Factorial Experimental Modelling of Ball Milling Response for Baban Tsauni (Nigeria) Lead-Gold Ore

Factorial Experimental Modelling of Ball Milling Response for Baban Tsauni (Nigeria) Lead-Gold Ore

E. A. P. Egbe1,1*

1Mechanial Engineering Department, Federal University of Technology, Minna, Nigeria.

Abstract

An empirical model of the ball milling response of Baban Tsauni (Nigeria) lead-gold ore was developed in this work by applying factorial experimental design. The four factors considered were, mill speed, steel ball diameter, grinding time and the ratio of mass of ball to mass of ore samples. The factors were treated at two levels and the measurable ball milling response was the size reduction ratio. The student t-test was used for determining the significance of the factors. All the factors were found to have both main and interactive effect on ball milling response. Predicted values calculated using model equation were in good

agreement with experimental values of the response (R2 = 0.997). The mathematical model indicated that ball milling output increased with increase in grinding time, ball mass to ore sample mass ratio and very slowly with increase in mill speed. The response decreased with increase in ball size. This work established a model for selecting the parameters required for a desired size reduction ratio. The work also indicated that ball milling response is a function of the feed particle size.

Key words: four factors at two levels, ball milling response, model equation, experimental design.

1.0 Introduction

Energy consumption in cumminution is the highest in mineral dressing processes [13], [19] and conservation of energy can be achieved by controlling the conditions that affect grinding response.

A well structured factorial experimental design makes it possible for effects of several variables on the performance of a system to be determined with a few

experimental runs and a line of action for product or process improvement is identified. The effects of interactions between factors can also be examined. The data generated from the designed experiment are used to develop a modelling equation which is subjected to significance test and analysis of variance to determine quality of the model before validation and optimization [1], [14], [11] [3].

2.0 Literature review

The resistance of materials to breakage is often expressed as its grindability and the Bonds Work index has generally been accepted as the measure of grindability in the mineral industry [2], [20], [4], [8]. The effects of operating conditions on the performance of mills have been observed over the years [15], [10], [19]. It has also been indicated that the product size, the power consumption and

production rate are influenced by the size of the ball utilised in a ball milling operation [6], [9], [18]. The number of balls and particle density have also been indicated as factors that affect the performance of a ball mill [15].

In a factorial design at two levels each of the factors are taken at two treatment levels: low (-1) and high (+1). The numbers of factors which are perceived to influence the process determine the number of

experimental runs for a full factorial design at two

levels. When three factors are considered to be of importance the factorial design denoted by 23 will require eight experimental runs [11], [15]. As a rule each of the runs must be unique and the order of run is usually random in order to avoid structural error. The modelling equation for the 23 factorial experiments above is given by:

Y=b0+b1X1+b2X2+b3X3+b12X1X2+b13X1X3+b23X2X3+b

123X1X2X3 1

where b0, b1, b2 to b123 are coefficients which are determined from measurable responses of eight runs; X1, X2, X3 are the coded values of the factors of interest and Y is a measurable performance indicator [15]. A well designed experiment provides the eight responses required to solve eight simultaneous equations that are obtainable from Equation 1.

It is indicated that a good modelling information can be obtained from a half factorial design of experiments[11]. Particularly when a large number of factors appear to be of interest, a half factorial design provides a means of screening out the insignificant factors by carrying out smaller number of test runs. It is a subset of full experimental design. Further improvement design at more levels with only significant factors from the screening stage experiment leads to optimum conditions.

3.0 Experimental Procedures

Y b0 b1X1 b2 X2 b3 X3 b4 X4

b12X1X2 b13X1X3 b14X1X4 b23X2 X3

b24X2 X4 b34X3X4 2

where all the parameters are same as defined for Equation 1. Table 1 shows the coded and actual values of the factors for the test runs. To avoid structured error, the experimental runs were performed in a random order. There are eleven unknown parameters in Equation 1 and twelve unique responses were obtained for the simulation of a modelling equation while the replicated test runs were used for analysis of variance. The response that was measured was breakage reduction ratio given by, 80% passing size of feed divided by 80% passing size of the product (F80/P80 [5]). The set of twelve unique responses were used with Equation 1 on a MATLAB programme to generate the coefficients for the modelling equation. The significance of main effects was tested with student t-test. The validation and optimum test runs were carried out and the responses were subjected to residual analysis.

Table 1: The coded and actual values of the factors

for the test runs.

The cylindrical mill that was used has internal diameter of 125mm and length of 175mm. The mill did not have lifters. An initial sample

Coded values

x1= Speed (rpm)

x2= Ball diameter (mm)

x3= Grinding time

x4= MB/MO

preparation aimed at ensuring a homogeneous feed for

(minutes)

batch milling operations were mixed thoroughly and passed through a Jones riffling sampler, until sets of 0.51kg samples were obtained.

Four factors were considered in this work, namely; mill speed (x1), ball diameter (x2), holding time (x3) and Mass of ball to mass of ore ratio (x4). The factors were considered at two levels. The high level was coded +1 and the low level -1. The form of the modelling equation adopted for this work is given by,

4.0 Results and Discussion

The results of particle size analyses for the feeds used for factorial experimental design of the effects of grinding conditions are presented in Figure

1. The 80% passing size was determined from the equation of the line. The equation of the line in Figure 1 is,

y = -0.00002×2 + 0.0958x-26.849 3

The eighty per cent passing size of the feed (F80) was calculated by substituting y =80 into Equation 3.

Therefore,

x2 -4790x + 5342450 = 0. 4

Solving the quadratic equation yielded, x = F80 = 1767.644m.

The products of each unique grinding condition were subjected to particle size analyses and the 80% passing sizes (P80) were calculated in the same way as for the fed above.

The summary of the responses in terms of ratio of F80/P80 for twelve unique grinding conditions, which

S 2 = (0.000441+0.000169+0.001089)/(3-1) = (0.001699)/2 = 0.0008495

1

1

p

p

By a similar calculation, the variance in the replicated test run condition G8 was found to be 0.0009. The pooled variance (S 2) is the mean of all variance from

replicated test runs. Thus,

p

p

S 2 = (0.0008495 +0.0009)/2

= 0.000875

According to Devor et al. (2007) the sample variance of an effect is calculated by,

were required for the calculation of the eleven coefficients of Equation 2, are presented in Table 2.

S 2

4(S 2

)

)

p

p

8

E N

where N = the total number of test runs used for the model. In this work N = 16 and hence,

There are eleven unknown parameters in Equation 2 and matrix methods yielded easy solution.

y =X 5

2 4(0.000875)

S

S

E 16

= 0.00021875

where y = column matrix of the responses (column 13 on Table 2 )

X = an m by n calculation matrix of coded values of xk deduced from Equation 1 (Table 3, column 2 to 12 by row 2 to 13 ), with b0 represented by one and = column matrix of the coefficients. Multiplying Equation 2 by XT and rearranging yielded,

= (XTX)-1XTy 6

The calculation matrix (Table 3) was used with Equation 6 in a MATLAB programme to generate the coefficients of the modelling equation (Equation 2) and these coefficients are presented in Table 4.

In order to test for the significance of the coefficients by using students t-test, the variance of the effect estimates and the average effect were calculated from the variance of the replicated observations [17], [12], [5]. The calculated variance for the replicated test runs are presented in Table 5.

.

The standard error (s.e) of the effects is the square root of the sample variance and hence,

2

2

s.e = (0.00021875) = 0.0147902

The sample variance of the average (Sav ) is given by,

Sav p

Sav p

2=S 2/N 9

= 0.000875/16 = 0.0000547.

Therefore,

sav = (0.0000547) = 0.0073959

1. Hypotheses test

   The t-statistics was calculated for each effect estimate (under the hypothesis; H0:effect =0) by,

   t=(Ei-effect)/s.e 10

   where Ei = effect estimate = 2xbi for i = 1, 2, 3,34 and the bi are the coefficients of the anticipated model equation.

   The alternative hypothesis is, H0:effect 0. Substituting for b1 = -0.1107 yielded E1 = -0.221 and by Equation 4.20,

   t= (E1-effect)/(0.0147902) = (-0.221- 0)/(0.0147902) = -14.942.

   For the average effect, b0= 2.2908 = E0 and the t-value is,

   Adopting the expression given by [3], the variance in the test run condition G1 became,

   E0 effect

   t=

   S

   2.2902 309.735 11

   0.0073959

   1

   1

   11

   11

   S 2 (u

   – u1

   2 n

   av.

   The null hypothesis requires that effect should be 0.

   u12 – u n

   2

   1

   1

   u – u 2 n

   7 The null hypothesis was rejected when the calculated t-value of an effect or average effect was greater than

   the corresponding t-distribution at a desired

   13 1

   where u1i = individual responses in G1i for i = a, b and c, 1 = mean of the responses in G1 and n = number of replicates. Substituting the responses yielded,

   confidence level [17], [12], [3]. A 95% confidence level was considered adequate in this work. The corresponding t-distribution value for four degrees of freedom (t4,0.975 or t4,0.025) was read from standard t-test

   table [11] as 2.776. Table 6 presents the calculated associated t-values for all the estimated effects.

   The results obtained when the estimated effects were subjected to T-test (Table 6) showed that the calculated absolute t values for all the effects were higher than t4,0.975 (or t4,0.025). This implied that the main effects and interaction effects affected the grinding outputs which were measured in terms of size reduction ratio- F80/P80 at a confidence level of 95% [3], [11]. This meant that interactions between the four factors under consideration were important in addition to the individual effects of the factors.

2. The equation for the model

   The implication of rejecting the null hypothesis for all the effects was that the model equation should contain all the coefficients of Table 5. The resulting modelling equation for predicting future grinding outputs then became Equation 12.

   Y 2.2908 0.1107 X1 0.0572X 2

   0.6941X 3 0.8131X 4 0.0258X1 X 2

   0.0663X1 X 3 0.1536 X1 X 4

   0.1424 X 2 X 3 0.2072 X 2 X 4 12

   0.4491X 3 X 4

   An empirical model is normally checked for possible presence of structured error before validation and adoption for future prediction. This was done by subjecting the model to residual analysis. The calculated residuals of all the responses used for generating the model and their percentages are presented on Table 7.

   The actual residuals and their percentages (as functions of the actual responses) are shown on the tenth and eleventh columns of Table 7 respectively. It was observed that the per cent residuals varied from – 0.737% to a maximum of 4.884%. The levels of these residuals were quite low and more so when the complex environment in a grinding mill is considered. Figure 2 shows a plot of predicted responses against the actual responses. A correlation of 0.997 was found column of Table 8). This observation supported the submission above on the effect of feed size. Moreover a considerable change in the grinding time from the initial mean value caused a shift in the mean particle size in the bulk sample. This explained the slight

   between the predicted values and the actual values.

   The modelling equation indicated that positive coded values of X1 (grinding speed), X3 (grinding time) and X4 (MB/MO) enhanced grinding output. This implied that grinding output increased as grinding speed, holding time and mass of ball to mass of ore ratio increased above 110.5rpm (or 0.849Vc), 10 minutes and ratio of 4 respectively. On the contrary negative values of X2 improved the grinding response. Positive (or negative) coded value is relative to the mean values used in the experiments.

   Figure 3 show the predicted responses as each factor was varied within a feasible range while the other three factors were kept at the highest level of their experimental values. The responses converged to 3.975 which marks the best response within the range of values used in the experiment.

3. Model validation

The results of breakage responses of nine model validation test runs are shown in Table 8.

The results of the first three validation test runs showed very small deviation from the predicted values with residuals ranging from 0.97% to 2.225%. The results showed reasonable deviation from the predicted values when the feed sizes (V4, V5,.V9) were altered from the value used in generating the model.

Two reasons were most likely responsible for this increase in deviation: (1) the difference in the feed size and (2) the effect of grinding time on the mean particle size in the bulk samples in the mill. It is fairly well known that the grindability of an ore increases with decrease in particle size [16], [7], [20]. A change in feed size directly implied a change in the bulk particle size if all other conditions (ball size, mill speed, grinding time and ball mass to sample mass ratio) were kept constant. The 80% feed size for the factorial grinding experiment was 1767.644m. The divergence between predicted and actual responses increased with decrease in the 80% passing size of the feed (last

difference in residuals recorded for samples ground for 25minutes and 20minutes.

When te complex environment in grinding mills was considered, the model gave a good prediction of ball milling response of Baban Tsauni ore.

5.0 Conclusion

This research work has established a mathematical relationship between the grinding response of Baban Tsauni lead gold ore and ball milling conditions. All the factors investigated in this work have direct and interactive effect on the grindability of the ore. The model equation will serve as a valuable tool for

comminution flow sheet design and production planning. The effect of feed size on breakage response was observed in this work. An adequate correction factor that will account for the effect of feed size on grindability is desired to make the model more versatile.

References

1. Chandra, M.J. Statistical Quality Control. Florida: CRC Press, P280(2001)..

2. Deniz, V. (Relationships Between Bond's Grindability (Gbg) and Breakage Parameters of Grinding Kinetic on Limestone. Proceedings of 18th International Mining Congress and Exhibition of Turkey-IMCET 2003. 2003, Pp 451-456,

3. Devor, R.E., Chang, T. and Sutherland, J.W. (Statistical Quality Design and Control, Contemporary Concepts and Methods. New Jersey: Pearson Prentice Hall, 2007, P942.

4. Doll, A., and Barratt, D. Grinding: Why So Many Tests? 43rd Annual Meeting of the,

   Canadian Mineral Processors, Ottawa, Ontario, Canada. January 18 to 20, 2011.

5. Egbe, E.A.P. (Characterization and Beneficiation of Baban Tsauni, Gwagwalada Ore,

   Nigeria, Thesis Submitted in Partial Fulfilment for the Award of the Degree of Doctor of Philosophy, Department of Mechanical Engineering, School of Engineering and Engineering Technology, Federal University of Technology, Minna, Nigeria. 2012).

6. Fuerstenau, D.W., Lutch, J.J. and De, A., The effect of ball size on the energy efficiency of hybrid high-pressure roll mill/ball mill grinding, Powder Technol. 105, 1999, 199204.

Comminution Circuits. Their Operation and Optimisation. JKMRC Monograph Series in Mining

1. Ross, S.M. Introduction to Probability and Statistics for Engineers and Scientists (Third Edition). Amsterdam: Elsevier Academic Press. 2004, P641.

2. Sahoo, A. and Roy, G. K., Correlations for the Grindability of the Ball Mill As a Measure of Its Performance, Asia Pacific Journal of Chemical Engineering, 3 (2) , 2008, 230-235.

3. Schneider, C.L. Measurement and Calculation of Liberation in Continuous Milling Circuits. Dissertation submitted to the Faculty of The

1. Gupta C.K. (Chemical Metallurgy: Principles and Practice. Weinheim: Wiley- VCH Verlag GmbH and Company, 2003, 1-222.

2. Gupta, A. and Yan, D.S. Mineral Processing Design and Operation- An Introduction. Amsterdam, Elsevier B.V., 2006, P693.

3. Kotake, N., Daibo, K., Yamamoto, T. and Kanda, Y., 2004, Experimental investigation on a grinding rate constant of solid materials by a ball milleffect of ball diameter and feed size, Powder Technol. 143 144, 196 203

4. Man, Y.T. Technical Note: Why is the Bond Ball Mill Grindability Test done the way it is done? The European Journal of Mineral Processing and Environmental Protection, Vol. 2, No. 1, 1303-0868, 2002, pp. 34-39.

5. Montgomery, D.C. & Runger, G.C. Applied Statistics and Probability for Engineers. New York: John Wiley and Sons, 2002, 2003, P822.

6. Naik, P.K., Sukla, L.B. & Das, S.CAqueous SO2 leaching studies on Nishikhal manganese ore through factorial experiment. Journal of Hydrometallurgy, Elsevier Science B.V., 54, 2000, 217-228.

7. Napier-Munn, T.J., Morell, S., Morrison, R.D. Kojovic, T Mineral and Mineral Processing 2, University of Queensland, Australia. (1996).

University of Utah in partial fulfilment of the requirements for the degree of Doctor of

Philosophy, Department of Metallurgical Engineering, The University of Utah. (1995).

1. Thella, J.S. and Venugopal, R. Modelling of iron ore pelletization using 3**(k-p) factorial design of experiments and polynomial surface regression methodology. Journal of Powder Technology, Elsevier, 211, 2011),, 54- 59.

2. Trumic, M., Magdalinovic, N., Trumic, G., , The model for optimal charge in the ball mill, Journal of Mining and Metallurgy A, 43, 2007, 1932.

3. Whittles, D.N., Kingman, S.W. & Reddish, D.J. (Application of numerica modelling for prediction of the influence of power density on microwave-assisted

   breakage. International Journal of Mineral Processing, Elservier, 68, 2003, 71-9.

4. Wills, B.A. and Napier-Munn, T. Wills Mineral Processing Technology-An Introduction to the Practical Aspects of Ore Treatment and Mineral Recovery. Seventh Edition. Amsterdam: Elsevier Science and Technology Books, P450, 2006.

Appendix: Figures and Tables.

100

y = -2E-05×2 + 0.0958x – 26.849

80

Cumulative mass passing(%).

Cumulative mass passing(%).

60

40

20

0

0 500 1000 1500 2000 2500

-20

Particle size of feed (microns).

Figure 1: Particle size analysis for factorial experimental design feeds

Table 2: The grinding conditions and summary of responses for 24** factorial grinding experiments.

Test Coded values Y= F80/ P80

Table 3: Calulation matrix for factorial experimental design of grinding conditions.

Test b0 X1 X2 X3 X4 X1X2 X1X3 X1X4 X2X3 X2X4 X3X4 Y=

F80/P80

Table 4: The calculated coefficients of Equation 1

Coefficients b0 b1 b2 b3 b4 b12 b13 b14 b23 b24 b34

Values 2.29 -0.111 -0.057 0.694 0.8131 0.0258 0.0663 0.1536 -0.142 – 0.4491

0.207

Table 6: Coefficients, effect estimates and associated t values for grinding experiment.

Coefficients Values Effects Effect

estimate

Associated t value

t4,0.025 t4,0.975

Table 7 Residuals of grinding experiment responses.

Test Run Coded values y= Y= F80/y Y' Residuals Per cent

6

5 R² = 0.997

Predicted responses

Predicted responses

4

3

2

1

0

0 1 2 3 4 5 6

Actual responses

Figure 2: Correlation between the predicted and the actual grinding responses

X2 5

4.5

Grinding response (Y)

Grinding response (Y)

4

3.5

X1 3

X3 2.5

2

X4 1.5

1

0.5

0

-1.5 -1 -0.5

Values0of factors 0.5 1 1.5

Figure 3: Response plot of the effect of each factor when others remain constant.

Table 8: Summary of grinding model validation and optimum test runs results.

Y' Residuals

Y-Y'

4.7373 0.046581

3.7053 0.084295

3.9753 -0.06934

8.74 -2.547

8.711 -2.618

8.761 -1.736

7.138 -1.462

8.171 -4.184

7.933 -3.603