About ...Moly-Cop Tools, Version 3.0About the BallParam_Batch Spreadsheet ...Moly-Cop Tools, Version 3.0About the BallParam_Batch Spreadsheet ...Moly-Cop Tools, Version 3.0About the BallParam_Batch Spreadsheet ...

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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.

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.

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Control_Panel

Moly-Cop Tools TM (Version 3.0)BALLPARAM_BATCH : Estimation of Grinding Parameters from Laboratory Data

Test N1

SELECTION FUNCTION :

alpha00.000001 alpha12.611 alpha22.5 Dcrit8029 Expanded Form alpha020.000000 alpha121.000

BREAKAGE FUNCTION :

beta00.20000 beta10.250 beta24.0 Expanded Form beta010.000

Objective Function1.16
Jaime E. Seplveda J.: To minimize this value, use Tools / Solver / Min, by changing any combination of Cells E10:E24.Note : Current calculations are not valid, if SOLVER has not been run after the last data modification.

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

Moly-Cop Tools TMBALLPARAM_Batch : Summary of Parameter Search Results

Test N1234

Ave.Std.Coeff.SELECTION FUNCTION :ValueDev.of Var.

alpha00.0069700.0071000.0065000.0072300.006950.0003180.046 alpha10.6930.6930.6930.6930.6930.00000.000 alpha22.52.52.52.52.500.0000.000 Dcrit597061006030582459811170.020 Expanded Form alpha020.0000000.0000000.0000000.0000000.000000.000000ERROR:#DIV/0! alpha121.0001.0001.0001.0001.0000.00000.000

BREAKAGE FUNCTION :

beta00.200000.230000.210000.240000.220000.0182570.083 beta10.2500.2500.2500.2500.2500.00000.000 beta24.04.04.04.04.000.0000.000 Expanded Form beta010.0000.0000.0000.0000.0000.0000ERROR:#DIV/0!

Objective Function2.873.152.172.65

(*) To bring values from the above Control_Panel table proceed as follows : 1. Select all cells in the range E5:E27. 2. Use the Copy or Control_C commands. 3. Select a convenient cell in the range E35:M35 for every different data set being analyzed. 4. Use the Paste Special_Paste Values command.

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Data_File

Moly-Cop Tools TM (Version 3.0)BALLPARAM_BATCH : Estimation of Grinding Parameters from Laboratory Data

RemarksComprobando caracteristicas del mineral Cerro verdeTest N1

Mill Dimensions and Operating ConditionsNetEff. Diam.Eff. LengthSpeedChargeBallsInterstitialPowerPower Decomposition, kWftft% CriticalFilling, %Filling, %Slurry Filling,%kWBallsOverfillingSlurry0.670.6774.0
Jaime E. Seplveda J.: Rotational Mill Speed, expressed as a percentage of the critical centrifugation speed of the mill.165.0
jsepulveda: See Mill Power_Ball Mills for further details.64.6
jsepulveda: See Mill Power_Ball Mills for further details.100.0
jsepulveda: See Mill Power_Ball Mills for further details.0.450
jsepulveda: Actual measurement of the Net Mill Power Draw, i. e. discounting all losses. 0.254
jsepulveda: See Mill Power_Ball Mills for further details.0.156
jsepulveda: See Mill Power_Ball Mills for further details.0.040
jsepulveda: See Mill Power_Ball Mills for further details.rpm69.25

Ore Weight,11.1 kgBalls Weight,20.1 kgApp.Grinding Time,9 minOre Density,2.70 ton/m3Balls Density,7.75 ton/m3Charge% Solids (by weight)72.0 %Voids Fraction,0.40 /1Density,Total Energy,6.06 kWh/ton1.0525400Slurry Density,1.829 ton/m3Balls Volume (app.),4.3 literston/m3Balls Energy,3.42
jsepulveda: See Mill Power_Ball Mills for further details. kWh/ton0.74219050Slurry Weight,15.5 kgMill Volume,6.7 liters3.2190.52512700Slurry Volume,8.5 liters0.3719500Overfilling,6.7
jsepulveda: See Mill Power_Ball Mills for further details. liters0.31257938Feed Size DistributionsObjective Function36700Mill FeedMill Disch. (exp)Mill Disch. (adj)% Passing%wiwi (error)^244750iMeshOpeningMid-Size% Ret% Pass% Ret% Pass% Ret% PassExpAdj.Error633508236011.0525400
jsepulveda: The top Mesh Opening must always be defined to allow 100% of the material passing through such screen.100.00
jsepulveda: The top Mesh Opening must always be defined to allow 100% of the material passing through such screen.100.00
jsepulveda: The top Mesh Opening must always be defined to allow 100% of the material passing through such screen.100.0010170020.74219050219970.00100.000.00100.000.00100.00100.00100.00(0.00)1
jsepulveda: User defined Weighting Factor, included to represent the relative quality and reliability of this particular measurement with respect to the rest of the available data.At the extreme, a value equal to zero means that this measurement is not included in the Objective Function calculation.0.0030.52512700155540.00100.000.00100.000.00100.00100.00100.00(0.00)10.0040.3719500109840.00100.000.00100.000.00100.00100.00100.00(0.00)10.00MallaAberturaFeedDischarge50.31257937.586840.00100.000.00100.000.00100.00100.00100.00(0.00)10.0010.001700.00100.00100.0063670072930.00100.000.00100.000.00100.00100.00100.00(0.00)10.0012.001400.0095.08100.0074475056410.00100.000.00100.000.00100.00100.00100.00(0.00)10.0020.00850.0076.42100.0086335039890.00100.000.00100.000.00100.00100.00100.00(0.00)10.0032.00500.0062.90100.0098236028120.00100.000.00100.000.00100.00100.00100.00(0.00)10.0048.00300.0051.58100.001010170020030.0899.92-0.00100.000.00100.00100.00100.000.0010.0065.00212.0045.24100.001112140015434.8495.080.00100.000.00100.00100.00100.000.0010.00100.00150.0040.00100.001220850109119.6675.420.00100.000.00100.00100.00100.000.0010.00150.00106.0034.9579.52133250065212.5262.900.00100.000.00100.00100.00100.000.0010.00200.0075.0030.3063.97144830038711.3251.580.00100.000.00100.00100.00100.000.0010.00270.0053.0025.8651.4015652122526.3445.240.00100.000.1299.88100.0099.880.1210.01400.0037.0022.4344.67161001501785.2440.000.00100.004.4395.45100.0095.454.77122.73-400.00171501061265.0534.9520.4879.5215.3280.1379.5280.13(0.76)10.581820075894.6530.3015.5563.9716.4363.7163.9763.710.4110.171927053634.4425.8612.5751.4011.5852.1351.4052.13(1.40)11.952040037443.4322.436.7344.677.5044.6344.6744.630.0910.0121-40001922.430.0044.670.0044.630.000.000.000.0000.00

sum19.0025.45

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ReportsMoly-Cop ToolsTM, Version 3.0Test N1BALL_PARAMBall Mill Grinding Model Parameter EstimatorRemarks :Comprobando caracteristicas del mineral Cerro verde DESIGN AND OPERATING CONDITIONSConfiguration : BATCH

Ore, kg11.1 Eff. Diameter, ft0.67 Water, lt4.3 Eff. Length, ft0.67 Slurry, kg15.5 Balls Filling, %64.6 Slurry, lt8.5 Speed, % Critical74.0 Slurry Dens., kg/lt1.829 App. Dens., ton/m33.219 % Solids (by weight)72.0 Power, kW0.45 Grinding Time, min9.0 Energy, kWh/ton6.06

Particle Size Distributions (Cumm. % Passing)

iMeshOpeningFeedDischargeExp.Adj.

11.0525400100.00 100.00 /100.00 00020.74219050100.00 100.00 /100.00 00030.52512700100.00 100.00 /100.00 00040.3719500100.00 100.00 /100.00 00050.31257937.5100.00 100.00 /100.00 000636700100.00 100.00 /100.00 000744750100.00 100.00 /100.00 000863350100.00 100.00 /100.00 000982360100.00 100.00 /100.00 0001010170099.92 100.00 /100.00 0001112140095.08 100.00 /100.00 965.09836298500122085075.42 100.00 /100.00 000133250062.90 100.00 /100.00 000144830051.58 100.00 /100.00 000156521245.24 100.00 /99.88 0001610015040.00 100.00 /95.45 0106.970907267801715010634.95 79.52 /80.13 00105.7384801995182007530.30 63.97 /63.71 000192705325.86 51.40 /52.13 000204003722.43 44.67 /44.63 000

D80, microns965 107 /106 965.098362985106.9709072678105.7384801995

MODEL PARAMETERSSelectionBreakage alpha010.000001 beta000.20000 alpha020.0000000 beta010.000 alpha112.611 beta10.250 alpha121.000 beta24.000 alpha22.500 Dcrit8029 Obj. Function1.16

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ModelSelection Function :Breakage Function :alpha01alpha02alpha11alpha12alpha2dcritBeta00Beta01Beta1Beta20.000000932302.61070539212.58028.61917792480.200.254iMeshOpeningMid-SizeSiEiMeshOpeningBeta0JCummulativeiMeshOpeningPartial11.052540021997.045256124715074.914803158811.05254000.2111.05254000150749.14803158820.7421905015554.259866673213156.655694168820.742190500.21120.742190500.66125490790131566.55694168830.5251270010984.079387914110353.649793096730.525127000.20.33874509211130.525127000.1211991430.56352331130103536.49793096740.37195008683.67721647928065.830197291640.37195000.20.21754594910.43647668871140.37195000.0327476110.13657843520.4189062428080658.301972915850.31257937.57292.54756583736344.954032434550.31257937.50.20.18479833810.29989825350.58109375721150.31257937.50.01853832170.06747897980.19989028810.4021776308063449.54032434546367005641.36508302734101.07469343246367000.20.16626001650.23241927370.38120346910.5978223692116367000.02183922730.060358740.1630241860.31931986190.6143799633041010.74693432367447503989.04750535761998.40683942647447500.20.14442078910.17206053370.21817928310.27850250730.3856200367117447500.01414152060.02485642780.05168867810.09191807550.16744075360.6187965309019984.06839426418633502811.7610140266877.77166067678633500.20.13027926850.14720410590.16649060490.18658443180.21817928310.3812034691118633500.01143632470.01493736010.0222462790.03264743160.05178663330.16454160490.619727926608777.71660676749823602002.9977533687376.67397626929823600.20.11884294380.13226674580.1442443260.15393700010.16639264970.21666186410.3802720734119823600.00948004780.01103610850.01334375240.01619661650.02113074410.04884415860.15841616570.600351132103766.7397626924101017001542.7248620542193.3105661091101017000.20.1093628960.12123063730.13090057360.13774038370.14526190560.16781770550.22185590780.399648867911101017000.00520658920.00587042360.00660619480.00735539490.00851605080.01441756470.03664858490.12505315260.441511020501933.105661091111214001090.871211463678.9464799344111214000.20.10415630680.11536021370.12429437880.13038498880.13674585480.15340014080.18520732280.27459571530.558488979511111214000.01223293140.01361773240.01485910160.0158698360.01717663490.02249956720.03994541720.10619589950.34030969650.71475009130789.46479934441220850651.920240520320.690168240812208500.20.09192337550.10174248120.10943527720.11451515280.11956921990.13090057360.14526190560.16839981580.21817928310.28524990871112208500.01142246360.01265207570.01363441350.01430628490.01501131550.01688258590.02055340870.03109902120.06490717210.11762346330.72906263160206.90168240781332500387.29833462075.320461648613325000.20.08050091190.08909040560.09580086380.10020886790.10455790440.11401798770.12470849680.13730079470.15327211090.16762644540.27093736841113325000.00965139940.01068233650.01149000630.01202332060.01255388760.0137431330.01524903840.01767045880.02286847950.02986466160.10436944720.7202976526053.20461648641448300252.19040425841.73650608114483000.20.07084951240.07840806910.08431085740.08818554720.09200401670.10027485460.10945945840.11963033590.13040363150.13776178380.16656792120.27970234741114483000.00589036780.00651890960.00701002440.00733271960.00765123890.00834521550.00913465770.01008534170.01135945440.01257924150.02213413560.09245873740.617125204017.36506080961565212178.32554500130.70270007715652120.20.06495914470.07188915950.07730083310.08085282770.08435277780.09192963910.10032480070.10954499410.11904417710.12518254230.14443378560.187243610.3828747961115652120.0053820820.00595628920.00640475450.00669918780.00698943310.00761878170.0083207840.00911087630.00999215190.0106521490.01403015410.03274704910.1646955130.616071933807.027000769616100150126.09520212920.2843401509161001500.20.05957706270.06593287030.07089607850.07415363990.07736334480.08431085740.09200401670.10043411780.10905202520.11453039330.13040363150.15449656090.21817928310.383928066211161001500.00495316040.00548158220.00589423950.00616510380.00643202050.00701002440.00765123890.00835867450.00909880820.0095922750.01135945440.01716993090.05151329530.16574878310.61712520402.84340150881715010689.16277250060.1150523924171501060.20.05462390240.06045128820.0650018390.06798853610.07093132430.07730083310.08435277780.09207544330.0999532170.10493811830.11904417710.137326630.16666598780.21817928310.38287479611171501060.00452576630.0050085860.00538561940.00563308550.00587692140.00640475450.00698943310.00763091630.0082895050.00871198350.00999215190.01245503450.02211963150.05140302450.1646955130.616071933801.1505239237182007563.04760106460.046553100318200750.20.05009813610.05544270220.05961621960.06235545050.06505440280.07089607850.07736334480.0844445270.0916637120.09622613490.10905202520.12487159550.14454635620.16677625860.21817928310.38392806621118200750.0041650930.00460943370.00495641590.00518415410.00540854550.00589423950.00643202050.00702115110.00762282290.00800451190.00909880820.01065199020.01410327810.02222990230.05151329530.16574878310.61712520400.4655310026192705344.2831796510.018509303419270530.20.04593304310.05083326850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