MEASUREMENT AND CALCULATION OF LIBERATION IN …

1 MEASUREMENT AND CALCULATION OF LIBERATIONIN CONTINUOUS MILLING CIRCUITSbyClaudio Luiz SchneiderA dissertation submitted to the Faculty ofThe University of Utahin partial fulfillment of the requirements for the degree ofDoctor of PhilosophyDepartment of Metallurgical EngineeringThe University of UtahAugust 1995 Copyright claudio Luiz Schneider 1995 All Rights ReservedABSTRACTThe LIBERATION phenomenon is of paramount importance in mineral processingscience; however, it is generally not well understood, primarily because the liberationspectrum in a particle population is very difficult to measure by any means.

2 In this work,a complete procedure based on Image Analysis and Stereology is proposed for themeasurement of the binary LIBERATION spectra in two-phase particles that are in narrow sizeranges. The procedure includes a technique for the MEASUREMENT of the transformationfunction that relates one- and three-dimensional spectra based on dense liquidfractionation, a model for the transformation function based on the incomplete betafunction and a solution to the Fredholm Integral equation of the first kind associated withthe stereological correction transformation function was successfullymeasured and calibrated for a Dolomite-Sphalerite ore.

3 The MEASUREMENT procedure isshown to be consistent throughout the work. A symmetrical transformation function forores with nearly symmetrical texture is derived from the test texture of the test a successful procedure for measuring the LIBERATION spectra, at least in thetest ore, it was possible to carry out the first experimental MEASUREMENT of thequadrivariate breakage, including its internal structure. The MEASUREMENT was successful,and a complete model, also based on the incomplete beta function, for the quadrivariatebreakage function is proposed and calibrated for the test model wasimplemented in MODSIM, a simulation package, with King s solution to the populationbalance model equation that includes the LIBERATION emulated closed continuous grinding experiment, under controlled labconditions.

4 Was carried out to verify the LIBERATION model and King s unitary operations, namely the ball mill and the elutriator, werecharacterized and modeled in detail, so that the results from the LIBERATION calculationwere not masked by any other phenomena in the experimental. The LIBERATION calculationwas successful; however it required a value for the geometrical texture parameter differentthan predicted, indicating that the parameter is probably dependent on particle , an industrial grinding circuit that process Taconite ore was characterizedwith respect to LIBERATION .

5 The Taconite texture is nearly symmetrical, and stereologicalcorrection using the symmetric transformation function proved to be consistent. Theliberation model had to be adjusted according to the Taconite texture, showing thatreparameterization alone was not sufficient. Circuit simulation with the adjusted modelwas successful, showing excellent agreement between measured and calculated memory of Andre Barreta GraffTABLE OF ivLIST OF ixLIST OF OF CORRECTION OF LINEAR GRADEDISTRIBUTIONS FOR MINERAL 12An Experimental 22A Model for the Dolomite-Sphalerite Transformation 44 The Dolomite-Sphalerite Transformation 52A Transformation Kernel for Symmetric 56 Inversion of the Transformation Equation for 62 Stereological Correction Procedure 67 Discussion and OF THE INTERNAL STRUCTURE OF

6 THEQUADRIVARIATE BREAKAGE 88 MEASUREMENT of the Discrete Quadrivariate Breakage 94 MEASUREMENT of the Geometrical Texture 129 Modelling the First Moment of the Conditional QuadrivariateBreakage 136 Modelling the Conditional Quadrivariate Breakage OF THE LIBERATION SPECTRA PRODUCED INA CONTINUOUS GRINDING 178 Objective and 178 Transport Characterization of the Experimental 181 Dolomite-Sphalerite Emulated Closed Continuous 258 MODSIM Simulation Setup and 295 MODSIM Simulation EMULATED CLOSED, CONTINUOUSGRINDING ORE SAMPLES FROM THE FAIRLANE PLANTSECONDARY GRINDING CIRCUIT 353viiiLIST OF separation results, density check with a HeliumPycnometer and measured volumetric grades by image volumetric grade distribution by volume, unliberatedfjvvolumetric grade distribution by volumeand correspondingfjvuvolumetric grade measured, conditional on size, linear grade distributions by length.

7 For each narrow grade sample produced by fractionation. Also shownare the % of apparent linear liberationandin rows i = 1 and(1)Aj(1)Bji=12 .. cumulative, conditional on size, unliberated linear gradedistributions by 43 Fju(giD) estimates, based on measured discrete fractions, of the first andsecond moments and the variance of the cumulative unliberated lineargrade 50 Fju(gD) calculated, conditional on size, volumetric grade distribution byvolume, for the g/cc Dolomite-Sphalerite sample produced byfjvfractionation.

8 Here, j is used as an index to a volumetric grade classrather than the volumetric grade range of the particles that generated shown, the measured and back-calculatedcumulative linear grade distributions and the final value of the objectivefunction after constrained optimization with Rosenbrock calculated, conditional on size, volumetric grade distribution byvolume, for the + g/cc Dolomite-Sphalerite sample producedfjvby fractionation. Here, j is used as an index to a volumetric grade classrather than the volumetric grade range of the particles that generated shown, the measured and back-calculatedcumulative linear grade distributions and the final value of the objectivefunction after constrained optimization with Rosenbrock calculated, conditional on size, volumetric grade distribution byvolume, for the + g/cc Dolomite-Sphalerite sample producedfjvby fractionation.

9 Here, j is used as an index to a volumetric grade classrather than the volumetric grade range that generated the shown, the measured and back-calculated cumulative linear gradedistributions and the final value of the objective function afterconstrained optimization with Rosenbrock calculated, conditional on size, volumetric grade distribution byvolume, for the + g/cc Dolomite-Sphalerite sample producedfjvby fractionation. Here, j is used as an index to a volumetric grade classrather than the volumetric grade range that generated the shown, the measured and back-calculated cumulative linear gradedistributions and the final value of the objective function afterconstrained optimization with Rosenbrock calculated, conditional on size, volumetric grade distribution byvolume, for the + g/cc Dolomite-Sphalerite sample producedfjvby fractionation.

10 Here, j is used as an index to a volumetric grade classrather than the volumetric grade range that generated the shown, the measured and back-calculated cumulative linear gradedistributions and the final value of the objective function afterconstrained optimization with Rosenbrock calculated, conditional on size, volumetric grade distribution byvolume, for the + g/cc Dolomite-Sphalerite sample producedfjvby fractionation. Here, j is used as an index to a volumetric grade classrat