Minerals Enghleerhlg, Vol. 3, No. I/2, pp, 199-206, 1990 0892-6875/90 $3.00 +0.00 Printed in Great Britain Pergamon Press plc

COMPUTER SIMULATION OF THE BOND GRINDABILITY TEST

K.A. LEWIS, M. PEARL and P. TUCKER

Warren Spring Laboratory, Gunnels Wood Rd, Stevenage, Herts., England

ABSTRACT

The Bond grindability test is widely used, in the minerals industry, to provide data fundamental to the design of commercial milling installations. The traditional Bond test method is, however, labour intensive and yields only a limited amount of information. Extrapolation of this information, for example to investigate a range of product sizes, can only be achieved through the application of empirical correction factors or through carrying out a separate Bond test at each product size.

The work presented in this paper was aimed at alleviating these limitations. A new Bond grindability test method was developed which not only significantly reduces the experimental testwork involved but also implicitly provides the necessary data for as many product sizes that are required. The new method is based on a computer simulation which closely parallels the traditional method. A population balance model of size reduction forms the kernel of the simulator. This model is based on a well-known mathematical representation of the comminution process which has been adapted specifically for the traditional Bond mill. The development of the model is discussed in detail and the simulation methodology outlined. The simulation results are presented and compared with results obtained using the traditional method.

Keywords Grindability; Bond test; simulation; modelling

INTRODUCTION

The object of the Bond grindability test is to determine the comminution parameter which expresses the resistance of a material to grinding. This parameter is known as the work index. By definition the work index is the kWh/short ton required to reduce the material from a theoretically infinite feed size to 80 percent passing 100 microns. However work indices are now usually calculated using the product specification required and in kWh/tonne. The primary function of the work index is predicting the size and capacity of new milling installations for commercial plants. The test was originally developed by Bond [1]. A full description is given by Deister [2]. In summary, the traditional test technique used a cyclic procedure of grinding and screening in order to determine the mill operating conditions which will produce a stable 250% recirculating load. At this point, the average amount of undersized material produced during each mill revolution (g/rev) remains constant. In practice, the operator reaches this point by a process of trial and error where the g/rev tends to climb initially and then go through damped oscillations, gradually converging on the endpoint. Often five, six or more cycles through the procedure may be required before the amount of undersized material, produced during each mill revolution,

199

200 K . A . LEWIS et al.

begins to stabilise. Then a further three cycles are normally carried out to confirm equilibrium. On each cycle, the undersized material is removed from the mill charge and the weight made up by adding a representative amount of original mill feed. The Bond mill itself is made to the standard specification: 0.31 x 0.31 metres with an average ball charge size of 25 mm and total ball weight of 20Kg. Rotational speed is specified as 70rpm.

Calculation of the work index (WI) is made using the following standard formula:

44.5 (I) WI (Kwh/short ton) =

(CCS) °'23 x (g/rev) °'82 x (i__00 - i__00

JP JF

Where CSS is the closing screen size, P is the 80% passing size of the undersized mill product in microns and F is the 80% passing size feed in microns, with g/ rev taken as the equilibrium value.

The test itself does, however, have some quite severe drawbacks, not least being the amount of time it takes to carry out a determination. (One test may take two or more shifts to complete depending upon the hardness of the ore and the operators experience). Because the g/ rev depends on the limiting screen size, the relationship between WI and CSS can not be directly determined from the experimental results, so to determine WI at different closing screen sizes would usually mean a separate test being carried out at each size, or by the use of empirical extrapolation factors. [3,4].

Warren Spring Laboratory has recently carried out some further research into the sensitivity of the Bond test. This research has led to the development of a new rapid test method, designed to minimise the time needed to conduct the test and to provide a fuller range of information (from a single determination) whilst still preserving an accuracy and reliability associated with the traditional method. This paper focuses primarily on the new test. Results from the full sensitivity analysis (e.g. ball charge weight, mill speed, size analysis etc.) are outside the scope of the paper and will be discussed in a separate publication. In developing the model,it has been assumed that the experimental and simulated Bond mills are set up exactly to specification.

The new test is based on the computer simulation which parallels the tradition test procedure. The core of the simulator is a standard selection/breakage function model of the grinding process. Models of this type have been developed over the last 30 years and are regularly reported in the technical literature. Details are now so well known that they need not be reiterated here. Simulation of the Bond test by computer simulation has previously been reported by Austin et al. [5] using selection and breakage functions derived from laboratory batch milling tests to simulate continuous closed circuit operation at 250% recirculating load. Comparisons with standard Bond test results were limited to just two materials, one of which showed close agreement whilst the other was at variance by about 25%. It was thought, in this study, that a simulation route that more closely mimicked the semi-batch nature of the experimental test might offer better potential for reliability.

MODEL DE VE L OPME NT

The first stage of the model development was the data collection. Two different ores were selected for the initial in-depth study. The first, a volcanic tuff , was a relatively soft material. The second, a gold ore, was much harder. In addition, a wide variety of other ores were used in the subsequent validation work.

In most respects, for data collection, the normal Bond test procedure was followed. However , in order to maximise the amount of information gathered, at the end of each grinding period a complete size analysis was carried out on the unscreened mill product i.e.

The Bond grindability test 201

total undersized and oversized material. Figures la and lb show the equilibrium, graphically, for the two ores.

)

2

I 0 0 - -

50

Period g/rev

I 2.5 2 2.2 :3 1.9 4 1.57 5 1.43 6 1.48 7 1.46 8 1 . 4 4

• ~ ~ ' .11.1.-

F e e d ~ . ~

P2

I I I I00 500 I000

I 00

50

c

-,"~ Period g/rev

I 0.99 2 i . i i 3 1.24 4 1.31 5 1 3 1 6 i .34 7 1.40 8 l . 36

I0

o

P4 o

P6 ~ e ~

P8 ~ •

Size microns

Fig.la Approach to equilibrium for Tuff C l o s i n g s c r e e n s i s e = 3 8 m i c r o n

" P 4 ~

P6 -----o

PS --m

Fig. I b

I I i I00 5 0 0 I000

Size microns

Approach to equilibrium for gold ore Closing screen sise = 125 micron

approach to

The applicability of a number of the published comminution models to the laboratory Bond mill was assessed for each of the above two ore. The model chosen as most reliable fol lowed the formulation of Reid [6]:

i wi(t) -7, ani. exp ( -k n. t ) (2)

n=l

202 K . A . LEWtS et al.

For n ¢ i

i-i

anl = Y, ( kj. blj. anj ) / ( k i k n ) (3) j=n

For n - i

i-i

aii = wi(O ) n=l

anl (4)

Where wi(t) is the weight in size fraction i after time t, k i is the selection function for size i and b . . is the breakage function from size j to size i Selection and breakage functions

1 . 1 " .

were chosen to be of the same form as those used by Austin et al. [5]:

k i = SIO00. ( X i / 1000 )~ (5)

bi j = Bij - Bi+l, j (6)

Bij = ~ . ( X i / Xj )~+ ( 1 - ~ l ) . ( X i / X j ) # (7)

where X: is the geometric mean for the size interval i in microns. S1000, or, el, 13 and ~t are all model' parameters. The mill residence time (t) is directly computed from the number of revolutions and the rpm.

Model parameters were derived by regression, of the model functions on the experimental mill feed and product size distributions. The simplex method of Nelder and Mead [7] was used for the regression, minimising the sum of squares of the residuals between the measured and the computed distribution values. Each cycle of the Bond test was treated separately thereby producing eight parameter sets for each of the two ores. Results are given in Table I. The least squares error (%E) confirms that acceptably good fits had been obtained for all data sets.

It should be noted that when parameter 3 has the value unity, parameter 5 becomes undefined. Further, parameter l is not constrained to be below unity in this application. Relaxing this constraint is found to provide better mathematical fits, though at the expense of obscuring the physical interpretation.

It is clear, f rom these results, that the model parameters evolve with each cycle of the test, eventually converging on an endpoint when the mill operation stabilises. Of the parameters, the first two (i.e. Sl000 and alpha) appear to be the most sensitive to ore type and to the grinding period. In order to achieve the aims of the investigation, it is necessary to be able to predict the way in which the parameters change as grinding progresses. The following empirical relationships were found to hold reasonably well, for all ores that have so far been studied.

The variation of SI000 from one grinding period to the next can be modelled as a function of the product of the g/ rev and the weight of undersized material produced.

SI000 = Function (g/rev x wt US) (8)

The trend of the parameter alpha, with each grinding period, was found to closely follow a hyperbolic tangent type curve (i.e. the curve increases sharply with the first few periods but levels out around a value of 1.0 as equilibrium is reached). Alpha appears closely linked to the value of Sl000. The value of S1000 determines the exact shape and position of the 'alpha relationship' curve. This is illustrated in Figure 2. Alpha is given by:

Alpha = Function (tanh(S1000 1. period)) (9)

The Bond grindability test

TABLE 1 Calculated Model Parameters

203

PARAMETER PI P2 P3 P4

Sl000 ~ ~ 7 P5 %E

Tuff

Period i 0.64 0.82 0.37 0.26 Period 2 1.31 0.15 0.67 0.84 Period 3 0.83 0.31 1.00 0.69 Period 4 0.88 0.55 1.00 0.66 Period 5 1.58 0.92 1.00 0.73 Period 6 2.23 1.03 1.00 0.72 Period 7 2.62 1.06 1.00 0.81 Period 8 2.92 1.07 1.00 0.89

5.97 3.72

0.31 0.84 0.73 i.i0 1.04 0.90 0.88 0.80

Gold

Period Period Period Period Period Period Period Period

i 0.38 0.20 0.53 0.67 2 0.56 0.18 0.79 1.48 3 0.46 0.65 1.00 0.99 4 0.40 0.72 1.00 1.01 5 0.06 1.00 1.00 0.42 6 0.39 0.99 0.97 .86 7 0.40 0.90 I000 1.00 8 0.40 0.94 1.00 1.01

4.59 5.07

(4.94)

0.85 2.06 1.74 0.77 2.35 1.68 2.29 1.88

=o

9

i I I I I 0 2 3 4 5

Period

Fig.2 Relationship between Alpha and SI000

204 K . A . LEwis et al.

The third parameter, Phi, rapidly approaches the value of 1.00 as the recirculating load builds up. It appears to stabilises at 1.0 after the first two or three periods. Within the model Phi is set to 1.0 after the third period, prior to that it is calculated using linear extrapolation between the starting value of Phi and 1.0.

Trends in gamma were the most diff icult relationship to define. Predicted model results, however, do not appear particularly sensitive to errors in gamma's value. At process equilibrium gamma stabilises and all the calculated values of gamma lie within a very narrow range of 0 - 1.5. The variation of gamma from one grinding period to the next is associated with mill residence time and is modelled using a function combining this with the g/ rev for the previous grinding period.

Beta remains unchanged from its initial value.

SIMULATION M E T H O D O L O G Y

The simulator is divided into two distinct parts. The first part uses experimental data from the first grinding period to regress to the initial set of model parameters. The calculated model parameters, together with a description of the feed material are then stored in a database, for use by the second part of the simulator, for prediction. The prediction method mimics the traditional procedure. For each cycle, material finer than the closing screen size is removed and replaced by a representative amount of new feed. Calculations proceed, using parameter values adjusted for period number. Four periods are automatically computed. On the fourth and subsequent periods, a check is made to assess whether the g/ rev has remained constant (to within 3%) for the last three periods. If so, then a stable endpoint is assumed, else computation proceeds for a further cycle. Once the endpoint is reached, the work index is calculated from equation (1). The method is shown schematically in Figure 3.

Period= Period +

RT I : S T A ~

T INPUT Se,(-up [

feed dis'( period d s t

1 .E~.ESS I

STOP

IModify CSS:Yes/No l=

Per iod '2

_l Screen at CSS I_ -I ~ odd new feed I-

I Adjos por= ors I=

CALCULATE: size dis,( g / rev new revs

Is period >=4 No

Yes ~ 4 IS g / rev s,(abLe

No over 3 periods Yes

CALCULATE WI

~ S T O P ~

Fig.3 Simulation Schematic

L parameters

The Bond grindability test 205

To simulate the effect of different closing screen sizes, part two of the simulator can be rerun at the new screen sizes. The experimental data and the parameter values derived in the first part of the simulator will still apply and need not be repeated for these predictions.

MODEL VALIDATION

The simulation technique was validated using a number of different ores and closing screen sizes. Computed results are given in Table 2 where they are compared with results obtained from the traditional method. These results demonstrate that the computer predictions compare well with those obtained experimentally. The results also confirm that the work index can be reliably predicted for different closing screen sizes f rom a single initial data set.

TABLE 2 Validation results

Css WI ~ G/rev No. Mill #m Kwh/Tonne error at equilbm revs at

equilbm.

Gold (development) measured 125 simulated 125

Tuff (development) measured 38 simulated 38

Gold2 (validation) measured I00 simulated I00

Limestone (validation) measured 50 simulated 50 measured i00 simulated i00 measured 250 simulated 250

Sulphide (validation) measured 90 simulated 90

Fluorspar (validation) measured i00 simulated I00

Granite (validation) measured i00 simulated i00 measured 250 simulated 250

15.58 1.40 217 15.85 +1.73 1.43 215

9.14 1.46 151 9.19 +0.55 1.44 154

16.39 1.36 225 16.39 **** 1.35 237

8.36 1.86 164 8.15 -2.51 1.92 169 7.77 2.85 104 8.25 +6.18 2.77 103 7.69 4.50 62 7.72 +0.39 4.39 69

18.78 0.986 352 18.48 -1.60 1.030 329

13.40 1.55 217 13.38 -0.15 1.59 210

16.67 1.23 267 15.98 -4.14 1.27 262 14.78 2.25 134 14.28 -3.38 2.19 136

M E 3 - 1 2 N

206 K.A. LEWIS et aI.

The discrepancies between the measured and calculated work indices are, for the most part, no greater than the reproducibility error of measurement. Reproducibility was assessed by repeating the traditional test 5 times (with the Gold ore). Three different operators were employed, two operators repeating the test twice. The results are given in Table 3.

TABLE 3 Reproducibility of Bond Test

Operator WI KWH/tonne

difference from simulation

la 16.31 +2.90 Ib 15.84 -0.06 2a 16.39 +3.41 2b 15.95 +0.63 3 15.58 -1.70 Simulation 15.85 *****

CONCLUSIONS

The validation work illustrates that the simulation technique for determining Bond's Grindability Work Index is a valid and accurate method which compares favourable with the traditional procedure. The amount of laboratory test work necessary is significantly reduced. Only one grinding cycle is needed compared with eight (typically) for the traditional test. An estimate of the total time, to undertake a determination, is that the new test can be accomplished in less than 40% of the time needed for the standard test method.

The new method has the added advantage of being able to directly predict the ore's Work Index at any number of different closing sizes with relatively little further investment of time (i.e. just a few minutes to run the new simulation) - rather than having to repeat the full test.

The new method simply provides a rapid alternative to the full Bond grindability test. It does not attempt (in the version presented here) to overcome any of the perceived inaccuracies (e.g. those associated with excessive fines or with very fine closing screen sizes) that have been noted in the literature [e.g Ref. 8]. Work is currently being undertaken to address these problems.

REFERENCES

.

2.

3.

4.

.

6.

7.

8.

Bond, F.C. Crushing and Grinding Calculations. British Chemical Engineering, 6 (1960), (Revised 1961 by Allis Chalmers Publications 07R923B). Deister R.J. How to Determine the Bond Work Index using Laboratory Lab Ball Mill Grindability Tests. Eng. and Min. J., 42-45 (Feb. 1987). Magdalinovic N.M. Calculations of Energy Required for Grinding in a Ball Mill. Int. J. Mineral Proc., 25, 41-46 (1989). Smith R.W. & Lee K.H. A Comparison of Data from Bond Type Simulated Closed- Circuit and Batch Type Grindability Tests. Trans. of the Soc. of Min. Engs., 91, 91- 101 (1968). Austin L.G., Bagga P. & Celik C. Breakage Properties of some Materials in a Laboratory Ball Mill. Powder Technology, 28, 235-241 (1981). Reid K.J. A Solution to the Batch Grinding Equation. Chem. Eng. Sci., 20,953-963 (1965). Nelder J.A. & Mead R.A. A Simplex Method for Function Minimisation, Computer Journal, 7, 308-313 (1965). Levin J. Observations on the Bond Standard Grindability Test, and a Proposal for a Standard Grindability Test for Fine materials. J. of the South African Ins. of Min. and Met., 89, 13-21 (1989).