 Open Access
 Total Downloads : 506
 Authors : E. A. P. Egbe
 Paper ID : IJERTV2IS4848
 Volume & Issue : Volume 02, Issue 04 (April 2013)
 Published (First Online): 01052013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Factorial Experimental Modelling of Ball Milling Response for Baban Tsauni (Nigeria) LeadGold 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) leadgold 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 ttest 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 ttest. 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)
subsequent test runs was carried out. Crushed ore 
1 
90.6 
20 
5 
2 
samples from jaw crusher and roll crusher ( in that 
+1 
131.1 
40 
15 
6 
order) were subjected to the same grinding conditions of mill speed, ore mass to ball mass ratio, ball size and grinding time of 10minutes [5]. The products of this 
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 + 0.0958x26.849 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)/(31) = (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 ttest, 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

Hypotheses test
The tstatistics was calculated for each effect estimate (under the hypothesis; H0:effect =0) by,
t=(Eieffect)/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= (E1effect)/(0.0147902) = (0.221 0)/(0.0147902) = 14.942.
For the average effect, b0= 2.2908 = E0 and the tvalue 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 tvalue of an effect or average effect was greater than
the corresponding tdistribution 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 tdistribution value for four degrees of freedom (t4,0.975 or t4,0.025) was read from standard ttest
table [11] as 2.776. Table 6 presents the calculated associated tvalues for all the estimated effects.
The results obtained when the estimated effects were subjected to Ttest (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.

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.

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

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

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 TurkeyIMCET 2003. 2003, Pp 451456,

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.

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.

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).

Fuerstenau, D.W., Lutch, J.J. and De, A., The effect of ball size on the energy efficiency of hybrid highpressure roll mill/ball mill grinding, Powder Technol. 105, 1999, 199204.
Comminution Circuits. Their Operation and Optimisation. JKMRC Monograph Series in Mining

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

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, 230235.

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

Gupta C.K. (Chemical Metallurgy: Principles and Practice. Weinheim: Wiley VCH Verlag GmbH and Company, 2003, 1222.

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

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

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, 13030868, 2002, pp. 3439.

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

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, 217228.

NapierMunn, 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).

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

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

Whittles, D.N., Kingman, S.W. & Reddish, D.J. (Application of numerica modelling for prediction of the influence of power density on microwaveassisted
breakage. International Journal of Mineral Processing, Elservier, 68, 2003, 719.

Wills, B.A. and NapierMunn, T. Wills Mineral Processing TechnologyAn 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 = 2E05×2 + 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
X1 
X2 (Ball 
X3 
X4 
P80 (microns) 

(speed) 
size) 
(grinding 
(MB/MO) 

Time) 

G1a 
1 
1 
1 
1 
1350.01 
1.309356 
G1b 
1 
1 
1 
1 
1342.51 
1.316671 
G1c 
1 
1 
1 
1 
1296.61 
1.363281 
G2 
1 
1 
1 
1 
883.97 
1.999665 
G3 
1 
1 
1 
1 
980.25 
1.803258 
G4 
1 
1 
1 
1 
1489.18 
1.186991 
G5 
1 
1 
1 
1 
389.6 
4.537074 
G6 
1 
1 
1 
1 
1212.02 
1.458428 
G7 
1 
1 
1 
1 
945.66 
1.869217 
G8a 
1 
1 
1 
1 
459.76 
3.84471 
G8b 
1 
1 
1 
1 
452.68 
3.904842 
G8c 
1 
1 
1 
1 
456.21 
3.874628 
G9 
1 
1 
1 
1 
1088.06 
1.624583 
G10 
1 
1 
1 
1 
926.24 
1.908408 
G11 
1 
1 
1 
1 
367.93 
4.804294 
G12 
1 
1 
1 
1 
468.63 
3.771939 
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
G1av 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1.33 
G2 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
2.000 
G3 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1.803 
G4 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1.187 
G5 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
4.537 
G6 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1.458 
G7 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1.869 
G8av 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
3.875 
G9 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1.625 
G10 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1.908 
G11 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
4.804 
G12 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
3.772 
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
b0 
2.2908 
E0 
2.291 
309.7392 2.776 
b1 
0.1107 
E1 
0.221 
14.942 or+2.776 
b2 
0.0572 
E2 
0.114 
7.7078 
b3 
0.6941 
E3 
1.388 
93.8459 
b4 
0.8131 
E4 
1.626 
109.9377 
b12 
0.0258 
E12 
0.052 
3.5158 
b13 
0.0663 
E13 
0.133 
8.9924 
b14 
0.1536 
E14 
0.307 
20.7570 
b23 
0.1424 
E23 
0.285 
19.2695 
b24 
0.2072 
E24 
0.414 
27.9915 
b34 
0.4491 
E34 
0.898 
60.7159 
Table 7 Residuals of grinding experiment responses.
Test Run Coded values y= Y= F80/y Y' Residuals Per cent
order 
X1 
X2 
X3 
X4 
(P80) 
(Actual response) 
(Predicted response) 
YY' residuals 

G1a 
5 
1 
1 
1 
1 
1350.01 
1.309 
1.3 
0.0127 0.967 
G1b 
4 
1 
1 
1 
1 
1342.51 
1.317 
1.3 
0.02 1.517 
G1c 
1 
1 
1 
1 
1 
1296.61 
1.363 
1.3 
0.0666 4.884 
G2 
9 
1 
1 
1 
1 
883.97 
2.000 
2.03 
0.034 1.692 
G3 
10 
1 
1 
1 
1 
980.25 
1.803 
1.84 
0.033 1.832 
G4 
3 
1 
1 
1 
1 
1489.18 
1.187 
1.22 
0.033 2.806 
G5 
8 
1 
1 
1 
1 
389.6 
4.537 
4.57 
0.033 0.737 
G6 
2 
1 
1 
1 
1 
1212.02 
1.458 
1.49 
0.033 2.254 
G7 
6 
1 
1 
1 
1 
945.66 
1.869 
1.9 
0.033 1.781 
G8a 
11 
1 
1 
1 
1 
459.76 
3.845 
3.98 
0.131 3.397 
G8b 
13 
1 
1 
1 
1 
452.68 
3.905 
3.98 
0.07 1.804 
G8c 
16 
1 
1 
1 
1 
456.21 
3.875 
3.98 
0.101 2.598 
G9 
7 
1 
1 
1 
1 
1088.06 
1.625 
1.56 
0.0665 4.092 
G10 
12 
1 
1 
1 
1 
926.24 
1.908 
1.84 
0.0673 3.527 
G11 
15 
1 
1 
1 
1 
367.93 
4.804 
4.74 
0.067 1.394 
G12 
14 
1 
1 
1 
1 
468.63 
3.772 
3.71 
0.0666 1.767 
6
5 RÂ² = 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.
Test 
Coded values 
(F80) 
y= (P80) 
Y= F80/y 

X1 
X2 
X3 
X4 

V1 
1 
1 
1 
1 
1767.644 
369.5 
4.783881 
V2 
1 
1 
1 
1 
1767.644 
466.447 
3.789595 
V3 
1 
1 
1 
1 
1767.644 
452.55 
3.905964 
V4 
1 
1 
3 
1.515 
1350 
218 
6.19271 
V5 
1 
1 
3 
1.504 
1342.5 
220.35 
6.09258 
V6 
1 
1 
3 
1.524 
1489.2 
211.98 
7.0251 
V7 
1 
1 
2 
1.506 
1296.6 
228.463 
5.67536 
V8 
1 
1 
2 
2.005 
883.97 
221.72 
3.98688 
V9 
1 
1 
2 
1.89 
926.24 
213.87 
4.33086 
Test 
Coded values 
(F80) 
y= (P80) 
Y= F80/y 

X1 
X2 
X3 
X4 

V1 
1 
1 
1 
1 
1767.644 
369.5 
4.783881 
V2 
1 
1 
1 
1 
1767.644 
466.447 
3.789595 
V3 
1 
1 
1 
1 
1767.644 
452.55 
3.905964 
V4 
1 
1 
3 
1.515 
1350 
218 
6.19271 
V5 
1 
1 
3 
1.504 
1342.5 
220.35 
6.09258 
V6 
1 
1 
3 
1.524 
1489.2 
211.98 
7.0251 
V7 
1 
1 
2 
1.506 
1296.6 
228.463 
5.67536 
V8 
1 
1 
2 
2.005 
883.97 
221.72 
3.98688 
V9 
1 
1 
2 
1.89 
926.24 
213.87 
4.33086 
Table 8: Summary of grinding model validation and optimum test runs results.
Y' Residuals
YY'
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