{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Materials Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"Aggregated Source Repository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Krishnamoorthy, Prashanth","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"Date Available","value":"2023-06-08T17:49:50Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"Date Issued","value":"2023","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree (Theses)","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"Degree Grantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Catalytic solutes have been reported to significantly enhance dissolution rates of copper from chalcopyrite. Ethylene thiourea (ETu), a thiocarbonyl compound which has been shown to have such a catalytic effect, is used as a model solute to study the distribution of solute in a column of crushed ore under a point source to simulate the effects of a single drip emitter. To best understand the transport of solutes the transport of water in a column should be quantified first. Using breakthrough curve analysis and a 2-D axisymmetric model for water transport the hydrology parameters are estimated and validated. Further, steady state infiltration tests of ETu solution are performed to generate ETu breakthrough curves which are compared with curves from metal ions such as copper and lithium to estimate solute transport properties such as adsorption and reaction. In the case of adsorption, the Redlich-Peterson isotherm is used successfully to describe the adsorption behavior of ETu in a column of crushed ore. Further, the kinetics of homogeneous reactions of ETu with other solutes in the leaching solution such as Cu\u00b2\u207a, Fe\u00b3\u207a and O\u2082 are estimated by analyzing the sensitivity of model breakthrough curves and comparing them to experimental results. Thus, a parameterized model for ETu transport though a bed of ore is obtained which is used to estimate ETu breakthrough curves resulting from varying input concentrations, column heights and infiltration rates. The catalytic effect of ETu in the leaching of copper from chalcopyrite ore is analyzed from the perspective of the amount of ETu used to achieve the outcome defined as consumption index (CI). The apparent increase of copper extraction resulting from the doubling the feed concentration of ETu was found to be less efficient than doubling the irrigation flux with half the ETu concentration. Thus, the model is useful tool in providing insight and run what-if scenarios.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"Digital Resource Original Record","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/84815?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"Full Text","value":"Modelling the transport of thiocarbonyl catalysts through a bed of ore by  Prashanth Krishnamoorthy  B.E (Hons.), Birla Institute of Technology and Science, 2009 M.S., Cleveland State University, 2014  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Materials Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  June 2023  \u00a9 Prashanth Krishnamoorthy, 2023   ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Modelling the transport of thiocarbonyl catalysts through a bed of ore  submitted by Prashanth Krishnamoorthy in partial fulfillment of the requirements for the degree of Doctor of Philosophy In Materials Engineering  Examining Committee: David G. Dixon, Professor, Materials Engineering, UBC  Supervisor   Edouard Asselin, Professor, Materials Engineering, UBC Supervisory Committee Member Marek Pawlik, Professor, Mining Engineering, UBC University Examiner Susan A. Baldwin, Professor, Chemical Engineering, UBC University Examiner  Saman Ilankoon, Senior Lecturer, Chemical Engineering, Monash University External Examiner   Additional Supervisory Committee Members: Wenying Liu, Associate Professor,  Materials Engineering, UBC Supervisory Committee Member     iii  Abstract Catalytic solutes have been reported to significantly enhance dissolution rates of copper from chalcopyrite. Ethylene thiourea (ETu), a thiocarbonyl compound which has been shown to have such a catalytic effect, is used as a model solute to study the distribution of solute in a column of crushed ore under a point source to simulate the effects of a single drip emitter. To best understand the transport of solutes the transport of water in a column should be quantified first. Using breakthrough curve analysis and a 2-D axisymmetric model for water transport the hydrology parameters are estimated and validated. Further, steady state infiltration tests of ETu solution are performed to generate ETu breakthrough curves which are compared with curves from metal ions such as copper and lithium to estimate solute transport properties such as adsorption and reaction. In the case of adsorption, the Redlich-Peterson isotherm is used successfully to describe the adsorption behavior of ETu in a column of crushed ore. Further, the kinetics of homogeneous reactions of ETu with other solutes in the leaching solution such as Cu2+, Fe3+ and O2 are estimated by analyzing the sensitivity of model breakthrough curves and comparing them to experimental results. Thus, a parameterized model for ETu transport though a bed of ore is obtained which is used to estimate ETu breakthrough curves resulting from varying input concentrations, column heights and infiltration rates. The catalytic effect of ETu in the leaching of copper from chalcopyrite ore is analyzed from the perspective of the amount of ETu used to achieve the outcome defined as consumption index (CI). The apparent increase of copper extraction resulting from the doubling the feed concentration of ETu was found to be less efficient than doubling the irrigation flux with half the ETu concentration. Thus, the model is useful tool in providing insight and run what-if scenarios.  iv  Lay Summary Copper is an essential material within the context of the energy transition. Keeping net zero targets in mind, the demand for copper will increase by more than 50% from 2022 to 2040 while the mine supply growth will peak by around 2024, with a dearth of new projects around the world and the existing sources drying up. Towards solving this supply problem, Jetti Resources, in partnership with University of British Columbia, has developed a catalyst that can liberate copper from low-grade chalcopyrite ores which can have copper contents well below 1%. This work uses mathematical modeling in conjunction with experimental work to estimate how a model catalytic solute is distributed when applied to columns of low-grade copper ore.  v  Preface The research presented in this thesis originated from a patent filed by the University of British Columbia on the impact of thiocarbonyl compounds in leaching copper from chalcopyrite, and was financially supported by Jetti Resources Inc.  The author of this work was responsible for designing and performing experiments, developing computer code in MATLAB to solve the 2-D axisymmetric model with modifications for adsorption, analyzing the results of the experiments, writing the thesis, journal articles and conference posters.  The journal papers and conference posters listed below have been prepared from the results shown in this thesis. The first article was published based on the experimental data in Chapter 4. The second article, which was prepared by Dr. Zihe Ren, was published using some of the data from Chapter 6. My contribution was in performing the experiments and analyzing the data. Peer-Reviewed papers:  1. P. Krishnamoorthy, D G. Dixon, Z. Ren, N. Mora, C. Chao, \u201cModeling the distribution of an adsorbing solute in a catalyzed column.\u201d Minerals Engineering, Volume 182, 2022, 107556. 2. Z. Ren, P. Krishnamoorthy, P. Z. Sanchez, E. Asselin, D. G. Dixon, N. Mora, \u201cCatalytic effect of ethylene thiourea on the leaching of chalcopyrite.\u201d Hydrometallurgy, Volume 196, 2020, 105410. 3. Z. Ren, C. Chao, P. Krishnamoorthy, E. Asselin, D. G. Dixon, N. Mora, \u201cThe overlooked mechanism of chalcopyrite passivation.\u201d Acta Materialia, Volume 236, 2022, 118111. Conference posters: P. Krishnamoorthy, D. G. Dixon, N. Mora, \u201cModeling Solute Distribution in a Column.\u201d Conference of Metallurgists, August 18\u221221, 2019, Vancouver, Canada. vi  Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ......................................................................................................................... vi List of Tables ..................................................................................................................................x List of Figures ............................................................................................................................... xi List of Symbols ........................................................................................................................... xvi List of Abbreviations ............................................................................................................... xviii Acknowledgements .................................................................................................................... xix Dedication .....................................................................................................................................xx Chapter 1: Introduction ................................................................................................................1 1.1 Heap Leaching .............................................................................................................. 2 1.2 Problem Definition........................................................................................................ 3 1.3 Objectives of this work ................................................................................................. 4 1.4 Thesis Outline ............................................................................................................... 5 Chapter 2: Transport in Unsaturated Beds (Literature Review) ..............................................7 2.1 Introduction of rate enhancing (catalytic) solutes ......................................................... 7 2.1.1 Chloride................................................................................................................... 8 2.1.2 Iodine ...................................................................................................................... 8 2.1.3 Thiocarbonyl compounds........................................................................................ 9 2.1.3.1 Ethylene Thiourea (ETu) ............................................................................. 10 2.2 Transport in Unsaturated beds .................................................................................... 11 2.2.1 Water Transport .................................................................................................... 12 2.2.1.1 Darcy\u2019s law and the equation of continuity ................................................. 12 2.2.1.2 The Richards equation ................................................................................. 14 2.2.1.3 The Water Retention Curve \u2013 capillary potential (\u03c8) and water content (\u03b8) 15 2.2.1.4 Hydraulic conductivity function \u2013 K(\u03b8) ....................................................... 17 2.2.2 Solute Transport .................................................................................................... 20 2.2.2.1 Solute Advection .......................................................................................... 20 2.2.2.2 Molecular Diffusion ..................................................................................... 21 2.2.2.3 Mechanical Dispersion................................................................................. 22 vii  2.2.2.4 Dispersion in 2D (longitudinal and transverse) ........................................... 24 2.2.2.5 Sorption ........................................................................................................ 26 2.2.2.6 Reaction ....................................................................................................... 28 2.3 Review of current modeling efforts towards column leaching ................................... 28 2.3.1 Measuring liquid hold-up to estimate water distribution in a porous bed ............ 29 2.3.2 Using a liquid hold-up model to estimate water distribution in a column ............ 32 2.3.3 Experimental approach to estimate parameters in the \u03b8-explicit form of the Richards equation.................................................................................................. 34 2.3.4 Modeling unsaturated flow using Computational Fluid Dynamics ...................... 36 2.4 Motivation for this work ............................................................................................. 38 2.5 Focus of this work ....................................................................................................... 39 Chapter 3: Water Model Development ......................................................................................42 3.1 The Water Model ........................................................................................................ 42 3.2 Steady State Infiltration Tests \u2013 Estimating Hydraulic Parameters ............................ 44 3.2.1 Estimation of Kw, m, n and \u03bb ................................................................................. 46 3.2.2 Estimation of hc,0 ................................................................................................... 55 3.3 Analysis of water transport through the column ......................................................... 61 3.4 Effect of column height and radius ............................................................................. 63 Chapter 4: Solute Model and Adsorption ..................................................................................68 4.1 Tracer Breakthrough Analysis .................................................................................... 69 4.2 Modeling Solute Transport ......................................................................................... 76 4.3 Adsorption Isotherm ................................................................................................... 84 4.4 Simulation of solute distribution under a drip emitter ................................................ 90 Chapter 5: Transport of an Adsorbing, Reacting Solute .........................................................94 5.1 Homogeneous Decay .................................................................................................. 94 5.2 Factors affecting Decay .............................................................................................. 95 5.2.1 Copper ................................................................................................................... 95 5.2.2 Oxygen .................................................................................................................. 96 5.2.3 Iron ........................................................................................................................ 97 5.2.4 Mineral Surfaces ................................................................................................... 98 5.3 Modeling ramp tracer input ...................................................................................... 101 5.4 Modeling homogeneous decay ................................................................................. 104 5.4.1 Copper ................................................................................................................. 105 viii  5.4.2 Oxygen ................................................................................................................ 114 5.4.3 Iron ...................................................................................................................... 119 5.4.4 Real lixiviant solution ......................................................................................... 124 5.5 Modeling a column controlled based on outlet ETu concentration .......................... 125 5.5.1 Concentration control by addition ...................................................................... 126 5.5.2 Impact of flow rate .............................................................................................. 129 5.5.3 Impact of height of column ................................................................................. 130 5.6 Impact of leaching on decay of ETu ......................................................................... 131 5.6.1 Copper ................................................................................................................. 131 5.6.2 Iron ...................................................................................................................... 132 5.7 Impact of sulfide mineral content ............................................................................. 133 Chapter 6: Catalytic Effect of ETu on Copper Leaching from Chalcopyrite ......................137 6.1 Compatibility of ETu in leaching conditions ............................................................ 137 6.1.1 Redox Properties ................................................................................................. 137 6.1.2 Biocompatibility Analysis .................................................................................. 138 6.2 Leaching copper sulfides with ETu .......................................................................... 141 6.2.1 Reactor leaching experiments with ETu ............................................................. 141 6.2.2 Rate Law of ETu-Assisted Leaching .................................................................. 144 6.3 Sensitivity of Cu leaching to [ETu] .......................................................................... 146 6.4 Effect of leaching progress on ETu consumption index ........................................... 149 6.5 Irrigation flux to improve ETu utilization ................................................................ 153 Chapter 7: Conclusion and Recommendations .......................................................................156 7.1 Describe and quantify water flow through a column ................................................ 156 7.2 Estimate adsorption and dispersion of a catalytic solute (ETu) ................................ 157 7.3 Quantify reactive decay of ETu in real lixiviant solutions ....................................... 157 7.4 Gauge ETu consumption as a function of copper extracted ..................................... 158 7.5 Recommendations ..................................................................................................... 158 Bibliography ...............................................................................................................................160 Appendices ..................................................................................................................................165 Appendix A ETU analysis on HPLC .................................................................................... 165 Appendix B MATLAB code written to solve the water model ............................................ 167 B.1 Main program...................................................................................................... 167 B.2 Initializing Water Model Coefficients ................................................................ 170 ix  B.3 Water Model Solving .......................................................................................... 171 B.4 Initializing Solute Model Coefficients ................................................................ 174 B.5 Solute Model Solving ......................................................................................... 177  x  List of Tables Table 3.1  Particle size distribution of chalcopyrite ore sample ............................................ 47 Table 3.2 Effective (steady state) saturation of the column at different water fluxes .......... 49 Table 3.3 Change in Kw(BC) and Kw(VGM) (m\/s) with fines added .......................................... 50 Table 3.4 Equilibrium moisture retention of dry ore with increasing flux; \u03b8initial = 0 and \u03b8r = 2.19% (R1) ............................................................................................................ 51 Table 3.5 Equilibrium moisture retention of partially wet ore with increasing flux; \u03b8initial = 2.19% and \u03b8r = 2.81% (R2) .................................................................................. 51 Table 3.6 Equilibrium moisture retention of fully wet ore with increasing flux; \u03b8initial = \u03b8r = 4.68% (R3) ............................................................................................................ 51 Table 3.7 Change in Kw(VGM) of dry ore, partially wet and fully wet ore .............................. 53 Table 3.8 Effective saturation of the column for different water fluxes ............................... 64 Table 4.1 Hydrology parameters estimated from water irrigation tests in Chapter 3 ........... 79 Table 4.2 Value of each parameter for each iteration of SSE minimization ........................ 85 Table 5.1 Volume of 1 g\/L Cu2+ solution in B1 .................................................................. 103 Table 5.2 Model parameters for the Redlich-Peterson model (Eqn (4.2)) .......................... 134 Table 6.1 Results of mineral phase analysis of the ore provided ........................................ 146  xi  List of Figures Figure 1.1  Chalcopyrite ore heap bioleaching process from \u00a9 Pradhan, N. Heap bioleaching of chalcopyrite: a review.  Minerals Engineering, 21, Page 357.  BACFOX is Bacterial Ferrous Oxidation. ................................................................................... 3 Figure 2.1  Structure of a generic thiocarbonyl compound ....................................................... 9 Figure 2.2 Structure of ethylene thiourea ............................................................................... 10 Figure 2.3  Example of a SWRC showing the components of the curve fit model developed by Rossi and Nimmo (Perkins, 2011; Rossi and Nimmo, 1994) .......................... 16 Figure 2.4  Typical relationship of hydraulic conductivity and matric potential for coarse and fine textured soils (Lal and Shukla, 2004) ............................................................ 18 Figure 2.5 Preferential flow induced by irrigation flux and segregating material (O\u2019Kane et al., 1999) ............................................................................................................... 19 Figure 2.6  Wetted and non-wetted regions of the ore bed from \u00a9 Ilankoon, I.M.S.K. and Neethling, S.J, Permission Received (Ilankoon and Neethling, 2016) ................. 32 Figure 2.7  Column test setup and experimental procedure (Cherkaev, 2019) ....................... 33 Figure 2.8  Experimental setup used to estimate hydrology parameters (Afewu and Dixon, 2008) ..................................................................................................................... 35 Figure 2.9  Comprehensive CFD model incorporating all possible contributions in a heap (McBride et al., 2018) ........................................................................................... 38 Figure 3.1  Column water breakthrough curve test apparatus ................................................. 47 Figure 3.2  Water flux vs effective saturation to estimate hydraulic conductivity (Kw) ......... 48 Figure 3.3  Water flux vs effective saturation to estimate hydraulic conductivity parameters (Kw(BC) and Kw(VGM)) ............................................................................................... 49 Figure 3.4 Water flux vs water content curves on the same ore bed but with different initial moisture levels ...................................................................................................... 52 Figure 3.5  Water flux vs effective saturation on the same ore bed but with different initial moisture levels ...................................................................................................... 53 Figure 3.6  Simulated moisture contents for Runs R1, R2 and R3 at 20.29 L\/m2\/h. Air entry head hc,0 assumed at 0.35 m. Simulated moisture levels: R1 = 4.50%, R2 = 7.08%, R3 = 8.78%. .......................................................................................................... 54 Figure 3.7  Discharge bucket mass (tared) as recorded by Scale 2 of Figure 3.1 ................... 56 Figure 3.8 Water breakthrough curve \u2013 the red dots are effluent; the blue dotted line represents an infiltration flux of 11.5 L\/m2\/h (1.5 mL\/min) ................................. 56 Figure 3.9 Sensitivity of water breakthrough curve to air-entry head hc,0 ............................. 57 Figure 3.10  SSE values plotted vs. hc,0 ..................................................................................... 58 Figure 3.11  Model generated and experimental water breakthrough curves for 11.6 L\/m2\/h . 58 xii  Figure 3.12  Model generated and experimental water breakthrough curves ........................... 59 Figure 3.13 Breakthrough curves of water from a 0.85-m column .......................................... 60 Figure 3.14  Experimental (R2) breakthrough curves vs model generated curves ................... 60 Figure 3.15 Sensitivity of water breakthrough to hydraulic conductivity ................................ 61 Figure 3.16  Sensitivity of steady-state water content to hydraulic conductivity ..................... 62 Figure 3.17  Sensitivity of steady-state water content to air-entry head ................................... 63 Figure 3.18  Water flux vs effective saturation to estimate hydraulic conductivity ................. 65 Figure 3.19  Sensitivity of hc,0 to water breakthrough curve in the large column ..................... 65 Figure 3.20  Sum of square errors plotted against hc,0 .............................................................. 66 Figure 3.21  Model and experimental water breakthrough curves at an irrigation flux of 10 L\/m2\/h ................................................................................................................... 67 Figure 4.1 Normalized tracer concentration feed and breakthrough curves .......................... 69 Figure 4.2  Breakthrough curve of 0.204 g\/L ETu from a 30 cm column at 17.54 L\/m2\/h .... 70 Figure 4.3  Breakthrough curves of ETu from a 30 cm column ............................................. 72 Figure 4.4  Normalized solute loss vs. normalized input concentration of ETu ..................... 73 Figure 4.5 Breakthrough curves of 1.02 g\/L ETu and Cu2+ under different conditions ........ 74 Figure 4.6  ETu breakthrough in the presence of added pyrite (DI water) ............................. 75 Figure 4.7  Breakthrough curves of Cu2+ tracer and ETu from a 1m tall column ................... 76 Figure 4.8  Section of the control volume used to apply the conservation principle .............. 77 Figure 4.9 Block diagram representation of proposed solute decomposition, where k* is the first-order decomposition rate constant ................................................................ 78 Figure 4.10  Sensitivity of the model to the slope of \u03b1L(v) ....................................................... 80 Figure 4.11  Sensitivity of the model to the intercept of \u03b1L(v) .................................................. 81 Figure 4.12 Sensitivity of the model to the ratio of \u03b1T to \u03b1L .................................................... 82 Figure 4.13  Cu Tracer curves in the 1 m column ..................................................................... 82 Figure 4.14 Sensitivity of the model to the rate constant k* in the 30-cm column (non-adsorbing solute, Cu) ............................................................................................ 83 Figure 4.15  The sum of square errors (SSE) of model and experimental data points plotted against model parameters K, k*, g and ar ............................................................. 85 Figure 4.16  Sensitivity of the model to the parameter K in the 30 cm column ........................ 86 Figure 4.17  Sensitivity of the model to the rate constant k* in the 30 cm column .................. 86 Figure 4.18  Sensitivity of the model to the parameter g in the 30 cm column ........................ 87 Figure 4.19  Sensitivity of the model to the parameter ar in the 30 cm column ....................... 87 xiii  Figure 4.20  Breakthrough curves for other ETu concentrations with the Redlich-Peterson model in the 30-cm column .................................................................................. 89 Figure 4.21 Using the Redlich-Peterson model to predict ETu breakthrough from a 1-m column................................................................................................................... 90 Figure 4.22 The concentration profile of a) Cu, b) ETu and c) ETu in the presence of SDDC in the 30 cm \u00d7 10 cm column at steady state corresponding to breakthrough curves from Figure 4.5 ..................................................................................................... 91 Figure 4.23 The concentration profile of a) Cu at 3 mL\/min, b) ETu at 3 mL\/min, c) ETu at 2 mL\/min and d) ETu at 1 mL\/min, in a 1 m \u00d7 8 cm column at steady state with different infiltration rates, corresponding to breakthrough curves from Figure 4.21............................................................................................................................... 91 Figure 4.24 Simulated concentration profiles of a) Cu and b) ETu in a 4 m x 50 cm column at steady state at a flowrate of 20 L\/m2\/h.................................................................. 92 Figure 5.1  Normalized ETu concentration as a function of time in acidic water alone ......... 95 Figure 5.2 a) Normalized ETu concentration as a function of time in acidic water with varying [Cu2+]; b) Normalized ETu concentration as a function of time in acidic water at constant [Cu2+] ........................................................................................ 96 Figure 5.3  Normalized ETu concentration as a function of time in acidic water sparged with N2, air and O2 ........................................................................................................ 97 Figure 5.4  a) Normalized ETu concentration as a function of time in acidic water with varying [Fe3+]; b) Normalized ETu concentration as a function of time in acidic water at constant [Fe3+] ......................................................................................... 98 Figure 5.5  Normalized ETu concentration as a function of time in neutral DI water sparged with gas and 1 g pulverized pyrite ........................................................................ 99 Figure 5.6  Normalized ETu concentration as a function of time in acidic water sparged with gas and 1 g pulverized pyrite .............................................................................. 100 Figure 5.7  Normalized ETu concentration as a function of time in various solutions and 1 g pulverized pyrite ................................................................................................. 101 Figure 5.8  Test apparatus for diluting feed concentration .................................................... 102 Figure 5.9 Normalized Cu2+ as a function of time for varying input fluxes and a steadily diluted feed.......................................................................................................... 104 Figure 5.10  Normalized ETu concentration as a function of time in acidic water sparged with gas ....................................................................................................................... 106 Figure 5.11  Normalized ETu concentration as a function of time in acidic medium with [Cu2+] = 1 g\/L and sparged with N2 gas ......................................................................... 107 Figure 5.12  Experimental apparatus to hold [ETu] steady while varying [Cu2+] .................. 108 Figure 5.13  Normalized ETu concentration as a function of time when [Cu2+] is varied while steady [ETu] is maintained ................................................................................. 110 Figure 5.14 Natural log of ETu not recovered vs natural log of [Cu2+] at [ETu] =1.02 g\/L .. 110 xiv  Figure 5.15  Logarithm of ETu not recovered vs logarithm of [ETu] at [Cu2+] = 1 g\/L ......... 111 Figure 5.16  Sensitivity of the Normalized ETu concentration to rate constant kCuETu ........... 112 Figure 5.17 Sum of squares between model and experimental breakthrough curves plotted against rate constant parameter kCuETu ................................................................ 113 Figure 5.18 Breakthrough of ETu when [Cu2+] is varied, model vs experimental ................ 113 Figure 5.19 Breakthrough of ETu when [ETu] is varied, model vs experimental ................. 114 Figure 5.20 ETu breakthrough curves when purged with 100 mL\/min gas ........................... 115 Figure 5.21 Logarithm of ETu loss plotted versus logarithm of [O2(aq)] at [ETu] = 1.02 g\/L 116 Figure 5.22  ETu breakthrough curves when [ETu] is varied and purged with 100 mL\/min of air in acidic medium............................................................................................ 117 Figure 5.23 Logarithm of amount of ETu not recovered against logarithm [ETu] and purged with 100 mL\/min of air in acidic medium .......................................................... 117 Figure 5.24 Sensitivity of ETu breakthrough to the rate constant kETuO2 ............................... 118 Figure 5.25  Sum of squares between model generated and experimental breakthrough curves plotted against rate parameter kETuO2 .................................................................. 118 Figure 5.26 ETu breakthrough with varying [Fe3+] and constant [ETu] input ....................... 119 Figure 5.27 ETu breakthrough with varying [ETu] and [Fe3+] = 1.1 g\/L .............................. 120 Figure 5.28 Logarithm of amount of ETu lost vs logarithm of [ETu] at [Fe3+] = 1.1 g\/L ..... 121 Figure 5.29 Logarithm of amount of ETu lost vs logarithm of [Fe3+] at [ETu] = 1.02 g\/L ... 121 Figure 5.30  Sensitivity of ETu breakthrough to rate constant kFeETu ..................................... 122 Figure 5.31 Distance between model and experimental curves plotted against rate constant kFeETu ................................................................................................................... 122 Figure 5.32 ETu breakthrough with varying [ETu] and [Fe3+] = 1.1 g\/L .............................. 123 Figure 5.33 ETu breakthrough with varying [Fe3+] and [ETu] = 1.02 g\/L ............................ 123 Figure 5.34 ETu breakthrough with lixiviant solution of [Fe3+] =1.1 g\/L, [Cu2+] =1 g\/L and  initial [ETu] = 1.02 g\/L ....................................................................................... 125 Figure 5.35 ETu curves generated by modeling target [ETu] ................................................ 127 Figure 5.36 ETu curves generated by setting each rate constant of ETu decay to zero independently ...................................................................................................... 128 Figure 5.37 ETu curve generated for an irrigation flux of 8.76 L\/m2\/h ................................. 129 Figure 5.38 ETu curve generated for a 1 m tall column with an irrigation rate of 17.52 L\/m2\/h............................................................................................................................. 130 Figure 5.39 ETu curve generated increasing [Cu2+] from 1.0 g\/L to 2.68 g\/L (concentration values in the legend are normalization constants) .............................................. 131 Figure 5.40 ETu curve generated increasing [Fe3+] from 1.0 g\/L to 3.87 g\/L (concentration values in the legend are normalization constants) .............................................. 132 xv  Figure 5.41 ETu curves in DI water for different Pyrite content ore ..................................... 134 Figure 5.42 Trends of K and k* with respect to % Pyrite content .......................................... 135 Figure 5.43 Model and Experimental ETu curve for 2% Pyrite ............................................ 135 Figure 6.1 Cyclic voltammetry study on 0.2 g\/L ETu at three scan rates ............................ 138 Figure 6.2 Effect of three ETu concentrations on ferrous oxidation by A. ferrooxidans ..... 140 Figure 6.3 ETu concentration in flasks with four initial concentrations during growth of A. ferrooxidans during a 744-hour period ............................................................... 141 Figure 6.4 XRD pattern with Rietveld refinement for the chalcopyrite sample used .......... 142 Figure 6.5 Oxidation-reduction potential (ORP) during chalcopyrite bioleaching with A. ferrooxidans and 0 (control and Phase 1 in all bioreactors), 10, 50, and 100 ppm ETu ...................................................................................................................... 143 Figure 6.6 Copper recovery during chalcopyrite bioleaching with A. ferrooxidans and 0 (control and Phase 1 in all bioreactors), 10, 50, and 100 ppm ETu .................... 144 Figure 6.7 Extraction of Cu from chalcopyrite ore in the case of ETu and Control ............ 148 Figure 6.8 ORP of the effluent from chalcopyrite ore column in ETu and Control cases ... 148 Figure 6.9 Model breakthrough curves and experimental data on Day 20 for the catalyzed column................................................................................................................. 150 Figure 6.10 Model breakthrough curves and experimental data on Day 106 for the catalyzed column................................................................................................................. 151 Figure 6.11 Model breakthrough curve and experimental data on Day 278 for the catalyzed column................................................................................................................. 152 Figure 6.12 Extraction of Cu from ore in the case of column at different [ETu] and fluxes . 154 Figure 6.13 Extraction of Cu from ore in the case of ETu fed at different fluxes ................. 155  xvi  List of Symbols Symbols Meaning Common Unit ar material specific constant  \u2212 ca solute concentration kmol\/m3 water DA,L longitudinal coefficient of hydrodynamic dispersion m2\/s DA,T transverse coefficient of hydrodynamic dispersion  m2\/s DA,0 molecular diffusivity m2\/s Ddiff effective coefficient of molecular diffusion m2\/s g material specific constant \u2212 h total pressure head m hc capillary head m hc,0 air-entry head m k*  rate constant of solute decomposition  s\u22121 Ka adsorption equilibrium constant m Kw hydraulic conductivity m\/s Kw(VGM) hydraulic conductivity van Genuchten model parameter m\/s Kw(BC) hydraulic conductivity Brooks-Corey model parameter m\/s krw intrinsic permeability m2 n reaction order \u2212 na advective molar flux kmol\/m2\/s ndiff diffusive molar flux kmol\/m2\/s q solid phase (adsorbed) concentration kmol\/kg solid Se effective saturation \u2212 vw water volume flux m2 water\/m2\/s z height above a certain datum m \u03b1L,W longitudinal dispersivity m \u03b1T,W transverse dispersivity m \u03b5D,W effective diffusivity factor \u2212 \u03b8 volumetric water content m3 water\/m3 \u03b8r residual (or fully drained) volumetric water content m3 water\/m3 \u03b8s saturated volumetric water content  m3 water\/m3 \u03bb material specific constant \u2212 xvii  \u03bcw water viscosity kg\/m\/s \u03c1w water density kg\/m3 \u03c6 total moisture potential Pa \u03c6m matric potential Pa \u03c6z gravitational potential Pa \u03c8 material specific constant  \u2212 xviii  List of Abbreviations Abbreviation     AEH Air Entry Head AFT Anodic Fenton Treatment BC Brooks-Corey CFT Classic Fenton Treatment CI Consumption Index ETu Ethylene Thiourea LD Longitudinal Dispersion PLS Pregnant Leach Solution ROM Run of Mine RP Redlich-Peterson  SWRC Soil Water Retention Curve TD Transverse Dispersion VGM van Genuchten-Maulem    xix  Acknowledgements It is hard to put into words the gratitude I feel for the support, insight, motivation and admonishment that Dr. David G. Dixon has provided me. Over multiple revisions, failed experiments, and erroneous code, his unwavering faith in my abilities and constant inspiration has been the north star in this work reaching its destination.  Dr. Zihe Ren has been a great partner with whom to bounce off ideas, discuss experiments, explain results and question conclusions, and this has added to the care and nuance in the experiments performed and results analyzed. This project has also been a lot of fun, which is a true testament to the support of Dr. Ren.  Dr. Nelson Mora and Jetti Resources have supported my work financially and offered industry perspective to me, which motivated a research approach that had to be scalable and would be a useful tool for operations.   Lab work is never fun without great lab mates. Thank you for all the great times, Monse Robelledo, Pablo Zuniga Sanchez, Chih Wei Chao, and Kush Shah.  Family support has been the foundation on which this work was built. I have had many personal challenges throughout the course of my project and without the support of Amma, Appa, Thimu, and all my friends this would not have been possible.  All I want to tell everyone involved is Thank you!  xx  Dedication  To all of the women who have shaped, and who continue to shape me 1  Chapter 1: Introduction The concept of sustainability as applied to the mining industry is considered an \u201coxymoron\u201d as exploitation of a finite, non-renewable resource has serious negative environmental impacts.  Based on the classic Brundtland Commission definition of sustainable development as \u201cdevelopment that meets the needs of the present without compromising the ability of future generations to meet their own,\u201d one of the attributes that could make mining sustainable is the ability of R&D programs to develop metal extraction technologies that can lead to extending mine life by processing ores previously considered refractory. In the copper industry, low-grade ores containing primary sulfides such as chalcopyrite are most abundant and recalcitrant (Singer, 2017). Hence, unlocking this stranded resource would constitute sustainable utilization of a copper resource based on the previous definition. Heap leaching is the most important method of hydrometallurgical copper extraction which is used to treat copper ores with oxides and secondary sulfide minerals (Petersen, 2019). However, for treating ores abundant in primary copper sulfides such as chalcopyrite, additional innovation is required owing to slow and incomplete leaching of the mineral (Dreisinger, 2006). Some of the proposed process improvements to heap leach technologies include heap design and construction to improve flow through the heap (Liu and Granata, 2018), operating heaps at elevated temperatures (\u201chot heaps\u201d) to improve leaching kinetics (Robertson et al., 2012) and introducing additives to the lixiviant to provide a catalytic effect to enhance oxidation rates (Granata et al., 2019; Ren et al., 2020; Wu et al., 2019). It is apparent that catalyzed oxidation rates would depend on the local concentration of such solutes. To this end, the current work is to study and model the propagation of such solutes through the ore bed. As an introduction to the study, an overview of 2  the heap leaching process will be given to set the background required to explain the problem, objectives, and significance of this study. 1.1 Heap Leaching  Heap leaching is a type of percolation leaching process which can be defined as the selective removal of metal from an ore by causing a suitable solvent or leaching agent to seep through a mass or pile of the ore (Ilankoon et al., 2018). A modern heap is constructed on an impervious base with a slight slope towards a series of drains. Depending on the operation, some heaps are built with ore sent directly from the mine, known as run-of-mine (ROM) also known as dump leaching. In many heap leach operations, the ore is crushed (typically 12\u221250 mm) and agglomerated to enhance permeability. The leaching solution or lixiviant is distributed over the surface of the heap through a network of pipes at the end of which there is either a drip emitter or sprinkler. Upon seeping through the constructed ore bed, the solution is collected in basins and further processed for metal extraction or recycle.  Ores containing oxide minerals of copper are relatively easily processed by acid heap leaching. Bluebird copper oxide mine was the first mine to implement this technology in 1968 for copper extraction (Bartlett, 2007). Modifications to acid heap leaching technology have been developed to allow its application to various types of ores, including those containing secondary copper sulfides.  One such modification is heap bioleaching, when the leaching solution contains sulfuric acid, ferric, and acidophilic iron- and sulfur-oxidizing microbes (Jia et al., 2021). The acidophilic microbes mediate the oxidative dissolution of sulfide minerals in the presence of ferric and sulfuric acid. This is being used commercially in many operations around the world for 3  secondary copper sulfide ores (Pradhan et al., 2008) The flow diagram of a typical heap bioleaching operation with solvent extraction is shown in Figure 1.1.  Figure 1.1  Chalcopyrite ore heap bioleaching process from \u00a9 Pradhan, N. Heap bioleaching of chalcopyrite: a review.  Minerals Engineering, 21, Page 357.  BACFOX is Bacterial Ferrous Oxidation.  1.2 Problem Definition In the copper industry, heap and dump bioleaching contribute 20\u221225% of the world\u2019s copper production (Christel et al., 2018). These are mainly secondary copper ores which contain chalcocite (Cu2S) and covellite (CuS). The dissolution of copper from primary copper sulfide ores containing chalcopyrite is little compared to secondary sulfides and oxides, thus rendering a similar approach nonviable.  The plethora of factors affecting heap leaching is broadly classified into the physical and chemical, biological, mineralogical, and operational. Examples of physical and chemical factors are temperature, pH, oxygen availability, and ferric concentration, to name a few. Biological 4  factors include microbial diversity, bacterial population, bacterial activity, and metal tolerance. Mineralogical factors that can influence heap leaching include minerology, mineral liberation, acid consumption, porosity, surface area, hydrophobicity, and particle size. Finally, operational factors are bulk ore density, heap geometry, mode of irrigation, and use of aeration. Optimizing each or a combination of all these factors would substantially improve heap leaching operations for oxide and secondary sulfide ores, but, unless the kinetics of chalcopyrite dissolution is substantially increased, this would have only a limited impact on heap leaching of low-grade primary sulfide ores. To enhance the rate of dissolution, novel technologies suggest addition of catalytic reagents to the heap as solutes in the lixiviant solution. Additives which have been used to enhance copper extraction from chalcopyrite ore include chloride (Muller et al., 2008), iodine (Sato, 2015) and thiocarbonyl compounds (Dixon et al., 2016). The rate of dissolution of copper from the ore would depend on the local concentration of these solutes as the lixiviant percolates through the ore bed. Therefore, the transport of these solutes governed by advection, dispersion, adsorption, and reaction must be described and quantified.  1.3 Objectives of this work The main objective of this work is to describe and quantify transport parameters of a reactive, adsorbing solute of interest as it percolates though a bed of ore in a cylindrical column, under a point source of irrigation. This principal objective is divided into the following: 1. To describe and quantify water flow through a column using water breakthrough curves  2. To identify the physical processes which may manifest as an interactive solute propagates through a porous bed of ore from tracer and solute breakthrough curves  3. To undertake laboratory tests under various operating conditions to calibrate and validate a 2-D axisymmetric model, thus quantifying specific solute interactions on the ore  5  4. To consider the interactions of the solute with other solutes in the lixiviant and use techniques of breakthrough curve analysis to estimate associated transport parameters  5. To simulate the solute distribution in a column of ore under real leaching conditions  The scope of this work will encompass the development and validation of a 2-D axisymmetric solute model to describe and quantify breakthrough curves of a solute having multiple simultaneous interactions under varying operating conditions in a column of ore. To isolate and quantify parallel interactions, the operating condition of each test is controlled such that only certain interactions of the solute are viable, resulting in model parameters that can be associated with corresponding physical interactions such as dispersion, adsorption, or reaction. The solute chosen in this work is ethylene thiourea (ETu). ETu is a simple derivative of thiourea (Tu) which is an effective lixiviant for gold leaching (Reinhold, 1990). It is used here as a model thiocarbonyl compound owing to the simplicity of its degradation mechanism, which has been the subject of various studies (Saltmiras and Lemley, 2000). Further details about ETu as a solute of interest will be discussed in the literature review in detail. This work is intended as a step towards using solute breakthrough curves to illustrate and quantify transport properties of reactive solutes. In addition, parameters estimated for ETu on ore are used to perform sensitivity analysis on factors such as lixiviant flow rates, column height and concentration of lixiviant that have a significant effect on solute transport, which would be useful as a tool to provide insight and predict what-if scenarios.  1.4 Thesis Outline  This thesis is divided into seven chapters and organized in the following manner: \uf0d8 The introduction, motivation, objectives, and scope of this thesis are given in Chapter 1 6  \uf0d8 A review of literature, comprising discussions of related research, existing knowledge and applicable theory related to lixiviant flow and solute transport in porous media, is given in Chapter 2 \uf0d8 The development of a 2-D axisymmetric water model and estimating the hydrology characteristics of the column using water breakthrough tests is discussed in Chapter 3 \uf0d8 The solute model is built on top of the water model with the addition of an adsorption term, and breakthrough curves are used to estimate the adsorption isotherm that best describes the behavior of ETu on ore in Chapter 4 \uf0d8 Chapter 5 investigates the impact of other solutes in the leaching system such as ferric, cupric, and O2 on ETu transport \uf0d8 Chapter 6 combines the ETu distribution and a generic rate law to estimate leaching of Cu from columns. This is followed by sensitivity analysis and discussions of what-if scenarios on leaching performance \uf0d8 Overall conclusions and recommendations for future work are stated in Chapter 7  7  Chapter 2: Transport in Unsaturated Beds (Literature Review) The objective of this chapter is to report on the existing literature and present the motivation for this thesis. Innovative technologies that use additives to leach copper from low grade chalcopyrite are introduced in the first section. The second section focuses on the theory of transport in unsaturated beds. The third section summarizes literature which attempts to explain and model leaching of copper from chalcopyrite in columns. The final section presents the motivation for this work based on the gaps in the models discussed which are a consequence of assumptions made regarding flow of pregnant leach solution (PLS or lixiviant) though the bed of ore. A concise introduction to novel technologies in heap leaching follows.    2.1 Introduction of rate enhancing (catalytic) solutes  Heap leaching is commonly applied to acid leaching of copper oxides, cyanide leaching of gold, and ferric leaching of uranium. Secondary and primary sulfides of copper can be leached using the same technique with adequate modifications. New technologies to leach copper from chalcopyrite add solutes to the lixiviant medium (PLS) to enhance leaching rates and render heap leaching of low-grade chalcopyrite viable. Some exotic solutes such as glycine and methane sulfonic acid are used to enhance leaching rates of chalcopyrite in stirred tanks (Barton and Hiskey, 2022). However, Barton and Hiskey acknowledge the impracticality of applying these to any industrial process at scale. This is because the cost of reagents such as glycine or methane sulfonic acid render the process nonviable. More realistic additives that are at various stages of commercial implementation are chloride (Cl\u2212) (Lu and Dreisinger, 2013) , iodine as potassium iodide (KI) (Manabe, 2012) and thiocarbonyl compounds (Dixon et al., 2016). Iodine and thiocarbonyl compounds are added at concentration levels orders of magnitude lower than chloride. In the case of Cl\u2212 heap leaching, the additive solute can be considered as the solution 8  medium, in contrast to I\u2212 and thiocarbonyl compounds which are catalytic solutes that are added only sparingly but must be delivered to every ore surface in order to be effective. 2.1.1 Chloride  Dissolution of chalcopyrite in chloride or chloride-sulfate systems has been discussed in detail (Elsherief, 2002; Hiroyoshi et al., 2000; Third et al., 2002; Vel\u00e1squez-Y\u00e9venes et al., 2010). In the work of Vel\u00e1squez-Y\u00e9venes et al. (2018), an ore sample with 2.1% (by weight) copper sulfides, of which 89.3% was chalcopyrite, was leached in columns. The ore was agglomerated with different amounts of NaCl under various curing conditions, which gave various concentrations of Cl\u2212 in the lixiviant. This led to a significant improvement in the dissolution rate of chalcopyrite and a higher overall recovery of copper from the ore. The concentration of Cl\u2212 ranged from 50 g\/L to about 160 g\/L.  Furthermore, sulfuric acid concentrations of 20 g\/L and 100 g\/L were also tested in combination with the Cl\u2212. The Cu2+ and Fe3+ concentrations were about 0.5 g\/L and 1.0 g\/L, respectively. While the ability of Cl\u2212 to enhance copper leaching is obvious, its high concentration in the leach solution makes it part of the medium rather than a dilute catalyst. Hence, delivery of Cl\u2212 ions to the surface of the ore is guaranteed as long as the ore is wetted by solution. This is not the case with the dilute catalysts discussed below.  2.1.2 Iodine The addition of iodine to the conventional ferric-assisted leaching of chalcopyrite enhances dissolution rates of copper as presented in the patented process of JX Nippon Mining and Metals Corporation (Manabe, 2012). The addition of I\u2212 as KI at a concentration of about 0.5 g\/L to the lixiviant increased copper extraction by 47% at 50\u00b0C (Granata et al., 2019). As mentioned in the literature, I\u2212 is oxidized by Fe3+ in solution to I2 which can in turn exert its oxidizing ability towards chalcopyrite. The innovation in this work is the use of low concentrations of KI to assist 9  chalcopyrite leaching in ferric sulfate media. The low concentrations of I\u2212 used warrants the need to further investigate the transport of this novel catalytic reagent. The complexity arises from the fact that I2 evaporates, and this needs to be quantified as a separate term in the overall mass balance of total iodine. Hence, estimation of the distribution of I\u2212 in the leaching system would require a three-phase mass balance (on the surface of the ore, in the lixiviant and in the gaseous phase). 2.1.3 Thiocarbonyl compounds Thiocarbonyl compounds are reagents that possess the thiocarbonyl functional group (C=S), e.g., thiourea (also known as thiocarbamide, abbreviated as Tu).  Figure 2.1 below shows the generic structure of a thiocarbonyl compound.   Figure 2.1  Structure of a generic thiocarbonyl compound   In an unexpected discovery (Dixon et al., 2016) it was found that the reagents possessing this functional group can be used to facilitate the leaching of copper from primary copper sulfide minerals such as chalcopyrite. Another such compound known as ethylene thiourea (ETu) was demonstrated to enhance dissolution of copper from chalcopyrite to about 90% extraction in 250 hours at 25\u00b0C (Ren et al., 2020). The amount of ETu used to achieve this level of extraction was about 50 ppm or 0.05 g\/L in a reactor with 2.2 g\/L Fe3+ at a pH of 1.7. 10  2.1.3.1 Ethylene Thiourea (ETu) ETu is a white to pale green crystalline solid with a faint odor. The chemical formula for ETu is C3H6N2S, and it has a molecular weight of 102.2 g\/mol. The solid powder is highly soluble in water with a solubility limit of 19 g\/L.  It is used in electroplating, dyes, pharmaceuticals, and synthetic resins. In the rubber industry, ETu is used extensively as an accelerator in the curing of polychloroprene (Neoprene) and other elastomers. ETu is also the decomposition product of dithiocarbamate pesticides. These are an important class of organic fungicides used all over the world to control many diseases in a variety of crops.   Figure 2.2 Structure of ethylene thiourea    ETu is synthesized by treating ethylenediamine with carbon disulfide. In the context of hydrometallurgy, ETu is an effective lixiviant for gold leaching (Reinhold, 1990). ETu is taken as a model thiocarbonyl compound owing to the simplicity of its degradation mechanism, which has been the subject of various studies. ETu is oxidized to produce ethylene urea (EU) using classic Fenton treatment (CFT) or anodic Fenton treatment (AFT) (Saltmiras and Lemley, 2000), both of which are irreversible. While ETu forms complexes with some transition metals such as cobalt (Carlin and Holt, 1963) and gold (Jones et al., 1976), there is little evidence to show that ETu complexes with iron, which is the main lixiviant used for copper leaching. Furthermore, it has been shown in previous literature that sulfide minerals can facilitate the decomposition of thiocarbonyl compounds (Zhu, 1992) and that ETu, being susceptible to Fenton treatment, 11  decomposes by a known and irreversible pathway which is accelerated under acidic conditions (Zhang et al., 2014). In the context of transport of interacting solutes, ETu allows all the intricacies of solute transport to be examined. In summary, catalytic solutes enhance leaching rates of copper from chalcopyrite but the differences in concentration of these solutes in the leach solution distinguish the dilute solutes from the matrix anions which are ubiquitous. In fact, in the case of the matrix anions no transport modeling is required as they are part of the medium. The additives which are added in lower concentrations require a nuanced and thorough approach to characterizing their transport behavior.  2.2 Transport in Unsaturated beds  The heap leaching process can be conceptualized as a packed-bed reactor under unsaturated flow. In hydrology, the term \u2018flow\u2019 refers to the movement of liquid and gaseous phases while \u2018transport\u2019 refers to the migration of solutes (Freeze, 1971). Unsaturated flow is a type of multiphase flow through a porous bed, with two phases, water and air, in the pore channels around the ore particles (Fairley et al., 2004). In the case of heap leaching, the water phase is the leaching solution, which displaces the air in the pores between the ore particles thus increasing the moisture content of the bed of ore. The additive solutes are carried to the surface of the ore by the leaching solution. Some solutes are transferred from the leaching solution onto the solid ore and some may be transferred to the vapor phase from leach solution. The propagation of solutes under unsaturated conditions is limited to the area wetted by the leach solution. In the next section the wetting of a bed of ore is discussed.  12  2.2.1 Water Transport  The law of conservation of mass, expressed mathematically as the equation of continuity, states that the net rate of water flow is equal to the change in the sum of water stored and generated in the bed of ore which reflects in the moisture content of the ore. Furthermore, the driving force for the rate of flow is the relative proportions of air and water in the pores between the ore particles, which is represented by the hydraulic properties of the bed of porous ore. Hence, to characterize water transport through a bed of ore it is vital to have a conservation law and a constitutive relationship.  2.2.1.1 Darcy\u2019s law and the equation of continuity Darcy (in 1856) originally developed a relationship between volume flux (also known as \u201csuperficial velocity\u201d) and the hydraulic gradient, as:  dxdhKv ww \u2212=   (2.1)  where vw = water volume flux (m3 water\/m2\/s)   Kw = hydraulic conductivity (m\/s)  x = distance (m)  h = total hydraulic head (m) This law is known as Darcy\u2019s law, where the negative sign indicates that the direction of flow is from higher head to lower. While Darcy originally developed the equation for saturated flow in a porous bed where Kw was the saturated hydraulic conductivity and was a constant, it was adapted to unsaturated flow where Kw is a varying function of water content. Expressing Darcy\u2019s law in a 2D axisymmetric configuration assuming no pressure gradients within the air phase, the water volume flux may be written in an isotropic, unsaturated porous medium thus (Dixon and Afewu, 2022): 13   )()( crwwcwwrww hzkKpzg\u03c1\u03bckk\u2207+\u2207=\u2207+\u2207=v  (2.2) where k = intrinsic permeability (m2)   \u03bcw = water viscocity (kg\/m\/s)  \u03c1w = water density (kg\/m3)  g = acceleration due to gravity (m\/s2)  Kw = hydraulic conductivity ww \u03bcg\u03c1k=  (m\/s)  krw = relative water premeability (\u2212)  pc = capillary pressure (Pa)  hc = capillary head g\u03c1p wc=  (m) The equation of continuity of water volume (which represents water mass assuming constant density) is written thus:  wwwrwzww s\u03c1Mt\u03b8rvrrzv\u2212\u2202\u2202=\u2202\u2202\u2212\u2202\u2202\u2212=\u22c5\u2207\u2212)(1 ,,v  (2.3) where vw = water volume flux (m3 water\/m2\/s)   z = depth (m)  r = radius (m)  \u03b8 = volumetric water content (m3 water\/m2)  t = time (s)  sw = net rate of water production (kmol water\/m3\/s)    Mw = water molecular mass  (kg water\/kmol water) Using the vw from Eqn (2.2) in Eqn (2.3), Eqn (2.3) can be solved for \u03b8 if the parameters of the bed of ore such as krw and hc can be cast in terms of \u03b8. 14  2.2.1.2 The Richards equation  The Richards equation was developed by Lorenzo A. Richards (Richards, 1931) to represent movement of water in unsaturated soils. The equation is the mathematical form of the law of conservation of mass which states that the rate of change of saturation of a closed volume is equal to the rate of change of net flux into the control volume.  qt\u03b8\u22c5\u2207=\u2202\u2202  (2.4) where q = net flux of water out of the control volume  A well-known and widely used 1-D expression (Farthing and Ogden, 2017) for Eqn (2.4) to describe vertical infiltration is   \uf8fa\uf8fb\uf8f9\uf8ef\uf8f0\uf8ee\uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb \u2212\u2202\u2202\u2202\u2202=\u2202\u2202 1)(z\u03c8\u03c8Kzt\u03b8  (2.5) where K(\u03c8) = hydraulic conductivity function   \u03c8 = capillary potential Another form of Eqn (2.5) results when K and \u03c8 are written as functions of \u03b8 to obtain an equation explicit in \u03b8:   \uf8fa\uf8fb\uf8f9\uf8ef\uf8f0\uf8ee \u2212\u2202\u2202\u2202\u2202=\u2202\u2202 )()( \u03b8\u03b8\u03b8\u03b8 KzDzt (2.6) where \uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb\u2202\u2202=\u03b8\u03b8\u03c8\u03b8\u03b8 )()()( KD  and is referred to as the soil-water diffusivity. In Eqn (2.6) the first term in the brackets captures the effect of capillarity while the second term represents the effect of gravity. This distinction between Equations (2.5) and (2.6) is significant as it delineates the contribution of each physical effect. The hydraulic conductivity function and the capillary potential are discussed in further detail in the next section.  15  2.2.1.3 The Water Retention Curve \u2013 capillary potential (\u03c8) and water content (\u03b8) The soil water characteristic curve or the soil water retention curve (SWCC or SWRC) illustrates the relationship between soil water content (\u03b8) and soil water potential (\u03c8) in a porous medium. The SWRC is considered one of the most fundamental and reliable depictions of soil hydraulic properties (Vogel and Cislerova, 1988), and accurate parameterization of the SWRC has great significance in understanding moisture dynamics and soil hydrology (Jarvis et al., 1991). To understand what the SWRC represents, consider the following thought experiment: An ore sample saturated with water is subjected to a slight negative pressure and little flow is observed. When the negative pressure (interchangeably used with the term suction pressure) is increased to a critical value the largest pore starts draining first, assuming all pores between the particles of ore are not of the same size owing to a disparity in particle sizes. The volume vacated by the water is duly filled by air. In soil hydrology the suction pressure which is just enough to overcome the ability of the soil to hold water (capillarity) is called the air entry head (AEH). In this thesis, the AEH is represented mathematically by the symbol hc,0. As can be deduced, AEH is generally very small in coarsely textured materials with highly varied size distributions, and is larger in dense, finely textured material owing to the coarse material having larger pore sizes. Alternatively, poorly stacked material may also present with a low AEH because of the presence of large voids. As suction increases, the intermediate size pores drain after the largest pores, finally followed by the small pores. At a very high suction, only the water that is adsorbed onto the soil surfaces remain. In the context of heap leaching the porous particles would have some moisture absorbed into the pores which is referred to as residual water content (Freeze, 1971). This residual water can only be removed by evaporation.  16  There are two main ways of obtaining SWRCs: the first is experimental determination and the second is derivation from basic soil properties by using pedotransfer functions (PTFs) (Bittelli and Flury, 2009; Ghanbarian-Alavijeh and Liaghat, 2009). The experimental approach is time consuming and costly but accurate (Pan et al., 2019) while the PTFs may be able to relate particle size distributions of the ore bed to hydraulic properties. In the field, SWRC determination consists of measurements of soil water pressure by tensiometers installed at depths of interest and of water content by gravimetric sampling or neutron or gamma attenuation techniques (Arya and Paris, 1981), which are costly and time consuming. However, once these measurements are made, a plethora of semi-empirical models are available to fit the discrete data points, of which the van Genuchten, Brooks-Corey, Rossi-Nimmo and Campbell (Shervin et al., 2016) models have been shown to have explicit physical significance and feasibility for application to a wide range of soils. A graphical representation of a SWRC with a Rossi-Nimmo parameter fit is shown in Figure 2.3.   Figure 2.3  Example of a SWRC showing the components of the curve fit model developed by Rossi and Nimmo (Perkins, 2011; Rossi and Nimmo, 1994) 17  To sum up, the determination of the air entry head is best accomplished using some experimental data and semi-empirical models to fit the data. While AEH is the driving force for flow (along with gravity), the hydraulic conductivity is the measure of permeability of the bed of ore. This is further discussed in the next section.       2.2.1.4 Hydraulic conductivity function \u2013 K(\u03b8)  The hydraulic conductivity function describes the relationship between the unsaturated hydraulic conductivity and the water content (Anderson, 1981; Hillel, 1998). The presence of air in the pores of the medium (the bed of ore) decreases the hydraulic conductivity as the level of saturation decreases. This implies that the hydraulic conductivity of a porous bed is at its maximum at 100% saturation. Under saturated conditions (with low matric potential) the coarsely textured material has a higher hydraulic conductivity than the finely textured material due the higher number of large pores in the coarse material. However, with a little suction, the large pores drain their water, and the hydraulic conductivity of the coarsely textured material decreases due to a decrease in the cross-sectional area available for water flow, increased tortuosity and viscosity of the system (O\u2019Kane et al., 1999).  This relationship is illustrated in Figure 2.4. 18   Figure 2.4  Typical relationship of hydraulic conductivity and matric potential for coarse and fine textured soils (Lal and Shukla, 2004) The hydraulic conductivity of the finely textured material will remain near its saturated permeability as the suction increases, due to the higher number of small pores in the finely textured material. In other words, the finely textured material has greater water retention ability than the coarse material. As the suction increases, the fine material will start draining its water. The rate at which hydraulic conductivity decreases for any unsaturated porous medium is a function of the gradation of the material and tends to decrease more rapidly for uniform materials than for well-graded materials (Hillel, 1998; O\u2019Kane et al., 1999). Another important observation from Figure 2.4 is that even though the saturated hydraulic conductivity of the coarsely textured material is higher than that of the finely textured material, the conductivities of both the coarse and fine materials decrease steeply with increasing suction. The implication of this trend in unsaturated bed hydrology is that transport processes taking place in wet conditions are faster than in dry conditions in the same porous medium. The permeability-saturation functions of the coarsely textured and finely textured materials cross at a 19  specific degree of saturation, as shown in Figure 2.4. Therefore, during the construction of a heap, segregation of coarse and fine materials may affect the entire hydrology of the heap. Solution application rates that are greater than the saturated permeability of the finely textured material will lead to preferential flow through the coarse material, thus creating a pressure profile leading to saturation conditions where the effective permeability of the coarse material is greater than that of the fine material (O\u2019Kane et al., 1999). For an irrigation rate that is lower than the saturated hydraulic conductivity of the fine material, preferential flow in the finely textured material will result. In other words, this is the basis for preferential flow, also known as \u201cchannelling\u201d in the common parlance. To demonstrate this behaviour, O\u2019Kane et al. [1999] filled a laboratory column with inert material intentionally segregated as shown in Figure 2.5.   Figure 2.5 Preferential flow induced by irrigation flux and segregating material (O\u2019Kane et al., 1999) The surface of the column material was uniformly irrigated with water using a peristaltic pump. At steady state, the amounts of water collected from below the fine and coarse fractions were 20  compared. When the applied flux was slightly higher than the saturated permeability of the finely textured material, approximately 5% and 95% of the water was collected from the fine and coarse compartments, respectively, as shown in Figure 2.5. On the other hand, when the flux applied at the surface was less than the saturated permeability of the finely textured material, approximately 68% and 32% of the water was collected from the fine and coarse compartments, respectively. The understanding of this complex flow behaviour under unsaturated conditions is required for meaningful operating strategies and modeling the flow of solution and transport of solutes in unsaturated porous media.  2.2.2 Solute Transport Similar to water transport, solute transport requires a mass conservation law and a constitutive relationship. The law of conservation of mass, in the case of solutes, states that the net rate of flux of solute out of a control volume is equal to the sum of the amount of solute accumulated and the amount of net solute generated within the control volume. The net rate of solute flux out of a control volume depends on the amount of solute carried through by the water (advection) and local variations in flow velocities and concentration of solute (dispersion). The accumulation term includes sorption and the generation term is a source or sink term which accounts for reaction of the solute. 2.2.2.1 Solute Advection Advection is the process whereby solutes are transported by the bulk mass of the flowing fluid (Freeze, 1971). In the absence of other transport processes, the solution and the solute will move at the same rate and in the same direction as the flow (Lal and Shukla, 2004; Schwartz and Zhang, 2002). The driving force for this type of process is the hydraulic gradient (sum of gravity and capillarity) and the mass flow of water (interchangeably used with the term lixiviant in this 21  context) in the porous medium. The water carries with it an advective flux of solutes proportional to their concentration (Zheng and Bennett, 2002) as given by Eqn (2.7):  awawa cdxdhKcvn \u2212==  (2.7) where na =  advective molar flux (kmol\/m2\/s)  ca =  solute concentration (kmol\/m3 water) In a highly permeable medium, advection is the most important transport mechanism by which solution travels to the discharge point (Freeze, 1971; Spitz and Moreno, 1996). In the previous sections the movement of solution through the path of least resistance has been discussed at length. Thus, the solute flux through preferred flow channels leads to uneven distribution of solutes through the bed of ore. In Eqn (2.7) the driving force represented by the hydraulic gradient is the sum of gravity and capillarity.  2.2.2.2 Molecular Diffusion  Molecular diffusion is a spontaneous process whereby dissolved mass is transported from a higher potential to a lower potential by random molecular motion, and so it does not depend on bulk flow of solution. The driving force responsible for diffusion of ions, atoms, or molecules is the gradient of chemical potential or, for dilute solution, concentration (Lim et al., 1998). Diffusion tends to occur to decrease concentration gradients and restore homogeneity. The movement of solutes occurs in the opposite direction of the concentration gradient, and may be expressed according to Fick\u2019s law (Schwartz and Zhang, 2002):  dxdc\u03b8Dn diffdiff )(\u2212=  (2.8) where ndiff  =  diffusive molar flux (kmol\/m2\/s)  Ddiff  =  effective coefficient of molecular diffusion (m2\/s) 22  Diffusion coefficients in porous media are generally smaller than those in pure solution because collisions with the porous ore bed due to the convoluted flow paths in the porous medium hinder the diffusion process (Freeze, 1971; Schwartz and Zhang, 2002). The convoluted pore passages in the porous medium render the actual diffusion length longer than the direct path length. In unsaturated porous media, such as heaps, the effective diffusion coefficient depends on the degree of saturation of the medium and decreases with a decrease in water content due to an increase in the diffusion path length (Lim et al., 1998). Other factors contributing to this decrease in diffusion with decrease in saturation include increased viscosity of the liquid adjacent to soil particles, increased ionic interaction along small pores and water films, interconnectivity between the water-filled pores and the water films (owing to less amount of water in the bed of ore) and a decreasing cross-sectional area normal to the diffusive flux (Hillel, 1998; Lim et al., 1998). However, diffusion is a slow process, and it is considered insignificant compared to dispersion, which is a function of the prevailing flow field (Dixon and Afewu, 2022) in the context of percolation leaching. In the model HeapSim 2D (which is the basis of this work), it is not ignored, but rather assigned a common value for all solutes in the lixiviant to simplify the mathematics of the problem.   2.2.2.3 Mechanical Dispersion  Mechanical dispersion is the movement of solute that occurs because of velocity variations in a porous medium and it is caused by three mechanisms. The first mechanism is variations in the travel velocities of molecules at different points across pores due to the drag exerted on the fluid by the roughness of the pore surfaces. According to Poiseuille\u2019s Law, the laminar flow velocity varies with radial distance across a capillary tube, thus (Hillel, 1998): 23   \uf8fa\uf8fb\uf8f9\uf8ef\uf8f0\uf8ee\u2212= 2212)(Rrvru  (2.9) where u  =  velocity (m\/s)   v  =  volume flux or average velocity (m\/s)   r  =  radial distance from the centre of tube (m)   R  =  tube radius (m)  At the tube wall, the velocity u = 0. At the center of the tube, the velocity u is at its maximum and equal to twice the average velocity v. The velocity of the solute molecule carried by advection therefore depends on its position within the pore. The second mechanism is the variation in pore sizes, shapes, and surface areas along which the solution flows, leading to different fluid bulk velocities. Another variant of Poiseuille\u2019s Law gives the volumetric flow through a capillary tube as a function of the pressure head gradient, thus (Hillel, 1998):  dxdh\u03bcg\u03c1R\u03c0vR\u03c0Qwwww 842 ==  (2.10) In any given cross-sectional area, the pore diameters may vary by orders of magnitude resulting in a proportional change in the volumetric flow rate as shown in Eqn (2.10). This results in widely varying bulk liquid velocities, thus causing mechanical dispersion. The third mechanism is fluctuations in the flow paths of a fluid element compared to the mean flow direction caused by tortuosity, branching and inter-fingering of pore channels. In an unsaturated porous medium, the tortuosity increases as the moisture content decreases. In heaps, fluid particles follow tortuous flow paths around ore particles, and therefore the velocities vary in direction from one point to another. Also, the magnitude of the velocity varies, and is greater in 24  the large pores than in the small pores. For isolated pores or \u2018dead-end\u2019 pores the velocity is even zero (Petersen and Dixon, 2003). Solute flux by mechanical dispersion is also expressed by Fick\u2019s Law, thus (Schwartz and Zhang, 2002; Spitz and Moreno, 1996):  dxdc\u03b8Dn dispdisp )(\u2212=   (2.11) where ndisp  =  molar flux by mechanical dispersion (kmol\/m2\/s)   Ddisp  =  coefficient of mechanical dispersion (m2\/s) Macroscopically, dispersion is like diffusion. However, unlike diffusion, it occurs only because of flow (Freeze, 1971; Lal and Shukla, 2004). As a result of this macroscopic similarity, the two processes are additive and the sum of mass fluxes due to molecular diffusion and mechanical dispersion yield the net mass flux due to hydrodynamic dispersion, thus:   dxdcDdxdcDdxdcDnnn dispdispdiffdispd )()()( \u03b8\u03b8\u03b8 \u2212=\u2212\u2212=+=  (2.12) where  nd  =  molar flux by hydrodynamic dispersion (kmol\/m2\/s)   D =  coefficient of hydrodynamic dispersion (m2\/s) Finally, the net molar flux of a solute is the sum of advective and dispersive fluxes, thus:  dxdcDcvnnn aawda \u2212=+=  (2.13) 2.2.2.4 Dispersion in 2D (longitudinal and transverse) In the previous sections we discussed the different factors contributing to dispersion, while in this section we resolve the dispersion vector in 2D, longitudinal (along the direction of flow, LD) and transverse (normal to the direction of flow, TD). LD is normally much stronger than TD (Freeze, 1971). The amount of mechanical dispersion is a function of the average linear velocity 25  in the direction of flow (Fetter, 2018; Schwartz and Zhang, 2002). Thus, the coefficients of hydrodynamic dispersion in the longitudinal and transverse directions are often taken as linear functions of water volume flux, with coefficients which are functions of volumetric water content, thus:  0,,,, )()( AwDwwLLA D\u03b8\u03b5v\u03b8\u03b1D +=  (2.14)  0,,,, )()( AwDwwTTA D\u03b8\u03b5v\u03b8\u03b1D +=  (2.15) where DA,L  =  longitudinal coefficient of hydrodynamic dispersion (m2\/s)   DA,T  = transverse coefficient of hydrodynamic dispersion (m2\/s)   \u03b1L,W  =  longitudinal dispersivity (m)  \u03b1T,W  =  transverse dispersivity (m)  \u03b5D,W =  effective diffusivity factor (\u2013)  DA,0  =  molecular diffusivity (m2\/s) Thus, if a solute is transported along the flow path, it spreads in all directions in the horizontal plane (Freeze, 1971) .The total mass of solute would be conserved in the domain of flow but the mass occupies an increasing volume of the porous medium. Even though the medium may be isotropic (directionally uniform) with respect to textural properties and hydraulic conductivity, mechanical dispersion is directionally dependent. Dispersion is thus anisotropic, being stronger in the direction of flow (longitudinal dispersion) than in directions normal to flow (transverse dispersion) (Freeze, 1971; Schwartz and Zhang, 2002). This implies that 1D transport models may only be useful in the interpretation of laboratory column data, but not in analyzing field data. At high solution flows, where mechanical dispersion is the dominant mechanism of dispersion, the migrating solute cloud is ellipsoidal. In the same light, at low solution flow rates, where molecular diffusion is the dominant mechanism of dispersion; the migrating solute cloud is spherical (Freeze, 1971). As noted above, microscopic heterogeneity and tortuosity of a porous 26  medium are responsible for dispersion. These factors determine the dispersivity in laboratory scale experiments, resulting in longitudinal dispersivities in the range of 0.1 to 1 cm. However, dispersivities two to four orders of magnitude larger are required to account for observed dispersion in field scale experiments. This disparity arises because dispersion at the field scale is caused by macroscopic heterogeneities rather than pore-scale heterogeneity. Thus, the macroscopic heterogeneities mask the microscopic heterogeneities, rendering the effect of the latter minor in comparison.  2.2.2.5 Sorption  When a porous medium is contacted with a solution containing dissolved matter, it frequently happens that certain solutes are removed from solution and immobilized within or onto the solid matrix by electrostatic or chemical forces (Zheng and Bennett, 2002). This process is referred to as sorption. The opposite process, whereby solute particles are detached from the solid matrix into solution, is called desorption. Sorption comprises both adsorption (onto solid surfaces) and absorption (into particle pores). Sorption of solute onto the solid phase during solute displacement in the heap decreases the concentration of the solute in solution. When a solution containing a particular solute is contacted with a porous solid matrix, the concentration in solution will fall as concentration on the solid phase rises, until equilibrium is established. A plot of the equilibrium concentration in solution against the equilibrium concentration on the solid is called an adsorption isotherm (Zheng and Bennett, 2002), and is often represented by a power-law equation, thus:  naacKq =  (2.16) with the following gradient: 27   1\u2212=\u2202\u2202 naancKcq   (2.17) where q  =  solid phase (adsorbed) concentration (kmol\/kg solid)   Ka  =  adsorption equilibrium constant (variable)   n  =  exponent (\u2013) The equilibrium constant and exponent must be determined for each species in each porous medium. Eqn (2.16) is called the Freundlich isotherm. For certain species, particularly at low concentrations, n may be taken as equal to one, thus giving a linear isotherm. The nature of the Freundlich isotherm assumes that the solid matrix has infinite sorption capacity, whereby the adsorbed concentration increases indefinitely with solute concentration. In practice this is not the case, and there is an upper limit to adsorption. A pseudo-linear sorption isotherm which accounts for the maximum sorption capacity of the solid matrix, called the Langmuir isotherm, is given thus (Zheng and Bennett, 2002) :   aaaacKcKqq+=1max  (2.18) with the following gradient:    ( )2max 1 aaacKKqcq+=\u2202\u2202   (2.19) where qmax  =  maximum sorption capacity (kmol\/kg solid) If sorption occurs during solute transport in a bed of ore, then the rate of mass accumulating on the solid matrix by sorption is equal to the rate at which the interstitial solution in the matrix loses its solute. In addition to the Freundlich and Langmuir isotherms, there are others which will be used in this work to describe the adsorption of certain catalytic solutes. These sorption 28  isotherm expressions are empirical, and the parameters can only be obtained by fitting them to experimental data.  2.2.2.6 Reaction  Reaction is the final piece of the puzzle regarding the transport of interactive solutes through a porous bed of ore. Any solute that interacts with other solutes in the lixiviant may not exhibit a stable concentration through the leaching process. A rate law can characterize the behavior of the concentration of these solutes in the lixiviant as follows:   rnakcr \u2212=   (2.20) where r  =  rate of change of solutes (kmol\/s)   k  =  rate constant (variable)   nr  =  order of the reaction (\u2212) This constitutes the consumption term in the solute mass balance. In the context of the solute of interest that is being accounted for, the reaction term is the source or sink term that needs to be considered.  To summarize, solute transport is a combination of advection, dispersion, adsorption and reaction, the basis for which has been discussed briefly. In combination with the section on water transport, a theoretical tool kit is provided to assist the reader in further literature review.  2.3 Review of current modeling efforts towards column leaching  In this section an attempt is made to summarize the existing literature towards modeling of columns of ore. There is a rich history of heap and dump leach modeling work dating back to early 1970s. Over the past decades there has been significant progress in computational capacity resulting in various different approaches to heap leach modeling. These range from spreadsheet-29  based approaches to complex computational fluid dynamics (CFD) models using specialized software. Each approach has its respective advantages and limitations detailed in a recent review article (Marsden and Botz, 2017)  .  In the context of understanding solute propagation through a column of ore, spreadsheet-based models offer little insight.  Hence this section focuses on the empirical modeling work done on columns.  2.3.1 Measuring liquid hold-up to estimate water distribution in a porous bed  Ilankoon and Neethling (2012) examined the unsaturated liquid flow through a packed bed of particles consisting of spherical glass beads simulating a bed of ore particles. In the motivation for the literature cited above, it is stated that the Bond number, which is a function of particle size, reflects the forces that govern fluid flow. The Bond number is a dimensionless number which describes the ratio of gravitational forces to capillary forces given by the equation   \u03b3\u03c1 2dgBo =   (2.21) where \u03c1  =  density of liquid (kg\/m3)   g  = acceleration due to gravity (9.81 m2\/s)    d  =  particle size (m)  \u03b3  =  surface tension (N\/m) A Bond number significantly above 1 implies gravitational force domination and significantly below 1 implies capillary force domination. The Bond number is a function of particle size of the porous bed, implying that in the case of heap leaching, especially for Run-Of-Mine (ROM) ores, the Bond number reflects the transition of flow regimes from gravity-dominated to capillary-dominated. Upon performing infiltration tests, hysteresis of liquid hold-up by the bed of particles was observed. It was concluded that the dominant cause of hysteresis is a change in the number of liquid rivulets flowing through the porous bed.   30  To better understand hysteresis, one should consider the following thought experiment. A column of ore is irrigated with water at a flux of 5 L\/m2\/h. In an effort to understand water flow, the flux is changed from 5 to 10 to 15 L\/m2\/h. The corresponding values of \u03b8 are \u03b85a < \u03b810a < \u03b815a  where the subscript \u2018a\u2019 denotes ascending flux. Subsequently, the flux is reduced from 15 L\/m2\/h to 10 to 5 L\/m2\/h. While \u03b815d > \u03b810d > \u03b85d (and \u2018d\u2019 denotes descending flux), it is observed that \u03b810a \u2260 \u03b810d and \u03b85a \u2260 \u03b85d; i.e., hysteresis is observed.  In a subsequent publication (Ilankoon and Neethling, 2013), a bed of ore is compared against the bed of glass beads. A distinction is made between the liquid that is held \u201cinternally\u201d (defined as the water held within the ore particles) and \u201cexternally\u201d (in the void space between ore particles). It is further argued that this distinction is fundamental to modeling liquid holdup in real ore systems. A \u201cpre-factor\u201d K is introduced as a parameter in the modeling efforts to account for the fact that the particles are irregular to distinguish them from the spherical glass beads. Further, a novel technique is used to measure dispersion coefficient (discussed in detail in section 2.2.2.4) using PEPT (positron emission particle tracking) as opposed to using salt as a tracer (Ilankoon et al., 2013). An advantage of this technique is that the radial dispersion coefficients have been calculated simultaneously with axial dispersion coefficients. It was demonstrated that the radial dispersion coefficient is smaller than the axial dispersion coefficient by a factor of 4.       It is clear that the wetting characteristics of a column are different from the drainage characteristics. The transient liquid hold up of columns during drainage is detailed in other work of Ilankoon and Neethling (2014). That study demonstrates that the wetting behavior of a packed bed is simple and progresses as a standing wave through the bed of ore. However, the drying behavior when the flux is removed is dependent on the porosity of the bed of particles being wetted. It is proved that, in the absence of liquid held within the particles, the relatively simple 31  behaviors can be used to explain wetting and drying cycles and the simulated and measured average liquid holdup and liquid flow rate out of the column is in good agreement.   To better understand and model the liquid spread mechanisms, an experiment was setup (Ilankoon and Neethling, 2016) which consisted of ore particles of two different PSDs, a narrow distribution (20\u221226.6 mm) and a more realistic ore distribution of (2\u221226.5 mm), in a rectangular bed of dimensions 80 cm \u00d7 60 cm \u00d7 10 cm. The following test procedure was outlined in the work:  1. Single drip emitter steady state infiltration tests to investigate the radial water spread and the development of flow paths through the bed of ore 2. Intermittent liquid addition tests to estimate the effect of packing on the external flow paths Based on the results, it was concluded that the existence of hysteresis provides the opportunity for heap leach operators to optimize the system via two parameters, liquid holdup and hysteresis, rather than just one. Further, it was concluded that flooding the column to measure void volume can yield inconsistent results when compared with wetting a bed of ore from the dry state. Hence, lab scale tests need to consider hysteresis when testing. The reason for hysteresis, according to the paper, is a change in the number of liquid rivulets flowing through the heap. In a subsequent paper (Ilankoon and Neethling, 2016) the mechanism of liquid spread is discussed. The experimental setup was a Perspex bed of dimensions 800 mm \u00d7 600 mm \u00d7 100 mm, which was referred to as \u201cpseudo 2D.\u201d Two load cells were used to measure the liquid holdup, which represented a relatively small proportion of the total weight of the column. The figure below shows the wetted and non-wetted regions of the ore bed when irrigated at 4.2 L\/h.  32   Figure 2.6  Wetted and non-wetted regions of the ore bed from \u00a9 Ilankoon, I.M.S.K. and Neethling, S.J, Permission Received (Ilankoon and Neethling, 2016) Based on the outflow and wetting behaviour it was concluded that the liquid flow around the particles establish itself quickly, but with limited horizontal spread, while the liquid within particles spreads much wider, but over a much longer time period. Further it is suggested that application of higher flow rates initially as well as a more uniform liquid addition to the top of the bed reduces channelling, but does not eliminate it. 2.3.2 Using a liquid hold-up model to estimate water distribution in a column  A different approach to estimating liquid holdup to estimate water distribution in a column is explored in the work of Cherkaev (Cherkaev, 2019). According to Cherkaev, applying the Richards equation to estimate hydraulic parameters that predict water distribution and flow out a column of ore is not very robust. From the experimental setup described in the work of Cherkaev, columns of 30 cm tall \u00d7 20 cm diameter were used to conduct infiltration studies. The particle size distribution of the ore was synthetically produced by mixing different fractions of coarse and fine ore based on the Gates-Gaudin-Schumann (GGS) distribution. It is reported that 33  the weight of fines (defined as \u22121.4 mm in diameter particles) was roughly 10% of the total weight of ore used in column tests. The ore sample was assembled, agglomerated and loaded into columns. Then the columns were flooded by applying a very high flux of water and then allowed to drain. Subsequently, multiple flow rates were applied to elicit flow data which could be modeled. An experiment was performed by using a step-up tracer and then a step-down tracer to glean solute transport experimental data. A schematic of the columns used in the test is shown in Figure 2.7.   Figure 2.7  Column test setup and experimental procedure (Cherkaev, 2019) To account for the limitations of the Richard equation, the work suggested the adoption of a liquid holdup model which estimated the holdup volume based on PSD (particle size distribution). The rationale for flooding the column was to overcome the issue of hysteresis of liquid holdup which was experienced. However, as described in previous sections, when preferential flow paths are created (by flooding the column in this case) the permeability of 34  certain zones in the bed of ore is higher than other zones. Moreover, the capillary forces are also distorted by pushing water to a zone of ore that would never otherwise experience that amount of moisture owing to the distance from the drip emitter. In the next section an experimental setup which attempts to calibrate the capillarity of a column of ore is discussed.  2.3.3 Experimental approach to estimate parameters in the \u03b8-explicit form of the Richards equation   The experimental setup used to estimate the hydraulic conductivity and air entry head (which is a measure of capillary action) is shown in Figure 2.8.  A column, 4 m in height and comprised of four 1 \u00d7 1 m cylindrical sections plus a bottom section which served as the stand was constructed of HDPE (Afewu and Dixon, 2008). The individual sections were bolted together using flange fittings with rubber gaskets between the flanges to seal the joints. The space above the stand section was filled with two layers of coarse silica gravel, each of a height of 100 mm, but of different particle size distributions. The particle size distributions were 80% passing 25 mm and 12.5 mm for the bottom and top layers, respectively. The silica gravel layer constituted a drainage layer for the column material during irrigation. The effluent solution exited the column from a spout on the bottom section through a flow-through conductivity probe connected to a meter. As each section was installed, it was filled with the ore to half of its height (0.5 m). Then, five TDR (Time Domain Reflectometry) probes to measure moisture were laid horizontally in a rectangular pattern on top of the ore bed, one across the center, and others to the east, west, north and south at 0.2, 0.2, 0.3 and 0.3 m from the center, respectively. Irrigation fluxes of 6, 9 and 12 L\/m2\/h were used to test and calibrate the capillary head and the hydraulic conductivity.  35   Figure 2.8  Experimental setup used to estimate hydrology parameters (Afewu and Dixon, 2008) The results of the tests showed that, within the limited water content range relevant to safe and successful operation of a heap, close agreement was found between model and experimental data. The test also observed that as irrigation flux increased the dispersivity also increased which resulted in higher volumetric water content. It is interesting to note that the authors observed no preferential flow or channeling even in large columns when \u201cas delivered\u201d ore was agglomerated and tested. While the experiment successfully demonstrated the ability of a water model incorporating advection and dispersion to closely agree with experimental results, it is impractical to execute this extensive column work for every ore sample. Given that a water model can explain wetting of an ore bed, a method to parameterize the model in columns using breakthrough curves would significantly improve the tedious nature of the extensive experimental work undertaken. 36  To summarize, the wetting of an ore bed is a result of unsaturated water transport through a porous medium. The movement of water through the pores can be conceptualized using mass conservation within a certain fixed volume; the mass input is equal to the sum of the mass output plus the mass accumulated within the volume. Also, the Richards equation provides a useful relationship for determining the water content in a porous medium by accounting for the pressure gradient associated with hydrostatic and capillary forces (which cannot be ignored). But which version of the Richards equation is applied matters, as it would reflect in the experimental techniques used to estimate the parameters. 2.3.4 Modeling unsaturated flow using Computational Fluid Dynamics    A generic computational method for solving variably saturated flow equations using CFD codes is presented (McBride et al., 2006) so that the solution may be utilized in comprehensive modelling of reactive porous-media-based processes, such as industrial heap leaching. In this work the h-based (pressure head) form of the Richards equation is used as a starting point.   zhKhhKthhC\u2202\u2202+\u2207\u2207=\u2202\u2202 )(])([)(   (2.22) where C(h) is the specific moisture capacity, defined as,  hhC\u2202\u2202=\u03b8)(   (2.23) In highly non-linear problems, such as infiltration into very dry heterogeneous beds, these methods can suffer from mass-balance error and convergence problems. It is stated (Celia et al., 1990) that \u2018The reason for poor mass balance resides in the time derivative term,\u2019 while d\u03b8\/dt and C(dh\/dt) are mathematically equivalent in the continuous partial differential equation, their discrete analogues are not. The inequality in the discrete forms is exacerbated by the highly nonlinear nature of the specific capacity term C(h). This leads to significant mass-balance errors 37  in the h-based formulations because the change in mass in the system is calculated using discrete values of d\u03b8\/dt while the approximating equations use the expansion C(h)(dh\/dt). Essentially, the hysteresis in the measured liquid hold-up does not align with the approximating equation. To overcome this problem, Pan and Wierenga (Pan and Wierenga, 1995) adopted an approach where h is transformed into a new dependent variable, ht:  \uf8f4\uf8f3\uf8f4\uf8f2\uf8f1\u2265<+=001hhhh\u03b2hht   (2.24)  where \u03b2  =  is a universal constant (~ \u22120.04 cm\u22121) The Richards equation and the continuity equation are solved with the transformed independent variable and a comprehensive model for copper sulphide heap leaching is presented (Bennett et al., 2012). The model covers a mixture of chalcocite and pyrite in a column under leach with a ferric raffinate and includes reaction kinetics of key minerals (such as chalcocite) within the context of a shrinking core algorithm. This work demonstrates how small column tests can be used to parameterize the model and larger scale column tests can be used to validate the model for scale up with a particular type of ore. Also, it is observed that the simulation results of the smaller and larger column tests are broadly in conformance with experimental measurements despite the differences in behavior of the two systems.  Furthermore, a comprehensive CFD model is detailed in the work by (McBride et al., 2018). Figure 2.9 incorporates all governing equations related to vapor and liquid phases in addition to heat balance. These equations are solved over a mesh which is generated by discretizing the domain using hexahedral individual elements. 38   Figure 2.9  Comprehensive CFD model incorporating all possible contributions in a heap (McBride et al., 2018) The numerical approach and algorithm used to solve the equations are detailed by McBride et al. (2006). The advantage of this approach is the ability of the model to capture complex myriad physical interactions and produce simulation results thus enabling engineers to design and optimize heaps. Given the complexity of heaps, the model requires a significant amount of data and calibration to provide accurate simulations.     2.4 Motivation for this work Heap leaching requires many months and sometimes years to realize the ultimate expected metal extractions and translation of these values into recovered metal production. In the column leaching tests conducted to showcase the effect of Jetti\u2019s catalytic heap leach technology (Rebolledo et al., 2019), 954 days of leaching was required to achieve 70% copper extraction from a 2-m column. By contrast, milling and flotation operations have total in-process retention times of a few hours, depending on stockpiling and surge capacities provided, to recover metal in 39  concentrate in a readily accountable form. Similarly, milling and agitated leaching operations typically have retention times of about 12 to 48 hours.  Therefore, forecasting of metal extraction rates and other process parameters in scale-up of lab or bench scale work to commercial operations is undemanding. The short time interval also hastens the ability of production to do metallurgical accounting on a comprehensive basis rather than in the case of heap leaching.  In heap leaching, representative sampling of feed grades is difficult to achieve, the process itself is harder to control, the results are slower, and the metal recoveries are delayed. The holdup of metal values within the heap itself is substantial in most cases, thereby contributing to delays between ore stacking and final metal production (Manning and Kappes, 2016; Marsden and Botz, 2017; Scheffel, 2006; Thenepalli et al., 2019).  Consequently, in the context of heap leach modeling, an ability to forecast metal extraction rates based on myriad input parameters of ore bed height, flow rate of lixiviant, aeration rate, solute concentrations, etc., would be considered as the holy grail. Such a model would incorporate and parameterize every physical and chemical process that happens within a heap. Significant calibration and validation efforts against actual column and heap data would be required to populate the model parameters. Thus, the model could then be used to simulate \u201cwhat-if\u201d scenarios regarding operations in a heap. 2.5 Focus of this work  To summarize, it is clear that the ferric leaching of copper from chalcopyrite is significantly enhanced when additives are added to the lixiviant. To apply the novel leach technology on a heap, the additives are dissolved in the lixiviant and delivered to the bed of ore using drip irrigation. Hence, to understand and estimate the impact of the additives on leaching, the 40  distribution of these additives through the bed of ore must be characterized. As described in earlier sections, the transport of additives dissolved in lixiviant through the bed of ore is a combination of advection, dispersion, adsorption, and reaction. An additive that affects any of the four aspects of transport resulting in better distribution of the lixiviant through the bed of ore might not significantly enhance the ability of the lixiviant to extract copper from chalcopyrite, given the passive nature of chalcopyrite. An example of this is the work of Yanez et al. (2019) using a leaching aid, LixTRATM, to extract copper. The study uses copper extraction as a metric to study the impact of an additive which decreases surface tension of the lixiviant and hence should improve wetting of the ore bed. This study assumes that better wetting results in better leaching, which is premature. As discussed earlier, the transport of water (advection) is only one of the modes of transport and the other modes of transport such as adsorption and reaction are overlooked. In the case of copper oxides, the impact of this assumption is limited as the material is readily leachable.  To this end, this work aims to better understand transport of lixiviant through a bed of ore using breakthrough curves of other measurable quantities generated by carefully tailored experiments, and comparing them with breakthrough curves generated using a 2D axisymmetric model. The following objectives are defined to characterize the transport and distribution of additives within a bed of ore. i. A technique to parameterize the nature of water flow through a column of ore The ability of a column of ore to conduct and spread water can be represented as hydraulic conductivity and capillarity. These parameters are estimated by comparing results of experimental water flow breakthrough curves to breakthrough curves generated using a 2D axisymmetric water model which describes the flow of water through a bed of crushed ore. 41  ii. Interaction of additives on the surface of ore under non-leaching conditions  The breakthrough curves generated by inert solutes such as Li, used as a tracer, is compared with ETu which is used as a model thiocarbonyl catalyst that aids the leaching of copper from chalcopyrite.  ETu curves deviate from those of the tracer. A solute model is used to generate a breakthrough curve of an adsorbing and reacting solute. The model curve is compared against the experimental curve to estimate specific adsorption parameters.  iii. The transport of interactive solutes through the ore bed under leaching conditions   Tracer breakthrough curves are generated to validate the solute model when the feed concentration is steadily reduced, thus simulating a decaying solute. ETu, which decays in the lixiviant, is used to compare the model generated ETu breakthrough curves to the experimental curves. Additionally, N2, air and O2 are purged into a column to study the impact of oxygen concentration of the rate of ETu decay. Also, pyrite is agglomerated with ore at different levels to simulate the varied concentration of sulfides in ore bodies. The resultant breakthrough curves of ETu are compared to the sensitivity of parameters in the solute model to discern the impact of pyrite on the decay of ETu.  iv. Breakthrough curve analysis to elicit kinetics of ETu assisted leaching of chalcopyrite Given the overall ETu distribution in a catalytic leaching column of chalcopyrite ore, breakthrough curve analysis is used to estimate kinetic parameters in ETu assisted leaching of chalcopyrite. The resulting extraction of rate of Cu provides context to the ETu distribution in the column under leach.   42  Chapter 3: Water Model Development  In this chapter the flow of water through a bed of ore is investigated in detail. As discussed earlier, in heap leaching, solution or lixiviant is fed to the bed the ore through a series of pipes at the end of which is a drip emitter. The lixiviant wets the bed of ore directly under a drip emitter by spreading vertically and radially. Hence, the best way to understand and quantify water flow is to describe water flow under a single drip emitter. A heap could then be conceived as a collection of adjoining cylinders under single drip emitters.  Hence, the rationale to develop a 2D axisymmetric model under a single point source in a column of ore is to have a tool that would describe all of the important phenomena occurring in columns (and, by extension, heaps) which runs easily on a PC or laptop. This formulation avoids the intricacies and significant computational resources of 3D modeling, while acting as a convenient tool for simulation and interpretation of column breakthrough data.   The percolation of water and solutes through a bed of ore is governed by the conservation of mass. Thus, for observing mass conservation in a fixed volume, the mass input is equal to the mass output plus the mass accumulated within the volume. Also, Darcy\u2019s Law provides an accurate relationship for determining the velocity of water through a porous medium by relating it to the attendant hydrostatic and capillary forces, the viscosity of water and the local permeability of the porous medium.  3.1 The Water Model  Assuming a constant density of water, the 2D axisymmetric equation of water volume continuity is given thus: 43  wwwrwzw s\u03c1Mt\u03b8rvrrzv\u2212\u2202\u2202=\u2202\u2202\u2212\u2202\u2202\u2212=\u22c5\u2207\u2212)(1 ,,wv                                         (3.1) where vw = water volume flux (a vector) (m3 water\/m2\/s)  z = depth (m)  r = radius (m)  \u03b8 = volumetric water content (m3 water\/m3)  t = time (s)  sw = net rate of water production (kmol\/m3\/s)  \u03c1w = water density (assumed to be 1000 kg\/m3 water)  Mw = water molecular mass (18.015 kg\/kmol) The water volume flux given by Darcy\u2019s Law may be written in isotropic, unsaturated water-air media as follows:  )()( crwwcwwrw hzkKpzg\u03c1\u03bckk\u2207+\u2207=\u2207+\u2207=wv  (3.2)  rhkKrp\u03bckkvzhkKzpg\u03c1\u03bckkv crwwcwrwrwcrwwcwwrwzw \u2202\u2202=\u2202\u2202=\uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb\u2202\u2202+=\uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb\u2202\u2202+= ,, 1  (3.3) where k = saturation permeability (m2)  \u03bcw = water viscosity (kg\/m\/s)  g = gravitational acceleration (9.81 m\/s2)  Kw = hydraulic conductivity ww \u03bcg\u03c1k=  (m\/s)  krw = relative water permeability (\u2013)  pc = capillary pressure (Pa)  hc = capillary head g\u03c1p wc=  (m) Darcy\u2019s Law may be recast in terms of \u03b8 thus:  \u03b8Dz\u03b8U\u03b8\u03b8ddhkKz\u03b8\u03b8kKwwcrwwrww \u2207\u2212\u2207=\u2207\uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb\u2212\u2212\u2207\uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb=wv  (3.4) 44   r\u03b8Dvz\u03b8D\u03b8Uv wrwwwzw \u2202\u2202\u2212=\u2202\u2202\u2212= ,,  (3.5) where \u03b8kKU rwww = and   \u03b8ddhkKD crwww \u2212=  (3.6) and the continuity equation (3.1) may be recast as follows:  wwwww s\u03c1Mt\u03b8\u03b8Dz\u03b8U \u2212\u2202\u2202=\u2207\u2212\u2207\u22c5\u2207\u2212 )(  (3.7) Equation 3.7 can be solved for \u03b8. The discretization and solution of the water model is given in detail in Dixon and Afewu (2022).  To estimate the hydraulic parameters, steady state infiltration tests were performed on columns filled with ore and the resulting breakthrough curves were analyzed. 3.2 Steady State Infiltration Tests \u2013 Estimating Hydraulic Parameters  In unsaturated flow the water content (\u03b8) varies with change in flux applied. According to Afewu and Dixon (2008), plotting the applied irrigation flux at steady state against effective saturation Se is well fitted by a power law function. This suggests that the Brooks-Corey model (Corey, 1977) is adequate to describe the hydraulic characteristics of the ore in the column, as would be expected at low saturation levels typical of beds of coarse ore. The Brooks-Corey model is defined thus:  \u03c8eww SKv =  (3.8)  \u03bbecc Shh10,\u2212=  (3.9) where \u03bb and \u03c8 are material-specific constants, and 45   ersrrsre S\u03b8\u03b8\u03b8\u03b8\u03b8\u03b8\u03b8\u03b8S )(or \u2212+=\u2212\u2212=  (3.10) where  \u03b8s = saturated volumetric water content (the effective macro-porosity) (m3 water\/m3)  \u03b8r = residual (or fully drained) volumetric water content (m3 water\/m3) Equations 3.8 and 3.9 relate to the conductivity and capillarity of the bed of ore. One limitation of the BC model is the sudden change of the Se curve at hc = hc,0 as can be deduced mathematically from Eqn (3.9), denoting the pressure at which the largest pores drain (Looney & Falta, 2002). Some continuously differentiable relationships have been proposed to improve the description of the soil moisture characteristic curve near saturation and one such equation is that of van Genuchten (van Genuchten, 1980):  mncce hhS])(1[10,+=  (3.11) where n  = an exponent related to the pore size distribution (\u2212)  m = an exponent with values over the range 0 < m < 1 (\u2212), often taken as m = 1 \u2212 1\/n  Further, according to the widely adopted formalism of van Genuchten and Mualem (VGM) (Mualem, 1976; van Genuchten, 1980), the relative water permeability and capillary head may be specified as unique functions of effective saturation Se (the fraction of macro-porosity filled with flowing water) thus:  meecceeerwmmnSySyhhSSySk )1(whereand111)1( 11210,2\u2212=\uf8f7\uf8f7\uf8f8\uf8f6\uf8ec\uf8ec\uf8ed\uf8eb=\uf8f4\uf8f3\uf8f4\uf8f2\uf8f1\u2265<\u2212=  (3.12) where m = an empirical constant, 0 < m < 1 (\u2013)  n = an empirical constant, often taken as (1\u2013m)\u20131 (\u2013)  hc,0 = the capillary head scaling factor or air-entry head (m) 46  The relationship between the Brooks-Corey (BC) parameters and the van Genuchten-Maulem (VGM) parameters are as follows:   124\u2212=\u03c8m  (3.13)  m\u03bbn =  (3.14)  2)()(mKKBCwVGMw =  (3.15) With these relationships, Eqn (3.7) can be solved for \u03b8 if \u03c8, Kw, m, n, \u03bb and hc,0 are known.  The conductivity parameter (Kw) estimation is discussed in the first subsection followed by capillarity.  3.2.1 Estimation of Kw, m, n and \u03bb A short transparent PVC cylinder with a diameter of 10 cm and a height of 40 cm was used as an ore column. A chalcopyrite ore sample with P80 = 6.75 mm was riffled and oven dried at 45\u00b0C. The weight of ore was 3.418 kg (on a dry basis). The dry ore was mixed with 100 g of DI water and loaded into the column and the height of ore in the column was measured at 30 cm. Figure 3.1 is a pictorial representation of the experimental setup used to estimate hydraulic parameters.  B1 and B2 are the respective buckets from which water was fed to the column and into which the effluent was collected. P1 is the peristaltic pump used to regulate a specific flow rate of solution into the column. The buckets and the column of ore each rest on scales. The sample was supplied by Jetti Services Canada. Chalcopyrite was the only copper bearing phase. Inductively coupled plasma atomic emission spectroscopy (ICP-AES) showed that the copper content was 0.344%. Table 3.1 gives the particle size distribution of the sample. 47    Figure 3.1  Column water breakthrough curve test apparatus Table 3.1  Particle size distribution of chalcopyrite ore sample  Sieve Size (mm) % Total Weight + 9.50 0 \u2212 9.50 + 6.75 20.16 \u2212 6.75 + 4.75 16.60 \u2212 4.75 + 2.36 25.79 \u2212 2.36 + 0.85 17.62 \u2212 0.85 + 0.60 3.79 \u2212 0.60 + 0.30 4.88 \u2212 0.30 11.12  The steady-state infiltration tests involved feeding a constant flow of solution to the 30 cm column until the mass of the column reached a constant value (Scale 3 as shown in Figure 3.1). Throughout the experiment the values of Scales 1 and 2 are recorded at 1-minute time intervals which gives the change of weight over time of the respective buckets B1 and B2. This procedure was repeated at different solution flow rates. The entire test involved stepping upward through four flow rates as follows: 7.6, 11.5, 15.3 and 22.9 L\/m2\/h. By noting the mass of the entire 48  column apparatus on a digital balance and subtracting the known tare weight and the mass of dry ore, the total amount of moisture in the column could be measured directly. Between each flow rate change 48 hours were allowed to pass to ensure complete draining of the column. For the purposes of this study the density of water is assumed to be 1000 kg\/m3. The bulk density of the ore filled into the column was calculated to be 1451.3 kg\/m3.  The residual moisture content \u03b8r is the amount of moisture in the column after it has drained completely. This was measured to be 0.290 kg (8.51% by mass or 12.35% by volume). Also, the saturated moisture content \u03b8s, estimated by flooding the column at the end of all experiments, was measured to be 0.894 kg (26.1% by mass or 37.9% by volume). The equilibrium moisture content \u03b8e is the amount of moisture at steady state for each flow rate, which is given in Table 3.2 along with the effective saturation Se as defined in Eqn (3.5).  Water flux is plotted against S3 in Figure 3.2.    Figure 3.2  Water flux vs effective saturation to estimate hydraulic conductivity (Kw)  49  Table 3.2 Effective (steady state) saturation of the column at different water fluxes  Water Flux (L\/m2\/h) Weights (Kg) Moisture content by Volume (\u03b8, %) Effective Saturation (Se, %) 7.6 0.364 15.45 12.25 11.5 0.368 15.62 12.91 15.3 0.376 15.96 14.24 22.9 0.382 16.22 15.23  From Figure 3.2, according to Eq (3.8) Kw (BC) = 151354 L\/m2\/h = 0.042 m\/s. Similarly, m = 0.4778 using Eqn (3.13) and n = 1.915. Therefore, Kw(VGM) from Eq (3.15) is 0.18 m\/s and \u03bb = 0.915 from Eqn (3.14). Further steady state infiltration tests were performed on the wet ore.  The fluxes applied were the same as before but the bed of ore was changed by adding fines in the form of pyrite of size 75 \u2013 102 \u00b5m (0.5% or 18 g, 1.2% or 43 g, and 1.7% or 60 g by weight).  Figure 3.3  Water flux vs effective saturation to estimate hydraulic conductivity parameters (Kw(BC) and Kw(VGM)) 50  Table 3.3 Change in Kw(BC) and Kw(VGM) (m\/s) with fines added  % Pyrite added as fines  Kw(BC) (m\/s) Kw(VGM) (m\/s) (Se, %)  at uw = 7.6 L\/m2\/h (Se, %)  at uw = 22.9 L\/m2\/h 0 0.0421 0.184 12.25 15.23 0.5  0.0174 0.0771 15.12 18.90 1.2 0.0069 0.0250 15.02 19.41 1.7 0.0024 0.0059 14.15 19.06 For the 0.5%, 1.2 % and 1.7% tests, the value of both Kw(BC) and Kw(VGM) changes significantly while the Se range is roughly between 14% and 19%. The Kw(BC) value is also significantly lower than the  Kw(VGM) implying that the respective Kw values denote the unsaturated hydraulic conductivity parameter based on the Brooks-Corey or van Genuchten-Maulem formalism and should not be construed as the value of hydraulic conductivity of the bed of ore. Hence, Kw should only be considered as a model parameter that can predict the effective saturation for an applied irrigation flux. Extrapolation of Kw as the saturated hydraulic conductivity is not valid as the tests have been performed under very low saturation levels.  Subsequently, a 1 m tall column was filled with 6.184 kg of dry ore. The height of the bed of ore was measured at 0.85 m. Similar tests were repeated at different irrigation fluxes (11.9, 20.3, 27.4 and 35.8 L\/m2\/h) to observe the behavior of Kw(VGM) and Se when an irrigation flux is applied on a dry bed of ore as compared to a bed of ore which has already been wetted. The irrigation fluxes were applied in ascending order to the dry bed of ore. For example, A flux of 11.9 L\/m2\/h was applied to the column and the steady state weight of the column measured to record the moisture content. Then the pump was stopped and the column allowed to drain fully before the next irrigation flux was applied. The final flux of 35.8 L\/m2\/h was allowed to irrigate the column for 12 hours (overnight) and then shut off the next day. The data set generated from applying the four irrigation fluxes to the column of ore was called Run 1 (R1). Upon fully 51  draining, the four irrigation fluxes were applied again to the column of (now) wet ore with an initial moisture content of 0.094 kg, and the data set was named Run 2 (R2). In R2, the column was not allowed to drain completely in between changing the irrigation fluxes. Once R2 was complete the column was allowed to drain overnight and the residual moisture content was measured. The next day, the column of ore was flooded and the saturated weight of water in the column measured at 1.44 kg. Upon allowing the column to drain for 24 hours (after saturation) the irrigation fluxes were applied again and this data set was named Run 3 (R3). The data form the experiments is presented in the tables below.  Table 3.4 Equilibrium moisture retention of dry ore with increasing flux; \u03b8initial = 0 and \u03b8r = 2.19% (R1) Flux L\/m2\/h   Weight of Water in column, kg \u03b8 (%) Se (%) 11.94 0.180 4.21 6.39 20.29 0.190 4.44 7.13 27.45 0.204 4.77 8.17 35.81 0.212 4.96 8.77 Table 3.5 Equilibrium moisture retention of partially wet ore with increasing flux; \u03b8initial = 2.19% and \u03b8r = 2.81% (R2) Flux L\/m2\/h   Weight of Water in column, kg \u03b8 (%) Se (%) 11.94 0.278 6.50 11.97 20.29 0.304 7.11 13.94 27.45 0.320 7.48 15.15 35.81 0.334 7.81 16.21 Table 3.6 Equilibrium moisture retention of fully wet ore with increasing flux; \u03b8initial = \u03b8r = 4.68% (R3) Flux L\/m2\/h   Weight of Water in column, kg \u03b8 (%) Se (%) 11.94 0.336 7.86 10.97 20.29 0.360 8.42 12.90 27.45 0.374 8.74 14.03 35.81 0.396 9.26 15.81 52  From the \u03b8 values of R1 (Table 3.4), it is obvious that the water content of the column at equilibrium for the fluxes from 11.94, 20.29 and 27.45 L\/m2\/h is lesser than the residual water content of R3. Even after irrigating the column for 24 hours at a flux of 35.81 L\/m2\/h, the residual moisture content is only 60% of the residual moisture of a fully wet ore. It is therefore likely that the \u03b8r measured in the first 2 runs R1 and R2 is not uniformly distributed through the bed of ore and doesn\u2019t represent the true ability of the ore to hold residual moisture. Figure 3.4 shows the change in water content of the bed of ore for the same set of irrigation fluxes but different initial moisture levels.    Figure 3.4 Water flux vs water content curves on the same ore bed but with different initial moisture levels Thus, \u03b8r as measured is not a material property of the ore, but rather a path-dependent attribute of a specific set of experiments. The hysteresis of the water volume holdup (as discussed in section 2.3.1) could be a result of this variation. Further, Se (from Eqn. 3.10) is a representation of the saturation of the bed of ore normalized against \u03b8r and \u03b8s. While \u03b8s is constant, the value of \u03b8r 53  (being path-dependent) results in the value of Se being path-dependent. Hence the Brooks-Corey relationship (Eqn 3.8) which relates the flux to the effective saturation should be viewed as a relationship that predicts the water content for a specific irrigation flux for the experiments performed rather than a tool with which to elicit material properties of the ore. Figure 3.5 shows the water flux vs effective saturation curves for all 3 runs.  Figure 3.5  Water flux vs effective saturation on the same ore bed but with different initial moisture levels Table 3.7 Change in Kw(VGM) of dry ore, partially wet and fully wet ore  Run   Kw(VGM) (m\/s) \u03b8inital (%) \u03b8r (%) 1 0.0591 0  2.19  2  0.0163 2.19 2.81 3 0.0043 4.68 4.68 As the \u03b8r is different for each run, the effective moisture levels (\u03b8e) of R1 and R2 are most likely quasi-equilibrium values and not a true representation. Therefore, it follows that for any further tests to be valid the change in \u03b8r should be less significant which can be achieved by ensuring 54  agglomeration and measuring the drain weight of the column after infiltration tests are concluded. While the nature of \u03b8r variation is beyond the subject of this work, the author recognizes the importance of this attribute in the measurement of hydrology parameters.  In the context of this work, the estimated Kw(VGM) parameter needs to provide a relationship to predict water content \u03b8 to irrigation flux vw for a given \u03b8initial and \u03b8r. In Figure 3.6 below, a simulation of the moisture content of the column is shown.  Figure 3.6  Simulated moisture contents for Runs R1, R2 and R3 at 20.29 L\/m2\/h. Air entry head hc,0 assumed at 0.35 m. Simulated moisture levels: R1 = 4.50%, R2 = 7.08%, R3 = 8.78%. The moisture levels predicted by the simulation is in close agreement with the measured moisture levels in Table 3.4, 3.5 and 3.6. However, the model suggests that the moisture is evenly spread throughout the column which is a consequence of the assumed hc,0 value of 0.34 m. This might not be case especially in R1 when the initial moisture content was zero. A novel 55  method to measure the hc,0 from the effluent of the column is discussed in the next section. To summarize, the hydrology parameters calculated using this technique should be viewed in context of the assumptions such as even spread of water and an insignificant change in \u03b8r throughout the test. 3.2.2 Estimation of hc,0  By measuring the weight of the column under irrigation the moisture content can be measured and the well-known Brooks-Corey (BC) and van Genuchten-Mualem (VGM) equations can be used to parameterize a relationship between the effective saturation and the applied flux. The only other parameter required to solve the water model, Eqn (3.9), is the air-entry head parameter, hc,0.  Air-entry head has the biggest impact in determining the degree by which water will spread out under a point source such as a drip emitter in a heap. Very low values of hc,0 associated with coarsely crushed solids ensure very little capillary head, and therefore, very little lateral spreading of water. The experimental method to estimate hc,0 would be akin to the estimation of Kw where the local capillary heads hc would be measured by using tensiometers and the resulting data would be fit to a power-law function as expressed in Eqn (3.11). However, the measurement of local capillary head is often noisy and unreliable and the very fact of inserting probes into the flow domain significantly changes the nature of capillary action. Hence, breakthrough curve analysis is used here as a novel approach to estimate hc,0.  56   Figure 3.7  Discharge bucket mass (tared) as recorded by Scale 2 of Figure 3.1 Water breakthrough curves are calculated by measuring the weight of the drainage bucket B2 in Figure 3.1. For example, for the irrigation flux at 11.5 L\/m2\/h in the 30 cm column the weight of water in bucket B2 is plotted against time in Figure 3.7.A weighted central difference approach is used to estimate the first derivative of the discharge mass with respect to time, which results in the water breakthrough curve shown in Figure 3.8.   Figure 3.8 Water breakthrough curve \u2013 the red dots are effluent; the blue dotted line represents an infiltration flux of 11.5 L\/m2\/h (1.5 mL\/min) 1 2 3 57  The portion of the graph marked as \u201c1\u201d is the point of first water breakthrough observed. Steady state is achieved at the portion of the graph labeled as \u201c2\u201d, where water flow into the column is equal to water flow out of the column. The section marked as \u201c3\u201d is the drainage of water from the column after irrigation ceases. The dashed blue line represents the amount of time for which a steady irrigation flux was applied to the column. Hence, using our prior calculated hydrology parameters Kw, m, n and \u03bb, values of air-entry head hc,0 are estimated starting from 0, 0.25, 0.5, 0.75 and 1.00. Using each of these values we generate model breakthrough curves which are in turn compared to the breakthrough curve observed experimentally. The sensitivity of the model breakthrough curve to the hc,0 value is represented in Figure 3.9.   Figure 3.9 Sensitivity of water breakthrough curve to air-entry head hc,0 The deviation between each model generated breakthrough curve and the actual data is expressed as a sum of square errors (SSE), resulting in an SSE value for each value of air-entry head guessed. This data is presented graphically in Figure 3.10. A parabolic function is used to fit the SSE data to the air-entry head and the minimum SSE is found by setting the derivative of the function to zero to estimate the best-fit hc,0 value from the experimental breakthrough curve.  58  The best-fit air-entry head value is estimated to be hc,0 = 0.35 m. To validate the model against experimental values obtained, the parameters Kw(VGM) = 0.18 m\/s, m = 0.4778, n = 1.915 and \u03bb = 0.915 and hc,0 = 0.35 m are used to generate a water breakthrough curve for the flux of 11.6 L\/m2\/h and the experimental values are plotted in Figure 3.11.   Figure 3.10  SSE values plotted vs. hc,0    Figure 3.11  Model generated and experimental water breakthrough curves for 11.6 L\/m2\/h 59  Subsequently, all irrigation fluxes in Table 3.2 are used to generate model breakthrough curves and their respective experimental values are plotted in Figure 3.12.    Figure 3.12  Model generated and experimental water breakthrough curves As can be observed, there is very good agreement between the experimental breakthrough curves and the model generated curves. The flow rate of 7.6 L\/m2\/h deviated from the model curve as the initial irrigation was on a partially wet bed of ore. Subsequently the effect of the breakthrough curve on starting form a dry bed of ore is analyzed.  The weight of the discharge bucket was measured for Runs R1 (initially dry) and R2 (wetted by R1). Breakthrough curves were generated for the column with an ore bed depth of 0.85 m.   Three of the Run 2 breakthrough curves (for 1.7, 2.3 and 3.0 mL\/min, or 20.3, 27.4 and 35.8 L\/m2\/h, respectively) do not start from zero because the column was not allowed to fully drain before the next flux was applied. It can be observed that the R1 breakthrough curves are steeper and bimodal when compared to the same fluxes applied in R2 but on a wet bed of ore.  60   Figure 3.13 Breakthrough curves of water from a 0.85-m column  Figure 3.14  Experimental (R2) breakthrough curves vs model generated curves As shown in Figure 3.14, good agreement between model generated and experimental breakthrough curves is observed for the R2 data set. Based on the observed behavior of dry and wet beds, subsequent hydrology measurements were only conducted on wetted beds of ore. 61  3.3 Analysis of water transport through the column  This section illustrates the impact of hydraulic parameters Kw, m, n, \u03bb and hc,0 on water flow and breakthrough through a column. Sensitivity analysis of unsaturated hydraulic conductivity (Kw) results in the following breakthrough curves of water from a column shown in Figure 3.15.  Figure 3.15 Sensitivity of water breakthrough to hydraulic conductivity As shown, the lower the conductivity the more delayed the water breakthrough, which is observed as a lateral shift of the breakthrough curve. The conductivity also has an impact on the steady-state water content of the column. It can be deduced that a lower conductivity would result in a higher steady-state moisture content of the column as shown in Figure 3.16.  62   Figure 3.16  Sensitivity of steady-state water content to hydraulic conductivity   The air-entry head hc,0 represents the spread of water in a column. A value of hc,0 = 0 implies no capillary head and hence no ability of the ore bed to wick water sideways. A rivulet of water is expected to flow along the drip emitter axis down the entire column, as might be observed is columns of glass beads or pebbles with no fines. An unevenly stacked column might also experience a similar effect of poor water spreading, which would be undesirable in a column leach or heap leach context. As the hc,0 value increases more water is spread to the periphery of the column. Figure 3.16 shows the sensitivity of the steady-state water content to the air-entry head or capillary head of the ore bed. In Figure 3.17, the hc,0 value is changed from 0 (glass beads or pebbles with no fines) to 1 m (a column of mostly fines) and the spread of water is correspondingly represented.   63   Figure 3.17  Sensitivity of steady-state water content to air-entry head As discussed above, a rivulet type of water distribution can be observed in the case of a hc,0 = 0 under the drip emitter source. Similarly, at hc,0 = 1.0 m a virtually uniform spreading of water is observed. 3.4 Effect of column height and radius  The breakthrough curve tests in the previous section were performed on columns of ore with dimensions 30 cm \u00d7 10 cm and 100 cm \u00d7 8 cm columns. To test the robustness of the technique, a breakthrough curve test was performed on a column with dimensions 200 cm \u00d7 20 cm. A commercial lab performed the tests based on operating instructions given by the author. An experimental setup like that described in Section 3.2 was used. Water was fed to the column from a closed feed bucket and the discharge from the column was collected into a bucket placed on an electronic scale with a data logger. The mass of the effluent bucket was recorded every 60 seconds and the electronic scale had a precision of 1 g. Dry ore with P80 = 2.54 cm was loaded in 64  the column and an initial bed height of 122 cm was measured. The ore mass loaded on a dry basis was 64.9 kg. The column was irrigated for 72 h with tap water at a flow rate of 5 L\/m2\/h. Breakthrough was observed after 22.3 h of irrigation. After the irrigation was stopped, the column was allowed to drain for 48 h. No change in weight was recorded for the effluent bucket during a 2-h interval and this criterion was used to determine that the column had drained fully. The difference between the weight of the feed bucket and the water collected in the effluent bucket was 3.03 kg. This was the residual moisture content of the ore column. The depth of the ore bed after draining decreased to 105 cm. The bulk density of the ore was calculated as 1787 kg\/m3.  Subsequently, the column of ore was irrigated with different flow rates: 3, 7.5, 10 and 15 L\/m2\/h. Table 3.8 shows the computed data.  Table 3.8 Effective saturation of the column for different water fluxes Run Water Flux (L\/m2\/h) Mass Water Content Ratio, \u03b8e Effective Saturation (Steady State) Test 3 3 0.098 0.124 Test 2 7.5 0.101 0.143 Test 4 10 0.106 0.171 Test 5 15 0.109 0.189 Test 1 was the test with 5 L\/m2\/h, however that is not included in the calculation as that was used as a trial run and to wet the ore. Figure 3.18 is plotted with effective saturation against water flux to obtain the power-law Brooks-Corey parameters for the column with (effective) dimensions of 105 cm \u00d7 20 cm.  65   Figure 3.18  Water flux vs effective saturation to estimate hydraulic conductivity   Tests 2 and 3 were performed in the order of descending flow rate which probably caused the data point of Test 2 (at ~ 7.5 L\/m2\/h) to be an outlier in the data set.  Subsequently, hc,0 was estimated from Test 5, at a flux of 15 L\/m2\/h. Sensitivity of the water breakthrough to the air-entry head is shown in Figure 3.19 below.   Figure 3.19  Sensitivity of hc,0 to water breakthrough curve in the large column 66  It bears repeating that the balance used in this experiment had a precision of only 1 g, which is why the data have a \u201cstepped\u201d appearance in the plot. The curve corresponding to hc,0 = 0.05 m is a bimodal breakthrough curve, suggesting that the ore lacks enough capillary head to spread the water, resulting in a sudden rush of water through the bed. Anecdotally, this type of behavior is observed in poorly stacked columns or columns with relatively large ore particles. To estimate hc,0, it is plotted against the sum of square errors and Figure 3.20 shows the parabolic curve.   Figure 3.20  Sum of square errors plotted against hc,0 From the tests the following model parameters are calculated: Kw = 0.0105 m\/s , m = 0.535, n = 2.15 and hc,0 = 0.31.  At z = 1.05 m, the model predicts the breakthrough curve, which is tested against Test 4, at a flux of 10 L\/m2\/h .  Figure 3.21 compares the model-generated and experimental water breakthrough curves. 67   Figure 3.21  Model and experimental water breakthrough curves at an irrigation flux of 10 L\/m2\/h The experimental data, after the column reached steady state, has some granularity as can be observed. This is again attributed to the scale used in measurement of the weight of the outlet bucket. The scale had a precision of 1 g, so any fractional change in weight was not observed. The oscillation of the calculated experimental flux values about 10 L\/m2\/h is a result of this limitation in measurement. Notwithstanding this limitation, satisfactory agreement is observed between the experimental and model generated breakthrough curves. Thus, breakthrough curve analysis can be used to estimate and explain the behavior of water distribution in a column of ore using simple measurement techniques. In the next chapter a similar technique is used to describe solute transport.   68  Chapter 4: Solute Model and Adsorption  In Chapter 3, water flow through a column of ore was discussed using modeling techniques and experimental breakthrough curves. In this chapter the physical parameters affecting the distribution of solutes in the bed of ore is investigated in detail.  Solute breakthrough curves using ethylene thiourea (ETu) under various operating conditions are reported. Further, a solute model is built to explain the breakthrough curves. The 30 cm column was used to obtain solute breakthrough curves. Cu and Li inert tracer solutions were prepared with ACS grade CuSO4\u00b75H2O and Li2SO4\u00b7H2O (Sigma-Aldrich), respectively. A single point source, representing a single drip emitter, was used to feed solution into the column using a peristaltic pump. The column was irrigated with acidic water at a steady flow rate of 17.54 L\/m2\/h. The column was operated in open loop mode with separate feed and effluent buckets. Any change in feed concentration was achieved by switching the inlet tube from one bucket to another, thus achieving a step change in inlet concentration. The flowrate was calibrated by allowing the feed to drip into a beaker for 30 minutes and measuring the net weight. At the one-hour mark, the feed tube was switched to the Cu feed bucket to achieve a step change in input concentration. All samples were collected from the effluent tube of the column at specific time points. Upon reaching a steady outlet concentration the feed was switched to the acidic DI water bucket. After rinsing the column for 24 hours the Li feed solution was fed to the column.  The Cu tracer test was repeated at flow rates of 13 L\/m2\/h and 21.3 L\/m2\/h. The column was rinsed further with DI water for 48 hours. Subsequently, ETu feed solution was prepared by adding a specific amount of reagent grade ETu (Sigma-Aldrich, 99.9% purity) to DI water and 69  allowing it to mix for one hour before initial samples were taken. This feed was used to perform ETu breakthrough tests using the same experimental setup as the tracer tests. Reagent grade SDDC (Sigma-Aldrich, 98% purity) was added to the ETu feed solution in subsequent tests to study the impact of a competing adsorbate on ETu breakthrough curves. These solutions were adjusted to pH 8 with 0.1M NaOH solution to prevent the oxidative decay of ETu in acidic medium. A detailed description of the decay is provided in the next section. Analysis of the outlet samples was either by ICP (for Cu and Li) or HPLC (for ETu and SDDC). The observed concentrations were plotted against time to generate breakthrough curves. The following section discusses the breakthrough curves observed from tracer and reactive solutes tests. 4.1 Tracer Breakthrough Analysis Figure 4.1 shows the breakthrough curves of Li and Cu used as tracers at different flowrates. The deformation of the breakthrough curve relative to the feed concentration curve is attributed to dispersion.   Figure 4.1 Normalized tracer concentration feed and breakthrough curves 70  The breakthrough curves of Li+ and Cu2+ are similar at the same flow rate suggesting that dispersion is not a function of the solute added, as expected. It is therefore concluded that all solutes in the feed solution would disperse similarly. The area under each curve in Figure 4.1, calculated using the trapezoid rule, represents the total amount of solute fed and recovered respectively. In the case of Cu2+ fed to the column at 17.54 L\/m2\/h, 40.6 g of was fed into the column and 38.5 g was recovered, amounting to 95% of the copper fed. In the case of Li+, 15.3 g was fed to the column and 14.9 g recovered, amounting to 97% of the lithium fed. To observe the breakthrough curve of ETu a solution of 0.204 g\/L at a neutral pH was fed to the column at a flow rate of 17.54 L\/m2\/h. Samples were taken from the feed and the effluent of the column and analyzed at specific time points. The data are plotted in Figure 4.2.  Figure 4.2  Breakthrough curve of 0.204 g\/L ETu from a 30 cm column at 17.54 L\/m2\/h A minor drop in concentration was observed at 14 hours in the ETu IN curve. A fresh 3-L batch of 0.204 g\/L solution was needed to sustain adequate feed for the duration of the experiment and 71  an inaccuracy in measurement of the ETu required to make up the batch resulted in the concentration variation. It is observed that beyond the 14-hour time point the ETu OUT curve tends to flatten, suggesting that a steady state condition had largely been achieved. Hence for further tests, keeping all other conditions the same, the column was only fed for 14 hours. Comparing the ETu OUT curve with the Cu OUT curve at the same flowrate shows that the ETu breakthrough curve is very different. A significant horizontal shift and vertical depression is observed for the ETu curve when compared with Cu2+. This behavior can only be explained by accumulation and consumption terms, respectively, as will be discussed in the solute modeling section (Section 4.2). To further analyze the plot, it is divided into 2 sections. The area between the two ETu curves, IN and OUT, in Figure 2 labeled as \u201c1\u201d represents the difference between the amount of solute fed and amount of ETu recovered until steady state was achieved. The area labeled as \u201c2\u201d represents the amount of the solute that was recovered after the feed was switched to DI water. If the total amount of ETu that was fed to the column was recovered, then the area of \u201c1\u201d should be equal to the area of \u201c2\u201d, but the actual ETu that was recovered was 81.2% of the total ETu fed to the column, which is significantly lower than 95% in the case of Cu and 97% in the case of Li. This difference in the amount of ETu must be attributed to the interaction of ETu with the ore. The effect of feed ETu concentration on the amount of ETu not recovered is studied by repeating the experiment using ETu concentrations of 0.051 g\/L, 0.51 g\/L and 1.02 g\/L. The column was fed for 14 hours with the 0.051 g\/L feed solution and switched to neutral DI water at the 14-hour mark. Samples were taken periodically, and subsequent concentrations of 0.51 g\/L and 1.02 g\/L ETu were fed to the column after thorough rinsing with DI water. The breakthrough curves are shown in Figure 4.3. The crisscrossing of the ETu curves suggest that the adsorption of ETu does 72  not follow ideal monolayer adsorption where there are no interactions between adsorbate molecules on adjacent sites regardless of surface occupancy. An adsorption isotherm which adequately represents this behavior is presented in Section 4.3.  The amount of ETu not recovered was calculated using trapezoid rule integration at each concentration. The highest amount of ETu not recovered and its corresponding inlet concentration of 1.02 g\/L was used to normalize both axes of the plot as shown in Figure 4.4.  Figure 4.3  Breakthrough curves of ETu from a 30 cm column 73   Figure 4.4  Normalized solute loss vs. normalized input concentration of ETu The linear function relating the amount of solute not recovered to the concentration of feed implies a pseudo first-order relationship for the interaction of ETu with the ore. This is consistent with the nature of auto-generated H2O2 in DI water on the surface of FeS2 (this is owing to the well-known Fenton reaction which generates S2O32\u2212, SO42\u2212 and Fe2+ from the adsorbed O2 and H2O on the surface of FeS2 while producing intermediates such as H2O2 and other reactive oxygen species (ROS)) (Gil-Lozano et al., 2017) which results in the oxidative decay of ETu. However, to isolate the effect on ETu adsorption and limit the ETu interaction (from the Fenton radicals produced) on the surface of the ore, the pH was adjusted to 8 by using 0.1 M NaOH solution and a 1.02 g\/L ETu solution was used to generate a breakthrough curve. According to Gil-Lozano et al. (2017), the Fenton reaction is minimal under alkaline conditions. Subsequently, SDDC was added to the ETu solution in different ratios to the concentration of ETu and the shift in the breakthrough curves was observed. The data are shown in Figure 4.5. 74    Figure 4.5 Breakthrough curves of 1.02 g\/L ETu and Cu2+ under different conditions As shown in Figure 4.5, in alkaline solution the ETu adsorbed but did not react, as shown by the elevation of the ETu curve. Also, when a second thiocarbonyl compound, SDDC, was added, it competed with ETu for adsorption sites, thus causing the ETu breakthrough curve to shift towards the Cu curve, suggesting less adsorption. To further establish the effect of sulfide mineral surfaces as adsorption sites, finely ground pyrite was agglomerated with the ore at different mass ratios. By increasing the available sites for adsorption, a shift of the breakthrough curves of ETu in DI water towards the right of the graph is expected. Moreover, depression of the breakthrough curve is also expected at higher pyrite content because of autogenerated H2O2 on pyrite surfaces in DI water as discussed before. The ETu breakthrough curves are shown in Figure 4.6. 75   Figure 4.6  ETu breakthrough in the presence of added pyrite (DI water) As expected, the ETu breakthrough curves shift to the right owing to more solute being adsorbed and are also depressed suggesting additional reaction on the added pyrite surface. However, in the case of 1.0% and 1.5% added pyrite mineral, there is a increase in the ETu breakthrough concentration at about the 11- to 13-hour mark suggesting a lower rate of decay.  This result could reflect the pyrite concentration dependency of ETu decay kinetics which is not considered in this chapter. Similar experiments were conducted in a 1-m column with the same ore at the same particle size distribution. The objective was to explain the behavior of ETu breakthrough using a model calibrated from the 30-cm column. Tracer tests using a 6.25 g\/L solution of Cu2+ at flow rates of 11.9 L\/m2\/h and 35.8 L\/m2\/h were performed. Subsequently, an ETu solution with a concentration of 1.02 g\/L was fed at 35.8 L\/m2\/h. The results are shown in Figure 4.7. Only the OUT curves are shown to avoid clutter. Time zero is defined for each curve as the time when the tracer solute was introduced into the column. 76   Figure 4.7  Breakthrough curves of Cu2+ tracer and ETu from a 1m tall column The breakthrough curves suggest a similar behavior to that of the 30-cm column tests. In the case of Cu tracer tests, nearly all of the solute fed to the column was recovered, and in the case of ETu tests there was a significant amount of solute not recovered, as expected. In the next section, a 2D axisymmetric solute transport model originally developed by Dixon and Afewu (2022) is used as the starting point to model the behavior of ETu. 4.2 Modeling Solute Transport The transport of solutes through a porous medium is governed by the principle of conservation of mass such that the net rate of flux from a certain fixed volume is equal to the sum of solute accumulated and generated within that volume. Also, Fick\u2019s law is a constitutive relationship that describes molar flux of solute as a sum of advection (driven by bulk flow) and dispersion (a flow-driven diffusion-like process) in terms of concentration of solute. 77  For a 2D axisymmetric system, the equation of solute continuity for the transport of a single aqueous solute is given by:  AAbArAzA sx\u03c1c\u03b8trnrrzn\u2212+\u2202\u2202=\u2202\u2202\u2212\u2202\u2202\u2212=\u22c5\u2207\u2212 )()(1 ,,An  (4.1) where nA = molar flux of solute A (a vector) (kmol\/m2\/s)  cA = molar concentration of solute A in solution (kmol\/m3 water)          xA = molar concentration of solute A on ore (kmol\/kg)          \u03b8 = volumetric water content (m3 water\/m3)          \u03c1b = solids bulk density (kg\/m3)  sA = net molar production rate of solute A (kmol\/m3\/s)  Figure 4.8  Section of the control volume used to apply the conservation principle If xA is the concentration of solute adsorbed onto the ore particles in the control volume, then adsorption isotherms may be used to express it as a function of solution concentration cA:  )( AA cfx =  (4.2) It follows that Eqn (4.1) may be recast as follows:  ( ) AAbArAzA scf\u03c1c\u03b8trnrrzn\u2212+\u2202\u2202=\u2202\u2202\u2212\u2202\u2202\u2212=\u22c5\u2207\u2212 )()(1 ,,An  (4.3) While Eqn (4.3) describes equilibrium adsorption, observations from the breakthrough curve tests suggest an obvious rate of decomposition associated with heterogeneous chemical reaction. Conc. of solute on ore particle, xA (kmol\/kg) Conc. of solute in solution, cA (kmol\/m3 water) nA 78  Hence it can be theorized that the solute A is adsorbed onto the ore and a certain proportion of it decomposes as shown below.    Figure 4.9 Block diagram representation of proposed solute decomposition, where k* is the first-order decomposition rate constant The reaction A \u2192 P represents the decay of solute A to products. From the law of mass action, the decomposition rate is also a function of concentration, and based on the results shown in Figure 4.4 we can safely assume pseudo first-order kinetics:   AA cks*\u2212=  (4.4) Eqn (4.1) can then be recast explicit in cA:  ( ) AAbArAzA ckcf\u03c1c\u03b8trnrrzn *,, )()(1\u2212+\u2202\u2202=\u2202\u2202\u2212\u2202\u2202\u2212=\u22c5\u2207\u2212 An  (4.5) Molar flux is given by Fick\u2019s law which, assuming dilute solution, may be written thus:  AA cc \u2207\u22c5\u2212= AwA Dvn  (4.6) where      DA = solute dispersivity of solute A (a symmetric tensor) (m3 water\/m\/s) The adsorption isotherm that best describes the behavior of ETu breakthrough can be determined only upon estimating dispersion, represented by DA in Eqn (4.6), and water content, represented by \u03b8 in Eqn (4.5). The calculation of \u03b8 has been detailed in Chapter 3. As the same column of ore is used, the VGM parameters (m,n), conductivity (Kw), and air-entry head (hc,0) previously estimated have been used to solve for \u03b8. That value is applied in Eqn (4.5). The method to solve the water and solute continuity equations is detailed in Dixon and Afewu (2022). A MATLAB k* K Liquid cA Solid xA  Solid   A        P 79  code was written by the author based on that solution technique to solve for \u03b8 and cA in the present study. The hydrology parameters used to solve for \u03b8 are given in Table 4.1. Table 4.1 Hydrology parameters estimated from water irrigation tests in Chapter 3 VGM parameters Kw (m\/s) hc,0 (m) m n  0.18 0.35 0.4778 1.915  Upon computing \u03b8 and setting the adsorption and reaction terms to zero, the solution of Eqn (4.5) gives the concentration profile and the breakthrough curve for an inert tracer solute (Li+ or Cu2+) affected by dispersion. This impact of dispersion is studied using sensitivity analysis of the dispersivity parameter \u03b1 on the breakthrough curve of the tracer solutes fed to the column. According to the well-known formalism of Bear (1972), the longitudinal and transverse dispersivities may be taken as linear functions of volume flux:  0,,,0,,, AwDwTTAAwDwLLA D\u03b5v\u03b1DD\u03b5v\u03b1D +=+=  (4.7) where \u03b1L  = aqueous longitudinal dispersivity coefficient (function of \u03b8) (m)  \u03b1T  = aqueous transverse dispersivity coefficient (function of \u03b8) (m)  vw  =  2 ,2 , rwzw vv +  = resultant water volume flux (m3 water\/m2\/s)         \u03b5D,w  = aqueous molecular diffusivity coefficient (function of \u03b8) (\u2013)   DA,0  = molecular diffusivity of solute A in water (m2\/s) According to (Afewu and Dixon, 2008), the transverse dispersivity coefficient \u03b1T may be taken as 1\/30 of the longitudinal dispersivity coefficient \u03b1L, which can be represented as a linear function of solution velocity in the form  06929043240 .v\u03b1 wL +=  (4.8) 80  The respective slope and intercept values were calibrated to the column experiments in that study. A sensitivity analysis on the effect of the slope and intercept of the linear function \u03b1(v) and the ratio of \u03b1T to \u03b1L is presented in Figures 4.10, 4.11 and 4.12.   Figure 4.10  Sensitivity of the model to the slope of \u03b1L(v) The value of the slope, for the linear relationship of dispersion with the volume flux, is very high because the units of v are m3 water\/m2\/s, which gives very small numerical values. For example, an infiltration rate of 7.6 L\/m2\/h for the test column translates to v = 2.1\u00d710\u20136 m3\/m2\/s. In Figure 4.10, an adequate fit for the slope is achieved from the family of curves. The same value of 43240 as determined by Afewu and Dixon is used in modeling the breakthrough curve from the 30-cm column experiments. In Figure 4.11, the sensitivity of the intercept value is tested at that slope value.   81   Figure 4.11  Sensitivity of the model to the intercept of \u03b1L(v) Again, an adequate fit is achieved for the same intercept value determined by Afewu and Dixon. Hence for all further modelling, Eqn (4.8) is used to describe the dispersivity of solutes.  Sensitivity of the ratio between the transverse and longitudinal dispersivity coefficients on ETu breakthrough curves is shown in Figure 4.12. From the model curves, it is observed that changing the ratio of \u03b1L to \u03b1T has no discernable impact on the breakthrough curves. This result is as expected since the column under test has a diameter of only 10 cm, and thus the dispersion is virtually all longitudinal and \u03b1T has little impact on the breakthrough curves generated. As suggested by Afewu and Dixon (2008), a ratio of 1\/30 is used in this work for a feed rate of 17.52 L\/m2\/h in a 30-cm column. In a 1-m column at flow rates of 11.9 L\/m2\/h and 35.8 L\/m2\/h, the dispersion parameters thus validated from the sensitivity tests are used again to obtain the breakthrough curves of Cu2+ as a tracer. The results are shown in Figure 4.13. An adequate fit of the experimental data suggests that the same dispersion parameters can be applied to a 1-m 82  column. As dispersion is a function of vw, all solutes in the test columns including ETu are assumed to have the same dispersion parameters.   Figure 4.12 Sensitivity of the model to the ratio of \u03b1T to \u03b1L  Figure 4.13  Cu Tracer curves in the 1 m column 83  Subsequently, the deviation of the ETu breakthrough curve, with respect to the Cu2+ curve, is analyzed in the 30-cm column. If there were no adsorption and only heterogeneous decay of ETu, then the expected breakthrough curves could be computed by observing the sensitivity of k*, which is the rate constant in Eqn (4.5), on the breakthrough curves generated. Assuming a linear relationship (as suggested in Figure 4.8), the units of k* would be s\u20131. The sensitivity analysis is shown in Figure 4.14.  Figure 4.14 Sensitivity of the model to the rate constant k* in the 30-cm column (non-adsorbing solute, Cu)  Clearly, such a model is unable to simulate the experimental data. The breakthrough curve of 1.02 g\/L ETu in the 30 cm column suggests that the lateral shift from the Cu curve can only be explained by an accumulation term, such as adsorption.  84  4.3 Adsorption Isotherm The adsorption of ETu on the ore is best described by the Redlich-Peterson adsorption isotherm, which has been used to describe the behavior of phenol in natural soils (Saadi et al., 2015). The Redlich-Peterson equation is:  gArAA caKccf+=1)(  (4.9)  This isotherm exhibits different types of behavior based on the values of ar and g. If ar >> 1, then Eqn (4.9) is reduced to a Freundlich isotherm. If ar = 1 and g = 1, then Eqn (4.9) becomes a Langmuir isotherm. This adsorption isotherm is used in Eqn (4.2) and the solute model is solved for cA. To estimate the parameters K, ar, g and k*, a variation of the Levenberg-Marquardt algorithm is applied. Arbitrary initial values of 1 are assigned to ar and g and the value of k* is set to 0. As the parameter K is changed, the distance between the model and experimental breakthrough curve is expressed as a sum of square errors, with a higher value indicating a greater deviation from the observed concentration values. A second-order polynomial is used to fit the data. The derivative of this function is set to zero and solved for K. That value of K is used to optimize the next parameter k*, followed by g and finally ar. This is the first iteration. Subsequent iterations are performed to obtain the values of the four parameters that provide the closest fit to the observed breakthrough curve. The reason for choosing K as the first parameter to optimize is the pronounced impact of K on the breakthrough curve. By the fourth iteration there was little change in the value of each parameter, suggesting that the best fit for the parameters had been achieved. The curves produced are shown in Figure 4.15. 85   Figure 4.15  The sum of square errors (SSE) of model and experimental data points plotted against model parameters K, k*, g and ar   The values of each parameter in the different iterations are shown in Table 4.2.  Table 4.2 Value of each parameter for each iteration of SSE minimization Iteration K k* (s\u20131) g ar Initial guess 0 0 1 1 1 3.73 \u00d7 10\u20134 2.03 \u00d7 10\u20136 0.35 1.34 2 3.86 \u00d7 10\u20134 2.60 \u00d7 10\u20136 0.46 1.42 3 3.84 \u00d7 10\u20134 2.78 \u00d7 10\u20136 0.54 1.51 4 3.83 \u00d7 10\u20134 2.82 \u00d7 10\u20136 0.55 1.52  Families of breakthrough curves produced by varying one parameter while using the values from the fourth iteration for the other three parameters are shown in Figures 4.16, 4.17, 4.18 and 4.19.   86   Figure 4.16  Sensitivity of the model to the parameter K in the 30 cm column The lateral shift of the model curve with the increase in K is consistent with what would be expected for a solute that adsorbs onto the ore surface. This indicates a delay which can be attributed to the accumulation term.   Figure 4.17  Sensitivity of the model to the rate constant k* in the 30 cm column 87  In Figure 4.17, the depression of curves increases with k* as expected.  Figure 4.18  Sensitivity of the model to the parameter g in the 30 cm column  Figure 4.19  Sensitivity of the model to the parameter ar in the 30 cm column   88  In Figure 4.18, the cases of Freundlich and Langmuir adsorption isotherms produce breakthrough curves that are significantly different from the curve observed in the experiments. The Freundlich isotherm gives a more exponential shape while the Langmuir isotherm gives a more sigmoidal shape.  The correct result lies between the two.  Finally, in Figure 4.19, the correct value of ar allows the subtle shape of the experimental breakthrough curve to be closely simulated.  Hence, the Redlich-Peterson isotherm which describes ETu breakthrough for this column with this specific ore is parameterized as follows:  55.052.11000383.0)(AAA cccf+=  (4.10) The g parameter of 0.55 suggests that the experimental curve is between the Freundlich and Langmuir adsorption isotherms. It can be theorized, therefore, that the mechanism of adsorption is mixed and does not follow ideal monolayer adsorption, which is expected given the non-uniformity of the adsorbent, the ore in this case. Eqn (4.10) is further applied to other concentrations of 0.51 g\/L and 0.051 g\/L ETu to check the validity of the parameters estimated through this exercise. The results are shown in Figure 4.20.  89   Figure 4.20  Breakthrough curves for other ETu concentrations with the Redlich-Peterson model in the 30-cm column An adequate fit for concentrations of 0.051 g\/L and 0.51 g\/L suggest that Eqn (4.10) can be used to explain the breakthrough curves generated from steady state infiltration tests of ETu in the 30 cm column for the concentration range of 0.051 g\/L to 1.02 g\/L.  In Figure 4.21, the Redlich-Peterson model parameters (Eqn (4.10)) from the 30-cm column tests, and the dispersion parameters (Eqn (4.8)) from the Cu2+ tracer test in the 30-cm column are used to predict ETu breakthrough for different flow rates from the 1-m column. The Cu model curve was generated by setting the K and k* values to 0 in Eqns (4.9) and (4.5), respectively.   90   Figure 4.21 Using the Redlich-Peterson model to predict ETu breakthrough from a 1-m column An adequate fit of the experimental data shows that the 2D model with a suitable adsorption isotherm can be used to explain the behavior of an interacting solute such as ETu. 4.4 Simulation of solute distribution under a drip emitter The focus of this section is simulating the solute distribution under a drip emitter. This information is crucial to anyone looking to implement percolation leaching of chalcopyrite using catalytic solutes. Figure 4.22 shows the concentration distribution of Cu, ETu and ETu + SDDC solution in a 30 cm x 10 cm column. Further, the concentration profiles of Cu and ETu corresponding to the breakthrough curves modelled in Figure 4.21 is shown in Figure 4.23.  91    Figure 4.22 The concentration profile of a) Cu, b) ETu and c) ETu in the presence of SDDC in the 30 cm \u00d7 10 cm column at steady state corresponding to breakthrough curves from Figure 4.5   Figure 4.23 The concentration profile of a) Cu at 3 mL\/min, b) ETu at 3 mL\/min, c) ETu at 2 mL\/min and d) ETu at 1 mL\/min, in a 1 m \u00d7 8 cm column at steady state with different infiltration rates, corresponding to breakthrough curves from Figure 4.21 92  From Figure 4.23 it can be observed that at higher flow rates the ETu shoots through the middle of the column while at lower flow rates a more gradual contour develops at various depths under a drip emitter. Using all adsorption and reaction parameters previously estimated and at a flux rate of 20 L\/m2\/h, ETu and Cu distribution under a single drip emitter in a 4 m \u00d7 50 cm column is simulated and the resulting concentration profiles are plotted in Figure 4.24.  Figure 4.24 Simulated concentration profiles of a) Cu and b) ETu in a 4 m x 50 cm column at steady state at a flowrate of 20 L\/m2\/h As can be observed, there is a significant concentration gradient in the distribution of ETu under a single drip emitter in a large column while the Cu concentration is similar in most parts of the column. With an inlet concentration of 0.08 g\/L of ETu in the feed stream a concentration of 0.05 g\/L is expected at the bottom of the column, which produces significant enhancement in the 93  leaching of copper under leaching conditions as described by Ren et al. (2020).When compared to the Cu distribution in the column  The breakthrough curves generated in a real leaching system will have significant deviations from the curves modelled in this work, as in a real system there are multiple reactions which involve the solute of interest, ETu. In the next chapter, the impact of solutes such as Cu2+, Fe3+ and O2, and varying sulfide mineral content on ETu breakthrough curves is examined. The other solutes may cause ETu to decay in the lixiviant and compete with the adsorption mechanism on the surface of the ore. To describe such scenarios, additional rate terms must be added to the solute model. These parameters must be quantified by calibrating them with corresponding operating conditions using the techniques described in this chapter. Hence a more realistic model which includes the various decay factors can describe the overall solute distribution in a real system.  94  Chapter 5: Transport of an Adsorbing, Reacting Solute   In this chapter the various types of interactions that impact the transport and distribution of ETu in a leaching column are discussed in detail. The technique of using solute breakthrough curves to estimate transport parameters is applied over a broad set of experiments to elicit specific parameters with regards to reaction of ETu with other solutes in the lixiviant. However, this is a challenging undertaking as the response of the ETu breakthrough curve should be attributable to one specific factor affecting the transport of the solute. In Chapter 4, some of the effects of adsorption and subsequent reaction were limited by either using a basic pH or using sacrificial solutes such as SDDC. While carefully curated experiments were successful in showcasing the ability of the breakthrough curve technique to estimate adsorption parameters, any meaningful application of the technique to real leaching systems would require the column operating conditions to resemble leaching conditions more closely. To this end, a rubric of a solute ETu model would need to be constructed whereby adsorption and reaction of ETu on the surface of the mineral occurs in parallel with reactions with other solutes in the lixiviant. The factors affecting the homogeneous reaction, defined as the interaction of ETu with dissolved solutes in the lixiviant, is estimated using closed stirred tank reactors in which the concentration of Cu2+, Fe3+ and O2 are changed and the drop in ETu concentration is measured using HPLC analysis. The next section discusses the homogeneous decay of ETu in lixiviant solution.  5.1 Homogeneous Decay A solution of ETu was prepared by adding 1.02 g of ETu to 1 L of DI water. The pH was adjusted to 1.8 with 98% sulfuric acid (ACS reagent grade). The solution was stored in a closed plastic bucket.  The concentration of ETu was analyzed by HPLC over a four-day period. The purpose of the test was to examine the stability of ETu in acidified water.  Figure 5.1 shows the 95  remaining ETu concentration over time. No decay of ETu was observed over the four-day test period.  Figure 5.1  Normalized ETu concentration as a function of time in acidic water alone  5.2  Factors affecting Decay As shown above, ETu in acidic water is stable. The following subsections deal with the decay of ETu in presence of Cu2+, Fe3+ and O2. 5.2.1 Copper  To observe the impact of ETu concentration in the presence of Cu2+, an experiment was conducted by varying the Cu2+ (0.5, 1 and 2 g\/L) at a constant ETu concentration (1.02 g\/L) and varying ETu concentration (0.204, 0.51 and 1.02 g\/L) at a constant Cu2+ concentration (1 g\/L). The ETu decay for either case is shown in Figure 5.2.  96   Figure 5.2 a) Normalized ETu concentration as a function of time in acidic water with varying [Cu2+]; b) Normalized ETu concentration as a function of time in acidic water at constant [Cu2+] From Figure 5.2, it is obvious that the decay of ETu in the presence of Cu2+ is a function of both [ETu] and [Cu2+]. By law of mass action, the Eqn (5.1) below should describe the rate of ETu decay in the presence of Cu2+:                                                              ETuCu ETuCuCuETuCuETunn ccks =                                            (5.1) 5.2.2 Oxygen To test the impact of ETu decay in the presence of dissolved O2, a reactor containing 1 L of 1.02 g\/L ETu solution at a pH of 1.8 was sparged with O2, air and N2.  No decay was observed over the six-day test period. Figure 5.3 shows the impact of O2 concentration on ETu decay.  97   Figure 5.3  Normalized ETu concentration as a function of time in acidic water sparged with N2, air and O2  A minor increase in concentration is observed on Day 3 uniformly across all samples. This is attributed to an inconsistency in measurement. The overall trend of the concentration of ETu suggests that no change was observed over a six-day period and the change in concentration of dissolved O2 has no effect on the concentration of ETu in solution. However, the effect of O2 on the decay of ETu in the presence of a mineral surface is significant and will be discussed in section 5.2.4. 5.2.3 Iron The effect of ETu concentration in the presence of Fe3+ in acidic media was observed by varying the Fe3+ concentration (1.10 ,1.65 and 2.20 g\/L) at a constant ETu concentration (1.02 g\/L) and varying the ETu concentration (0.204, 0.51 and 1.02 g\/L) at a constant Fe3+ concentration (1.1 g\/L). The ETu decay for either case is shown in Figure 5.4.   98   Figure 5.4  a) Normalized ETu concentration as a function of time in acidic water with varying [Fe3+]; b) Normalized ETu concentration as a function of time in acidic water at constant [Fe3+]  As in the case with Cu2+, the decay of ETu in an acidic solution of Fe3+ and ETu is dependent on both concentrations. Hence, by the law of mass action, a suitable rate law may be written as follows:   ETuFe ETuFeFeETuFeETunn ccks =  (5.2) 5.2.4 Mineral Surfaces  It has been shown in previous literature that sulfide minerals can facilitate the decomposition of thiocarbonyl compounds (Zhu, 1992). To test this observation, 1 g of pulverized pure FeS2 powder (75 \u2013 102 \u00b5m) was added to the reactor containing neutral DI water on Day 0 after sampling for analysis. The purpose of this experiment was to examine the effect of a mineral surface on the formation of hydroxyl radicals which would facilitate the decomposition of ETu.  99   Figure 5.5  Normalized ETu concentration as a function of time in neutral DI water sparged with gas and 1 g pulverized pyrite  It is observed that change in the concentration of dissolved O2 has an impact on the decay of ETu in solution. This suggests that, in the absence of an acidic pH, the mineral surface of pyrite produces radicals that may affect the decomposition of ETu. As discussed in Chapter 4, this observation is consistent with the nature of auto-generated H2O2 in DI water conditions on the surface of FeS2 (Gil-Lozano et al., 2017) which results in the oxidative decay of ETu. Subsequently, a 1.02 g\/L ETu solution was acidified to pH = 1.8 with 98% sulfuric acid. 1 g of pure FeS2 powder (75 \u2013 102 \u00b5m) was added to the reactor on Day 0 after sampling for analysis. Oxygen, nitrogen and air were sparged into the reactor at the rate of 100 mL\/min.  100   Figure 5.6  Normalized ETu concentration as a function of time in acidic water sparged with gas and 1 g pulverized pyrite   A substantial decay (~ 8% and 20% respectively) of ETu during the 6-day test period was observed in acidic water with 1 g of pulverized pyrite when sparged with air and oxygen. By comparison, in a similar test under neutral pH conditions, less decay was observed. However, when no pyrite was added, no decay was observed (Figure 5.3) suggesting that a mineral surface is essential for the decay.  Thus, the reaction of dissolved oxygen with ETu may be represented by the following equation:  ETu2O222 ETuOETuOETuOnn ccks =  (5.3) Finally, another stability test was performed with 1.02 g\/L ETu, 1.1 g\/L Fe3+ and 1 g\/L Cu2+ in acidic 1.8 pH solution with 1 g of pulverized pyrite and sparged with 100 mL\/min of air in a closed reactor.  101   Figure 5.7  Normalized ETu concentration as a function of time in various solutions and 1 g pulverized pyrite  As shown in Figure 5.7, the effect of all solutes in the presence of a mineral surface is additive, hence the overall decay of ETu in a real leaching condition may be expressed as follows:  ETuFeFe2ETuO2O22ETuCuCu2ETuFeFeETuETuOETuOETuCuCuETuFeETuETuOCuETuETunnnnnnTotalcckcckcckssss++=++= (5.4) 5.3 Modeling ramp tracer input  In Chapter 4, ETu was the only interactive solute in the lixiviant. The operating conditions of the column tests in that chapter were carefully designed to allow only certain interactions with ETu. Using neutral DI water eliminated any instability of ETu in the feed solution, but this is clearly not a replication of real lixiviant systems. In real lixiviant solutions the concentration of ETu fed to the column is not steady with time owing to the reactions of ETu with other solutes in the feed as portrayed in Section 5.2. Hence, modeling a step input of feed concentration to a column of 102  ore is not relevant to ETu addition to a column of ore under leaching conditions. A decaying ramp function closely simulates a feed that is decaying with time in the feed bucket. Towards this end, the robustness of the model to simulate a decaying ramp tracer input needs to be tested first.  The 30 cm column was used to obtain breakthrough curve results; a schematic of the experiment is shown in Figure 5.8.  Figure 5.8  Test apparatus for diluting feed concentration  The experimental setup for this test was modified from the previous setup (Figure 3.1) to include a third bucket of acidic water at pH 1.8. Bucket 1 (B1) had the feed solution of 1 g\/L Cu2+ at 1.8 pH, Bucket 2 (B2) had acidic DI water at pH 1.8 and Bucket 3 (B3) was the effluent collection bucket. The initial volume of Cu2+ feed in Bucket 1 was adjusted as per Table 5.1 below. 103  Table 5.1 Volume of 1 g\/L Cu2+ solution in B1 Feed Flux (L\/m2\/h) Volume of B1 (L) 17.52 1.23 8.76 0.61 5.84 0.41 The reason for specific volumes of solution in B1 is the need to achieve a target final concentration of [Cu2+]. For example, a flux of 17.52 L\/m2\/h translates to 2.29 mL\/min of solution being pumped to the column by P1. To achieve a final concentration (at 9 hours from the start of the experiment) of Cu2+ which is 50% of [Cu2+]initial a second pump P2 needed to feed water to the bucket B1 at the same rate (2.29 mL\/min) at which pump P1 was removing solution from the bucket. So, the volume of solution in B1 is calculated to be 2.29 (mL\/min) x 60 (min\/h) x 9(h) = 1.23 L. The final volume of B1 after the 9-hour test was measured to confirm that the volume had not changed from the start of the test (by weighing B1).  The experiment was started with acidic DI water fed to the 30 cm column through peristaltic pump P1 at the desired flux of 17.52 L\/m2\/h. At the one-hour mark the feed was switched from Bucket 2 to Bucket 1 and a second peristaltic pump (P2), which was pre calibrated and set to the same flow rate as the first (P1), was started and fed solution from Bucket 2 to Bucket 1. Thus, the volume of solution in Bucket 1 was unchanged through the experiment while [Cu2+] steadily decreased by dilution. The purpose of this setup was to simulate the decay of feed at a steady rate. Hence the breakthrough curve generated from this test for Cu2+ tracer would challenge the ability of the solute model as developed in Chapter 4 to accurately trace the path of Cu2+ breakthrough. Further, the flux of feed input was changed to 8.76 L\/m2\/h and 5.84 L\/m2\/h respectively to test the ability of the model to conform to the breakthrough curves measured. These values were arbitrary chosen at 1\/2 and 1\/3 of the initial flow rate of 17.52 L\/m2\/h. Figure 104  5.8 shows the experimental and model generated breakthrough curves. Only one feed curve is shown, as a solid line, so as not to clutter the plot.   Figure 5.9 Normalized Cu2+ as a function of time for varying input fluxes and a steadily diluted feed   As shown in Figure 5.9, very good agreement between the experimental and model-generated tracer breakthrough curves is obtained for a decaying feed. The values of the advection (Kw and hc,0) and dispersion (\u03b1) parameters used to generate the model curves are the same as those estimated in Chapter 4, further demonstrating the robustness of the model.  5.4 Modeling homogeneous decay  In Section 5.2 it was deduced that the decay of ETu in the lixiviant can be ascribed to multiple reactions occurring simultaneously with other solutes in the lixiviant, which is in addition to the adsorption and reaction of ETu on the mineral surface in the ore as discussed in Chapter 4.  Hence a decaying solute input as feed to a column of ore needs to be modeled to estimate the 105  reaction parameters associated with the solute. To elaborate, a feed with Cu2+ concentration decay simulated by dilution simulates the decay of ETu in the feed bucket (in the case of a real lixiviant solution) but cannot simulate the decay of ETu as it propagates through the column as the Cu2+ does not react while it moves through the column. Hence, the simulated decay Cu2+ curve is a \u201cbaseline\u201d against which a decaying ETu curve should be compared. Section 5.3 demonstrated the ability of the solute model to trace the path of a decaying tracer input to establish a baseline output of a tracer. As the inert tracer does not react within the column (the rate of consumption can be set to 0), the deviation of the breakthrough curve generated from the simulated decaying solute (Cu2+) to the actual decaying solute (ETu) would help in isolating the effect of reaction on the solute. The following subsections uses this principle to populate the rate law parameters (Eqns 5.1, 5.2 and 5.3) deduced from reactor tests of ETu with Cu2+, O2 and Fe3+. 5.4.1 Copper  The experiment to estimate the effect of Cu concentration on ETu breakthrough was designed carefully to observe the effect of decay of ETu as a function of both Cu and ETu concentrations in the lixiviant. To this end, an ETu feed simulating a decay rate of 15% [ETu] per day was applied to the 30 cm column for 24 hours. A similar experimental setup to the one explained in Section 5.3 (Figure 5.8) was used to achieve the dilution of ETu feed, which was initially at a concentration of 1.02 g\/L. Nitrogen gas was fed at 100 mL\/min to purge the column of any air in order to eliminate any effect of dissolved O2. The breakthrough curve obtained from this experiment was the \u201cETu Simulated\u201d breakthrough curve which should consist of the adsorption and reaction effects discussed in Chapter 4. Subsequently, the column was rinsed with acidic DI water and no N2 flow. A 1 g\/L Cu2+ solution at pH 1.8 was fed to the column for a period of 24 hours. The objective of using a 1 g\/L Cu2+ 106  solution to feed the column for 24 hours was to allow the column to achieve uniform concentration distribution. Therefore, while modeling the ETu breakthrough, it can be safely assumed that the results being observed were produced under a constant background of Cu2+ concentration distributed well within the column.  The feed was switched to a 4th feed bucket which contained 1.02 g\/L ETu and 1 g\/L Cu2+ at pH 1.8 and N2 flow at 100 mL\/min was initiated. This feed solution was prepared 15 minutes before the switching of feed by adding 3.06 g of ETu to 3 L of 1 g\/L Cu2+ solution at pH 1.8. A sample of the feed was taken when the feed was switched to ensure the concentration of ETu was 1.02 g\/L.  The breakthrough curve of ETu for either case is shown in Figure 5.10  Figure 5.10  Normalized ETu concentration as a function of time in acidic water sparged with gas  In Figure 5.10, the area between the red and yellow curves is the additional decay of ETu owing to the reaction of ETu with Cu in the column. From Eqn (5.2) the decomposition rate of ETu can be expressed as a function of both ETu and Cu concentrations. The parameters that must be 107  estimated are the orders nCu and nETu and the rate constant kCuETu in order to establish a rate law that would explain the decay rate of ETu. The following set of breakthrough curves of ETu are generated by using a constant Cu concentration at 1.0 g\/L in the feed and setting the ETu concentration to 0.204 g\/L, 0.51 g\/L and 1.02 g\/L.    Figure 5.11  Normalized ETu concentration as a function of time in acidic medium with [Cu2+] = 1 g\/L and sparged with N2 gas  The test to supply the column with a constant and steady ETu concentration is complicated to achieve in acidic media in the presence of Cu2+ owing to the decay of ETu in the feed bucket as shown in Figure 5.2. Using basic media or SDDC as an additional reagent would not serve the purpose of simulating a realistic operating condition. Hence, a modification to the experimental setup was employed. A schematic of the modification is shown in Figure 5.12.  Bucket 1 contained feed with 2.04 g\/L ETu, Bucket 2 contained feed with 1 g\/L Cu2+ solution, Bucket 3 contained acidic DI water, and a 100 mL beaker on a stir plate was used as a premixing chamber. Pump P1 was used to feed solution into the column initially from Bucket B3. Pump P2 108  was set up with 2 pump heads to move solution from B2 (Cu2+) and B3 (acidic DI water) initially and was later switched to B1 (ETu).  Figure 5.12  Experimental apparatus to hold [ETu] steady while varying [Cu2+]  The idea for the experimental setup was based on the premise that, for a fresh mix of ETu + Cu2+ solution it could be assumed that the [ETu] is constant. The definition of \u201cfresh mix\u201d is a solution made up under 10 minutes before it is introduced into the column of ore. This operating condition is achieved by controlling the flow rate of feed from two buckets B1 and B2 mixed in a 109  premixing beaker from which the column is fed. The beaker is set up on a stir plate (to ensure proper mixing) and the volume of the solution in the beaker monitored through the experiment to ensure that the volume of solution flowing in is equal to the volume of solution flowing out.  For example, in order to achieve a composition of solution at 0.5 g\/L Cu2+ and 1.02 g\/L ETu fed to the column at a flux of 17.52 L\/m2\/h (~2.29 mL\/min), solution of 1 g\/L Cu2+ (from B2) and 2.04 g\/L of ETu (from B1) was moved to a premixing beaker with the pump P2. P2 was operated with two pump heads at the same flow setting calibrated to achieve a flow rate of 1.145 mL\/min (half of the total intended flow rate to the column). The [ETu] was corroborated by 3 measurements of the solution from the premixing beaker made during the 24-hour test. The column was continuously fed N2 at 100 mL\/min when the test started.  Similar to the previous test, a 0.5 g\/L Cu2+ solution at pH 1.8 was fed to the column for a period of 24 hours. This was achieved by using acidic DI water (from B1) mixed with 1 g\/L Cu2+(from B2) in the proportion discussed above. The objective of using a 0.5 g\/L Cu2+ solution to feed the column for 24 hours was to allow the column to achieve uniform concentration distribution. Therefore, while modeling the ETu breakthrough, it can be safely assumed that the results being observed were produced under a constant background of Cu2+ concentration distributed well within the column.  Subsequently, the Cu2+ concentration in B2 was changed to 2.0 g\/L to achieve a solution composition of [ETu] = 1.02 g\/L and [Cu2+] = 1 g\/L, and so on. The ETu breakthrough curve is shown in Figure 5.13. The time axis shows 0 at the 24-hour mark of the experiment. 110   Figure 5.13  Normalized ETu concentration as a function of time when [Cu2+] is varied while steady [ETu] is maintained  Figure 5.14 Natural log of ETu not recovered vs natural log of [Cu2+] at [ETu] =1.02 g\/L The area under the curve of ETu recovered is subtracted from the total ETu fed to the column. From the ETu not recovered, the amount of ETu adsorbed (estimated from previous modeling as shown in Chapter 4) is subtracted and the natural logarithm of the remaining amount is plotted against the natural logarithm of [Cu2+] in Figure 5.14.  111  Similarly, an experiment varying the [ETu] at constant [Cu2+] =1.0 g\/L was conducted (as described in Figure 5.11) and the area under the breakthrough curves (calculated using the trapezoidal rule) was used to estimate the amount of ETu not recovered. The total ETu not recovered was plotted against the [ETu] which is shown in Figure 5.15.   Figure 5.15  Logarithm of ETu not recovered vs logarithm of [ETu] at [Cu2+] = 1 g\/L Form Figure 5.14 and 5.15, nCu = 1.7 and nETuCu =0.985. Eqn (5.1) has 3 parameters of which 2 have been estimated. These values are used in the solute model (including adsorption and reaction) of ETu in the presence of [Cu2+] and a sensitivity analysis on the breakthrough curve is generated at [ETu] = 1.02 g\/L and [Cu2+] = 1 g\/L by varying the parameter kCuETu. Figure 5.16 shows the breakthrough curves generated and the corresponding experimental values.  112   Figure 5.16  Sensitivity of the Normalized ETu concentration to rate constant kCuETu  To estimate the kCuETu the distance between the experimental and model generated curves is plotted against kCuETu, and a curve is fitted. The second order polynomial which describes the ETu curve is differentiated with respect to kCuETu , and the value is estimated at 3.15\u00d710\u22126 s\u22121. Figure 5.17 shows the minimization of distance between the model generated and experimental curve.  113   Figure 5.17 Sum of squares between model and experimental breakthrough curves plotted against rate constant parameter kCuETu  The next step is to validate the reaction parameters obtained using the tests. Figure 5.18 is the case where the Cu2+ is varied from 0.5 g\/L, 1.0 g\/L and 2.0 g\/L and the ETu is fed at 1.02 g\/L.  Figure 5.18 Breakthrough of ETu when [Cu2+] is varied, model vs experimental   Figure 5.19 is the case where [Cu2+] = 1 g\/L and [ETu] is changed from 0.204 g\/L to 1.02 g\/L.  114   Figure 5.19 Breakthrough of ETu when [ETu] is varied, model vs experimental   As can be observed there is very good agreement of the experimental and model-generated breakthrough curves. As expected, when [Cu2+] is held steady the variation in the breakthrough curve is very similar to the ones generated in neutral DI water. Hence it can be deduced that the effect of the Redlich-Peterson adsorption model is manifesting in the variation of the breakthrough curves when the [Cu2+] is steady. When feed [ETu] is constant there is significantly more decay owing to the change in [Cu2+] with more [Cu2+] resulting in more decay of ETu. 5.4.2 Oxygen The reaction with dissolved O2 is relatively less complex to design experiments for and to model owing to the stability of ETu in acidic medium in the absence of mineral surface. ETu of 1.02 g\/L in acidic water at pH 1.8 is used to feed a 30 cm column at 17.52 L\/m2\/h. 100 mL\/min of N2, air and O2 are fed to the column at the time when the feed is switched from acidic DI water to the 115  prepared feed. Figure 5.20 shows the breakthrough curves of ETu when different gases are purged through the column.  Figure 5.20 ETu breakthrough curves when purged with 100 mL\/min gas  As can be observed the final [ETu] at the outlet is significantly lower in the case of air and O2. Again, the trapezoid rule is used to estimate the area under the curve and the amount of ETu not recovered in the case of N2, air and O2 is estimated. The amount of ETu lost in the case of N2 purge is further subtracted from the totals of air and O2 as that decay is attributed to adsorption-related consumption. Hence, the contribution to the total ETu lost from the N2 test is set to 0 and the amount with air or O2 is measured against the N2-based ETu decay. As the logarithm of 0 is not defined the value of ETu lost to N2 is assigned a value one order of magnitude lower than that estimated for air and O2. Figure 5.21 shows the plot of the logarithm of the ETu loss plotted against [O2] dissolved in the water. Tromans\u2019 model (Tromans, 2000) was used to estimate the concentration of O2 in pure water: 116  ( )( )23144.810591.20298092.0378.299298ln35.203046.0exp32Oaq PTTTTTTc\uf8f4\uf8f4\uf8f3\uf8f4\uf8f4\uf8f2\uf8f1\uf8f4\uf8f4\uf8fe\uf8f4\uf8f4\uf8fd\uf8fc\u00d7\u2212\u2212+\u2212\uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb+=  (5.5) where caq  =  aqueous concentration of O2 (moles of O2 \/ kg of H2O)   PO2  =  Partial pressure of O2 (atm) For a temperature to 298 K, the oxygen solubility caq is calculated to be 0.00127 \u00d7 PO2.  Figure 5.21 Logarithm of ETu loss plotted versus logarithm of [O2(aq)] at [ETu] = 1.02 g\/L To test the sensitivity of the ETu reaction with the concentration of ETu, the column was purged with air at 100 mL\/min and the 3 different concentrations of ETu (0.204 g\/L, 0.51 g\/L and 1.02 g\/L) were fed to the column. The respective breakthrough curves are shown in Figure 5.22. The amount of ETu not recovered is plotted against [ETu] in Figure 5.23. The reaction orders with respect to O2 and ETu towards the decay of ETu are estimated to be nO2 = 0.45 and nETuO2 = 0.981. Using these parameters in Eqn (5.3) , a sensitivity analysis was performed on the breakthrough of ETu when purged with air by changing kETuO2. Figure 5.24 shows the sensitivity analysis. 117  As in the previous section, the distance between the experimental and model generated breakthrough curves are plotted against kETuO2 and the resulting plot is fit to a second order polynomial which, when differentiated and set to zero, gives the value of kETuO2 = 2.42\u00d710\u22123 s\u22121. Figure 5.25 shows the graphical representation of this analysis.  Figure 5.22  ETu breakthrough curves when [ETu] is varied and purged with 100 mL\/min of air in acidic medium   Figure 5.23 Logarithm of amount of ETu not recovered against logarithm [ETu] and purged with 100 mL\/min of air in acidic medium  118    Figure 5.24 Sensitivity of ETu breakthrough to the rate constant kETuO2   Figure 5.25  Sum of squares between model generated and experimental breakthrough curves plotted against rate parameter kETuO2  119  When compared with kCuETu, the value of kETuO2 is three orders of magnitude higher, which suggests that the impact of O2 on the decay of ETu is three orders of magnitude stronger than the effect of Cu2+ on this ore. However, this effect is ameliorated by the low concentration of O2 in the aqueous medium. Hence the magnitude of the rate constant is a qualitative assessment on the potency of O2 as a dissolved solute. 5.4.3 Iron The experimental setup discussed in Section 5.4.1 (Figure 5.12) was used to maintain a steady input concentration of 1.02 g\/L ETu while varying [Fe3+]. Figure 5.26 shows the breakthrough curves of ETu.   Figure 5.26 ETu breakthrough with varying [Fe3+] and constant [ETu] input Similarly, a modification to the test setup used in Figure 5.8 to generate the simulated ETu breakthrough curve, as applied in Section 5.4.1, is used to generate breakthrough curves of ETu for different [ETu] while maintaining a steady background of [Fe3+]. In the case of [Fe3+], a complication makes it differ from the [Cu2+] experiment is that [Fe3+] leaches from the ore under 120  acidic conditions. This effect is mitigated in shorter columns owing to the small quantity of ore loaded. The variation during the experiments was on the order of 0.05\u22120.075 g\/L Fe3+ (analyzed by ICP) which is considered negligible. This is also the case regarding the leaching of Cu2+ from the ore in the presence of Fe3+ and ETu.  Figure 5.27 ETu breakthrough with varying [ETu] and [Fe3+] = 1.1 g\/L As can be seen in Figure 5.27, the ETu breakthrough curve with varying [ETu] is shown. The next step is to find the area under the curve using the trapezoid rule and estimate the amount of ETu not recovered. Figure 5.28 and Figure 5.29 are plotted to estimate the corresponding orders of the reaction.  121   Figure 5.28 Logarithm of amount of ETu lost vs logarithm of [ETu] at [Fe3+] = 1.1 g\/L  Figure 5.29 Logarithm of amount of ETu lost vs logarithm of [Fe3+] at [ETu] = 1.02 g\/L From the figures, nETuFe and nFe are estimated to be 0.979 and 1.49 respectively. These parameters are used in the solute model to generate a family of breakthrough curves by changing kFeETu. Figure 5.30 shows the sensitivity of the ETu curve to the rate parameter.  122   Figure 5.30  Sensitivity of ETu breakthrough to rate constant kFeETu   Figure 5.31 Distance between model and experimental curves plotted against rate constant kFeETu  From Figure 5.31, kFeETu = 0.0000103 s\u22121.  123   Figure 5.32 ETu breakthrough with varying [ETu] and [Fe3+] = 1.1 g\/L  Figure 5.33 ETu breakthrough with varying [Fe3+] and [ETu] = 1.02 g\/L From Figures 5.32 and 5.33, close agreement between the experimental and model-generated breakthrough curves is observed. The next section discusses the combination of all solutes Cu2+, 124  Fe3+ and O2 together in a lixiviant solution which resembles leaching conditions in a heap leach process.  5.4.4 Real lixiviant solution The equation of the ETu solute consumption in a real lixiviant, Eqn (5.4), is populated using parameters estimated from the previous sections, which results in Eqn (5.7):  ( ) ETu49.1Fe545.0O37.1Cu6ETu 1003.11042.21015.3 2 ccccsTotal \u2212\u2212\u2212 \u00d7+\u00d7+\u00d7=  (5.7) In the solute model, as explained in Chapter 4, the consumption of ETu is defined as a function of [ETu]. In a realistic lixiviant system, that is modified, and Eqn (5.7) is appended to the solute generation term. Hence, the model, including adsorption (Redlich-Peterson model), heterogenous reaction (defined as the amount of solute that reacts upon adsorption, which has pseudo-first-order kinetics), and homogeneous reaction (reaction of ETu with other solutes in the lixiviant in the presence of mineral surface) is the overall model that can explain the transport of ETu through a bed of ore.  (Note that the reaction orders with respect to [ETu] for all of the solute reaction terms have been rounded up to 1.) To test the model, a 30 cm column was subjected to the same flux of 17.52 L\/m2\/h using a lixiviant solution with [Fe3+] = 1.1 g\/L, [Cu2+] = 1 g\/L, and [ETu] = 1.02 g\/L, aerated with 100 mL\/min of air. It bears noting that the Tromans model (Tromans, 2000) for determining [O2(aq)] is different in the case of ionic lixiviant as shown in the equation below.   { ( ) } 223.0116.1232.212\u2212\u2217 += IHOaq CKPc (5.8) where KH*  = the Henrian saturation constant for pure water as given in Eqn (5.5) 125   Figure 5.34 ETu breakthrough with lixiviant solution of [Fe3+] =1.1 g\/L, [Cu2+] =1 g\/L and  initial [ETu] = 1.02 g\/L  The curve shows that if 1.02 g\/L ETu is fed to the column then about 0.2 g\/L of [ETu] is observed at the bottom of a 30\u00d710 cm cylindrical column of ore. This implies that roughly about 80% of the ETu fed to the column is consumed non-productively. It is interesting, however, to note that the amount of ETu adsorbed does desorb once the ETu feed is cut off. Heterogeneous reactions consume about 18% of the total ETu consumed within the column, and homogeneous reactions consume about 82%. Of the 82% consumed homogeneously, about 10% is attributed to Cu2+ at 1.0 g\/L, 43% is attributed to Fe3+ at 1.1 g\/L, and 29% is attributed to oxygen from air.  5.5 Modeling a column controlled based on outlet ETu concentration The previous sections discussed the various forms of non-productive ETu consumption when fed to a column of ore. As the concentration of ETu in the effluent from the column is the minimum [ETu] that is available throughout the bed of ore in the column, setting a target outlet concentration of ETu in lixiviant solution ensures that a value greater than the target is available 126  to catalyze the leaching of chalcopyrite everywhere within the column. In heap leach operations, it is not practical to control myriad factors such as [Fe3+] or pH, which are a consequence of the composition of minerals in the bed of ore. A practical approach for efficient delivery of ETu through the bed of ore is to control the rate of addition by setting a desired target of [ETu] in the effluent. Intuitively, targeting an outlet concentration of ETu will help deliver the maximum amount of ETu to the bed of ore but might lead to higher non-productive ETu consumption.  In this section, modeling is used as a tool to estimate how much and how frequently ETu must be added to arrive at a steady outlet concentration. Further, the impact of irrigation flux is also explored using the model to estimate what rate of ETu addition is required to achieve a steady outlet concentration. Subsequently, the model-predicted solute addition is tested on 30 cm columns to validate the simulations. Additionally, simulations are performed on a taller column of height 1 m to test the robustness of the model.  5.5.1 Concentration control by addition  A target outlet [ETu] = 0.1 g\/L (100 ppm) was arbitrarily set to test the ability of the solute model to predict addition rate. The column was irrigated at a flux of 17.52 L\/m2\/h using a lixiviant feed solution with [Fe3+] = 1.1 g\/L and [Cu2+] = 1.0 g\/L, and air was fed to the column at 100 mL\/min. The model was run multiple times to estimate what the initial [ETu] would need to be in order to achieve 0.1 g\/L in the effluent. The model predicted that a feed [ETu] of 0.51 g\/L (510 ppm) would achieve an outlet [ETu] of 0.1 g\/L (100 ppm). [ETu] in the feed bucket was made up every two days to account for decay. Subsequently, a column test was set up with 15 L of feed solution with the above mentioned ferric and cupric concentrations by adding anhydrous Fe2(SO4)3 and CuSO4.5H2O to acidified DI water at pH 1.8. 10.2 g of ETu was added to 5 L of acidified DI water and allowed to mix for 30 mins. The peristaltic pump was started with the 127  irrigation flux of 17.52 L\/m2\/h from an acidic DI water feed bucket. At the 45-minute mark, the 5 L ETu solution was added to the 15 L feed and mixed manually with a plastic rod. At the one-hour mark the feed tube was switched to the lixiviant solution bucket. The experiment was run for a total of 120 hours. Figure 5.35 shows the model feed, the model ETu breakthrough curve, and the actual measurements of ETu from the feed bucket and the effluent of the column.  Figure 5.35 ETu curves generated by modeling target [ETu]  The experiment started with 10.2 g of ETu initially added to the feed. Subsequently 2.16 and 1.18 g of ETu were added at 48 h and 96 h to restore the feed concentration of ETu. The total ETu left over in the feed bucket at the end of the experiment was 1.56 g and the total ETu effluent from the column was 1.635 g (area under the curve was calculated using the trapezoidal rule). Therefore, the total non-productive loss of ETu in this experiment was 10.365 g (~ 86 % of the total ETu fed). About 1.8 g (~17.3% of the total 10.35 g) was lost in the feed bucket. Of the total non-productive loss, the total amount of ETu that contacted the ore was 8.55 g. Figure 5.36 shows the impact of the absence of each individual component of [ETu] decay discussed in the 128  previous sections on the breakthrough of ETu, and consequently the distribution of ETu though the bed of ore in the column. For example, in the case of Fe3+ contribution to ETu decay, kFeETu was set to zero while keeping all other rate constants (kO2ETu, kCuETu etc.)  at their original value.   Figure 5.36 ETu curves generated by setting each rate constant of ETu decay to zero independently  From the figure it is obvious that the impact of [Fe3+] is significantly higher than next biggest contributor to the decay of ETu which is O2. The impact of Cu2+ and the decay owing to adsorption are considerably lower. Quantifying the individual solute contributions for the specific case of ETu decay on the ore in the 30 cm column, of the 8.55 g of ETu that actually entered the column, Fe3+ consumed 3.21 g, Cu2+ consumed 1.53 g, O2 consumed 2.41 g, and adsorption consumed 1.4 g.  Hence, the contribution of Fe3+ is roughly 33% higher than that of O2 and about 200% higher than the contribution of Cu2+. In this test and subsequent analysis, the concentration of solute remains relatively steady throughout the bed of ore. This may not be the case in an actual leaching scenario where the solutes, especially Cu2+, would increase significantly as the lixiviant percolates through the bed of ore due to leaching. This aspect of 129  change in solute concentration is dealt with in Section 5.6. In the next section, the model is tested at different flow rates and the amount of ETu addition is calibrated based on that.  5.5.2 Impact of flow rate   The model was initially run to estimate the amount of solute concentration needed in the feed lixiviant to achieve a relatively steady 0.1 g\/L concentration of ETu in the outlet flow for an irrigation flux of 8.76 L\/m2\/h. The model suggested a concentration of 1.02 g\/L and a daily addition frequency to achieve the 0.1 g\/L ETu concentration.  Figure 5.37 shows the model and experimental data.   Figure 5.37 ETu curve generated for an irrigation flux of 8.76 L\/m2\/h  Intuitively, doubling the feed concentration and adding ETu every day doubles the non-productive loss of ETu. However, to achieve a 0.1 g\/L effluent ETu the doubling of non-productive loss is necessary. The outlet concentration of ETu is closely linked to the amount of Cu2+ leached from the ore (as will be discussed in chapter 6) and hence the tradeoff might be 130  worth the extra loss of ETu.  In the next section the height of the column is changed to 1 m with the original 17.52 L\/m2\/h solution flux. 5.5.3 Impact of height of column The impact on the height of the column is tested in this section. The model was run to estimate the feed concentration of ETu to achieve an [ETu] of 0.1 g\/L at the outlet. The model suggested to have an initial concentration of 3.5 g\/L in the feed. Hence, feed was prepared, and the experiment was performed with a concentration top up at the 24-hour mark. Figure 5.38 shows the effluent curve.   Figure 5.38 ETu curve generated for a 1 m tall column with an irrigation rate of 17.52 L\/m2\/h In this case, there is significant loss of ETu suggesting that as the depth increases there is significant non-productive loss of ETu from all the sources of decay computed earlier.  131  5.6 Impact of leaching on decay of ETu In all the above sections the other solutes in the lixiviant, Fe3+ , Cu2+ and O2 are relatively steady, or assumed to be close enough to the initial feed rate, as the columns are not tall enough to have a significant concentration gradient. This, however, would not be the case in an actual heap leach situation. Even with no significant variation in [Fe3+], there would significant variation in [Cu2+] as Cu will leach into the solution. Hence the sensitivity of ETu consumption to increasing [Fe3+] and [Cu2+] must be analyzed.  5.6.1 Copper To achieve the above mentioned increase in [Cu2+], 100 g of CuSO4 was added to the ore in the 30 cm column  and mixed well. In addition, every 24 hours the [Cu2+] in the feed bucket was adjusted by 0.25 g\/L by adding appropriate amounts of CuSO4. In the figure below, ETu feed and effluent are modeled and the experimental data are also plotted which are normalized against 1 g\/L ETu. The [Cu2+] curve is normalized against 3 g\/L Cu2+.   Figure 5.39 ETu curve generated increasing [Cu2+] from 1.0 g\/L to 2.68 g\/L (concentration values in the legend are normalization constants) 132  It can be observed that even a threefold increase in [Cu2+] doesn\u2019t not significantly affect [ETu] at the bottom of the column. There is a slight decrease of about 0.02 g\/L from the case where there was no change to [Cu2+]. 5.6.2 Iron  Similarly, 100 g of anhydrous Fe2(SO4)3 was added to the ore in 30 cm column and mixed well before loading the column. While [ETu] was normalized against 1 g\/L of ETu, [Fe3+] was normalized against 4.5 g\/L of Fe3+. The feed bucket was adjusted to reflect an increase of 0.5 g\/L of Fe3+ every 24 hours.  Figure 5.40 ETu curve generated increasing [Fe3+] from 1.0 g\/L to 3.87 g\/L (concentration values in the legend are normalization constants) In this case it is clearly observed that the initial outlet [ETu] is about twice that of the final outlet [ETu] which in turn shows the sensitivity of [Fe3+] in affecting the decay of ETu. This reflects the results in Section 5.5.1, which show that a significant portion of the decay of ETu can be 133  attributed to Fe3+. The rate constant kFeETu is an order of magnitude higher than kCuETu, which explains the higher sensitivity of the model to [Fe3+] over [Cu2+]. 5.7 Impact of sulfide mineral content    In Chapter 4 (Figure 4.6), the behavior of ETu breakthrough curves in DI water on ore with additional pyrite suggested that pyrite concentration in the ore would affect ETu decay kinetics. In this section, that effect is expanded upon.  The total consumption of ETu from adsorption (Eqn (4.5)) and solute effects (Eqn (5.4)) has been parametrized. However, in the above-mentioned equations, the corresponding rate constants of ETu decay \u2013 k*, kCuETu, kO2ETu and kFeETu \u2013 are a lumped representation of the mineral surface effects and the thermal effect. Only the concentration effect is delineated as a function of [ETu] or [Cu2+] or [O2] or [Fe3+] respectively. A correlation that would explain the variation of the corresponding adsorption and rate factors with the concentration of pyrite is attempted.  Breakthrough curves for [ETu] = 0.51 g\/L in DI water were generated for columns in which finely ground pure pyrite was agglomerated with ore at different mass ratios of 0.5%, 1.0% and 1.5%. Figure 5.41 shows the breakthrough curves and the corresponding model fits.   134   Figure 5.41 ETu curves in DI water for different Pyrite content ore The model parameters K and k* (from the Redlich-Peterson adsorption isotherm) were adjusted to model the ETu breakthrough for each % of pyrite content. Table 5.2 shows the values used.  Table 5.2 Model parameters for the Redlich-Peterson model (Eqn (4.2)) Pyrite % K \u00d7 10\u22124 k* \u00d7 10\u22126 0.5 1.2 4.2 1.0 7.5 5.3 1.5 14.5 6.6 The trend of this variation in the model parameters is shown in Figure 5.42.   135   Figure 5.42 Trends of K and k* with respect to % Pyrite content  A linear trend is observed in the case of both K and k*. To validate this trend, a column with 2.0% pyrite mass ratio is used with all the other operational parameters held the same. Figure 5.43 shows the corresponding model and experimental ETu breakthrough curves.  Figure 5.43 Model and Experimental ETu curve for 2% Pyrite  136  It is observed that the linear function that describes the change of model parameters K and k* is not valid in the case of 2.0% pyrite content. The actual model parameters that fit the experimental breakthrough curves are 20% (in the case of K) and 27% (in the case of k*) higher than their linear fit estimations. The sensitivity of the ETu breakthrough curve to the % Pyrite content in the ore is not captured well by the Redlich-Peterson adsorption isotherm. The parameters K and k* are not linear functions of %Pyrite which may be one of the reasons for the deviation. The limitation of using SSE minimization of breakthrough curves to parameterize the model is the fundamental assumption that the model used to produce the breakthrough curves is valid and can capture the sensitivity of the breakthrough curve. In any case, a real ore body has multiple minerals which are exposed in varying proportions. It is the author\u2019s view that delineation of the individual mineral surface effects on the decay of ETu is unrealistic to develop. Hence, while the Redlich-Peterson adsorption isotherm describes the ETu adsorption, the parameters quantified in Chapters 4 and 5 should be viewed in the prism of the underlying ore analyzed. If a different ore with different mineral composition is used to estimate ETu adsorption or reaction, the form of the equations and models used in this study will hold well but the actual magnitude of the parameterized values would not necessarily behave in a characteristic pattern. In Chapter 6, the catalytic effect of ETu on the leaching of Cu from the ore is discussed in the context of associated decay of ETu, which has been characterized.   137  Chapter 6: Catalytic Effect of ETu on Copper Leaching from Chalcopyrite  In Chapters 3, 4 and 5, a model was presented that describes the transport and distribution of ETu through a bed of ore under leaching conditions. In this chapter the beneficial effect of ETu as a catalytic reagent in the leaching of chalcopyrite is demonstrated. However, this catalytic effect should be viewed in the context of distribution and non-productive consumption of ETu. A rate law for the kinetics of ETu-assisted leaching of chalcopyrite is used to model copper extraction as a function of ETu distribution in a column. The sensitivity of copper extraction rate to variations in ETu feed concentrations and irrigation fluxes is demonstrated experimentally. These variations are captured by the model. Experimental observations and a column leaching model are used to optimize operating parameters of the column. First, the compatibility of ETu as a catalytic reagent is established by analyzing the stability and the biocompatibility of ETu in lixiviant solutions.  6.1 Compatibility of ETu in leaching conditions  ETu, being an organosulfur compound, is not commonly present in leaching systems. Hence properties such as redox behavior and biocompatibility in the leach solution must be examined. 6.1.1 Redox Properties The electrochemical behavior of ETu was studied using cyclic voltammetry on a potentiostat (VersaSTAT 3) with a chalcopyrite working electrode and a platinum counter electrode. The electrolyte was an acidic ETu solution at pH 1.8. ETu demonstrated irreversible redox behavior, and no reduction peak was observed within the potential range studied (Figure 6.1).  138   Figure 6.1 Cyclic voltammetry study on 0.2 g\/L ETu at three scan rates  The linear increase of current when the potential is increased to over 500 mV (vs Ag\/AgCl reference electrode) suggests the oxidation of ETu on the surface of the chalcopyrite electrode.  The potential implies that ETu can be oxidized at potentials higher than 500 mV which may be present in chalcopyrite leaching owing to the presence of ferric.  6.1.2 Biocompatibility Analysis In acidic ferric sulfate leaching, iron and sulfur oxidizing bacteria such as Acidithiobacillus ferrooxidans and Leptospirillum ferrooxidans are often used to oxidize ferrous to ferric in aerated environments through the reaction:  4 Fe2+  +  O2  +  4 H+  \u2192  4 Fe3+  +  2 H2O (6.1) 139  The consortia commonly used in heap bioleaching are not naturally acclimatized to chemical compounds such as ETu. Therefore, determining the compatibility of ETu and iron\/sulfur oxidizing bacteria is the first and most critical step for further application of ETu to heap bioleaching.  Shake flask incubation tests were used to determine the influence of ETu on the ability of A. ferrooxidans to oxidize ferrous to ferric. The A. ferrooxidans used in these tests was harvested from an active bioleaching column. The typical 9K medium for bacterial cultures was not used to avoid too rapid growth and potassium jarosite precipitation. Identical amounts of bacterial culture (1 mL) were inoculated in 40 mM ferrous sulfate solutions (pH 1.8) containing 0, 0.01 (10 ppm) , 0.05 (50 ppm), and 0.1 g\/L (100 ppm) of ETu. The oxidation-reduction potential (ORP) and ETu concentration were the response variables. All tests were carried out at ambient temperature and atmosphere for 750 hours. A. ferrooxidans was able to adapt to solutions containing at least 0.1 g\/L of ETu. The exponential growth (log) phase was delayed by less than 48 hours by the addition of 0, 0.01, 0.05, and 0.1 g\/L of ETu (Figure 6.2). The ORP plateau was lower for the 0.05 and 0.1 g\/L treatments due to the influence of ETu on the mixed potential of the solution. The ORP plateaus suggest the oxidation of all ferrous to ferric.  140   Figure 6.2 Effect of three ETu concentrations on ferrous oxidation by A. ferrooxidans   Over 70% of the ETu remained in solution when bacterial growth entered the stationary phase at 408 hours (Figure 6.3). These results suggest that ETu does not participate in bacterial metabolism during ferrous oxidation. It is likely that the oxidation of ETu by ferric is responsible for the small decrease in ETu concentration. No oxidation product was detected by the current reverse phase HPLC method. According to previous literature, it is believed that ethylene urea is the oxidation product, which is eventually oxidized to CO2, H2O and NO3\u2013.  141   Figure 6.3 ETu concentration in flasks with four initial concentrations during growth of A. ferrooxidans during a 744-hour period  6.2 Leaching copper sulfides with ETu Acidic ferric sulfate solution (2.2 g\/L Fe3+; pH 1.7) was used as a lixiviant to examine the catalytic effect of ETu on chalcopyrite leaching in the presence of A. ferrooxidans at room temperature in stirred bioreactors. This bacterial consortium is the same as the one used in the biocompatibility analysis. A benefit of leaching in a bioreactor is that the iron-oxidizing bacteria provide sufficient oxidant (ferric), so no addition of oxidant such as peroxide or permanganate is required to maintain the ORP. 6.2.1 Reactor leaching experiments with ETu One gram of synthetic chalcopyrite mineral with P80 = 75 \u03bcm was added to each of the bioreactors containing 1 L of ferric sulfate solution and A. ferrooxidans. The chalcopyrite 142  mineral sample was characterized by X-ray powder diffraction. The XRD pattern with Rietveld refinement is shown in the figure below.  Figure 6.4 XRD pattern with Rietveld refinement for the chalcopyrite sample used   The XRD analysis was performed with the use of the software Match 3. No mineral phases other than chalcopyrite were observed.  Four bioreactors set at a stirring speed of 500 rpm were maintained at room temperature under identical conditions for 138.5 h so the bacteria could acclimate to the solution matrix, agitation, and mineral. ETu was then added to each reactor to a final concentration of 0, 0.01, 0.05, and 0.1 g\/L . The ETu concentration was measured daily and maintained at the designated concentration through manual make-up.  During the acclimation period (Phase 1), the ORP remained stable at > 600 mV within the first 100 hours and climbed to > 700 mV thereafter, indicating that the bacteria had adjusted to the 143  operating conditions (Figure 6.5) and converted all ferrous to ferric. During Phase 2 (504 h leaching period), the ORP was generally lower in the bioreactors containing of 0, 0.01, 0.05, and 0.1 g\/L ETu. The other purpose of Phase 1 was to establish the passivation layer. During Phase 1, 4.18% Cu was leached compared to 10.01% in the control during Phase 2 (Figure 6.6). This passivation behavior is consistent with previous literature on chalcopyrite leaching in acidic ferric sulfate media. In Phase 2, copper extraction was accelerated relative to the control in all reactors containing ETu, at a rate that increased with ETu concentration (Figure 6.6). Curves for the two highest ETu concentrations showed power functions and achieved 100% Cu extraction after 642 hours, whereas the reactor maintained at 10 ppm ETu showed a linear increase in Cu recovery and reached 45.5% Cu extraction. The control maintained a high ORP throughout the leaching period but only achieved 14.2% Cu extraction.   Figure 6.5 Oxidation-reduction potential (ORP) during chalcopyrite bioleaching with A. ferrooxidans and 0 (control and Phase 1 in all bioreactors), 10, 50, and 100 ppm ETu 144   Figure 6.6 Copper recovery during chalcopyrite bioleaching with A. ferrooxidans and 0 (control and Phase 1 in all bioreactors), 10, 50, and 100 ppm ETu 6.2.2 Rate Law of ETu-Assisted Leaching  A kinetic model that explains the leaching of chalcopyrite is of the following form:  )1()()( XgCfTkdtdX\u2212\u22c5\u22c5=  (6.2) where X  = chalcopyrite conversion  The function k(T) is a thermal function, f(C) is a chemical function and g(1\u2212X) is a topological function representing the changing chalcopyrite topology. The thermal function k(T) is usually expressed as an Arrhenius function of temperature:  \uf8fa\uf8fa\uf8fb\uf8f9\uf8ef\uf8ef\uf8f0\uf8ee\uf8f7\uf8f7\uf8f8\uf8f6\uf8ec\uf8ec\uf8ed\uf8eb\u2212=refaref TTRETkTk 11exp)()(  (6.3) where Ea =  activation energy (kJ\/mol)  R  =  universal gas constant (8.3145 J\/mol\/K)  T  =  absolute temperature (K) 145  Tref  =  reference temperature (typically the baseline temperature) (K) Generally, metal sulfide minerals in ferric sulfate media are dissolved by the oxidative action of iron in solution as follows:  MS + 2 Fe3+  \u2192  M2+ + 2 Fe2+ + S  (6.4) where M2+ is a divalent metal ion. The leaching reaction of metal sulfides is electrochemical in nature and Eqn (6.4) is the result of two half-cell reactions which occur simultaneously on the surface of the mineral at the mixed potential, Em:  Anodic:   MS  \u2192  M2+ + S + 2e\u2212 (6.5)  Cathodic:   Fe3+ + e\u2212  \u2192  Fe2+  (6.6) When chemical reactions control the rate of reaction, and this rate is unaffected by the presence of a product layer, then the conversion rate of the particle is proportional to the available surface area of the unreacted core (Dixon and Liu, 2019). In this case, assuming spherical particles, the fraction unreacted can be written as:  3011 \uf8fa\uf8fb\uf8f9\uf8ef\uf8f0\uf8ee\uf8f7\uf8f8\uf8f6\uf8ec\uf8ed\uf8eb\u2202\u2202\u2212=\u2212dttdX   (6.7) where d  = particle diameter (\u03bcm)  d0 = initial particle diameter (\u03bcm) The shrinking rate of the particle, denoted by the partial derivative of diameter with respect to time, can be further expanded:  baTktd ]ETu[][Fe][Fe)( 23\uf8f7\uf8f7\uf8f8\uf8f6\uf8ec\uf8ec\uf8ed\uf8eb=\u2202\u2202++  (6.8) Hence the overall leaching rate of the mineral can be expressed as follows:  146   3230]ETu[][Fe][Fe11exp)(11\uf8fa\uf8fa\uf8fb\uf8f9\uf8ef\uf8ef\uf8f0\uf8ee\u22c5\uf8f7\uf8f7\uf8f8\uf8f6\uf8ec\uf8ec\uf8ed\uf8eb\uf8fa\uf8fa\uf8fb\uf8f9\uf8ef\uf8ef\uf8f0\uf8ee\uf8f7\uf8f7\uf8f8\uf8f6\uf8ec\uf8ec\uf8ed\uf8eb\u2212\u2212=\u2212 ++tTTREdTkX barefaref   (6.9) From Eqn (6.9) the extraction rate of copper is a function of temperature, initial particle size, ferric-ferrous ratio, and ETu concentration. In the context of this chapter, the interesting part is the impact of ETu concentration. Hence, the other aspects of the rate law are lumped into one factor klump and Eqn (6.9) is simplified thus:   ( )3]ETu[11 tkX blump \u22c5\u2212=\u2212  (6.10) From Eqn (6.10), if [ETu] is changed, keeping all other factors the same, then the impact of ETu on the copper leaching rate can be discerned. 6.3 Sensitivity of Cu leaching to [ETu] An experiment was set up with chalcopyrite ore which was previously used to estimate the adsorption and reaction parameters of ETu. Two columns of 30 cm height and 10 cm diameter were set up with ore weighing 3.396 kg and 3.584 kg. Chalcopyrite was the only copper-bearing phase. Table 6.1 shows the mineral composition of the low-grade copper ore used.  Table 6.1 Results of mineral phase analysis of the ore provided Mineral Ideal Formula Wt % Chalcopyrite CuFeS2 1.4 Kaolinite Al2Si2O5(OH)4 2.3 K-Feldspar KAlSi3O8 17.9 Molybdenite MoS2 < 0.1 Muscovite KAl2AlSi3O10(OH)2 21.9 Plagioclase NaAlSi3O8 \u2013 CaAl2Si2O8 13.6 Pyrite FeS2 2.3 Quartz SiO2 40.0 Rutile TiO2 0.5 Total  100.0 147  Inductively coupled plasma atomic emission spectroscopy (ICP) gave a copper content of 0.344%. The P80 of the ore was 6.75 mm. 2 mL of the same A. ferrooxidans culture that was used in the biocompatibility test was used to inoculate each of the columns. The columns were irrigated at a rate of 1.4 L\/day with acidic ferric sulfate solution (2.2 g\/L Fe3+; pH 1.8). Columns were maintained in closed-loop operation for 10 days to allow bacteria to acclimate to experimental conditions. From Day 10 to Day 19, one column was operated in open loop conditions with the addition of freshly pre-made feed with acidified ferric sulfate lixiviant at a pH of 1.7 and 0.5 g\/L Cu2+. Batches of 20 L of the stock feed lixiviant (2.2 g\/L Fe3+, 0.5 g\/L Cu2+ and pH 1.7) were prepared periodically to refill the feed bucket. The effluent from the column on closed loop was periodically sampled and the ORP was checked as a measure to indicate the ferric availability for leaching. Samples were sent for analysis of Fe and Cu by ICP. On Day 20, 4 g of ETu was added to the feed bucket, which contained 4 L of feed solution resulting in a 1 g\/L ETu feed lixiviant. The feed to the ETu leaching column was adjusted for ETu concentration at regular intervals. The duration of the total test was about 300 days. The copper extraction, represented as a fraction of the total theoretical copper available in the head ore (calculated from Table 6.1), is plotted against the time in days in Figure 6.6. 148   Figure 6.7 Extraction of Cu from chalcopyrite ore in the case of ETu and Control    Figure 6.8 ORP of the effluent from chalcopyrite ore column in ETu and Control cases  The data points which indicate a reduction in % extraction could be the result of errors in analysis. The beneficial effect of ETu as a catalyst to leach copper from chalcopyrite is obvious from the respective curves. However, the focus of this chapter is to weigh the productive use of ETu as a catalyst versus the non-productive consumption of ETu. To this end, the ratio of grams 149  of ETu used (in non-productive consumption) to the grams of Cu extracted is useful. For the purposes of this work, this metric is defined as the ETu consumption index. The higher the ETu consumption index, the less efficient the leaching. The next section discusses how this consumption rate is estimated using the solute transport model.  6.4 Effect of leaching progress on ETu consumption index  It is well known that the bed of ore is heterogeneous and as leaching progresses the bed of ore continues to change in both physical and chemical aspects. Hence, three time points were arbitrarily chosen \u2013 Day 20, Day 106, and Day 278 \u2013 to estimate the ETu consumption index. In the case of Day 20, 1 g\/L of ETu was fed to the column, and samples were taken from the feed and effluent at intervals of time 20 days + 0, 1, 3, 5, 8, 10 and 24 hours. It is assumed that within the span of 24 hours the bed of ore does not undergo significant change. Using the solute model, a function that represents the concentration of ETu fed to the column was used as input and the model generated the Cu and ETu curves that were expected from the effluent. The sensitivity analysis of breakthrough curves is used to estimate the lumped kinetics rate constant, klump. The differential of [Fe3+] and [Cu2+] from the feed to the outlet is not substantial enough to impact the adsorption and homogeneous reactions associated with ETu in the lixiviant. Eqns (4.10) (the Redlich-Peterson adsorption isotherm) and (5.7) (the ETu consumption source term) were used to generate the ETu breakthrough curve. For the Cu breakthrough curve the source term is the rate law Eqn (6.10) in addition to the inlet Cu concentration of 0.5 g\/L. The model-generated and experimental breakthrough curves are shown in Figure 6.9. 150   Figure 6.9 Model breakthrough curves and experimental data on Day 20 for the catalyzed column   The actual analyzed value of \u2206[Cu2+] is 0.015 g\/L (averaged). At a flow rate of 1.4 L\/day, 21 mg of Cu were extracted per day. The head ore had 11.6 g of total Cu as chalcopyrite, which equates to 0.18% of total Cu extracted over a 24-hour period when the inlet concentration of ETu was adjusted to 1 g\/L. As ETu concentration at the outlet of the column is measured, an estimate of ETu consumption is obtained by subtracting the effluent ETu from the total ETu fed to the column. In this case, 0.89 g of ETu was consumed non-productively. Hence, as defined earlier, the ETu consumption index for this ore at Day 20 for a feed concentration of 1 g\/L fed at 1.4 L\/day is 42.3. It is obvious that, the higher the number, the lower the viability of using ETu as a catalytic lixiviant. Using this number as a starting metric or baseline, the ETu consumption index at the preselected time points is gauged.  In Figure 6.9 the breakthrough curves, model-generated and experimental, are plotted.  151   Figure 6.10 Model breakthrough curves and experimental data on Day 106 for the catalyzed column In this case the feed ETu in the bucket decays at the same rate as that of the feed on Day 20. While there are small deviations from the values observed at specific timepoints the overall trend of feed [ETu] to the column on Day 106 is the same as that on Day 20 to the first decimal point. In that context, a slight bump in \u2206[Cu2+] = 0.018 g\/L (averaged) is observed. It is also observed that the outlet [ETu] is generally higher on Day 108 than on Day 20. To compensate for this upward shift in ETu decay, the k* value (the ETu consumption attributed to adsorption) is adjusted in the solute model. A good agreement between the model and experiments is observed with that modification. Based on the model and experimental curves it was calculated that non-productive consumption of ETu was 0.81 g. The adjusted Cu extraction for the day was 25.2 mg which results in an ETu consumption index of 32.1 \u2013 a reduction of almost 24% from the value on Day 20. The rationale of using k* as the parameter that needs to be adjusted is owing to the sensitivity of ETu breakthrough to k*. Adjusting any of the Redlich-Peterson parameters or other 152  rate constants relating to Cu, Fe or O2 had an exaggerated effect (owing to concentration of the same in the lixiviant; hence the need to feed 0.5 g\/L Cu as inlet to the column) on ETu breakthrough. Moreover, any adjustment of the Redlich-Peterson parameters would mean delay in ETu breakthrough, which would present itself as a bump more towards the right in Figure 6.9. To confirm this trend of less decay from adsorption, the analysis of ETu consumption index was done again on Day 278. Figure 6.11 shows the model-generated and experimental breakthrough curves.   Figure 6.11 Model breakthrough curve and experimental data on Day 278 for the catalyzed column   Using similar analysis, the non-productive loss for ETu was 0.77 g of ETu and the corresponding ETu consumption index was 27.5. The clear trend that shows the reduction in ETu consumption index and similar Cu extraction from the ore reflects the reduction in non-productive consumption of ETu as the leaching progresses. Perhaps the most meaningful conclusion that can be drawn is more qualitative than quantitative in nature: the benefit of adding ETu to a column of 153  chalcopyrite ore can best be utilized later in the leaching cycle than earlier. It may turn out that the more prudent option is not to increase the concentration ETu in the leach solution but, rather, to use a higher irrigation flux to improve ETu utilization. The next section explores this idea. 6.5 Irrigation flux to improve ETu utilization As the non-productive consumption of ETu is proportional to [ETu], increasing the irrigation flux could be a better solution to improve extraction rates. However, if higher [ETu] yields more Cu the higher non-productive loss might be justified. Essentially, if the denominator of the ETu consumption index increases more than the numerator, then the utilization of ETu is improved. This is important because Cu is the source of revenue and the cost of ETu must be expressed as a function of Cu recovered.   To understand the impact of irrigation fluxes, fresh batches of the same ore were leached in three 30 cm columns. The test procedure was very similar to the previous section, except that the columns were run for much longer times under control conditions, without any addition of ETu. One column involved no addition of ETu (the control).  ETu was added to the second column at 1 g\/L following Day 168 and to the third column at 2 g\/L following Day 224, and the results are shown in Figure 6.12.  154   Figure 6.12 Extraction of Cu from ore in the case of column at different [ETu] and fluxes  A drastic increase in Cu leaching rate is observed in the case of 2 g\/L ETu. However, the ETu consumption index of this test was 95.4, which was significantly higher than test with 1 g\/L ETu which was 47.2. Hence, as leaching progressed, a different strategy was adopted. At day 400, for a period of 50 days, the 1 g\/L and 2 g\/L ETu columns were run at the same total ETu addition. This was achieved by reducing the irrigation flux of the 2 g\/L column by half (0.5 mL\/min ~ 0.72 L\/day). Results are shown in Figure 6.13. 155   Figure 6.13 Extraction of Cu from ore in the case of ETu fed at different fluxes  In this test the 2 g\/L fed at 0.72 L\/day broke through the column and produced extraction curves similar to 1 g\/L ETu. The ETu consumption index for the 2 g\/L and 1 g\/L tests were 49.75 and 46.37 respectively. The similar slopes of the trendlines show that the same catalytic effect was observed at the same ETu addition rate, irrespective of flow rate or concentration.  To summarize, in the chapter the model is used to estimate the ETu consumption index, a metric that quantifies the viability of ETu as a catalytic solute. Changes in operating conditions such as irrigation flux and inlet feed concentration affect copper extraction and the ETu consumption index, which are captured well by the model. Hence the model can be used as a useful tool to optimize operating parameters to leach copper from chalcopyrite in a column of ore.  156  Chapter 7: Conclusion and Recommendations Catalytic thiocarbonyl compounds have been demonstrated to enhance the leaching kinetics of copper from chalcopyrite in acidic ferric sulfate media at ambient temperatures. This discovery makes the heap leaching of low-grade chalcopyritic ore viable. However, adding these organic solutes in acidic ferric sulfate lixiviant has not been investigated in detail. Given that these thiocarbonyl catalysts are involved in parallel reactions with other solutes in the lixiviant media and on the surface of the ore, the fundamental question this thesis has sought to answer is this: how much of a reactive catalyst is transported (and thus distributed) though the bed of ore if fed as a dissolved solute in the lixiviant. ETu is taken as a model thiocarbonyl reactive catalyst.  In this thesis, this transport problem is delineated into describing and quantifying the flow of water thorough a bed of ore in a column (Objective I) followed by estimating the amount of solute (ETu) resident on the surface of the ore (adsorbed) and dispersed (Objective II). As the ETu decays in lixiviant solution, the reactive decay of ETu is quantified (Objective III) to present the overall distribution of ETu through the bed of ore. This information is only useful if viewed in the context of leaching of copper, so copper leaching tests are used to estimate ETu consumption as a function of copper extracted (Objective IV).  Each of these objectives was achieved and the conclusions are presented in this chapter.  7.1   Describe and quantify water flow through a column Breakthrough curve analysis is an effective tool to calibrate a 2D axisymmetric water model under a single point source in a column of ore. The hydraulic parameters Kw, m, n and \u03c8 provide a relationship to estimate the level of saturation of a column under irrigation at a specific flux (within a specified interval) from an initial moisture level. The air entry head parameter hc,0 is 157  confirmed as the master parameter that determines the degree of radial spreading of water in the column. A value of hc,0 = 0.05 m implies very limited horizontal spread, and results in a water breakthrough curve which is bimodal, whereas a value of hc,0 = 1 m represents plug flow.  The breakthrough curve technique demonstrates good scalability as it describes flow in columns of size 40 \u00d7 10 cm, 100 \u00d7 8 cm, and 200 \u00d7 20 cm reasonably well. Given the path dependency of the hydrology parameters to the applied flux the water infiltration tests to elicit them need to be performed on either a well agglomerated bed or an already wet bed of ore. If dry ore is used to perform the tests, the parameters are significantly different.    7.2 Estimate adsorption and dispersion of a catalytic solute (ETu)   The breakthrough of a model catalytic solute, ETu, through a column of ore is used to calibrate and discern specific adsorption parameters on the ore under non-leaching conditions. The Redlich-Peterson adsorption isotherm used to model the ETu breakthrough curves suggests that ETu adsorption does not follow ideal monolayer adsorption, where the adsorbed molecules have no interaction with adjacent adsorption sites irrespective of surface occupancy. This inference is supported by the ETu behavior which tends towards non-interacting solutes such as Cu2+ when competing for adsorption sites in the presence of other adsorbing solutes under alkaline conditions where the ETu is non-reactive. The concentration profile computed by the model is used to determine the local concentration of the solute at any depth and radius in the 30 \u00d7 10 cm and 100 \u00d7 8 cm columns, and simulation of a 400 \u00d7 50 cm column of the solute of interest using the same solute parameters is also provided.  7.3 Quantify reactive decay of ETu in real lixiviant solutions  The impact of solutes such as Cu2+, Fe3+ and O2 and varying sulfide mineral content on ETu breakthrough curves was captured well by rate terms added to the solute model. Reaction with 158  O2 has the highest rate constant, but Fe3+ causes the most decay of ETu because of its significantly higher concentration. The individual contribution of each factor towards a real lixiviant solution is delineated, which establishes the expediency of this technique. While the effect of mineral composition of ores on the ETu decay might not be linear (as in the case of pyrite), the tool still provides a useful way to compare the decay attributed to changes in minerology versus changes in lixiviant composition.  7.4 Gauge ETu consumption as a function of copper extracted  The model is applied to estimate ETu consumption versus amount of copper extracted. It is observed that the non-productive consumption of ETu is lesser as leaching progresses. It is postulated that the nature of the ore changes as the column is leached, resulting in less ETu consumption. This observation is used to operate a column which results in the same extraction rate but used 50% to total ETu as compared to the non-optimized column. Hence this tool can be very useful in comparing dosing strategies for operations looking to implement novel catalytic solutes to leach low grade chalcopyrite ore.  7.5 Recommendations The ultimate focus of this thesis is the transport and decay of a catalytic solute, ETu, in the context of copper extraction in columns and heaps. However, the hydrological measurements used to estimate the flow of lixiviant, and the kinetic measurements used to arrive at the rates of copper extraction, are not sufficiently rigorous. Concerning hydrology, the path dependency of infiltration tests affected the hydrological parameters estimated. Agglomeration and\/or pre-saturation of the ore significantly increased both equilibrium and residual moisture contents relative to initially dry ore. The impact of 159  different wetting strategies on the residual moisture content should be explored much more systematically than was attempted in this thesis. Further, it is important to measure the air entry head parameter hc,0 under every wetting condition, as this parameter may increase with increasing residual saturation. Hence the evolution of the hydrological parameters estimated in this thesis needs to be investigated in greater detail.  Concerning kinetics, the factor klump is essentially a scaling factor used to simplify a very complex rate law. A comprehensive rate law including [ETu] must be added to the solute transport model in order to predict copper extraction rates. However, ETu is probably not an economically viable catalyst for leaching copper from chalcopyrite. The ETu consumption index shows that even in the best case, the utilization of ETu is about 27 times more (by mass) than the copper extracted. The ETu consumption index can be optimized by manipulating operating conditions. However, unless the index can be significantly decreased, it remains unlikely that ETu would be a viable catalytic lixiviant for low grade chalcopyrite ores. In any case, there are other more suitable thiocarbonyl compounds which could be employed as catalysts for chalcopyrite heap leaching, and much more rigorous kinetic study would be warranted on those catalysts than was attempted in this thesis. However, by utilizing the specific set of experimental and analytical techniques detailed in this thesis, a model for a cheaper or more effective alternative catalytic solute could be calibrated and validated. Thus, this thesis provides a roadmap for model-based experiments in the catalytic heap leaching field.   160  Bibliography Afewu, K.I. and Dixon, D.G., 2008. Calibrating a 3D axisymmetric water and solute transport model for heap leaching. Hydrometallurgy 2008: Proceedings of the 6th International Symposium: 955\u2212966. Anderson, M.G., 1981. Book reviews: Freeze, R.A. and Cherry, J.A. 1979: Groundwater. Hemel Hempstead: Prentice-Hall International. Progress in Physical Geography: Earth and Environment, 5(4): 616\u2212617. Arya, L.M. and Paris, J.F., 1981. A Physicoempirical Model to Predict the Soil Moisture Characteristic from Particle-Size Distribution and Bulk Density Data. Soil Science Society of America Journal, 45(6): 1023\u22121030. Bartlett, R.W., 2007. Solution mining: leaching and fluid recovery of materials. Taylor & Francis, London. Barton, I.F. and Hiskey, J.B., 2022. Chalcopyrite leaching in novel lixiviants. Hydrometallurgy, 207: 105775. Bear, J., 1972. Dynamics of Fluids in Porous Media. Soil Science, 120: 162\u2212163. Bennett, C.R., McBride, D., Cross, M. and Gebhardt, J.E., 2012. A comprehensive model for copper sulphide heap leaching: Part 1 Basic formulation and validation through column test simulation. Hydrometallurgy, 127\u2212128: 150\u2212161. Bittelli, M. and Flury, M., 2009. Errors in water retention curves determined with pressure plates. Soil Science Society of America Journal, 73(5): 1453\u22121460. Carlin, R.L. and Holt, S.L., 1963. Some Transition Metal Complexes of Ethylenethiourea. I. Cobalt(II). Inorganic Chemistry, 2(4): 849\u2212853. Celia, M.A., Bouloutas, E.T. and Zarba, R.L., 1990. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resources Research, 26(7): 1483\u22121496. Cherkaev, A., 2019. The art of soaking rocks: systematic study of liquid and solute flow in packed rock beds in the context of heap leaching. Christel, S. et al., 2018. Weak Iron Oxidation by Sulfobacillus thermosulfidooxidans Maintains a Favorable Redox Potential for Chalcopyrite Bioleaching. Frontiers in microbiology, 9: 3059. Corey, A.T., 1977. Mechanics of Heterogenous Fluids in Porous Media. Water Resources Publications. Dixon, D., G. and Liu, W., 2019. Copper Sulfide Leaching Kinetics \u2013 Fundamentals and Case    Studies, in Copper Hydrometallurgy: Principles and Practice. Canadian     Institute of Mining, Metallurgy and Petroleum, pp. 24\u221251. Dixon, D.G. and Afewu, K.I., 2022. HeapSim2D \u2013 A 2D axisymmetric model of heap leaching under drip emitters I. Hydrology and solute transport. Minerals Engineering, 183: 107555. 161  Dixon, D.G., Olmedo, O., Asselin, E., Ghahreman, A. and Ren, Z., 2019. Process for leaching metal sulfides with reagents having thiocarbonyl functional groups. University of British Columbia, Canadian Patent 2973612. Dreisinger, D., 2006. Copper leaching from primary sulfides: Options for biological and chemical extraction of copper. Hydrometallurgy, 83(1): 10\u221220. Elsherief, A., 2002. The influence of cathodic reduction, Fe2+ and Cu2+ ions on the electrochemical dissolution of chalcopyrite in acidic solution. Minerals Engineering, 15(4): 215\u2212223. Fairley, J., Podgorney, R. and Wood, T., 2004. Unsaturated Flow through a Small Fracture-Matrix Network: Part 2. Uncertainty in Modeling Flow Processes. Vadose Zone Journal, 3: 101\u2212108. Farthing, M.W. and Ogden, F.L., 2017. Numerical Solution of Richards' Equation: A Review of Advances and Challenges. Soil Science Society of America Journal, 81(6): 1257\u22121269. Fetter, C.W, Thomas.B, David .K, 2017. Contaminant hydrogeology: Third Edition. Waveland Press, Long Grove. Freeze, R.A., 1971. Three-dimensional, transient, saturated-unsaturated flow in a groundwater basin. Water Resources Research, 7(2): 347\u2212366. Ghanbarian-Alavijeh, B. and Liaghat, A., 2009. Evaluation of soil texture data for estimating soil water retention curve. Canadian Journal of soil science, 89(4): 461\u2212471. Gil-Lozano, C., Davila, A.F., Losa-Adams, E., Fair\u00e9n, A.G. and Gago-Duport, L., 2017. Quantifying Fenton reaction pathways driven by self-generated H2O2 on pyrite surfaces. Scientific Reports, 7(1): 43703. Godoy, B., Braslavsky, J. and Aguero, J., 2008. A Simulation Study on Model Predictive Control and Extremum Seeking Control for Heap Bioleaching Processes, IFAC Proceegings Volumes,Volume 41, Issue 2, Pages 9368-9373. Granata, G., Miura, A., Liu, W., Pagnanelli, F. and Tokoro, C., 2019. Iodide-assisted leaching of chalcopyrite in acidic ferric sulfate media. Hydrometallurgy, 186: 244\u2212251. Hillel, D., 1998. Environmental Soil Physics: Fundamentals, Applications, and Environmental Considerations. Elsevier Science. Hiroyoshi, N., Miki, H., Hirajima, T. and Tsunekawa, M., 2000. A model for ferrous-promoted chalcopyrite leaching. Hydrometallurgy, 57(1): 31\u221238. Ilankoon, I.M.S.K., Cole, K.E. and Neethling, S.J., 2013. Measuring hydrodynamic dispersion coefficients in unsaturated packed beds: Comparison of PEPT with conventional tracer tests. Chemical Engineering Science, 89: 152\u2212157. Ilankoon, I.M.S.K. and Neethling, S.J., 2012. Hysteresis in unsaturated flow in packed beds and heaps. Minerals Engineering, 35: 1\u22128. Ilankoon, I.M.S.K. and Neethling, S.J., 2013. The effect of particle porosity on liquid holdup in heap leaching. Minerals Engineering, 45: 73\u221280. 162  Ilankoon, I.M.S.K. and Neethling, S.J., 2014. Transient liquid holdup and drainage variations in gravity dominated non-porous and porous packed beds. Chemical Engineering Science, 116: 398\u2212405. Ilankoon, I.M.S.K. and Neethling, S.J., 2016. Liquid spread mechanisms in packed beds and heaps. The separation of length and time scales due to particle porosity. Minerals Engineering, 86: 130\u2212139. Ilankoon, I.M.S.K. et al., 2018. The current state and future directions of percolation leaching in the Chinese mining industry: Challenges and opportunities. Minerals Engineering, 125: 206\u2212222. Jarvis, N., Bergstr\u00f6m, L. and Dik, P., 1991. Modelling water and solute transport in macroporous soil. II. Chloride breakthrough under non-steady flow. Journal of Soil Science, 42(1): 71\u221281. Jia, Y. et al., 2021. Industrial Heap Bioleaching of Copper Sulfide Ore Started with Only Water Irrigation. Minerals, 11(11). Jones, P.G., Guy, J.J. and Sheldrick, G.M., 1976. Bis(ethylenethiourea)gold(I) chloride hydrate. Acta Crystallographica Section B, 32(12): 3321\u22123322. Lal, R. and Shukla, M., 2004. Principles of soil physics. Marcel Dekker, Inc., New York. Lim, P.C., Barbour, S.L. and Fredlund, D.G., 1998. The influence of degree of saturation on the coefficient of aqueous diffusion. Canadian Geotechnical Journal, 35(5): 811\u2212827. Liu, W. and Granata, G., 2018. The Effect of Aeration on Chalcocite Heap Leaching: Proceedings of the First Global Conference on Extractive Metallurgy, pp. 1353\u22121359. Lu, J. and Dreisinger, D., 2013. Copper leaching from chalcopyrite concentrate in Cu(II)\/Fe(III) chloride system. Minerals Engineering, 45: 185\u2212190. Manabe, M., 2012. Method of leaching copper sulfide ore with the use of iodine. US Patent US8163063B2. Manning, T.J. and Kappes, D.W., 2016. Chapter 25 - Heap Leaching of Gold and Silver Ores. In: M.D. Adams (Ed.), Gold Ore Processing (Second Edition). Elsevier, pp. 413\u2212428. Marsden, J.O. and Botz, M.M., 2017. Heap Leach Modeling \u2013 A Review of Approaches to Metal Production Forecasting. Mining, Metallurgy & Exploration 34, 53-64. McBride, D., Cross, M., Croft, N., Bennett, C. and Gebhardt, J., 2006. Computational modelling of variably saturated flow in porous media with complex three-dimensional geometries. International Journal for Numerical Methods in Fluids, 50: 1085\u22121117. McBride, D., Gebhardt, J., Croft, N. and Cross, M., 2018. Heap Leaching: Modelling and Forecasting Using CFD Technology, Minerals , 8,9 . Mualem, Y., 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resources Research, 12(3): 513\u2212522. Muller, E.L., Basson, P. and Nicol, M.J., 2007. Chloride heap leaching. US Patent WO2007134343A2. 163  Pradhan, N., Nathsarma, K.C., Srinivasa Rao, K., Sukla, L.B. and Mishra, B.K., 2008. Heap bioleaching of chalcopyrite: A review. Minerals Engineering, 21(5): 355-365 O\u2019Kane, M., Barbour, S. and Haug, M., 1999. A framework for improving the ability to understand and predict the performance of heap leach piles. In Proceedings of the Copper 99\u2014Cobre 99 International Conference. Vol. 4, Hydrometallurgy of Copper. Edited by S.K. Young, D.B. Dreisinger, R.P. Hackl, and D.G. Dixon. Warrendale, PA: Minerals, Metals and Materials Society. pp. 409\u2013419 . Pan, L. and Wierenga, P.J., 1995. A Transformed Pressure Head-Based Approach to Solve Richards' Equation for Variably Saturated Soils. Water Resources Research, 31(4): 925\u2212931. Pan, T., Hou, S., Liu, Y. and Tan, Q., 2019. Comparison of three models fitting the soil water retention curves in a degraded alpine meadow region. Scientific Reports, 9(1): 18407. Perkins, K.S., 2011. Measurement and modeling of unsaturated hydraulic conductivity: Chapter 21. Hydraulic Conductivity - Issues, Determination and Applications: 419-434 Petersen, J., 2019. Heap Leaching of Copper Ores \u2013 State of the Science. Paper presented in Canadian Institute of Mining, Metallurgy and Petroleum Copper 2019 Conference, Vancouver, British Columbia, 597297. Petersen, J. and Dixon, D.G., 2003. The dynamics of chalcocite heap bioleaching. Hydrometallurgy, 1: 351\u2212364. Pradhan, N., Nathsarma, K.C., Rao, K.S., Sukla, L.B. and Mishra, B., 2008. Heap bioleaching of chalcopyrite : A review. Minerals Engineering, 21: 355\u2212365. Rebolledo, M., Zarate, G. and Mora, N., 2019. Catalytic Heap Leaching of Chalcopyrite Ores Using Jetti\u2019s Technology. Canadian Institute of Mining, Metallurgy and Petroleum, Copper 2019, Vancouver, British Columbia, 595316. Reinhold, S., 1990. Process for the isolation of noble metals. US Patent 4925485A. Ren, Z. et al., 2020. Catalytic effect of ethylene thiourea on the leaching of chalcopyrite. Hydrometallurgy, 196: 105410. Richards, L.A., 1931. Capillary conduction of liquids through porous mediums. Physics, 1(5): 318\u2212333. Robertson, S.W., van Staden, P.J. and Seyedbagheri, A., 2012. Advances in high-temperature heap leaching of refractory copper sulphide ores. Journal of the Southern African Institute of Mining and Metallurgy, 112: 1045\u22121050. Rossi, C. and Nimmo, J.R., 1994. Modeling of soil water retention from saturation to oven dryness. Water Resources Research, 30: 701\u2212708. Saadi, R., Saadi, Z., Fazaeli, R. and Fard, N.E., 2015. Monolayer and multilayer adsorption isotherm models for sorption from aqueous media. Korean Journal of Chemical Engineering, 32(5): 787\u2212799. Saltmiras, D.A. and Lemley, A.T., 2000. Degradation of Ethylene Thiourea (ETU) with Three Fenton Treatment Processes. Journal of Agricultural and Food Chemistry, 48(12): 6149\u22126157. 164  Sato, K.H., JP), Manabe, Manabu (Tokyo, JP), 2015. Method of leaching copper ore. JX Nippon Mining & Metals Corporation (Tokyo, JP), US Patent US8992666B2. Scheffel, R.E., 2006. The rewards of patience, Paper presented at the Society of Mining Engineers Annual Meeting, St. Louis, Missouri 27-29 March 2006, Preprint 06-014   Schwartz, F.W. and Zhang, H., 2002. Fundamentals of Ground Water. Wiley. Shervin, H., Farid, H., Siamak, B. and Seth, E., 2016. Comparing the Applicability of Soil Water Retention Models. Int. J. Environ. Sci. Technol, 1: 114\u2212118. Singer, D.A., 2017. Future copper resources. Ore Geology Reviews, 86: 271\u2212279. Spitz, K. and Moreno, J., 1996. A Practical Guide to Groundwater and Solute Transport Modeling. Wiley. Thenepalli, T., Chilakala, R., Habte, L., Tuan, L.Q. and Kim, C.S., 2019. A Brief Note on the Heap Leaching Technologies for the Recovery of Valuable Metals. Sustainability, 11(12): 3347. Third, K., Cord\u2010Ruwisch, R. and Watling, H., 2002. Control of the redox potential by oxygen limitation improves bacterial leaching of chalcopyrite. Biotechnology and Bioengineering, 78(4): 433\u2212441. Tromans, D., 2000. Modeling Oxygen Solubility in Water and Electrolyte Solutions. Industrial & Engineering Chemistry Research, 39(3): 805\u2212812. van Genuchten, M.T., 1980. A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils. Soil Science Society of America Journal, 44(5): 892\u2212898. Vel\u00e1squez-Y\u00e9venes, L., Nicol, M. and Miki, H., 2010. The dissolution of chalcopyrite in chloride solutions: Part 1. The effect of solution potential. Hydrometallurgy, 103(1): 108\u2212113. Vel\u00e1squez-Y\u00e9venes, L., Torres, D. and Toro, N., 2018. Leaching of chalcopyrite ore agglomerated with high chloride concentration and high curing periods. Hydrometallurgy, 181: 215\u2212220. Vogel, T. and Cislerova, M., 1988. On the reliability of unsaturated hydraulic conductivity calculated from the moisture retention curve. Transport in porous media, 3(1): 1\u221215. Wu, J., Ahn, J. and Lee, J., 2019. Copper Leaching From Low Grade Copper Ores Using Alternative Leaching Systems. Canadian Institute of Mining, Metallurgy and Petroleum, Copper 2019, Vancouver, British Columbia, 595448. Yanez, H., Ardiles, L. and Bender, J., 2019. Heap Leaching Improvements Using a New Leaching Aid Reagent, Paper presenrted at Hydroprocess 2019, Chile. Zhang, Y., Zhang, K., Dai, C., Zhou, X. and Si, H., 2014. An enhanced Fenton reaction catalyzed by natural heterogeneous pyrite for nitrobenzene degradation in an aqueous solution. Chemical Engineering Journal, 244: 438\u2212445. Zheng, C. and Bennett, G.D., 2002. Applied Contaminant Transport Modeling. Wiley. Zhu, T., 1992. The redox reaction between thiourea and ferric iron and catalysis of sulphide ores. Hydrometallurgy, 28(3): 381\u2212397. 165  Appendices Appendix A  ETU analysis on HPLC  Standards were made by adding ETu to 0.5 L DI water solution adjusted to a pH of 1.7 with H2SO4. The standards were analyzed and the areas of the peaks observed were recorded. For example, ETu in DI water (pH =1.7) produced the data in Table A.1. Table A.1 Area under peaks for ETu at 233 nm detection level  Concentration of ETu, mM Area under Peak  0 0 0.5 2000905 1.0 4007850 1.5 5803252 2 7429602   Figure A.1 Calibration curve for ETu in DI water   The calibration curve was used to generate concentration values for unknown ETu samples in DI water.  In the case of solutions with Fe3+ or Cu2+ , standards of ETu were prepared with the specific solution matrix and the ETu peaks were analyzed. As ETu decays in the presence of these solutes the solutions were prepared with the Fe3+ and Cu2+ concentrations and ETu added to them later. The time difference between ETu addition and analysis was restricted to less than 166  15 minutes and any decay in that time was considered negligible. An example of ETu area under the peak with a 1 g\/L [Fe3+] background is given in Table A.2. Table A.2 Area under peaks for ETu at 233 nm detection level in the presence of 1 g\/L Fe3+ Concentration of ETu, mM Area under Peak  0 0 0.25 1477437 0.5 2192300 1 3278928 1.5 4208776 2.0 5082297   Figure A.2 Calibration curve for ETu with [Fe3+] = 1.0 g\/L background pH = 1.7    167  Appendix B  MATLAB code written to solve the water model B.1 Main program close all;  clc; clear; twc = 1;   for twc_cnt=1:4 Z= .85 ; time = 12000; Z_diff = .05; R_diff =.01; R = .04;   theta_r = .028 ; theta_s= .3348 ;theta_0 = .028;  II=int8(Z\/Z_diff); II = (double(II))+1; JJ=(R\/R_diff)+1;  rho_b= (6.184\/(pi*(R^2)*Z));    % kg\/m^3  delta_tw =1; delta_tc = 15*delta_tw;  delta_z = Z\/II; delta_r = R\/JJ; Mod_tw = 1\/delta_tw; Mod_tc = 1\/delta_tc; s(II,JJ)= 0; % s_A(II,JJ)=0; K_w(II,JJ) = 0; theta(II,JJ)=0; v_wz(II,JJ)= 0; v_wr(II,JJ)=0; v_wz_in(JJ)=0; % t_vector1(time)=0; % v_wz_time(time)=0; sumv_z(II+1) = 0; beta_wz(II,JJ)=0; beta_wr(II,JJ)=0; gamma_w(II,JJ)=0;  [s,K_w,theta,v_wz,v_wr,sigma_A,sumv_z,beta_wz,beta_wr,gamma_w,v_wz_time,t_vector1] = initializetozero(II,JJ,time,delta_tw);    if twc_cnt == 1          waterflux = 11.9\/3600000;    elseif twc_cnt == 2 168            waterflux = 20.3\/3600000;   elseif twc_cnt == 3          waterflux = 27.4\/3600000;  elseif twc_cnt == 4          waterflux = 35.8\/3600000;   elseif twc_cnt == 5           else      end    ratio = delta_tc\/delta_tw; % react =0; cnt=1;  m_w = 0.644268; n_w = 2.811; h_c_0 = .12;     % initialize theta and conductivity for i = 1:II      for j = 1:JJ    theta(i,j)= theta_0;             K_w(i,j) =0.0163 ;                              sigma_A(i,j)=theta_0; %             c(II,JJ)=c_0;                              end end   for time1=1:delta_tw:time             %ponding criterion for j=1:JJ     v_wz_in(j) = (JJ ^ 2) * waterflux \/ (2 *j - 1);         for n = 1:j - 1             v_wz_in(j) = v_wz_in(j) - ((2 *n- 1) * K_w(1, n)) \/ (2 * j - 1);                   end          169          if v_wz_in(j) > K_w(1, j)             v_wz_in(j) = K_w(1, j);         end         if v_wz_in(j) < 0              v_wz_in(j) = 0;         end      end            %  compute beta and gamma [beta_wr,beta_wz,gamma_w,omega_w]=water_coeffecients(theta,theta_r,theta_s,s,v_wz_in,m_w,n_w,h_c_0,K_w,delta_z,delta_r,II,JJ);  theta = watermodel_solve(theta,beta_wz,beta_wr,gamma_w,omega_w,Mod_tw,II,JJ);  [v_wr,v_wz]=calculate_water_fluxes(gamma_w,beta_wr,beta_wz,theta,delta_z,delta_r,II,JJ);  sumv_z=integrate_Z_fluxes(v_wz,II,JJ);     t_vector1(cnt)=time1;    v_wz_time(cnt)=sum(sumv_z(II)).*(3600000);          cnt= cnt+1; end     %    figure(twc_cnt+1);    Z1=theta(1:II,2:JJ); Z2=[fliplr(theta),Z1]; XX=(0:Z_diff:Z); YY=(-R:R_diff:R); [XX,YY]=meshgrid(YY,XX);  figure(1);  plot(t_vector1,v_wz_time,'LineWidth',1);  xlabel('Time in seconds'); ylabel('Sum of all z water fluxes at the Bottom'); title('Water flux vs Time');  hold on; grid on;          twc_cnt =twc_cnt+1;     170  B.2 Initializing Water Model Coefficients function [beta_wr,beta_wz,gamma_w,omega_w] = water_coeffecients(theta,theta_r,theta_s,s,v_wz_in,m_w,n_w,h_c_0,K_w,delta_z,delta_r,II,JJ) Mod_Z = 1\/delta_z; Mod_R = 1\/delta_r; Mod_ZZ = Mod_Z\/delta_z; Mod_RR = Mod_R\/delta_r; omega_w(II,JJ)=0; gamma_w(II,JJ)=0; beta_wz(II,JJ)=0; beta_wr(II,JJ)=0; S_e(II,JJ)=0; U_w(II,JJ)=0; D_w(II,JJ)=0;  for i = 1:II      for j = 1:JJ    S_e(i,j) = (theta(i,j) - theta_r) \/ (theta_s - theta_r); %Effective Saturation             U_w(i, j) = K_w(i, j) * krel_w(S_e(i,j), m_w) \/ theta(i, j);             D_w(i, j) = K_w(i, j) * h_c_0 * Krwhdc(S_e(i,j), m_w, n_w) \/ (theta_s - theta_r);                                         end end   for j = 1:JJ        for i = 1:II             gamma_w(i, j) = Mod_Z * U_w(i, j);             omega_w(i, j) = 0.018015 * s(i, j);         end         omega_w(1, j) = omega_w(1, j) + double(Mod_Z * v_wz_in(j));          end     for j = 1:JJ %         beta_wz(1, j) = 0;         for i = 1:II-1             beta_wz(i, j) = Mod_ZZ * 0.5 * (D_w(i, j) + D_w(i + 1, j));         end         beta_wz(II, j) = 0;     end      for i = 1:II %         beta_wr(i, 1) = 0;         for j = 1:JJ-1             beta_wr(i, j) = Mod_RR * 0.5 * (D_w(i, j) + D_w(i, j + 1));         end         beta_wr(i, JJ) = 0;     end           171  end  B.3 Water Model Solving function val=watermodel_solve(theta,beta_wz,beta_wr,gamma_w,omega_w,Mod_tw,II,JJ) BL(II*JJ)=0; BM(II*JJ)=0; BR(II*JJ)=0; RV(II*JJ)=0; SV(II*JJ)=0; delta_theta(II,JJ)=0;  for j=1:JJ     for i=1:II         ctr = (j-1)*II + i;         if j==1             if i==1                 BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j)+ beta_wz(i,j);                BR(ctr) = -beta_wz(i,j);                RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(-(gamma_w(i,j)+ beta_wz(i,j)) * theta(i,j) + beta_wz(i,j) * theta(i + 1,j));                RV(ctr) =RV(ctr)+ jfac(j) * (-j* beta_wr(i,j)* theta(i,j) + j* beta_wr(i,j) * theta(i,j + 1));                RV(ctr)= Mod_tw*RV(ctr);              elseif i == II                BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j);                BR(ctr) = -beta_wz(i,j);                RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(gamma_w(i - 1,j) + beta_wz(i - 1,j))*theta(i - 1,j) - (gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j)) * theta(i,j);                RV(ctr) =RV(ctr)+ jfac(j) * (-j* beta_wr(i,j)* theta(i,j) + j* beta_wr(i,j) * theta(i,j + 1));                RV(ctr)= Mod_tw*RV(ctr);              else                   BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j);                BR(ctr) = -beta_wz(i,j);                RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(gamma_w(i - 1,j) + beta_wz(i - 1,j))*theta(i - 1,j) - (gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j)) * theta(i,j) + beta_wz(i,j) * theta(i + 1,j);                RV(ctr) =RV(ctr)+ jfac(j) * ( - j*beta_wr(i,j) * theta(i,j) + j* beta_wr(i,j) * theta(i,j + 1));                RV(ctr)= Mod_tw*RV(ctr);              end         elseif j == JJ              if i==1                 BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j) + beta_wz(i,j);                BR(ctr) = -beta_wz(i,j); 172                 RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(-(gamma_w(i,j)+ beta_wz(i,j)) * theta(i,j) + beta_wz(i,j) * theta(i + 1,j));                RV(ctr) =RV(ctr)+ jfac(j) * ((j-1) * beta_wr(i,j - 1) * theta(i,j - 1) - ((j-1) * beta_wr(i,j - 1) +(j) * beta_wr(i,j)) * theta(i,j));                RV(ctr)= Mod_tw*RV(ctr);              elseif i == II                BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j);                BR(ctr) = -beta_wz(i,j);                RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(gamma_w(i - 1,j) + beta_wz(i - 1,j))*theta(i - 1,j) - (gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j)) * theta(i,j) ;                RV(ctr) =RV(ctr)+ jfac(j) * ((j-1) * beta_wr(i,j - 1) * theta(i,j - 1) - ((j-1) * beta_wr(i,j - 1) +(j) * beta_wr(i,j)) * theta(i,j));                RV(ctr)= Mod_tw*RV(ctr);              else                   BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j);                BR(ctr) = -beta_wz(i,j);                RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(gamma_w(i - 1,j) + beta_wz(i - 1,j))*theta(i - 1,j) - (gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j)) * theta(i,j) + beta_wz(i,j) * theta(i + 1,j);                RV(ctr) =RV(ctr)+ jfac(j) * ((j-1) * beta_wr(i,j - 1) * theta(i,j - 1) - ((j-1) * beta_wr(i,j - 1) +(j) * beta_wr(i,j)) * theta(i,j));                RV(ctr)= Mod_tw*RV(ctr);              end         else              if i==1                 BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j)+ beta_wz(i,j);                BR(ctr) = -beta_wz(i,j);                RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(-(gamma_w(i,j) + beta_wz(i,j)) * theta(i,j) + beta_wz(i,j) * theta(i + 1,j));                RV(ctr) =RV(ctr)+ jfac(j) * ((j-1) * beta_wr(i,j - 1) * theta(i,j - 1) - ((j-1) * beta_wr(i,j - 1) +(j) * beta_wr(i,j)) * theta(i,j) + (j)* beta_wr(i,j) * theta(i,j + 1));                RV(ctr)= Mod_tw*RV(ctr);              elseif i == II                BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j);                BR(ctr) = -beta_wz(i,j);                RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(gamma_w(i - 1,j) + beta_wz(i - 1,j))*theta(i - 1,j) - (gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j)) * theta(i,j);                RV(ctr) =RV(ctr)+ jfac(j) * ((j-1) * beta_wr(i,j - 1) * theta(i,j - 1) - ((j-1) * beta_wr(i,j - 1) +(j) * beta_wr(i,j)) * theta(i,j) + (j)* beta_wr(i,j) * theta(i,j + 1));                RV(ctr)= Mod_tw*RV(ctr);              else                   BL(ctr) = -(gamma_w(i,j) + beta_wz(i,j));                        BM(ctr) = Mod_tw + gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j);                BR(ctr) = -beta_wz(i,j); 173                 RV(ctr) = omega_w(i,j);                RV(ctr) =RV(ctr) +(gamma_w(i - 1,j) + beta_wz(i - 1,j))*theta(i - 1,j) - (gamma_w(i,j) + beta_wz(i - 1,j) + beta_wz(i,j)) * theta(i,j) + beta_wz(i,j) * theta(i + 1,j);                RV(ctr) =RV(ctr)+ jfac(j) * ((j-1) * beta_wr(i,j - 1) * theta(i,j - 1) - ((j-1) * beta_wr(i,j - 1) +(j) * beta_wr(i,j)) * theta(i,j) + (j)* beta_wr(i,j) * theta(i,j + 1));                RV(ctr)= Mod_tw*RV(ctr);              end         end     end end  SV= tridiagonal(II*JJ,BL, BM, BR, RV);  for j = 1:JJ  for i = 1:II   ctr = (j - 1)*II + i;   delta_theta(i,j) = SV(ctr);     end end  for i=1:II     for j=1:JJ         ctr = (i-1)*JJ + j;             if j==1                 BL(ctr) = -jfac(j+1)*(j)*beta_wr(i,j);           BM(ctr) = Mod_tw + jfac(j)*j*beta_wr(i,j);           BR(ctr) = -jfac(j)*j*beta_wr(i,j);           RV(ctr) = delta_theta(i,j);             elseif j== JJ                 BL(ctr) = -jfac(j+1)*(j)*beta_wr(i,j);           BM(ctr) = Mod_tw + jfac(j)*((j - 1)*beta_wr(i,j - 1) + (j)*beta_wr(i,j));           BR(ctr) = -jfac(j)*(j)*beta_wr(i,j);           RV(ctr) = delta_theta(i,j);             else                BL(ctr) = -jfac(j+1)*(j)*beta_wr(i,j);          BM(ctr) = Mod_tw + jfac(j)*((j - 1)*beta_wr(i,j - 1) + (j)*beta_wr(i,j));          BR(ctr) = -jfac(j)*(j)*beta_wr(i,j);          RV(ctr) = delta_theta(i,j);             end      end end   SV = tridiagonal(II*JJ,BL, BM, BR, RV);  for i = 1:II  for j = 1:JJ   ctr = (i-1)*JJ + (j);   delta_theta(i,j)= SV(ctr);   theta(i,j) = theta(i,j) + delta_theta(i,j);     end 174  end val = theta; end  B.4  Initializing Solute Model Coefficients function [gamma_A_z,gamma_A_r,beta_A_zz,beta_A_zr,beta_A_rr,beta_A_rz,omega_A,omega_prime_A,sigma_A,sigma_A_old] ...     =solute_coeffecients(theta,c_in,sigma_A,sigma_A_old,s_A,s_A_prime,v_wz,v_wz_in,v_wr,a_w_slope,a_w_inter,a_w_ratio,D_A_0,e_w_coeff,e_w_exp,delta_z,delta_r,II,JJ,time1)  Mod_Z = 1\/delta_z; Mod_R = 1\/delta_r; Mod_ZZ = Mod_Z\/delta_z; Mod_RR = Mod_R\/delta_r; Mod_ZR = Mod_Z\/delta_r; D_A_zz(II,JJ)=0; D_A_rr(II,JJ)=0; D_A_zr(II,JJ)=0; gamma_A_z(II,JJ)=0; beta_A_zz(II,JJ) = 0; beta_A_zr(II,JJ) = 0; gamma_A_r(II,JJ)=0; beta_A_rr(II,JJ) = 0; beta_A_rz(II,JJ) = 0; omega_A(II,JJ)=0; omega_prime_A(II,JJ)=0; a1 = 1; vres(II,JJ)=0; vcos(II,JJ)=0; vsin(II,JJ)=0;   %    dispersion     for i = 1:II         for j = 1:JJ             if i ==1                  if j ==1                   vres(i,j) = 0.5 * ( v_wz(i, j)^ 2 + v_wr(i, j) ^ 2) ^ 0.5 ;                     if vres(i,j) == 0                        vcos(i,j) = 0;                       vsin(i,j) = 0;                     else                         vcos(i,j) = 0.5 * (v_wz(i, j)) \/ vres(i,j) ;                         vsin(i,j) = 0.5 * (v_wr(i, j)) \/ vres(i,j);                     end                   Lfac = a_w_slope * vres(i,j) + a_w_inter;                   Tfac = Lfac * a_w_ratio;                   Dfac = D_A_0 * e_w_coeff * theta(i, j) ^ e_w_exp;                   D_A_zz(i, j) = (vres(i,j) * (Lfac * vcos(i,j) ^ 2 + Tfac * vsin(i,j) ^ 2) + Dfac)*a1; 175                    D_A_rr(i, j) = (vres(i,j) * (Tfac * vcos(i,j) ^ 2 + Lfac * vsin(i,j) ^ 2) + Dfac)*a1;                   D_A_zr(i, j) = (vres(i,j) * (Lfac - Tfac) * vcos(i,j) * vsin(i,j))*a1;                   sigma_A_old(i,j) = sigma_A(i,j);                   sigma_A(i,j) = theta(i,j);                                     else                     vres(i,j) = 0.5 * ((v_wz(i, j)) ^ 2 + (v_wr(i, j - 1) + v_wr(i, j)) ^ 2) ^ 0.5 ;                     if vres(i,j) == 0                        vcos(i,j) = 0;                       vsin(i,j) = 0;                     else                         vcos(i,j) = 0.5 * (v_wz(i, j)) \/ vres(i,j) ;                         vsin(i,j) = 0.5 * (v_wr(i, j - 1) + v_wr(i, j)) \/ vres(i,j);                     end                   Lfac = a_w_slope * vres(i,j) + a_w_inter;                   Tfac = Lfac * a_w_ratio;                   Dfac = D_A_0 * e_w_coeff * theta(i, j) ^ e_w_exp;                   D_A_zz(i, j) = (vres(i,j) * (Lfac * vcos(i,j) ^ 2 + Tfac * vsin(i,j) ^ 2) + Dfac)*a1;                   D_A_rr(i, j) = (vres(i,j) * (Tfac * vcos(i,j) ^ 2 + Lfac * vsin(i,j) ^ 2) + Dfac)*a1;                   D_A_zr(i, j) = (vres(i,j) * (Lfac - Tfac) * vcos(i,j) * vsin(i,j))*a1;                   sigma_A_old(i,j) = sigma_A(i,j);                   sigma_A(i,j) = theta(i,j);                 end                              else                 if j ==1                   vres(i,j) = 0.5 * ((v_wz(i - 1, j) + v_wz(i, j)) ^ 2 + (v_wr(i, j)) ^ 2) ^ 0.5 ;                     if vres(i,j) == 0                        vcos(i,j) = 0;                       vsin(i,j) = 0;                     else                         vcos(i,j) = 0.5 * (v_wz(i - 1, j) + v_wz(i, j)) \/ vres(i,j) ;                         vsin(i,j) = 0.5 * (v_wr(i, j)) \/ vres(i,j);                     end                   Lfac = a_w_slope * vres(i,j) + a_w_inter;                   Tfac = Lfac * a_w_ratio;                   Dfac = D_A_0 * e_w_coeff * theta(i, j) ^ e_w_exp;                   D_A_zz(i, j) = (vres(i,j) * (Lfac * vcos(i,j) ^ 2 + Tfac * vsin(i,j) ^ 2) + Dfac)*a1;                   D_A_rr(i, j) = (vres(i,j) * (Tfac * vcos(i,j) ^ 2 + Lfac * vsin(i,j) ^ 2) + Dfac)*a1;                   D_A_zr(i, j) = (vres(i,j) * (Lfac - Tfac) * vcos(i,j) * vsin(i,j))*a1;                   sigma_A_old(i,j) = sigma_A(i,j);                   sigma_A(i,j) = theta(i,j);                 else                     vres(i,j) = 0.5 * ((v_wz(i - 1, j) + v_wz(i, j)) ^ 2 + (v_wr(i, j - 1) + v_wr(i, j)) ^ 2) ^ 0.5 ; 176                      if vres(i,j) == 0                        vcos(i,j) = 0;                       vsin(i,j) = 0;                     else                         vcos(i,j) = 0.5 * (v_wz(i - 1, j) + v_wz(i, j)) \/ vres(i,j) ;                         vsin(i,j) = 0.5 * (v_wr(i, j - 1) + v_wr(i, j)) \/ vres(i,j);                     end                   Lfac = a_w_slope * vres(i,j) + a_w_inter;                   Tfac = Lfac * a_w_ratio;                   Dfac = D_A_0 * e_w_coeff * theta(i, j) ^ e_w_exp;                   D_A_zz(i, j) = (vres(i,j) * (Lfac * vcos(i,j) ^ 2 + Tfac * vsin(i,j) ^ 2) + Dfac)*a1;                   D_A_rr(i, j) = (vres(i,j) * (Tfac * vcos(i,j) ^ 2 + Lfac * vsin(i,j) ^ 2) + Dfac)*a1;                   D_A_zr(i, j) = (vres(i,j) * (Lfac - Tfac) * vcos(i,j) * vsin(i,j))*a1;                   sigma_A_old(i,j) = sigma_A(i,j);                   sigma_A(i,j) = theta(i,j);                 end                                               end         end    end %    gamma Z    for j = 1:JJ %       gamma_A_z(1, j) = 0;         for i = 1:II             gamma_A_z(i, j) = Mod_Z * v_wz(i, j);         end       end     %   beta ZZ ZR            for j = 1:JJ %         beta_A_zz(1, j) = 0; %         beta_A_zr(1, j) = 0;         for i = 1:II-1             beta_A_zz(i, j) = Mod_ZZ * 0.5 *(D_A_zz(i, j) + D_A_zz(i + 1, j));             if beta_A_zz(i, j) == 0                  beta_A_zz(i, j) = -0.5 * (gamma_A_z(i, j) - abs(gamma_A_z(i, j)));             else                 beta_A_zz(i, j) = UPWIND(gamma_A_z(i, j) \/ beta_A_zz(i, j)) * beta_A_zz(i, j);             end             beta_A_zr(i, j) = Mod_ZR * 0.125 * (D_A_zr(i, j) + D_A_zr(i + 1, j));         end         beta_A_zz(II, j) = 0;         beta_A_zr(II, j) = 0;            end %   gamma R   for i = 1:II         for j = 1:JJ             gamma_A_r(i, j) = Mod_R * v_wr(i, j);         end  177    end  % beta RR and RZ     for i = 1:II %         beta_A_rr(i,1) = 0; %         beta_A_rz(i,1) = 0;         for j =1:JJ-1             beta_A_rr(i, j) = Mod_RR * 0.5 * (D_A_rr(i, j) + D_A_rr(i, j + 1));             if beta_A_rr(i, j) == 0                 beta_A_rr(i, j) = -0.5 * (gamma_A_r(i, j) - abs(gamma_A_r(i, j)));             else                 beta_A_rr(i, j) = UPWIND(gamma_A_r(i, j) \/ beta_A_rr(i, j)) * beta_A_rr(i, j);             end             beta_A_rz(i, j) = Mod_ZR * 0.125 * (D_A_zr(i, j) + D_A_zr(i, j + 1));         end         beta_A_rr(i,JJ) = 0;         beta_A_rz(i,JJ) = 0;     end        for i= 2:II     for j = 1:JJ            omega_A(i,j)=s_A(i,j);            omega_prime_A(i,j)=s_A_prime(i,j);            if ( (time1 >= 3600)) %                &&(time1 <=50400))            omega_A(1,j)=s_A(1,j)+Mod_Z*v_wz_in(j)*c_in;            else            omega_A(1,j)=0;                end            omega_prime_A(1,j)=s_A_prime(1,j);     end   end    end   B.5 Solute Model Solving function [c,delta_c] = solutemodel_solve(omega,omega_prime,sigma_A,sigma_A_old,kappa_prime,gamma_A_z,gamma_A_r,beta_A_zz,beta_A_zr,beta_A_rz,beta_A_rr,c,min_conc_val,Mod_t_c,II,JJ) BL(II*JJ)=0; BM(II*JJ)=0; BR(II*JJ)=0; RV(II*JJ)=0; SV(II*JJ)=0; delta_c(II,JJ)=0;   for j =1:JJ        for i = 1:II             n = (j - 1) * II + i;             if j == 1                if i ==1 178                   BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j)+ gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + ( - (gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j) + beta_A_zz(i, j) * c(i + 1, j));                  RV(n) = RV(n) + jfac(j) * ( - (j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j) + j * beta_A_rr(i, j) * c(i, j + 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (c(i, j + 1) + c(i + 1, j + 1));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (c(i + 1, j) + c(i + 1, j + 1));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                elseif i == II                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j) + beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + (gamma_A_z(i - 1, j) + beta_A_zz(i - 1, j)) * c(i - 1, j) - (beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j);                  RV(n) = RV(n) + jfac(j) * (- (j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j) + j * beta_A_rr(i, j) * c(i, j + 1));                  RV(n) = RV(n) - beta_A_zr(i - 1, j) * (c(i - 1, j + 1) + c(i, j + 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (c(i, j + 1));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (- c(i - 1, j) - c(i - 1, j + 1));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                else                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j) + beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + (gamma_A_z(i - 1, j) + beta_A_zz(i - 1, j)) * c(i - 1, j) - (beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j) + beta_A_zz(i, j) * c(i + 1, j);                  RV(n) = RV(n) + jfac(j) * (- (j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j) + j * beta_A_rr(i, j) * c(i, j + 1));                  RV(n) = RV(n) - beta_A_zr(i - 1, j) * (c(i - 1, j + 1) + c(i, j + 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (c(i, j + 1) + c(i + 1, j + 1));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (c(i + 1, j) + c(i + 1, j + 1) - c(i - 1, j) - c(i - 1, j + 1));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n); 179                 end             elseif j== JJ                if i ==1                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j)+ gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + ( - (gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j) + beta_A_zz(i, j) * c(i + 1, j));                  RV(n) = RV(n) + jfac(j) * ((j - 1) * (gamma_A_r(i, j - 1) + beta_A_rr(i, j - 1)) * c(i, j - 1) - ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j));                  RV(n) = RV(n) + beta_A_zr(i, j) * (- c(i, j - 1) - c(i + 1, j - 1));                  RV(n) = RV(n) - jfac(j) * (j - 1) * beta_A_rz(i, j - 1) * (c(i + 1, j - 1) + c(i + 1, j));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (c(i + 1, j));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                elseif i == II                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j) + beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + (gamma_A_z(i - 1, j) + beta_A_zz(i - 1, j)) * c(i - 1, j) - (beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j);                  RV(n) = RV(n) + jfac(j) * ((j - 1) * (gamma_A_r(i, j - 1) + beta_A_rr(i, j - 1)) * c(i, j - 1) - ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j) );                  RV(n) = RV(n) - beta_A_zr(i - 1, j) * (- c(i - 1, j - 1) - c(i, j - 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (- c(i, j - 1));                  RV(n) = RV(n) - jfac(j) * (j - 1) * beta_A_rz(i, j - 1) * (- c(i - 1, j - 1) - c(i - 1, j));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (- c(i - 1, j));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                else                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j) + beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + (gamma_A_z(i - 1, j) + beta_A_zz(i - 1, j)) * c(i - 1, j) - (beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j) + beta_A_zz(i, j) * c(i + 1, j); 180                   RV(n) = RV(n) + jfac(j) * ((j - 1) * (gamma_A_r(i, j - 1) + beta_A_rr(i, j - 1)) * c(i, j - 1) - ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j));                  RV(n) = RV(n) - beta_A_zr(i - 1, j) * (- c(i - 1, j - 1) - c(i, j - 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (- c(i, j - 1) - c(i + 1, j - 1));                  RV(n) = RV(n) - jfac(j) * (j - 1) * beta_A_rz(i, j - 1) * (c(i + 1, j - 1) + c(i + 1, j) - c(i - 1, j - 1) - c(i - 1, j));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (c(i + 1, j) - c(i - 1, j));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                end             else                if i ==1                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j)+ gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + ( - (gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j) + beta_A_zz(i, j) * c(i + 1, j));                  RV(n) = RV(n) + jfac(j) * ((j - 1) * (gamma_A_r(i, j - 1) + beta_A_rr(i, j - 1)) * c(i, j - 1) - ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j) + j * beta_A_rr(i, j) * c(i, j + 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (c(i, j + 1) + c(i + 1, j + 1) - c(i, j - 1) - c(i + 1, j - 1));                  RV(n) = RV(n) - jfac(j) * (j - 1) * beta_A_rz(i, j - 1) * (c(i + 1, j - 1) + c(i + 1, j));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (c(i + 1, j) + c(i + 1, j + 1));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                elseif i == II                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j) + beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + (gamma_A_z(i - 1, j) + beta_A_zz(i - 1, j)) * c(i - 1, j) - (beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j);                  RV(n) = RV(n) + jfac(j) * ((j - 1) * (gamma_A_r(i, j - 1) + beta_A_rr(i, j - 1)) * c(i, j - 1) - ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j) + j * beta_A_rr(i, j) * c(i, j + 1));                  RV(n) = RV(n) - beta_A_zr(i - 1, j) * (c(i - 1, j + 1) + c(i, j + 1) - c(i - 1, j - 1) - c(i, j - 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (c(i, j + 1) - c(i, j - 1));                  RV(n) = RV(n) - jfac(j) * (j - 1) * beta_A_rz(i, j - 1) * (- c(i - 1, j - 1) - c(i - 1, j));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (- c(i - 1, j) - c(i - 1, j + 1)); 181                   RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                else                  BL(n) = -(gamma_A_z(i, j) + beta_A_zz(i, j));                  BM(n) = Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j))- omega_prime(i, j) + beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j);                  BR(n) = -beta_A_zz(i, j);                  RV(n) = omega(i, j);                  RV(n) = RV(n) - Mod_t_c * (sigma_A(i, j) - sigma_A_old(i, j)) * c(i, j);                  RV(n) = RV(n) + (gamma_A_z(i - 1, j) + beta_A_zz(i - 1, j)) * c(i - 1, j) - (beta_A_zz(i - 1, j) + gamma_A_z(i, j) + beta_A_zz(i, j)) * c(i, j) + beta_A_zz(i, j) * c(i + 1, j);                  RV(n) = RV(n) + jfac(j) * ((j - 1) * (gamma_A_r(i, j - 1) + beta_A_rr(i, j - 1)) * c(i, j - 1) - ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j))) * c(i, j) + j * beta_A_rr(i, j) * c(i, j + 1));                  RV(n) = RV(n) - beta_A_zr(i - 1, j) * (c(i - 1, j + 1) + c(i, j + 1) - c(i - 1, j - 1) - c(i, j - 1));                  RV(n) = RV(n) + beta_A_zr(i, j) * (c(i, j + 1) + c(i + 1, j + 1) - c(i, j - 1) - c(i + 1, j - 1));                  RV(n) = RV(n) - jfac(j) * (j - 1) * beta_A_rz(i, j - 1) * (c(i + 1, j - 1) + c(i + 1, j) - c(i - 1, j - 1) - c(i - 1, j));                  RV(n) = RV(n) + jfac(j) * j * beta_A_rz(i, j) * (c(i + 1, j) + c(i + 1, j + 1) - c(i - 1, j) - c(i - 1, j + 1));                  RV(n) = (Mod_t_c * (sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j)) * RV(n);                end             end                     end   end      SV =tridiagonal(II*JJ,BL, BM, BR, RV);      for j = 1:JJ         for i = 1:II             n = (j - 1) * II + i;             delta_c(i, j) = SV(n);         end     end         for i = 1:II         for j = 1:JJ             n = (i - 1) * JJ + j;             if j == 1               BL(n) = -jfac(j + 1) * j * (gamma_A_r(i, j) + beta_A_rr(i, j));               BM(n) = Mod_t_c *(sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j) + jfac(j) * (j * (gamma_A_r(i, j) + beta_A_rr(i, j)));               BR(n) = -jfac(j) * j * beta_A_rr(i, j);               RV(n) = delta_c(i, j);             elseif j == JJ                                BL(n) = -jfac(j + 1) * j * (gamma_A_r(i, j) + beta_A_rr(i, j));               BM(n) = Mod_t_c *(sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j) + jfac(j) * ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j)));               BR(n) = -jfac(j) * j * beta_A_rr(i, j); 182                RV(n) = delta_c(i, j);                            else                                BL(n) = -jfac(j + 1) * j * (gamma_A_r(i, j) + beta_A_rr(i, j));               BM(n) = Mod_t_c *(sigma_A(i, j)+kappa_prime(i,j)) - omega_prime(i, j) + jfac(j) * ((j - 1) * beta_A_rr(i, j - 1) + j * (gamma_A_r(i, j) + beta_A_rr(i, j)));               BR(n) = -jfac(j) * j * beta_A_rr(i, j);               RV(n) = delta_c(i, j);             end                      end     end     SV = tridiagonal(II*JJ,BL, BM, BR, RV);      for i = 1:II         for j = 1:JJ             n = (i - 1) * JJ + j ;             delta_c(i, j) = SV(n);              c(i, j) = c(i, j) + delta_c(i, j);             if c(i, j) <= min_conc_val                 c(i, j) = min_conc_val;             elseif c(i,j) > 1                 c(i,j) = 1;             end         end     end end ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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