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Stochastic modeling of rock fragment flow under gravity
Int. J. Rock Mech. Min. Sci. Vol. 34, No. 2, pp. 323-331, 1997
Pergamon PII:
S0148-9062(96)00051-4
© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/97 $17.00 + 0.00
Technical Note Stochastic Modeling of Rock Fragment Flow Under Gravity GANG CHEN'~
INTRODUCTION The caving mining methods are widely used due to their advantages of high productivity, high efficiency, low cost, operational safety and breadth of geologic application. Following some rock fragmentation by blasting, the methods utilize gravitational flow of rock fragments for ore material extraction. The removal of ore materials also induces movements of overlying waste rock, causing possible dilution in the ore material. The flow pattern of ore and rock fragments is, therefore, of great importance to mine design and control of caving. It significantly affects the degree of dilution in the drawn ore, the recovery of the orebody and eventually, the efficiency of the caving method. Over the years, significant efforts have been devoted to the study of rock fragment or granular material flow under gravity. Based on in-mine observations and physical modeling studies performed by various researchers [1-4], a number of mathematical models have been proposed to describe the flow patterns of rock fragments under gravity. The proposed models can be generally categorized into three groups: (i) analog models involving the identification of draw volume geometry [5-7]; (ii) force-acceleration and stress-strain analysis based on fundamental physical laws [8-10]; (iii) stochastic theories and probability studies [11-14]. An analog model assuming a ellipsoidal draw shape for granular material flow was first proposed by Kvapil [5]. Due to its simplicity, the model has been widely applied in caving mine design and draw volume estimation [15-17]. However, the ellipsoid theory deals only with the geometry of the gravity flow and provides little insight into the caving mechanism. In addition, experiments and field observations showed that the draw shape is not exactly ellipsoidal [16,17]. A stochastic model was proposed by Litwiniszyn [18] to describe the flow of dry sand. This stochastic model was used to predict the draw shape of granular material flow [13]. The model is a good description of the stochastic processes of granular material flow. However, there is some discrepancy between the draw shape
predicted by the stochastic model and that obtained with physical modeling studies. Although the model has been utilized by many to study the phenomena of random movements in granular material flow, few applications of the theory have been found in practical caving mine designs. In this study, the stochastic model was modified to better predict the draw shape of rock fragment flow. The model was implemented in a computer program and the development of the flow zone was investigated. The modeling results were compared with the experimental results of a large scale physical model as well as the ellipsoid theory. The outcomes were analyzed and discussed. STOCHASTIC FLOW OF GRANULAR MATERIAL UNDER GRAVITY
According to Litwiniszyn's theory [18], the stochastic flow of granular material can be illustrated by a two dimensional system of blocks (Fig. 1). Each block contains a ball subjected to the force of gravity. Removal of a ball from block a~ in the bottom layer will cause either one of the two balls immediately above it to take its place, assuming equal probability of both events, i.e. 1/2. As a result of the displacement of the ball from block a2 to a~, block a2 will be occupied by a ball from block a3 or b3. Similarly, removal of the ball from block b2 will cause its place to be taken by a ball from block b3 or c3. The process continues and the resulting distribution of probabilities is illustrated in Fig. 1. Given a sufficiently dense network of blocks, removal of a sufficient number of balls from block a~ will give a distribution curve approaching Gauss' curve, symmetrical with respect to a perpendicular straight line passing through the center of block a~. A more precise description of the above model can be obtained using a planar system of blocks in the Cartesian coordinate system {X, Z}, with the Z axis directed perpendicularly upward, as shown in Fig. 2. Each block contains material particles subject to the force of gravity, acting parallel to the Z axis, but in the opposite direction. Under the influence of gravity, the particles can only displace downward (e.g. from C to A or B). The tUniversity of Alaska-Fairbanks, Department of Mining and Geological Engineering, School of Mineral Engineering, P.O. Box downward migration of particles corresponds with an oppositely directed upward migration of empty spaces. 755800, Fairbanks, U.S.A. 323
324
CHEN:
TECHNICAL NOTE
Letting p = q = produces:
1/2 and rearranging the equation
b P(x, z + b) - P(x, z) b a 2 P(x + a, z) - 2P(x, z) + P(x - a, z)
= ~-
[r I! U _J rrr-lrrc I U I I II II I; I ,1°,1, r-lVr-lr,
r~7
Fig. 1. 2-D system of blocks [18].
These two migration processes are two attributes of draw motion in a granular medium. With the system of blocks as illustrated in Fig. 2, a void migrates from block A to C with a probability p or from B to C with a probability q and p + q = 1. In the case being studied, it can be further assumed that there is no bias along horizontal direction (x direction), i.e. the void has an equal chance to migrate from A to C, or from B to C and, therefore, p = q = 1/2. To the centers of blocks A, B and C, the corresponding coordinates (Fig. 2) may be assigned as follows:
(2)
Letting A = aZ/2b results in:
OP(x, z) OZ
b3 ] c3
a 3
i7r-
aS
O2p(x, z) -- A
(3)
OX 2
Being a function of the block width a and the block height b, the parameter A relates to the ratio of horizontal to vertical displacements of the voids at each motion step, which depends on the properties of the material and the condition of flow. The equation is formally identical to the diffusion equation. The function P(x, z) being a solution of equation 3 describes the probability of the appearance of an empty block at the coordinate (x, z). Assuming that N particles are removed from the draw point during the time period At, an equivalent of N voids will be migrating upward, originating at the draw point. If N is sufficiently large, in accordance with the law of large number, there will be N*P(x, z) voids going up through the block at the coordinate (x, z). Defining J- to be the flux of voids in the direction of the Z axis (number of voids going through a unit area perpendicular to Z axis per unit time) produces:
J-(x, z) = NP(x, z) a2At
(4)
Substituting equation 4 into equation 3 produces:
(x-a,z),
(x+a,z),
(x,z+b).
~ ( x , z)
Let P(x, z) denote the probability of the occurrence of an empty block with the coordinate (x,z). The probabilities that voids occur at block A or B are P(x - a, z) and P(x + a, z), respectively. The void has probabilities p and q, respectively, to migrate from A to C, or from B to C. Thus, P ( x , z + b) which is the probability that a void occurs at block C can be represented as:
P(x, z + b) = pP(x - a, z) + qP(x + a, z).
(1)
Oz
A ~J_-(x, z)
w h e r e a 2 = projection area of a block on the plane
perpendicular to the Z axis; Z = the z direction component of the flux of voids at the coordinate (x, z); N -- the total number of voids generated from the draw point (or the total number of particles removed from the draw point); At = the time period during which the N voids are generated. For a point opening at the coordinate (0, 0), Mullins obtained a solution in which the coefficient A in equation 5 was taken as a constant [12]: J:(x, z) - 2(nAz) Q l/2 exp
fI ! I i
x-a
- ~
•
(6)
I I I I
c3Jy OJt~~- ÷ ~ = 0.
I
I
( x2)
It is assumed that there is no discontinuity of the flux, i.e. the inflow of voids equals the outflow at every block during any time period. This continuity condition is mathematically expressed as:
--- z + b
/
(5)
Ox2
(7)
Comparing equations 5 and 7 produces: X I x+a
Fig. 2. The coordinates of blocks []8].
t~J:Qx
J,(x,z)=
-Aox-4(nAz3)
( X 2) l/2exp - ~ . (8)
CHEN:
TECHNICAL NOTE
If n is the number of voids (or particles) per unit volume, the vertical and horizontal velocities of flowing voids are V:=J~/n and V,=J~/n, respectively. By taking integration, the time required for a void to travel from the draw point (0, 0) to location (x, z) is given by:
T(x,z)
-
4n(nA)'/2 3Q exp ( ~x2 ) z 3/2.
(9)
Note that T(x, z) is also the time required for a particle to travel from location (x, z) to the draw point, since a particle travels at the same speed, but in the opposite direction, as the corresponding void. The above discussion may be applied to steady-state flow where the flow pattern is independent of time. This happens in bins having the same filling rate at the top as the drawing rate at the bottom. In caving mines, however, the flow zone is not in a steady state. It is in a process of expanding or growing, that is, in a nonsteady state. Experimental work shows that the density of granular material in the flow zone is lower than that outside the flow zone, i.e. p~ = p0 - Ap, where p= is the density in the flow zone, p0 the density outside the flow zone and Ap the density change. Assuming the boundary of the flow zone is advancing at a speed of U = J/Ap, where J is the flux at the same point as U, the time required for the flow zone boundary to reach (x, z) is then:
325
Ap T(x, z).
(10)
The particle at (x, z) starts moving toward the draw point after the flow zone boundary reaches the point (x, z). The total time required for the particle initially at (x, z) to reach the draw point at (0, O) is, therefore, given by:
T*(x,z)=(1 +-~)T(x,z)
(11)
where T(x, z) is the time required for a particle at (x, z) to reach the draw point at (0, 0) in steady state. A given time T* permits determination of all the points reaching the draw point in time T*. Connecting all these points produces the draw envelope defined by the stochastic theory. DISCREPANCY IN DRAW ENVELOPES AND MODEL MODIFICATION Based on Mullins' results [13], the above stochastic model generated a draw envelope that had a narrow top and a relatively wide bottom (Fig. 3). Most experimental rcsults [4,16,17], however, showed a wide top and a relatively narrow bottom (Fig. 4). This discrepancy can . .:,"'~:... •
Z
•" .:"
..." ....• jl
"'1°
POSSII~i-E "-
-"
....
I
,
DRAW
" - , "",... / •
SHAPE
";
i
/
ELLIPSOIDAL SHAPE
I I
I
l
J
"I :I
I, I:
!
I:
I i
$ I
I :l *t
f. |:
3 :! :#
I*,
i".. l: Ik
:l :l :I
I" -i :t :I
l ~. ~'.:
,° i .° I .l
,~'.
*I • ".
--
°p
x
Fig. 3. Draw envelope from Mullins' results [13].
Fig. 4. Possible draw shape.
326
CHEN:
TECHNICAL NOTE
be corrected by introducing a variable coefficient A in equation 5, instead of treating the coefficient A as a constant. As mentioned previously, the coefficient A is a parameter related to the properties of the granular material and the flowing conditions. The coefficient A is mathematically defined as a function of the block width a and the block height b (refer to equation 3), where a can be interpreted as the displacement distance in the X direction during a particle movement increment and b the displacement distance in the Z direction. Experimental work shows that the displacement occurring in the flow zone is unevenly distributed. In most cases, the horizontal displacement distance during a particle movement increment increases as the flow zone develops upwards. This behavior of particle movements in the flow zone produces a slightly wider top and relatively narrower bottom. Therefore, it is reasonable to assume that the coefficient A is a function of z--the vertical distance from the draw point. The actual relation between A and z is not precisely quantified at present, but for simplicity, it is assumed that A is a linear function of z, as expressed below:
A(z) = g(1 + C'z)
(12)
where K and C are constants which relate to granular material properties and may be determined through
where coefficient A (z) is no longer a constant but a linear function of z. FINITE DIFFERENCE MODELING AND RESULTS
The finite difference method is employed to solve equation 13. A FORTRAN program is implemented and a draw point with a finite width instead of a point opening is considered in the model. The program computes the flux line trajectories and the velocities of flow along the trajectories. The moving cones can be generated by setting a number of points on a horizontal line before initiating ore drawing, computing the distances that the points pass after drawing and connecting all these points at new positions. The time required for a particle to reach the draw point can also be estimated. Connecting the initial coordinates of all the points having the same exit time produces the draw envelope for any given time period. Figure 5 shows the sequence of moving cones at various times after the initiation of ore drawing. The moving cone changes during the drawing process. At the moment the base of a moving cone reaches the draw point, its volume approximately equals the volume of draw. With this information it is possible to estimate the
7,
~gJ=(x, z) O2J.(x, z) d2J..(x, z) Oz - A(z) ~ - K(1 + C'z) dx 2 (13)
_/,3
× Fig. 5. Computer output of moving cones. (1) Draw point. (2) Moving cones at consecutive times. (3) Horizontal line before drawing.
--X Fig. 6. Draw shapes at different draw heights.
CHEN:
TECHNICAL NOTE
327
7,
volume of ore that can be drawn before barren rock reaches the draw point. The draw envelopes at three different draw heights are shown in Fig. 6. It can be seen that the draw envelope becomes more elongated as the height of draw increases. This phenomenon has been observed in many physical models, but the conventionally accepted ellipsoid theory can not describe it. This relationship is more clearly illustrated in Fig. 7, where draw height to draw point width ratio (horizontal axis) is compared with draw height to draw width ratio (vertical axis). With a fixed draw point width, the curve shows a clear pattern of increasing draw height to draw width ratio as the draw height increases. The constants K and C are important parameters in this model and represent flow material properties. The two parameters significantly affect the shape of draw envelopes. The value of K affects the slimness of draw envelope (Fig. 8). By defining the slimness of draw envelope S as: draw height S - draw width
(14)
the relation between S and K can be illustrated as in Fig. 9, which shows that as K increases the draw envelope becomes wider and vice versa. The value of C affects the shape of the draw envelope (Fig. 10). With a horizontal line at 50% of draw height dividing the draw envelope into the upper and lower halves, the shape factor R is defined as: R = area of upper half draw envelope area of lower half draw envelope "
(15)
The relation between R and C can be depicted as in Fig. 11. The curve shows that the greater the value of C, the wider the top of the draw envelope (greater R value) and vice versa.
--
X
Fig. 8. The effect of parameter K upon the slimness of draw envelope. (1)/~
= 0.5. ( 2 ) / ~
= 0.3.
The draw point width also affects the slimness and shape of the draw envelope. Figure 12 shows draw envelopes with different draw point widths and Fig. 13 gives the relationship between the width of draw envelope and width of draw point.
4.0
3.0
5.0
2.0
4.0
3.0 1.0 D r a w Height D r a w Point Width
0
I
10
I
20
I
30
I
40
Fig. 7. The relation between the ratio of draw height to draw width and the ratio of draw height to draw point width.
2"o 1I 0
! 0.2
i 0.4
~ 0.6
I 0.8
I 1.0
K
Fig. 9. The relation between parameter K and function S (C = 0.05).
328
CHEN:
TECHNICAL NOTE R
2.0
1.5
1.0
0.5
0
I
0.05
I
I
0.1
I
0.15
0.2
c
Fig. 11. The relation between parameter C and function R (K = 0.6).
X Fig. 10. The effect of parameter C upon draw shape. (1) C = 0.05, (2) C = 0.00.
COMPARISONS AND DISCUSSIONS Large scale physical modeling experiments were performed by Peters [3]. The results of the experiments are compared with the stochastic model developed in this study. Data computed by ellipsoid theory are also presented as a reference. Three items are compared in the study to evaluate the stochastic model and they are: (1) The ratio of draw height to draw width at different stages of the draw envelope development. (2) The area (volume) of draw at different draw heights. (3) The ratio of upper half of the draw to lower half of the draw. The first two items are considered important because these are the key parameters for caving mine design. The ratio of the height to the width of the draw shape determines the horizontal spacing of the draw points. The volume of draw at different draw heights determines how much ore can be drawn before barren rocks reach the draw point. Since a variable coefficient is used in equation 13 to simulate the phenomena of a relative wider top and narrower bottom of draw shape, the third comparison is introduced to evaluate this effect.
/ j l
19.
X Fig. 12. The effect of draw point width upon draw shape. (1) Wide draw point. (2) Narrow draw point.
CHEN:
T E C H N I C A L NOTE
329
30--
20 "
Z
(cm)
0
30
~
10-Draw
0
i
I
,
10
Pointwidth 2I0
Fig. 13. The relation between draw width and draw point width (draw height = 254 cm).
The physical model developed by Peters [4] was 4.6 m (15 It) high and 1.8 m (6 ft) wide. It was considered the largest physical model ever built at that time. For the stochastic computer simulation, the material constants K -60
Z (cm)
-30
60
X (cm)
Fig. 15. Draw envelopecomparison. Solidline--computersimulation. Dashed line---experiment. Draw point width = 30 cm (12 in).
X -60 -30
0
30
(cm)
60
Fig. 14. Draw envelopecomparison. Solid line--~omputersimulation. Dashed line--experiment. Draw point width = 20 cm (8 in).
and C were selected based on the draw shape of the physical model tests at one draw height and used for different draw heights throughout the comparison. The same process was used to select material constant for the ellipsoid theory. Typical draw shape comparisons between the stochastic model and the physical model are shown in Figs 14 and 15. Figure 14 is the draw shape with a draw point width of 20 cm (8 in) and Fig. 15 is the draw shape with a draw point width of 30 cm (12 in). Detailed comparisons among the physical model, stochastic model and ellipsoid theory are given in Tables 1 and 2. Both Figs 14 and 15 show excellent agreement between the draw shapes generated by the physical model and the computer model. Both models display a relatively cylindrical midsection of the draw envelope, which can not be predicted by the ellipsoid theory. Data presented in Tables 1 and 2 show that the differences in the area of the draw envelope between the physical model and computer model are generally small, most within 8%. The differences between the physical model and the ellipsoid are significantly larger. The physical model and the computer model agree well on the draw height to draw width ratio at various stages of draw height development (Figs 16 and 17). Both models show the trend that at small draw heights,
330
CHEN: TECHNICALNOTE Table 1. Comparison between computer simulation and physical modeling experiments (draw point width = 20 cm) Draw height Stochastic model Ellipsoidtheory (cm) Experiment (K = 0.33, C = 0.05) (E = 0.968) 427 Area of draw 39,206 37,123 (-5.3%) 35,755(-9.0%) 396 envelope (cm2) 33,723 31,993 (-5.1%) 30,826(-9.0%) 366 29,264 27,858 (-4.8%) 26,271( - 10.2%) 297 20,813 19,264 (-7.4%) 17,342( - 16.7%) 264 17,929 16,000 ( - 10.8%) 13,703(-23.6%) 427 Draw height to 4.0 4.0 4.0 396 draw width ratio 3.76 3.9 4.0 366 3.6 3.85 4.0 297 3.8 3.8 4.0 264 3.4 3.6 4.0 427 Upper half to lower 1.09 1.04 1.0 396 half ratio 1.10 1.05 1.0 366 1.07 1.03 1.0 297 1.06 1.06 1.0 264 1.05 1.04 1.0 Percentage in parentheses indicates the difference from the experiment.
the draw height to draw width ratio is small (wider draw envelope) and becomes larger with increasing draw height (narrower draw envelope). This phenomenon can not be described by the ellipsoid theory. Both the physical model and the stochastic model show a slightly larger upper half as compared with the lower half of the draw shape. By contrast, the ellipsoid theory predicts the upper and lower halves to have equal areas. In general, the stochastic model agrees well with the physical model. It provides a better description of the flow pattern of rock fragments in caving mines than the conventionally accepted ellipsoid theory.
CONCLUDING REMARKS
As a result of this study, the following concluding remarks are given: (1) The granular material flow under gravity can be well described with the proposed stochastic model. The model predicts the flow pattern better than the
ellipsoid theory and successfully simulates some phenomena that can not be explained by the ellipsoid theory. (2) The material flow conditions vary with the distance from the draw point. The coefficient in the differential equation describing the stochastic flow should, therefore, be a function of this distance (the z coordinate). A proposed linear relation between the coefficient value and the z coordinate works well, resulting in an excellent agreement between the stochastic model and the physical model. (3) The stochastic flow model is based on the random flow pattern of granular materials and is derived from the phenomena of the flow instead of fundamental physical laws. At the present time, there is no technique available to obtain the model parameters directly from mechanical testing in the laboratory. It will require a small scale physical modeling to acquire the needed parameters. However, the link between the stochastic model parameters and the physical properties of the material can be revealed by further studies.
Table 2. Comparison between computer simulation and physical modeling experiments (draw point width = 30 cm) Draw height Stochastic model Ellipsoidtheory (cm) Experiment (K = 0.33, C = 0.05) (e = 0.968) 442 Area of draw 41,987 40,922 (-2.5%) 38,355(-8.7%) 401 envelope (cm2) 36,413 34,664 (-4.8%) 31,626( - 13.2%) 361 30,193 29,013 (-3.9%) 25,542( - 15.4%) 284 19,974 19,316 (-3.3%) 15,890(-20.4%) 234 15,510 17,626 ( - 11.3%) 10,723( - 30.9%) 442 Draw height to 4.0 4.0 4.0 401 draw width ratio 3.85 3.9 4.0 361 3.56 3.8 4.0 284 3.6 3.6 4.0 234 3.1 3.3 4.0 442 Upper half to lower 1.0 1.13 1.0 401 half ratio 1.01 1.11 1.0 361 0.98 1.10 1.0 284 0.87 0.90 1.0 234 0.92 0.89 1.0 Percentage in parentheses indicates the difference from the experiment.
CHEN: 4,0
O
3.0
"~
TECHNICAL NOTE
additional studies will expand the knowledge acquired from this 2D model study and provide a solid basis for more vigorous mine designs.
/
IIt
Accepted for publication 29 July 1996.
2.0
REFERENCES
1.O
Draw Height I
0
t 200
100
(cm) t 400
300
Fig. 16. Draw height to draw width ratio vs draw height. Solid line---computer simulation. Dashed line---experiment. Draw point width = 20 cm (8 in).
O
3.0
1.G
Draw ge£ght 0
331
I 100
l 200
I 300
(am)
I 400
Fig. 17. Draw height to draw width ratio vs draw height. Solid line---computer simulation. Dashed line--experiment. Draw point width = 30 cm (12 in).
(4) The current 2D model may be further developed to simulate 3D flow patterns. The interaction of adjacent draw points may also be included. These
1. Janelid I. Study of the gravity flow process in sublevel caving. In Proceedings of Sublevel Caving Symposium, Atlas Copco, Stockholm, 65 p (1975). 2. Yenge L. I. Analysis of bulk flow of materials under gravity caving process---Part 2: Theoretical and physical modeling of gravity flow of broken rock. CSM Quarterly 76, 67 (1981). 3. Su H. and Wei S. The properties of ore flow. Nonferrous Metals 35, China, 7-12 (1983). 4. Peters D. C. Physical modeling of the draw behavior of broken rock in caving. CSM Quarterly 79(1) (1984). 5. Kvapil R. Gravity flow of granular materials in hoppers and bins in mines. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 2, 25-41 (1965). 6. Kvapil R. Gravity flow of granular materials in hoppers and bins in mines--II, coarse material. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 2, 277-304 (1965). 7. Wang C. et al. Study of Caving, Metallurgy Industry, Beijing, 168 p (1982). 8. Johanson J. R. Stress and velocity fields in the gravity flow of bulk solids. J. Appl. Mech. 31, 499-506 (1964). 9. Jenike A. W. Gravity flow of frictional-cohesive solids--convergence to radial stress fields. J. Appl. Mech. 32, 205-207 (1965). 10. Enstad G. G. A novel theory on the arching and doming in mass flow hoppers. Thesis, the Chr. Michelsen Institute (Norway), 95 pp. (1983). 11. Jolley D. Computer simulation of the movement of ore and waste in underground mining pillar. Canad. Min. Met. Bull. 61, 854-859 (1968). 12. Mullins W. W. Stochastic theory of particle flow under gravity. J. Appl. Physics 43, 665-678 (1972). 13. Mullins W. W. Nonsteady state particle flow under gravity--an extension of the stochastic theory. J. Appl. Physics 4, 867-872 (1974). 14. Huo Z. and Xiong G. Computer simulation of oredraw. Nonferrous Metals (China) 34, 1-6 (1982). 15. Coates D. F. Rock Mechanics Principles, CANMET Monograph 874, pp. 5.3-5.8 (1981). 16. Just G. D. The significance of material flow in mine design and production. Design and Operation of Caving and Sublevel Stoping Mines (Edited by Stewart D. R.), pp. 715-728. SME-AIME (1981). 17. Kvapil R. Sublevel caving. In SME Mining Engineering Handbook, (Edited by Hartman H. L.) 2nd Edn, Vol. 2, pp. 1789-1814. SME (1992). 18. Litwiniszyn J. Stochastic Methods in Mechanics of Granular Bodies, Int. Center for Mech. Sci., Lectures No. 93. Springer (1974).
Pergamon PII:
S0148-9062(96)00051-4
© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/97 $17.00 + 0.00
Technical Note Stochastic Modeling of Rock Fragment Flow Under Gravity GANG CHEN'~
INTRODUCTION The caving mining methods are widely used due to their advantages of high productivity, high efficiency, low cost, operational safety and breadth of geologic application. Following some rock fragmentation by blasting, the methods utilize gravitational flow of rock fragments for ore material extraction. The removal of ore materials also induces movements of overlying waste rock, causing possible dilution in the ore material. The flow pattern of ore and rock fragments is, therefore, of great importance to mine design and control of caving. It significantly affects the degree of dilution in the drawn ore, the recovery of the orebody and eventually, the efficiency of the caving method. Over the years, significant efforts have been devoted to the study of rock fragment or granular material flow under gravity. Based on in-mine observations and physical modeling studies performed by various researchers [1-4], a number of mathematical models have been proposed to describe the flow patterns of rock fragments under gravity. The proposed models can be generally categorized into three groups: (i) analog models involving the identification of draw volume geometry [5-7]; (ii) force-acceleration and stress-strain analysis based on fundamental physical laws [8-10]; (iii) stochastic theories and probability studies [11-14]. An analog model assuming a ellipsoidal draw shape for granular material flow was first proposed by Kvapil [5]. Due to its simplicity, the model has been widely applied in caving mine design and draw volume estimation [15-17]. However, the ellipsoid theory deals only with the geometry of the gravity flow and provides little insight into the caving mechanism. In addition, experiments and field observations showed that the draw shape is not exactly ellipsoidal [16,17]. A stochastic model was proposed by Litwiniszyn [18] to describe the flow of dry sand. This stochastic model was used to predict the draw shape of granular material flow [13]. The model is a good description of the stochastic processes of granular material flow. However, there is some discrepancy between the draw shape
predicted by the stochastic model and that obtained with physical modeling studies. Although the model has been utilized by many to study the phenomena of random movements in granular material flow, few applications of the theory have been found in practical caving mine designs. In this study, the stochastic model was modified to better predict the draw shape of rock fragment flow. The model was implemented in a computer program and the development of the flow zone was investigated. The modeling results were compared with the experimental results of a large scale physical model as well as the ellipsoid theory. The outcomes were analyzed and discussed. STOCHASTIC FLOW OF GRANULAR MATERIAL UNDER GRAVITY
According to Litwiniszyn's theory [18], the stochastic flow of granular material can be illustrated by a two dimensional system of blocks (Fig. 1). Each block contains a ball subjected to the force of gravity. Removal of a ball from block a~ in the bottom layer will cause either one of the two balls immediately above it to take its place, assuming equal probability of both events, i.e. 1/2. As a result of the displacement of the ball from block a2 to a~, block a2 will be occupied by a ball from block a3 or b3. Similarly, removal of the ball from block b2 will cause its place to be taken by a ball from block b3 or c3. The process continues and the resulting distribution of probabilities is illustrated in Fig. 1. Given a sufficiently dense network of blocks, removal of a sufficient number of balls from block a~ will give a distribution curve approaching Gauss' curve, symmetrical with respect to a perpendicular straight line passing through the center of block a~. A more precise description of the above model can be obtained using a planar system of blocks in the Cartesian coordinate system {X, Z}, with the Z axis directed perpendicularly upward, as shown in Fig. 2. Each block contains material particles subject to the force of gravity, acting parallel to the Z axis, but in the opposite direction. Under the influence of gravity, the particles can only displace downward (e.g. from C to A or B). The tUniversity of Alaska-Fairbanks, Department of Mining and Geological Engineering, School of Mineral Engineering, P.O. Box downward migration of particles corresponds with an oppositely directed upward migration of empty spaces. 755800, Fairbanks, U.S.A. 323
324
CHEN:
TECHNICAL NOTE
Letting p = q = produces:
1/2 and rearranging the equation
b P(x, z + b) - P(x, z) b a 2 P(x + a, z) - 2P(x, z) + P(x - a, z)
= ~-
[r I! U _J rrr-lrrc I U I I II II I; I ,1°,1, r-lVr-lr,
r~7
Fig. 1. 2-D system of blocks [18].
These two migration processes are two attributes of draw motion in a granular medium. With the system of blocks as illustrated in Fig. 2, a void migrates from block A to C with a probability p or from B to C with a probability q and p + q = 1. In the case being studied, it can be further assumed that there is no bias along horizontal direction (x direction), i.e. the void has an equal chance to migrate from A to C, or from B to C and, therefore, p = q = 1/2. To the centers of blocks A, B and C, the corresponding coordinates (Fig. 2) may be assigned as follows:
(2)
Letting A = aZ/2b results in:
OP(x, z) OZ
b3 ] c3
a 3
i7r-
aS
O2p(x, z) -- A
(3)
OX 2
Being a function of the block width a and the block height b, the parameter A relates to the ratio of horizontal to vertical displacements of the voids at each motion step, which depends on the properties of the material and the condition of flow. The equation is formally identical to the diffusion equation. The function P(x, z) being a solution of equation 3 describes the probability of the appearance of an empty block at the coordinate (x, z). Assuming that N particles are removed from the draw point during the time period At, an equivalent of N voids will be migrating upward, originating at the draw point. If N is sufficiently large, in accordance with the law of large number, there will be N*P(x, z) voids going up through the block at the coordinate (x, z). Defining J- to be the flux of voids in the direction of the Z axis (number of voids going through a unit area perpendicular to Z axis per unit time) produces:
J-(x, z) = NP(x, z) a2At
(4)
Substituting equation 4 into equation 3 produces:
(x-a,z),
(x+a,z),
(x,z+b).
~ ( x , z)
Let P(x, z) denote the probability of the occurrence of an empty block with the coordinate (x,z). The probabilities that voids occur at block A or B are P(x - a, z) and P(x + a, z), respectively. The void has probabilities p and q, respectively, to migrate from A to C, or from B to C. Thus, P ( x , z + b) which is the probability that a void occurs at block C can be represented as:
P(x, z + b) = pP(x - a, z) + qP(x + a, z).
(1)
Oz
A ~J_-(x, z)
w h e r e a 2 = projection area of a block on the plane
perpendicular to the Z axis; Z = the z direction component of the flux of voids at the coordinate (x, z); N -- the total number of voids generated from the draw point (or the total number of particles removed from the draw point); At = the time period during which the N voids are generated. For a point opening at the coordinate (0, 0), Mullins obtained a solution in which the coefficient A in equation 5 was taken as a constant [12]: J:(x, z) - 2(nAz) Q l/2 exp
fI ! I i
x-a
- ~
•
(6)
I I I I
c3Jy OJt~~- ÷ ~ = 0.
I
I
( x2)
It is assumed that there is no discontinuity of the flux, i.e. the inflow of voids equals the outflow at every block during any time period. This continuity condition is mathematically expressed as:
--- z + b
/
(5)
Ox2
(7)
Comparing equations 5 and 7 produces: X I x+a
Fig. 2. The coordinates of blocks []8].
t~J:Qx
J,(x,z)=
-Aox-4(nAz3)
( X 2) l/2exp - ~ . (8)
CHEN:
TECHNICAL NOTE
If n is the number of voids (or particles) per unit volume, the vertical and horizontal velocities of flowing voids are V:=J~/n and V,=J~/n, respectively. By taking integration, the time required for a void to travel from the draw point (0, 0) to location (x, z) is given by:
T(x,z)
-
4n(nA)'/2 3Q exp ( ~x2 ) z 3/2.
(9)
Note that T(x, z) is also the time required for a particle to travel from location (x, z) to the draw point, since a particle travels at the same speed, but in the opposite direction, as the corresponding void. The above discussion may be applied to steady-state flow where the flow pattern is independent of time. This happens in bins having the same filling rate at the top as the drawing rate at the bottom. In caving mines, however, the flow zone is not in a steady state. It is in a process of expanding or growing, that is, in a nonsteady state. Experimental work shows that the density of granular material in the flow zone is lower than that outside the flow zone, i.e. p~ = p0 - Ap, where p= is the density in the flow zone, p0 the density outside the flow zone and Ap the density change. Assuming the boundary of the flow zone is advancing at a speed of U = J/Ap, where J is the flux at the same point as U, the time required for the flow zone boundary to reach (x, z) is then:
325
Ap T(x, z).
(10)
The particle at (x, z) starts moving toward the draw point after the flow zone boundary reaches the point (x, z). The total time required for the particle initially at (x, z) to reach the draw point at (0, O) is, therefore, given by:
T*(x,z)=(1 +-~)T(x,z)
(11)
where T(x, z) is the time required for a particle at (x, z) to reach the draw point at (0, 0) in steady state. A given time T* permits determination of all the points reaching the draw point in time T*. Connecting all these points produces the draw envelope defined by the stochastic theory. DISCREPANCY IN DRAW ENVELOPES AND MODEL MODIFICATION Based on Mullins' results [13], the above stochastic model generated a draw envelope that had a narrow top and a relatively wide bottom (Fig. 3). Most experimental rcsults [4,16,17], however, showed a wide top and a relatively narrow bottom (Fig. 4). This discrepancy can . .:,"'~:... •
Z
•" .:"
..." ....• jl
"'1°
POSSII~i-E "-
-"
....
I
,
DRAW
" - , "",... / •
SHAPE
";
i
/
ELLIPSOIDAL SHAPE
I I
I
l
J
"I :I
I, I:
!
I:
I i
$ I
I :l *t
f. |:
3 :! :#
I*,
i".. l: Ik
:l :l :I
I" -i :t :I
l ~. ~'.:
,° i .° I .l
,~'.
*I • ".
--
°p
x
Fig. 3. Draw envelope from Mullins' results [13].
Fig. 4. Possible draw shape.
326
CHEN:
TECHNICAL NOTE
be corrected by introducing a variable coefficient A in equation 5, instead of treating the coefficient A as a constant. As mentioned previously, the coefficient A is a parameter related to the properties of the granular material and the flowing conditions. The coefficient A is mathematically defined as a function of the block width a and the block height b (refer to equation 3), where a can be interpreted as the displacement distance in the X direction during a particle movement increment and b the displacement distance in the Z direction. Experimental work shows that the displacement occurring in the flow zone is unevenly distributed. In most cases, the horizontal displacement distance during a particle movement increment increases as the flow zone develops upwards. This behavior of particle movements in the flow zone produces a slightly wider top and relatively narrower bottom. Therefore, it is reasonable to assume that the coefficient A is a function of z--the vertical distance from the draw point. The actual relation between A and z is not precisely quantified at present, but for simplicity, it is assumed that A is a linear function of z, as expressed below:
A(z) = g(1 + C'z)
(12)
where K and C are constants which relate to granular material properties and may be determined through
where coefficient A (z) is no longer a constant but a linear function of z. FINITE DIFFERENCE MODELING AND RESULTS
The finite difference method is employed to solve equation 13. A FORTRAN program is implemented and a draw point with a finite width instead of a point opening is considered in the model. The program computes the flux line trajectories and the velocities of flow along the trajectories. The moving cones can be generated by setting a number of points on a horizontal line before initiating ore drawing, computing the distances that the points pass after drawing and connecting all these points at new positions. The time required for a particle to reach the draw point can also be estimated. Connecting the initial coordinates of all the points having the same exit time produces the draw envelope for any given time period. Figure 5 shows the sequence of moving cones at various times after the initiation of ore drawing. The moving cone changes during the drawing process. At the moment the base of a moving cone reaches the draw point, its volume approximately equals the volume of draw. With this information it is possible to estimate the
7,
~gJ=(x, z) O2J.(x, z) d2J..(x, z) Oz - A(z) ~ - K(1 + C'z) dx 2 (13)
_/,3
× Fig. 5. Computer output of moving cones. (1) Draw point. (2) Moving cones at consecutive times. (3) Horizontal line before drawing.
--X Fig. 6. Draw shapes at different draw heights.
CHEN:
TECHNICAL NOTE
327
7,
volume of ore that can be drawn before barren rock reaches the draw point. The draw envelopes at three different draw heights are shown in Fig. 6. It can be seen that the draw envelope becomes more elongated as the height of draw increases. This phenomenon has been observed in many physical models, but the conventionally accepted ellipsoid theory can not describe it. This relationship is more clearly illustrated in Fig. 7, where draw height to draw point width ratio (horizontal axis) is compared with draw height to draw width ratio (vertical axis). With a fixed draw point width, the curve shows a clear pattern of increasing draw height to draw width ratio as the draw height increases. The constants K and C are important parameters in this model and represent flow material properties. The two parameters significantly affect the shape of draw envelopes. The value of K affects the slimness of draw envelope (Fig. 8). By defining the slimness of draw envelope S as: draw height S - draw width
(14)
the relation between S and K can be illustrated as in Fig. 9, which shows that as K increases the draw envelope becomes wider and vice versa. The value of C affects the shape of the draw envelope (Fig. 10). With a horizontal line at 50% of draw height dividing the draw envelope into the upper and lower halves, the shape factor R is defined as: R = area of upper half draw envelope area of lower half draw envelope "
(15)
The relation between R and C can be depicted as in Fig. 11. The curve shows that the greater the value of C, the wider the top of the draw envelope (greater R value) and vice versa.
--
X
Fig. 8. The effect of parameter K upon the slimness of draw envelope. (1)/~
= 0.5. ( 2 ) / ~
= 0.3.
The draw point width also affects the slimness and shape of the draw envelope. Figure 12 shows draw envelopes with different draw point widths and Fig. 13 gives the relationship between the width of draw envelope and width of draw point.
4.0
3.0
5.0
2.0
4.0
3.0 1.0 D r a w Height D r a w Point Width
0
I
10
I
20
I
30
I
40
Fig. 7. The relation between the ratio of draw height to draw width and the ratio of draw height to draw point width.
2"o 1I 0
! 0.2
i 0.4
~ 0.6
I 0.8
I 1.0
K
Fig. 9. The relation between parameter K and function S (C = 0.05).
328
CHEN:
TECHNICAL NOTE R
2.0
1.5
1.0
0.5
0
I
0.05
I
I
0.1
I
0.15
0.2
c
Fig. 11. The relation between parameter C and function R (K = 0.6).
X Fig. 10. The effect of parameter C upon draw shape. (1) C = 0.05, (2) C = 0.00.
COMPARISONS AND DISCUSSIONS Large scale physical modeling experiments were performed by Peters [3]. The results of the experiments are compared with the stochastic model developed in this study. Data computed by ellipsoid theory are also presented as a reference. Three items are compared in the study to evaluate the stochastic model and they are: (1) The ratio of draw height to draw width at different stages of the draw envelope development. (2) The area (volume) of draw at different draw heights. (3) The ratio of upper half of the draw to lower half of the draw. The first two items are considered important because these are the key parameters for caving mine design. The ratio of the height to the width of the draw shape determines the horizontal spacing of the draw points. The volume of draw at different draw heights determines how much ore can be drawn before barren rocks reach the draw point. Since a variable coefficient is used in equation 13 to simulate the phenomena of a relative wider top and narrower bottom of draw shape, the third comparison is introduced to evaluate this effect.
/ j l
19.
X Fig. 12. The effect of draw point width upon draw shape. (1) Wide draw point. (2) Narrow draw point.
CHEN:
T E C H N I C A L NOTE
329
30--
20 "
Z
(cm)
0
30
~
10-Draw
0
i
I
,
10
Pointwidth 2I0
Fig. 13. The relation between draw width and draw point width (draw height = 254 cm).
The physical model developed by Peters [4] was 4.6 m (15 It) high and 1.8 m (6 ft) wide. It was considered the largest physical model ever built at that time. For the stochastic computer simulation, the material constants K -60
Z (cm)
-30
60
X (cm)
Fig. 15. Draw envelopecomparison. Solidline--computersimulation. Dashed line---experiment. Draw point width = 30 cm (12 in).
X -60 -30
0
30
(cm)
60
Fig. 14. Draw envelopecomparison. Solid line--~omputersimulation. Dashed line--experiment. Draw point width = 20 cm (8 in).
and C were selected based on the draw shape of the physical model tests at one draw height and used for different draw heights throughout the comparison. The same process was used to select material constant for the ellipsoid theory. Typical draw shape comparisons between the stochastic model and the physical model are shown in Figs 14 and 15. Figure 14 is the draw shape with a draw point width of 20 cm (8 in) and Fig. 15 is the draw shape with a draw point width of 30 cm (12 in). Detailed comparisons among the physical model, stochastic model and ellipsoid theory are given in Tables 1 and 2. Both Figs 14 and 15 show excellent agreement between the draw shapes generated by the physical model and the computer model. Both models display a relatively cylindrical midsection of the draw envelope, which can not be predicted by the ellipsoid theory. Data presented in Tables 1 and 2 show that the differences in the area of the draw envelope between the physical model and computer model are generally small, most within 8%. The differences between the physical model and the ellipsoid are significantly larger. The physical model and the computer model agree well on the draw height to draw width ratio at various stages of draw height development (Figs 16 and 17). Both models show the trend that at small draw heights,
330
CHEN: TECHNICALNOTE Table 1. Comparison between computer simulation and physical modeling experiments (draw point width = 20 cm) Draw height Stochastic model Ellipsoidtheory (cm) Experiment (K = 0.33, C = 0.05) (E = 0.968) 427 Area of draw 39,206 37,123 (-5.3%) 35,755(-9.0%) 396 envelope (cm2) 33,723 31,993 (-5.1%) 30,826(-9.0%) 366 29,264 27,858 (-4.8%) 26,271( - 10.2%) 297 20,813 19,264 (-7.4%) 17,342( - 16.7%) 264 17,929 16,000 ( - 10.8%) 13,703(-23.6%) 427 Draw height to 4.0 4.0 4.0 396 draw width ratio 3.76 3.9 4.0 366 3.6 3.85 4.0 297 3.8 3.8 4.0 264 3.4 3.6 4.0 427 Upper half to lower 1.09 1.04 1.0 396 half ratio 1.10 1.05 1.0 366 1.07 1.03 1.0 297 1.06 1.06 1.0 264 1.05 1.04 1.0 Percentage in parentheses indicates the difference from the experiment.
the draw height to draw width ratio is small (wider draw envelope) and becomes larger with increasing draw height (narrower draw envelope). This phenomenon can not be described by the ellipsoid theory. Both the physical model and the stochastic model show a slightly larger upper half as compared with the lower half of the draw shape. By contrast, the ellipsoid theory predicts the upper and lower halves to have equal areas. In general, the stochastic model agrees well with the physical model. It provides a better description of the flow pattern of rock fragments in caving mines than the conventionally accepted ellipsoid theory.
CONCLUDING REMARKS
As a result of this study, the following concluding remarks are given: (1) The granular material flow under gravity can be well described with the proposed stochastic model. The model predicts the flow pattern better than the
ellipsoid theory and successfully simulates some phenomena that can not be explained by the ellipsoid theory. (2) The material flow conditions vary with the distance from the draw point. The coefficient in the differential equation describing the stochastic flow should, therefore, be a function of this distance (the z coordinate). A proposed linear relation between the coefficient value and the z coordinate works well, resulting in an excellent agreement between the stochastic model and the physical model. (3) The stochastic flow model is based on the random flow pattern of granular materials and is derived from the phenomena of the flow instead of fundamental physical laws. At the present time, there is no technique available to obtain the model parameters directly from mechanical testing in the laboratory. It will require a small scale physical modeling to acquire the needed parameters. However, the link between the stochastic model parameters and the physical properties of the material can be revealed by further studies.
Table 2. Comparison between computer simulation and physical modeling experiments (draw point width = 30 cm) Draw height Stochastic model Ellipsoidtheory (cm) Experiment (K = 0.33, C = 0.05) (e = 0.968) 442 Area of draw 41,987 40,922 (-2.5%) 38,355(-8.7%) 401 envelope (cm2) 36,413 34,664 (-4.8%) 31,626( - 13.2%) 361 30,193 29,013 (-3.9%) 25,542( - 15.4%) 284 19,974 19,316 (-3.3%) 15,890(-20.4%) 234 15,510 17,626 ( - 11.3%) 10,723( - 30.9%) 442 Draw height to 4.0 4.0 4.0 401 draw width ratio 3.85 3.9 4.0 361 3.56 3.8 4.0 284 3.6 3.6 4.0 234 3.1 3.3 4.0 442 Upper half to lower 1.0 1.13 1.0 401 half ratio 1.01 1.11 1.0 361 0.98 1.10 1.0 284 0.87 0.90 1.0 234 0.92 0.89 1.0 Percentage in parentheses indicates the difference from the experiment.
CHEN: 4,0
O
3.0
"~
TECHNICAL NOTE
additional studies will expand the knowledge acquired from this 2D model study and provide a solid basis for more vigorous mine designs.
/
IIt
Accepted for publication 29 July 1996.
2.0
REFERENCES
1.O
Draw Height I
0
t 200
100
(cm) t 400
300
Fig. 16. Draw height to draw width ratio vs draw height. Solid line---computer simulation. Dashed line---experiment. Draw point width = 20 cm (8 in).
O
3.0
1.G
Draw ge£ght 0
331
I 100
l 200
I 300
(am)
I 400
Fig. 17. Draw height to draw width ratio vs draw height. Solid line---computer simulation. Dashed line--experiment. Draw point width = 30 cm (12 in).
(4) The current 2D model may be further developed to simulate 3D flow patterns. The interaction of adjacent draw points may also be included. These
1. Janelid I. Study of the gravity flow process in sublevel caving. In Proceedings of Sublevel Caving Symposium, Atlas Copco, Stockholm, 65 p (1975). 2. Yenge L. I. Analysis of bulk flow of materials under gravity caving process---Part 2: Theoretical and physical modeling of gravity flow of broken rock. CSM Quarterly 76, 67 (1981). 3. Su H. and Wei S. The properties of ore flow. Nonferrous Metals 35, China, 7-12 (1983). 4. Peters D. C. Physical modeling of the draw behavior of broken rock in caving. CSM Quarterly 79(1) (1984). 5. Kvapil R. Gravity flow of granular materials in hoppers and bins in mines. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 2, 25-41 (1965). 6. Kvapil R. Gravity flow of granular materials in hoppers and bins in mines--II, coarse material. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 2, 277-304 (1965). 7. Wang C. et al. Study of Caving, Metallurgy Industry, Beijing, 168 p (1982). 8. Johanson J. R. Stress and velocity fields in the gravity flow of bulk solids. J. Appl. Mech. 31, 499-506 (1964). 9. Jenike A. W. Gravity flow of frictional-cohesive solids--convergence to radial stress fields. J. Appl. Mech. 32, 205-207 (1965). 10. Enstad G. G. A novel theory on the arching and doming in mass flow hoppers. Thesis, the Chr. Michelsen Institute (Norway), 95 pp. (1983). 11. Jolley D. Computer simulation of the movement of ore and waste in underground mining pillar. Canad. Min. Met. Bull. 61, 854-859 (1968). 12. Mullins W. W. Stochastic theory of particle flow under gravity. J. Appl. Physics 43, 665-678 (1972). 13. Mullins W. W. Nonsteady state particle flow under gravity--an extension of the stochastic theory. J. Appl. Physics 4, 867-872 (1974). 14. Huo Z. and Xiong G. Computer simulation of oredraw. Nonferrous Metals (China) 34, 1-6 (1982). 15. Coates D. F. Rock Mechanics Principles, CANMET Monograph 874, pp. 5.3-5.8 (1981). 16. Just G. D. The significance of material flow in mine design and production. Design and Operation of Caving and Sublevel Stoping Mines (Edited by Stewart D. R.), pp. 715-728. SME-AIME (1981). 17. Kvapil R. Sublevel caving. In SME Mining Engineering Handbook, (Edited by Hartman H. L.) 2nd Edn, Vol. 2, pp. 1789-1814. SME (1992). 18. Litwiniszyn J. Stochastic Methods in Mechanics of Granular Bodies, Int. Center for Mech. Sci., Lectures No. 93. Springer (1974).