OPTIMIZATION of Ring Burden

Int. J. Rink Mech. Min. Sci. & Geomech. Abstr. VoL I0, pp, 119-131. Pergamon Pr¢~ 1973. Printed in Great Britain

OPTIMIZATION OF RING BURDEN IN SUB-LEVEL CAVING

G. D. JUST and G. D. FREE

Department of Mining and Metallurgical Engineering, University of Queensland, Australia

and

G. A. BnsHoe Peko Mines, Tennant Creek, N.T.

( Recei~'ed 24 June 1972)

Abstract--Sub-level caving mining-system design involves the optimization of a wide range of variables. Mathematical descriptions of the relationships between these variables requires a knowledge of the flow characteristics of ore and waste material. This paper describes the development of a mathematical model to optimize the ring burdens. The model is based on the results of studies of the flow characteristics of granular material. Comparisons between theoretical and actual restdts, which are briefly described, show encouraging correlations.

I. INTRODUCTION

TIlE use of trial and error design techniques in sub-level caving mining systems is a costly procedure. This is due to the generally large scale of such operations and the need to inte- grate sub-levels in the overall design. Successful simulation of this mining method, either by physical or mathematical models, would reduce these costs. Investigations into these simulation problems has been carried out in the Department of Mining and Metallurgical Engineering at the University of Queensland since 1966. The gravity flow of material, which is one of the most important factors in sub-level caving design, is discussed in detail in a recent paper by Jus t and FREE [1]. One significant conclusion of this research and work conducted by other authors, for example JANELID and KVAPIL [2], is that the flow of material can be described in terms of an ellipsoidal draw shape.

On the basis of these conclusions mathematical models were developed to determine the recovery and dilution relationships for different designs. A series of such models was developed to account for different sources of waste dilution. This paper describes the models which enable the calculation of dilution and recovery values when the only source of dilution is from the waste material in the previous ring. The major design parameters affecting these values are the ring burden and the ring gradient. Both of these dimensions can be changed during actual operations so that it should be possible to evaluate the accuracy of the mathe- matical model during production.

The lack of accurate knowledge concerning flow parameters in actual operations may introduce inaccuracies in the quantitative calculations using these mathematical models. For this reason it should be accepted that full-scale tests are always required during produc- tion. However these models can be used to prepare design graphs indicating the relative advantages of different geometrical layouts. The model calculations may also assist in the evaluation of results from field trials.

119

120 G.D. JUST, G. D. FREE AND G. A. BISHOP

2. DESIGN FUNCTIONS

Design functions include those parameters which affect the efficiency of the sub-level caving system and the parameters which may be used to measure the extraction efficiency. These functions have been described in detail by JANELID [3] and FREE [4]. These include geometrical factors such as extraction heading width, height and spacing, sub-level interval, height of draw, and ring burden and ring inclination. Factors dependent upon the flow properties of the material include, eccentricity of the draw ellipsoid, deviation of the axis of the draw ellipsoid, width and depth o f the draw opening, angle of internal friction of the ore and waste rock, and relative swell factors. Operational factors include the degree of ore fragmentation, and the method of extraction.

Functions which have been used to measure the extraction efficiency include the extraction percentage (E), recovery percentage (R) and dilution percentage (D). These functions are defined as follows:

where 1"+ T/ To 1".

T~ E - - "~I x I00 (1)

To R = y , x 100 (2)

T. o = ~ × 100 O)

= tons of ore plus tons of waste rock drawn = tons o f ore fired = tons of pure ore drawn, and ----- tons of waste rock drawn.

Equations (i)-(3)can be combined to give the following relationship:

R = E(! -- O/lO0). (4)

Figure I is a graphical representation of equation (4). This figure and tit , above equations provide a measure of these operational parameters but do not provide a ~ of~omlmring relative extraction ¢fi~ciencies. The ideal efficiency is achieved if recovery is l ~ per cent and the dilution is 0 per cent. This cannot be achieved in the sub-level caving system where generally acceptable recovery values are from 60 to 90 per cent with dilution values from 5 to 35 per cent. An accurate measure of relative efliciencies should involve cost data. However, for the purpose of optimizing ring burden in this analysis a parameter defined as extraction efficiency ~o) has been used. This is defined as follows:

To e /~, --'-- X 100.

T,z.r/

Substituting from equations (I) and (2) it is realized that

~, = RZ/E. (5)

This may also be expressed as follows by combining equations (4) and (5):

~,, = R(l - D / l O 0 ) . (6)

OPTIMIZATION OF RING BURDEN IN SUB-LEVEL CAVING 121

0

I 00

8 0

6 0

4 0

2 0

0 20 40 60 130 I00

Recovery, %

Fl~;. I. Graphical relationship between extraction, recovery and dilution percentages.

Examination of equation (6) shows that tL is high if recovery is high and dilution is low Conversely if dilution is high and recovery is low then/~ will be extremely low. A low value of t~ will also be derived if both recovery and dilution are low. An illustration of how IL varies as extraction proceeds is given in Fig. 2. This shows that at some point in the extrac- tion a maximum etliciency is reached. It is at this point that drawing of ore should be terminated.

3. ASSUMPTIONS IN DEVELOPMENT OF MATHEMATICAL MODEL

The initial development of a mathematical model to optimize the ring burden required a number of assumptions. These were necessary in order to avoid the development of extremely complex relationships. These initial studies were subsequently modified to enable the derivation of more complex equations which will be discussed later in this paper.

Most of the assumptions related to the flow characteristics of the ore and waste rock and including the following:

(i) The caving material draw shape is eilipsoidai. (ii) Only half of the ellipsoid is formed because of the presence of the solid ring face.

(iii) Drawing occurs evenly over the full width of the extraction heading. (iv) The source of waste dilution is from the back of the ellipsoid and not from the

sides or the top. (v) The ore and waste rock have similar flow characteristics.

(vi) The bulk densities of ore and waste are equal. (vii) There is no deviation between the axis of the draw ellipsoid and the ring face.

12." G. D. JUST, G. D. FREE A N D G. A. BISHOP

I101

I00

9 0

o~ 80

70 E o ~,so ¢:¢

co so .u

4 0 T3 o: "= 30 o al

zo

E x t r o c t i 0 n ,

Recove ry ,

x t r a c t i o n ef fsciency

D i l u t i on , %

IOO 200 3 0 0 4 0 0 500 6 0 0 700 8 0 0 9 0 0 1000

Tons of ore d r o w n / r i n g

FIG. 2. Graph showing [he varialion of the opcratiorml ©lf~:iency pararr~b:rs as a ring is drawn.

The first assumption was shown to be valid for model experimenLs with cr',~hed ~ k a n d glass beads (JUST and FREE [!]). Ellipsoidal draw shapes have also been ~ s ~ l by J^NEUD and KVAmL [2]. The validity of such an assumption in full-scale operations will depend upon ma~rial characteristics and would have to be tested by full-septic exp~iments.

The second assumption is valid only if the ring face is smooth and vert~al and the flow characleristics and bulk densities of d~ ore and w~sl~ are similar [assumptions(v)and (vi)]. A rough ring face or inclination of the ring face will cause deviation of the axis oi't~c draw ellipsoid to form. Later development of the model has enabled this varial3on tobe taken into account.

Drawing will occur evenly over the full width of the extraction heading i f the loading operations are carefully controlled. Variation from this ideal could be taken into ~ :ount by substituting an effective heading width for the real he',~ling width.

It has been assumed that there is only one source of wast~ rock dilution, that is from the back of the ring. Ot3~r dilution sources are neglected, for example from the top and sides, of the ellipsoid. M~thcmatical models, including petameeers such as heading widd% spacing and sub-level interval, were developed to calculate these dilution values.

The above assumptions were necessary in order to develop a simple madn~'mm~cal model. However, due to the significant etTcct of inclination of the draw ellipsoid axisa more complex set of equations was later developed to include this paran'~er. Oevelol~nnem o f ~ equa- tions will be described later.

OPTIMIZATION OF RING BURDEN IN SUB-LEVEL CAVING 123

4. EQUATIONS FOR LNITIAL MODEL

The basic equation for an ellipse with the origin at the centre, position O in Fig. 3 is:

x ' y2 b--. ~ + ~ = 1 (7)

where a = semi-major axis dimension b = semi-minor axis dimension.

\

°1: --×

= ×"

FI(;. 3. Elliptical section of draw ellipsoid.

Ifthe origin is shifted to the position shown by O' in Fig. 3, i.e. thecentre of the back of the extraction heading, the transformed elliptical equation is:

x z ( y - - h + a) z

b -q + a-" = I. (8)

Solving equation (8) for a and b in terms of h, the height ofdraw w the extraction heading width and E M , a measure of the ellipse eccentricity produces the following:

w z + 4h" E M a = (9)

8h E M

w z + 4h 2 E M b = (I0)

8h v ' E M

where

E M = l - - ~2 = b2/a z

and c = ellipse eccentricity.

The volume of an ellipsoid (IF) is given by the equation:

(11)

V = 4 / 3 Ir ab 2.

124 G. D. JUST, G. D. FREE A N D G. A. BISHOP

If only the volume above the plane AB (shown in Fig. 4) is required, the equation is as follows:

, rh[ . 4 h Z E M ) v=ytw'+g

f

\

[

Integrotion slice

B

FIG. 4. Elliptical section showing volume integration slice.

Since only half of this volume is the volume drawn (lid) in sub-level caving this equation reduces to the following expression:

~-h I'd = -[~ (w z +'~ h" EM). (12)

Eq nation (I 2) may be rearranged to relate eccentricity to Vd, w arvJ h as fo l lows:

3w z 12Vd ( z = + I -- (13)

4h z' ~th 3 •

Eq uation (13) can be used to determine the value ofthe eccentricity from model or full-scale tests.

In order to calculate dilution from waste at the back oftbe ellipsoid, as shown in Fig. 5,

r k -

~ of ring

m

FiG. 5. Section showing the source of di lution from the back of the ellipsoid.

OPTIMIZATION OF RING BURDEN IN SUB-LEVEL CAVING 125

integration of the volumeequation is necessary. The ring burden (r) is used in this equation and is expressed as a function of the semi-minor axis dimension (b) as follows:

K = r ib .

The resultant expression for dilution is

This relationship is illustrated graphically in Fig. 6.

I00

90

80

70 .D=100-300/2 K ( I - K Z / 3 )

60

5o

L3 40

30

20

I0

OI 0.2 0.~ 0 4 0 5 0 6 0.? 0.8 0 9 IO Roho K ( = r / b )

F,(;. 6. Graphical relatkmship between dilution percentage and the ring burden to semi-minor axis ratio.

,f t-w °1

Fu;. 7. El l ip l ical sections illustrating how the volume o f a ring is ddermincd.

126 G. D. JUST. G. D. FREE A N D G. A. BISHOP

Ore recovery estimation requires the calculation of the volume of ore fired in each ring (Vs). Assuming the geometrical layout shown in Fig. 7 this may be calculated from the following equation:

V/ ---- r(is vw) (15)

g here

i = sub-level interval s = extraction heading spacing

v ---- ex trac t ion h e a d i n g he ight .

Optimization equations relating dilution to head grade (gb) and average grade (go), and extraction efficiency to recovery and extraction percentage are as follows:

D = 100 (i -- gdgo) (16)

and

~ . = R2/E.

These relationships are illustrated graphically in Figs 8 and 9.

(17)

IOC

9C

6C

o~

o '-- 4C

3C

2C

0 I 2 3 4 5 ~ r u ~ rO

~ , v e r o q e o s s o y q r o d e , q0

FJo. 8. Graphical relationship between dilution ~ 8 = , cut-off head grade and avcr-~lp: assay grade.

OPTIMIZATION OF RING BURDEN IN SUB-LEVEL CAVING 127

8

ql

8 g

tOO

90

80

70

60

5(:

4C

3C

2C

20 4 0 60 80 I00 i20 ~40 160 ~80 200

E~ f rochon~ °/o

Fl(;. 9. Graphical relationship between extraction efficiency, recovery and extraction percentages.

5. F O R M U I , A T I O N O F M A T I I E M A T I C A L M O D E l ,

Equations (I)-(I 7) can be used to calculate the relationship between ring burden, extrac- tion percentage, recovery, dilution and extraction efficiency. Geometrical data, such as extraction heading, width, height and spacing, height of draw and sub-level interval may be used in these equations for a range of values of ring burden. The eccentricity of the draw ellipsoid should z, lso be known. The various stages of the calculations for the complete mathematical model are as follows:

(i) (ii)

(iii)

(iv)

(vi) (vii)

(viii)

Calculate the semi-minor axis of the draw ellipsoid from equation (10). Use equation (12) to determine the volume of ore drawn. Select a low value of ring burden to commence the iteration procedure (suggested value of K = 0-5b). Calculate the dilution percentage using equation (14). Determine the extraction percentage from equation (I) and equation (15). Calculate recovery percentage from equation (4). Calculate extraction efficiency from equations (5) or (6). Increment the ring burden and repeat calculations from statement (iv) onwards. The iteration process should cease when K equals unity.

An example of the type of computer output for these calculations is shown in Fig. 10. These results may be used to present results in graphical form as shown in Fig. 2.

If the optimum ring burden is to be determined directly, a different approach to the

128 G. D. JUST, G. D. FREE A N D G. A. BISHOP

DATA REQUIRED FOR THIS RUN

Extraction heading width :13 Extraction heading height :10 Extraction heading spacing :38 Sob-level interval :40 Height of draw :74 Eccentricity of ellipsoid :0.950

DATA CALCULATED FOR THIS RUN

Eccentricity factor = 0"0975 Semi-minor axis = 12.4675

RING BURDEN OPTIMIZATION TABLE

Ring burden Ratio K Ext. ("(,) Re¢. ( " , ) Dit. (%) Ext. eft.

: 7"0 0"5615 131.54 99"14 24"63 74"72 : 7-5 0"6016 10'7.77 97"4] 2~(15 77"30 :8"0 0"6417 115"10 g~56 16-96 79.37 :8"5 0"6818 101"33 93"62 13"58 80~)1 : 9-0 0-7219 1~'31 91.54 10"53 81 .go : 0"5 0 " / 6 ~ g6"93 89"34 7"8~ 82"35 : 10"0 0"80el m'(]8 87"(~ 5"49 8~25 : 10"5 0"8422 87-70 64"56 3"54 81"00 : 11 "0 0. _m~__ 83" 71 8~-04 2"00 80"40 : 11 "5 0"9~4 80"07 79-36 0"88 78"67

FIG. I0 Example of primed output from Program SLC RB I.

problem may be adopted. The stages of such a calculation are as follows:

(i) From known values of head grade and average ore grade select the required value of dilution from Fig. 8.

(ii) Select the corresponding value for K from Fig. 6 and calculate the ring burden by substituting b from equation (10).

(iii) Calculate the volume drawn and volume of ore fired from equations (12) and (15) to determine the extraction percentage [equation (I)].

(iv) Finally the values of recovery percentage and extraction efficiency can be deter- mined from Figs ! and 9.

It should be noted that the above calculation of optimum ring burden is a very simplified method of optimizing this parameter. The result should give a reasonable estimate of the best value for the burden. More complex relationships involving other de~-~t and ~ t i n g parameters would be required to give an accurate value of the optimum ring burden.

6. EXTENSIONS TO ORiGiNAL MODEL

The initial mathematical model developed by Free required a number of simplifying assumptions. Subsequent physical model tes~ng by FUE [4], which coaflrmed many of JANELID'S [3] results, showed that the eccentricity of the ellipsoid of draw varied ~ height

OPTIMIZATION OF RING BURDEN IN SUB-LEVEL CAVING 129

of draw and the deviation of the ellipsoid axis varied with the ring inclination. Modifica- tions were therefore made in the original model in order to include these variables in the mathematical calculations.

The following relationship derived from experimental results was used to include eccentri- city variations (JUST and FREE [! 1):

log, (1 -- ~) = mh 2 + nh z_ q (18)

where

m ~ - +0-00549 4- 0.00037

n = --0.23850 4- 0.00819

q = --0"06400 4- 0-03529.

The variation in deviation of the axis ofdraw (0) could not be included as a mathematical relationship (Just and FREE [I]). Consequently separate calculations for a range of these values are required. Inclusion of this variable complicated the mathematical derivation of intersecting volumes since the planar ring surfaces were no longer parallel to the axis of the ellipsoid (Fig. I1). A method was developed which involved reduction of the ellipsoid

r = Ring burden h , Draw height 0 = Oeviarion of draw axis

I I

t

I I

I

//J 2_

, %'%.

FtG. I 1. Ellipsoid of draw intersecting back of ring.

is as follows:

= {( v , = abc l

where

130 G . D . JUST, G: D. FREE A N D G. A. BISHOP

to an equivalent sphere, derivation of the appropriate equations and s u ~ u e n t re-conver- sion of the sphere to the ~lipsoid shape. The equation derived for volume of m a t e ~ l drawn

-1"- h..~]-- a~ ;= (2 119)

C 2 A - - a 2 J r -

t a n z 0

and a, b and c are the ellipsoid semi-axis dimensions. The equation for volume of waste material drawn (V.) is:

V,, = -~ abc I + h -- - - r/sin O) 2 -- h --.~a - - r/sin 0~./ . 120)

A typical set of results using equations 118), 09) and (20) is illustrated in Fig, 12. These computed values show how increased deviation of the axis of draw causes a reduction in the metal recovery for a given dilution. Physical model experiments with dense ore (magnetite)

o

2 i5

60

40

20

25"

%, / / / / , 6 - % / / / / / , . . o,,.

20 to 60 80 iO0

Recovery. %

F~. 12. Theoretical dilution-recovery curves lya.-sed on the mathematical model for dilution from the I:mck of the ring.

and less dense waste (granite) showed close correspondence to the calculated values for an axis deviation of 0"5 °. (Fig. 13). Similarly for less dense ore and dense waste, tests corres- ponded to an axis deviation of 12 °. (Fig. 13). These results would appear to indicate that the relative densities of ore and waste have a major effect on the optimum ring burden.

7. CONCLUSION

The optimum ring burden in sub.level caving operations may b¢ determined from full- scale tests during production. For exampt¢, a range of ring burdens may be selected and

OPTIMIZATION OF RING BURDEN IN SUB-LEVEL CAVING 131

I

°1 "5

g ~ S_2-

P h y s i c a ! m o d e l M a m e m a r i c a l m o d e l

" 3 - I - - 12 ° O e v i a h o n o Series D ~- 3-2 of axis exDeriments • 3-3 o / •

3-4 - : - 0-5* Deviation .j~'~'% ~: 3-5 of axis . /

' 3-6 o / ~ "

,, [

,2) 4 0 60 80

Recovery. %

FIG. 13. Comparison o f physical and mathematical model results.

[ I 0 0

metal recovery and waste dilution may be monitored during extraction from the rings. However. in such a situation it may be dilficult to avoid waste dilution from sources other than the back of the ring. This will complicate the evaluation of results. The use of mathe- matical models to describe the probable trend of recovery and dilution values as design dimensions are changed will assist in this evaluation. Such models should also reduce the number of tests required and may indicate the relative significance of different design parameters .

Theoret ical est imates based on the mathemat ica l models descr ibed in this paper have shown encouraging correht t ions with results from physical model testing. Al though assump- tions were necessary in the deve lopment o f these models , subsequent modif icat ions based on the results o f physical tests, have improved the accuracy o f theoret ical predict ions. Fur the r deve lopment o f this concept o f sub-level caving design should enable more effective design and evaluat ion procedures to be appl ied to ac tual mining systems.

Acknowledgement--The authors wish to acknowledge the tinancial assistance provided for project rcsearch at Ihe University of Queensland by the Australian Mineral Industries Research Association.

REFERENCES I. JUST G. D. and FRE~ G. D. The Gravity Flow of Material in the Sub-Level Caving Mining System,

Proceedin.¢s o f the First Australia-New Zealand Conference on Geomechanics, Melbourne, pp. 88-97 (1971).

2. JANEUU I. and KVAPIL R. Sub-level caving. Int. J. Rock Mech. Min. ScL 3, 129-153 (1966). 3. JANEUL) I. Guide on Sub-Level Caving, Swedish Mining Association, B. Series No. 75 (1965). 4. FaEE G. D. Mathematical anti Model Studies o f the Flow o f Material in the Sub-Level Caving Mining

Ah.thod, Unpublished M.Eng.Sc. Thesis, University of Queensland (1970).