BallParam Batch Curso

Moly-Cop Tools, Version 3.0

About the BallParam_Batch Spreadsheet ...

Scope : The BallParam_Batch spreadsheet was designed to be used - in conjunction with the Ball Mill Grinding Simulators BallSim in the "tuning" of such simulators to any specific actual grinding system, via the estimation of the various Model Parameters that characterize the grindability of any given ore. In other words, the attached spreadsheets provide an effective algorithm to search for the set of parameter values that best approximate the model response to the actual experimental measurements available (obtained from Batch, Laboratory Scale Mills), based on a typical non-linear, least-squares criterion. Theoretical Framework : The reference ball mill model is described with further details in the About ... worksheets of the BallSim_Direct or BallSim_Reverse files. Such model is based on the so called Modern Theory of Comminution. This theory introduced two new sets of parameters : the Selection Function S and the Breakage Function B. The first set – also referred to as Grindability – relates to the grinding kinetics of each independent particle and the second set – also referred to as Distribution of Primary Fragments – characterizes the size distribution of the fragments produced as a result of breakage events. The size-dependence of the Selection and Breakage Parameters is represented by the following relationships : - For the Selection Function :

SiE = a0 (di*)

a1 / [ 1 + (di* / dcrit)a2 ] (1)

with : di* = (di · di+1)

0.5 = average particle size of the ‘i-th’ fraction. An expanded form of this expression - also available in this simulation routine - is given by :

SiE = [1/(1+a02/a01)] { a01 (di*)

a11 / [ 1 + (di* / dcrit)a2 ] + a02 (di*)

a12 } (1e) - For the Breakage Function :

Bij = b0 (di/dj+1)b1 + (1- b0) (di/dj+1)

b2 (2)

An expanded form of this expression - also available in this estimation routine - is obtained by replacing b0 in Equation 2 by :

b0j = b00(dj+1/1000) -b01 , never > 1 (2e) These expanded forms for Si and Bij are intended to provide the model greater descriptive flexibility when tuning the model to actual specific grinding systems but, since they increment the total number of ore characteristic parameters to be estimated, its use should be avoided whenever possible. In any case, the

proposed expanded forms reduce to the normal forms if both a02 and b01 are set equal to zero.

Moly-Cop Tools / 237047277.xlsx.ms_office 7/4/2014 / 11:05 PM



On the basis of Laboratory Scale Data, like those typically obtained from batch grinding tests in small diameter ball

mills, the BallParam_Batch routine allows for the calculation of all a's and b's (including dcrit) that minimize the least-squares Objective Function : n

f = S wi [(fi - fi)/fi]2 (3)

i = 1 where the fi's represent the experimental size distribution of the mill product (as % retained on screen 'i' ) and the fi's

represent the model response for each corresponding fi, for a given set of model parameters. The a's and b's

(including dcrit) that yield the minimum possible value of f are so considered to be representative of the particular ore

and test conditions under analysis. Also in Equation 3, the wi's represent user defined Weighting Factors that quantify the relative quality and reliability of each particular mesh value with respect to the other screens data. Relatively high values of such weighting factors indicate more reliable measurements. At the extreme, a wi factor equal to zero means that this particular measurement is not being included in the Objective Function. The minimization problem stated above may be readily solved with the aid of the Excel Subroutine Solver. Data Input and Program Execution : Most of the data required by the algorithm must be defined in each corresponding unprotected white background cell - inside the red double-lined border - of the here attached Data_File worksheet. Gray background cells contain the results of the corresponding formula there defined and are protected to avoid any accidental editing. The remaining information required to run the program is entered in the Control_Panel worksheet, where the user is requested to provide initial guesses of the grinding parameters listed above. To run the program, select the objective function Cell E27 in Control_Panel and then, from the Tools Menu, select Solver ..., then Min and then By Changing any combination of Cells E10:E24 (see Useful Hints below). Clicking on the Solve button will execute the desired calculations. Important Notice : Solver ... must be run every time any element of input data gets to be modified. Otherwise, the current outputs are not valid. Calculation results are summarized in the Reports worksheet. New Moly-Cop Tools users are invited to explore the brief comments inserted in each relevant cell, rendering the whole utilization of the worksheets self-explanatory. Eventually, the user may wish to remove the view of the comments by selecting Tools / Options / View / Comments / None.




Useful Hints : The proper utilization of the parameter estimation spreadsheets - like BallParam_Batch - is by far the most complex task to be undertaken with Moly-Cop Tools. There is no set of firm recommendations to obey, but the following hints may help guiding the user in the search of the most representative set of grinding parameters : 1. It is not required to run the Solver search for all parameters at the same time. In fact, this is strongly not recommended, as the search algorithm will most likely fail to find the minimizing optimum when dealing with too many parameters at once. To exclude any given parameter off the search, remove the corresponding cell reference (in the E10:E24 range in Control_Panel) off the By Changing list of cells in Solver and that parameter value will then remain constant and equal to its original guess during the whole search. 2. When multiple experimental data sets are available (the most desirable condition) for various ore types (hard, soft, etc.) or grinding conditions (mill filling, ball size, % solids, etc.), always attempt to obtain a reasonable model fit of these data with the same average parameter values for all sets. If that is not acceptable, proceed to allow differences

in the most critical parameters in the following sequence : a0, dcrit, b0, a1and b1. Keep in mind that a larger number of adjustable parameters in the search (that is, included in the By Changing list) will necessarily yield lower Objective Function (Cell E27) values, but the values of the parameters that achieve such minimum will be less reliable in statistical terms and therefore, from one data set to another, these parameters will exhibit significant variability and covariance amongst them, rendering them meaningless. 3. An advisable starting run of the algorithm for each independent available data set should considered a search for

parameters a0, a1, dcrit and b0; leaving the others fix at nominal values : a2 = 3.0, b1 = 0.5 and b2 = 4.0. Use the table below the Control_Panel area to summarize the results of your search with the various available data sets. Be careful to follow the instructions provided at the bottom of such table. If the model fit appears to be reasonable, then you should attempt to reduce the number of adjustable parameters. Look for the one parameter that exhibit the lowest Coefficient of Variation (the ratio of its standard deviation to its average value, for all data sets) and set it fix at its average value and equal for all data sets, for the next runs of the search. Continue removing the remaining adjustable parameters off the By Changing list, one at a time, until the model fit is no longer judged acceptable. The concept behind this parameter screening process is to lump the impact of whatever ore properties or process variable are being investigated into the least possible number of parameters in order to improve the statistical significance of the estimations and facilitate the derivation of valid conclusions regarding such

effects. For instance, it is believed that the distribution of ball sizes in the mill only affects the parameters a0 and dcrit.

Similarly, hardness variations of the same ore source may be well described by changes in the a0 parameter only. Not all grinding systems require a finite value of dcrit; whenever the estimated value of this parameter tends to be greater than the top mesh opening, set it fix at 100,000 and remove it from the By Changing list.

4. If the step above does not provide a satisfactory model fit, then you should add b1 to the list of adjustable

parameters and eventually, include b01, a02 and a12 (one at a time, in that order). If the model fit is still not satisfactory, there is a good chance the data contains significant experimental errors and should be discarded.

5. Never run a search for a2 or b2, as these parameters have little effect on the objective function and may just confuse the search algorithm. When in doubt, simply try assigning them values between 2 and 4 and observe the response of the Objective Function (Cell E27). In other words, these two parameters are better optimized by a very basic trial-and-error procedure, not via the Solver routine.

green background


Moly-Cop Tools TM (Version 3.0)

Test N° 1

SELECTION FUNCTION :

alpha0 0.010675

alpha1 0.551

alpha2 3.2

Dcrit 5555

Expanded Form

alpha02 0.000000

alpha12 1.000

BREAKAGE FUNCTION :

beta0 0.40931

beta1 0.390

beta2 3.3

Expanded Form

beta01 0.000

Objective Function 0.96

(*) Suggested Default Values : alpha02 = 0 and beta01 = 0.

Note : Current calculations are not valid, if SOLVER has not been run after the last data modification.

BALLPARAM_BATCH : Estimation of Grinding Parameters from Laboratory Data

1

10

100

10 100 1000 10000 100000

% P

assin

g

Particle Size, microns

Feed

Discharge (Exp.)

Discharge (Adjusted)

SiE * 10


Moly-Cop Tools TM

Test N° 1 2 3 4

Ave. Std. Coeff.

SELECTION FUNCTION : Value Dev. of Var.

alpha0 0.006970 0.007100 0.006500 0.007230 0.00695 0.000318 0.046

alpha1 0.693 0.693 0.693 0.693 0.693 0.0000 0.000

alpha2 2.5 2.5 2.5 2.5 2.50 0.000 0.000

Dcrit 5970 6100 6030 5824 5981 117 0.020

Expanded Form

alpha02 0.000000 0.000000 0.000000 0.000000 0.00000 0.000000 #DIV/0!

alpha12 1.000 1.000 1.000 1.000 1.000 0.0000 0.000

BREAKAGE FUNCTION :

beta0 0.20000 0.23000 0.21000 0.24000 0.22000 0.018257 0.083

beta1 0.250 0.250 0.250 0.250 0.250 0.0000 0.000

beta2 4.0 4.0 4.0 4.0 4.00 0.000 0.000

Expanded Form

beta01 0.000 0.000 0.000 0.000 0.000 0.0000 #DIV/0!

Objective Function 2.87 3.15 2.17 2.65

BALLPARAM_Batch : Summary of Parameter Search Results


Moly-Cop Tools TM (Version 3.0)

Remarks Test N° 1

Mill Dimensions and Operating Conditions Net

Eff. Diam. Eff. Length Speed Charge Balls Power

ft ft % Critical Filling, % Filling, % kW Balls Overfilling Slurry

1.05 1.15 75.0 36.3 30.4 0.413 0.326 0.028 0.059

rpm 56.07

Ore Weight, 8.0 kg Balls Weight, 40.0 kg App. Grinding Time, 5 min

Ore Density, 3.30 ton/m3 Balls Density, 7.75 ton/m3 Charge

% Solids (by weight) 75.0 % Voids Fraction, 0.40 °/1 Density, Total Energy, 4.30 kWh/ton

Slurry Density, 2.095 ton/m3 Balls Volume (app.), 8.6 liters ton/m3 Balls Energy, 3.40 kWh/ton

Slurry Weight, 10.7 kg Mill Volume, 28.3 liters 4.942

Slurry Volume, 5.1 liters

Overfilling, 1.7 liters

% wi wi (error)^2

i Mesh Opening Mid-Size % Ret % Pass % Ret % Pass % Ret % Pass Exp Adj. Error

1 1.05 25400 100.00 100.00 100.00

2 0.742 19050 21997 0.00 100.00 0.00 100.00 0.00 100.00 100.00 100.00 (0.00) 1 0.00

3 0.525 12700 15554 0.00 100.00 0.00 100.00 0.00 100.00 100.00 100.00 (0.00) 1 0.00

4 0.371 9500 10984 8.76 91.24 1.32 98.68 4.73 95.27 98.68 95.27 3.58 1 12.80

5 3 6700 7978 28.58 62.66 14.61 84.07 9.46 85.81 84.07 85.81 (2.03) 1 4.13

6 4 4750 5641 25.75 36.91 4.92 79.15 6.93 78.88 79.15 78.88 0.34 1 0.11

7 6 3350 3989 11.06 25.85 5.48 73.67 5.62 73.26 73.67 73.26 0.56 1 0.31

8 8 2360 2812 4.94 20.91 5.80 67.87 5.51 67.75 67.87 67.75 0.18 1 0.03

9 10 1700 2003 2.95 17.96 5.55 62.32 5.45 62.29 62.32 62.29 0.04 1 0.00

10 14 1180 1416 2.66 15.30 6.13 56.19 5.95 56.34 56.19 56.34 (0.26) 1 0.07

11 20 850 1001 2.09 13.21 5.36 50.83 5.44 50.90 50.83 50.90 (0.13) 1 0.02

12 28 600 714 1.97 11.24 5.42 45.41 5.43 45.47 45.41 45.47 (0.13) 1 0.02

13 35 425 505 1.72 9.52 5.03 40.38 5.07 40.40 40.38 40.40 (0.05) 1 0.00

14 48 300 357 1.51 8.01 4.70 35.68 4.73 35.68 35.68 35.68 0.01 1 0.00

15 65 212 252 1.29 6.72 4.28 31.40 4.30 31.37 31.40 31.37 0.09 1 0.01

16 100 150 178 1.11 5.61 3.86 27.54 3.88 27.50 27.54 27.50 0.16 1 0.02

17 150 106 126 0.94 4.67 3.47 24.07 3.47 24.02 24.07 24.02 0.19 1 0.04

18 200 75 89 0.79 3.88 3.09 20.98 3.07 20.96 20.98 20.96 0.11 1 0.01

19 270 53 63 0.66 3.22 2.74 18.24 2.71 18.25 18.24 18.25 (0.04) 1 0.00

20 400 38 45 0.54 2.68 2.32 15.92 2.29 15.95 15.92 15.95 (0.20) 1 0.04

21 -400 0 19 2.68 0.00 15.92 0.00 15.95 0.00 0.00 0.00 0.00 0 0.00

sum 19.00 17.63

100.0

Power Decomposition, kW

BALLPARAM_BATCH : Estimation of Grinding Parameters from Laboratory Data

% Passing

Objective Function

Base Case Example

Feed Size Distributions

Mill Feed Mill Disch. (exp) Mill Disch. (adj)

Interstitial

Slurry Filling,%


Test N° 1

Remarks : Base Case Example

Ore, kg 8.0 Eff. Diameter, ft 1.05

Water, lt 2.7 Eff. Length, ft 1.15

Slurry, kg 10.7 Balls Filling, % 30.4

Slurry, lt 5.1 Speed, % Critical 75.0

Slurry Dens., kg/lt 2.095 App. Dens., ton/m3 4.942

% Solids (by weight) 75.0 Power, kW 0.41

Grinding Time, min 5.0 Energy, kWh/ton 4.30

i Mesh Opening Feed

Exp. Adj.

1 1.05 25400 100.00 100.00 / 100.00

2 0.742 19050 100.00 100.00 / 100.00

3 0.525 12700 100.00 100.00 / 100.00

4 0.371 9500 91.24 98.68 / 95.27

5 3 6700 62.66 84.07 / 85.81

6 4 4750 36.91 79.15 / 78.88

7 6 3350 25.85 73.67 / 73.26

8 8 2360 20.91 67.87 / 67.75

9 10 1700 17.96 62.32 / 62.29

10 14 1180 15.30 56.19 / 56.34

11 20 850 13.21 50.83 / 50.90

12 28 600 11.24 45.41 / 45.47

13 35 425 9.52 40.38 / 40.40

14 48 300 8.01 35.68 / 35.68

15 65 212 6.72 31.40 / 31.37

16 100 150 5.61 27.54 / 27.50

17 150 106 4.67 24.07 / 24.02

18 200 75 3.88 20.98 / 20.96

19 270 53 3.22 18.24 / 18.25

20 400 38 2.68 15.92 / 15.95

D80, microns 8408 5048 / 5031

alpha01 beta00 0.40931

alpha02 beta01 0.000

alpha11 beta1 0.390

alpha12 beta2 3.337

alpha2

Dcrit Obj. Function 0.96

Moly-Cop ToolsTM

, Version 3.0

5555

0.0000000

MODEL PARAMETERS

BreakageSelection

0.010675

0.551

1.000

3.200

Discharge

Particle Size Distributions (Cumm. % Passing)

BALL_PARAMBall Mill Grinding Model Parameter Estimator

DESIGN AND OPERATING CONDITIONSConfiguration : BATCH


Model

Selection Function : Breakage Function :

alpha01 alpha02 alpha11 alpha12 alpha2 dcrit

0.010675 0 0.550504491 1 3.199979816 5555.170763

i Mesh Opening Mid-Size SiE i

1 1.05 25400 21997.04526 0.031699688 1

2 0.742 19050 15554.25987 0.07750129 2

3 0.525 12700 10984.07939 0.181540022 3

4 0.371 9500 7978.095011 0.358702445 4

5 3 6700 5641.365083 0.604865973 5

6 4 4750 3989.047505 0.761099548 6

7 6 3350 2811.761014 0.759425428 7

8 8 2360 2002.997753 0.675560176 8

9 10 1700 1416.333294 0.572340476 9

10 14 1180 1001.498877 0.476908018 10

11 20 850 714.1428429 0.396986391 11

12 28 600 504.9752469 0.328341887 12

13 35 425 357.0714214 0.271395346 13

14 48 300 252.1904043 0.224133271 14

15 65 212 178.325545 0.185208885 15

16 100 150 126.0952021 0.153040888 16

17 150 106 89.1627725 0.126459043 17

18 200 75 63.04760106 0.104493971 18

19 270 53 44.87761134 0.086659378 19

20 400 38 19 0 20

Ebar 3.40

Matrix J

1 2 3 4 5 6

1 0.897929 0 0 0 0 0

2 0 0.768569754 0 0 0 0

3 0 0 0.539787796 0 0 0

4 0 0 0 0.295735331 0 0

5 0 0 0 0 0.128175399 0

6 0 0 0 0 0 0.075397249

7 0 0 0 0 0 0

8 0 0 0 0 0 0

9 0 0 0 0 0 0

10 0 0 0 0 0 0

11 0 0 0 0 0 0

12 0 0 0 0 0 0

13 0 0 0 0 0 0

14 0 0 0 0 0 0

15 0 0 0 0 0 0

16 0 0 0 0 0 0

17 0 0 0 0 0 0

Model

18 0 0 0 0 0 0

19 0 0 0 0 0 0

20 0 0 0 0 0 0

Matrix T

1 2 3 4 5 6

1 1 0 0 0 0 0

2 0.344555 1 0 0 0 0

3 0.101062 0.305615321 1 0 0 0

4 0.049838 0.145809252 0.469872442 1 0 0

5 0.026299 0.075923998 0.254133618 0.662396256 1 0

6 0.019637 0.056404516 0.197427702 0.6121455 1.775268243 1

7 0.018894 0.054615712 0.203607757 0.778393916 4.691217781 -208.8468066

8 0.019918 0.05870218 0.239194846 1.213513282 26.0789166 818.2821829

9 0.025162 0.076398722 0.349797878 2.661409565 -279.6763358 -1220.506035

10 0.028546 0.090940474 0.494699196 7.546163913 528.44671 820.2964666

11 0.036546 0.123672638 0.844429129 53.65708155 -423.1932757 -243.8210902

12 0.045675 0.168014402 1.611778084 -346.7100338 164.2745606 31.85777159

13 0.058649 0.240869577 3.911505378 549.6708976 -25.83646671 0.134288494

14 0.075944 0.361964723 15.02231852 -375.8652804 0.982506216 0.205747305

15 0.100092 0.584965012 492.204623 118.3965239 0.231621967 0.182596683

16 0.135711 1.058778228 -1487.516439 -13.98558277 0.166448835 0.158993679

17 0.188941 2.273214185 1593.606371 0.064224131 0.139094084 0.137009667

18 0.275346 6.743744577 -781.54388 0.040842309 0.120093691 0.119135228

19 0.409812 42.67749203 166.9724527 0.032600365 0.099516742 0.099064597

20 -2.960626 -56.09712554 -8.321892591 0.224704617 0.700123474 0.700675244

Vector SiE

1 2 3 4 5 6

1 0.0317 0.07750129 0.181540022 0.358702445 0.604865973 0.761099548

Matrix bij

1 2 3 4 5 6

1 0 0 0 0 0 0

2 0.497833 0 0 0 0 0

3 0.132108 0.410261952 0 0 0 0

4 0.079584 0.200785919 0.458542085 0 0 0

5 0.046508 0.087752401 0.17056839 0.454576771 0 0

6 0.034254 0.050765041 0.079975843 0.174533703 0.458542085 0

7 0.027764 0.035842101 0.047360904 0.080161913 0.172561801 0.45938098

8 0.022136 0.026922802 0.032314584 0.044791766 0.075479671 0.164957884

9 0.021301 0.025300161 0.029098238 0.036091253 0.049863664 0.084933524

10 0.016651 0.019599727 0.022161781 0.026160852 0.032314584 0.044791766

11 0.015477 0.018160343 0.020409658 0.02365036 0.027872752 0.034620489

12 0.01338 0.015680894 0.017582155 0.020227657 0.023387267 0.027631617

13 0.011805 0.013829237 0.015492985 0.017777587 0.020409658 0.02365036

14 0.010275 0.012035369 0.013479207 0.015452188 0.017694213 0.020356349

15 0.008945 0.01047632 0.011731849 0.013444491 0.015380927 0.017648896

16 0.007843 0.009185568 0.010286004 0.011786135 0.013479207 0.015452188

Model

17 0.006828 0.007996239 0.008954067 0.010259493 0.011731849 0.013444491

18 0.005987 0.007011226 0.007851024 0.008995493 0.010286004 0.011786135

19 0.005025 0.005884635 0.006589479 0.007550006 0.008633011 0.009891633

20 0.036297 0.042510064 0.047601748 0.05454033 0.062363308 0.071453689

1 1 1 1 1 1

Matrix Tij (inverse)

1 2 3 4 5 6

1 1 -2.32498E-15 3.59996E-15 -2.39997E-15 -2.99997E-15 -1.46248E-15

2 -0.344555 1 -9.5003E-16 5.54118E-16 3.58547E-16 2.40886E-16

3 0.004239 -0.305615321 1 -4.99009E-16 -7.95294E-17 -2.69654E-16

4 -0.001591 -0.002209035 -0.469872442 1 -9.3301E-16 -1.04982E-15

5 -0.000162 0.003206386 0.057108128 -0.662396256 1 1.26517E-15

6 0.000223 -0.000407529 -0.011179647 0.563785537 -1.775268243 1

7 0.047545 -0.090823622 -2.440602145 120.0738602 -375.4503211 208.8468066

8 0.013667 -0.096675387 -1.781109812 35.43857218 -76.51599153 17.83160576

9 -0.014878 0.686720825 12.4858731 -115.9403613 145.0283487 14.02756182

10 0.024842 0.492604141 11.17783784 -55.56639574 33.77065344 11.71731696

11 0.146371 0.943675919 42.50399562 -121.7793579 21.60292842 11.2794157

12 -0.227561 1.088344223 -75.94315148 124.0358746 15.99015717 10.39312863

13 -0.066903 3.115208377 -44.26562408 38.61047697 13.32069587 9.871333587

14 -0.029275 9.743761996 -62.5735953 23.24123007 11.64205604 9.390539753

15 -2.003607 168.7489052 -586.3494362 17.22924838 10.58969756 9.041843584

16 1.292415 -33.42233586 66.15373532 14.10049773 9.859051028 8.757717658

17 2.388305 -28.34094683 31.26616036 12.18559559 9.29733303 8.494762231

18 5.972384 -38.71442228 21.02139447 10.95465289 8.89190655 8.292434361

19 25.04455 -97.12588576 16.41851917 10.16247852 8.625176163 8.164521193

20 1 1 1 1 1 1

Matrix J

1 2 3 4 5 6

1 0.897929 0 0 0 0 0

2 0 0.768569754 0 0 0 0

3 0 0 0.539787796 0 0 0

4 0 0 0 0.295735331 0 0

5 0 0 0 0 0.128175399 0

6 0 0 0 0 0 0.075397249

7 0 0 0 0 0 0

8 0 0 0 0 0 0

9 0 0 0 0 0 0

10 0 0 0 0 0 0

11 0 0 0 0 0 0

12 0 0 0 0 0 0

13 0 0 0 0 0 0

14 0 0 0 0 0 0

15 0 0 0 0 0 0

16 0 0 0 0 0 0

17 0 0 0 0 0 0

18 0 0 0 0 0 0

Model

19 0 0 0 0 0 0

20 0 0 0 0 0 0

Matrix T

1 2 3 4 5 6

1 1 0 0 0 0 0

2 0.344555 1 0 0 0 0

3 0.101062 0.305615321 1 0 0 0

4 0.049838 0.145809252 0.469872442 1 0 0

5 0.026299 0.075923998 0.254133618 0.662396256 1 0

6 0.019637 0.056404516 0.197427702 0.6121455 1.775268243 1

7 0.018894 0.054615712 0.203607757 0.778393916 4.691217781 -208.8468066

8 0.019918 0.05870218 0.239194846 1.213513282 26.0789166 818.2821829

9 0.025162 0.076398722 0.349797878 2.661409565 -279.6763358 -1220.506035

10 0.028546 0.090940474 0.494699196 7.546163913 528.44671 820.2964666

11 0.036546 0.123672638 0.844429129 53.65708155 -423.1932757 -243.8210902

12 0.045675 0.168014402 1.611778084 -346.7100338 164.2745606 31.85777159

13 0.058649 0.240869577 3.911505378 549.6708976 -25.83646671 0.134288494

14 0.075944 0.361964723 15.02231852 -375.8652804 0.982506216 0.205747305

15 0.100092 0.584965012 492.204623 118.3965239 0.231621967 0.182596683

16 0.135711 1.058778228 -1487.516439 -13.98558277 0.166448835 0.158993679

17 0.188941 2.273214185 1593.606371 0.064224131 0.139094084 0.137009667

18 0.275346 6.743744577 -781.54388 0.040842309 0.120093691 0.119135228

19 0.409812 42.67749203 166.9724527 0.032600365 0.099516742 0.099064597

20 -2.960626 -56.09712554 -8.321892591 0.224704617 0.700123474 0.700675244

Matrix T (inverse)

1 2 3 4 5 6

1 1 -2.32498E-15 3.59996E-15 -2.39997E-15 -2.99997E-15 -1.46248E-15

2 -0.344555 1 -9.5003E-16 5.54118E-16 3.58547E-16 2.40886E-16

3 0.004239 -0.305615321 1 -4.99009E-16 -7.95294E-17 -2.69654E-16

4 -0.001591 -0.002209035 -0.469872442 1 -9.3301E-16 -1.04982E-15

5 -0.000162 0.003206386 0.057108128 -0.662396256 1 1.26517E-15

6 0.000223 -0.000407529 -0.011179647 0.563785537 -1.775268243 1

7 0.047545 -0.090823622 -2.440602145 120.0738602 -375.4503211 208.8468066

8 0.013667 -0.096675387 -1.781109812 35.43857218 -76.51599153 17.83160576

9 -0.014878 0.686720825 12.4858731 -115.9403613 145.0283487 14.02756182

10 0.024842 0.492604141 11.17783784 -55.56639574 33.77065344 11.71731696

11 0.146371 0.943675919 42.50399562 -121.7793579 21.60292842 11.2794157

12 -0.227561 1.088344223 -75.94315148 124.0358746 15.99015717 10.39312863

13 -0.066903 3.115208377 -44.26562408 38.61047697 13.32069587 9.871333587

14 -0.029275 9.743761996 -62.5735953 23.24123007 11.64205604 9.390539753

15 -2.003607 168.7489052 -586.3494362 17.22924838 10.58969756 9.041843584

16 1.292415 -33.42233586 66.15373532 14.10049773 9.859051028 8.757717658

17 2.388305 -28.34094683 31.26616036 12.18559559 9.29733303 8.494762231

18 5.972384 -38.71442228 21.02139447 10.95465289 8.89190655 8.292434361

19 25.04455 -97.12588576 16.41851917 10.16247852 8.625176163 8.164521193

20 1 1 1 1 1 1

Model

Matrix T*J

1 2 3 4 5 6

1 0.897929 0 0 0 0 0

2 0.309386 0.768569754 0 0 0 0

3 0.090747 0.234886692 0.539787796 0 0 0

4 0.044751 0.112064581 0.25363141 0.295735331 0 0

5 0.023614 0.058352889 0.137178226 0.195893976 0.128175399 0

6 0.017632 0.043350805 0.106569064 0.181033052 0.227545715 0.075397249

7 0.016966 0.041975985 0.109904982 0.230198582 0.601298711 -15.74647478

8 0.017885 0.04511672 0.129114459 0.358878752 3.342675543 61.69622589

9 0.022594 0.058717747 0.188816626 0.787072839 -35.84762596 -92.02279803

10 0.025632 0.069894098 0.267032589 2.231667284 67.73386795 61.84809734

11 0.032816 0.095051049 0.455812538 15.86829478 -54.242967 -18.38343956

12 0.041013 0.129130788 0.87001814 -102.5344067 21.05595736 2.401988353

13 0.052663 0.185125072 2.111382868 162.5571049 -3.311599432 0.010124983

14 0.068192 0.278195139 8.108864204 -111.1566432 0.125933126 0.015512781

15 0.089875 0.449586415 265.6860487 35.0140352 0.029688238 0.013767288

16 0.121859 0.813744923 -802.9432202 -4.136030951 0.021334646 0.011987686

17 0.169656 1.747123668 860.209271 0.018993345 0.01782844 0.010330152

18 0.247242 5.183038112 -421.8678486 0.012078514 0.015393057 0.008982469

19 0.367982 32.80062956 90.12969225 0.00964108 0.012755598 0.007469198

20 -2.658432 -43.11455399 -4.492056061 0.066453094 0.089738606 0.052828986

Matrix [T*J*T(inv.)]

1 2 3 4 5 6

1 0.897929 -2.08766E-15 3.23251E-15 -2.15501E-15 -2.69376E-15 -1.31321E-15

2 0.044571 0.768569754 3.83612E-16 -3.1664E-16 -6.52579E-16 -2.67334E-16

3 0.012103 0.069919271 0.539787796 -3.56993E-16 -2.30948E-16 -2.2169E-16

4 0.006744 0.033897647 0.114673528 0.295735331 -3.90167E-16 -4.17315E-16

5 0.003758 0.016407364 0.052452902 0.110991072 0.128175399 -1.0096E-16

6 0.002839 0.011080631 0.033658417 0.072815501 0.093695373 0.075397249

7 0.002605 0.009337069 0.027056118 0.059127516 0.086177873 0.089788709

8 0.00245 0.008264618 0.022974814 0.04978914 0.077696208 0.09378646

9 0.002646 0.008540931 0.02286498 0.048180434 0.076315863 0.097543006

10 0.002373 0.007441518 0.019386459 0.039702917 0.062363126 0.080691881

11 0.00236 0.007273673 0.018597699 0.037090568 0.056838323 0.071961431

12 0.002179 0.006643309 0.016793894 0.032883479 0.049248742 0.06051598

13 0.00201 0.006095486 0.015314246 0.02965941 0.043691697 0.052287835

14 0.001813 0.005480065 0.013727491 0.026441938 0.038588277 0.045368461

15 0.00162 0.004890124 0.012234693 0.023513923 0.034165482 0.039773837

16 0.001449 0.004369212 0.010927058 0.02098763 0.030447333 0.03527803

17 0.001282 0.003863359 0.009661816 0.018560167 0.026923009 0.031137319

18 0.001137 0.003428297 0.008574749 0.016478215 0.023913138 0.027642622

19 0.000966 0.002910534 0.007281105 0.013999705 0.020331416 0.023506894

20 0.007165 0.021587138 0.054032236 0.104043055 0.15142874 0.175320286

Model

Breakage Function :

Beta00 Beta01 Beta1 Beta2

0.409313626 0 0.389681706 3.33683955

Mesh Opening Beta0J Cummulative

1.05 25400 0.409313626 1

0.742 19050 0.409313626 1 1

0.525 12700 0.409313626 0.502166797 1 1

0.371 9500 0.409313626 0.37005928 0.589738048 1

3 6700 0.409313626 0.290475373 0.388952128 0.541457915

4 4750 0.409313626 0.243967256 0.301199727 0.370889526

6 3350 0.409313626 0.209712856 0.250434686 0.290913683

8 2360 0.409313626 0.181949196 0.214592585 0.243552779

10 1700 0.409313626 0.159812975 0.187669783 0.211238195

14 1180 0.409313626 0.138512239 0.162369622 0.182139958

20 850 0.409313626 0.121861355 0.142769896 0.159978176

28 600 0.409313626 0.106384327 0.124609552 0.139568519

35 425 0.409313626 0.093004297 0.108928658 0.121986364

48 300 0.409313626 0.081199118 0.095099421 0.106493379

65 212 0.409313626 0.070923645 0.083064052 0.093014172

100 150 0.409313626 0.061978744 0.072587732 0.081282323

150 106 0.409313626 0.054135757 0.063402164 0.070996319

200 75 0.409313626 0.047308213 0.055405925 0.062042252

270 53 0.409313626 0.041321702 0.048394699 0.054191227

400 38 0.409313626 0.036297123 0.042510064 0.047601748

7 8 9 10 11 12

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0.075827176 0 0 0 0 0

0 0.100815805 0 0 0 0

0 0 0.143146507 0 0 0

0 0 0 0.197945882 0 0

0 0 0 0 0.259676861 0

0 0 0 0 0 0.327858116

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

Model

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

7 8 9 10 11 12

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

-4.003478924 1 0 0 0 0

6.039660742 -3.077983353 1 0 0 0

-4.094392146 3.380577354 -2.651507226 1 0 0

1.229524081 -1.695309549 2.709260359 -2.732257673 1 0

-0.162804403 0.411364586 -1.311214887 2.774059786 -2.632796095 1

-0.000480732 -0.028474476 0.277163782 -1.296962543 2.602065034 -2.640038634

-0.001019118 0.00066194 -0.018019818 0.269698597 -1.190072175 2.600544844

-0.000914525 0.001146452 -0.001307422 -0.01466863 0.235408851 -1.177374942

-0.000797477 0.001045253 -0.000637075 -0.000522398 -0.011856842 0.230615273

-0.000687386 0.000906187 -0.000497984 2.89599E-05 -0.000796595 -0.011728569

-0.000597736 0.000788377 -0.000424812 7.31642E-05 -0.000305531 -0.000721704

-0.000497033 0.000655036 -0.000350963 6.72754E-05 -0.000214146 -0.000211995

-0.003515342 0.004622193 -0.002463954 0.000483462 -0.001432501 -0.001084273

7 8 9 10 11 12

0.759425428 0.675560176 0.572340476 0.476908018 0.396986391 0.328341887

7 8 9 10 11 12

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0.4421142 0 0 0 0 0

0.186996274 0.470289 0 0 0 0

0.076385805 0.158821474 0.4421142 0 0 0

0.048341068 0.079829108 0.181570989 0.457879656 0 0

0.034366454 0.046798537 0.08181109 0.171230818 0.455247294 0

0.027966272 0.034243736 0.048341068 0.079829108 0.17386318 0.457879656

0.02360122 0.027588847 0.034574174 0.047062656 0.079643158 0.17197523

0.020314067 0.023325148 0.027758551 0.033979618 0.046984488 0.079084695

0.017738714 0.020233683 0.02360122 0.027588847 0.034146898 0.047062656

Model

0.015419187 0.017545581 0.020314067 0.023325148 0.02751418 0.033979618

0.013512603 0.015362694 0.017738714 0.020233683 0.0234272 0.027588847

0.011339193 0.012887698 0.014866509 0.016917044 0.019453653 0.022494654

0.081904944 0.093074493 0.107309419 0.121953422 0.139719949 0.159934643

1 1 1 1 1 1

7 8 9 10 11 12

-3.74996E-16 -3.07497E-15 -2.02498E-15 -2.66247E-15 -3.37496E-16 -2.24998E-15

1.13686E-16 3.29132E-16 2.47246E-16 2.48041E-16 1.21636E-16 2.24191E-16

8.17595E-17 -1.29208E-17 -2.69315E-16 -2.14048E-16 -3.57461E-16 3.49704E-18

-8.10694E-16 -1.07165E-15 8.95966E-16 -1.29204E-15 1.94018E-16 -8.21828E-16

-2.03895E-16 1.12804E-15 -2.25535E-15 1.82441E-15 3.53824E-16 4.14203E-16

-2.46589E-16 -2.66957E-15 3.56525E-15 -2.44152E-15 -8.35324E-16 -5.44114E-16

1 -5.60361E-13 7.5453E-13 -5.17385E-13 -1.76192E-13 -1.14763E-13

4.003478924 1 1.61111E-13 -1.19532E-13 -3.11676E-14 -2.39597E-14

6.282980741 3.077983353 1 2.90697E-13 4.724E-14 7.33035E-14

7.219690794 4.780717749 2.651507226 1 6.19562E-15 3.91619E-14

8.26143688 6.418404018 4.535340606 2.732257673 1 9.57042E-14

8.477077697 7.260885189 5.896402355 4.419417547 2.632796095 1

8.619748106 7.843693562 6.927220584 5.854883452 4.348618373 2.640038634

8.573472561 8.107158558 7.546774849 6.855005266 5.756221659 4.328178801

8.500717222 8.235843036 7.924405042 7.524632784 6.801217895 5.756641388

8.395806446 8.258845051 8.112404036 7.918437608 7.49275279 6.810685856

8.250532818 8.19169968 8.147944471 8.087864521 7.875340098 7.482724209

8.124654481 8.11083225 8.126030007 8.145551687 8.072014813 7.884086881

8.045765103 8.055949635 8.101519484 8.163384154 8.172559498 8.115741711

1 1 1 1 1 1

7 8 9 10 11 12

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0.075827176 0 0 0 0 0

0 0.100815805 0 0 0 0

0 0 0.143146507 0 0 0

0 0 0 0.197945882 0 0

0 0 0 0 0.259676861 0

0 0 0 0 0 0.327858116

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

Model

0 0 0 0 0 0

0 0 0 0 0 0

7 8 9 10 11 12

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

-4.003478924 1 0 0 0 0

6.039660742 -3.077983353 1 0 0 0

-4.094392146 3.380577354 -2.651507226 1 0 0

1.229524081 -1.695309549 2.709260359 -2.732257673 1 0

-0.162804403 0.411364586 -1.311214887 2.774059786 -2.632796095 1

-0.000480732 -0.028474476 0.277163782 -1.296962543 2.602065034 -2.640038634

-0.001019118 0.00066194 -0.018019818 0.269698597 -1.190072175 2.600544844

-0.000914525 0.001146452 -0.001307422 -0.01466863 0.235408851 -1.177374942

-0.000797477 0.001045253 -0.000637075 -0.000522398 -0.011856842 0.230615273

-0.000687386 0.000906187 -0.000497984 2.89599E-05 -0.000796595 -0.011728569

-0.000597736 0.000788377 -0.000424812 7.31642E-05 -0.000305531 -0.000721704

-0.000497033 0.000655036 -0.000350963 6.72754E-05 -0.000214146 -0.000211995

-0.003515342 0.004622193 -0.002463954 0.000483462 -0.001432501 -0.001084273

7 8 9 10 11 12

-3.74996E-16 -3.07497E-15 -2.02498E-15 -2.66247E-15 -3.37496E-16 -2.24998E-15

1.13686E-16 3.29132E-16 2.47246E-16 2.48041E-16 1.21636E-16 2.24191E-16

8.17595E-17 -1.29208E-17 -2.69315E-16 -2.14048E-16 -3.57461E-16 3.49704E-18

-8.10694E-16 -1.07165E-15 8.95966E-16 -1.29204E-15 1.94018E-16 -8.21828E-16

-2.03895E-16 1.12804E-15 -2.25535E-15 1.82441E-15 3.53824E-16 4.14203E-16

-2.46589E-16 -2.66957E-15 3.56525E-15 -2.44152E-15 -8.35324E-16 -5.44114E-16

1 -5.60361E-13 7.5453E-13 -5.17385E-13 -1.76192E-13 -1.14763E-13

4.003478924 1 1.61111E-13 -1.19532E-13 -3.11676E-14 -2.39597E-14

6.282980741 3.077983353 1 2.90697E-13 4.724E-14 7.33035E-14

7.219690794 4.780717749 2.651507226 1 6.19562E-15 3.91619E-14

8.26143688 6.418404018 4.535340606 2.732257673 1 9.57042E-14

8.477077697 7.260885189 5.896402355 4.419417547 2.632796095 1

8.619748106 7.843693562 6.927220584 5.854883452 4.348618373 2.640038634

8.573472561 8.107158558 7.546774849 6.855005266 5.756221659 4.328178801

8.500717222 8.235843036 7.924405042 7.524632784 6.801217895 5.756641388

8.395806446 8.258845051 8.112404036 7.918437608 7.49275279 6.810685856

8.250532818 8.19169968 8.147944471 8.087864521 7.875340098 7.482724209

8.124654481 8.11083225 8.126030007 8.145551687 8.072014813 7.884086881

8.045765103 8.055949635 8.101519484 8.163384154 8.172559498 8.115741711

1 1 1 1 1 1

Model

7 8 9 10 11 12

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0.075827176 0 0 0 0 0

-0.3035725 0.100815805 0 0 0 0

0.457970416 -0.31030937 0.143146507 0 0 0

-0.310466192 0.340815628 -0.379553998 0.197945882 0 0

0.093231338 -0.170913997 0.387821157 -0.540839156 0.259676861 0

-0.012344998 0.041472052 -0.187695831 0.549113712 -0.683676225 0.327858116

-3.64526E-05 -0.002870677 0.039675027 -0.256728395 0.675696079 -0.865558092

-7.72769E-05 6.67341E-05 -0.002579474 0.053385727 -0.309034206 0.852609733

-6.93459E-05 0.00011558 -0.000187153 -0.002903595 0.061130231 -0.38601193

-6.04704E-05 0.000105378 -9.11951E-05 -0.000103407 -0.003078948 0.075609089

-5.21225E-05 9.1358E-05 -7.12847E-05 5.7325E-06 -0.000206857 -0.003845306

-4.53246E-05 7.94809E-05 -6.08103E-05 1.44825E-05 -7.93394E-05 -0.000236617

-3.76886E-05 6.6038E-05 -5.02391E-05 1.33169E-05 -5.56087E-05 -6.95043E-05

-0.000266558 0.00046599 -0.000352706 9.56993E-05 -0.000371987 -0.000355488

7 8 9 10 11 12

-3.3672E-16 -2.7611E-15 -1.81829E-15 -2.39071E-15 -3.03048E-16 -2.02032E-15

-2.86431E-17 -6.9839E-16 -4.36473E-16 -6.33094E-16 -1.09311E-17 -5.23804E-16

3.68064E-17 -2.08709E-16 -2.71058E-16 -2.98889E-16 -1.95009E-16 -1.4963E-16

-2.23055E-16 -4.20928E-16 1.33749E-16 -5.27743E-16 -3.47577E-17 -3.17722E-16

-1.7595E-16 -1.20524E-16 -1.83901E-16 -9.70196E-17 3.34506E-17 -1.4747E-16

-2.04721E-16 -1.79931E-16 -1.35873E-16 -6.18536E-17 1.38815E-17 -1.25133E-16

0.075827176 -6.24758E-17 -1.29702E-16 -4.51639E-17 1.07873E-17 -1.02769E-16

0.100041452 0.100815805 -1.26791E-16 -4.88959E-17 1.41175E-17 -8.65055E-17

0.11504014 0.130293196 0.143146507 -7.08835E-17 1.30455E-17 -9.10965E-17

0.098359598 0.118878135 0.14530094 0.197945882 1.54673E-17 -7.85115E-17

0.086266131 0.103904721 0.131505237 0.168664939 0.259676861 -6.07495E-17

0.069960152 0.081334987 0.100761967 0.130076011 0.179507342 0.327858116

0.058151817 0.064432745 0.075559865 0.093359342 0.126814205 0.184695235

0.04897118 0.05197272 0.057264584 0.066071627 0.084131382 0.117874108

0.042139069 0.043405095 0.045467669 0.049032684 0.057291941 0.073790039

0.037001465 0.037454793 0.037974097 0.038951478 0.042094306 0.049049029

0.032501117 0.032605452 0.03246938 0.032318917 0.033095545 0.035440453

0.028792841 0.028763316 0.028391397 0.027822929 0.027638711 0.028024281

0.024466718 0.024396518 0.023984818 0.023333875 0.022837424 0.022496315

0.182481144 0.181742516 0.178173541 0.172422316 0.166912283 0.160772424

Model

1

1 1

0.545423229 1 1

0.370889526 0.541457915 1 1

0.290727612 0.368896115 0.54061902 1 1

0.245935846 0.293416444 0.375661136 0.5578858 1 1

0.209844593 0.243552779 0.290727612 0.370889526 0.529711 1 1

0.183683741 0.211238195 0.245935846 0.294503721 0.370889526 0.5578858 1

0.160033381 0.183365443 0.211315358 0.246162653 0.291060418 0.376314811 0.54212

0.139805723 0.159978176 0.183683741 0.2117962 0.24426188 0.294503721 0.37089

0.122028136 0.139568519 0.160033381 0.183829928 0.210018144 0.246162653 0.29106

0.106575947 0.121874306 0.139677032 0.160228708 0.182429297 0.211588479 0.243998

0.093131457 0.106493379 0.122028136 0.139914642 0.159104149 0.183829928 0.210018

0.081345322 0.093014172 0.106575947 0.122175928 0.138870466 0.160228708 0.182429

0.071085829 0.081282323 0.093131457 0.106756741 0.121324885 0.139914642 0.159104

0.062090336 0.070996319 0.081345322 0.093244138 0.105962191 0.122175928 0.13887

0.05454033 0.062363308 0.071453689 0.081904944 0.093074493 0.107309419 0.121953

13 14 15 16 17 18 19

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

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Model

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Model

1

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0.13972 0.159935 0.183569 0.211384 0.245757 0.293701 0.375148042 0.554172 1

20

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0

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1

20 SiE

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1

Mill Mill

20 Feed Discharge

0 1 0.00 1 -1.49407E-13

0 2 0.00 2 -3.36415E-14

0 3 8.76 3 4.728541094

0 4 28.58 4 9.456655866

0 5 25.75 5 6.932128776

0 6 11.06 6 5.622464185

0 7 4.94 7 5.513605589

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0 10 2.09 10 5.439631756

0 11 1.97 11 5.428995181

0 12 1.72 12 5.067358651

0 13 1.51 13 4.726844324

0 14 1.29 14 4.302568841

0 15 1.11 15 3.875562122

0 16 0.94 16 3.472330069

0 17 0.79 17 3.068862234

0 18 0.66 18 2.709239027

0 19 0.54 19 2.294995284

1 20 2.68 20 15.95173094

100.00 100.00

Model

i Mesh Opening Partial

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Model

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Model

0

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0.032877 0.045489 0.077189 0.167813 0.445828 0

0.211384 0.245757 0.293701 0.375148 0.554172 1 0

Model

0.316997

0.775013

1.8154

3.587024

6.04866

7.610995

7.594254

6.755602

5.723405

4.76908

3.969864

3.283419

2.713953

2.241333

1.852089

1.530409

1.26459

1.04494

0.866594

100 100

100 100

100 100

91.24 98.68

62.66 84.07

36.91 79.15

25.85 73.67

20.91 67.87

17.96 62.32

15.3 56.19

13.21 50.83

11.24 45.41

9.52 40.38

8.01 35.68

6.72 31.4

5.61 27.54

4.67 24.07

3.88 20.98

3.22 18.24

2.68 15.92

BallParam Batch Curso

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