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The kinetics of zinc extraction in the di(2-ethylhexyl) phosphoric acid, n-heptane-zinc perchlorate,.. 1991

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THE KINETICS OF ZINC EXTRACTION IN THE DK2-ETHYLHEXYL) PHOSPHORIC ACID, n-HEPTANE - ZINC PERCHLORATE, PERCHLORIC ACID, WATER SYSTEM by D O N A L D WILLIAM JOHN MacLEAN B.A.Sc, The University of British Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Metals and Materials Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Apri l 1991 © Donald William John MacLean, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Metals and Materials Engineering The University of British Columbia Vancouver, Canada Date A p r i l 9. 1991 DE-6 (2/88) ABSTRACT The kinetics of zinc extraction from perchlorate solutions with di(2-ethylhexyl) phosphoric acid in n-heptane have been measured using the rotating diffusion cell technique. The extraction of zinc is controlled by the mass transfer of reactants (Zn 2 + and D2EHPA) to the interface. At low zinc concentrations, the system is controlled by the aqueous transport of Zn 2 + to the interface; at higher zinc concentrations transport of D2EHPA becomes rate controlling. For the range of D2EHPA concentrations examined, the transport of D2EHPA is rate controlling. Bulk p H has a negligible effect, except perhaps at the lowest p H values examined, where there may be a slight decrease in extraction rate. This decrease was attributed to less favourable thermodynamics at low interfacial p H values. It appears that the chemical reaction rate is fast enough that it has a negligible effect on the overall extraction rate. A basic mathematical model was developed which is adequate for predicting the extraction rate under variable conditions of zinc concentration, D2EHPA concentration, and pH. The effect of using a partially loaded organic extractant was also investigated, and the system was found to be mass transfer controlled. A n extended mathematical model was developed which predicts that the speciation of organic complexed zinc changes with increasing preload, and at high loadings the direction of Z n L 2 H L and Zn^-(HL) 2 flux reverses, with these species providing extractant to the interface. At very high loadings, Z n L 2 H L provides almost all the extractant to the interface. Experimental studies of the effect of temperature on the rate of zinc extraction resulted in a calculation of the activation energy which was consistent with a diffusion controlled mechanism. Finally, the effect of different filter pore sizes on extraction was examined. The extraction rate decreases significantly with a very small filter pore size, while there appeared to be little or no effect for larger filter pore sizes. For the filter pore size used in this study, it was therefore concluded that the filter pores do not pose an additional resistance to mass transfer. i i Table of Contents Abstract ii List of Tables vi List of Figures vii List of Symbols xiii Acknowledgements xv CHAPTER 1 - Introduction 1 CHAPTER 2 - Literature Review 3 2.1 Industrial Solvent Extraction Practice 3 2.1.1 Solvent Extraction Reagents 3 2.1.2 Industrial Solvent Extraction Circuits 4 2.1.2.1 Uranium 4 2.1.2.2 Copper 5 2.2 Di(2-ethylhexyl) Phosphoric Acid 6 2.2.1 Applications 7 2.2.2 Chemistry 8 2.2.3 Parameter Estimation 10 2.3 Solvent Extraction Kinetics 11 2.3.1 Interfacial Mechanisms 13 2.3.2 Mass Transfer Control 16 2.3.3 Mixed Regime (Mass Transfer with Chemical Reaction) 18 2.3.4 Other considerations 21 2.3.5 Summary 22 2.4 Kinetic Contactors 22 iii 2.4.1 Lewis Cell (constant interface area cells) 22 2.4.2 AKUFVE 24 2.4.3 Single Drop Cell (Moving Drop Cell) 24 2.4.4 Growing Drop Cell 25 2.4.5 Laminar Liquid Jet (Liquid Jet Recycle Reactor) 26 2.5 Rotating Diffusion Cell 27 2.5.1 Apparatus 27 2.5.2 Theory 28 2.5.3 Previous Work using the RDC 29 2.5.4 Summary 32 2.6 Zn-D2EHPA equilibrium and kinetics 32 2.6.1 Zn-D2EHPA equiUbrium 32 2.6.2 Zn-D2EHPA Kinetics 34 2.7 Supported Liquid Membranes 37 2.8 Summary 39 CHAPTER 3 - Experimental Methods 41 3.1 Reagents 41 3.1.1 D2EHPA Purification 41 3.1.2 Preparation of Preloaded Zinc-D2EHPA 42 3.2 Solution Analysis 43 3.3 Rotating Diffusion Cell Apparatus 44 3.4 Filter Preparation 45 3.5 Experimental Procedure 46 iv CHAPTER 4 - Results and Discussion 49 4.1 Initial Data from Test Runs 49 4.2 Basic Mathematical Model 54 4.3 Basic Model Predictions 61 4.3.1 Basic Model Verification 61 4.3.2 Basic Model Predictions 70 4.4 Extended Mathematical Model 79 4.5 Preload Results 84 4.5.1 Extended Model Verification 84 4.5.2 Extended Model Predictions 88 4.6 Variable Temperature 96 4.7 Filter Characterization 98 4.8 SLM Applications 100 CHAPTER 5 - Conclusions 101 CHAPTER 6 - Recommendations for Further Work 102 REFERENCES 103 APPENDIX A - Optical Tachometer 109 APPENDIX B - Data Acquisition Hardware and Software 112 APPENDIX C - Raw Experimental Data 121 APPENDIX D - Basic Mathematical Model 123 APPENDIX E - Extended Mathematical Model 130 v List of Tables Table 2.1 Physical and Chemical Properties of D2EHPA 9 Table 2.2 Diffusion Coefficients predicted by the Wilke-Chang Relationship 11 Table 2.3 Results of Patel's study of Ni, Co, Cu and Zn extraction 31 Table 2.4 Survey of Equilibrium Studies using Zinc and D2EHPA 33 Table 2.5 Survey of Kinetic Studies using Zinc and D2EHPA 34 Table 4.1 Experimental Conditions 50 Table 4.2 Membrane Filter Characteristics 98 Table B.l Pascal Listing of Data Acquisition Computer Program 113 Table D. l Pascal Listing of Basic Mathematical Model 123 Table E. l Pascal Listing of Extended Mathematical Model 130 vi List of Figures Figure 2.1 Flow Sheet of the DAPEX Process 4 Figure 2.2 Bluebird Mine Solvent Extraction and Electrowinning Flowsheet 6 Figure 2.3 Structural diagram of a D2EHPA dimer 8 Figure 2.4 Viscosity of solutions of the Znai)-HDEHP complex at 0.01M total Zn(H) in dodecane at 20 °C as a function of the excess HDEHP concentration 10 Figure 2.5 Schematic diagram of the reaction zone and aqueous/organic boundary layers 12 Figure 2.6 Concentration profile for an interfacial reaction mechanism 14 Figure 2.7 Interfacial Profile : Mass transfer limited reaction 16 Figure 2.8 Schematic diagram of the reaction zone and aqueous/organic boundary layers - Mass transfer with chemical reaction model 18 Figure 2.9 The original Lewis Cell 23 Figure 2.10 The moving drop cell 25 Figure 2.11 The growing drop cell 26 Figure 2.12 The inner portion of the Liquid-Jet Recycle Reactor 27 Figure 2.13 The Rotating Diffusion Cell 28 Figure 2.14 Fluid flow patterns near the Rotating Diffusion Cell 29 Figure 2.15 The extraction of Co, Ni, Cu, and Zn by D2EHPA 31 Figure 2.16 Extraction processes in a SLM 38 Figure 2.17 Axial and cross-sectional view of a hollow-fibre SLM module 38 Figure 3.1 Sample Gran Plots 44 Figure 3.2 The Rotating Diffusion Cell 45 Figure 3.3 A Sample RDC Plot showing lines from three different experiments : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um 47 vii Figure 4.1 Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, © = 100 rpm 52 Figure 4.2 Effect of changing D2EHPA concentration on zinc flux : [Zn] = 0.05 M , p H = 4.5, T = 25 "C, filter = 0.45um, © = 100 rpm 52 Figure 4.3 Effect of changing bulk pH on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 100 rpm 53 Figure 4.4 Effect of changing temperature on zinc flux: [Zn] = 0.05 M , Formal [D2EHPA] = 0.05 M , pH = 4.5, filter = 0.45pm, © = 100 rpm 53 Figure 4.5 Diagram of species and flux direction definitions for the VMR model 55 Figure 4.6 VMR predictions and experimental data for changes in bulk zinc concentration : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, © = 100 rpm 57 Figure 4.7 VMR predictions and experimental data for changes in bulk D2EHPA concentration: [Zn] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm 57 Figure 4.8 Basic program structure of the simple mathematical model 61 Figure 4.9 Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, © = 100 rpm 64 Figure 4.10 Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um, © = 300 rpm 64 Figure 4.11 Effect of changing D2EHPA concentration on zinc flux : [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm 65 Figure 4.12 Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05M, pH = 4.5, T = 25 'C, filter = 0.45pm, © = 300 rpm 65 Figure 4.13 Effect of changing bulk pH on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 100 rpm 66 Figure 4.14 Effect of changing bulk pH on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 300 rpm 66 Figure 4.15 Zinc flux vs. zinc concentration for changes in the organic species diffusion coefficients : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm 67 viii Figure 4.16 Zinc flux vs. D2EHPA concentration for changes in the organic species diffusion coefficients : [Zn] = 0.05M, pH = 4.5, T = 25 "C, filter = 0.45pm, co = 100 rpm 67 Figure 4.17 Zinc flux vs. zinc concentration for changes in the equilibrium constant : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 68 Figure 4.18 Zinc flux vs. D2EHPA concentration for changes in the equifibrium constant Ka: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 68 Figure 4.19 Zinc flux vs. zinc concentration for changes in the filter equivalent thickness L/cc: Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 69 Figure 4.20 Zinc flux vs. D2EHPA concentration for changes in the filter equivalent thickness L/cc: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 69 Figure 4.21 Predicted change in association factor (« a v g ) with bulk zinc concentration: Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25"C, filter = 0.45pm, co = 100 rpm 73 Figure 4.22 Predicted change in interfacial p H with bulk zinc concentration : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 73 Figure 4.23 Predicted change in interfacial zinc concentration with bulk zinc concentration: Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm 74 Figure 4.24 Predicted change in interfacial D2EHPA concentration with bulk zinc concentration: Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um, co = 100 rpm 74 Figure 4.25 Predicted change in association factor (navg) with bulk D2EHPA concentration: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 75 Figure 4.26 Predicted change in interfacial pH with bulk D2EHPA concentration: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 75 Figure 4.27 Predicted change in interfacial zinc concentration with bulk D2EHPA concentration : [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 76 ix Figure 4.28 Predicted change in interfacial D2EHPA concentration with bulk D2EHPA concentration: [Zn] = 0.05M, p H = 4.5, T = 25X1, filter = 0.45um, © = 100 rpm 76 Figure 4.29 Predicted change in association factor (navg) with bulk pH : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45um, © = 100 rpm 77 Figure 4.30 Predicted change in interfacial pH with bulk p H : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 100 rpm 77 Figure 4.31 Predicted change in interfacial zinc concentration with bulk p H : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45um, © = 100 rpm 78 Figure 4.32 Predicted change in interfacial D2EHPA concentration with bulk p H : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 100 rpm 78 Figure 4.33 Basic program structure of the extended mathematical model 83 Figure 4.34 Comparison of experimental data and extended model predictions for flux vs. preload for selected values of p\2/Pi3: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, © = 100 rpm 85 Figure 4.35 Effect of preload on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm, p\2/p1 3 = 6 x 10"5 86 Figure 4.36 Effect of preload on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25°C, filter = 0.45pm, © = 300 rpm, p 1 2 /p 1 3 = 6 x 10 s 86 Figure 4.37 Zinc flux vs. preload for changes in the organic species diffusion coefficients : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, © = 100 rpm, pVPis = 6 x 10 s 87 Figure 4.38 Zinc flux vs. preload for changes in the equilibrium constant : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm, (VPu = 6 x 10"5 87 Figure 4.39 Zinc flux vs. preload for changes in the filter equivalent thickness L / a : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm, pVPia = 6 x 10 s 88 x Figure 4.40 Predicted change in interfacial zinc concentration (aqueous) with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, p V P u = 6 x 10"s 91 Figure 4.41 Predicted change in interfacial p H with preload: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, pVPi3 = 6xl0- 5.... 91 Figure 4.42 Predicted change in bulk D2EHPA concentration with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 XL, filter = 0.45pm, co = 100 rpm, p V P i s = 6 x 10 s 92 Figure 4.43 Predicted change in interfacial D2EHPA concentration with preload: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, f W P u = 6 x 10 s 92 Figure 4.44 Predicted association factors vs. preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 X, filter = 0.45pm, co = 100 rpm, p\ 2/Pi3=6xlO- 5 93 Figure 4.45 Predicted fractions of zinc species vs. preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 XL, filter = 0.45pm, co = 100 rpm, p 1 2 /p 1 3 = 6xl0- 5 93 Figure 4.46 Predicted change in bulk zinc concentrations (organic) with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 XL, filter = 0.45pm, co = 100 rpm, p\ 2/p\ 3 = 6 x 10 s 94 Figure 4.47 Predicted change in interfacial zinc concentrations (organic) with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a ) = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, p V P u = 6 x 10 s 94 Figure 4.48 Predicted change in organic zinc concentrations with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, p V P u = 6 x 10 s 95 Figure 4.49 Predicted flux of organic species with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, P12/P13 = 6 x 10"5 95 Figure 4.50 VMR predictions and experimental data for changes in temperature: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , pH = 4.5, filter = 0.45pm, co = 100 rpm 97 xi Figure 4.51 Arrhenius Plot for experimental temperature data with linear regression fit: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , pH = 4.5, filter = 0.45pm, co = 100 rpm 97 Figure 4.52 Effect of changing filter pore size on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, co = 100 rpm 99 Figure 4.53 Effect of changing filter pore size on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, co = 300 rpm 99 Figure A . l The REXZ and the Optical RPM Sensor 109 Figure A.2 The Tachometer Circle 110 Figure A.3 Schematic Diagram of Optical Counter Signal Conditioner I l l Figure B.l Chart Feed pin-out specifications 112 Figure B.2 Major elements of the data acquisition program 113 xii List of Symbols c organic species concentration (kmol/m 3) C aqueous species concentration (kmol/m 3) D diffusion coefficient (m 2 / s ) F formal D2EHPA concentration (kmol/m 3) / species flux (kmol/m 2 /s) k mass transfer coefficient of specified species (m/s) concentration-based equilibrium constant ((kmol/m3)02""') KD dimerization constant ((kmol/m3)"1) L effective filter thickness (m) M solvent molecular weight (kg/kmol) m valence of metal cation n association factor (number of dimerized extractant molecules complexing the divalent metal) P Partition coefficient T temperature ( ° C ) VB molar volume of the solute at the normal boiling point (m 3/kmol) x solvent association factor ZDAH aqueous equivalent boundary layer thickness (m) zD/OTX organic equivalent boundary layer thickness (m ) Subscripts for c, C Z n 2 + aqueous zinc H L organic D2EHPA monomer (HL)2 organic D2EHPA dimer (HL)2,total total organic D2EHPA concentration, expressed as dimer xiii Znlq organic n=1 zinc-D2EHPA complex Znl^-HL organic »=1.5 zinc-D2EHPA complex ZnL2-(HL)2 organic n=2 zinc-D2EHPA complex Znl̂ CHLĴ tota, total organic zinc-extractant complex, all forms Zn,total total organic zinc, all forms Superscripts for c, C bulk bulk species i interfacial species Superscripts (overbar) property or species in the organic phase Greek Symbols a filter porosity P12 Znl^ metal-extractant complex formation constant Pl3 ZnLj-HL metal-extractant complex formation constant ((kmol/m3)"1) ZnLjCHDj metal-extractant complex formation constant ((kmol/m3)"2) 5aq aqueous diffusion boundary layer thickness 5 o r g organic diffusion boundary layer thickness 5r reaction zone thickness M- absolute viscosity (N-s/m 2) V kinematic viscosity (m 2 / s ) CO RDC angular velocity (radians/s) xiv Acknowledgements I would like to take this opportunity to thank some of the many special people who have helped me in this endeavour. Thanks to my research supervisor, Dr. David Dreisinger, who has not only been an academic guide but an encourager; constantly exhorting me to press on towards excellence, and leading by example. With much appreciation I acknowlege the many "rays of sunshine" that Joan, Merlin, Moyra, and Celeste have brought my way. You have brightened up more than a few "rainy" days. Thanks to Ken Scholey, Mike Lo, and Ish Grewal for your friendship and companionship; I know that I am richer for having known you, and I hope that you feel the same way. Thanks to Fred Rink, who has demonstrated to me in his life the true meaning of the word perseverance, and who has been a burden bearer and pray-er. And last, but very definitely not least, I would like to thank my parents, my sister Heather, and my aunts Win and Dot for their unconditional love and support through this difficult time. Thanks are also extended to the Cy and Emerald Keyes Foundation and the Natural Sciences and Engineering Research Council of Canada for their financial support. xv CHAPTER 1 - Introduction Solvent extraction is a metals purification process that has grown from its roots in analytical chemistry to become a major unit operation in industrial practice. The first large-scale use of solvent extraction processes was in the early 1940's, when it was used to produce super high-purity uranium by extracting uranyl nitrate with diethyl ether; this Manhattan Project production of uranium was used to produce both the and ^ P u that was used in the first atomic bombs. Since that time, the number of applications involving solvent extraction techniques has grown dramatically. Currently, metals that are produced (or purified) by solvent extraction techniques include, but are not limited to, copper, nickel, cobalt, uranium, tungsten, vanadium, chromium, niobium, tantalum, platinum group metals, and rare-earths. Copper is one of the highest volume metals processed by solvent extraction techniques; world production by solvent extraction has been estimated at 282,000 tpy (1984 figures).111 The rapid growth of solvent extraction has now slowed somewhat, with the industry entering a more mature phase of development. As a result, research is now being directed at more fundamental topics, with new discoveries more likely in the realm of incremental improvements rather than radical breakthroughs. Modern plant practice is still fraught with many problems. Poor mixing results in larger-than-necessary contactors to provide long residence times. A poorly mixed system can also result in excessive solvent entrainment in a third phase which does not easily disengage, resulting in high solvent losses and low settler capacity. Extractants can coat particulates carried over from earlier processing stages, resulting in a "crud" formation which interferes with phase separation.121 Finally, since solvent extraction circuits run at low temperatures (unlike pyrometallurgical systems), acceleration of slow kinetics is important in ensuring the economical operation of a solvent extraction plant. Research is therefore being focused on the above as well as other areas. Chemists are constantly searching for new reagents which will extract metals faster, with greater selectivity, and at a lower overall cost. The use of a higher priced extractant can often be justified if it has a lower solubility in the aqueous phase (and therefore lower losses) or a higher rate of extraction (reducing equipment capital costs). New applications for solvent extraction technology are 1 constantly being explored. For example, research is under way on techniques for extracting impurities from copper refinery electrolytes. Strong extractants which remove heavy metals from acidic waste waters are currently being marketed. Efforts are also being expended to develop new types of contactors which operate more efficiently or with lower capital costs than present designs. One example is a supported liquid membrane (SLM), where an extractant-impregnated membrane is contacted with a feed solution on one side and a strip solution on the other side. Metal ions in the feed react with the organic in the membrane, the metal-organic complex diffuses through the membrane, and the metal is then transferred to the strip solution on the opposite side. Such systems look very promising for the treatment of very dilute metal-containing waste streams. The objective of this thesis is to study the kinetics of zinc extraction from perchlorate solutions by di(2-ethylhexyl) phosphoric acid (D2EHPA) in heptane. This particular topic was selected to increase our understanding of the rate controlling processes at or near the aqueous/organic interface. The results of this work should contribute to 1) the design of extractants with superior metal extraction kinetics, 2) the optimization of current solvent extraction processes using D2EHPA, and 3) the development of supported liquid membrane contactors. This work is divided into six major sections. Chapter 2 consists of a summary of a literature search of solvent extraction kinetics with a critical discussion of present theory and practice. Chapter 3 describes the experimental procedures used in the current study, and Chapter 4 is a discussion of the experimental results. The results are summarized in Chapter 5, and recommendations for further work are presented in Chapter 6. 2 CHAPTER 2 - Literature Review 2.1 Industrial Solvent Extraction Practice This introduction to industrial solvent extraction practice will consist of two sections. First, the types of solvent extraction reagents and their general properties will be discussed. Two examples of industrial solvent extraction circuits will then be given. 2.1.1 Solvent Extraction Reagents There are three general types of solvent extractants: acid, basic, and solvating.121 Acid extractants can be further divided into two subcategories: acidic extractants and acid chelating extractants. Acidic extractants exchange a hydrogen ion for a metal cation; their reactions are thus pH dependent. The alkylphosphoric acids (including di(2-ethylhexyl) phosphoric acid, or D2EHPA) are the most commonly used members of this group, and are widely used for extracting many metals including uranium, cobalt/nickel, and rare-earths. Acid chelating extractants typically form bidentate (2-membered) complexes with the target metal cation; examples are the 8-hydroxyquinoUne and hydroxyoxime derivatives. The LIX series of reagents, including LIX63 and LIX65N, are hydroxyoximes. They are primarily used for copper extraction, and have enjoyed great commercial success. Basic extractants are amines or quaternary ammonium salts which extract anionic metal species. In the case of primary, secondary, and tertiary amines, the organic extractant molecule is first protonated forming an amine salt which is then able to anion exchange with the target anionic metal species. The resulting metal-amine is then transferred to the organic phase. An example of a basic extractant currently used in industrial practice is Alamine 336, a tertiary amine which is used for uranium, cobalt, tungsten, and vanadium extraction. Solvating extractants extract neutral complexes from the aqueous phase. The most common donor groups are oxygen bonded to carbon, or oxygen or sulphur bonded to phosphorous. A solvating extractant widely used in industrial practice is tributylphosphate 3 (TBP), which is an oxygen-phosphorous bond extractant. This particular extractant is commonly used in nuclear metallurgy where it is used to refine uranium and reprocess spent nuclear reactor fuel.11,21 2.1.2 Industrial Solvent Extraction Circuits 2.1.2.1 Uranium The DAPEX (Di-AlkylPhosphoric acid Extraction) process is used to concentrate and purify uranium extracted from ore leach solutions. In this process, extraction is achieved by using both D2EHPA and TBP (there is a synergistic effect). The uranium feed is clarified and injected into the extraction system - 5 stages are typically used. The loaded organic solution is stripped using a 15% N a 2 C 0 3 solution, forming a uranium tricarbonate complex and D2EHPA sodium salt. In the extraction stages, acid is added to replenish the acid lost due to the presence of the D2EHPA sodium salt. The purified aqueous solution is filtered to remove any iron or titanium precipitated during stripping. Uranium is recovered after solution neutralization by adding peroxide and precipitating uranyl peroxide, which is then dried to produce yellow cake (Na 2U0 4-2H 20). [ 1 1 A typical process flowsheet is shown in Figure 2.1. Feed 7 g/t U 3 0 8 CLARIFIER HDEHP 4% TBP 4% EXTRACTION 5 STAGES H 2 0 N a 2 C Q 3 STRIPPING 3 STAGES Tailing NaDEHP 4% TBP 4% 1 N H 3 H 2 0 2 1 1 FILTER PRECIPITATION • YELLOW C A K E Figure 2.1 Row sheet of the DAPEX process (after Alegret111) 4 2.1.2.2 Copper The most common copper solvent extraction system is found in a leach- extraction-electrowinning system. In this process, low-grade oxidized copper ores are first leached with sulphuric acid. The dilute leach solution is then processed through a solvent extraction circuit, where the solution is purified and the copper concentration increases to a level which is practical for electrowinning. The first copper plant to use this process was the Ranchers Exploration and Development Corporation Bluebird plant at Miami, Arizona, which started operation in 1968; a process flowsheet is shown in Figure 2.2. The copper ore is sequentially leached in 9 ponds, with a total leaching time of -135 days. The solution is then filtered, heated, and sent to the solvent extraction plant where the copper is extracted by a 9.5% LIX64N solution in a Napoleum 470 diluent. Three mixer settlers are used in a countercurrent configuration with an organic/aqueous ratio of 2.5:1, resulting in a throughput of -22000 1/min. The copper feed concentration is 1.8 to 2.4 gpl Cu; the raffinate contains -250 ppm Cu and is returned to the leaching ponds. The loaded organic is stripped in two mixer settlers; the strip solution contains -140 gpl H 2 S0 4 and -30 gpl Cu, and the final copper solution sent for electrowinning contains -34 gpl Cu. The electrowinning plant produces -18.2 tpd Cu. 1 1 , 3 1 Since the Ranchers plant was built, many other copper solvent extraction plants have been constructed. As of 1984, the largest plant constructed is the Nchanga plant in Zambia, which operates an acid leach of mine tailings. This plant produces 182 tpd Cu using four solvent extraction streams, one using SME529 and the other three using LD(64N.m 5 FROM HEAPS TO HEAPS A - LEACH LIQUOR STORAGE POND B - PUMP BOXES C - 100,000 GAL. TANK D - ACID STORAGE TANK E - HEAT EXCHANGER F - Cu EXTRACTION G - Cu STRIP H - FLOTATION I " Cu ELECTROWINNING Figure 2.2 Bluebird Mine Solvent Extraction and Electrowinning Flowsheet (after Ritcey and Ashbrook131) 2.2 Di(2-ethylhexyl) Phosphoric A c i d EH(2-ethylhexyl) phosphoric acid (hereafter referred to as D2EHPA in the text and as H L in formulae) is an alkylphosphoric acid that has become widely used in solvent extraction practice. First used in 1949, this general-purpose extractant is used for extracting a wide variety of metals, including uranium, cobalt/nickel, and rare-earths. It has adequate kinetics, moderate separation capability, low aqueous solubility, and is chemically stable.111 Its wide availability and low cost keep it popular when compared to other reagents. 6 2.2.1 Applications As mentioned above, D2EHPA is used extensively in nuclear applications, particularly the DAPEX process. Although it is used to extract many other metals, this discussion will focus exclusively on the extraction of zinc with D2EHPA. The traditional pyrometallurgical/hydrometallurgical methods for the primary production of zinc from zinc ores (and more advanced hydrometallurgical methods such as the zinc pressure leach) are already quite efficient and well established. Therefore, there is little need for the use of solvent extraction technology in primary metal production. However, solvent extraction is a particularly effective unit process for the recovery and concentration of zinc from dilute aqueous process streams, and several processes have been developed. A significant impetus for the further development and implementation of these processes is developing as a result of legislation requiring either proper waste disposal or metal recovery.'41 The Zincex process12,5'71 uses two solvent extraction circuits to recover zinc from pyrite cinder leach solutions. An amine extractant is used in the first circuit to produce a purified zinc chloride solution; D2EHPA is then used in the second circuit to extract zinc. The zinc is then stripped with spent sulphuric acid electrolyte and recovered in a conventional electrowinning plant. Even after passing through the first circuit, some iron remains in the purified solution; it is removed by bleeding of some of the D2EHPA and treating it with strong HC1. A Zincex plant producing 22 tpd of 99.99% pure zinc is currently in operation at Bilbao, Spain; the value of recovered zinc pays the plant operating costs. The METSEP process18,91 was developed to recover zinc and iron from galvanizing pickling solutions. Zinc is removed using an ion exchange column; the column is then eluted with HC1 and the zinc is extracted with D2EHPA. Zinc is stripped using sulphuric acid, and then electrowon. The iron is removed from the pickling effluent by pyrohydrolysis, which recovers Fe as iron oxide. HC1 is then recovered in an absorber along with the HC1 from the SX circuit, and recycled. A plant was in operation in South Africa, but was closed due to poor process economics.191 7 Finally, the Valberg process17'9,101 is used to extract zinc from rayon manufacturing waste waters. A plant in Sweden uses D2EHPA in kerosene in a two-step countercurrent process to reduce the zinc concentration from ~0.2 gpl to less than 2 ppm. Sulphuric acid is used to strip the zinc from the organic solution; the resulting 80 gpl zinc solution is recycled back to the rayon spinning bath. Although process installation was required by environmental legislation, the value of recovered zinc pays all operating costs. 2.2.2 Chemistry The basic physiochemical properties of D2EHPA are outlined in Table 2.1. It is a weak acid, and is relatively insoluble in water. The reaction between a metal cation and the D2EHPA dimer can be expressed by the generalized formula: M m + + nWJi -> ML m (HL)^_ m ) + mtT [2-1] where M represents the metal being extracted, L is the extractant (D2EHPA), m is the valence of the metal cation, n represents the number of dimerized extractant molecules participating in the reaction, and a bar indicates a compound in the organic phase. The D2EHPA molecule usually exists as a dimer in nonpolar media (i.e. most aromatic and v R O • • • OH R aliphatic solvents'111), and is monomelic in highly \ ^ \ / P P polar media (i.e. alcohols, carboxylic acids, and • / \ ^ \ water).'121 The D2EHPA dimer consists of two R OH • • • O R D2EHPA monomers, joined by hydrogen bonds Figure 2.3 Structural diagram of a between adjacent P=0 and P-OH groups, as shown D2EHP A dimer in Figure 2.3. The non-ideal behaviour of D2EHPA at high extractant concentrations was investigated by Danesi and Vandegrift'131, who examined results from equihbrium studies of Eu 3*, Tm 3*, and Ca 2 + extraction by D2EHPA in n-dodecane. They concluded that the D2EHPA dimer activity coefficient yD is fitted fairly well by the expression: logyD = 0.83 - ^ l " where F is the formal D2EHPA concentration. [2-2] 8 Table 2.1 Physical and Chemical Properties of D2EHPA chemical name: Bis(2-ethylhexyl) phosphoric acid CAS registry number: [298-07-7] formula: (C 8 H ] 7 0) 2 POOH* structure: (monomer) CH^CH^aCHCHsO 0 1 \ s C 2 H 5 P / \ CH 3(CH 2) 3CHCH 20 OH C 2 H 5 molecular weight 322 g/mole (monomer) * specific gravity (20/20 °C) 0.977 ± 0.003* viscosity (cps/25°C) 35* Uquid molar volume, V B 422.9 mVkmol (monomer/ 845.8 mVkmol (dimer)+ 1.003 x 10"9 m2/sec (dimer)* pit, 0.1M(H,Na)NO3/heptane 0.1M(H,Na)ClO4/hexane 0.0001-lM(H,Na)ClO4/octane 1.49 Komasawa et al. [14] 1.30 Komasawa et al. [14] 1.36 to 1.44 Ul'yanov et al. [15] distribution constant (H,Na)C104/octane 0.1M(H,Na)NO3/heptane log = 3.44 Ul'yanov et al. [15] log K,, = 3.20 Komasawa et al. [14] dimerization constant (H,Na)C104/octane 0.1 M(H,Na)N03/heptane (H,Na)C104/Isopar-H log K D = 4.47 Ul'yanov et al. [15] log K D = 4.50 Komasawa et al. [14] log K D = 4.70 Sastre et al. [1] 'source: manufacturer's information sheet Calculated by the method of Le Bas as reported in Perry1 Calculated by the Wilke-Chang relationship1171 9 The extracted metal-D2EHPA complex can have a variable number of additional D2EHPA molecules associated with it. At high metal loadings, the metal:D2EHPA ratio approaches the theoretical limit of 1:2; as this Umit is approached the mixture polymerizes, resulting in an increase in viscosity. For zinc/D2EHPA in dodecane, there is a threefold increase in viscosity as the loading increases from 75% to 100%, while for cobalt/D2EHPA in dodecane, there is a fourfold increase in viscosity as the loading increases from -85% to 100%; Figure 2.4 shows this change in viscosity for Zn-D2EHPA complexes. Evidence suggests that the polymers formed at high metal loadings can be very large, perhaps even approaching molecules with thousands of monomelic elements.1181 1 1 1 1 1— 1 0.00 001 0O2 0.03 O04 F HDEHP Figure 2.4 Viscosity of solutions of the ZnflD-HDEHP complex at 0.01M total Zn(II) in dodecane at 20 °C as a function of the excess HDEFEP concentration (after Kolarik & Grimm1181) 2.2.3 Parameter Estimation The Wilke-Chang1171 relationship, shown in equation [2-3], was used to estimate the diffusion coefficients of the various organic species. 10 D\i = 7AxlQ-12(xMfs [2-3] T ~ V°B6 Dreisinger1191 has previously shown that this correlation is acceptable for predicting diffusion coefficients - a predicted value for the diffusion coefficient of HEHEHP (mono 2-ethylhexyl phosphonic acid mono 2-ethylhexyl ester) in heptane was within 2% of the measured value. The diffusion coefficients calculated using equation [2-3] are shown in Table 2.2. Data 1 1 6 , 2 0 1 used in the calculations are: absolute viscosity, p(heptane) = 0.386 cP; solvent association factor, x(heptane) = 1.0; solvent molecular weight, M(heptane) = 100.21; and temperature, T = 298 K. The molar volume of the solute at the normal boiling point, V B , was estimated by using the method of Le Bas1161; these values for different organic species are also shown in Table 2.2. Table 2.2 Diffusion Coefficients predicted by the Wilke-Chang Relationship. Species D (heptane) ( n ^ k m o l 1 ) ( m V ) H L 422.9 1.520 x 10"9 (HL)2 845.8 1.003 x 10"9 Znl^ 858.4 0.993 x 10'9 ZnLz-HL 1281.7 0.781 x 10"9 ZnL2-(HL)2 1704.2 0.659 x 10'9 2.3 Solvent Extraction Kinetics In pyrometallurgical systems, thermodynamics and mass transfer are the primary considerations when determining the feasibility of a process; systems are generally near equiUbrium since chemical reaction kinetics are fast at elevated temperature. However, in hydrometallurgical systems reactions occur at much lower temperatures and chemical reaction kinetics can become very important. It follows that the study and evaluation of the different 11 reaction mechanisms are important to consider if hydrometallurgical systems are to be applied effectively and economically in industrial practice. Danesi and Chiarizia121' have written an excellent review of solvent extraction kinetics, and it is not the purpose of this section to repeat their conclusions. Instead, there will be a brief introduction to the subject of solvent extraction kinetics, followed by a discussion of a few reaction models which have been proposed by researchers. The rate controlling step in solvent extraction systems can be mass transfer, chemical reaction, or both. In most solvent extraction contactors, the aqueous and organic phases are well mixed, and therefore any resistance to mass transfer generally occurs in the diffusion layers between the bulk (stirred) phases and the reaction zone. If the extraction of a metal ion by D2EHPA occurs according to the reaction stoichiometry given in Equation 2-1, then a simplified diagram of the region near the reaction zone can be drawn as shown in Figure 2.5. Extractant diffuses through the organic boundary layer to the reaction zone, where it reacts with the metal species which has diffused through the aqueous boundary layer. The product species then diffuse back through the organic and aqueous boundary layers and into the bulk phases. Interfacial Reaction Zone MLm(HL) (2n-m) mH + Organic Bulk Aqueous Bulk n(HL)2 M m+ 6, o r g <5 aq Figure 2.5 Schematic diagram of the reaction zone and aqueous/organic boundary layers 12 In some systems, the rate of diffusion of reactants to, and products from, the reaction zone determines the rate of reaction; such a system is said to be operating in the "diffusional regime", or "mass transfer controlled with instantaneous chemical reaction." Such systems tend to be either those that are not well agitated (i.e. the diffusion layer thickness is large) or those in which one or more reactants have small concentrations. In contrast, systems in which some chemical step is very slow are said to be in the "kinetic regime", where the contribution to slow reaction rates by mass transfer resistances can be ignored. In such systems, either the thickness of the diffusion layer approaches zero, or the diffusion coefficient is large enough relative to the rate constant that there are no significant diffusional gradients between the bulk phases and the reaction zone.1211 Most solvent extraction systems operate neither in the purely diffusional regime nor in the purely kinetic regime; instead, they operate in a mixed regime where both mass transfer and chemical reaction kinetics are significant. Many experimenters have worked on the problem of accurately characterizing the mechanisms and methods by which these reactions occur. In spite of all their work, the field of solvent extraction kinetics is still largely empirical, with a large body of isolated (and sometimes conflicting) data having being gathered on many systems using many types of apparatus. The determination of the rate controlling step for most solvent extraction systems is further complicated since the observed rate controlling step is often influenced by the measurement technique. Finally, some researchers have shown some bias in explaining the mechanisms of extraction; chemists tend to assume that rates are governed by purely kinetic mechanisms, while engineers tend to assume that mass transfer is dominant. The following discussion on kinetic models will focus on several relevant mechanisms with examples from the literature. 2.3.1 Interfacial Mechanisms If we consider the case where the system is operating in the kinetic regime; that is, there is effectively no mass transfer resistance, then one possibility is an interfacial mechanism. Although there is still a stagnant boundary layer on either side of the interface, there is no concentration gradient of either reactants or products through the boundary layer since the mass transfer rate is much greater than the chemical reaction rate, i.e. [ ] i n t = [ ]b u l k. A reactant/product concentration profile can then be drawn as shown in Figure 2.6. 13 [HR] bulk Organic Bulk [MR 2] bulk 0 /org laq [HI bulk [M2+] bulk Aqueous Bulk Figure 2.6 Concentration profile for an interfacial reaction mechanism Ajawin et al.1 m investigated the extraction of zinc by D2EHPA from sulphuric acid solutions. Using a constant interfacial area contactor, the mixing speed was increased until a plateau region was reached where the reaction rate did not change with increasing speed (only to a point - very high mixing rates perturb the interface and change the interfacial area). In this plateau region, they concluded that the reaction rate was controlled by a slow chemical reaction. It was found that the rate of reaction was proportional to the interfacial contact area and independent of solution volume, thus indicating that extraction occurs at the interface. By varying the aqueous zinc concentration, pH, and extractant concentration, it was shown that the extraction of zinc is first order with respect to zinc and D2EHPA, and inverse first order with respect to p H (hydrogen ion concentration). It was proposed that the reaction occurs in several steps. In the first step, dimeric D2EHPA ionizes and dissociates as follows: ionization ( H R ) 2 <_> HR" + iT (0 dissociation (HR)2 <-> 2HR (ii) The species HR2" and HR then rapidly adsorb at the interface, and react with zinc ions to form an extractable zinc-D2EHPA compound through the following two reactions: Zn 2 + + HRj <-> ZnR2 + FT (iii) ZnR2 + HR <-> ZnR2 • HR (iv) 14 In the reaction scheme given above, only reaction (iii) agrees with experimental findings, that is, first order with respect to Z n 2 + and D2EHPA, and inverse first order with respect to H + . They therefore concluded that reaction (iii) is the rate controlling step. While the analysis that Ajawin et al. have made with respect to the rate controlling mechanism may be correct, Hughes and Rod 1 2 3 1 have shown that it is possible that the system may be operating in a "pseudokinetic" regime. They demonstrated that the flattening of the rate-rpm curve in the plateau region may be due to the attainment of a minimum value of the diffusion layer thickness rather than the chemical reaction becoming rate controlling. Further, Danesi et al.[24] have demonstrated that diffusion coupled with a fast chemical reaction at the interface can mimic a multi-step interfacial process. It is therefore possible that the results found in Ajawin's study can be explained by mass transfer processes rather than a multi-step interfacial reaction. Other authors have postulated interfacial mechanisms to explain the results of various kinetic studies. Roddy et al.[2S] studied the extraction of Fe3* from acid perchlorate solutions using D2EHPA in n-octane. They concluded that the extractable species FeR3-3HR is formed by a multi-step reaction which occurs at the interface. They also demonstrated that reaction rates could be increased by introducing species which replace water as a solvating ligand and allow faster ligand exchange. Komasawa and Otake'261 concluded that the extraction of Co, Cu, and Ni from nitrate media with D2EHPA proceeded by a two-step interfacial mechanism. The effects of aliphatic (heptane) and aromatic (benzene, toluene) diluents on the overall extraction rate were studied, and it was found that the extraction rate was much higher when an aliphatic diluent was employed. However, diluent type had little effect on the stripping rate, indicating that the enhancement of extraction was due to a change in the forward reaction rate constant. Albery et al.l27] found that the extraction of copper from sulphate media by Acorga P50 (an oxime extractant) in n-heptane was governed by an interfacial reaction. At low reagent concentrations, extraction was first order for both copper and extractant; at higher concentrations the reaction approached a Umiting rate which suggests a saturated interface. 15 Danesi and Vandegrift1281 examined the extraction of Eu 3* and Am3* from a chloride media using D2EHPA in n-dodecane. They attempted to fit their data to two models: a pure interfacial model assuming no diffusional resistances and a mass transfer model coupled to a fast chemical reaction. Both models gave a qualitative fit, but the kinetic model yielded the best quantitative fit. The combined body of research suggests that interfacial mechanisms are generally dominant in the extraction of metal ions using alkylphosphoric acids. However, in those cases when reaction kinetics have been evaluated using constant interface stirred cells operating at low mixing rates, some doubt remains as to whether the results have been correctly interpreted. 2.3.2 Mass Transfer Control In the case where bulk phase mixing is complete but the mass transfer resistance is much greater than the kinetic resistance, then the system is mass transfer controlled. Since the chemical reaction is fast, it occurs instantaneously at or immediately adjacent to the interface. Diffusion is therefore the only important process, and the rate of reaction is determined solely by the rate of transport of reactants and products. Under these circumstances, the interfacial concentrations must be computed from the flux equations and chemical equilibria at the interface. A typical profile showing concentration gradients near the interface for a mass transfer Umited reaction is shown in Figure 2.7. Figure 2.7 Interfacial Profile : Mass transfer limited reaction 16 The extraction eqxuhbrium for a divalent metal reacting with an organic acid can be written as: 2 + — ^ [MR2],.[Hlf M 2 + + 2HR <—> MR 2 + 21? = — , , (i) [M2 +] ; [HR]2 If we assume steady-state conditions and no chemical reactions in the boundary layer, then the fluxes to and from the interface must obey equation (i), and therefore, JM>+ = ~2Jm = J m * = ~~2Jrt ^ Expressions can be written for the fluxes to and from the interface for each species by Fick's first law, i.e. V = ir"1^ {P^+u-rM 2!} aii) M2\aq Jm = I T 5 ^ {[HRJ.-tHRU} (iv) ° H R , o r « ^ = ^ ^ f ^ - ^ U (v) DJl If the diffusion coefficients, bulk concentrations, and boundary layer thicknesses are known, and if a value for K^x is available, then the mass transfer limited flux of species M 2 + can be found by substituting equations (ii) - (vi) into equation (i) and solving for / m 2 + : Heming et al.l29] found that the extraction of copper with LIX63 and LDC64N in chloroform from aqueous nitrate media was controlled by mass transfer in the organic phase. They determined that the extraction reaction occurred at the interface, and that while stirring in the aqueous phase had no effect, changes in the stirring rate of the organic phase caused a proportional shift in the rate of extraction. J * = 5 ^ {[HWrKT],} (vi) 17 Miyake et al. studied the extraction of copper (LT) and cobalt(II) by 2-ethylhexyl phosphonic acid mono-2-ethylhexyl ester. They found that the rate of extraction is limited by the transport of the metal-extractant complex from the interface to the organic bulk. 2.3.3 Mixed Regime (Mass Transfer with Chemical Reaction) The above two rate models can be combined to create the mass transfer with chemical reaction (MTWCR) model. In this model, it is assumed that the system operates in a mixed regime, with neither kinetic nor diffusional mechanisms dominant. The reactants diffuse from the well mixed bulk through the stagnant boundary layer to the reaction zone, where a slow chemical reaction occurs. Since the chemical reaction is slow, the assumption that the reaction occurs solely at the interface is no longer valid, and the reaction may occur in a zone which extends from the interface into the aqueous phase (and perhaps even into the aqueous bulk). I I !_I I I interface ^ ® z< z<* Figure 2.8 Schematic diagram of the reaction zone and aqueous/organic boundary layers - Mass transfer with chemical reaction model (after Hughes and Rod1311) Hughes and Rod 1 3 1 1 have developed a general model incorporating interfacial film diffusion and a chemical reaction in the aqueous phase which describes the extraction of metals in a mixed regime. Distribution of the extractant and the extracted metal complex between the organic and the aqueous phase is incorporated into the model through the partition coefficient (PHR and P^ respectively). For finite values of the partition coefficient and for mixed reaction conditions, extraction occurs in a reaction zone of thickness zr. 18 For the purposes of their model, Hughes and Rod suggested that the reaction occurs in five steps: HR -» HR step 1 HR -» H* + R~ step 2 M 2 + + R" MR + + R" 4 MR + MR, MR 2 -» MR 2 step 3 step 4 step 5 with the overall reaction expressed by the equation: M 2 + + 2HR <-> MR 2 + 2fT Ka = c^c^c^c^ Either step 3 or step 4 may be rate controlling. If step 3 is assumed to be rate controlling, then the flux of extractant can be given by the equation: 1 — (CHR4 — ^ H R C H R ) where (i) (ii) and C H R j C H R t, HR.org (iii) -MRj, i — MRj 2>t (iv) CUR — [ PMR/^HRCH ^HR ^PHRDMRJ(EXC M V P M R ^ H R ^ H 4P H R D M R j 7ST E X C M +- ^ E X C M ^ H R ^ M R J C H R , ! MRj-f HR y (v) 19 ( c ^ \ C« J J r P2 \ ^ M R j J H R "M,i - H 4 \ iCHR,i C H R ^ H R ) CH+T ( £ > H R V ^ H ^ H R J ( C H R , i C H R ^ * H R ) (Vi) (vii) (viii) Special cases of the model occur under the following circumstances: particular case h •f H R £ R £ * H R © J = 2 reaction in the film finite finite finite equiUbrium reaction in the film oo finite oo reaction at the interface oo oo finite instantaneous reaction at the interface oo oo oo Hughes and Rod extended the model and used it to analyze results reported in the literature for the LDC64N/CuS0 4/H 2S0 4 system using a rising drop and a constant interface (gauze cell) contactor. The extraction of Cu from slightly acidic media with a rotating diffusion cell was also analyzed. In each case the model was able to adequately fit the results, and the addition of the first ligand (step 3) was found to be rate controlling. Dreisinger and Cooper1 3 2 , 3 3 1 used a modified version of the MTWCR model to model the extraction of Co and Ni from sulphate solutions by HEHEHP (mono 2-ethylhexyl phosphonic acid mono 2-ethylhexyl ester) and Co from perchlorate solutions by D2EHPA. The modified MTWCR model was able to fit the experimental results. However, Ni extraction from perchlorate solutions by D2EHPA was so slow that a significant portion of the extractant partitioned into the aqueous phase as either ionized extractant anion (L) or as a N i H x L c + x y ) + complex, and the model was unable to accommodate these conditions. The extraction of zinc from perchlorate solutions by D2EHPA was so fast that the reaction was completely mass transfer controlled; thus, the model could not be applied. 20 2.3.4 Other considerations A mixed or kinetic regime may exist if the rate of chemical reaction is slow. Different metals have different ligand exchange rates and thus reaction rates in systems with identical extractant, diluent, and aqueous media but with different metals can vary substantially. For example, if a comparison is made between the water exchange rate constants'" for zinc, cobalt, and nickel, Jtj(Zn2+) = 2.5 x 107 s'1 Jti(Co2+) = 2.6 x 10s s 1 rCi(Ni2+) = 1.3xl0 4 s 1 it can be seen that there is a difference of over three orders of magnitude.1341 Since there is a general relationship between the rate of complex formation from aqueous ions and the water exchange rate, a qualitative comparison of metal chemical reaction rates may be made.1341 Thus, a system extracting zinc might be mass transfer controlled, while under the same conditions nickel might be extracted by a MTWCR mechanism. The solubility of extractant in aqueous solutions affects not only solvent losses to the raffinate, but can also determine the location of the chemical reaction. Since water is a polar solvent, molecules which have a polar component will tend to be more soluble. Increasing the extractant carbon chain length and chain branching tends to decrease solubility. Increasing salt content in the aqueous phase has the effect of decreasing reagent solubility, while increasing temperature and pH increases solubility. High solubility can result in a significant extractant concentration in the aqueous phase, and thus metal extraction may occur in the aqueous bulk.131 The effect of pH on the aqueous solubility of acidic extractants must be considered when attempting to determine the reaction location. Since acidic extractants dissociate in water according to the equihbrium: 1 Water exchange rates are a measure of the speed at which bulk water molecules replace water molecules in the metal ion coordination shell. 21 decreasing the pH can substantially decrease the aqueous phase solubility of the extractant and therefore move the location of the chemical reaction from the aqueous bulk (or aqueous film) to the interface. 2.3.5 Summary In an alkylphosphoric acid extraction system, all three regimes can be found under different conditions. If the aqueous phase metal concentration is high and the region near the interface is well mixed (i.e. the boundary layers are thin), then it is likely that the system is operating in the kinetic regime and an interfacial mechanism will be dominant. If the same system is operated with low metal concentrations or poor mixing near the interface (but with adequate bulk phase mixing), then the system may operate in the diffusional regime with a mass transfer mechanism dominant. Under these conditions, although the chemical reaction rate may be finite, the rate of metal diffusion is much slower and therefore dominant. Extraction may occur in the mixed regime if the mass transfer and chemical reaction resistances are of the same order of magnitude. This particular situation is most likely to occur when mixing is poor (thick boundary layer) and chemical reactions are slow. 2.4 Kinet ic Contactors The earliest research on solvent extraction kinetics grew out of analytical chemistry methods for equiUbrium studies, and were little more than shakeout tests, informing the experimenter of the time it took for a particular system to reach equiUbrium. Since these early experiments, many researchers have developed different types of apparatus for the study of solvent extraction reaction kinetics. 2.4.1 Lewis Cell (constant interface area cells) The Lewis cell is a constant area kinetic contactor which has proved to be very popular for investigating the kinetic mechanisms of solvent extraction reactions.12,351 The contact area between the two immiscible phases is defined by the geometry of the reaction vessel, and each phase is separately stirred by an impeller. Various researchers have improved upon the basic design of the Lewis cell, generally either by introducing some sort of phase 22 separating media in the interface area, by adding baffles which modify phase mixing, or by attempting to improve the stirring in either (or both) phase.12,211 For the purposes of this discussion, the different Lewis cell variants including the Nitsch cell and the ARMOLLEX contactor will be grouped under a common heading. Problems with Lewis-type cells include the accumulation of surfactants at the organic/aqueous interface, the formation of an agitated, unstable interface, and scatter in collected data of ±30%. 1 3 6 1 Figure 2.9 The original Lewis Cell (after Hanna and Noble'371) The major difficulty in the use of the Lewis cell is the ehmination (or accurate characterization) of diffusion barriers. At low mixing speeds, the diffusion layer thickness is large, and mixing of the bulk solution is incomplete. At high mixing speeds, the aqueous/organic interface is disturbed and eddies form; this type of interface is extremely difficult to characterize, and thus results obtained under these conditions are invalid. A compromise must thus be obtained between poor mixing at low speeds and interface disruption at high speeds. Experimenters have attempted to find mixing speeds somewhere between these two extremes where satisfactory mixing is obtained and yet the interface is not disrupted.1381 Researchers working in this range claim to have removed all mass transfer limitations from the system, i.e. the rate of reaction is solely determined by chemical reaction kinetics. However, as Hughes and Rod 1 2 3 1 have demonstrated, it is more likely that they are operating in a pseudokinetic regime, where the diffusion layer thickness is small, but is still limited to a finite value which may have an effect on the reaction rate. This effect depends on the relative magnitudes of the diffusional resistances and kinetic resistances. 23 2.4.2 AKTJFVE The AKTJFVE apparatus is a highly stirred tank system which operates near equilibrium conditions.1391 The aqueous/organic media are continuous, with one phase completely dispersed in the other; this dispersion makes it impossible to define the surface area. A high stirring rate is used so that mixing is turbulent and complete, removing most mass transfer resistances. The aqueous and organic phases are continuously sampled to detenrvine concentration changes over time. The complete mixing found in the AKUFVE contactor makes it unsuitable for studying kinetic systems where the chemical rate constants are large and therefore reactions are primarily mass transfer limited. However, in systems which have slow reaction kinetics, the AKUFVE apparatus can be used to study the rate of chemical reaction. Since the interfacial contact area in the AKUFVE apparatus is unknown1361, the interfacial flux cannot be measured. Thus, reaction rates can only be qualitatively compared between metal systems; a quantitative evaluation of absolute reaction rates is not possible. However, using this technique it is possible to gain valuable information about the reaction orders of different extractant systems.1211 2.4.3 Single Drop Cell (Moving Drop Cell) In the single drop cell, drops of one phase are formed on the tip of a small capillary and are then allowed to fall (or rise, depending on relative densities) through the other continuous phase.1381 While the drop is travelling to the collector, metal extraction takes place through the drop surface. If the drops are assumed to be spherical, the contact area can be computed. After many drops have been reacted, the drop phase is collected and analyzed for metal content. Thus, the metal extraction rate and interfacial flux can be calculated. There are several difficulties with the single drop method. First, the drop may not be spherical. Also, internal drop circulation may be poor; that is, the bulk solution in the drop interior may not be well mixed.121 ,381 The drop may oscillate as it moves, and there may be a stagnant region in the drop's wake.1401 Finally, in fast systems some extraction may occur during drop formation and when the drop is in the coalesced stagnant pool at the end of the cell.1401 24 Figure 2.10 The moving drop cell (after Danesi and Chiarizia1 !) Left side: falling drop; Right side: rising drop 2.4.4 Growing Drop Cell In a typical mixer-settler type contactor, extraction generally occurs between drops of one phase dispersed in a continuous second phase. Single-phase drops are sheared away from large drop aggregates (drop formation), they travel through the second phase, and then coalesce back into drop aggregates. Since researchers believe that extraction occurring in the drop formation phase is a significant part of the total extraction occurring over the lifetime of the drop, the growing drop method was designed to study extraction occurring during drop formation.'411 As in the single drop cell, drops grow at the end of a small needle. After they detach, they rise a short distance and then are sucked into a collection system. When a sufficient number of drops have been created, the collected drops are analyzed for metal content. Drop formation times are typically on the order of 1 to 20 seconds, and about 100 to 200 drops are required for each data point. While the hydrodynamics around and inside the growing drop are complex1361, researchers claim to have been able to at least partially model the behaviour of the growing drop.1411 25 BURETTE SAMPLING S TUBE INTERFACE VACUUM o AQUEOUS NEEDLE ORGANIC Figure 2.11 The growing drop cell (after Hughes and Zhu1411) 2.4.5 Laminar Liquid Jet (Liquid Jet Recycle Reactor) In the liquid jet recycle reactor, developed by Freeman and Tavlarides1421, one phase flows as a jet through a continuous second phase. New surface is continuously created by the jet action, and the surface area is fairly well-defined. The circulating phases can be continuously monitored to determine concentration profiles, and the hydrodynamics of the jet are understood.12,361 However, the hydrodynamic conditions cannot be widely varied if jet stability is to be maintained. 26 Figure 2.12 The inner portion of the Liquid-Jet Recycle Reactor (after Hanna and Noble1371) 2.5 Rotating Diffusion Cell In order to evaluate solvent extraction kinetics in a practical way, the interfacial area must be known and the contribution of diffusion must either be understood and taken account of or be eh'minated. The rotating diffusion cell fulfills both of these criteria: the interfacial area can be measured and the thickness of the diffusion layers can be determined. 2.5.1 Apparatus The rotating diffusion cell (RDC) technique was first developed by Albery et fl/.1431 as a method for studying interfacial reaction kinetics. The RDC, which is based on the rotating disc electrode, consists of a permeable filter mounted on a rotating hollow cylinder (see Figure 2.13). The hydrodynamics on both sides of the filter are well-defined, with the action of the rotating disk producing a uniform equivalent diffusion boundary layer across the surface of the disk. Furthermore, the thickness of the diffusion layers can be calculated if the rotational speed, kinematic viscosity, and diffusivity are known. 27 Figure 2.13 The Rotating Diffusion Cell (after Albery et A / . ' 4 3 1 ) When adapted to solvent extraction kinetic study, one compartment of the RDC is filled with a metal-bearing aqueous solution, and the other compartment is filled with the extractant-containing organic solution. A thin (-O.lmrn) microporous filter of known area separates the two compartments. The speed of the rotating RDC cylinder is measured and a stationary baffle ensures that correct hydrodynamics are maintained in the inner compartment. 2.5.2 Theory • The rotating motion of the filter produces flow patterns which are well-defined according to rotating disk hydrodynamics. A cross-sectional view of the flow is shown in Figure 2.14a, and a view of the flow below the filter is shown in Figure 2.14b. Fluid in the inner cell is pulled down through the baffle to the surface of the filter, where it is radially thrown out in a circular motion. New fluid is pulled in through holes in the baffle - the gap between the baffle and the cylinder is sufficient to ensure no disruption in flow. On the underside of the filter, fluid circulates in a similar fashion. Levich1 4 5 1 solved the Navier-Stokes equations to give the velocity profiles for flow adjacent to a rotating disk The convective-diffusion equation may then be solved if it is assumed that concentration only depends on the distance from the disk (i.e. radial and tangential symmetry) and there are no chemical reactions in the diffusion layer. Once the mass flux has been determined, an equation may be developed which describes the thickness of the equivalent diffusion layer, i.e. 28 Membrane (a) Cross-section through the RDC (b) Flow below the filter Figure 2.14 Fluid flow patterns near the Rotating Diffusion Cell (after Patel1441) zD = 0.643 <&mvv6Dm [2-4] where z D is the equivalent diffusion layer thickness in m, co is the disk's angular velocity in radians/second, v is the kinematic viscosity in m 2/s, and D is the diffusion coefficient in m 2/s. If the interfacial flux is measured and plotted with respect to co~1/2, the intercept is equal to the flux at mfinite angular velocity. According to the Levich equation, at infinite velocity the thickness of the diffusion layer is equal to zero. At this point, the measured flux contains only the mass transfer resistances of diffusion through the filter and interfacial chemical reactions. The mass transfer resistance through the membrane may be calculated and subtracted, leaving only the interfacial chemical reaction. 2.5.3 Previous Work using the RDC The RDC has been used to study interfacial transfer mechanisms in pharmaceutical systems, since by impregnating the filter with certain organic compounds it can effectively simulate drug absorption through biological membranes, especially skin. 1 4 6 , 4 7 1 More recently, the RDC has been applied to hydrometallurgical systems by several researchers in an attempt to study solvent extraction kinetics. 29 Albery and Fisk1481 examined copper extraction and stripping with Acorga P50, and determined rate constants for both the forward and reverse reactions. Albery et a/.1271 placed a ring electrode on the RDC membrane surface in order to follow the rate of copper stripping. Using this method, the copper flux was determined by measuring the electrode current. Dreisinger and Cooper1321 studied the extraction of cobalt and nickel with HEHEHP and found that a simplified MTWCR model adequately fit their results. The fitted parameters in the model indicated that both systems were operating in the mixed regime. Dreisinger and Cooper1331 later expanded on their earlier work by examining zinc, cobalt and nickel extraction from perchlorate solutions using D2EHPA in heptane. The effect of metal concentration, extractant concentration, pH and temperature on the extraction rate were examined in an attempt to determine the rate controlling steps. As indicated earlier, they found that the MTWCR model was able to fit the data for cobalt extraction, but nickel extraction was too slow to be adequately modelled. The extraction of zinc was fast enough that mass transfer became the rate limiting step; at low zinc concentrations aqueous mass transfer was rate hmiting, while at higher zinc concentrations the rate limiting step became extractant transfer in the organic phase. Most recently, Dreisinger et al.m examined the extraction of cobalt and nickel with D2EHPA. A baseline study involving the extraction of cobalt from sulphate media was carried out, and then the coextraction of cobalt and nickel from perchlorate solution was examined. In addition, the effect of buffering interfacial pH with a weak acid was examined with cobalt extraction from perchlorate solution. Patel1441 examined the extraction of zinc, cobalt, copper, and nickel from sulphate solutions by either D2EHPA or D2EHDTPA in n-heptane using the RDC. He first conducted a series of experiments for all four metals, in which he varied the concentration of D2EHPA from 0.015M to 0.4M; the results of this experiment are shown in Figure 2.15. Under identical conditions, the reaction rate increases in the order Ni < Co < Cu < Zn. He then examined the effect of varying the metal concentration, pH, and D2EHPA concentration over a wider range for cobalt. Finally, he fitted all of his experimental results to a modified MTWCR model; estimated rate constants calculated by the model are given in Table 2.3. 30 [ M 2 > 1 0 m M , pH=4.5, T=25°C,CJ=3Hz [D2EHPA] , mol/dm 3 Figure 2.15 The extraction of Co, Ni, Cu and Zn by D2EHPA (after Patel1441) Table 2.3 Results of Patel's study of Ni, Co, Cu, and Zn extraction [44] Measured Flux Second Order Rate 8 r Metal (kmol m"2 s"1) Constant kR (urn) ( r ^ k m o r V 1 ) Nickel 9.674 x 10"9 1.325 x 107 8.02 Cobalt 2.110 xlO"8 9.460 x 107 3.56 Copper 4.936 xlO"8 1.213 x 108 1.39 Zinc 1.247 x 10"7 2.436 x 109 0.42 Using parameters calculated by the MTWCR model, Patel estimated the thickness of the reaction zone near the interface. Values of 8 r for each metal are shown in Table 2.3; the thickness of the RDC aqueous diffusion layer (8 a q) was estimated to be 32.59 um. For zinc, 5 r « 8 a q ; thus, as an approximation it can be assumed that the reaction occurs at the interface. 31 2.5.4 Summary A number of different techniques for the study of solvent extraction kinetics have been examined. There is some uncertainty in the nature of diffusion barriers in the Lewis cell, and the interfacial contact area in the highly stirred AKUFVE cell is unknown. The single drop technique is useful for studying the behaviour of individual drops, but there is some uncertainty surrounding the hydrodynamics of the drop and mixing in its interior. The growing drop also has complex hydrodynamics, although claims have been made that at least partial solutions have been obtained. Finally, the laminar liquid jet can only be operated under certain conditions if jet stability is to be maintained, limiting its usefulness for studying a wide range of systems. The rotating diffusion cell appears to be useful for the study of solvent extraction kinetics due to the well-defined interfacial area and its ability to ehminate diffusional contributions. Therefore, it was selected as the contactor of choice for this study. 2.6 Z n - D 2 E H P A equi l ibr ium and kinetics 2.6.1 Zn-D2EHPA equilibrium The stoichiometry of zinc extraction by D2EHPA can be expressed by an equation of the form: Zn 2 + + «(HL)2 -> ZnL2(HL)(2„_2) + 2KT [2-5] where the value of n varies depending on the nature of the organic solvent and the aqueous media. For example, Huang and Juang1501 have shown that the extracted species (in kerosene at low metal loadings) has one associated D2EHPA monomer (i.e. n=1.5). Li et al.1 5 1 1 have demonstrated that as the organic phase becomes more fully loaded, viscosity increases, suggesting the formation of metal-extractant polymers. The equilibria of the above reaction have been examined by different researchers17'8'22'5^581 under various experimental conditions. Their results are summarized in Table 2.4. In general, the composition of the extracted species was found to be ZnR 2 HR (i.e. n=l .5) for aliphatic diluents, and ZnR 2(HR) 2 (n=2) for aromatic diluents. Exceptions are the studies by Grimm and Kolarik1551 in N03/n-dodecane, Sato et al.[52] in Cl/kerosene, and 32 Table 2.4 Survey of Equilibrium Studies using Zinc and D2EHPA aqueous phase diluent composition of extracted species source (Na,H)S04 Shellsol T ZnR 2(HR) 2 N/A Rice and Smith [8] (Na,H)S04 n-heptane ZnR 2 HR 7.35 x10 3 Ajawin et al. [22] (Na,H)S04 n-heptane ZnR 2 HR 2.6 x 10"3 Ajawin et al. [84] (Na,H)S04 kerosene ZnR 2HR 9.5 x 10 3 Huang and Juang [50] (Na,H)CI kerosene ZnR 2(HR) 2 N/A Sato et al. [52] (Na,H)N03 n-hexane ZnR 2HR 8.0 x 10"2 Smelov et al. [53] (Na,H)N03 n-heptane ZnR 2HR 6.3 x 10" 2 Teramoto era/. [54] (Na,H)N03 n-octane ZnR 2 HR 8.5 x 10"2 Smelov et al. [53] (Na,H)N03 n-decane ZnR 2HR 9.0 x 10" 2 Smelov et al. [53] (Na,H)N03 n-dodecane ZnR 2HR &ZnR 2(HR) 2 N/A Grimm and Kolarik [55] (Na,H)N03 benzene ZnR 2(HR) 2 5.0 x 10" 2 Smelov er al. [56] (Na,H)CI04 lsopar-Hf ZnR 2 HR & ZnR 2(HR) 2 4.9 x 1 0 ' 2 7.6 x 10" 2 Sastre and Muhammed [57] (Na,H)CI04 Escaid 100* ZnR 2 HR 10.1 x 10"1 Li eta/. [51] Tlsopar-H is odourless aliphatic kerosene (ESSO). *Escaid 100 is an aliphatic diluent which contains 19wt% aromatics (ESSO). Sastre and Muhammed1571 in C104/Isopar-H. Sato et al. found that the dominant species was ZnR2-(HR)2, while Grimm and Kolarik and Sastre and Muhammed concluded that the extracted species was a mixture of ZnR 2 HR and ZnR 2(HR) 2. Grimm and Kolarik1551 found that the log D vs. log [(HL)2] plot curved as the concentration of extractant increased. Li et al.m suggested that this was due to the co-extraction of sodium (present to maintain the aqueous phase at constant ionic strength). If this is indeed the case, then a higher value of n would be measured, erroneously implying that the extracted species was mixed. Another possibility is non-ideal behaviour of D2EHPA at high extractant concentrations (c.f. Section 2.2.2). 33 Trends may be observed with respect to the effects of changes in diluent and aqueous phase on the value of the extraction equiUbrium constant, K^. From Table 2.4, Ka decreases with increasing tendency for complex formation in the aqueous phase (complexing ability is in the order C104 < N 0 3 < CI < SO4). Thus, the largest values for Ka are found with species which have little or no aqueous complexing ability (nitrate and perchlorate). Also, at moderate to high ionic strengths, the ionic environment can have a significant effect on the activities of aqueous species, changing K^. The data in Table 2.4 also suggest that aromatic diluents tend to lower K^. 2.6.2 Zn-D2EHPA Kinetics As implied earlier, the kinetics of zinc extraction by D2EHPA have been studied by many different researchers using many different types of kinetic contactors. Some relevant results are summarized in Table 2.5. Table 2.5 Survey of Kinetic Studies using Zinc and D2EHPA Aqueous / Diluent Contactor Rate Controlling Mechanism Source (Na,H)S04 / heptane Lewis Cell interfacial chemical reaction Ajawin et al. [7,22] (K,H)N03 / dodecane Lewis Cell interfacial reaction Cianetti & Danesi [59] (Na,H)S04 / heptane Lewis Cell mixed regimet Ajawin et al. [60] (Na,H)S04 / kerosene Lewis Cell interfacial 2-step chemical reaction Huang & Juang [61] (Na,H)S04 / heptane Rotating Diffusion Cell mass transfer with chemical reaction Patel [44] (Na,H)CI04 / Isopar-H* Lewis Cell interfacial Aparicio & Muhammed [62] HCI04 / heptane Rotating Diffusion Cell mass transfer Dreisinger & Cooper [33] ^ote: intentional selection of parameters to provide mixed regime. *lsopar-H is odourless aliphatic kerosene (ESSO). 34 Ajawin et al.' attempted to evaluate the reaction kinetics of zinc extraction from sulphate solutions by D2EHPA in heptane using a modified Nitsch cell (a variant of the Lewis cell). As indicated in Section 2.3.1, they found that the rate controlling mechanism was an interfacial chemical reaction. They also found that increasing the temperature or decreasing the aqueous ionic strength increased the rate of chemical reaction. An expression using the Debye-Huckel equation was derived to predict the rate of reaction under different conditions of ionic strength and temperature. Although this equation gave a good general fit to experimental results, Hughes and Zhu [ 4 1 1 have questioned its theoretical validity since Ajawin used the simple form of the Debye-Huckel equation which is valid only for low ionic strengths. Hughes and Zhu also suggested that at the high extractant concentrations used in Ajawin's experiments, the interface would be saturated. Cianetti and Danesi1591 studied the kinetics and reaction mechanism of zinc, cobalt, and nickel extraction from nitrate solutions by D2EHPA in n-dodecane using an ARMOLLEX cell (a Lewis Cell variant). After analyzing their experimental results in terms of both interfacial two-step chemical reaction and interfacial film diffusion mechanisms, they concluded that since the rate constants for reactions between either Zn 2 + , Co 2 + , or N i 2 + and D2EHPA were very similar, it was unlikely that a chemical reaction was rate controlling since the water exchange rate constants for these three metals are quite dissimilar. It was therefore proposed that the rate of reaction was diffusion controlled. However, Cianetti and Danesi suggested that, although macroscopic interfacial film diffusion could not be completely excluded, it was more likely that the rate controlling step was the microscopic diffusion of the solvated metal ion through a structured water layer at the aqueous/organic interface. Ajawin etal.m expanded on their earlier work by studying zinc extraction under mixed rate controlling conditions. The bulk aqueous zinc concentration was decreased so that the concentration gradient across the boundary layer would be much less, creating conditions where the rate of film diffusion was approximately equal to the chemical reaction rate. Using the reaction mechanisms found in their earlier paper1221, a simple model was developed which 35 incorporated both diffusion and chemical reaction terms, and the effect of stirring speed on the rate of reaction was examined. Both the aqueous and organic mass transfer coefficients were found to be directly proportional to the stirring speed. Huang and Juang1611 studied both the extraction and stripping of zinc in sulphate media by D2EHPA in a kerosene diluent. The constant interfacial area cell design used in their study appears to be rather simplistic, with no baffles or other modifications evident. This perhaps provides an explanation for the rather low mixing speeds used (the plateau region was between 90 and 120 rpm). They determined that the extraction reaction was interfacial and that increasing ionic strength caused a decrease in extraction rate at high sulphate concentrations. There was no corresponding effect during stripping. The initial extraction rate was predicted by the equation: R = ^[Zn ' i t t i^jtiri '^i+^tso 2 : ] 1 - 2 )" 1 where the rate constant, k, is equal to 3.21±0.20 x 10"7 (mol/dm)1^"1. The reaction sequence that they developed is the same as that developed by Ajawin1221 (c.f. Section 2.3.1), but instead they favoured reaction (iv) as the rate controlling step, i.e. ZnR2 + HR <—> ZnRjHR (iv) Patel1441 studied the extraction of Co, Ni, Cu, and Zn from sulphate solutions by D2EHPA in n-heptane using the rotating diffusion cell. He developed a MTWCR model which adequately described the extraction rate under different experimental conditions, although there were some discrepancies. In general, the model provided a fairly good fit under rate controlling conditions varying from almost completely chemical reaction controlled (Ni) to almost completely diffusion controlled (Zn). Aparicio and Muhammed1621 examined the extraction of zinc from perchlorate solutions by D2EHPA in Isopar-H. A modified Lewis cell with baffles and screens near the interface allowed fast mixing speeds (plateau region in the range 450 - 575 rpm), while an automated sampling system was used to track the variation in zinc concentration during the test run. They used an interfacial two-step chemical reaction, and incorporated the formation of both Z n R 2 H R and ZnR 2(HR) 2 by using a fast equihbrium at the interface. The results 36 reported in this study are questionable due to the very low zinc concentrations used (0.05 - 0.5mM), and it is entirely probable that, although cell mixing was rapid, the system may have been operating in the mixed regime. Finally, Dreisinger and Cooper1331 also studied the extraction kinetics of Zn, Co, and Ni from perchlorate solutions by D2EHPA in n-heptane using the rotating diffusion cell. They found that Zn extraction was controlled by interfacial film diffusion, and Co extraction could be modelled by a MTWCR model. However, as indicated in Section 2.3.3, Ni extraction was too slow to be modelled by either technique. To summarize, different experimenters have used different types of apparatus under different experimental conditions in an attempt to elucidate the extraction mechanism of zinc from aqueous solutions by D2EHPA. At this point, it appears that the extraction of zinc is very rapid, and, when chemical reaction is the rate hmiting step, it occurs as a multiple-step reaction either at or in a very narrow zone adjacent to the interface. However, due to the extremely fast ligand exchange rate of zinc, some experiments which have been interpreted as operating in the chemical control regime may have instead been operating in either a mixed or diffusional regime. 2.7 Supported Liquid Membranes Supported liquid membranes (SLM's) have been proposed as an alternative process for the extraction of metals from dilute process streams. An extractant impregnated support is placed between two aqueous solutions. Metal is extracted from the feed solution into the membrane, migrates through the membrane, and is stripped into the second solution. SLM technology appears attractive when compared to conventional hquid-liquid extraction methods: possible advantages include lower capital and operating costs, lower energy costs, and higher separation factors. Also, since the amount of extractant used is small and since it is continuously regenerated in the SLM, very expensive extractants optimized for the particular metal system may be used.'1,631 If we consider the case of metal extraction in a SLM system by an acidic extractant, then the reaction may be expressed by the following equilibria: feed side M* + HX (membrane) <—> MX (membrane) + I-T strip side 37 A general diagram of the transport and reaction processes in a SLM are shown in Figure 2.16. As shown in the figure, the extraction proceeds in four steps: metal extraction from the feed solution into the SLM, transport of the metal-extractant complex through the membrane, stripping of the metal into the strip solution and the corresponding regeneration of the extractant, and diffusion of the extractant back through the membrane to the feed side. An example of a simple hollow-fibre SLM module is shown in Figure 2.17. SLM STRIP FLUX H" FLUXM + Figure 2.16 Extraction processes in a SLM (after Tavlarides et al.m) Stripping Solution Metal Ions Out .A Membrane Extractant Impregnated FeLd Solution Figure 2.17 Axial and cross-sectional view of a hollow-fibre SLM module Since in a SLM contactor the feed and strip phases are not particularly well agitated (they flow slowly past the membrane), the boundary layers are likely to be thick and therefore mass transfer may play a significant role in the overall reaction kinetics. In order to understand and effectively model the SLM system, it is therefore necessary to have a thorough understanding of not only the chemical reaction kinetics, but also mass transfer kinetics and transport processes through the membrane. Fernandez et al[M\ Teramoto et al.l54\ and Huang and Juang1651 have investigated the transport of zinc through a D2EHPA impregnated SLM. 38 Investigating zinc extraction from a sulphate system with a kerosene diluent, Huang and Juang1651 found that the rate controlling step was in most cases membrane diffusion. However, under certain circumstances (a combination of lower [M 2 +] f e e d , higher [(HR)2] tota l and lower [H+] f ced) aqueous film diffusion became either significant or dominant. Qualitatively, this is reasonable since when the metal concentration is low the concentration gradient is smaller so the total diffusion flux is less, while higher membrane extractant concentrations decrease the membrane resistance, which increases the possibility that aqueous film diffusion may have an influence on the reaction rate. The work of Teramoto et a/.1541 examined the extraction of zinc in a spiral SLM under industrial-type conditions. Experiments were performed with a n-dodecane diluent, a 0.7 mol/dm 3 (HL)2 concentration, an input pH of 5.6, and an input zinc concentration of 100 ppm. A long-term (1 month) test was performed to evaluate the behaviour of the SLM contactor with time. Output zinc concentration was ~1 ppm, with some degradation in membrane performance after 32 days. The zinc concentration in the recirculated strip solution eventually reached a concentration of 40 gpl, at which point it was replaced; this value represents a concentration factor of -40,000. Regeneration of the membrane was accomplished by passing the extractant phase and the strip solution through the strip side of the membrane - performance was restored without interrupting continuous operation. It was concluded that the rate determining step was zinc diffusion in the aqueous feed solution. 2.8 Summary The understanding of reaction mechanisms and chemical reaction kinetics is essential for efficient process design and accurate modelling of existing industrial processes. In particular, the emerging technology of supported liquid membranes appears promising for the extraction of metals from dilute solution. Initial attempts have been made to investigate the extraction of zinc in an SLM system with D2EHPA. One particular method for evaluating solvent extraction kinetics which is particularly well suited for this purpose is the rotating diffusion cell. The RDC operates under conditions 39 which are similar to SLM operating conditions, with the filter in the RDC in some circumstances simulating the SLM membrane. Thus, investigations of basic kinetic mechanisms by the RDC technique may be most helpful in measuring the rate controlling steps in SLM contactors. 40 CHAPTER 3 - Experimental Methods 3.1 Reagents All solutions were prepared from reagent grade chemicals with the exception of D2EHPA, which was purified by the technique outlined below (Section 3.1.1). Zinc, cobalt, and nickel aqueous solutions were prepared by dissolving the required amount of the perchlorate salt in deionized water. The NaOH solution used for pH stabilization during the RDC run was prepared by dissolving the required amount of NaOH in deionized water. The solution was then standardized (see Section 3.2). D2EHPA/heptane stock solution was prepared by adding a weighted amount of purified D2EHPA to heptane. These solutions were not titrated. 3.1.1 D2EHPA Purification The D2EHPA supplied by Albright & Wilson was contaminated with small amounts of mono- and tri-(2-ethylhexyl) phosphoric acid and contained trace amounts of metals such as iron. Since the objective of this research project was to investigate the extraction kinetics of D2EHPA, the removal of impurities was necessary to ensure that the results obtained were correct. It was therefore purified by a modified version of Partridge and Jensen's copper salt method.1661 Copper hydroxide ( Cu(OH) 2 ) was prepared by adding a concentrated sodium hydroxide solution to a strong copper sulphate solution. The Cu(OH)2/water slurry produced was quite gelatinous, and filtration had to be done in several steps. The filter cake was washed several times, however, the final cake still had a significant water content. The Cu(OH) 2 filter cake was then added to a 1:1 mixture of impure D2EHPA and toluene. Since Cu (as hydroxide) was in excess, the result was an organic phase which was fully loaded with copper (i.e. all the free D2EHPA had combined with Cu to form Cul^). The three-phase mixture was placed in a separatory funnel and allowed to settle overnight, and the final 41 product was a solution containing three zones - a fully loaded D2EHPA-organic zone, a mixed organic/aqueous zone, and an aqueous zone containing suspended Cu(OH)2 particles. The organic solution was decanted and filtered with PS (phase separator) paper to remove any entrained water. The mixed organic/aqueous solution was centrifuged, and the organic phase was recovered and filtered. Acetone was then added to the Cu-D2EHPA/toluene mixture, precipitating Cu-D2EHPA salt. The solution was filtered, and the Cu-D2EHPA precipitate was washed with further acetone and then recovered. It was then re-dissolved in fresh toluene, and precipitated a second time with acetone. The final product was a virtually pure Cu-D2EHPA salt, with most of the impurities remaining in the toluene. The pure Cu-D2EHPA salt was contacted twice with strong sulphuric acid, extracting the Cu into the aqueous phase as CuSCv Trace sulphuric acid was removed from the organic by repeated washing. The purified D2EHPA was then decanted and PS-paper filtered. The final product was colourless, with no trace of the yellowish tinge that characterizes impure D2EHPA. The yield was -55 percent, and a Gran Plot (see Section 3.2) gave a purity of 97.9% (the balance was assumed to be solvated water). 3.1.2 Preparation of Preloaded Zinc-D2EHPA A bulk solution containing 0.205M D2EHPA in heptane was prepared for use in the preloading experiments. A first attempt to load the D2EHPA with zinc was made by contacting the D2EHPA-heptane solution with aqueous zinc perchlorate solutions. An addition of NaOH was required to maintain a high pH, however, this NaOH addition resulted in some sort of emulsion formation. After several attempts, this method was abandoned in favour of another, in which the D2EHPA-heptane solution was directly contacted with a stoichiometric amount of solid ZnO. The advantage of this method was that no NaOH addition was required to maintain a high pH, since the reaction proceeded according to the equation: ZnO + n(HL)2 -> Z n l ^ H L ) ^ ^ + H 2 0 [3-1] 42 producing a very small amount of water. After stirring the D2EHPA-heptane/ZnO mixture on low heat for approximately 1/2 hour, it was filtered with PS-paper and the partially loaded Zn-D2EHPA solution was recovered. 3.2 Solution Analysis As mentioned above, the NaOH solution was standardized by using potassium hydrogen phthalate (KHP) as a primary standard. Standardization was accomplished by titrating weighed amounts of KHP (dried for 3-4 hours at > 100 °C) with the unknown NaOH solution. Phenolphthalein was used as an indicator. The initial and final aqueous solutions for each RDC run were analyzed for zinc (and, where appropriate, for either nickel or cobalt) by diluting the unknown to a metal concentration appropriate for atomic absorption spectrophotometry. Each metal was analyzed for on its strongest characteristic wavelength: 213.9,232.0, and 240.7 nm for Zn, Ni, and Co, respectively. The final organic solution from each RDC run was stripped by three contacts with O.5MH2SO4. The resulting strip solution was then analyzed by atomic absorption spectrophotometry as described above. The initial zinc content of the preloaded Zn-D2EHP A strip solutions was also determined by this method. The Gran Plot method1671 was used to determine D2EHPA content, both to check the purity of the purified D2EHPA, and also to verify the loading of the preloaded Zn-D2EHPA strip solution. This method is effective since D2EHPA is a weak acid. An aliquot of the unknown organic sample was mixed with 2-propanol, and then titrated with a standardized 0.1M NaOH/75% 2-propanol solution. When the titration was near the equivalence point, the titrant was added dropwise, and the change in p H was recorded. The plot of 1/delta pH vs. Number of Drops has an inflection at the equivalence point, yielding the exact volume of titrant which was required to completely neutralize the D2EHPA. This type of plot is called a Gran plot; two sample plots are shown in Figures 3.1a and 3.2a. The D2EHPA concentration in the unknown solution could then be determined. 43 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Number of Drops Number of Drops (a) Gran Plot of purified D2EHPA (b) Gran plot of 60% preloaded D2EHPA Figure 3.1 Sample Gran Plots 3.3 Rotating Diffusion C e l l Apparatus A diagram of the RDC apparatus used in this study is shown in Figure 3.2. The RDC was mounted on a lab stand anchored at each end to prevent vibration. The RDC was rotated by a pulley connected to a variable speed motor, and the rotational speed of the RDC was measured by an optical sensor (see Appendix A for details). The thermostatted beaker below the RDC was filled with the aqueous solution, while the inner compartment of the cell was filled with the organic solution. A lid with holes for the RDC, a p H probe, a gas purge, and titrant addition was attached to the beaker; nitrogen was injected into the aqueous chamber above the solution level to prevent atmospheric C 0 2 from reacting with the aqueous solution and changing the pH. The beaker assembly was mounted on a labjack so that the height of the beaker could be adjusted so that at all times the net flux due to hydrostatic pressure differences through the filter would be equal to zero. A constant temperature circulator was used to pump water through the double-walled beaker, maintaining the aqueous phase (and organic phase by conduction) at the desired temperature. An autotitration system consisting of a Radiometer PHM82 pH meter, ABU80 autoburette, and TTT80 titrator was used to keep the p H at a constant level during a RDC run. A p H electrode was inserted into the outer compartment through the hole in the lid, and the system was preset at the desired pH. The volume of titrant dispensed was recorded by connecting the chart recorder 44 1. Hollow Mounting Shaft Thermostatted Beaker Figure 3.2 - The Rotating Diffusion Cell Acrylic Filter Mount Filter output of the autoburette to a Data Translation DT2805 analogs-digital converter which was attached to an IBM PC-XT. Existing data acquisition software was considered to be inadequate, so a Pascal computer program was written to record both the data from the autotitrator and the elapsed time. Program details and a flowchart are given in Appendix B. 3.4 Filter Preparation The filter used for the majority of the RDC runs was a Millipore cellulose-acetate / cellulose-nitrate membrane filter with a pore size of 0.45pm. The acrylic RDC cylinder was prepared for mounting by sanding with fine sandpaper, and then washing to remove any acrylic particles. It was then allowed to dry thoroughly. The filter was then mounted on the cylinder by using an acrylic cement which was prepared by dissolving scrap acrylic in chloroform. A good 45 bond between the filter and the cylinder was achieved by applying pressure with a small piece of acrylic sheeting; sticking was avoided by using a sheet of Teflon between the two pieces. The filter was allowed to dry thoroughly before "clearing". The process of defining the interface surface, or clearing, requires the application of a solvent to the filter surface. The solvent partially dissolves the filter, collapsing the pores and rendering the affected area impermeable. This solvent, or clearing solution, consists of a mixture of equal parts of 1,4-dioxane, 1,2-dichloroethane, and hexanes. The mounted filter was rotated at approximately 200 rpm and the clearing solution was applied using a brush, starting at the outer edge and moving towards the center. A region approximately 12 mm in diameter was left uncleared; across this area metal transfer occurred during the RDC test. Once the filter had dried, the cleared area was transparent while the active filter area remained opaque. The active filter area was then determined by measuring the diameter of the uncleared circle and calculating the area. 3.5 Experimental Procedure In a typical RDC run, heptane was pipetted through the hollow mounting rod (shown in Figure 3.2), wetting the filter and allowing the organic/aqueous interface to form on the aqueous side of the filter. The cell was then lowered into the aqueous solution until the organic solution level was just below the aqueous solution level. Additional heptane was pipetted into the interior of the cell, and the height of the cell was adjusted to the vertical position where there is no net convective flow due to hydrostatic forces. A pH electrode was placed in the solution and the pH of the solution was adjusted by adding dilute NaOH until the p H reached the desired value. A volume of heptane containing the extractant (D2EHPA) was pipetted into the inner compartment and the data acquisition system and the pH-stat were started. A time-NaOH addition profile was recorded by the data acquisition system since the pH of the system was maintained at a constant value. Since NaOH titrant of known concentration was added to the system to maintain the pH, and since the rate of H + release is directly proportional (2:1 ratio) to the amount of zinc extracted (c.f. reaction stoichiometry), the rate of zinc extraction could be determined. 46 A single test run consisted of operating the RDC at a certain rotational speed until steady-state had been attained and enough data had been recorded. This usually took only a few minutes since zinc extraction was quite rapid. The rotational speed was then changed and data collection occurred again. Six rotational speeds were tested in random order: 60, 100, 150, 200, 250, and 300 rpm. After the experiment had been concluded, the raw datafile was analyzed to determine the steady-state extraction rate at each rotational speed. Since the diffusion boundary layer is proportional to the inverse square root of the rotational speed (according to equation 2-4), a RDC plot, normalized with respect to interfacial area could be constructed. A sample RDC plot is shown in Figure 3.3. RDC PLOT 4.0 -i «= i . o - 0.5- 0 i i i i i i i i i i 0 0.2 0.4 0.6 0.8 1.0 U"2 (s1 / z) Figure 3.3 - A Sample RDC Plot showing lines from three different experiments Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45am As well as furnishing the flux of the system containing only chemical reaction resistances and diffusional resistance through the membrane, the RDC plot also provides qualitative information about the relative importance of diffusion and chemical reaction. For a system which 47 is operating completely in the chemical reaction regime the slope of the line will be equal to zero. As mass transfer processes in the aqueous and organic phases become more important, the slope increases, with a steep slope signifying strong mass transfer control. In general, the extraction rate (flux) for a particular set of test conditions was characterized by the flux at a rotational speed of 1.67 Hz, i.e. 100 rpm. Plots which examined the variation in the rate of metal extraction under different conditions could then be drawn. 48 CHAPTER 4 - Results and Discussion This discussion of the results of the experimental program will take the following form. First the raw data from the system characterization studies will be presented, with some discussion of errors. Next the mass transfer mathematical model which was formulated to describe and predict extraction in the RDC system will be introduced, with a full theoretical derivation of relevant equations. The results of the major kinetic studies will then be shown, along with results from the mathematical model. The second mathematical model which was produced to describe the results from the preloading tests will then be introduced, again with theoretical derivations. The results of the preload experiments and model predictions will then be shown and discussed. The effect of temperature on extraction will be analyzed, although no model was developed which examined this parameter. Finally, results of the experiments which were conducted to examine the properties of the filters used in the RDC will be presented and discussed. 4.1 Init ial Data from Test Runs The initial work involved developing a "standard condition" baseline against which subsequent experiments could be compared. A series of experiments in which the basic system parameters (zinc concentration, D2EHPA concentration, pH, and temperature) were varied were then performed. A summary of the baseline conditions and the range of parameters examined is given in Table 4.1. Figures 4.1 through 4.4 show the experimental data collected for the four parameters. For each parameter, representative figures were constructed for a RDC rotational speed of 100 rpm. The effect of zinc concentration on the zinc flux is shown in Figure 4.1. The flux is highly dependent on the zinc concentration at low zinc concentrations, and is independent of bulk aqueous zinc concentration at higher concentrations. This suggests that the rate controlling step at low aqueous zinc concentrations may be mass transfer of zinc from the aqueous bulk to the interface, whereas at higher zinc concentrations some other step is rate controlling. 49 Table 4.1 Experimental Conditions Baseline RDC conditions 0.05M Zn(C104)2 0.05F D2EHPA Bulk pH = 4.5 T = 25°C 0.45pm Millipore filter Parameters examined: [Zn(C104)2]: 0.001 - 0.20M [D2EHPA]: 0.0025-0.10F BulkpH: 3.25-5.5 temperature: 15 - 50 °C In Figure 4.2 the effect of D2EHPA concentration on the zinc flux is examined. The rate of zinc extraction is approximately proportional to the formal D2EHPA concentration, which again suggests a mass transfer mechanism. However, in this case the rate controlling mechanism may be the transfer of D2EHPA from the organic bulk to the interface. There is little dependence of zinc flux on pH as shown in Figure 4.3, indicating that this parameter is not particularly significant under the conditions examined. It is likely that the system is rate controlled by the flux of D2EHPA to the interface under all p H conditions, so changes in the bulk pH (and thus the resulting H + concentration gradient) would have little effect on the flux until a critical value was reached. As expected, the extraction rate increases with temperature, as shown in Figure 4.4. The increase in rate may be due to enhanced chemical reaction kinetics, or the increase in temperature may change solution properties so that resistances to mass transfer decrease. These results will be analyzed at a later time. As can be seen in Figures 4.1 through 4.4, there is some scatter in the experimental data. Although it is difficult to make an accurate estimation of the errors involved, an examination of the results obtained from various tests of the standard conditions baseline indicated that in seven trials the average flux was 5.49 x 10"* kmol/m2/sec, with a sample standard deviation of 0.32 x 10"8. 50 Possible sources of error are numerous. Although attempts were made to maintain the concentrations of all reagents at constant values, it was necessary to prepare new batches of stock solutions. The sodium hydroxide used for acid neutralization was prepared on a biweekly basis to avoid errors due to the absorption of atmospheric C0 2 . Many batches of zinc perchlorate stock solution were used; in addition the zinc perchlorate source was changed. Finally, a new batch of purified D2EHPA was prepared when the stock of previously purified D2EHPA was consumed. One of the major sources of error had to do with the measurement of the area of the cleared filter. Although a simple optical system was used, this method alone had a high systematic error; a measurement error of approximately 3 percent could be attributed to this step. Also, some of the cleared filters were not exactly circular; as much as possible these filters were ehminated. Finally, a random source of error is the consistency of the filter membrane. The type of filter used in this study is a cellulose acetate-nitrate membrane which is typically used in microbiological applications. As with any manufactured product, it is likely that there is some variation in the porosity and filter thickness from filter to filter, which would cause changes in the flux. Also, the pore size distribution is not a "spike", but rather a normal distribution with some pores larger than the specified value and some pores smaller. 51 Zinc Flux vs. Zinc Concentration 0.05 0.10 [Zn] (kmol/m3) 0.20 Figure 4.1 Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm Zinc Flux vs. D2EHPA Concentration o CD CO O E 00 O X 13 o c N 0.10 Formal [D2EHPA] (kmol/mJ) Figure 4.2 Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 52 Zinc Flux vs. pH pH Figure 4.3 Effect of changing bulk pH on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, co = 100 rpm o CD CO O E CO O X J 3 LL. O C fsl Zinc Flux vs. Temperature 25 30 35 40 Temperature (°C) 50 Figure 4.4 Effect of changing temperature on zinc flux : [Zn] = 0.05 M , Formal [D2EHPA] = 0.05 M , pH = 4.5, filter = 0.45pm, co = 100 rpm 53 4.2 Basic Mathematical M o d e l A mathematical model was developed to predict the rate of extraction with changes in bulk zinc concentration, bulk D2EHPA concentration, and pH. In order to evaluate the relative importance of mass transfer as opposed to chemical reaction, initial calculations were made of the maximum flux (virtual maximum rate, or VMR) allowed by the system conditions. An initial estimation of the effect of mass transfer on zinc extraction was made by considering the extreme cases in which either aqueous phase mass transfer (of Z n 2 + to the interface) or organic phase mass transfer (of D2EHPA to the interface) is dominant. Since it is likely that over the range of concentrations examined the extracted species exists as both ZnLj-HL (i.e. n=l .5) and Znl^-fl-IL^ («=2) G ) , VMR rates for both species will be calculated. The association factor, n, is defined as the ratio of D2EHPA dimer to extracted zinc. For the general extraction equilibrium Zn 2 + + n (HL)2 <—> ZnU-HL^_ 2 , + 2rT K- = — - — — [4-1] with equihbrium constant K e x / the fluxes of Zn 2 + and (HL)2 may be related by a simple mass balance, i.e. • J - l-J C4'2] Note that the fluxes are defined as shown in Figure 4.5. The aqueous mass transfer rate limited flux can be directly determined from Fick's first law, i.e. v - K'-~CLA [4-31 where C ^ 2 + and Cffi. are the interfacial and bulk concentrations of Zn 2 + , respectively. The mass transfer coefficient, k^, is defined by the equation D , t [4-4] v - 2 0 2 It is unlikely that there would be much ZnL, present under standard operating conditions, and therefore this species was not considered. 54 Organic INNER FILTER OUTER Aqueous Bulk Bulk J(HL)2 [ J z n 2 * J z n M H L ) p n . 2 ) 1 J H - *" zD,org *" "* ZD,aq *" interface ^ Figure 4.5 Diagram of species and flux direction definitions for the VMR model where is the aqueous diffusion coefficient of Zn , and z0jaq is the diffusion boundary layer thickness as given by the Levich equation, zD = 0.643 <o-V2vmDm [4-5] The organic mass transfer rate Kmited flux can be similarly derived from Fick's first law, i.e. (̂HL), = (̂HL), [ C ( H L ^ _ C ^ L J W-6] where and are the interfacial and bulk concentrations of (HL)2, respectively. Note that in this derivation, C denotes an aqueous species, and c denotes an organic species. Furthermore, the superscript i indicates an interfacial species, while either b or bulk indicates a species in the bulk phase. The mass transfer coefficient, k^a^, is defined by the equation1331 [4-7] k, where is the organic diffusion coefficient of (HL)2, zDjBrg is the diffusion boundary layer thickness as given by equation [4-5], L is the effective filter length, and a is the filter porosity. The 55 additional term L/a is due to the resistance of the filter to diffusion; this term is only present for organic phase species since it has been established that the aqueous/organic interface exists on the aqueous side of the filter surface. The virtual maximum rates of Z n 2 + transport (aqueous phase mass transfer control) and (HL)2 transport (organic phase mass transfer control) may be calculated from equations [4-3] and [4-7]. The results were overlaid on the experimental plots for zinc flux vs. zinc concentration, and zinc flux vs. D2EHPA concentration, as shown in Figures 4.6 and 4.7. In Figure 4.6, for low zinc concentrations, it can be seen that zinc transport to the interface appears to be rate controlling. At higher zinc concentrations, the rate controlling step appears to become the transport of D2EHPA to the interface. If a value of n=1.5 is used, the VMR is greater than the observed flux; for n=2, the VMR is lower than the observed results. Therefore, it is probable that the value of n is neither 1.5 nor 2, but rather extraction of both species occurs simultaneously. Figure 4.7, showing VMR plots for flux vs. D2EHPA concentration, implies that the association factor is near 1.5 at low D2EHPA concentrations, and increases as the concentration of D2EHPA increases. This is reasonable, since the amount of additional D2EHPA associated with an extracted zinc-extractant molecule is governed by an equihbrium which favours higher values of n with increasing D2EHPA concentration. From the plot, it appears that the assumption that the rate controlling step at moderate to high D2EHPA concentrations is mass transfer of D2EHPA to the interface is correct. The VMR calculations imply that the chemical reactions occur fast enough that the system can be simulated by a simple mass transfer model. The following discussion will focus on describing the theoretical basis for the mathematical model; source code is given in Appendix D. For the aqueous zinc flux and the hydrogen ion flux only two species were assumed to be present: Z n 2 + and H + . It was assumed that there were no chemical reactions in the boundary layer, and therefore diffusion profiles were linear as stated by Fick's first law. All D2EHPA present in the organic bulk was assumed to be dimeric, and the extractant diffusing to the interface was also assumed to be entirely in the dimeric form. Once at the interface, the partition between monomelic and dimeric D2EHPA was computed based on the total D2EHPA concentration at the interface. The zinc-extractant complex formed at the interface was assumed 56 Zinc Flux vs. Zinc Concentration o CD CO O E CO O X CJ c N • Experimental Data V M R : Z n 2 + VMR:(HL) 2 -n=1.5 VMR : (HL)2 - n=2 0.05 0.10 [Zn] (kmol/m3) 0.20 Figure 4.6 VMR predictions and experimental data for changes in bulk zinc concentration: Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 "C, filter = 0.45um, co = 100 rpm Zinc Flux vs. D2EHPA Concentration o CD CO O E CO O X _Z5 LL. O c Ki 12 10 8 6^ 4 2 0 Experimental Data VMR :(HL)2 -n=1.5 VMR : (HL)2 - n=2 0.02 0.04 0.06 0.08 Formal [D2EHPA] (kmol/m3) 0.10 Figure 4.7 VMR predictions and experimental data for changes in bulk D2EHPA concentration: [Zn] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 57 to exist as ZnLj-HL and ZnL^HL)^ and no zinc-extractant complex was present in the organic bulk. It was assumed that for the HL, (HL)^ Z n l ^ H L and ZnL2- (HL)2 species that reactions occurred only at the interface and in the bulk phase, and that no speciation adjustment occurred in the boundary layer. The activity coefficients of all species were assumed to be constant. The viscosity of the D2EHPA/heptane mixture was assumed to be equal to the viscosity of pure heptane; this will probably result in the computed diffusion coefficients for the organic species being too large. Since viscosity is also a parameter in the Levich equation, the viscosity term will cause a small increase in the equivalent boundary layer thickness; however, the increase will not be enough to offset the error in the diffusion coefficient. The extraction stoichiometry for zinc reacting with monomelic D2EHPA to form a zinc-extractant complex with one or two associated D2EHPA molecules can be expressed as: [4-8] Zn 2 + + 3HL <—> ZnL, • HL + 2ft p 1 3 and Zn 2 + + 4HL <—> ZnLj • (HL^ + 2Yt p14 HL c r2 [4-9] where (313 and p 1 4 are the formation constants for the species ZnLj-HL and ZnL 2-(HL) 2 respectively. A similar expression can be developed for the dimerization equilibrium between monomelic and dimeric D2EHPA: c ( H L ) 2 [4-10] 2HL <—* (HL)2 KD = C H L where KD is the dimerization constant for D2EHPA. An expression for the extraction of zinc (as Znl^HL) by D2EHPA in the dimeric form can be developed by combining equations [4-8] and [4-10]: Z n 2 + + - ( H L ) 2 *—» ZnL, • HL + 2lT Ka 2J+ C{WL\ 1.5 58 From equations [4-8] and [4-9] the distribution of the two zinc species Znl^-HL, and ZnL2-(HL)2 can be interrelated as follows: ZnLj • (HL^ <—> ZnLj • HL + HL p13/p14 CznLj-fHL̂ The quotient p 1 3 /p 1 4 is equal to the stepwise formation constant K*. At any point in the organic, the monomelic and dimeric concentrations of D2EHPA are related by the following equations: 1 [4-13] C(HL)2,total " 2 C H L C ( H L ^ substituting from [4-10], I v 2 [4-14] C(HL)j,totaI — 2 C , I L + DCHL rearranging, r 2 I _ n [4-15] ^DCHL + 2 C h L C(HL)j,totia — " This can be solved using the quadratic equation if the total concentration of D2EHPA is known. From equation [4-1], the fluxes of the aqueous and organic species diffusing to and from the interface may be related by a mass balance, i.e. I r - - l 4 " 1 6 ] JZx?* ~ n " ' C H L ) , - • / ZdL,- (HL) , ,«x< i ; - 2JJf¥ Once again, the fluxes are defined as shown in Figure 4.5. From Fick's first law, 59 J~Znl2-Q1L.)ll,tot<il ' -̂ ZnLj-HL + ẐnLj-CHL̂  [4-20] = ẐoLj-HL [cZn̂ -HL^ZnLj.HL.] + fcznLj.flfl.^ (̂ ZaLj-CHLfe— CZnI,.<HL)J where for aqueous species, A [4-21] = 1 and for organic species, A [4-22] By substituting c^.^-^ from equation [4-12] into equation [4-20], and assuming that the bulk concentrations of Znl^HLand ZnL 2 (HL) 2 are equal to zero, an expression for/^LJ-CHL),u*ai solely in terms of C ^ - H L may be developed, i.e. •̂ ZnL2-(HL)l.»o«j; < 4*. \ . [4-23] ^ . H L + ^.(HL) J p i 3 / p i 4 ^ ''ZnLj-HL If a value is assumed for Jz,,, then the concentration of the interfacial species may be computed by using equations [4-16], [4-17], [4-18], [4-19], and [4-23]. The expression for (equation [4-11]) can be re-written as: cL^ # [4-24] - r — — - K - = <-Zn C(HL)j where/(/zn) should equal 0. If the concentrations of the four interfacial species are substituted into equation [4-24], then the correct value for the flux can be deteirnined by iterating until fQ^) converges. A general flowchart showing the basic structure of the simple mathematical model is shown in Figure 4.8. 60 Input Data Iterate Flux Guess Zinc Flux Compute Interfacial Concentrations Compute Equilibrium Constant Figure 4.8 Basic program structure of the simple mathematical model 4.3 Basic M o d e l Predictions 4.3.1 Basic Model Verification The basic mathematical model fits the results surprisingly well when the number of assumptions and the accuracy of some critical physiochemical parameters (including the diffusion coefficients of D2EHPA and zinc-D2EHPA species in heptane) are considered. Figures 4.9 through 4.14 show the measured rates of zinc flux and the model predictions for changes in aqueous zinc concentration, formal D2EHPA concentration, and bulk pH. The model fits the very low zinc concentration regions of Figures 4.9 and 4.10 well, but overpredicts the zinc flux at higher zinc concentrations. Similarly, the fit of the model is good in Figures 4.11 and 4.12 at low D2EHPA concentrations, but again diverges at higher D2EHPA concentrations. 61 As shown in Figures 4.13 and 4.14, the model overpredicts the zinc flux for all values of p H examined. The model curve is fairly flat, indicating that pH has little to no effect on the zinc flux in the range examined; the predicted curve dips slightly around pH=3, indicating that at this point the effect of pH may start becoming significant. The model compares well with experimental data only at low zinc concentrations, where the rate controlling step is not the diffusion of (HL)2 in the organic phase, but rather the diffusion of Z n 2 + in the aqueous phase. At high zinc concentrations, moderate to high D2EHPA concentrations, and at all values of pH, the model overpredicts the rate of zinc extraction. This indicates that the inaccuracy in the model most likely lies on the organic side. There are three likely explanations. The first is that since most of the physiochemical parameters for the organic species were not available as experimentally verified data, empirical correlations were used. The most critical values derived using correlations are the diffusion coefficients of (HL)2, Znl^-HL, and ZnI^-(HL)2. In particular, the model is most sensitive to the diffusion coefficient of (HL)2 since, under conditions of organic rate control, the diffusion of this species is rate controlling. The second possibility for error is that the thickness and porosity of the filters used in this study are not as specified by the manufacturer. The final possibility is that some of the assumptions incorporated in the model are not completely valid. The assumption that all D2EHPA present in the bulk organic is dimeric is valid at high D2EHPA concentrations; examination of the monomer/dimer equiUbrium indicates that at a total D2EHPA concentration of 0.05 F, 98.6% of the D2EHPA would be in the dimeric form; even at a total D2EHPA concentration of 0.0005 F, 86.8% of the D2EHPA is dimeric. Thus, the assumption that aU D2EHPA diffusing to the interface is dimeric is not too bad, although there is some error since H L diffuses faster than (HL)2. The greatest error would be from the assumption that there is no readjustment of speciation as the zinc-extractant species migrate to the .bulk. Under conditions where the system is (HL)2 mass transfer controUed, there would be very Uttle free (HL)2 at the interface, resulting in a low value of n at the interface. However, as these low n zinc-extractant complexes migrate 62 through the boundary layer towards the bulk, the concentration of D2EHPA increases. If the chemical reactions are fast, some of the D2EHPA reacts and further sovlates the zinc-extractant complexes, creating a non-linear (HL)2 diffusion profile. To better understand the effect of different parameters on the model, a sensitivity analysis of the model was performed. The effect of three different parameters was examined: the organic side diffusion coefficients, the equiUbrium constant K^, and the filter equivalent thickness L / a . Figures 4.15 and 4.16 show the effect of changing the organic species diffusion coefficients for different zinc and D2EHPA concentrations. For this analysis, all three diffusion coefficients (for (HQ2, Z n L 2 H L , and ZnI^(HL)2) were altered by up to ± 20%. As can be seen in these figures, small changes in the organic species diffusion coefficients can have a significant effect on the extraction rate. The effect of changing the equiUbrium constant, K^, was examined in Figures 4.17 and 4.18. Small changes of the order of 10 or 20 percent had little effect on the extraction rate, so larger steps were used in order that the effect of different K^. values could be seen. For larger values of Ka/ the transition from one mass transfer regime to another occurs faster (c.f. Figure 4.17); smaller values cause the mass transfer limited rate to be approached more slowly (c.f. Figures 4.17 and 4.18). The final parameter exarnined is the filter equivalent thickness L/cc; plots are shown in Figures 4.19 and 4.20. Since the thickness of the filter is larger than the organic side diffusion boundary layer, increasing or decreasing the filter thickness has a significant effect on the flux if the system is in the organic mass transport controlled regime. 63 o CD <0 o E CO O 7 6 5 ^ 3 x L L O C N 2 1 0 Zinc Flux vs. Zinc Concentration w = 100 rpm Experimental Data Mathematical Model 0 0.05 0.10 [Zn] (kmol/m 3) 0.15 0.20 Figure 4.9 Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm o CD CO o E co O X _g LL. O C Kl 8 7 6 5 4 3 2 1 0 Zinc Flux vs. Zinc Concentration w = 300 rpm Experimental Data Mathematical Model 0 0.05 0.10 0.15 [Zn] (kmol/m 3) 0.20 Figure 4.10 Effect of changing bulk zinc concentration on zinc flux: Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45um, co = 300 rpm 64 12 Zinc Flux vs. D2EHPA Concentration w= 100 rpm o CD w o E CO O X ID o c N 10 Experimental Data Mathematical Model 0.10 Formal [D2EHPA] (kmol/m3) Figure 4.11 Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm Zinc Flux vs. D2EHPA Concentration w = 300 rpm o CO o E GO o X J2 Ll_ O c Kl 14 12 10 Experimental Data Mathematical Model Formal [D2EHPA] (kmol/m3) 0.10 Figure 4.12 Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 300 rpm 65 o CD co o E CO o X 3 O c "N 7 6 5 4 3 1 0 3.0 Zinc Flux vs. pH w = 100 rpm Experimental Data Mathematical Model 3.5 4.0 4.5 5.0 5.5 PH Figure 4.13 Effect of changing bulk pH on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, co = 100 rpm o CD CO O E co O X _3 L L O c N 8 7- 6- 5 4-\ 3 2 1 0 3.0 Zinc Flux vs. pH w = 300 rpm Experimental Data Mathematical Model 3.5 4.0 4.5 5.0 5.5 pH Figure 4.14 Effect of changing bulk p H on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, co = 300 rpm 66 o CD o E 00 o O c kl Sensitivity Analysis - Zinc Flux vs. [Zn] Vary Organic Diffusion Coefficients 8 L L I 7- 6- 5- 4- 3 2 1 H 0 f Experimental Data +20 % +10% Model Value -10% -20 % 0 0.05 0.10 [Zn] (kmol/m 3) 0.15 0.20 Figure 4.15 Zinc flux vs. zinc concentration for changes in the organic species diffusion coefficients : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25"C, filter = 0.45pm, co = 100 rpm Sensitivity Analysis - Zinc Flux vs. [D2EHPA] Vary Organic Diffusion Coefficients  12 o CD CO O E CO O X o c N 10 8 6 4H 2 • Experimental Data +20% +10% Model Value -10% —- -20% 0.02 0.04 0.06 0.08 Formal [D2EHPA] (kmol/m3) 0.10 Figure 4.16 Zinc flux vs. D2EHPA concentration for changes in the organic species diffusion coefficients : [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 67 o CD SO o E CO O X _z> LL O c N Sensitivity Analysis - Zinc Flux vs. [Zn] Vary Equilibrium Constant  8 6 5 4 3 2 | 1 - — »- -I 'if o Experimental Data x4 x2 Model Value x0.5 x0.25 0 0.05 0.10 [Zn] (kmol/m 3) 0.15 0.20 Figure 4.17 Zinc flux vs. zinc concentration for changes in the equiUbrium constant : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45um, co = 100 rpm Sensitivity Analysis - Zinc Flux vs. [D2EHPA] o CD CO O E co O X _3 LL O c 12 10 8H 6 4 2 0 Vary Equilibrium Constant • Experimental Data x4 . . . . . . x 2 Model Value x0.5 x0.25 0.02 0.04 0.06 0.08 Formal [D2EHPA] (kmol/m3) 0.10 Figure 4.18 Zinc flux vs. D2EHPA concentration for changes in the equiUbrium constant K«: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 68 o CD W O E GO o o c k l 8 Sensitivity Analysis - Zinc Flux vs. [Zn] Vary L/or 1 1 = 2 " UL 7H 6 5 4 3 2 1 H 0 r 0 Experimental Data +20 % +10% Model Value -10% -20 % 0.05 0.10 [Zn] (kmol/m 3) 0.15 0.20 Figure 4.19 Zinc flux vs. zinc concentration for changes in the filter equivalent thickness L/a: Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm Sensitivity Analysis - Zinc Flux vs. [D2EHPA] Vary Ua  12 o CD CO o E 00 O X J2 LL- CS c k l 10 H 8 6 4 2 • Experimental Data — +20% - - +10% Model Value — -10% —- -20% y y..-\' yy. •'.yy 0 0.02 0.04 0.06 0.08 Formal [D2EHPA] (kmol/m3) 0.10 Figure 4.20 Zinc flux vs. D2EHPA concentration for changes in the filter equivalent thickness L/a: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 69 4.3.2 Basic Model Predictions When the model was used to predict the zinc flux for any given set of conditions (bulk zinc concentration, formal bulk D2EHPA concentration, and bulk pH), the interfacial concentrations of the various species in the system were computed. In this section, some of the data produced by the model will be examined to see if they are consistent with the current understanding of the rate controlling processes. Figures 4.21 - 4.24 examine the effect that bulk zinc concentration has on the association factor n, the interfacial pH, and the interfacial zinc and D2EHPA concentrations. At low bulk zinc concentrations aqueous mass transfer of Z n 2 + is the rate controlling step, and as the zinc concentration increases the organic mass transfer of D2EHPA becomes rate controlling. In Figure 4.21, at low bulk zinc concentrations the association factor is large, but it decreases rapidly as zinc concentration increases. This may be attributed to the transition from aqueous mass transfer of Z n 2 + as the rate controlling step to organic mass transfer of D2EHPA. If D2EHPA transport is not rate controlling, then there exists an excess concentration of D2EHPA at the interface which is available for complexing zinc-extractant molecules. Figure 4.22 examines the effect of zinc concentration on the interfacial pH. In the low zinc concentration region, where zinc transport is rate controlling, the flux increases with increasing zinc concentration, which causes the interfacial p H to decrease. Again, as the transition is made from zinc mass transfer control to D2EHPA mass transfer control, the zinc concentration has a decreasing effect on the flux, and therefore a decreasing effect on the interfacial pH. When D2EHPA transport is rate controlling, only a small concentration gradient is required to maintain the zinc flux at the organic rate controlled limit. Thus, for concentrations above approximately 0.01 kmol/m 3 (from Figure 4.23), the interfacial zinc concentration essentially remains at a fixed value below the bulk zinc concentration. 70 Figure 4.24 shows the predicted effect of bulk zinc concentration on the concentration of the various interfacial D2EHPA species. The D2EHPA concentration decreases with increasing zinc concentration, first rapidly, and then more slowly as D2EHPA mass transport becomes dominant. Figures 4.25 - 4.28 examine the effect that bulk D2EHPA concentration has on the average association factor n a v g , the interfacial pH, and the interfacial zinc and D2EHPA concentrations. The average association factor, n a v g , is defined as the computed total flux of D2EHPA (expressed as dimer) divided by the computed zinc flux. In Figure 4.25, the association factor increases with increasing D2EHPA concentration. This may be once again attributed to the amount of free D2EHPA available for complexation. In the entire concentration region examined, the rate controlling step is the organic mass transfer of D2EHPA. Thus, with increasing overall D2EHPA concentration there will be a small increase in the amount of free D2EHPA at the interface, resulting in some increase in the value of n a v g . The decrease in interfacial pH with increasing D2EHPA concentration, as shown in Figure 4.26, is directly related to the increase in zinc flux. As the zinc flux increases, more FT ions are generated at the interface, resulting in a larger concentration gradient across the aqueous boundary layer. The same mechanism results in a decrease in the interfacial zinc concentration, as seen in Figure 4.27, due to a faster rate of consumption of Z n 2 + ions. The decrease in interfacial Z n 2 + concentration and the increase in interfacial FT concentration with increasing bulk D2EHPA concentration result, for fixed equihbrium constant in a larger interfacial D2EHPA concentration at equihbrium. Thus, although the zinc (and thus the D2EHPA) flux is increasing, the interfacial D2EHPA concentration also increases slightly to maintain equihbrium. This effect can be seen in Figure 4.28. Figures 4.29 - 4.32 examine the effect that the bulk pH has on the association factor n a v g , the interfacial pH, and the interfacial zinc and D2EHPA concentrations. In the pH region examined, D2EHPA transport is the rate controlling step. 71 Figure 4.29 shows the effect of changes in pH on the value of n a v g . For low values of pH, the interfacial p H will decrease, resulting in a larger interfacial D2EHPA concentration, which will result in a slight increase in n a v g . The interfacial p H increases slightly as the bulk pH is increased in order to keep the concentration gradient across the boundary layer at a relatively constant value (Figure 4.30). Since the flux is relatively constant across the entire pH range, there is little effect on either the interfacial zinc concentration or the interfacial D2EHPA concentration (Figures 4.31 and 4.32). The slight dip in flux at low pH values (c.f. Figure 4.13) is most likely caused by low interfacial p H values causing a shift in the extraction equiUbrium. 72 Association Factor (navg) vs. Zinc Cone. 1.75 > CD 1.70- C o o 1.65- 03 LL c g 1.60- co "o o to w 1.55- < 1.50 0.05 0.10 [Zn] (kmol/m3) 0.20 Figure 4.21 Predicted change in association factor (navg) with bulk zinc concentration : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25*C, filter = 0.45pm, co = 100 rpm Interfacial pH vs. Zinc Concentration 0.20 [Zn] (kmol/m3) Figure 4.22 Predicted change in interfacial pH with bulk zinc concentration : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 "C, filter = 0.45pm, co = 100 rpm 73 Interfacial Zinc Cone. vs. Bulk Zinc Cone. [Zn] (kmol/m3) Figure 4.23 Predicted change in interfacial zinc concentration with bulk zinc concentration: Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm Interfacial D2EHPA Cone. vs. Zinc Cone. o E < I L U CM Q "5 "8 CP 0.05 0.10 [Zn] (kmol/m3) 0.20 Figure 4.24 Predicted change in interfacial D2EHPA concentration with bulk zinc concentration : Formal [D2EHPA] = 0.05 M , pH = 4.5, T = 25 "C, filter = 0.45pm, co = 100 rpm 74 Association Factor (navg) vs. D2EHPA Cone. 1.65-j _ « o 1.60- Fa d  c o A ss o ci at  1.55- 1.50 i 1 1 1 1 1 1 1 1 1 0 0.02 0.04 0.06 0.08 0.10 Formal [D2EHPA] ( kmol/m3) Figure 4.25 Predicted change in association factor (« a v g ) with bulk D2EHPA concentration: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm Interfacial pH vs. D2EHPA Concentration 2.2 ~\ 1 1 1 1 1 1 1 1 1 0 0.02 0.04 0.06 0.08 0.10 Formal [D2EHPA] (kmol/m3) Figure 4.26 Predicted change in interfacial pH with bulk D2EHPA concentration: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm 75 Interfacial Zinc Cone. vs. D2EHPA Cone. 0.050 E 0.048 o E 0.040 0.10 Formal [D2EHPA] ( kmol/m J) Figure 4.27 Predicted change in interfacial zinc concentration with bulk D2EHPA concentration : [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45um, co = 100 rpm Interfacial D2EHPA Cone. vs. D2EHPA Cone. ^ 0.007 *E ^ 0.006 E ^ 0.005 2 0.004 X CM 0.003- Q « 0.002 o CO t= 0.001 CD K^tota, , interface / / yy // yy 0.02 0.04 0.06 0.08 0.10 Formal [D2EHPA] ( kmol/m3) Figure 4.28 Predicted change in interfacial D2EHPA concentration with bulk D2EHPA concentration: [Zn] = 0.05M, pH = 4.5, T = 25 °C, filter = 0.45um, co = 100 rpm 76 Association Factor (navg) vs. pH 1.70i 1.65- 1.60 o CO o 1.55-to CO < 1.50 -I 1 1 1 1 1 1 1 1 1 3.0 3.5 4.0 4.5 5.0 5.5 PH Figure 4.29 Predicted change in association factor (« a v g ) with bulk pH : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45um, co = 100 rpm O CO u_ X Q_ "CO "o co CD 3.0 Interfacial pH vs. Bulk pH 4.0 4.5 Bulk pH 5.5 Figure 4.30 Predicted change in interfacial pH with bulk p H : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °Q filter = 0.45pm, co = 100 rpm 77 Interfacial Zinc Concentration vs. pH 0.050 * | 0.049 o E N . 0.048 - 0.047 co o co t £ 0.046- 0.045 3.0 3.5 -i 1— 4.0 4.5 P H — i r 5.0 5.5 Figure 4.31 Predicted change in interfacial zinc concentration with bulk pH : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, co = 100 rpm Interfacial D2EHPA Concentration vs. pH 0.0030 co E ^ 0.0025 E j * : ~ 0.0020 < X 0.0015 LL! C M ^ 0.0010 co 'o ' ^ 0.0005 CD -*—> C [(HL).] '2'total. Intarfaca I(HL). 2'interlace [HL] Interlace 3.0 3.5 ~i 1 1 1— 4.0 4.5 -i 1— 5.0 PH 5.5 Figure 4.32 Predicted change in interfacial D2EHPA concentration with bulk pH : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45um, co = 100 rpm 78 4.4 Extended Mathematical Model The mathematical model presented in Section 4.2 was extended to incorporate the case where some of the extractant in the bulk organic phase is "tied up" with zinc. The amount of extractant tied up is defined by a percentage preload, where the preload is equal to the moles of zinc divided by the total molar content of D2EHPA (expressed as dimer). At low loadings, there will actually be less free D2EHPA than predicted by the preload, since the preload calculation assumes n=l, while n will be approximately equal to 1.7 at low loadings and only approach 1 near 100% preload. The conditions and equations for the aqueous phase are the same as those presented in Section 4.2. In the organic phase, five possible species are now considered: HL, (HL)^ Znl^, Z n L 2 H L , and ZnL2-(HL)2. Each of these species exists both at the interface and in the organic bulk. Furthermore, at both the bulk and the interface, the concentrations of these five species are interrelated by a series of equilibria. The values input into the model are the total concentration of zinc in the organic bulk (expressed as preload) and the total concentration of D2EHPA in the bulk. The model must then compute the speciation of zinc and D2EHPA; the computation of the amount of free D2EHPA is particularly critical, since D2EHPA beyond the stoichiometric requirement is tied up with Znl^-HL and ZnL^HL)^ It is still assumed that there is no redistribution of speciation across the boundary layer; however, this model accommodates the speciation and diffusion of both D2EHPA species. The following derivation of the basic model equations will where necessary re-state those equations which are presented in Section 4.2 so that continuity of the derivation will be maintained. Again, the following discussion only focuses on the theoretical basis for the extended mathematical model; source code is given in Appendix E. The extraction stoichiometry for zinc reacting with monomelic D2EHPA to form a zinc-extractant complex with zero to two associated D2EHPA ligands can be expressed as: [4-25] Zn + + 2HL ZnLj + 21T [4-26] Zn + + 3HL ZnLj • HL + 2FT 79 r r2 [4-27] Zn 2 + + 4HL <—> ZnLj • (HL)2 + 2H+ p14 = ^ 4 C2J> CHL where P^ Pi3, and p : 4 are the formation constants for the species ZnL?, ZnL^-HL, and ZnLj-dTL^ respectively. A similar expression can be developed for the dimerization equihbrium between the monomelic and dimeric form of D2EHPA: cc™* [4-28] 2HL <—> (HL)2 KD = 2 C HL where KD is the dimerization constant for D2EHPA. Using this expression, if either the concentration of monomeric or dimeric D2EHPA is known at any point in the organic, then the concentration of dimeric or monomeric D2EHPA, respectively, may be calculated. An expression for the extraction of zinc (as ZnLj-HL) by D2EHPA in the dimeric form can be developed by combining equations [4-26] and [4-28]: 3 _ „ _ C ^ . H L C ^ [4-29] Zn + + - (HL^ <—> ZnLj • HL + 2fT K, ,1.5 Equations can be developed from equations [4-25], [4-26], and [4-27] which interrelate the speciation of Znl^, Zn l^HL, and ZnL^HL^as follows: C2aLj CHL [4-30] ZnLj • HL <—> ZnLj + HL P12/p13 = ZnLj • (HL)2 <—> ZnLj • HL + HL p13/p14 = CZnLj-HL CZnLj-HL CHL [4-31] The quotient Pim/pi(m +D is equal to the stepwise formation constant Km+V At any point in the organic, the zinc species ZnLj, ZnLjHL, and ZnL 2 (HL) 2 are related by the following equations: CZn,total — CZnLj + CZnLj-HL + CZnLj-(HL)j [4 32] and 80 -ZnLj-HL Pu/Pis t-ZnLj-CHLjj CHL CZnLj-HL CHL PVPu [4-33] [4-34] Combining equations [4-32], [4-33], and [4-34], £zn,tt>tal CZnLj-HL Pl2/Pl3 + CZnLj-HL + C Z D L J - H L C H L '•Zn.total PIS/PU ^ZnLj-HL M>14 [4-35] [4-36] Performing a mass balance over the entire organic/interface region for zinc and "L", for zinc, •̂ Zn,total + ^ZnLj + ̂ ZnLj-HL + ẐaLj-CHL̂ Ẑn.toul — ^ Z I I L J + ẐnLj-HL + ẐnLj-CMJj and for "L", •̂ HL + 2/(HL)j + + 3/2^. HL + 4/ Z n L j . ( H L ) j = 0 Expanding equation [4-38], " ẐnLj-f.HL)̂ ' •'zn.total — ^ZnLjCzaLj + ̂ ZnLj-HL̂ ZnLj-HL + ^^(HL^ZoLj-rHLJj [4-37] [4-38] [4-39] [4-40] Z Q L J - C . H L J J '-ZnLj-rHLJjl [4-41] — ^^ZnLjCznLj + ̂ ZnL2-HLCZnL2-HL + ^^(HL^ZnLj - fHLjJ let KCZnbulk — ^ZriL^ZriL, + ẐnLj-HL̂ ZnLj-HL + ẐnLj-CHL̂ L̂j-CHL̂ 'Za.total — ^ZnLjCZnLj + ̂ ZnL,-HLCZnL,-HL + ẐnL,-(HL>,C; Lj-HL̂ ZnLj-l nLj-CHL̂ ZaLj-CHLJj - KCZnbulk [4-42] Substitute in values of and c ^ - m L ) , from equations [4-33] and [4-34], 81 'Zn .total ẐnL 13 C H L + ẐnLj-HL CZnLj-HL [4-43] ZnLj-HL + ẐnLj-CHLJj /za.to.ai + KCZnbulk ^ e' + ẐnLj.CHLJj ( J ^ CZnLj-HL CHL Pl3/PM KCZnbulk [4-44] Note that in this equation C ^ L J - H L is a function of J^uu KCZnbulk, and c^. Expanding equation [4-39], * H L ( C H L - 4 L ) + 2 / : ( H L ) 2 ( C ( H L ) 2 - C ( I H L ) 2 ) + 2^^^-C^j + 3 / ^ n L 2 . H L ^ c Z n L j . H L - c Z N L 2 . H L ] + 4£ z ^. ( H L ) 2 ( c Z n L 2 . ( H L ) 2 - c^.pn.^j [4-45] = 0 let ĤLCHL + 2̂ (HL)ij''(HL)2 + 2&ZnLJCZNLJ + ^^hLj-HL^Lj-HL + ^̂ hLj.fHL̂ ZnLj-CHL̂  [4-46] — "̂HL̂ HL 2̂ (HL)2̂ '(HL)2 2&2nL2CznL2 3/*2nî .j|LCznLi.HL ^̂ Zol̂ .n̂ L̂ ZnLj'CHL̂  = 0 KCbulk = £ H L C H L + 2 £ ( H L ^ C ( I ^ + 2/^^2^+3/^^ ĤLCHL + 2A^HLjIC(HL)i + 2 £ Z N L J C Z l l L z + SfcznL̂ .jjLCznLj.HL ^ From equations [4-28], [4-33], and [4-34], equation [4-47] can be rewritten as: HL (VP 13 ĤLCHL + 2£(HLv[^DCHL ] + 2&ZnL C H L [4-48] CZnLj-HL CHL Pl3/Pl4 - KCbulk = 0 From equation [4-44], an expression has been developed for c^vm. a s a function of cl^ and J^^. If a value is assumed for /̂ .totai/ then equation [4-48] can be solved by iterating c'^. The other 82 variables can then be solved for via equations [4-28], [4-44], [4-33], and [4-34]. The method of solution is then the same as in the simple mathematical model; the interfacial concentrations are substituted into expression for Kex (equation [4-49]): [4-49] Ok C(HL)j 1.5 - Km /(•̂ Zn,total) where/̂ zntota,) should equal 0, and the correct value for the flux Jẑ totai can be determined by iterating /zn,totai u n t i l /(/zn,totai) converges. A general flowchart showing the basic program structure of the extended mathematical model is shown in Figure 4.33. Iterate Flux Input Data Compute Bulk Speciation Guess Zinc Flux Compute Interfacial Concentrations Compute Equilbrium Constant Figure 4.33 Basic program structure of the extended mathematical model 83 4.5 Preload Results An important part of this study was the examination of the effects that loading of the organic had on the overall extraction rate. Most academic studies of solvent extraction kinetics have used unloaded organic phases, producing values for the initial extraction rate. However, in industrial solvent extraction processes, circuits typically operate to high organic loadings. Extraction kinetics are likely quite different, since as loading increases the number of additional D2EHPA molecules complexing the species will change. 4.5.1 Extended Model Verification In the preload experiments, the amount of zinc contained in the organic bulk was varied from 10% to 90% preload; total D2EHPA concentration was maintained at 0.05 F. For the mathematical model, it was necessary to assume a value for the ratio of the formation constants of the ZnLj and ZnLjHL species, p\2/p\3, as a suitable value could not be obtained from the hterature. The best value of P12/P13 was evaluated by substituting different values into the extended mathematical model; a graphical representation of the effects of different values of P12/P13 are shown in Figure 4.34. For the purposes of evaluating the behaviour of the system, P12/P13 was set equal to 6 x 10s, which gave a fairly good fit to the experimental results in the high preload range. For any value of p\2/Pi3 selected, the model fit is rather poor over the preload range. A small value of p 1 2/Pi3 provides a fairly good fit at low preload values, but prevents significant amounts of the Znl^ complex from forming at high preload values: A larger value of P12/P13 nts better in the intermediate to high preload concentration range, but overpredicts the initial zinc flux. This effect is due to the predicted formation of a significant amount of ZnLj, which has a higher diffusion coefficient than either Z n L 2 H L or ZnL2-(HL)2, and requires no additional H L molecules for complexation. Thus, in the case where the system is rate controlled by the diffusion of HL, the overall zinc flux will be higher. No correction in the mathematical model is made for the change in viscosity which occurs as the extractant polymerizes at high loadings (c.f Figure 2.4). Since the diffusion 84 Zinc Flux vs. Preload o CD CO O E 0 0 O X ZD O c "N 6 5 4 3 2 1 Experimental Data #2/013 = 1 X 1 0 " 4 ftZ//?i3=6x10-5 #2/63 = 3x10"5 ( W l 3=1 x10"5 A 2 / / ? 1 3 =1 x lO* 0% 20% 40% 60% 80% Preload (% Loading) 100% Figure 4.34 Comparison of experimental data and extended model predictions for flux vs. preload for selected values of p\ 2/Pi3: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm coefficient is a function of the viscosity, it will also change, creating another source of error. Also, the assumption that there is no speciation redistribution in the boundary layer is highly suspect, as will be shown later. The predicted zinc flux vs. preload is compared with experimental data in Figures 4.35 and 4.36. As outlined earlier, the model overpredicts the zinc flux at low preloads, and fits the data acceptably at high preloads for the given (WP^. The effect of different values of the organic side diffusion coefficients, the equihbrium constant K^, and the filter equivalent thickness L/a on the predictions of the extended mathematical model were examined, with the results shown in Figures 4.37 through 4.39. Again, the extraction rate is quite sensitive to both the organic side diffusion coefficients and the filter equivalent thickness, but relatively insensitive to changes in Ka. 85 Zinc Flux vs. Preload w = 100 rpm Preload (% Loading) Figure 4.35 Effect of preload on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, $12/p\3 = 6 x 10 s Zinc Flux vs. Preload 0% 20% 40% 60% 80% 100% Preload (% Loading) Figure 4.36 Effect of preload on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] t o t a I = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 300 rpm, pVp^ = 6 x 10 s 86 Sensitivity Analysis - Zinc Flux vs. Preload vary Organic Diffusion Coefficients o CD CO o E oo O X X o c N 6 - i 5- 4 3 2 A 1 \ \ W \ \ \ \ \ " - ? \ W \ W W • Experimental Data — +20% +10% — Model Value —- -10% —- -20% 0% 20% 40% 60% 80% Preload (% Loading) 100% Figure 4.37 Zinc flux vs. preload for changes in the organic species diffusion coefficients : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm, pVpa3 = 6 x 10 s Sensitivity Analysis - Zinc Flux vs. Preload Vary "Equilibrium Constant o CD CO o E co O X 3 o c N 6 5 4 3 H 2 1 A W \ N ' W W - N S N w w - w Experimental Data x4 x2 Model Value x0.5 xO.25 \ \ \ \ \ 0% 20% 40% 60% Preload (% Loading) 80% 100% Figure 4.38 Zinc flux vs. preload for changes in the equilibrium constant : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 'C, filter = 0.45um, co = 100 rpm, p 1 2 /p 1 3 = 6 x 10"5 87 Sensitivity Analysis - Zinc Flux vs. Preload 7 . . • V a r V U a o CD o E 0 0 O X 13 o cr N 6 5 4 3 j 2 1 0 S > ^ X \ \ \ < - X \ sX>- • Experimental Data +20% ------ +10% Model Value — - -10% — - -20% 0% 20% 40% 60% 80% Preload (% Loading) 100% Figiure 4.39 Zinc flux vs. preload for changes in the filter equivalent thickness L / a : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5,T = 25°C, filter = 0.45pm, co = 100 rpm, p 1 2 /p 1 3 = 6 x 10 s 4.5.2 Extended Model Predictions Examination of the interfacial and bulk concentration values computed by the extended model provides some valuable insights into the nature of the reactions which are occurring at the interface and in the bulk phases. The aqueous interfacial zinc concentration and the interfacial pH both increase as expected for increasing preload CFigures 4.40 and 4.41). As zinc flux decreases, less Z n 2 + is required and fewer H + ions are produced at the interface, resulting in a shallower concentration gradient for both species. Due to the multiple species equihbrium in both the organic bulk and at the interface, the concentration of D2EHPA monomer was computed at both points. Thus, the total D2EHPA flux to the interface now included both D2EHPA monomer and dimer terms. The predicted concentration of the various D2EHPA species, including the total D2EHPA 88 concentration in both regions, is shown in Figures 4.42 and 4.43. Although the diffusion of both species was included in the model, no provision was made for the readjustment of speciation in the boundary layer. The association factor, showing the average amount of complexation of organic zinc by (HL)2, was computed in both the bulk organic and at the interface. As can be seen in Figure 4.44, the increase in free D2EHPA in the bulk phase causes a significant increase in the amount of additional H L associated with the zinc-D2EHPA species. Qualitatively, the difference between the two curves is a measure of the amount of readjustment of speciation which occurs across the boundary layer, and thus the deviation from linearity of the diffusion profile for the three zinc species. This deviation can be quite significant due to the difference in diffusion coefficients of the three species (c.f. Table 2.2). The amount of each zinc species, expressed as a fraction of the total amount of zinc, is shown in Figure 4.45 for both the bulk phase and the interface. Figures 4.46 and 4.47 show the absolute concentrations of all organic zinc species and the total organic zinc concentration for the bulk and interface, respectively. The total bulk zinc concentration, shown in Figure 4.46, is directly related to the preload, and thus increases with increasing preload. As preload increases, the amount of free D2EHPA in the bulk decreases, resulting in a shift from high n species (ZnI^-(HL)2) to low n species (Znl^). From low to intermediate preload, the increase in organic zinc concentration results in increased levels of all organic species; in this range the decrease in free H L has little effect. However, at intermediate preload values the amount of free H L becomes significant, and the formation of Znl^-CHL^ is less favourable than the formation of ZnL 2 -HL. Finally, for high preload values the amount of free D2EHPA becomes so low (c.f. Figure 4.42) that only the formation of ZnLj is favoured. Increasing preload has little effect on the total organic zinc concentration at the interface (Figure 4.47). The decrease in the amount of free D2EHPA (c.f. Figure 4.43) causes a change in speciation, with lower n species favoured. The concentration profiles of the interfacial and bulk species were overlaid in Figure 4.48. Initially, the bulk concentration of each zinc species is greater than that of its 89 interfacial counterpart, but as the preload increases the interfacial concentrations of first ZnL2-(HL)2 and then ZnLjHL become greater than their bulk counterparts. Thus, the direction of zinc transport reverses for both Znl^-CHL^ and ZnLjHL, and these two species diffuse to the interface where their additional complexed H L extracts Z n 2 + and forms ZnL 2 according to the equilibria: Zn 2 + + Znlv(HL) 2 <—> 2ZnL2 + 2fT [4-50] and Zn 2 + + 2ZIUVHL <—> 3 2 ^ + 2H+ t4"51] The calculated fluxes for all the zinc species and the total D2EHPA flux are shown in Figure 4.49. Of particular interest is the upper preload range (>50%) where a significant fraction of the "L" supplied to the interface arrives as ZnL2-(HL)2 and Z n L 2 H L ; at loadings greater than ~70%, free D2EHPA is no longer the major extractant. 90 Aqueous Interfacial Zinc Cone. vs. Preload 0.050 i : 1 E 0.049- "o E -* 0.048- 8 | 0.047- 5, 0.046- 0.045 -I 1 1 1 1 1 1 1 1 1 0% 20% 40% 60% 80% 100% Preload (% Loading) Figure 4.40 Predicted change in interfacial zinc concentration (aqueous) with preload: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm, p 1 2 /p 1 3 = 6 x 10 s Interfacial pH vs. Preload 2.4 -I 1 1 1 1 1 1 1 1 1 0% 20% 40% 60% 80% 100% Preload (% Loading) Figure 4.41 Predicted change in interfacial pH with preload: [Zn] = 0.05M, Formal [D2EHPA] t o t a ] = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm, p 1 2 /p\ 3 = 6 x 10 s 91 Bulk D2EHPA Cone. vs. Preload 100% Preload (% Loading) Figure 4.42 Predicted change in bulk D2EHPA concentration with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm, p V P i s = 6 x 10 s Interfacial D2EHPA Cone. vs. Preload o E < Q_ I UJ CM Q I..., .4 TO 'o 0.0022 0.0020 0.0018-1 0.0016 0.0014 0.0012 0.0010 0.0008 -\ 0.0006 0.0004 0.0002 [(HL)2jtotaJ inigrface [HL] Interface \ o% 20% 40% 60% 80% 100% Preload (% Loading) Figure 4.43 Predicted change in interfacial D2EHPA concentration with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , pH = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, p V P u = 6x 10"5 92 Association Factor vs. Preload 0% 20% 40% 60% 80% Preload (% Loading) 100% Figure 4.44 Predicted association factors vs. preload: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, p V P o = 6 x 10 s Fractional Zinc Speciation vs. Preload Preload (% Loading) Figure 4.45 Predicted fractions of zinc species vs. preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, p 1 2 /p 1 3 = 6 x 10"5 93 Organic Bulk Zinc Cone. vs. Preload ZnLpHl 20% 40% 60% Preload (% Loading) 80% 100% Figure 4.46 Predicted change in bulk zinc concentrations (organic) with preload: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25'C, filter = 0.45pm, co = 100 rpm, p 1 2 /p\ 3 = 6 x 10 s _ Organic Interfacial Zinc Cone. vs. Preload 20% 40% 60% Preload (% Loading) 100% Figure 4.47 Predicted change in interfacial zinc concentrations (organic) with preload: [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25'C, filter = 0.45pm, co = 100 rpm, p V P i a = 6 x 10"5 94 ~ Organic Zinc Concentration vs. Preload E 0.022 0 0.020 0.018 H c 0.016 1 0.014 c 0.012 o 0.010 — _ ZnL2HLin, g 0.008 £ 0.006 2,bulk ZnLg-HL bulk 0% 20% 40% 60% 80% Preload (% Loading) 100% Figure 4.48 Predicted change in organic zinc concentrations with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm, p\ 2/p\ 3 = 6 x 10"5 Species Flux vs. Preload 10 "o 8 cu CO o E CO O X 3 6 4 2H 0 -2 -4 -6 A ('""-^.effective B ZnLg C—- ZnL2-HL D—- ZnL2-(HL)2 E Z n total . c__ - B ' 0% 20% 40% 60% 80% Preload (% Loading) 100% Figure 4.49 Predicted flux of organic species with preload : [Zn] = 0.05M, Formal [D2EHPA] t o t a l = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45um, co = 100 rpm, p V P i s = 6 x 10'5 95 4.6 Variable Temperature Although no attempt was made to model the effect of temperature on the extraction processes, some preliminary work was done to investigate its effect. The zinc flux was measured over the temperature range 15 to 50 °C, and the observed rate was compared with a VMR model. Assuming a mass transfer controlled system, a change in temperature will change such parameters as the diffusion coefficient of the various species and the viscosity of the organic and aqueous phases. An attempt was made to include these parameters by incorporating the temperature dependence of the viscosity of heptane into the VMR model. Since the diffusion coefficient calculated by the Wilke-Chang equation is a function of both temperature and viscosity (c.f. equation [2-3]), this method will probably give a reasonable estimation of the variation in diffusion coefficient over the moderate temperature range under consideration. Data1201 were obtained for the viscosity of heptane at different temperatures, and the diffusion coefficients and Levich equivalent diffusion boundary layer thickness at each temperature were calculated. A VMR model was constructed for organic mass transfer control for both «=1.5 and n=2. The model with experimental data for comparison is shown in Figure 4.50; the VMR curves appear to trace the general shape of the observed values. An Arrhenius plot, shown in Figure 4.51, was constructed in order to determine the activation energy of the system. The points are generally linear, and the slope of the fitted line is -1497 K - this slope is equivalent to an activation energy E„ = 12.4 kj/mole. Activation energies of this magnitude are characteristic of diffusion controlled reactions. 96 Zinc Flux vs. Temperature 9 - i /s ec ) 8- • • CM E "o E 7 - 6- 5 ; • CO o 4- X X 33 3 - • Experimental Data L L 2- VMR:(HL)2-n=1.5 O c 1 - o- VMR : (HL)2 - n=2 N i i i i i i 15 20 25 30 35 40 45 50 Temperature (°C) Figure 4.50 VMR predictions and experimental data for changes in temperature : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , pH = 4.5, filter = 0.45pm, co = 100 rpm Arrhenius Plot - Ln flux vs 1/T -16.2 -16.9 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 1/Temperature x 103 (1/K) Figure 4.51 Arrhenius Plot for experimental temperature data with linear regression fit: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , pH = 4.5, filter = 0.45pm, co = 100 rpm 97 4.7 Filter Characterization The effect of different filter pore sizes on the overall zinc extraction rate was examined. Three different filters were tested in addition to the standard 0.45pm Milhpore filter; 0.05pm, 0.22pm, and 0.80pm. For the mathematical model, porosity and filter length values supplied by Milhpore were used to compute the predicted flux at each discrete filter size. Table 4.2 Membrane Filter Characteristics Average Pore Filter Length, Porosity, L/a Size (pm) L (pm) oc (pm) 0.05 150 0.72 208 0.22 150 0.75 200 0.45 150 0.79 190 0.80 150 0.82 183 As can be seen in Figures 4.52 and 4.53, there is only a slight decrease in zinc flux as the filter pore size is decreased from 0.80pm to 0.22pm; however, there is a substantial drop in flux with the 0.05pm filter. The diameter of the (HL) 2 and ZnLj molecules was estimated to be approximately 0.0014pm.(3) Thus, for the 0.05pm filter the (HL) 2 and Zn l^ molecules are only 35 times smaller than the filter pore size; it is likely that there is some interference between the filter and the organic "L" molecules which retards the flux. It can therefore be concluded that for the filter pore size used in the baseline studies (0.45pm), the filter has no effect on the rate of diffusion of the organic species. 3 The diameter of the molecules were calculated by computing the molecular volume at the normal boiling point using the method of Le Bas.[16) By assuming that the molecules are spherical, the diameter could then be determined. However, since the molecules are most likely elliptical, this method may have under-estimated the diameter of the (HL)2 and Znl^ molecules. 98 o CD CO 6- ^ 5 o E A CO O X L L O c Ki 1 - Zinc Flux vs. Filter Pore Size w = 100 rpm Experimental Data Mathematical Model 0.20 0.40 0.60 Filter Pore Size (/iim) 0.80 Figure 4.52 Effect of changing filter pore size on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25°C, co = 100 rpm Zinc Flux vs. Filter Pore Size g w = 300 rpm o CD CO O E CO O X 3 O c KJ 7H 6 5 4H 3 2 1 0 Experimental Data Mathematical Model 0 0.20 0.40 0.60 Filter Pore Size (/im) 0.80 Figure 4.53 Effect of changing filter pore size on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, co = 300 rpm 99 4.8 S L M Applications The results of this work have significant implications for the area of supported liquid membranes since the diffusion of extractant through the filter is analogous to the diffusion of extractant through the SLM. Some types of SLM's currently being investigated are similar to the membrane filter used in this study. For example, Huang and Juang1651 reported using Durapor microporous PVDF film with a thickness of 125pm, a porosity of -70%, and an average pore size of 0.45pm. This support is comparable to the Millipore membrane filter (L = 150pm, a = 79%, and average pore size = 0.45pm). Also, since the average pore sizes are equal, the conclusion that the pore size has no effect on the extraction rate is valid. The feed side interfacial behaviour in a SLM system will be very similar to that encountered in this study. Under most conditions, it is likely that aqueous mass transport of Z n 2 + to the interface will not be rate controlling, and the feed side extraction will be governed by the diffusion of (HL) 2 through the membrane to the interface. On the strip side of the SLM, it is probable that the situation wil l be similar to that encountered on the feed side. The chemical reaction will be fast, and since the only product diffusing through the aqueous to the interface would be H + , it is likely that mass transport of extracted complex through the membrane to the interface would be the rate controlling step. Thus, the rate controlling steps in the S L M system would probably be the diffusion of (HL) 2 to the interface on the feed side, and perhaps the diffusion of ZnL 2-(HL) x to the interface on the strip side. This hypothesis must still be subjected to experimental verification. 100 CHAPTER 5 - Conclusions The extraction of zinc is controlled by the mass transfer of reactants (Zn and (HL) 2) to the interface. At low zinc concentrations, the system is controlled by the aqueous transport of Z n 2 + to the interface; at higher zinc concentrations transport of D2EHPA becomes rate controlling. For the range of D2EHPA concentrations examined, the transport of D2EHPA was rate controlling. Bulk p H had a negligible effect, except perhaps at the lowest pH values examined, where there may be a slight decrease in extraction rate. This decrease can be most likely attributed to less favourable thermodynamics at low interfacial pH values. It appears that the chemical reaction rate is fast enough that it has a negligible effect on the overall extraction rate. The basic mathematical model was adequate for predicting the extraction rate under variable conditions of zinc concentration, D2EHPA concentration, and pH. Although the model tends to overpredict the extraction rate, this effect is probably due to an error in one of the model parameters. The most likely sources of error are either the values for the organic diffusion coefficients or the filter equivalent thickness, L / a . The extraction of zinc with a partially loaded organic phase is also mass transfer controlled. The extended mathematical model predicts that the speciation of organic complexed zinc changes with increasing preload, and at high loadings the direction of ZnLj-HL and ZnLjdTL^ flux reverses, with these species providing extractant to the interface. At very high loadings, Z n l ^ H L provides almost all the extractant to the interface. Experimental studies of the effect of temperature on the rate of zinc extraction resulted in a calculation of the activation energy, Et, equal to 12.4 kj/mole. This value is consistent with a diffusion mechanism. Pore size had little effect on extraction rate, except for the 0.05pm filter, which caused a significant decrease in the extraction rate. It can therefore be concluded that the filter pores do not pose an additional resistance to mass transfer. This conclusion is important in SLM applications, because it shows that there is a minimum filter pore size below which the diffusion of species will be retarded. 101 CHAPTER 6 - Recommendations for Further Work The recommendations for further work can be divided into two sections: those that clarify some of the questions posed by this thesis, and others that continue the thrust of this work. Experimental viscosity measurements of D2EHPA/heptane mixtures, and metal-loaded D2EHPA/heptane mixtures could be used by the Wilke-Chang relationship to determine more accurate organic species diffusion coefficients. Also, more extensive study could be done on the properties of the Millipore filter membranes, especially the filter thickness, tortuosity, and filter porosity. This information would result in more accurate predictions from the mathematical model. For the extended ma thematical model, it would also be useful to determine an experimental value for the ratio of the formation constants Pn/p\3- Finally, the extended mathematical model could be re-written to accommodate changes in speciation across the organic boundary layer. An investigation could be conducted into the properties of membranes being used for SLM applications. Determinations of thickness, tortuosity, and porosity could then be included in a mathematical model which could be used to optimize SLM design. The techniques developed in this work could be used in the study of stripping reactions. By placing a loaded organic solution in the RDC inner compartment and an aqueous strip solution in the outer compartment, similar procedures could be used to determine the rate controlling steps. The extraction behaviour of other metals could be studied, especially cobalt and nickel. 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Ed., 42 (7), pp. 375-378,1965. 108 APPENDIX A - Optical Tachometer A n optical tachometer was used to continuously measure the rotational speed of the R D C A diagram of the physical setup is shown in Figure A.1. A circle with six alternating black and white sections was glued onto the top of the filter mount (Figure A.2). An infra-red photodiode-phototransistor pair was used to sense the black (light absorbing) to white (light reflecting) transitions, producing a signal output consisting of a series of pulses. Figure A.1 - The RDC and the Optical RPM Sensor A frequency counter (Advance mstruments TC9A Timer/Counter) with adjustable sensitivity input (lOmV, lOOmV, or IV - the IV range was used) was used to process the signal. The advantage of this particular counter lay in its ability to acquire a signal for a fixed period (in this particular case, 10 seconds). By acquiring the signal for ten seconds, a reading for RPM could be directly determined according to the following equation: for Belt I j 109 Figure A.2 - The Tachometer Circle 6 counts , ,. . 1 minute x 10 second sampling time x revolution 60 seconds = 1 count/(revolution per minute) = 1 count/rpm i.e. the observed count reading on the counter display was equal to the rotational speed of the RDC in RPM. The photodiode/phototransistor pair was connected to the counter by a simple signal conditioner. The schematic diagram for the signal conditioner and its power supply is shown in Figure A.3. As can be seen in this figure, the entire signal conditioner (lower half of the schematic) consists of exactly four components: D3/Q1, R l , R2, and C4. The power supply design shown in the upper half of Figure A.3 is a conventional design using a three-terminal voltage regular (Ul), and will not be discussed further. Photodiode D3 is connected to the 5 volt supply via resistor R l , which hmits the current flow. Phototransistor Q l conducts when it is exposed to infra-red light (reflected by the white portions of the tachometer circle) and makes the signal output positive. When Q l is not conducting (i.e. a black portion of the tachometer circle), resistor R2 pulls the output signal to ground. Capacitor C4 is a noise filter. The signal conditioner was modified from an existing unit which had previously been built for another experiment. The modification, which was necessary to reduce noise, involved placing capacitor C4 across the signal output terminals. No other changes were made to the existing unit. 110 1 U1 + C1 ~ C2 3 C3 — 5V Regulated — ^ o- 5V Regulated R1 i D3 J T F1 -1/2 Amp fuse D1, D2-1N4005 diodes C1 - 470uf, 40V electrolytic capacitor C2 - 0.333uf, 100V capacitor C3 -10ut 63V electrolytic capacitor C4 - 0.1 uf ceramic disc capacitor R1 - 15Q, 1/2W resistor R2-2.8K, 1/2W resistor U1 - LM7805 5 volt voltage regulator D3/Q1 - infra-red photodiode/phototransistor F104 607 Figure A.3 - Schematic Diagram of Optical Counter Signal Conditioner 111 APPENDIX B - Data Acquisition Hardware and Software This appendix will describe the hardware and software used for acquiring the NaOH volume/time profiles from the autoburette in each RDC run. The chart recorder feed output from the Radiometer ABU80 autoburette was connected to a Data Translation DT2805 data aquisiton board via a DT707-T screw terminal panel. The board was configured with differential inputs, bipolar operation, and a range of +10/-10V; analog channel 1 was used for input. Due to the unusual nature of the chart recorder feed output which was used for tracking the volume dispensed from the autotitrator, a custom computer program was written in Pascal to record the data. The chart recorder feed output was well suited for this application, because for every 1 /4000th of total burette volume (10ml for the current study) one half of a +5V/GND square wave is output. Thus, by counting the number of positive/ground and ground/positive transitions, the total volume dispensed may be determined. OV when sending Print Command 0V when 10ml burette assembly is used 0V Pulse output - 2000 pulses for 10ml assembly Decimal point position: A Decimal Point position: B Figure B. l Chart Feed pin-out specifications Data acquisition software was evaluated and found unsuitable. Most software records the input signal at discrete time intervals, a method which is clumsy for this particular application since the values of interest was not the value of the input signal at any particular time, but rather the time of each voltage transition, and the total number of voltage transitions. A program was therefore written which checked the chart recorder feed voltage, and incremented a counter if it changed. Once every second, if the number of counts had changed, the elapsed time, number of counts, and total number of counts were written to a file. A flowsheet of the data acquisition program is shown in Figure B.2, and the Pascal program listing follows in Table B.l. 112 Begin Makescreen Draw text windows OpenDataFile Get Filename and create file Acquire Data initialize port (inline code) read current time set pointers read initial level of signal set elapsed time to 0 write initialization -*- read A/D channel if transition then increment count if count change then write to screen if count change then write to file 1— Repeat until keypressed Reset real-time clock I End 1 Figure B.2 Major elements of the data acquisition program Table B.l: Pascal Listing of Data Acquisition Computer Program Program AD_Data_Acquisition; const CountMAX = 12001; { Max. number of array elements - l i m i t i s imposed by memory r e s t r i c t i o n s } type t i m e s t r i n g = s t r i n g [ 8 ] ; F i l e s t r i n g = s t r i n g [ 1 4 ] ; timearray = array[0..countmax] of byte; countarray = array[0..countmax] of integer; Maxstring = s t r i n g [ 2 5 5 ] ; Regpack = record AX,BX,CX,DX,BP,SI,DI,DS,ES,Flags: integer; end; i : intege r ; n : intege r ; A, B, C : timearray; tcount : countarray; D a t a f i l e : tex t ; dummy : char; Base_Address, Data_Reglster : Integer; miscellaneous counter index } index f o r arrays A, B, C, and tcount } arrays f o r hour, minute, and second values } array f o r count values } Output text f i l e } Function Keyboardlnput : char; { Waits u n t i l a key i s pressed, and then reports the value of the key. } var Key : char; begin { Keyboardlnput } Repeat Delay(50) u n t i l KeyPressed; 113 Read(Kbd.Key); Keyboardlnput := Key; end; { Keyboardlnput } F u n c t i o n Q u e s t i o n ( p r o m p t : M a x S t r i n g ) : b o o l e a n ; { Asks the Yes or No question contained i n the s t r i n g prompt and then waits u n t i l a Y or N ) ( answer has been received. The value o f the answer i s then reported by the value of t h i s ) { boolean f u n c t i o n - True = Yes and False = No. } var query : char; begin ( Question } Repeat Write( prompt + ' <Y/N> ' + chr(8) ); query := K e y b o a r d l n p u t ; w r i t e l n ( q u e r y ) ; U n t i l (query i n ['Y','y','N','n']); { Keep asking u n t i l a Y or N answer i s received } I f (query i n ['Y','y']) then Question := true e l s e Question := f a l s e ; end; { Question } P r o c e d u r e B e e p ; ( Beeps the speaker - used when user has made a data entry e r r o r ) { or to a l e r t the user to a p o s s i b l e problem. } begin ( Beep } sound(880); { Beep at 880 Hz ) Delay(125); { f o r 125 m i l l i s e c o n d s . ) Nosound; { Turn o f f sound. } end; { Beep } P r o c e d u r e M a k e S c r e e n ; { This procedure draws the text screen which contains a status window, a window f o r } ( user input, and a window f o r d i s p l a y i n g acquired data. The windows are drawn using ) { IBM extended graphics characters. ) var i : Integer; begin { MakeScreen } C l r S c r ; GotoXYd, 1); Write (chr (201) ) ,- f o r i := 2 to 78 do Write(chr(205)); Write(chr(187)); GotoXYd , 2); Write (chr (186) ) ; GotoXY(79, 2); Write(chr(186)); GotoXYd, 3); Write(chr(204)); fo r i := 2 to 78 do Write(chr(205)); Write(chr(185)); for i := 4 to 10 do begin GotoXYd , i ) ; Write(chr(186)); GotoXY(79, i ) ; Write (chr (186) ) ; end; GotoXYd, 11); Write(chr(204)); fo r i := 2 to 78 do Write(chr(205)); Write(chr(185)); fo r i := 12 to 24 do begin GotoXYd , i ) ; Write (chr (186) ) ; GotoXY(79, i ) ; Write (chr (186) ) ; end; loop v a r i a b l e upper l e f t - h a n d corner } draw upper l e f t corner character ) draw l i n e ) draw upper r i g h t corner character } l e f t margin, row 2 } r i g h t margin, row 2 } row 3 ) l e f t side l i n e j o i n character } draw l i n e ) r i g h t side l i n e j o i n character ) rows 4 through 10 } draw l e f t margin border ) draw r i g h t margin border } row 11 } l e f t side l i n e j o i n character } draw l i n e ) r i g h t side l i n e j o i n character } rows 12 through 24 } draw l e f t margin border } draw r i g h t margin border } 114 GotoXYd, 25); Write(chr(200)); fo r i := 2 to 78 do Write(chr(205)) ; Write(chr(188)); LowVideo; end; {MakeScreen) ( bottom l i n e ) { draw bottom l e f t corner character ) ( draw bottom l i n e ) ( draw bottom r i g h t corner character ) ( se l e c t low i n t e n s i t y video ) P r o c e d u r e ResetWindow; ( Resets the text screen by s e t t i n g the window back to f u l l screen. } { Also moves the cursor to the bottom left-hand corner of the screen. ) begin { ResetWindow ) Window(l,l, 80,25); GotoXYd,24) ; end; ( ResetWindow } Pr o c e d u r e T i m e ( v a r h o u r , m i n , s e c , f r a c : b y t e ; v a r t i m e : t i m e s t r i n g ) ; { Reads the real-time clock and sets the values hour, min, sec, frac (hundredths of seconds) ) ( using DOS Inte r r u p t 21h. The v a r i a b l e s t r i n g time i s also set equal to the time, expressed ) ( i n the form HH:MM:SS. ) Regs : Regpack; AH, AL : byte; tempi, temp2, temp3 s t r i n g [ 2 ] ; { r e g i s t e r v a r i a b l e passed to i n t e r r u p t handler ) { 8 b i t r e g i s t e r s } ( temporary v a r i a b l e s used f o r holding time d i g i t s begin { Time } AL := $00; AH := $2C; Regs.AX := AH s h l + AL; not required, just be neat s e l e c t "get system time" f u n c t i o n convert AH and AL to 16 b i t AX value ) Intr($21,Regs) hour min sec f r a c Regs.CX shr 8; = Regs.CX mod 256; = Regs.DX shr 8; = Regs.DX mod 256; { c a l l i n t e r r u p t 21 hexadecimal } ( get value f o r hour ) { get value for min ) ( get value f o r sec ) { get value f o r f r a c ) str(hour,tempi); str(min, temp2); s t r ( s e c , temp3); i f hour < 10 then tempi i f min < 10 then temp2 i f sec < 10 then temp3 time := tempi + end; ( Time } ( convert numeric hour value i n t o s t r i n g } { convert numeric min value i n t o s t r i n g } { convert numeric sec value i n t o s t r i n g } '0' + tempi; { add a leading zero i f hour < 10 } '0' + temp2; ( add a leading zero i f min < 10 ) '0' + temp3; { add a leading zero i f sec < 10 } + temp2 + ' :' + temp3; { create time s t r i n g i n the form HH:MM:SS } P r o c e d u r e S e t T i m e ( h o u r , m i n , s e c : b y t e ) ; { Sets the time to the values s p e c i f i e d by the arguments hour, min, sec using DOS Interrupt { 21h. This p a r t i c u l a r routine i s not p a r t i c u l a r l y e f f i c i e n t , but i t i s not i n a t i m e - c r i t i c a l ( se c t i o n , and i s much c l e a r e r the way i t i s str u c t u r e d here. } Regs : Regpack; AH, AL, CH, CL, DH, DL : byte; ( r e g i s t e r v a r i a b l e passed to i n t e r r u p t handler ( 8 b i t r e g i s t e r s ) begin { SetTime } AL = $00; { not required, just be neat AH = $2D; { s e l e c t "set system time" f u n c t i o n CH = hour; { set hour CL = min; { set minutes DH = sec; { set seconds DL = 0; { set hundredths of seconds equal to 115 Regs.AX := AH s h l 8 + AL; Regs.CX := CH s h l 8 + CL; Regs.DX := DH s h l 8 + DL; Intr($21,Regs); end; { SetTime ) ( convert AH and AL to 16 b i t AX value } { convert CH and CL to 16 b i t CX value ) { convert DH and DL to 16 b i t DX value ) { c a l l i n t e r r u p t 21 hexadecimal ) P r o c e d u r e E r r o r ; { Catastrophic e r r o r handler. This procedure i s invoked i f the A/D converter reports } ( an e r r o r , something which should not happen under normal circumstances. When run, i t } ( resets the text window and immediately h a l t s program execution. } begin ( E r r o r } Writeln; Writeln('A/D CONVERSION ERROR'); Writeln; Re s e t W i n d o w ; Halt; end; { E r r o r } P r o c e d u r e R e s e t T i m e ( o l d h o u r , o l d m i n , o l d s e c , A , B , C : b y t e ) ; ( This procedure resets the real-time clock at the end of the a c q u i s i t i o n run. The time } { at the s t a r t of a c q u i s i t i o n was stored i n oldhour, oldmin, £ oldsec, and A, B, & C hold ) ( the elapsed time of the run (hours, minutes, and seconds, r e s p e c t i v e l y . The clock time ) ( i s obtained by summing the i n i t i a l time and the elapsed time, with c a r r y . ) var hour, min, sec : r e a l ; begin { ResetTime ) sec := oldsec + C; min := oldmin + B; hour := oldhour + A; i f (sec >= 60) then begin sec := sec - 60; min := min + 1; end; i f (min >= 60) then begin min := min - 60; hour := hour + 1; end; i f (hour >= 24) then hour := hour - 24; S e t T i m e(hour,min,sec); end; { ResetTime } P r o c e d u r e A c q u i r e D a t a ( D a t a R e g i s t e r : i n t e g e r ; v a r A , B , C : t i m e a r r a y ; v a r f c o u n t : c o u n t a r r a y ; v a r n : i n t e g e r ; v a r D a t a f i l e : t e x t ) ; ( AcquireData i s the master data a c q u i s i t i o n r o u t i n e . I t i s d i v i d e d i n t o two se c t i o n s : i n i t i a l i z a t i o n and a c q u i s i t i o n . Two sections are writ t e n i n IBM assembler (via i n l i n e statements) - one s e c t i o n h a l t s any a c q u i s i t i o n a c t i v i t y and prepares the board f o r command input, and the second s e c t i o n (duplicated twice i n t h i s procedure) acquires the A/D reading from channel 1. ) { The program constantly checks f o r a voltage t r a n s i t i o n i n the chart feed output (read into the A/D converter through analog channel #1. I f a t r a n s i t i o n i s detected, then countinc i s incremented and count flag i s set. The d e t e c t i o n loop then continues u n t i l ( a f t e r reading the clock) the new hundredths second value i s l e s s than the o l d hundredths second value ( i . e . a new second). At t h i s point the array tcount[n] i s set equal to tcount[n-l] plus the number of t r a n s i t i o n s (countinc) detected i n the previous second. The arrays A[n], B[n], & C[n] are set equal to the elapsed time (hours, minutes, and seconds). The time and the count values are output to the screen so that program operation may be v e r i f i e d ; the values of n, A[n], B[n], C[n], and tcount are al s o outputted to the f i l e . Thus, at the end of a c q u i s i t i o n , the arrays A, B, C, and tcount contain the count/time p r o f i l e , with a new array element whenever at l e a s t one voltage t r a n s i t i o n occurred i n a one second i n t e r v a l . } 116 var A f l a g , B f l a g : boo lean; H igh , Low : b y t e ; c o u n t f l a g : boo lean; c o u n t i n c : i n t e g e r ; timex : t i m e s t r i n g ; D : b y t e ; o l d h o u r , o l d m i n , o l d s e c Dold : b y t e ; dummy : c h a r ; k e y f l a g : boo lean ; { a p a i r of f l a g s which i n d i c a t e when a t r a n s i t i o n occur rs } { High and Low bytes of A/D conver ted value } { i n d i c a t e s i f a t r a n s i t i o n has occur red } { number of t r a n s i t i o n s i n the cur rent second } { s t r i n g i n d i c a t i n g the cur rent time i n HH:MM:SS format } { hundredths of a second d i g i t from r e a l - t i m e c l o c k } by te ; { O ld time read from c lock p r i o r to a c q u i s i t i o n } { o l d hundredths of a second va lue - f o r second d e t e c t i o n } { dummy charac te r f o r keypressed f u n c t i o n } { f l a g which i n d i c a t e s when a key has been pressed } { t o t a l volume which has been d ispensed } volume : r e a l ; begin { Acqui reData } { I n l i n e machine code which i n i t i a l i z e s the A/D conver te r . } i n l i n e <$BA/$ED/$02/$B0/$0F/$EE/ $BA/$EC/$02/$EC/ $BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $EC/$24/$04/ $3C/$04/$75/$F9/ $B0/$00 /$EE/ $EC/$24/$01/ $3C/$01/$75/$F9/ $BA/$EC/$02 /$EC/ $BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $EC/$24/$04/ $3C/$04/$75/$F9) ; T i m e ( o l d h o u r , o l d m i n , o l d s e c , D , t i m e x ) , n := 0; c o u n t i n c := 0; k e y f l a g := f a l s e ; { Port[Command_Register] := Command_Stop { TEMP := Por t [Data_Register ] ( Repeat { u n t i l ( (Por t [S ta tus_Regis ter ] and $02) = 0) { Repeat { u n t i l ( (Por t [S ta tus_Regis ter ] and $04) = $4) ( Port[Command_Register] := Command_Reset { Repeat { u n t i l ( (Por t [S ta tus_Regis ter ] and $1) = $1) { TEMP := Por t [Data_Regis ter ] ( Repeat ( u n t i l ( (Por t [S ta tus_Regis ter ] and $02) = 0) { Repeat { u n t i l ( (Por t [S ta tus_Regis ter ] and $04) = $4) { get time from c l o c k p r i o r to s t a r t of a c q u i s i t i o n } { set the count index = 0 } { set the current count increment = 0 } { no key has been pressed to terminate execut ion ) { Read A/D channel to get i n i t i a l s ta te of the chart feed } { LOW c o n t a i n s the low by te , and HIGH conta ins the h igh byte ) i n l i n e ($BA/$ED/$02/$B0/$0C/$EE/ $EC/ $24/$02/$75/$FB/ $BA/$EC/$02 /$B0/$00 /$EE/ $BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $BA/$EC/$02 /$B0/$01 /$EE/ $BA/$ED/$02/$EC/$24/$01/ $3C/$01/$75/$F9) ; LOW := Por t [Data_Regis te r ] ; i n l i n e ($BA/$ED/$02/$EC/$24/$01/ $3C/$01/$75/$F9) ; HIGH := Por t [Data_Regis te r ] ; i n l i n e ($BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $EC/$24/$04/ $3C/$04/$75/$F9) ; IF ( (Por t [Data_Regis ter ] AND $80) S e t T i m e ( 0 , 0 , 0 ) ; T i m e (A[0] ,B[0] ,C[0] , D, timex) ; tcount[0] := 0; c o u n t f l a g := f a l s e ; A f l a g := (HIGH > 9); B f l a g := A f l a g ; Window(5,2 ,75,3) ; G o t o X Y U . l ) ; W r i t e ( ' I n i t i a l i z a t i o n : ' , t i m e x ) ; Window( 5 , 4 , 7 5 , 1 0 ) ; C l r s c r ; Command ADIN Port[Command_Register] Repeat u n t i l ( (Por t [S ta tus_Regis ter ] and $02) = 0) Por t [Data_Register ] := ADgain Repeat u n t i l ( (Por t [S ta tus_Regis ter ] and $02) = 0) Por t [Data_Register ] := ADchannel Repeat u n t i l ( (Por t [S ta tus_Regis ter ] and $01) = $1) Repeat u n t i l ( (Por t [S ta tus_Regis ter ] and $01) = $1) Repeat u n t i l ( (Por t [S ta tus_Regis ter ] and $02) = 0) Repeat u n t i l ( (Por t [S ta tus_Regis ter ] and $04) = $4) $80) then E r r o r ; Set c l o c k to 0,0,0 - zero e lapsed time } Read back the time from the c l o c k ) T o t a l counts at time zero = 0 ) No t r a n s i t i o n s have occur red ) Set Aflag i f t h r e s h o l d output i s exceeded ("HIGH") ) Set Bflag so i t i s the same as AF lag ) S e l e c t s ta tus window } D i s p l a y " I n i t i a l i z a t i o n " and time to show a c t i v a t i o n ) S e l e c t data output window ) Volume := t coun t [0 ] / 400 ; { Compute the d ispensed volume ) { Wr i te the i n i t i a l i z a t i o n in fo rmat ion to the f i l e } W r i t e l n ( D a t a f i l e , A [ 0 ] : 2 , ' ' , B [ 0 ] : 2 , ' ' , C [ 0 ] : 2 , ' ' , t c o u n t [ 0 ] : 5 , ' ' , V o l u m e : 7 : 4 ) ; n + 1; Increment the array index 117 { Master a c q u i s i t i o n repeat loop. This loop repeats u n t i l an overflow occurs { i n the number of counts or a key i s pressed to end a c q u i s i t i o n . ) Repeat i f Keypressed then Keyflag := true; { Read A/D channel to get state of the chart feed. ) i n l i n e ($BA/$ED/$02/$B0/$0C/$EE/ $EC/ $24/$02/$75/$FB/ $BA/$EC/$02/$B0/$00/$EE/ $BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $BA/$EC/$02/$B0/$01/$EE/ $BA/$ED/$02/$EC/$24/$01/ $3C/$01/$75/$F9); LOW := Port[Data_Register]; i n l i n e ($BA/$ED/$02/$EC/$24/$01/ $3C/$01/$75/$F9); HIGH := Port[Data_Register]; i n l i n e ($BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $EC/$24/$04/ $3C/$04/$75/$F9); IF ((Port[Data_Register] AND $80) Dold := d; T i m e (A [n] ,B[n],C[n] ,D,timex) ; Port[Command_Register] : = Command_ADIN Repeat u n t i l ((Port[Status_Register] and $02) = 0) Port[Data_Register] := ADgain Repeat u n t i l ((Port[Status_Register] and $02) = 0) Port[Data_Register] := ADchannel Repeat u n t i l ((Port[Status_Register] and $01) = $1) Repeat u n t i l ((Port[Status_Register] and $01) = $1) Repeat u n t i l ((Port[Status_Register] and $02) = 0) Repeat u n t i l ((Port[Status_Register] and $04) = $4) $80) then E r r o r ; Keep the o l d hundredths second value f o r comparison } Read the elapsed time } ( i f a count has been measured and there i s a new second then compute } ( the t o t a l number of counts, c l e a r count flag, write the data to the } ( screen and to the f i l e , increment n, and reset countinc. } i f c o u n t f l a g and (D < Dold) then begin tcount[n] := tcount[n-1] + countinc; co u n t f l a g := f a l s e ; writeln(timex,' ',countinc:5,' counts; ',tcount[n]:5,' t o t a l counts'); Volume := tcount[n]/400; w r i t e l n ( D a t a f i l e , A [ n ] : 2 , ' ',B[n]:2,' ',C[n]:2,' ',tcount[n]:5,' ',Volume:7:4), n := n + 1; countinc := 0; end; A f l a g := (HIGH > 9); ( i f HIGH i s greataer than voltage threshold, set Aflag ) { i f A f l a g i s not equal to Bf l a g , then a voltage t r a n s i t i o n has occurred. } { Increment countinc, set countflag, and reset Bflag so i t i s equal to Aflag. } I f not (Aflag = Bflag) then begin countinc := countinc + 1; co u n t f l a g := true; b f l a g := a f l a g ; end; { I f array space l i m i t i s h i t or a key has been pressed then terminate a c q u i s i t i o n } u n t i l ((n = CountMAX) OR Keyflag); while keypressed read(kbd,dummy); { c l e a r keyboard b u f f e r ) T i m e (A [n] ,B[n] ,C[n] ,D, timex) ; ( read the time } Write l n ; Window(5, 14, 75, 24); { Select the output window ) C l r s c r ; W riteln('Execution terminated: '.timex); { Display status, termination time (elapsed) } R e s e t T i m e(oldhour,oldmin,oldsec,A[n],B[n],C[n]); { Reset the real-time clock } w r i t e l n ; w r i t e l n ( ' T o t a l volume dispensed: ',volume:7:4,' ml'); end; { AcquireData } P r o c e d u r e O p e n D a t a F i l e ( v a r D a t a f i l e : t e x t ) ; ( This procedure prompts the user f o r the d e s i r e d d a t a f i l e filename, f i l l s i n the ( extension "dat" i f i t i s not s p e c i f i e d , and then checks to see i f t h i s name i s { already present on the d i s k . I f i t i s , the user i s given the choice of over w r i t i n g { i t . I f the user d e c l i n e s , then they are prompted to enter the filename again. { The f i l e v a r i a b l e Datafile, i s used i n other routines for accessing the f i l e . 118 var Fi lename : F i l e s t r i n g ; E x i s t : boo lean ; OKf lag : b o o l e a n ; begin { OpenDataFi le \ Window(5,12,75,24); OKf lag := f a l s e ; repeat C l r s c r ; { name of data f i l e } { t rue i f s p e c i f i e d f i lename a l ready e x i s t s on d i s k } { f l a g i n d i c a t e s when f i lename has been s u c c e s s f u l l y s e l e c t e d ) ( Set and s e l e c t the user input window ) ( f i lename has not been s u c c e s s f u l l y s e l e c t e d yet } { repeat u n t i l OKFlag } { c l e a r the screen i n s i d e the window } { Read f i lenames u n t i l at l e a s t one c h a r a c t e r i s entered repeat W r i t e ( ' S a v e f i lename: ' ) ; Read ln (F i lename) ; u n t i l ( length( f i lename) > 0) ; ( i f the f i lename i s longer than 4 c h a r a c t e r s , check to see i f the ex tens ion ) ( i s " . d a t " . I f so , s t r i p i t o f f ). i f ( length( f i lename) > 4) then i f ( c o p y ( f i l e n a m e , l e n g t h ( f i l e n a m e ) - 3 , l e n g t h ( f i l e n a m e ) ) = ' . d a t ' ) then f i lename := copy( f i lename, 1 , l e n g t h ( f i l e n a m e ) - 4 ) ; { i f the f i lename i s longer than 8 c h a r a c t e r s , f i lename i s equal to f i r s t 8 charac te rs i f ( length( f i lename) > 8) then f i lename := c o p y ( f i l e n a m e , 1 , 8 ) ; Fi lename := Fi lename + ' . d a t ' ; A s s i g n ( D a t a f i l e , F i l e n a m e ) ; { Add the ex tens ion " .da t" } { Def ine the f i l e v a r i a b l e } { Turn o f f e r r o r check ing . Reset d a t a f i l e , and then re -enab le e r r o r c h e c k i n g . I f } { D a t a f i l e e x i s t s , then IOresu l t w i l l be equal to 0. } ($1-} R e s e t ( D a t a f i l e ) ($1+) ; E x i s t := ( IOresul t = 0 ); { I f the f i l e a l ready e x i s t s on the d i s k . E x i s t = t rue OKf lag := not E x i s t ; { The f i lename i s OK to use i f i t d o e s n ' t E x i s t ) { I f the f i lename e x i s t s , then warn the user and ask whether or not to overwr i te } i f E x i s t then begin B e e p ; Beep; W r i t e l n ; Writeln('WARNING: F i l e " ' , F i l e n a m e , ' " a l ready e x i s t s ! ' ) ; ( I f they answer yes to the q u e s t i o n , then the f i lename i s OK, so set } { OKf lag=true. C lose the D a t a f i l e and Erase i t to s t a r t with a c lean s l a t e . } I f q u e s t i o n ( ' D o you want to erase i t ? ' ) then begin OKf lag := t r u e ; C lose (Dataf i le ) ; E r a s e ( D a t a f i l e ) ; end; end u n t i l O K f l a g ; C l r s c r ; w r i t e l n ; w r i t e l n ( ' O u t p u t d a t a f i l e : ' . F i l e n a m e ) , { Repeat ask ing f o r f i lenames u n t i l one i s OK ( C l e a r the window ) ( I d e n t i f y the output f i lename s e l e c t e d ) end; { OpenDataFi le } { M a i n - M a i n p r o g r a m c o d e } Begin { Main } Base_Address := $2EC; Data_Register := Base_Address; M a k e s c r e e n ; O p e n D a t a F i l e ( D a t a f i l e ) ; R e w r i t e ( D a t a f i l e ) ; { Memory Address of the Data A c q u i s i t i o n board } { Data R e g i s t e r Address i s the same as the Base Address } { Draw the text windows ) ( Get the f i l e Fi lename and prepare f o r f i l e wr i te ) ( Open and empty the f i l e - set p o i n t e r to the beginn ing } A c q u i r e D a t a ( D a t a _ R e g i s t e r , A , B , C , t c o u n t , n , D a t a F i l e ) ; { Master A c q u i s i t i o n Rout ine ) C l o s e ( D a t a f i l e ) ; { C lose the f i l e } 119 f o r i := 1 to 20000 do; { do -noth ing wait loop to pause program - i t i s h e l p f u l to ) { see f i n a l count va lues before te rmina t ing program. ) W r i t e l n ; W r i t e ( ' P r e s s any key to c o n t i n u e . . . ' ) ; dummy := K e y b o a r d l n p u t ; ( Wait f o r a keypress before te rmina t ing execut ion } R e s e t W i n d o w ; ( Reset text screen } C l r s c r ; end. { Main ) ( t h a t ' s a l l f o l k s } 120 APPENDIX C - Raw Experimental Data NOTE: An asterisk (*) signifies that the marked parameter is being varied from the baseline. [Zn] [D2EHPA] pH Temp. Intercept Slope J(100rpm) J(300rpm) ( M ) ( F ) ( 'C ) ( x 10' ) ( x 10' ) kmol/mVs kmol/mVs STUDYA01 0 .05 0 .05 5. 50* 25 1. .20225 0, .48573 6 .34E-08 7. . 04E-08 STUDYA02 0 .05 0 .05 4 .50 25 1. .27801 0. .64266 5 .63E-08 6. .39E-08 STUDYA03 0 .05 0 .05 4 .50 25 1. .60495 0. .52152 4 .98E-08 5. .44E-08 STUDYA04 0 .05 0 .05 4 .50 25 1. .44678 0. .53018 5 .38E-08 5. . 94E-08 STUDYA05 0. 001* 0 .05 4 .50 25 0. .46758 5. .80803 2 .01E-08 3. .26E-08 STODYA06 0. 03* 0 .05 4 .50 25 1. .56962 0. .58608 4 . 94E-08 5. . 46E-08 STUDYA07 0. 01* 0 .05 4 .50 25 1. .62316 0. .81708 4 .43E-08 5. .03E-08 STUDYA08 0. 005* 0 .05 4 .50 25 1. .87731 1. .31886 3 . 45E-08 4 . 05E-08 STUDYA09 0. 003* 0 .05 4 .50 25 1. .44404 1. .59222 3 .74E-08 4 . 64E-08 STUDYA10 0. 10* 0 .05 4 .50 25 1. .56957 0. .51383 5 .08E-08 5. . 56E-08 STUDYA11 0 .05 0. 10* 4 .50 25 0. .87816 0. .33828 8 .77E-08 9. .71E-08 STUDYA12 0 .05 0. 025* 4 .50 25 2. .78002 1. .22596 2 .68E-08 3. .00E-08 STUDYA13 0 .05 0. 010* 4 .50 25 5. .63237 1. .98924 1 . 39E-08 1. . 53E-08 STODYA14 0 .05 0. 005* 4 .50 25 0. .93892 5. .15378 2 .03E-08 3. .08E-08 STUDYA15 0 .05 0.0025* 4 .50 25 14 , .39475 11. .31931 4 .32E-09 5. .14E-09 STUDYA16 0 .05 0. 005* 4 .50 25 12 . 06940 4 . 45671 6 .44E-09 7. .11E-09 STUDYA17 0 .05 0 .05 5. 00* 25 1. .38820 0. .58543 5 .43E-08 6. .06E-08 STUDYA18 0 .05 0 .05 3. 25* 25 1. .88770 0. . 12249 5 .04E-08 5. .15E-08 STUDYA19 0 .05 0 .05 4 . 00* 25 1. .56045 0. .53104 5 .07E-08 5. .56E-08 STUDYA20 0 .05 0 .05 3. 75* 25 1. .33529 0. .68410 5 .36E-08 6. .09E-08 STUDYA21 0 .05 0 .05 3. 50* 25 1. .41360 0. .77253 4 .97E-08 5. . 68E-08 STUDYA22 0 .05 0 .05 5. 50* 25 1. .40901 0, .63507 5 .26E-08 5. .91E-08 STUDYA25 0 .05 0 .05 4 . 50 30* 1. .15114 0. .49328 6 . 52E-08 7 . 29E-08 STUDYA26 0 .05 0 .05 4 .50 15* 1. .62001 0. .63318 4 . 74E-08 5. .25E-08 STUDYA27 0 .05 0 .05 4 .50 20* 1. .63941 0, .28731 5 .37E-08 5. . 66E-08 STUDYA28 0 .05 0 .05 4 . 50 35* 1. .07196 0. .44648 7 .05E-08 7. . 86E-08 STUDYA30 0. 005* 0 .05 4 . 50 25 1. .27783 1. .03012 4 .82E-08 5. .75E-08 STUDYA31 0. 10* 0 .05 4 .50 25 1. .25067 0 , .56321 5 .93E-08 6. . 66E-08 STUDYA32 0. 005* 0 .05 4 .50 25 1. .31220 1, .09904 4 .62E-08 5. . 54E-08 STUDYA33 0 .05 0 .05 4 .50 25 1. .21560 0. .59272 5 . 97E-08 6. .75E-08 STUDYA34 0 .05 0 .05 4 .50 25 1. .44421 0. .57371 5 .29E-08 5. .88E-08 STUDYA35 0 .05 0 .05 4 .50 25 1. .35095 0. . 61098 5 .48E-08 6. . 16E-08 STUDYA36 0. 005* 0 .05 4 .50 25 1. .30828 1, . 12605 4 .59E-08 5. . 52E-08 STUDYA37 0. 10* 0 .05 4 .50 25 1. .01799 0. .84444 5 .98E-08 7. .17E-08 STUDYA38 0. 20* 0 .05 4 .50 25 1. .23857 0. .49103 6 .18E-08 6. .86E-08 STUDYA39 0. 20* 0 .05 4 .50 25 1. .33987 0, .43416 5 .97E-08 6. .52E-08 STUDYA40 0. 02* 0 .05 4 .50 25 1. .34332 0. . 64144 5 .43E-08 6. . 13E-08 STUDYA41 0. 04* 0 .05 4 .50 25 1. .40675 0. .54474 5 .47E-08 6. ,06E-08 STUDYA42 0 .05 0. 10* 4 .50 25 0. .71916 0. ,30998 1 .04E-07 1. . 17E-07 STUDYA43 0 .05 0. 075* 4 .50 25 0. .92083 0. .36531 8 .31E-08 9. .22E-08 STUDYA44 0 .05 0. 035* 4 .50 25 1. .87262 0, . 68720 4 .16E-08 4 . 59E-08 STUDYA45 0 .05 0 .05 4 .50 35* 1. .09105 0. .41156 7 .09E-08 7. .84E-08 STUDYA4 8 0 .05 0 .05 4 .50 50* 0. .92457 0. .47012 7 .76E-08 8. .81E-08 STUDYA50 0 .05 0 .05 4 .50 50* 0. .88095 0. .43449 8 .21E-08 9. •30E-08 STUDYA51 0 .05 0 .05 4 .50 40* 1, ,03295 0, .38181 7 .53E-08 8. •31E-08 STUDYA52 0 .05 0 .05 4 .50 45* 0, .95741 0, .39778 7 .90E-08 8. .81E-08 121 [Zn] ( M ) [D2EHPA] % ( F ) Loading Intercept Slope J(100rpm) J(300rpm) ( x 107 ) ( x 107 ) kmol/mVs kmol/mVs STUDYB01 0. .05 0. .05 8.2 1. ,44008 0, ,62974 5. , 19E-08 5. .81E-08 STUDYB02 0. .05 0. .05 24 .3 1. .87071 0. .75522 4. .07E-08 . 4, .53E-08 STUDYB03 0. .05 0. .05 0 1, ,29804 0, .58299 5. , 72E-08 6. .42E-08 STUDYB04 0. .05 0. .05 18.7 1. .77354 0. .87059 4 . 09E-08 4 .  62E-08 STUDYB05 0. .05 0. .05 40.0 2, ,45371 1. ,32527 2, ,87E-08 3. •28E-08 STUDYB06 0. .05 0. .05 49.2 3. .29994 1. .49803 2. .24E-08 2. . 52E-08 STUDYB07 0. .05 0. ,05 28.5 2. .02465 0. .95048 3. ,62E-08 4 . 08E-08 STUDYB08 0, .05 0. .05 57 .7 4. .03017 2. .09985 1. ,77E-08 2. .01E-08 STUDYB09 0. .05 0. .05 82.2 12. .01515 3. .40072 6, .83E-09 7. .39E-09 STUDYB10 0. .05 0. .05 67.6 6, .45097 2. ,35808 1. .21E-08 1. •33E-08 STUDYB11 0, .05 0, .05 88.1 28 .52638 14 .30439 2. . 52E-09 2. .86E-09 [Zn] [D2EHPA] f i l t e r Intercept Slope J(100rpm) J(300rpm) ( M ) ( F ) ( x 107 ) ( x 107 ) kmol/mVs kmol/m2/s STUDYD01 0. .05 0. ,05 0, . 22|lm 1, .34584 0. ,61641 5, . 48E-08 6. , 17E-08 STUDYD02 0. .05 0. ,05 0. .80|im' 1, .27965 0. ,47415 6. . 07E-08 6. , 70E-08 STUDYD03 0. .05 0. ,05 0, . 05u.m 2, .86257 0. ,56463 3. .03E-08 3 . 21E-08 STUDYD04 0. .05 0. ,05 0, . 80u.m 1. .25882 0. .54710 5. .94E-08 6. .65E-08 STUDYD05 0. .05 0. .05 0. ,22u.m 1. .27255 0. ,54201 5. .91E-08 6. ,60E-08 STUDYD06 0. ,05 0. ,05 0, . 05u.m 2, .87390 0. .50528 3. .06E-08 3. .23E-08 STUDYD07 0. .05 0. .05 0, . 22U-m 1, .38042 0. .54523 5. .55E-08 6. . 16E-08 122 APPENDIX D - Basic Mathematical Model The mathematical derivation and program flowchart for the basic mathematical model have been given in Section 4.2, and will not be reproduced here. The program employs an incremental search / bisection routine to determine the steady-state flux under different bulk conditions. The source code for the program follows as Table D. l . Table D. l : Pascal Listing of Basic Mathematical Model Program Model; { guide to v a r i a b l e n o t a t i o n : Zn = Zn , a q ) . Hp s _ H \ , q l . HL = HL ( o t g ) , HL2 = (HL) 2 ( o r „ , ZnL2HLl = Z n L 2 - H L ( o t g ) , and ZnL2HL2 s ZnL 2 -(HL) 2 , „ „ } { a s u f f i x of b i n d i c a t e s a bulk s p e c i e s , and i i n d i c a t e s an i n t e r f a c i a l s p e c i e s const be ta l3_over_beta l4 = l e - 3 ; Kd = 5.012e4; K e x = 3 .236e-2; c_ZL_bulk = 0; w = 100; ( r a t i o of ZnL2-HL and ZnL 2 -(HL) 2 format ion c o n s t a n t s , = 10 " -3 } ( D imer i za t ion constant f o r D2EHPA, = 10 A 4.7 ) { E q u i l i b r i u m constant e x t r a c t i o n r e a c t i o n ) { T o t a l o rgan ic bulk z i n c c o n c e n t r a t i o n = 0 ) { RDC r o t a t i o n a l speed, i n rpm } type F i l e s t r i n g MaxStr ing Ex tens ion s t r i n g [ 1 4 ] ; s t r i n g [ 2 5 5 ] ; s t r i n g [ 4 ] ; aq_Zn, k_aq_Hp, k_or_HL2, k _ o r _ Z L l , k_or_ZL2, HL e f f _ b u l k , c_Zn_bulk, pH, c_Hp_bulk, c _ H L 2 t o t _ i , c _ H p _ i , c _ Z L t o t _ i , c HL2 i , c ZnL2HLl i , c ZnL2HL2 i . c _ Z n _ i , c _ H L _ i , JZn , n. V a r y _ i , V a r y _ f , V a r y _ i n c , Vary_value : r e a l ; mass t r a n s f e r c o e f f i c i e n t s } spec ies bulk c o n c e n t r a t i o n s } spec ies i n t e r f a c i a l concent ra t ions } spec ies i n t e r f a c i a l concent ra t ions } z i n c f l u x i n kmol /m 2 /sec } a s s o c i a t i o n f a c t o r ) i n i t i a l , f i n a l , increment, and current parameter va lues } P r o c e d u r e Beep; { Beeps the speaker - used when user has made a data entry e r r o r ) { or to a l e r t the user to a p o s s i b l e problem. } begin { Beep } sound(880); { Beep at 880 Hz } Delay(125) ; ( f o r 125 m i l l i s e c o n d s . } Nosound; { Turn o f f sound. } end; ( Beep } F u n c t i o n K e y b o a r d l n p u t : c h a r ; { Waits u n t i l a key i s p r e s s e d , and then r e p o r t s the value of the key. } var Key : char ; begin ( Keyboardlnput ) Repeat Delay(50) u n t i l KeyPressed; Read(Kbd,Key); Keyboardlnput := Key; end; ( Keyboardlnput } 123 F u n c t i o n Q u e s t i o n ( p r o m p t : M a x S t r i n g ) : b o o l e a n ; { Asks the Yes or No ques t ion conta ined i n the s t r i n g prompt and then wai ts u n t i l a Y or N } { answer has been r e c e i v e d . The value of the answer i s then repor ted by the va lue of t h i s } { boolean f u n c t i o n - True s Yes and F a l s e s No. ) var query : char ; begin { Quest ion ) Repeat Wr i te ( prompt + ' <Y/N> ' + chr(8) ) ; query := Keyboardlnput; w r i t e l n ( q u e r y ) ; U n t i l (query i n [ ' Y ' , ' y ' , ' N ' , ' n ' ] ) ; ( Keep ask ing u n t i l a Y or N answer i s rece ived ) I f (query i n [ ' Y ' , ' y ' ] ) then Quest ion := t rue e l s e Quest ion := f a l s e ; end; { Quest ion ) P r o c e d u r e O p e n D a t a F i l e ( v a r D a t a f i l e : t e x t ; f i l e n a m e _ e x t e n s i o n : e x t e n s i o n ) ; { Th is procedure prompts the user fo r the d e s i r e d d a t a f i l e f i lename, adds the ex tens ion } ( s p e c i f i e d , and then checks to see i f t h i s name i s a l ready present on the d i s k . I f i t ) { i s , the user i s g i v e n the cho ice of o v e r w r i t i n g i t . I f the user d e c l i n e s , then they } { are prompted to en ter the f i lename a g a i n . The f i l e v a r i a b l e Datafile, i s used i n } ( o ther r o u t i n e s f o r a c c e s s i n g the f i l e . ) var Fi lename : F i l e s t r i n g ; { name of data f i l e ) E x i s t : boo lean; { t rue i f s p e c i f i e d f i lename a l ready e x i s t s on d i s k ) OKf lag : boo lean; { f l a g i n d i c a t e s when f i lename has been s u c c e s s f u l l y s e l e c t e d ) begin { OpenDataFi le ) OKf lag := f a l s e ; ( f i lename has not been s u c c e s s f u l l y s e l e c t e d yet } repeat { repeat u n t i l OKFlag ) repeat ( Read f i lenames u n t i l at l e a s t one c h a r a c t e r i s entered ) W r i t e ( ' S a v e f i l ename: ' ) ; Read ln (F i lename) ; u n t i l ( length( f i lename) > 0) ; { i f the f i lename i s longer than 4 c h a r a c t e r s , check to see i f the ex tens ion ) { matches filename_extension. I f so , s t r i p i t o f f . ) i f ( length( f i lename) > 4) then i f ( copy ( f i l ename, length ( f i l ename) -3 , l eng th ( f i l ename) )= f i l ename_ex tens ion ) then f i lename := c o p y ( f i l e n a m e , 1 , l e n g t h ( f i l e n a m e ) - 4 ) ; { i f the f i lename i s longer than 8 c h a r a c t e r s , f i lename i s equal to f i r s t 8 charac te rs } I f ( length( f i lename) > 8) then f i lename := c o p y ( f i l e n a m e , 1 , 8 ) ; Fi lename := Fi lename + f i l ename_extens ion; ( Add the extens ion ) A s s i g n ( D a t a f i l e , F i l e n a m e ) ; { Def ine the f i l e v a r i a b l e } { Turn o f f e r r o r check ing . Reset d a t a f i l e , and then re -enab le e r r o r c h e c k i n g . I f } { D a t a f i l e e x i s t s , then IOresu l t w i l l be equal to 0. } {$1-} R e s e t ( D a t a f i l e ) ($1+) ; E x i s t := ( IOresul t = 0 ); ( I f the f i l e a l ready e x i s t s on the d i s k . E x i s t = t rue ) OKf lag := not E x i s t ; ( The f i lename i s OK to use i f i t d o e s n ' t E x i s t } { I f the f i lename e x i s t s , then warn the user and ask whether or not to overwr i te } i f E x i s t then begin Beep; Beep; W r i t e l n ; Writeln('WARNING: F i l e " ' , F i l e n a m e , " • a l ready e x i s t s ! ' ) ; { I f they answer yes to the q u e s t i o n , then the f i lename i s OK, so set ) ( OKf lag=true. C lose the D a t a f i l e and Erase i t to s t a r t wi th a c lean s l a t e . ) I f QuestionCDo you want to erase i t ? ' ) then begin OKf lag := t r u e ; C l o s e ( D a t a f i l e ) ; E r a s e ( D a t a f i l e ) ; 124 end; end u n t i l OKflag; { Repeat asking for filenames u n t i l one i s OK } Writeln('Output d a t a f i l e : '.Filename); { I d e n t i f y the output filename s e l e c t e d } Wri t e l n ; R e w r i t e ( D a t a f i l e ) ; ( Open and empty the f i l e - set p o i n t e r to the beginning ) { Write f i l e header } W r i t e l n ( D a t a f i l e , ' w = ',w:4); Write ( D a t a f i l e , ' [ Z n ] b u l k [HL]eff,b [ZnL]b pH,b JZn n [Zn]i [HL]i ' ) ; W r i t e l n ( D a t a f i l e , ' [ ( H L ) 2 ] i [HL2]ieff [ Z n L l ] i [ZnL2]i [ZnL]i,t [H+]i'); end; { OpenDataFile } F u n c t i o n P o w e r ( x , n : r e a l ) : r e a l ; { Computes the value of x" and returns the value as Power. } begin { Power ) power := e x p ( n * l n ( x ) ) ; end; ( Power } P r o c e d u r e Q u a d r a t i c ( a , b , c : r e a l ; v a r r o o t l , r o o t 2 : r e a l ) ; { Solves the quadratic equation and returns the two roots as r o o t l and root2. ) begin { Quadratic } r o o t l := (-b + sqrt(b*b-4*a*c))/(2*a); root2 := (-b - sqrt<b*b-4*a*c))/(2*a); end; ( Quadratic ) P r o c e d u r e C a l c u l a t e _ C o n s t a n t s ; { Computes the mass t r a n s f e r c o e f f i c i e n t s f o r each species. The Levich } ( equivalent boundary l a y e r thickness i s f i r s t c a l c u l a t e d , and then the ) { mass t r a n s f e r c o e f f i c i e n t i s computed using the appropriate formula. ) const D_HL2 D_ZnL2HLl D_ZnL2HL2 D_HC104 D_Zn nu H20 1.003e-9 0.781e-9 0.659e-9 2.854e-9 1.038e-9 8.93e-7; nu_heptane = 5.64e-7; L_over_alpha = 1.90e-4; var zD_aq_Zn, zD_aq_Hp, zD_or_HL2, zD_or_ZLl, zD_or_ZL2 : r e a l ; begin { C a l c u l a t e Constants } zD_aq_Zn zD_aq_Hp zD_or_HL2 zD_or_ZLl zD or ZL2 0.643/sqrt(w/60)*Power(nu_H2O, (1/6) ) * P o w e r (D_Zn, (1/3) ) ; 0.643/sqrt(w/60)*Power(nu_H20,(1/6) ) * Power(D_HC104, (1/3)); 0.643/sqrt(w/60 )*Power(nu_heptane,(1/6)) * P o w e r ( D_HL2 ,(1/3)); 0.643/sqrt(w/60)* P o w e r(nuheptane,(1/6)) * P o w e r ( D_ZnL2HLl,(1/3)); 0.643/sqrt(w/60)* P o w e r(nuheptane, (1/6) ) * P o w e r f D ZnL2HL2, (1/3)), k_aq_Zn k_aq_Hp k_or_HL2 k_or_ZLl k or ZL2 D_Zn D_HC104 D_HL2 D_ZnL2HLl D ZnL2HL2 zD_aq_Zn; zD_aq_Hp; (zD_or_HL2 (zD_or_ZLl (zD or ZL2 L_over_alpha); L_over_alpha); L_over_alpha); end; { C a l c u l a t e Constants P r o c e d u r e V a r y _ S e t u p ; { This procedure j u s t gets the i n i t i a l , f i n a l , and increment value when } { a range of concentrations i s examined. } 125 begin { Vary_Setup } Wr i te (' I n i t i a l Value Readln ( V a r y i ) ; Wri te (' F i n a l Value Readln (Vary_f ) ; Wr i te (' Increment Readln (Vary_ inc ) ; i f V a r y _ i > Vary_f then Vary_Setup e l s e i f V a r y _ i n c <= 0 then Vary_Setup; end; { Vary_Setup } recurse to get c o r r e c t va lues i f i n i t i a l i s > than f i n a l { recurse to get c o r r e c t va lues i f increment i s < 0 } Procedure Input_Parameters ( v a r c _ Z n _ b u l k , c _ H L e f f _ b u l k , p H : r e a l ; v a r c o d e : i n t e g e r " ) " ; { T h i s procedure prompts the user f o r va lues f o r the bulk z i n c c o n c e n t r a t i o n , bulk D2EHPA t c o n c e n t r a t i o n (expressed as d imer ) , and bulk pH. The user can examine a range of va lues ( by e n t e r i n g a va lue of - 1 . In t h i s case , the subrout ine Vary_Setup i s c a l l e d which ( prompts the user f o r the i n i t i a l , f i n a l , and incremental v a l u e s . code t e l l s the program { whether a range has been s e l e c t e d or not ; a value of zero i n d i c a t e s no range, whi le 1, { 2, and 3 i n d i c a t e z i n c , HL, and H + , r e s p e c t i v e l y . } var input : r e a l ; begin ( Input_Parameters } code := 0; W r i t e l n ; W r i t e l n (' En ter model parameters: Wr i te (' [Zn]bulk = ' ) ; Readln ( Input) ; i f (Input = -1) then begin code := 1; Vary_setup; end e l se c_Zn_bulk := i n p u t ; Wri te (' [HL]bulk = ' ) ; Readln ( Input) ; i f (Input = -1) then begin code := 2; Vary_setup; end e l s e c_HL_ef f_bulk := i n p u t ; Wri te (' pH = ' ) ; Readln ( Input) ; i f (Input = -1) then begin code := 3; Vary_eetup; end e l s e pH := i n p u t ; end; ( Input_Parameters } { temporary storage v a r i a b l e ) ( no range yet } (to vary , enter -1 at the p r o m p t ) ' ) ; { s e l e c t a range of va lues ) ( range i s fo r Zn ) ( get the range ) ( s e l e c t a range of va lues } ( range i s fo r HL ) ( get the range ) ( s e l e c t a range of va lues ) ( range i s f o r H* ) { get the range ) Procedure Compute_Interface_Concentrations ( J Z n : R e a l ) ; { Th is procedure computes the i n t e r f a c i a l concent ra t ions from F i c k ' s f i r s t law f o r the s p e c i f i e d f l u x JZn. The e x i s t e n c e o f both ZnL2-HL and ZnL2-(HL) 2 compl icates the procedure somewhat, so that 126 an i t e r a t i v e approach i s needed to f i n d the c o r r e c t value f o r n, the a s s o c i a t i o n f a c t o r . The i t e r a t i o n i s c a r r i e d out u n t i l the f r a c t i o n a l d i f f e r e n c e between the n_new and n_old i s l e s s tha the e r r o r c r i t e r i a eps. } const eps = le-4; var JHL2eff, JHp, JZL : Real; a, b, c, r o o t l , root2 : r e a l ; n_old, n_new : r e a l ; begin { Compute_Interface_Concentrations ] c_Zn_i := - JZn / k_aq_Zn + c_Zn_bulk; i f c _ z n _ i < 0 then c _ z n _ i := le-10; JHp := -2 * JZn; c_Hp_i := - JHp / k_aq_Hp + c_Hp_bulk; i f c_hp_i < 0 then c_hp_i := le-10; n_new := n; repeat n_old := n; n := n_new; JHL2eff := JZn * n; ( Compute the i n t e r f a c i a l [Zn2*] ) { i f i t ' s negative, make i t very small } { from the mass balance } { Compute the i n t e r f a c i a l [H+] ) { i f i t ' s negative, make i t very small } { set up i n i t i a l value } { i t e r a t e n's u n t i l within e r r o r eps ) { from the mass balance } c_HL2tot_i := - JHL2eff / k_or_HL2 + 0.5 * c_HL_eff_bulk; i f c_HL2tot_i < 0 then c_HL2tot_i := le-10; { i f i t ' s negative, make i t very small { solve f o r p a r t i t i o n between HL and (HL) 2 using the quadratic equation } a := Kd; b := 1/2; c := -c_HL2tot_i; Q u a d r a t i c ( a , b, c , r o o t l , root2) ; c_HL_i := r o o t l ; c HL2 i := c.HL2tot i - l / 2 * c HL i ; { i n t e r f a c i a l HL i s equal to the f i r s t root } ( compute i n t e r f a c i a l (HL) 2 from mass balance JZL := JZn; { from the mass balance } { compute the i n t e r f a c i a l concentrations of ZnL2-HL and ZnL 2-(HL) 2 } c_ZnL2HLl_i := (JZL) / (k_or_ZLl + k_or_ZL2 * c_HL_i / betal3_over_betal4); c_ZnL2HL2_i := (JZL) / (k_or_ZLl * betal3_over_betal4 / c_HL_i + k_or_ZL2); c _ Z L t o t _ i := c_ZnL2HLl_i + c_ZnL2HL2_i; n_new := (1.5 * c_ZnL2HLl_i + 2 * c_ZnL2HL2_i) / c _ Z L t o t _ i ; U n t i l ( (abs(n_new-n_old)/n_new) < eps ); n := n_new; end; { Compute_Interface_Concentrations } F u n c t i o n F ( x : r e a l ) : r e a l ; { The i n t e r f a c i a l concentrations are computed f o r the given f l u x X by the f u n c t i o n Compute_Interface_Concentrations. The e q u i l i b r i u m constant f o r the given i n t e r f a c i a l concentrations i s then computed and compared with the actual e q u i l i b r i u m constant. Note that i n t e r f a c i a l concentrations are used i n the Kex expression rather than a c t i v i t i e s , i . e . a l l a c t i v i t y c o e f f i c i e n t s are assumed to be equal to 1. ) begin { F } C o m p u t e _ I n t e r f a c e _ C o n c e n t r a t i o n s ( X ) ; F := c_ Z n L 2 H L l_i * Power(c_Hp_i,2) / c_Zn_i / P o w e r(c_ H L 2_i,1.5) - K_ex; end | F ); Procedure Solve( X i n i t i a l , X f i n a l , d X , e p s : r e a l ; v a r X3 : r e a l ) ; { t h i s procedure uses an incremental search followed by a b i s e c t i o n routine to f i n d the root of the f u n c t i o n F ( X ) . F(X) i s defined as the d i f f e r e n c e between the e q u i l i b r i u m constant computed 127 from the i n t e r f a c i a l c o n c e n t r a t i o n s and the ac tua l va lue of the e q u i l i b r i u m c o n s t a n t . In the f u n c t i o n , the i n t e r f a c i a l c o n c e n t r a t i o n s are f i r s t computed f o r a z i n c f l u x X, and then the d i f f e r e n c e between the two e q u i l i b r i u m constants i s computed. } var XI , YI , X2, Y2, X30LD, Y3 : REAL; LABEL LOOP; begin { Solve } XI := X i n i t i a l ; YI := F(X1) ; IF (YI = 0) then begin X3 := XI ; E x i t ; end; repeat X2 := XI + dX; i f (X2 > X f i n a l ) then begin W r i t e l n ( ' S o l u t i o n not found on search i n t e r v a l ' ) ; E x i t ; end; Y2 := F (X2 ) ; i f (Y1*Y2) > 0 then begin XI := X2; YI := Y2; end e l s e i f (Y1*Y2) = 0 then begin X3 := X2; E x i t ; end; u n t i l (Y1*Y2) < 0; { S ta r t B i s e c t i o n u n t i l accuracy i n X f a l l s w i th in e r r o r eps } LOOP: X3 := (XI + X2) / 2; i f (abs((X3 - XI) / XI) < eps) then E x i t ; Y3 := F (X3 ) ; i f (YI * Y3) = 0 then E x i t ; i f (YI * Y3) < 0 then begin ( y l * y 3 < 0 } X2 := X3; Y2 := Y3; end e l s e begin { y l * y 3 > 0 } XI := X3; YI := Y3; end; Goto LOOP; end; { Solve } P r o c e d u r e W r i t e _ C o n s o l e _ D a t a ; ( This procedure writes the z i n c f l u x , the a s s o c i a t i o n factor, the bulk concentrations, ) ( and the computed i n t e r f a c i a l concentrations to the screen. } begin { Write_Console_Data } W r i t e l n ( ' F l u x = ',JZn:8,' kmol/m A2/sec n = ',n:6:4); Write ('[Zn],b = ' , c_Zn_bulk: 6: 4,' ..- ' ) ; Write (' [HL]eff,b = ' , c_HL_ef f_bulk: 6: 4, ' ' ) ; Writeln('[ZnL2],b = ',c_ZL_bulk:6,'. pH = ',pH:4:2); Wr i t e l n (' [Zn],i = ',c_Zn_i:8:6); Write ('[HL]2eff,i = ' , c_HL2tot_i : 8: 6) ; W r i t e l n C [HL] , i = ' , c_HL_i: 8 : 6, ' [HL]2,i = ' , c_HL2_i : 8 : 6) ; Write ('[ZnL]tot,i = ' , c _ Z L t o t _ i : 8 : 6) ; W r i t e l n C [ZnL2HL],i = ' , c_ZnL2HLl_i : 8: 6,' [ZnL2HL2],i = ', c_ZnL2HL2_i: 8: 6) ; Wr i t e l n ; end; { Write_Console_Data } 128 P r o c e d u r e W r i t e _ F i l e _ D a t a ( v a r D a t a f i l e : t e x t ) ; { Th is procedure wr i t es the va lues f o r the bulk c o n c e n t r a t i o n s , the z i n c f l u x , the } { a s s o c i a t i o n f a c t o r , and i n t e r f a c i a l concen t ra t ions to the D a t a f i l e . } begin { W r i t e _ F i l e _ D a t a } Wri te ( D a t a f i l e , c _ Z n _ b u l k : 9 , ' ' , c _ H L _ e f f _ b u l k : 9 , ' ' , c _ Z L _ b u l k : 9 , ' ' ) ; Wri te ( D a t a f i l e , p H : 4 : 2 , ' ' , J Z n : 9 , ' ' , n : 6 : 4 , ' ' , c _ Z n _ i : 9 , ' ' , c _ H L _ i : 9 , ' ' ) ; Wri te ( D a t a f i l e , c _ H L 2 _ i : 9 , ' ' , c _ H L 2 t o t _ i : 9 , ' ' , c _ Z n L 2 H L l _ i : 9 , ' ' ) ; W r i t e l n ( D a t a f i l e , c _ Z n L 2 H L 2 _ i : 9 , ' ' , c _ Z L t o t _ i : 9 , ' ' , c _ H p _ i : 9 ) ; end; { Wri te F i l e Data } { M a i n - M a i n p r o g r a m c o d e } ( Th is i s the main program. Procedures are c a l l e d which c a l c u l a t e the mass t r a n s f e r c o e f f i c i e n t s , o b t a i n a d a t a f i l e f i lename and set up the f i l e f o r w r i t i n g , and o b t a i n the bulk c o n c e n t r a t i o n s ( or c o n c e n t r a t i o n ranges ) . A loop i s then executed which f i n d s the z i n c f lux f o r the c o n d i t i o n s s p e c i f i e d , and wr i tes va lues to the screen and the d a t a f i l e . Th is loop cont inues u n t i l the f l u x f o r a l l concen t ra t ion va lues s p e c i f i e d has been computed. The d a t a f i l e i s then c l o s e d and execut ion i s terminated . } ( The v a r i a b l e vary i n i t s v a r i o u s permutat ions i s used to accomodate the c i rcumstance where the z i n c f l u x i s computed f o r a range of v a l u e s . In t h i s case , code i s set to some value other than zero , i t s p a r t i c u l a r va lue s p e c i f y i n g the parameter to be a l t e r e d . Vary_i, vary_f, vary_inc, and vary_value are then used to c y c l e the parameter through the range. } X i = l e - 9 ; Xf = 5e -7 ; dX = 5e-9 ; eps = l e - 4 ; D a t a f i l e : t e x t ; code : i n t e g e r ; { I n i t i a l f l u x value f o r i t e r a t i v e search rou t ine ) { F i n a l f l u x value f o r i t e r a t i v e search r o u t i n e } ( Incremental f l u x value f o r i t e r a t i v e search r o u t i n e ) ( E r r o r c r i t e r i a f o r i t e r a t i v e search rou t ine } ( Output D a t a f i l e ) { Parameter code } begin { Main ) C l r s c r ; C a l c u l a t e _ C o n s t a n t s ; O p e n D a t a F i l e ( D a t a f i l e , ' . o u t ' ) ; { Compute mass t r a n s f e r c o e f f i c i e n t s ) { Get the f i l e Fi lename and prepare for f i l e wr i te ) { Execute I/O r o u t i n e to get va lues f o r the aqueous z i n c c o n c e n t r a t i o n , D2EHPA concent ra t ion ) { (expressed as monomer), and aqueous pH. The rout ine re turns these v a l u e s , as we l l as a ) ( code which i n d i c a t e s i f a parameter has been s e l e c t e d to be v a r i e d over a range. } I n p u t _ P a r a m e t e r s ( c _ Z n _ b u l k , c_HL_ef f_bulk , pH, code) ; Vary_value := V a r y _ i ; { set the range v a r i a b l e equal to the range i n i t i a l value } { i f code i s nonzero, set the appropr ia te parameter equal to the range v a r i a b l e } i f code = 1 then c_Zn_bulk := Vary_va lue ; i f code = 2 then c_HL_eff_bulk := Vary_va lue ; i f code = 3 then pH := Vary_va lue; Repeat n := 1.5; c_Hp_bulk := exp(-pH * 2 .303); S o l v e ( X i , X f , d X , e p s , J Z n ) ; W r i t e _ C o n s o l e _ D a t a ; W r i t e F i l e D a t a ( D a t a F i l e ) , { Loop once, or u n t i l range maximum i s exceeded ) { An i n i t i a l value f o r n ) { convert pH i n t o H* c o n c e n t r a t i o n } ( Conduct an i t e r a t i v e search f o r JZn over the range Xi - Xf, i n t e r v a l dX, and e r r o r c r i t e r i a eps. ( D i s p l a y the s o l u t i o n on the screen } ( Output va lues to s p e c i f i e d D a t a f i l e } { I f code i s not equal to zero , i . e . run f o r a range of v a l u e s , increment the range ) { v a r i a b l e and then set the appropr ia te parameter equal to the new range v a l u e . } i f not (code = 0) then begin Vary_value := Vary_value + V a r y _ i n c ; i f code = 1 then c_Zn_bu'lk := Vary_va lue ; i f code = 2 then c_HL_eff_bulk := Vary_va lue ; i f code = 3 then pH := Vary_va lue ; end; u n t i l ((code = 0) or (Vary_value > Vary_f ) ) , C l o s e ( D a t a f i l e ) ; end. { Main } ( C lose the f i l e } 129 APPENDIX E - Extended Mathematical Model The mathematical derivation and program flowchart for the extended mathematical model have been given in Section 4.4, and will not be reproduced here. The source code for the program follows as Table E. l . Table E . l : Pascal Listing of Extended Mathematical Model P r o g r a m P r e l o a d ; { guide to v a r i a b l e n o t a t i o n : Zn = Z n 2 * ( , q l , Hp 3 _ H * ( a q l , HL = HL ( o r g | , HL2 = (HL) 2 , „ „ , ZL = t o t a l o rgan ic z i n c , ZnL2HL0 = Z n L 2 ( o r g l , ZnL2HLl = Z n L 2 - H L ( o r g ] , and ZnL2HL2 H ZnL 2 -(HL) 2 ( o r g , } ( a s u f f i x of b i n d i c a t e s a bulk s p e c i e s , and i i n d i c a t e s an i n t e r f a c i a l s p e c i e s } const b l2_over_b l3 = 6e-5; b l3_over_b l4 = l e - 3 ; Kd = 5.012e4; K_ex = 3 .236e-2; c_Zn_bulk = 0.05; pH = 4 .5 ; c_HL_to ta l_bu lk = 0.05; w = 100; { r a t i o of ZnL 2 and ZnL 2-HL 2 format ion constants ) { r a t i o of ZnL2-HL and ZnL 2 -(HL) 2 format ion c o n s t a n t s , = 10 A -3 } { D imer i za t ion constant f o r D2EHPA, = 10 A 4.7 ) { E q u i l i b r i u m constant e x t r a c t i o n r e a c t i o n } { Aqueous z i n c concen t ra t ion } { Bulk pH } { T o t a l D2EHPA (as monomer) i n o rgan ic bulk ) { RDC r o t a t i o n a l speed, i n rpm } type F i l e s t r i n g = s t r i n g [ 1 4 ] ; MaxStr ing = s t r i n g [ 2 5 5 ] ; Ex tens ion = s t r i n g [ 4 ] ; k_aq_Zn, k_aq_Hp, k_or_HL, k_or_HL2, k_or_ZL0, k _ o r _ Z L l , k_or_ZL2, c_Hp_bulk, l o a d i n g , c_HL_ef f_bulk , c_HL_bulk, c_HL2_bulk, c_ZL_bulk , c_ZnL2HL0_b, c_ZnL2HLl_b, c_ZnL2HL2_ c _ Z n _ i , c Hp i , c HL2tot i , c _ H L _ i , c_HL2 i . c _ Z L t o t _ i , c_ZnL2HL0_ JZn , J Z L , n, KCZnbulk, KCbulk, V a r y _ i , V a r y _ f , Vary i n c c ZnL2HLl i , c ZnL2HL2 i . Vary_value : r e a l ; mass t r a n s f e r c o e f f i c i e n t s ) mass t r a n s f e r c o e f f i c i e n t s } spec ies bulk c o n c e n t r a t i o n s ) spec ies bulk c o n c e n t r a t i o n s ) spec ies bulk c o n c e n t r a t i o n s ) spec ies i n t e r f a c i a l concen t ra t ions ) spec ies i n t e r f a c i a l concen t ra t ions ) z i n c f l u x i n kmol /m 2 /sec ) a s s o c i a t i o n f a c t o r } in termedia te mass t r a n s f e r va lues } i n i t i a l , f i n a l , increment , and cur rent parameter va lues } P r o c e d u r e B e e p ; { Beeps the speaker - used when user has made a data entry e r r o r ) { or to a l e r t the user to a p o s s i b l e problem. ) begin { Beep } sound(880); { Beep at 880 Hz } Delay(125); { f o r 125 m i l l i s e c o n d s . } Nosound; { Turn o f f sound. } end; { Beep } 130 F u n c t i o n K e y b o a r d l n p u t : c h a r ; { Waits u n t i l a key i s p r e s s e d , and then repor ts the value of the key. ) var Key : char ; begin { Keyboardlnput ) Repeat Delay(50) u n t i l KeyPressed; Read(Kbd,Key); Keyboardlnput := Key; end; ( Keyboardlnput ) F u n c t i o n Q u e s t i o n ( p r o m p t : M a x S t r i n g ) : b o o l e a n ; ( Asks the Yes or No ques t ion conta ined i n the s t r i n g prompt and then wai ts u n t i l a Y or N ) ( answer has been r e c e i v e d . The value of the answer i s then repor ted by the va lue of t h i s ) { boolean f u n c t i o n - True s Yes and F a l s e = No. ) var query : char ; begin { Quest ion } Repeat Wr i te ( prompt + ' <Y/N> ' + chr(8) ); query := K e y b o a r d l n p u t ; w r i t e l n ( q u e r y ) ; U n t i l (query i n [ ' Y ' , ' y ' , ' N ' , ' n ' ] ) ; ( Keep asking u n t i l a Y or N answer i s rece ived ) I f (query i n [ ' Y ' , ' y ' ] ) then Quest ion := t rue e l s e Quest ion := f a l s e ; end; { Quest ion ) P r o c e d u r e O p e n D a t a F i l e ( v a r D a t a f i l e : t e x t ; f i l e n a m e _ e x t e n s i o n : e x t e n s i o n ) ; ( T h i s procedure prompts the user f o r the d e s i r e d d a t a f i l e f i lename, adds the ex tens ion } { s p e c i f i e d , and then checks to see i f t h i s name i s a l ready present on the d i s k . I f i t } ( i s , the user i s g iven the cho ice of o v e r w r i t i n g i t . I f the user d e c l i n e s , then they } { are prompted to enter the f i lename a g a i n . The f i l e v a r i a b l e Datafile, i s used i n } { o ther r o u t i n e s f o r a c c e s s i n g the f i l e . } var Fi lename : F i l e s t r i n g ; ( name of data f i l e } E x i s t : boo lean; ( t rue i f s p e c i f i e d f i lename a l ready e x i s t s on d i s k ) OKf lag : boo lean; { f l a g i n d i c a t e s when f i lename has been s u c c e s s f u l l y s e l e c t e d ) begin { OpenDataFi le ) OKf lag := f a l s e ; { f i lename has not been s u c c e s s f u l l y s e l e c t e d yet } repeat { repeat u n t i l OKFlag ) repeat { Read f i lenames u n t i l at l e a s t one charac te r i s entered } Wr i t e ( 'Save f i l ename: ' ) ; Readln (F i lename) ; u n t i l ( length( f i lename) > 0) ; { i f the f i lename i s longer than 4 c h a r a c t e r s , check to see i f the ex tens ion ) { matches filename_extension. I f so , s t r i p i t o f f . ) i f ( length( f i lename) > 4) then i f ( copy ( f i l ename, length ( f i l ename) -3 , l ength ( f i l ename) )= f i l ename_ex tens ion ) then f i lename := c o p y ( f i l e n a m e , 1 , l e n g t h ( f i l e n a m e ) - 4 ) ; { i f the f i lename i s longer than 8 c h a r a c t e r s , f i lename i s equal to f i r s t 8 charac te rs } I f ( length( f i lename) > 8) then f i lename := c o p y ( f i l e n a m e , 1 , 8 ) ; Fi lename := Fi lename + f i l ename_extens ion; ( Add the extension ) A s s i g n ( D a t a f i l e , F i lename) ; { Def ine the f i l e v a r i a b l e ) { Turn o f f e r r o r check ing . Reset d a t a f i l e , and then re -enab le e r r o r c h e c k i n g . I f ) { D a t a f i l e e x i s t s , then IOresu l t w i l l be equal to 0. } ($1-) R e s e t ( D a t a f i l e ) ($1+) ; E x i s t := ( IOresul t = 0 ); { I f the f i l e a l ready e x i s t s on the d i s k . E x i s t = t rue ) OKf lag := not E x i s t ; { The f i lename i s OK to use i f i t d o e s n ' t E x i s t ) 131 { I f the f i lename e x i s t s , then warn the user and ask whether or not to overwr i te ) i f E x i s t then begin B e e p ; B e e p ; W r i t e l n ; Writeln('WARNING: F i l e " ' . F i l e n a m e , ' " a l ready e x i s t s ! ' ) ; { I f they answer yes to the q u e s t i o n , then the f i lename i s OK, so set } { OKf lag=true . C lose the D a t a f i l e and Erase i t to s t a r t wi th a c lean s l a t e . ) I f Q u e s t i o n ( ' D o you want to erase i t ? ' ) then begin OKf lag := t r u e ; C l o s e ( D a t a f i l e ) ; E r a s e ( D a t a f i l e ) ; end; end u n t i l O K f l a g ; { Repeat ask ing fo r f i lenames u n t i l one i s OK } W r i t e l n ( ' O u t p u t d a t a f i l e : ' , F i l e n a m e ) ; { I den t i f y the output f i lename s e l e c t e d } W r i t e l n ; R e w r i t e ( D a t a f i l e ) ; ( Open and empty the f i l e - set p o i n t e r to the beginn ing ) ( Wri te f i l e header } W r i t e l n ( D a t a f i l e , ' w = ' , w : 4 ) ; Write ( D a t a f i l e , ' l o a d i n g [Znjbulk [HL ]e f f ,b [ZnL]b '),- Wri te ( D a t a f i l e , ' [ H L ] b u l k [HL2]bulk [ZnL0]b [ZnLl]b [ZnL2]b ' ) ; Wri te ( D a t a f i l e , ' pH,b JZn n ' ) ; Wri te ( D a t a f i l e , ' [Zn] i [HL] i [ (HL)2] i [ H L 2 ] i e f f ) ; W r i t e l n ( D a t a f i l e , ' [ZnL0] i [ Z n L l ] i [ZnL2]i [ Z n L ] i , t [H+] i ' ) ; end; { OpenDataFi le } F u n c t i o n P o w e r ( x , n : r e a l ) : r e a l ; f Computes the va lue of x" and re turns the value as Power. } begin { Power } power := e x p ( n * l n ( x ) ) ; end; { Power } P r o c e d u r e Q u a d r a t i c ( a , b , c : r e a l ; v a r r o o t l , r o o t 2 : r e a l ) ; ( Solves the q u a d r a t i c equat ion and re turns the two roots as rootl and root2. ) begin ( Quadra t ic } r o o t l := (-b + s q r t ( b * b - 4 * a * c ) ) / ( 2 * a ) ; root2 := <-b - sqr t (b*b-4*a*c) ) / (2*a) ,- end; { Quadra t ic } P r o c e d u r e C a l c u l a t e C o n s t a n t s ; ( Computes the mass t r a n s f e r c o e f f i c i e n t s f o r each s p e c i e s . The L e v i c h } { equ iva len t boundary l a y e r t h i c k n e s s i s f i r s t c a l c u l a t e d , and then the ) ( mass t r a n s f e r c o e f f i c i e n t i s computed u s i n g the appropr ia te formula . ) const D_HL D_HL2 D_ZnL2HL0 D_ZnL2HLl D_ZnL2HL2 D_HC104 D_Zn nu H20 1.520e 1.003e 0.993e 0.781e 0.659e 2.854e 1.038e 8.93e--7; nu_heptane = 5 .64e-7; L_over_alpha = 1.90e-4; var zD_aq_Zn, zD_aq_Hp, zD_or_HL, zD_or_HL2, zD_or_ZL0, zD_or_ZL l , zD_or_ZL2 : r e a l ; begin { C a l c u l a t e _ C o n s t a n t s ) zD_aq_Zn := 0 .643/sqrt (w/60)*Power (nu_H20, (1/6)) * Power (D_Zn, (1 /3 ) ) ; 132 zD_aq_Hp zD_or_HL zD_or_HL2 zD_or_ZLO zD_or_ZLl zD or ZL2 0. 643/sqrt (w/60)*Power (nu_H20, (1/6)) * Power(D_HC104, (1 /3 ) ) ; 0 .643 /sqr t ( w / 60 )* P o w e r (nu_heptane , (1 /6 ) ] 0 .643 /sqr t ( w / 60 )* P o w e r (nu_heptane , (1/6)) 0 .643 /sqr t ( w / 60 )* P o w e r (nu_heptane , (1/6)) 0 .643 /sqr t ( w / 60 )* P o w e r (nu_heptane , (1/6)) 0 .643 /sqr t ( w / 60 )* P o w e r (nu_heptane , (1/6)) Power (D_HL , ( 1 / 3 ) ) ; Power(D_HL2 , (1/3) ) ; Power(D_ZnL2HL0, (1 /3) ) , Power (D_ZnL2HLl , (1/3)) , Power (D ZnL2HL2, (1/3)); k_aq_Zn k_aq_Hp k_or_HL k_or_HL2 k_or_ZL0 k_or_ZLl k or ZL2 D_Zn D_HC104 D_HL D HL2 D ZnL2HL0 / D_ZnL2HLl D ZnL2HL2 zD_aq_Zn; zD_aq_Hp; (zD_or_HL (zD_or_HL2 (zD_or_ZL0 (zD_or_ZLl (zD or ZL2 L_over_alpha) ; L_over_alpha) ; L_over_alpha) ; L_over_alpha) ; L_over a l p h a ) ; end; { C a l c u l a t e _ C o n s t a n t s } P r o c e d u r e V a r y _ S e t u p ; { Th is procedure just gets the i n i t i a l , f i n a l , and increment va lue when ) ( a range of c o n c e n t r a t i o n s i s examined. } begin { Vary_Setup } Wri te (' I n i t i a l V a l u e : ' ) ; Readln (Vary_ i ) ; Wri te (' F i n a l Va lue: ' ) ; Readln (Vary_f ) ; Wri te (' Increment : ' ) ; Readln (Vary_ inc ) ; i f Vary i > Vary_f { recurse to get c o r r e c t va lues i f i n i t i a l i s > than f i n a l ) then V a r y _ S e t u p e l s e i f V a r y _ i n c <= 0 { recurse to get c o r r e c t va lues i f increment i s < 0 } then V a r y _ S e t u p ; end; ( Vary_Setup } P r o c e d u r e I n p u t _ P a r a m e t e r s ( v a r l o a d i n g : r e a l ; v a r v a r y c o d e : i n t e g e r ) ; { Th is procedure prompts the user f o r a va lue f o r the percentage z i n c l o a d i n g i n the organ ic } ( phase. The user can examine a range of va lues by e n t e r i n g a va lue of - 1 . In t h i s case , ) ( the subrout ine Vary_Setup i s c a l l e d which prompts the user f o r the i n i t i a l , f i n a l , and ) { incrementa l v a l u e s . varycode t e l l s the program whether a range has been s e l e c t e d or not ; ) ( a va lue of zero i n d i c a t e s no range, whi le 1 i n d i c a t e s a l o a d i n g range. } var input : r e a l ; { temporary storage v a r i a b l e } begin { Input_Parameters ) varycode := 0; W r i t e l n ; W r i t e l n C Enter model parameters: Wr i te (' l o a d i n g = ' ) ; Readln ( Input) ; i f (Input = -1) then begin varycode := 1; V a r y _ s e t u p ; end e l s e l o a d i n g := i n p u t ; end; { Input_Parameters } (to vary , enter -1 at the p r o m p t ) ' ) ; ( s e l e c t a range of va lues } { get the range } P r o c e d u r e S o l v e ( X i n i t i a l , X f i n a l , d X , e p s : r e a l ; v a r X 3 : r e a l ; f u n c t i o n c o d e : i n t e g e r ) ; f o r w a r d ; { Master root f i n d i n g r o u t i n e . S ince Solve re fe rences S o l v e f u n c t i o n , which re fe rences } ( Compute_ Inter face_Concentrat ions, which re fe rences S o l v e , a forward d e c l a r a t i o n i s ) { r e q u i r e d . functioncode i n d i c a t e s which f u n c t i o n i s to be so lved f o r ; i t a l s o ) ( s e l e c t s the increment method; i f functioncode i s equal to 1 then dx i s added to the ) { cur ren t va lue of X f o r each i n t e r v a l , but i f functioncode i s not equal to 1, then X ) I i s m u l t i p l i e d by dX f o r each i n t e r v a l . ) 133 P r o c e d u r e C o m p u t e _ B u l k _ C o n c e n t r a t i o n s ; ( Th is procedure computes the c o n c e n t r a t i o n of HL, (HL) 2 , ZnL 2 , ZnL 2-HL, and ZnL 2 -(HL) 2 } { based on the c o n d i t i o n s i n the bulk ( t o t a l D2EHPA concen t ra t ion and l o a d i n g ) . ) const Xi= l e - 1 0 ; dX = 2; eps = l e - 4 ; var X f , temp : r e a l ; begin { Compute_Bulk_Concentrat ions ) c_Hp_bulk := exp(-pH * 2 .303); ( compute [H*] from pH ) c_ZL_bulk := c_HL_tota l_bulk 12* l o a d i n g ; { compute z i n c c o n c e n t r a t i o n from l o a d i n g } { i f the bulk z i n c c o n c e n t r a t i o n i s equal to zero , then there i s no s p e c i a t i o n o f ) { o rgan ic z i n c i n the b u l k . There fo re , the e f f e c t i v e concen t ra t ion of HL i s equal ) { to the t o t a l c o n c e n t r a t i o n of HL i n the b u l k . The p a r t i t i o n between HL and } { (HL) 2 can t h e r e f o r e be c a l c u l a t e d . ) i f (c_ZL_bulk = 0) then begin c_HL_eff_bulk := c_HL_to ta l_bu lk ; Q u a d r a t i o ( K d , 0 . 5 , - c _ H L _ e f f bu lk /2 ,c_HL_bu lk , t emp) ; c_HL2_bulk := (c_HL_ef f _ b u T k -c_HL_bulk ) /2 ; c_ZnL2HL0_b := 0; c_ZnL2HLl_b := 0; c_ZnL2HL2_b := 0; end e l s e begin Xf := 2 * c_HL_to ta l_bu lk ; S o l v e ( X i , X f , d X , e p s , c _ H L _ b u l k , 3 ) ; { Solve over the i n t e r v a l , SOLVEFUNCTION #3 ) c_HL2_bulk := Kd * c_HL_bulk * c_HL_bulk; c_HL_eff_bulk := c_HL_bulk + 2 * c_HL2_bulk; c_ZnL2HLl_b := c_ZL_bulk / (b l2_over_b l3 /c HL_bulk+l+c_HL_bulk /b l3_over_bl4) ; c_ZnL2HL0_b := c_ZnL2HLl_b * b l2_over_bl3 7 c_HL_bulk; c_ZnL2HL2_b := c_ZnL2HLl_b * c_HL_bulk / b l3_over_b l4 ; end; end; { Compute_Bulk_Concentrat ions } P r o c e d u r e C o m p u t e _ I n t e r f a c e _ C o n c e n t r a t i o n s ( x : r e a l ) ; ( T h i s procedure computes the i n t e r f a c i a l concent ra t ions f o r the s p e c i f i e d f l u x JZn. The i n t e r f a c i a l c o n c e n t r a t i o n s f o r z i n c and the hydrogen i o n can be c a l c u l a t e d d i r e c t l y from F i c k ' s f i r s t law, but an i t e r a t i v e approach must be used to compute the i n t e r f a c i a l concen t ra t ions of the f i v e o rgan ic s p e c i e s . The procedure Solve i s used to compute the c o n c e n t r a t i o n of HL at the i n t e r f a c e . The upper l i m i t f o r the search i n t e r v a l , Yf, i s set at 1.5 t imes the t o t a l HL c o n c e n t r a t i o n i n the b u l k ; a m u l t i p l i e r o f 1.5 i s used s ince dY i s m u l t i p l i c a t i v e and a l s o equal to 1 .5 . Thus, complete coverage of the e n t i r e i n t e r v a l i s ensured . The i n t e r f a c i a l dimer c o n c e n t r a t i o n i s then c a l c u l a t e d , and the e f f e c t i v e t o t a l dimer c o n c e n t r a t i o n at the i n t e r f a c e i s found by adding the dimer and monomer c o n t r i b u t i o n s . F i n a l l y , the c o n c e n t r a t i o n s of the organ ic z i n c spec ies may be computed. ) const Y i = l e - 1 0 ; dY = 1.5; eps = l e - 4 ; var JHL2ef f , JHp, JZn : r e a l ; Yf : r e a l ; begin { Compute_Inter face_Concentrat ions ) JZn ':= x; c_Zn_i := - Jzn / k_aq_Zn + c_Zn_bulk; i f c_zn_ i < 0 then c_zn_i := l e - 1 0 ; JHp := -2 * J z n ; c_Hp_i := - JHp / k_aq_Hp + c_Hp_bulk; i f c_hp_i < 0 then c_hp_i := l e - 1 0 ; JZL := JZn; Yf := c_HL_eff_bulk * 1.5; S o l v e ( Y i , Y f , d Y , e p s , c _ H L _ i , 2 ) ; { Solve over the i n t e r v a l , SOLVEFUNCTION #2 } c_HL2_i := Kd * c_HL i * c _ H L _ i ; 134 c_HL2tot_ i := c_HL_i / 2 + c_HL2_i ; c_ZnL2HLl_ i := (JZL + KCZnbulk) / (k or_ZLO * b l2_over_bl3 / c_HL_i + k_or_ZLl + k_or_ZL2 * c_HL_i 7 b l 3_over_bl4) ; c_ZnL2HL0_i := c_ZnL2HLl_ i * b l2_over_b l3 / c _ H L _ i ; c_ZnL2HL2_i := c_ZnL2HLl_ i * c_HL_i / b l3_over_b l4 ; c _ Z L t o t _ i := c_ZnL2HL0_i + c_ZnL2HLl_i + c_ZnL2HL2_i; { compute the a s s o c i a t i o n f a c t o r at the i n t e r f a c e ) n := (c_ZnL2HL0_i + 1.5 * c_ZnL2HLl_i + 2 * c_ZnL2HL2_i) / c _ Z L t o t _ i ; end; { Compute_Inter face_Concentrat ions ) Function SolveFunction ( x : r e a l ; F u n c t i o n C o d e : i n t e g e r ) : r e a l ; { T h i s f u n c t i o n i s a g e n e r a l i z e d f u n c t i o n which i s c a l l e d by the r o o t - f i n d i n g procedure SOLVE. SOLVE i s used to f i n d the roots of three d i f f e r e n t f u n c t i o n s ; the p a r t i c u l a r f u n c t i o n i s s e l e c t e d by the va lue of Functioncode. I f Functioncode i s equal to 1, then SOLVE has been c a l l e d from Main, where i t i s used to f i n d the Z inc f l u x , JZn. Th is p a r t i c u l a r f u n c t i o n c a l l s the procedure Compute_ Inter face_Concentrat ions, which f i n d s the i n t e r f a c i a l c o n c e n t r a t i o n of HL v i a an i t e r a t i v e approach. It the re fo re c a l l s SOLVE aga in , t h i s time with a Functioncode equal to 2. F i n a l l y , SOLVE i s used by the procedure Compute_Bulk_Concentrations to f i n d the bulk c o n c e n t r a t i o n of HL, and S o l v e f u n c t i o n i s c a l l e d with Functioncode equal to 3. ) var C Z N L l _ i , tempi , temp2 : r e a l ; begin i f (FunctionCode = 1) then begin C o m p u t e _ I n t e r f a c e _ C o n c e n t r a t i o n s ( X ) ; So lveFunc t ion := c_ZnL2HLl_ i * P o w e r (c_Hp_i ,2 ) / c_Zn_i / P o w e r ( c _ H L 2 _ i , 1 . 5 ) - K_ex; end; i f (FunctionCode = 2) then begin CZNLl_ i := (JZL+KCZnbulk) / (k_or_ZL0*bl2_over_bl3 /x + k_or_ZLl + k_or_ZL2*x/bl3_over_bl4) ; S o l v e F u n c t i o n := k_or HL*x + 2*k_or_HL2*Kd*x*x + 2 *k_or_ZL0*CZNLl_ i *b l2_over_b l3 /x + 3*k"_or_ZLl*CZNLl_i + 4 *k_or_ZL2*CZNLl_ i *x /b l3_over_b l4 - KCbulk; end; i f (FunctionCode = 3) then So lveFunct ion := c_HL_tota l_bulk - x - 2 * Kd * x*x - ( 2 *b l2_over_b l3 /x + 3 + 4 *x /b l3_over_b l4 ) * c_ZL_bulk / (b l2_over_bl3 /x + 1 + x / b l 3 _ o v e r _ b l 4 ) ; end; Procedure Solve{ ( X i n i t i a l , X f i n a l < d X , e p s : r e a l ; v a r X 3 : r e a l ; f u n c t i o n c o d e : i n t e g e r ) } ; { Master root f i n d i n g r o u t i n e . Th is procedure uses an incremental search fo l lowed by a b i s e c t i o n method to f i n d the root of the f u n c t i o n SOLVEFUNCTION(x,functioncode). functioncode i n d i c a t e s which f u n c t i o n i s to be so lved f o r ; i t a l s o s e l e c t s the increment method; i f functioncode i s equal to 1 then dX i s added to the cur rent va lue of X fo r each i n t e r v a l , but i f functioncode i s not equal to 1, then X i s m u l t i p l i e d by dX f o r each i n t e r v a l . For f u n c t i o n #1, the va lue of the f u n c t i o n changes very r a p i d l y , and t h e r e f o r e smal l s teps were r e q u i r e d . Converse ly , f o r f u n c t i o n s #2 and 3, the e q u i l i b r i u m changes s lowly with x and a l a rge i n t e r v a l must be searched, and t h e r e f o r e i n s t e a d of adding dX to X, a m u l t i p l i c a t i v e approach i s used . ) var X I , Y l , X2, Y2, X30LD, Y3 : REAL; LABEL LOOP; begin { Solve } XI := X i n i t i a l ; Y l := S o l v e F u n c t i o n ( X I , F u n c t i o n C o d e ) ; IF (Yl = 0) then begin X3 := XI ; E x i t ; end; 135 Repeat i f (Funct ioncode = 1) then X2 := XI + dx { s e l e c t i t e r a t i o n method } e l s e X2 := XI * dx; i f (X2 > X f i n a l ) then begin W r i t e l n ( ' S o l u t i o n not found on search i n t e r v a l ' ) ; E x i t ; end; Y2 := S o l v e F u n c t i o n ( X 2 , F u n c t i o n C o d e ) ; i f <Y1*Y2) > 0 then begin XI := X2; YI := Y2; end e l s e IF (Y1*Y2) = 0 then begin X3 := X2; E x i t ; end; u n t i l (Y1*Y2) < 0; ( S ta r t B i s e c t i o n u n t i l accuracy i n X f a l l s w i th in e r r o r eps } LOOP: X3 := (XI + X2) 12; i f (abs((X3 - XI) / XI) < eps) then E x i t ; Y3 := S o l v e F u n c t i o n ( X 3 , F u n c t i o n C o d e ) ; IF (YI * Y3) = 0 then E x i t ; IF (YI * Y3) < 0 then beg in ( y l * y 3 < 0 } X2 := X3; Y2 := Y3; end e l s e beg in { y l * y 3 > 0 ) XI := X3; Y l := Y3; end; GOTO'LOOP; end; { Solve } P r o c e d u r e W r i t e _ C o n s o l e _ D a t a ; { Th is procedure w r i t e s the z i n c f l u x , the a s s o c i a t i o n f a c t o r , and the } ( computed bulk and i n t e r f a c i a l concent ra t ions to the sc reen . } begin { Wri te_Console_Data } W r i t e l n C F l u x = ' , J Z n : 8 , ' kmol /m / , 2/sec n = ' , n : 6 : 4 ) ; Wri te ( ' [ Z n ] , b = ' , c _ Z n _ b u l k : 6 : 4 , ' ' ) ; W r i t e l n ( ' [ Z n L 2 ] , b = ' , c _ Z L _ b u l k : 6 : 4 , ' pH = ' , p H : 4 : 2 ) ; W r i t e l n (' [Zn] , i = ' , c_Zn_i : 8: 6) ; Wri te ( ' [ H L ] e f f , b = ' , c _ H L _ e f f _ b u l k : 6 : 4 , ' ' ) ; W r i t e l n C [HL] , b = ' , c_HL_bulk: 8: 6 , ' [HL]2,b = ' , c_HL2_bulk: 8: 6) ; Wri te ( ' [ H L ] 2 e f f , i = ' , c_HL2tot_ i : 8: 6) ; W r i t e l n C [HL] , i = ' , c_HL_i : 8: 6 , ' [HL]2 , i = ' , c_HL2_i : 8: 6) ; W r i t e l n ( ' [ Z n L ] t o t , i = ' , c _ Z L t o t _ i : 8 : 6 ) ; W r i t e l n C [ Z n L 2 ] , i = ' , c_ZnL2HL0_i: 8: 6 , ' [ZnL2HL], i = ' , c_ZnL2HLl_ i : 8: 6, ' [ZnL2HL2], i = ' , c _ Z n L 2 H L 2 _ i : 8 : 6 ) ; W r i t e l n ; end; { Wri te Console Data } P r o c e d u r e W r i t e _ F i l e _ D a t a ( v a r D a t a f i l e : t e x t ) ; ( T h i s procedure w r i t e s the va lues f o r the bu lk c o n c e n t r a t i o n s , the z i n c f l u x , the } ( a s s o c i a t i o n f a c t o r , and i n t e r f a c i a l concen t ra t ions to the D a t a f i l e . } 136 begin { Wri t e _ F i l e _ D a t a } Write (Datafile,loading:6:4,' ',c_Zn_bulk:9,' ',c_HL_eff_bulk:9,' ',c_ZL_bulk:9,' ' ) ; Write (Datafile,c_HL_bulk:9,' ',c_HL2_bulk:9,' ' ) ; Write (Datafile,c_ZnL2HL0_b:9,' ',c_ZnL2HLl_b:9,' ',c_ZnL2HL2_b:9, ' ' ) ; Write (Datafile,pH:4:2,' ',JZn:9,' ',n:6:4,' ',c_Zn_i:9,' ',c_HL_i:9,' ' ) ; Write (Datafile,c_HL2_i:9,' ',c_HL2tot_i:9,' ',c_ZnL2HL0_i:9, ' ',c_ZnL2HLl_i:9,' ' ) ; Writeln(Datafile,c_ZnL2HL2_i:9,' ' , c _ Z L t o t _ i : 9 , ' ',c_Hp_i:9); end; { Wri t e _ F i l e _ D a t a ) { M a i n - M a i n p r o g r a m c o d e } { This i s the main program. Procedures are c a l l e d which c a l c u l a t e the mass t r a n s f e r c o e f f i c i e n t s , obtain a d a t a f i l e filename and set up the f i l e f o r w r i t i n g , and obtain the loading value (or range). A loop i s executed which then computes the bulk species concentrations, f i n d s the z i n c f l u x f o r the conditions s p e c i f i e d , and writes values to the screen and the d a t a f i l e . This loop continues u n t i l the zi n c f l u x f o r a l l loading values s p e c i f i e d has been computed. The d a t a f i l e i s then c l o s e d and execution i s terminated. } ( The v a r i a b l e vary i n i t s various permutations i s used to accomodate the circumstance where the zin c f l u x i s computed f o r a range of loading values. varycode equal to 1 i n d i c a t e s that the fl u x search routine i s to be executed for loadings i n the i n t e r v a l Vary_i - vary_f, with step s i z e vary_inc. ) const X i = le-11; { I n i t i a l f l u x value f o r i t e r a t i v e search routine ) Xf = 5e-7; ( F i n a l f l u x value f o r i t e r a t i v e search routine } dX = 5e-9; ( Incremental f l u x value f o r i t e r a t i v e search routine } eps = le-4; ( E r r o r c r i t e r i a f o r i t e r a t i v e search routine } var D a t a f i l e : tex t ; ( Output D a t a f i l e ) varycode : in t e g e r ; ( Parameter code ) begin { Main } C l r s c r ; C a l c u l a t e _ C o n s t a n t s ; { Compute mass t r a n s f e r c o e f f i c i e n t s } O p e n D a t a F i l e ( D a t a f i l e , ' . o u t ' ) ; ( Get the f i l e Filename and prepare f o r f i l e write ) { Execute I/O routine to get a value f o r the loading. The routine returns the loading, as } { well as varycode which i n d i c a t e s i f the user wishes to vary the parameter over a range. } I n p u t _ P a r a m e t e r s ( l o a d i n g , varycode); i f varycode = 1 ( I f a range i s selected, i n i t i a l i z e vary_value } then { and loading ) begin Vary_value := V a r y _ i ; loading := Vary_value; end; Repeat { Loop once, or u n t i l range maximum i s exceeded } C o m p u t e _ B u l k _ C o n c e n t r a t i o n s ; ( Compute constants which are used by Compute_Interface_Concentrations and Solvefunction } KCZnbulk := k_or_ZL0*c_ZnL2HL0_b + k_or_ZLl*c_ZnL2HLl_b + k_or_ZL2*c_ZnL2HL2_b; KCbulk := k_or_HL*c_HL_bulk + 2*k_or_HL2*c_HL2_bulk + 2*k_or_ZL0*c_ZnL2HL0_b + 3*k_or_ZLl*c_ZnL2HLl_b + 4*k_or_ZL2*c_ZnL2HL2_b; S o l v e(Xi,Xf,dX,eps,JZn,1); { Conduct an i t e r a t i v e search f o r JZn over the range Xi - Xf, i n t e r v a l dX, and e r r o r c r i t e r i a eps. ) W r i t e _ C o n s o l e _ D a t a ; ( Display the s o l u t i o n on the screen } W r i t e _ F i l e _ D a t a ( D a t a F i l e ) ; { Output values to s p e c i f i e d D a t a f i l e } ( I f varycode i s equal to one, increment the range v a r i a b l e and then set } { loading equal to the new range value. ) i f varycode = 1 then begin Vary_value := Vary_value + vary_in c ; l o a d i n g := Vary_value; end; u n t i l ((varycode = 0) or (Vary_value > V a r y _ f ) ) ; C l o s e ( D a t a f i l e ) ; ( Close the f i l e ) end. { Main } 137

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