ballparam open

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About the BallParam_Open Spreadsheet ...

Scope :

The BallParam_Open spreadsheet was designed to be used - in conjunction with the Ball Mill Grinding SimulatorsBallSim_Direct and BallSim_Reverse - in the "tuning" of such simulators to any specific actual grinding system, via theestimation of the various Model Parameters that characterize the grindability of any given ore. In other words, theattached spreadsheets provide an effective algorithm to search for the set of parameter values that best approximate themodel response to the actual experimental measurements available (obtained from Continuous, Pilot or Full PlantScale 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 orBallSim_Reverse files.

Such model is based on the so called Modern Theory of Comminution. This theory introduced two new sets ofparameters : the Selection Function S and the Breakage Function B. The first set also referred to as Grindabilityrelates to the grinding kinetics of each independent particle and the second set also referred to as Distribution ofPrimary 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 inEquation 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 tuningthe model to actual specific grinding systems but, since they increment the total number of ore characteristicparameters to be estimated, its use should be avoided whenever possible . In any case, the proposed expanded

forms reduce to the normal forms ifa02 and b01 are set equal to zero.


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On the basis ofPilot or Industrial Scale Data, like those typically obtained from plant sampling or audit campaigns, the

BallParam_Open routine allows for the calculation of all a's and b's (including dcrit) that minimize the least-squaresObjective 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 off are so considered to be representative of the particular ore under analysis.

Also in Equation 3, the w i's represent user defined Weighting Factors that quantify the relative quality and reliability ofeach particular mesh value with respect to the other screens data. Relatively high values of such weighting factorsindicate more reliable measurements. At the extreme, a w i factor equal to zero means that this particular measurement isnot 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 Turquesa backgroundcell - inside the red double-lined border - of the here attached Data_File worksheet. blue background cells contain theresults of the corresponding formulas 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 isrequested 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 ofCells E10:E24 (see Useful Hints below). Clicking on theSolve button will execute the desired calculations.Important Notice : Solver ... must be run every time any element of input data gets to be modified . Otherwise, thecurrent outputs are not valid.

Calculation results are summarized in the Reports worksheet.

An interesting feature of this routine is that the user has the option to save for later reference every analyzed data set bycopying the Data_File worksheet into as many as required Test 1, Test 2, etc. worksheets. For reprocessing these data,

simply copy the information back to the Data_File and re-run the Solverroutine.

New Moly-Cop Tools users are invited to explore the brief comments inserted in each relevant cell, rendering the wholeutilization of the worksheets self-explanatory. Eventually, the user may wish to remove the view of the comments byselecting Tools / Options / View / Comments / None.


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Useful Hints :

The proper utilization of the parameter estimation spreadsheets - like BallParam_Open - is by far the most complex taskto 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 Solversearch 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 inControl_Panel) off the By Changing list of cells in Solverand that parameter value will then remain constant and equalto 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.)

orgrinding conditions (mill filling, ball size, % solids, etc.), always attempt to obtain a reasonable model fit of these datawith 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 adjustableparameters in the search (that is, included in the By Changing list) will necessarily yield lowerObjective Function (CellE27) values, but the values of the parameters that achieve such minimum will be less reliable in statistical terms andtherefore, 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 thegreen background table below the Control_Panel area to summarize the results of your search with the variousavailable data sets. Be careful to follow the instructions provided at the bottom of such table. If the model fits appearto be reasonable, then you should attempt to reduce the number of adjustable parameters. Look for the one parameterthat exhibit the least Coefficient of Variation (the ratio of its standard deviation to its average value, for all data sets) andset it fix at its average value and equal for all data sets, for the next runs of the search. Continue removing the remainingadjustable parameters off the By Changing list, one at a time, until the model fit is no longer judged acceptable. Theconcept behind this parameter screening process is to lump the impact of whatever ore properties orprocess variableare being investigated into the least possible number of parameters in order to improve the statistical significance of theestimations 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 oresource may be well described by changes in the a0 parameter only. Not all grinding systems require a finite value ofdcrit; whenever the estimated value of this parameter tends to be greater than the top mesh opening, set it fix at 100,000and remove it from the By Changing list.

4. If the step above does not provide satisfactory model fits, 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 fits are still not satisfactory, there is agood chance the data contains significant experimental errors and should be discarded.

5. Never run a search fora2 orb2, as these parameters have little effect on the objective function and may just confuse thesearch algorithm. When in doubt, simply try assigning them values between 2 and 4 and observe the response of theObjective Function (Cell E27). In other words, these two parameters are better optimized by a very basic trial-and-errorprocedure, not via the Solverroutine.


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Moly-Cop Tools TM

Test N 1

SELECTION FUNCTION :

alpha0 0.009180

alpha1 0.650

alpha2 2.5

Dcrit 6532

Expanded Form

alpha02 0.0000000

alpha12 1.000

BREAKAGE FUNCTION :

beta0 0.20000

beta1 0.250

beta2 4.0

Expanded Form

beta01 0.000

Objective Function 0.00

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

BALLPARAM_OPEN : Estimation of Grinding Parameters from Plant Scale Data

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

1

10

100

10 100 1000 10000 100000

%P

assing

Particle Size, microns

Feed

Discharge (Exp.)

Discharge (Adjusted)

SiE * 10

Moly-Cop Tools / 138336330.xls.ms_office 4/6/2013 / 2:24 PM


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Moly-Cop Tools TM

Remarks Test N 1

Mill Dimensions and Operating Conditions 3348 BallsDiameter Length Speed Charge Balls Lift 0 Overfilling

ft ft % Critical Filling,% Filling,% Angle, () 536 Slurry

18.50 22.00 72.0 38.00 38.00 35.00 3885 Net Power

rpm 12.82 10.0 % Losses

4316 Gross kW

% Solids (by weight) 72.0 Charge Apparent Mill Flowrate, tph (dry) 1622.8

Ore Density, ton/m3 2.80 Volume, Ball Density

Slurry Density, ton/m3 1.86 m3 Charge Interstitial Excess ton/m3 Total Energy, kWh/ton 2.39

Balls Density, ton/m3 7.75 63.76 296.48 47.48 0.00 5.395 Balls Energy, kWh/ton 2.06

% wi wiABS(error)

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 3.68 3.68 (0.00) 3 0.00

3 0.525 12700 15554 2.35 97.65 1.12 98.88 1.12 98.88 6.19 6.19 (0.00) 4 0.00

4 0.371 9500 10984 6.65 91.00 2.56 96.32 2.56 96.32 8.47 8.47 (0.00) 5 0.00

5 3 6700 7978 5.98 85.02 2.51 93.81 2.51 93.81 10.84 10.84 (0.00) 5 0.006 4 4750 5641 4.83 80.19 2.28 91.53 2.28 91.53 12.44 12.44 (0.00) 5 0.00

7 6 3350 3989 4.43 75.76 2.37 89.16 2.37 89.16 13.04 13.04 (0.00) 5 0.00

8 8 2360 2812 4.41 71.36 2.72 86.44 2.72 86.44 14.66 14.66 (0.00) 5 0.00

9 10 1700 2003 4.49 66.87 3.16 83.28 3.16 83.28 17.23 17.23 (0.00) 5 0.00

10 14 1180 1416 5.38 61.49 4.13 79.16 4.13 79.16 21.06 21.06 0.00 5 0.00

11 20 850 1001 5.82 55.67 4.86 74.30 4.86 74.30 25.91 25.91 (0.00) 5 0.00

12 28 600 714 7.07 48.60 6.20 68.10 6.20 68.10 31.51 31.51 (0.00) 5 0.00

13 35 425 505 8.19 40.41 7.57 60.53 7.57 60.53 36.23 36.23 (0.00) 5 0.00

14 48 300 357 8.72 31.68 8.76 51.77 8.76 51.77 39.12 39.12 0.00 5 0.00

15 65 212 252 7.72 23.97 8.85 42.92 8.85 42.92 39.02 39.02 (0.00) 5 0.00

16 100 150 178 5.75 18.22 7.74 35.18 7.74 35.18 35.96 35.96 (0.00) 5 0.0017 150 106 126 3.91 14.31 6.10 29.08 6.10 29.08 30.53 30.53 (0.00) 5 0.00

18 200 75 89 2.59 11.73 4.51 24.57 4.51 24.57 24.08 24.08 (0.00) 5 0.00

19 270 53 63 1.78 9.95 3.33 21.24 3.33 21.24 35.18 35.18 (0.00) 5 0.00

20 400 38 45 1.21 8.74 2.40 18.84 2.40 18.84 29.08 29.08 (0.00) 4 0.00

21 -400 0 19 8.74 0.00 18.84 0.00 18.84 0.00 24.57 24.57 (0.00) 3 0.00

sum 94.00 0.00

Slurry

Interstitial

Slurry Filling,%

100.00


%Ret. (smoothed)

Objective Function

Base Case Example

Feed Size Distributions

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

Mill Charge Weight, tons



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Moly-Cop Tools TM Test N 1

Remarks : Base Case Example

Throughput, ton/hr 1622.8 Diameter, ft 18.50

Water, m3/hr 631.1 Length, ft 22.00

Slurry, ton/hr 2254.0 Balls Filling, % 38.0

Slurry, m3/hr 1210.7 Speed, % Critical 72.0Slurry Dens., ton/m3 1.862 App. Dens., ton/m3 5.395

% Solids (by weight) 72.0 Net Power, kW 3884.53

Energy, kWh/ton 2.39

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 97.65 98.88 / 98.88

4 0.371 9500 91.00 96.32 / 96.32

5 3 6700 85.02 93.81 / 93.81

6 4 4750 80.19 91.53 / 91.53

7 6 3350 75.76 89.16 / 89.16

8 8 2360 71.36 86.44 / 86.449 10 1700 66.87 83.28 / 83.28

10 14 1180 61.49 79.16 / 79.16

11 20 850 55.67 74.30 / 74.30

12 28 600 48.60 68.10 / 68.10

13 35 425 40.41 60.53 / 60.53

14 48 300 31.68 51.77 / 51.77

15 65 212 23.97 42.92 / 42.92

16 100 150 18.22 35.18 / 35.18

17 150 106 14.31 29.08 / 29.0818 200 75 11.73 24.57 / 24.57

19 270 53 9.95 21.24 / 21.24

20 400 38 8.74 18.84 / 18.84

D80, microns 4680 1273 / 1273

alpha01 beta00 0.20000

alpha02 beta01 0.000

alpha11 beta1 0.250

0.0000000

MODEL PARAMETERS

BreakageSelection0.009180

0.650

Discharge

Particle Size Distributions (Cumm. % Passing)

BALL_PARAMConventional Grinding Model Parameter Estimator

DESIGN AND OPERATING CONDITIONSConfiguration : OPEN


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Moly-Cop Tools TM

Remarks Test N 1

Mill Dimensions and Operating Conditions 3348 BallsDiameter Length Speed Charge Balls Lift 0 Overfilling

ft ft % Critical Filling,% Filling,% Angle, () 536 Slurry

18.50 22.00 72.0 38.00 38.00 35.00 3885 Net Power

rpm 12.82 10.0 % Losses

4316 Gross kW

% Solids (by weight) 72.0 Charge Apparent Mill Flowrate, tph (dry) 1622.8

Ore Density, ton/m3 2.80 Volume, Ball Density

Slurry Density, ton/m3 1.86 m3 Charge Interstitial Excess ton/m3 Total Energy, kWh/ton 2.39

Balls Density, ton/m3 7.75 63.76 296.48 47.48 0.00 5.395 Balls Energy, kWh/ton 2.06

% wi wiABS(error)

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 3.68 3.68 (0.00) 3 0.00

3 0.525 12700 15554 2.35 97.65 1.12 98.88 1.12 98.88 6.19 6.19 (0.00) 4 0.00

4 0.371 9500 10984 6.65 91.00 2.56 96.32 2.56 96.32 8.47 8.47 (0.00) 5 0.00

5 3 6700 7978 5.98 85.02 2.51 93.81 2.51 93.81 10.84 10.84 (0.00) 5 0.006 4 4750 5641 4.83 80.19 2.28 91.53 2.28 91.53 12.44 12.44 (0.00) 5 0.00

7 6 3350 3989 4.43 75.76 2.37 89.16 2.37 89.16 13.04 13.04 (0.00) 5 0.00

8 8 2360 2812 4.41 71.36 2.72 86.44 2.72 86.44 14.66 14.66 (0.00) 5 0.00

9 10 1700 2003 4.49 66.87 3.16 83.28 3.16 83.28 17.23 17.23 (0.00) 5 0.00

10 14 1180 1416 5.38 61.49 4.13 79.16 4.13 79.16 21.06 21.06 0.00 5 0.00

11 20 850 1001 5.82 55.67 4.86 74.30 4.86 74.30 25.91 25.91 (0.00) 5 0.00

12 28 600 714 7.07 48.60 6.20 68.10 6.20 68.10 31.51 31.51 (0.00) 5 0.00

13 35 425 505 8.19 40.41 7.57 60.53 7.57 60.53 36.23 36.23 (0.00) 5 0.00

14 48 300 357 8.72 31.68 8.76 51.77 8.76 51.77 39.12 39.12 0.00 5 0.00

15 65 212 252 7.72 23.97 8.85 42.92 8.85 42.92 39.02 39.02 (0.00) 5 0.00

16 100 150 178 5.75 18.22 7.74 35.18 7.74 35.18 35.96 35.96 (0.00) 5 0.0017 150 106 126 3.91 14.31 6.10 29.08 6.10 29.08 30.53 30.53 (0.00) 5 0.00

18 200 75 89 2.59 11.73 4.51 24.57 4.51 24.57 24.08 24.08 (0.00) 5 0.00

19 270 53 63 1.78 9.95 3.33 21.24 3.33 21.24 35.18 35.18 (0.00) 5 0.00

20 400 38 45 1.21 8.74 2.40 18.84 2.40 18.84 29.08 29.08 (0.00) 4 0.00

Mill Charge Weight, tons

Slurry

Feed Size Distributions Objective Function

Mill Feed Mill Disch. (exp) Mill Disch. (adj) %Ret. (smoothed)


Base Case Example

Interstitial

Slurry Filling,%

100.00



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21 -400 0 19 8.74 0.00 18.84 0.00 18.84 0.00 24.57 24.57 (0.00) 3 0.00



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ballparam open

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