14)

(15)

k = 0.4.

where In the physical sense, 2, means that for an ideal placement pieces greater than 2a constitute the skeleton of the loose mass, whereas pieces smaller than £~ fill the free spaces among the former (the quality of mixture is assumed to be ideal). The value of £~ shows how the material reacts to compression: greater pieces are inferior in strength, but, on the other hand, superior in roughness, so the greater the value of £a, the lesser the possibility of their slipping through and the greater the probability of crushing. As :~, is fully dependent on n, the behaviour of an ideally placed medium under compression is essentially determined by its size distribution. Now we shall examine the same process, i.e. filling space with pieces of decreasing sizes, for packings. For

AF = fraction x particle volume; l = effective size of particles in the medium used for the calculation of fraction x displacement; Vpor = voids volume in the medium consisting of pieces greater than x (i.e. volume of the existing voids). Figure 4 shows the influence of the mixture quality p on the coefficient of loosening. It demonstrates that the mixture quality influence is especially important for that coefficient when both n and p values are small. On the whole, the deduction is rather obvious: in the beginning the effect of several fraction mixing is stronger than at

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LOOSE MEDIA DENSITY CALCULATION

271

Table 2. Coefficients of loosening for the law of small numbers K = (V~/V)

ifr 1.5-1.4--

~

.

g

o

1.3 --

~

:

"~"~'~'-,,,~,.~,,

,2-

1

0.5

I 1.0

I 1.5

I 2.0 n-1

I 2.5

I 3.0

3.5

Fig. 5. Dependence of Kp on n different distributions: 1--A-G-Scb; 2 - - R - R - W ; 3--L-N; 4---data according to equation (3).

the following stages. The more fractions there are the more energetic must be the process of mixing to reach the state of their homogeneous distribution in the mixture. The comparison of theoretical and experimental data has shown that the mixture quality for fragmented materials is characterized by rather stable values, these being 0.7~).75 on average; when the mixture is spilled over several times p can decrease to 0.4-0.5. Let us consider some examples of calculations on the basis of our methodology. The comparison of the coefficients of loosening for granular materials with equal values of the index of the uniformity of fragmentation under different functions of size distribution (Fig. 5) shows that these are fairly close to each other, with minimum values of Kp corresponding to the distribution A-G-Sch. Calculations for experimental fragment size distribution (Table 1) testify to more significant discrepancies between the value of the coefficient of loosening when their n values are equal. However, they usually do not exceed the errors in industrial Kp measurements. As we have shown in ref. [34], the uniformity of fragment size distribution in the mass of crushed material depends on the mode of fragmentation which can be either volumetric or superficial. The volumetric mode means segmentation of pieces into comparable parts; whereas the superficial mode means the appear-

1.10 1.20 1.30 1.40 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00 4.50 5.00 7.50 10.00 12.50

2

10

30

100

300

1.869 1.833 1.808 1.790 1.775 1.748 1.728 1.713 1.701 1.682 1.668 1.657 1.647 1.640 1.613 1.597 1.585

1.875 1.814 1.778 1.754 1.735 1.702 1.677 1.658 1.643 1.619 1.602 1.588 1.577 1.568 1.536 1.517 1.503

1.878 1.802 1.752 1.716 1.688 1.642 1.612 1.590 1.573 1.546 1.526 1.511 1.498 1.487 1.453 1.432 1.418

1.881 1.798 1.737 1.691 1.655 1.591 1.550 1.520 1.499 1.468 1.447 1.431 1.418 1.407 1.372 1.351 1.337

1.885 1.800 1.735 1.684 1.643 1.570 1.521 1.486 1.461 1.426 1.402 1.385 1.371 1.360 1.326 1.306 1.293

ti

' (t~il ' F ( x ° K - i / 3 e - ~ ) = \ K J i! e-f¢,

trachyliparite granitporphyry granodiorite granite granite sandstone sandstone sandstone limestone and dolomite limestone limestone

1.892 1.810 1.745 1.692 1.645 1.565 1.512 1.475 1.446 1.406 1.380 1.361 1.347 1.335 1.300 1.281 1.269

1.890 1.807 1.741 1.689 1.644 1.565 1.512 1.475 1.446 1.406 1.380 1.361 1.347 1.335 1.300 1.282 1.269

Klnif~ t - 4/3 -- K -t/3

(16)

where x0 = original size of fragments; ifr = degree of fragmentation (ratio of mean size pieces before and after fragmentation); K = ratio of the original fragment volume to that of the least fragment resulting from an individual set of fragmentation.

Packing

Mineralovodski Ermentauski Novo-Smolenski Rovnoe Kamenogorski Kuchurovski Sadkinski Skuchareevski Sokski Sokski Karablinski Gurovski

1.888 1.804 1.738 1.686 1.643 1.565 1.512 1.475 1.447 1.409 1.383 1.365 1.351 1.340 1.305 1.286 1.274

In irt t - 1 - 2 -1/3;

F(xo2 -i/3) = ~ e-',

Coefficient of loosening

Rock

10000

ance in the process of segmentation of small pieces only (i.e. small when they are compared with the original size of dividing parent fragment). Let us examine the situation when the original mass consists of approximately equal pieces or blocks in the rock mass, approximately equal in size. Then, in the extreme cases such as bisection or the appearance of small pieces (detrition), the cumulative mass of fragments is nearing the law of small numbers. So, respectively,

Table 1. Coefficients of loosening of blasted rocks at several quarries

Quarry

1000 3000

dAy

do

mm

mm

n

Placement

p = 0.7

p = 0.5

332 348 413 284 348 339 334 302 238 278 275 324

122 216 297 102 103 135 181 103 107 125 106 158

0.597 0.837 0.961 0.467 0.504 0.511 0.682 0.458 0.515 0.587 0.507 0.598

1.242 1.394 1.427 1.227 1.188 1.196 1.265 1.212 1.247 1.239 1.240 1.287

1.334 1.476 1.513 1.343 1.349 1.291 1.387 1.285 1.347 1.326 1.298 1.406

1.436 1.552 1.581 1.431 1.459 1.404 1.504 1.392 1.441 1.431 1.412 1.490

272

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LOOSE MEDIA DENSITY CALCULATION

10.01

1.40

1.30

/ I I I I I I I I I 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 n Fig. 6. Dependence of Kp on n for the full and truncated size distributions. Numbers by curves show the truncation border in fractions of the maximum fragment size.

It follows that the values of the coefficient of loosening for the law of small numbers are of some interest. As may be inferred from Table 2, when using volumetric crushing, even with a rather high degree of fragmentation, it is impossible to obtain a great package density of the crushed mass. In contrast, superficial crushing results in a high package density with the value of K being of hardly any importance when the linear size of the obtained fragments does not exceed 10% of the parent fragment size. Thus, it is possible to obtain an appreciably compact rock mass only when superficial fragmentation participates in the mass forming. As stated above, when dealing with packing, cases v2 < Vmax and v2 > Vmax are indistinguishable, and the addition of any fraction leads to some part of it falling into voids and the rest of it creating an extra volume. Yet if p is not too small, there exists a domain of sizes in which a transition from a predominant increase of the medium volume to a predominant filling of voids takes place. Consequently, the conclusion that particle size distribution influences the real behaviour of crushed or granular rocks under compression is basically true. When the n values are small, it is the finest fractions that exercise the greatest influence on Kp. Figure 6 represents the dependencies of the coefficient of loosening from n for the distribution A--G-Sch when all the fragments smaller than 0.001Xm~x, 0.002X~x, 0.005x .... 0.01Xmax are replaced with fragments of corresponding cut-off sizes. Apparently, the deduction of Rzhevski [35] about the existence of a size distribution leading to a minimum coefficient of loosening is related to the dependencies shown in Fig. 6. However, the deduction remains, to a certain degree, valid because in natural crushed materials the content of fractions whose sizes are smaller than those of grains and microporosities of the rock is small. In artificially created compositions, it is rather difficult to create a given size distribution in the domain of the finest fractions. In practice, size distributions with n < 0.4-0.45 do not exist, because the range of fragment sizes covers nearly the whole range of sizes of blocks forming a rock massif. In consequence, when blasting, crushed materials with a coefficient of loosening less than 1.25 have never been met with. Accepted for publication 7 June 1996.

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