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The kinetics of zinc extraction in the di(2-ethylhexyl) phosphoric acid, n-heptane-zinc perchlorate,.. MacLean, Donald William John 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 DONALD 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 F A C U L T Y OF G R A D U A T E STUDIES Department of Metals and Materials Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 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 M e t a l s and M a t e r i a l s E n g i n e e r i n g 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 and D2EHPA) to 2+  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 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 p H . 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 H L and Zn^-(HL) flux reverses, with these species providing 2  2  extractant to the interface. At very high loadings, Z n L H L provides almost all the extractant to 2  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.  ii  Table of Contents Abstract  ii  List of Tables  vi  List of Figures  vii  List of Symbols  xiii  Acknowledgements  xv  C H A P T E R 1 - Introduction  1  C H A P T E R 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 A K U F V E  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 S L M 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  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  9  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  Figure 2.5  10  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 , p H = 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 , p H = 4.5, T = 25 °C, filter = 0.45pm, © = 100 rpm  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  Figure 4.3  52  Effect of changing bulk p H on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 100 rpm  Figure 4.4  52  53  Effect of changing temperature on zinc flux: [Zn] = 0.05 M , Formal [D2EHPA] = 0.05 M , p H = 4.5, filter = 0.45pm, © = 100 rpm  53  Figure 4.5  Diagram of species and flux direction definitions for the V M R model  55  Figure 4.6  V M R predictions and experimental data for changes in bulk zinc concentration : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, © = 100 rpm  Figure 4.7  57  V M R predictions and experimental data for changes in bulk D2EHPA concentration: [Zn] = 0.05 M , p H = 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 , p H = 4.5, T = 25°C, filter = 0.45pm, © = 100 rpm  Figure 4.10  Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45um, © = 300 rpm  Figure 4.11  66  Effect of changing bulk p H on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 300 rpm  Figure 4.15  65  Effect of changing bulk p H on zinc flux: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 100 rpm  Figure 4.14  65  Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05M, p H = 4.5, T = 25 'C, filter = 0.45pm, © = 300 rpm  Figure 4.13  64  Effect of changing D2EHPA concentration on zinc flux : [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm  Figure 4.12  64  66  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 = 0.45um, © = 100 rpm  67  viii  Figure 4.16  Zinc flux vs. D2EHPA concentration for changes in the organic species diffusion coefficients : [Zn] = 0.05M, p H = 4.5, T = 25 "C, filter = 0.45pm, co = 100 rpm  Figure 4.17  67  Zinc flux vs. zinc concentration for changes in the equilibrium constant  :  Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm Figure 4.18  Zinc flux vs. D2EHPA concentration for changes in the equifibrium constant K : [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm a  Figure 4.19  68  68  Zinc flux vs. zinc concentration for changes in the filter equivalent thickness L/cc: Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  Figure 4.20  69  Zinc flux vs. D2EHPA concentration for changes in the filter equivalent thickness L/cc: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  Figure 4.21  69  Predicted change in association factor ( « ) with bulk zinc concentration: avg  Formal [D2EHPA] = 0.05 M , p H = 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  Figure 4.23  73  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  Figure 4.24  74  Predicted change in interfacial D2EHPA concentration with bulk zinc concentration: Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45um, co = 100 rpm  Figure 4.25  74  Predicted change in association factor (n ) with bulk D2EHPA avg  concentration: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm Figure 4.26  75  Predicted change in interfacial p H with bulk D2EHPA concentration: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  Figure 4.27  75  Predicted change in interfacial zinc concentration with bulk D2EHPA concentration : [Zn] = 0.05M, p H = 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  Figure 4.29  76  Predicted change in association factor (n ) with bulk p H : [Zn] = 0.05M, avg  Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45um, © = 100 rpm Figure 4.30  77  Predicted change in interfacial p H with bulk p H : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, © = 100 rpm  Figure 4.31  77  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  Figure 4.32  78  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\ /Pi3: [Zn] = 0.05M, Formal 2  [D2EHPA] Figure 4.35  total  = 0.05 M , p H = 4.5, T = 25°C, filter = 0.45pm, © = 100 rpm  Effect of preload on zinc flux: [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M ,  p H = 4.5, T = 25 °C, filter = 0.45um, © = 100 rpm, p\ /p = 6 x 10"  86  5  2  Figure 4.36  13  Effect of preload on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] p H = 4.5, T = 25°C, filter = 0.45pm, © = 300 rpm, p / p = 6 x 10 12  Figure 4.37  total  = 0.05 M , 86  s  13  Zinc flux vs. preload for changes in the organic species diffusion coefficients : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 4.5,  T = 25°C, filter = 0.45pm, © = 100 rpm, pVPis = 6 x 10 Figure 4.38  total  87  s  Zinc flux vs. preload for changes in the equilibrium constant [Zn] = 0.05M, Formal [D2EHPA]  :  = 0.05 M , p H = 4.5, T = 25 °C,  filter = 0.45um, © = 100 rpm, (VPu = 6 x 10"  5  Figure 4.39  85  87  Zinc flux vs. preload for changes in the filter equivalent thickness L / a : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 4.5, T = 25 °C,  filter = 0.45um, © = 100 rpm, pVPia = 6 x 10  x  s  88  Figure 4.40  Predicted change in interfacial zinc concentration (aqueous) with preload : [Zn] = 0.05M, Formal [D2EHPA]  = 0.05 M , p H = 4.5, T = 25 °C,  total  filter = 0.45pm, co = 100 rpm, p V P u = 6 x 10"  s  Figure 4.41  Predicted change in interfacial p H with preload: [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm,  pVPi3 = 6xl0- ....  91  5  Figure 4.42  Predicted change in bulk D2EHPA concentration with preload : [Zn] = 0.05M, Formal [D2EHPA] filter = 0.45pm, co = 100 rpm,  Figure 4.43  = 0.05 M , p H = 4.5, T = 25 XL,  total  = 6 x 10  pVPis  s  = 0.05 M , p H = 4.5, T = 25 °C,  total  filter = 0.45pm, co = 100 rpm, f W P u = 6 x 10  s  total  = 0.05 M , p H = 4.5, T = 25 X, filter = 0.45pm, co = 100 rpm,  p\ /Pi3=6xlO-  93  5  2  Predicted fractions of zinc species vs. preload : [Zn] = 0.05M, Formal [D2EHPA] p /p 12  Figure 4.46  13  total  = 0.05 M , p H = 4.5, T = 25 XL, filter = 0.45pm, co = 100 rpm,  = 6xl0-  93  5  Predicted change in bulk zinc concentrations (organic) with preload : [Zn] = 0.05M, Formal [D2EHPA]  = 0.05 M , p H = 4.5, T = 25 XL,  total  filter = 0.45pm, co = 100 rpm, p\ /p\ = 6 x 10 2  Figure 4.47  tota)  s  94  Predicted change in organic zinc concentrations with preload : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm,  co = 100 rpm, p V P u = 6 x 10  95  s  Predicted flux of organic species with preload : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm,  P12/P13 = 6 x 10"  95  5  Figure 4.50  94  = 0.05 M , p H = 4.5, T = 25 °C,  filter = 0.45pm, co = 100 rpm, p V P u = 6 x 10  Figure 4.49  s  3  Predicted change in interfacial zinc concentrations (organic) with preload : [Zn] = 0.05M, Formal [D2EHPA]  Figure 4.48  92  Predicted association factors vs. preload : [Zn] = 0.05M, Formal [D2EHPA]  Figure 4.45  92  Predicted change in interfacial D2EHPA concentration with preload: [Zn] = 0.05M, Formal [D2EHPA]  Figure 4.44  91  VMR predictions and experimental data for changes in temperature: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 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 , p H = 4.5, filter = 0.45pm, co = 100 rpm  Figure 4.52  97  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  Figure 4.53  99  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  Ill  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 )  C  aqueous species concentration (kmol/m )  D  diffusion coefficient ( m / s )  F  formal D2EHPA concentration (kmol/m )  /  species flux ( k m o l / m / s )  k  mass transfer coefficient of specified species ( m / s )  3  3  2  3  2  concentration-based equilibrium constant ((kmol/m ) ""') 3  K  02  dimerization constant ((kmol/m )" ) 3  D  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 )  V  molar volume of the solute at the normal boiling point (m /kmol) 3  B  x  solvent association factor  Z  aqueous equivalent boundary layer thickness ( m )  DAH  z  organic equivalent boundary layer thickness (m )  D/OTX  Subscripts for c, C Zn  aqueous zinc  2+  HL (HL)  organic D2EHPA monomer 2  (HL) ,total 2  organic D2EHPA dimer 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  Znl^CHLJ^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/m )" ) 3  1  ZnLjCHDj metal-extractant complex formation constant ((kmol/m )" ) 3  5  aq  5  org  aqueous diffusion boundary layer thickness organic diffusion boundary layer thickness  5  reaction zone thickness  M-  absolute viscosity (N-s/m )  V  kinematic viscosity ( m / s )  CO  RDC angular velocity (radians/s)  r  2  2  xiv  2  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 p H 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. A n 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 C 0 solution, forming a 2  3  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 U0 -2H 0). 2  4  [11  2  A typical process flowsheet is shown in Figure 2.1.  Feed  7 g/t U 0 3  HDEHP 4% TBP 4%  8  CLARIFIER  EXTRACTION 5 STAGES  Tailing  H 0 Na CQ 2  2  NaDEHP 4% TBP 4%  STRIPPING 3 STAGES  NH  Figure 2.1  3  1  1  FILTER  3  H 0 2  2  1  PRECIPITATION  •  YELLOW CAKE  Row sheet of the DAPEX process (after Alegret ) 111  4  2.1.2.2 Copper The most common copper solvent extraction system is found in a leachextraction-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 S 0 and -30 gpl Cu, and the final copper solution sent for electrowinning 2  4  contains -34 gpl Cu. The electrowinning plant produces -18.2 tpd Cu.  11,31  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 Ashbrook ) 131  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  keep it popular when compared to other reagents.  6  Its wide availability and low cost  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 process ' uses two solvent extraction circuits to recover zinc from pyrite 12,5 71  cinder leach solutions. A n 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 process  18,91  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 process ' 17  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  nWJi  +  m +  ->  ML (HL)^_ m  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  aliphatic solvents' ), and is monomelic in highly  O • • • OH \  111  ^  R \  P  polar media (i.e. alcohols, carboxylic acids, and  • /  water).'  R  The D2EHPA dimer consists of two  121  D2EHPA monomers, joined by hydrogen bonds between adjacent P=0 and P-OH groups, as shown  / P  \  ^  \  OH • • • O  R  Figure 2.3 Structural diagram of a D2EHP A dimer  in Figure 2.3. The non-ideal behaviour of D2EHPA at high extractant concentrations  was  investigated by Danesi and Vandegrift' , who examined results from equihbrium studies 131  of Eu *, Tm *, and C a 3  3  2+  extraction by D2EHPA in n-dodecane. They concluded that the  D2EHPA dimer activity coefficient y is fitted fairly well by the expression: D  logy  D  =  [2-2]  0.83 - ^ l "  where F is the formal D2EHPA concentration.  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 H 0) POOH* 8  structure: (monomer)  ]7  2  CH^CH^aCHCHsO C H 2  0  \  1  5  /  s P  CH (CH ) CHCH 0 3  2  3  OH  2  C H 2  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  \  mVkmol (monomer/ 845.8 mVkmol (dimer)  422.9  B  +  1.003 x 10" m /sec (dimer)* 9  pit,  2  0.1M(H,Na)NO /heptane  1.49  Komasawa et al. [14]  0.1M(H,Na)ClO /hexane  1.30  Komasawa et al. [14]  1.36 to 1.44  Ul'yanov et al. [15]  (H,Na)C10 /octane  log  Ul'yanov et al. [15]  0.1M(H,Na)NO /heptane  log K,, = 3.20  Komasawa et al. [14]  (H,Na)C10 /octane  log K = 4.47  Ul'yanov et al. [15]  0.1 M(H,Na)N0 /heptane  log K = 4.50  Komasawa et al. [14]  (H,Na)C10 /Isopar-H  log K = 4.70  Sastre et al. [1]  3  4  0.0001-lM(H,Na)ClO /octane 4  distribution constant 4  3  = 3.44  dimerization constant D  4  3  D  4  D  'source: manufacturer's information sheet Calculated by the method of Le Bas as reported in Perry  1  Calculated by the Wilke-Chang relationship  1171  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  1 001  0.00  Figure 2.4  1 0O2  1 0.03  F HDEHP  1— O04  1  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 & Grimm ) 1181  2.2.3 Parameter Estimation The Wilke-Chang  1171  relationship, shown in equation [2-3], was used to estimate the  diffusion coefficients of the various organic species.  10  D\i  7AxlQ- (xMf 12  =  T  V°  ~  Dreisinger  1191  [2-3]  s  6 B  has previously shown that this correlation is acceptable for predicting diffusion  coefficients - a predicted value for the diffusion coefficient of H E H E H P (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  116,201  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 , was B  estimated by using the method of Le Bas ; these values for different organic species are 1161  also shown in Table 2.2.  Table 2.2 Diffusion Coefficients predicted by the Wilke-Chang Relationship. Species  D (heptane)  (n^kmol )  (mV)  422.9  1.520 x 10"  845.8  1.003 x 10"  Znl^  858.4  0.993 x 10'  ZnLz-HL  1281.7  0.781 x 10"  1704.2  0.659 x 10'  1  HL (HL)  2  ZnL2-(HL)  2  9  9  9  9  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 Chiarizia ' have written an 121  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 ML (HL) (2n-m) m  mH  +  Organic Bulk  Aqueous Bulk M  n(HL)  2  6,  <5aq  org  Figure 2.5  m+  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. [ ]  int  =[ ]  A reactant/product concentration profile can then be drawn as shown in Figure 2.6.  13  bulk  .  [HR]  bulk  [HI bulk [M ] bulk 2+  Organic Bulk  Aqueous Bulk [MR 2] bulk 0 /org  Figure 2.6  laq  Concentration profile for an interfacial reaction mechanism  Ajawin et al.  1m  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, p H , 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  dissociation  )  2  (HR)  2  <_>  HR" +  <->  2HR  iT  (0 (ii)  The species HR " and HR then rapidly adsorb at the interface, and react with zinc ions to 2  form an extractable zinc-D2EHPA compound through the following two reactions: Zn  +  2+  ZnR  2  +  HRj  <->  ZnR  HR  <->  ZnR • HR  2  +  FT  (iii) (iv)  2  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  1231  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 Fe * from acid perchlorate solutions 3  using D2EHPA in n-octane. They concluded that the extractable species FeR -3HR is formed 3  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' concluded that the extraction of Co, Cu, and N i from nitrate 261  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 Vandegrift  1281  examined the extraction of Eu * and Am * from a chloride 3  3  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:  M  2 + 2 +  — + 2HR <—>  ^ MR + 21?  [MR ],.[Hlf = —, , [M ] [HR] 2  2  2+  (i)  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,  M>+  J  ~2  =  Jm  =  J  m  *  =  ~~2 rt  ^  J  Expressions can be written for the fluxes to and from the interface for each species by Fick's first law, i.e.  V  = ir" ^ {P^+u-rM !} 1  aii)  2  M \aq 2  J  = I T ^ {[HRJ.-tHRU}  (iv)  5  m  °HR,or«  ^ ^^f^-^U  (v)  =  J  *  Jl = 5 ^ {[HWrKT],}  (vi)  D  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 can be found by substituting equations (ii) - (vi) into equation (i) and solving for /  Heming et al.  l29]  m 2 +  2 +  :  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.  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  interface ^ Figure 2.8  ®  I  !_I I  <  z  <*  z  Schematic diagram of the reaction zone and aqueous/organic boundary layers - Mass transfer with chemical reaction model (after Hughes and Rod ) 1311  Hughes and Rod  1311  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 z . r  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  MR  + R"  2 +  4  MR + R" +  MR  -»  2  MR  step 3  +  MR,  step 4  2  step 5  with the overall reaction expressed by the equation:  M  2 +  + 2HR <-> MR + 2fT  K  2  =  a  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 —  (i)  ( HR4 ^ H R C H R ) C  —  where  (ii) and  C  HRj  C  -MRj,i  —  HR  (iii)  t, HR.org  (iv)  MRj 2>t  PMR/^HR H C  CUR  — ^HR  V  [  (v)  ^PHRDMRJ(EX M C  ^HR^MRJCHR,!  P MR^HR^H +4P  H R  D  M R j  7ST  E X  C  M  ^ E X  C  M  19  MRj-f HR  y  r2  (c^ \  C  \ ^ M R jJ  «  J  J  "M,i  HR  i HR,i C  (  £>HR  C  H R ^ H R )  (vii)  HR^*HR)  (viii)  \  C +T  -H4  (Vi)  P  (  H  V ^ H ^ H R  C  HR,i  C  J  Special cases of the model occur under the following circumstances: particular case  h  •f H R  £R£*HR © J  reaction in the film  =  2  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 /H S0 system using a rising drop and a constant interface 4  2  4  (gauze cell) contactor. The extraction of C u 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 Cooper  132,331  used a modified version of the MTWCR model to model  the extraction of Co and N i from sulphate solutions by H E H E H P (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, N i  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 L x  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(Zn ) = 2.5 x 10 s' 2+  7  1  Jti(Co ) = 2.6 x 10 s 2+  s  rC (Ni ) = 1 . 3 x l 0 s 2+  4  1  1  i  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 p H 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 p H 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 p H 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 Kinetic 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%.  Figure 2.9  1361  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  1231  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 A K U F V E 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 A K U F V E apparatus can be used to study the rate of chemical reaction. Since the interfacial contact area in the AKUFVE apparatus is unknown be measured.  1361  , the interfacial flux cannot  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 Chiarizia ) 1  !  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 complex , researchers claim to have been able to at least partially model the behaviour 1361  of the growing drop.  1411  25  BURETTE  SAMPLING S TUBE  VACUUM INTERFACE  o  NEEDLE  AQUEOUS  ORGANIC  Figure 2.11  The growing drop cell (after Hughes and Zhu  1411  )  2.4.5 Laminar Liquid Jet (Liquid Jet Recycle Reactor) In the liquid jet recycle reactor, developed by Freeman and Tavlarides , one phase 1421  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 Noble ) 1371  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. Levich  1451  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 Figure 2.14  (b) Flow below the filter  Fluid flow patterns near the Rotating Diffusion Cell (after Patel ) 1441  z  =  D  0.643 <& v D m  v6  [2-4]  m  where z is the equivalent diffusion layer thickness in m, co is the disk's angular velocity in D  radians/second, v is the kinematic viscosity in m /s, and D is the diffusion coefficient in 2  m /s. 2  If the interfacial flux is measured and plotted with respect to co~ , the intercept is equal 1/2  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 R D C 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.  146,471  More recently,  the RDC has been applied to hydrometallurgical systems by several researchers in an attempt to study solvent extraction kinetics.  29  Albery and Fisk  1481  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 Cooper  1321  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 Cooper  1331  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, p H 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 p H with a weak acid was examined with cobalt extraction from perchlorate solution. Patel  1441  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 N i < Co < Cu < Zn. He then examined the effect of varying the metal concentration, p H , 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 > 1 0 m M , pH=4.5, T=25°C,CJ=3Hz 2  [D2EHPA], mol/dm  3  Figure 2.15 The extraction of Co, Ni, C u and Zn by D2EHPA (after Patel ) 1441  Table 2.3 Results of Patel's study of Ni, Co, Cu, and Zn extraction [44]  Metal  Measured Flux  Second Order Rate  8  (kmol m" s" )  Constant k  (urn)  2  1  R  r  (r^kmorV ) 1  Nickel  9.674 x 10"  9  1.325 x 10  8.02  Cobalt  2.110 xlO"  8  9.460 x 10  3.56  Copper  4.936 xlO"  8  1.213 x 10  1.39  Zinc  1.247 x 10"  2.436 x 10  0.42  7  7  8  7  9  Using parameters calculated by the MTWCR model, Patel estimated the thickness of the reaction zone near the interface. Values of 8 for each metal are shown in Table 2.3; the r  thickness of the RDC aqueous diffusion layer (8 ) was estimated to be 32.59 um. For zinc, aq  5 « 8 ; thus, as an approximation it can be assumed that the reaction occurs at the interface. r  aq  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 e q u i l i b r i u m 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  ->  ZnL (HL) 2„_2) + 2KT 2  [2-5]  (  where the value of n varies depending on the nature of the organic solvent and the aqueous media. For example, Huang and Juang  1501  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.  1511  have  demonstrated that as the organic phase becomes more fully loaded, viscosity increases, suggesting the formation of metal-extractant polymers. The  equilibria of  researchers ' ' ' ^ 17 8 22 5  581  the  above  reaction have  been  examined  by  different  under various experimental conditions. Their results are summarized  in Table 2.4. In general, the composition of the extracted species was found to be Z n R H R 2  (i.e. n=l .5) for aliphatic diluents, and ZnR (HR) (n=2) for aromatic diluents. Exceptions are 2  the studies by Grimm and Kolarik  1551  2  in N0 /n-dodecane, Sato et al. 3  32  [52]  in Cl/kerosene, and  Table 2.4 Survey of Equilibrium Studies using Zinc and D2EHPA diluent  aqueous  composition of  phase  extracted species  (Na,H)S0  4  Shellsol T  ZnR (HR)  (Na,H)S0  4  n-heptane  ZnR HR  7.35 x 1 0  (Na,H)S0  n-heptane  ZnR HR  3  4  2.6 x 10"  Ajawin et al. [84]  (Na,H)S0  kerosene  ZnR HR  9.5 x 10  3  4  Huang and Juang [50]  kerosene  ZnR (HR)  (Na,H)CI  2  N/A  2  2  2  2  2  Rice and Smith [8] Ajawin et al. [22]  3  N/A  2  Sato et al. [52]  (Na,H)N0  3  n-hexane  ZnR HR  8.0 x 10"  (Na,H)N0  3  n-heptane  ZnR HR  6.3 x 10"  (Na,H)N0  3  n-octane  ZnR HR  8.5 x 10"  Smelov et al. [53]  (Na,H)N0  3  n-decane  ZnR HR  9.0 x 10"  Smelov et al. [53]  (Na,H)N0  3  n-dodecane  ZnR HR &ZnR (HR)  (Na,H)N0  3  benzene  ZnR (HR)  (Na,H)CI0  4  lsopar-H  2  2  2  f  (Na,H)CI0  4  2  2  N/A  2  5.0 x 10"  2  ZnR HR & 2  4.9 x 1 0 '  ZnR (HR)  7.6 x 10"  2  ZnR HR  Escaid 100*  Teramoto era/. [54]  2  2  2  2  Smelov et al. [53]  2  2  2  T  source  Grimm and Kolarik [55] Smelov er al. [56]  2  Sastre and Muhammed  2  [57]  2  10.1 x 10"  1  2  Li eta/. [51]  lsopar-H is odourless aliphatic kerosene (ESSO).  *Escaid 100 is an aliphatic diluent which contains 19wt% aromatics (ESSO).  Sastre and Muhammed  1571  in C10 /Isopar-H. Sato et al. found that the dominant species was 4  ZnR -(HR) , while Grimm and Kolarik and Sastre and Muhammed concluded that the 2  2  extracted species was a mixture of Z n R H R and ZnR (HR) . 2  Grimm and Kolarik  1551  2  2  found that the log D vs. log [(HL) ] plot curved as the  concentration of extractant increased.  2  L i 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, K decreases a  with increasing tendency for complex formation in the aqueous phase (complexing ability is in the order C10 < N 0 < CI < SO4). Thus, the largest values for K are found with species 4  3  a  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  (Na,H)S0 / heptane  Lewis Cell  interfacial chemical reaction  Ajawin et al. [7,22]  (K,H)N0 / dodecane  Lewis Cell  interfacial reaction  Cianetti & Danesi [59]  (Na,H)S0 / heptane  Lewis Cell  mixed regime  (Na,H)S0 / kerosene  Lewis Cell  interfacial 2-step  4  3  4  4  t  Source  Ajawin et al. [60] Huang & Juang [61]  chemical reaction (Na,H)S0 / heptane 4  (Na,H)CI0 / Isopar-H* 4  HCI0 / heptane 4  Rotating  mass transfer with  Diffusion Cell  chemical reaction  Lewis Cell  interfacial  Rotating  mass transfer  Diffusion Cell ^ote: intentional selection of parameters to provide mixed regime. *lsopar-H is odourless aliphatic kerosene (ESSO).  34  Patel [44] Aparicio & Muhammed [62] Dreisinger & Cooper [33]  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. A n 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  [411  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 Danesi  1591  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 Z n , Co , or N i 2+  2+  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. expanded on their earlier work by studying zinc extraction under mixed m  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 paper , a simple model was developed which 1221  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 Juang  1611  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'itti^jtiri'^i+^tso :] - )" 2  1  2  1  where the rate constant, k, is equal to 3.21±0.20 x 10" (mol/dm) ^" . The reaction sequence 7  that they developed is the same as that developed by Ajawin  1  1221  1  (c.f. Section 2.3.1), but instead  they favoured reaction (iv) as the rate controlling step, i.e. ZnR  2  Patel  + 1441  HR  <—>  ZnRjHR  (iv)  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 Muhammed  1621  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 H R and ZnR (HR) by using a fast equihbrium at the interface. The results 2  2  2  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 Cooper  1331  also studied the extraction kinetics of Zn, Co, and  N i 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, N i 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. A n 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)  <—>  strip side  MX (membrane)  37  + I-T  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  Extractant Impregnated Membrane  Stripping Solution Metal Ions Out .A  FeLd  FLUXM  FLUX H" Figure 2.16  Solution  +  Extraction processes in a SLM (after  Figure 2.17  Tavlarides et al. )  Axial and cross-sectional view of a hollow-fibre SLM  m  module  Since in a S L M 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 \ Teramoto et al. \ and Huang and Juang [M  l54  of zinc through a D2EHPA impregnated SLM.  38  1651  have investigated the transport  Investigating zinc extraction from a sulphate system with a kerosene diluent, Huang and Juang  1651  found that the rate controlling step was in most cases membrane diffusion. However,  under certain circumstances (a combination of lower [M ] 2+  feed  , higher [(HR) ] 2  and lower [H ] ) +  total  fced  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) concentration, an input p H of 5.6, and an input zinc concentration of 100 ppm. A long-term 2  (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 S L M operating conditions, with the filter in the RDC in some circumstances simulating the S L M membrane. Thus, investigations of basic kinetic mechanisms by the RDC technique may be most helpful in measuring the rate controlling steps in S L M 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 N a O H solution used for p H stabilization during the RDC run was prepared by dissolving the required amount of N a O H 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) ) was prepared by adding a concentrated sodium 2  hydroxide solution to a strong copper sulphate solution.  The Cu(OH) /water slurry 2  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) filter cake was then added to a 1:1 mixture of impure D2EHPA and toluene. Since 2  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. A n addition of N a O H was required to maintain a high p H , however, this N a O H 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 N a O H addition was required to maintain a high p H , since the reaction proceeded according to the equation: ZnO + n(HL)2 ->  Znl^HL)^^ + H 0 2  42  [3-1]  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 A n a l y s i s 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 K H P (dried for 3-4 hours at > 100 °C) with the unknown N a O H 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 method  1671  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. A n 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 p H 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  Number of Drops (a) Gran Plot of purified D2EHPA  Figure 3.1  2  4  6  8  10  12 14 16  Number of Drops (b) Gran plot of 60% preloaded D2EHPA  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 from reacting with 2  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. A n autotitration system consisting of a Radiometer PHM82 p H 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  Acrylic Filter Mount  Thermostatted Beaker  Filter  Figure 3.2 - The Rotating Diffusion Cell  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 p H electrode was placed in the solution and the p H 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 p H 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.o0.50 i 0  i  i  0.2  i  i  0.4 U"  2  i  i  0.6  i  i  0.8  i  1.0  (s ) 1/z  Figure 3.3 - A Sample RDC Plot showing lines from three different experiments Formal [D2EHPA] = 0.05 M , p H = 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 Initial 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(C10 ) 4  2  0.05F D2EHPA Bulk p H = 4.5 T = 25°C 0.45pm Millipore filter Parameters examined: [Zn(C10 ) ]:  0.001 - 0.20M  [D2EHPA]:  0.0025-0.10F  BulkpH:  3.25-5.5  temperature:  15 - 50 °C  4  2  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 p H 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 p H (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/m /sec, with a sample standard deviation of 0.32 x 10". 2  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 C 0 . Many batches of zinc perchlorate stock 2  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  0.20  0.10 [Zn] (kmol/m ) 3  Figure 4.1  Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , p H = 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/m ) J  Figure 4.2  Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  52  Zinc Flux vs. pH  pH Figure 4.3  Effect of changing bulk p H on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, co = 100 rpm  Zinc Flux vs. Temperature o  CD CO  O  E CO O X  J3  LL. O C  fsl  25  30  35  40  50  Temperature (°C) Figure 4.4  Effect of changing temperature on zinc flux : [Zn] = 0.05 M , Formal [D2EHPA] = 0.05 M , p H = 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. A n 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) , VMR rates for both species will be calculated. The association factor, n, is defined as the G)  ratio of D2EHPA dimer to extracted zinc. For the general extraction equilibrium  Zn  2+  + n (HL)  2  <—>  ZnU-HL^_ , + 2rT 2  with equihbrium constant K  K-  =  —-——  [4-1]  the fluxes of Z n and (HL) may be related by a simple mass balance, 2+  e x /  2  i.e. •  J  -  -J  '  l  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'-~ LA  -  C  [4  31  where C ^ and Cffi. are the interfacial and bulk concentrations of Z n , respectively. The mass 2+  2 +  transfer coefficient, k^, is defined by the equation  v  2  -  D,  [4-4]  t  2 0  It is unlikely that there would be much ZnL, present under standard operating conditions, and therefore this species was not considered. 54  INNER  Organic Bulk  J(HL)  2  JznMHL) . p n  *"  z  D,org  2 )  [  Jzn *  1  J -  where  Aqueous Bulk  2  H  *"  interface Figure 4.5  OUTER  FILTER  "*  Z  D,aq  *"  ^  Diagram of species and flux direction definitions for the V M R model  is the aqueous diffusion coefficient of Zn , and z  0jaq  is the diffusion boundary layer  thickness as given by the Levich equation, z  = 0.643 <o- v D V2  D  m  [4-5]  m  The organic mass transfer rate Kmited flux can be similarly derived from Fick's first law, i.e. ^(HL),  where  =  ^(HL), [ ( H L ^ C ^ L J  and  C  W-6]  _  are the interfacial and bulk concentrations of (HL) , respectively. Note that 2  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 equation  1331  [4-7] k, where  is the organic diffusion coefficient of (HL) , z 2  DjBrg  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) transport (organic phase mass transfer control) may be calculated from equations [4-3] and 2  [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 V M R 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 V M R 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 and H . It was assumed that there were no chemical reactions in the boundary layer, 2+  +  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 •  Experimental Data  X  VMR:Zn  2 +  VMR:(HL) -n=1.5 2  VMR : (HL) - n=2  CJ  2  c N  0.05  0.10  0.20  [Zn] (kmol/m ) 3  Figure 4.6 V M R 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 12  o CD CO  10  Experimental Data VMR :(HL) -n=1.5 2  VMR : (HL) - n=2 2  O  E CO  O  X  8 6^ 4  _Z5 LL.  O  c Ki  2 0  0.02  0.04  0.06  0.08  0.10  Formal [D2EHPA] (kmol/m ) 3  Figure 4.7 V M R predictions and experimental data for changes in bulk D2EHPA concentration: [Zn] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  57  to exist as ZnLj-HL and Z n L ^ H L ) ^ and no zinc-extractant complex was present in the organic bulk. It was assumed that for the H L , (HL)^ Z n l ^ H L and ZnL - (HL) species that reactions occurred 2  2  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  13  HL  and  r  c Zn  2+  + 4HL <—>  ZnLj • (HL^ + 2Yt  p  [4-9]  2  14  where (3 and p are the formation constants for the species ZnLj-HL and ZnL -(HL) respectively. 13  14  2  2  A similar expression can be developed for the dimerization equilibrium between monomelic and dimeric D2EHPA:  2HL  <—*  (HL)  K  =  D  2  c C  [4-10]  (HL)2  HL  where K is the dimerization constant for D2EHPA. D  A n 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]:  Zn +-(HL) 2 +  2  *—» ZnL, • HL + 2lT  K  a  1.5  2J+ {WL\ C  58  From equations [4-8] and [4-9] the distribution of the two zinc species Znl^-HL, and ZnL2-(HL) can be interrelated as follows: 2  ZnLj • (HL^  <—>  ZnLj • HL + HL  p /p 13  14  CznLj-fHL^  The quotient p / p is equal to the stepwise formation constant K*. 13  14  At any point in the organic, the monomelic and dimeric concentrations of D2EHPA are related by the following equations: 1 2  (HL)2,total "  C  [4-13] CHL  C(HL  ^  substituting from [4-10],  (HL)j,totaI  C  I 2  —  v C,IL  +  [4-14]  2  D HL C  rearranging, r 2 ^D HL C  +  I 2  _ C h L  (HL)j,totia  C  [4-15]  n  —  "  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.  Ir Zx?*  J  ~  n  "'CHL),  -  • ZdL,-(HL),,«x< ; /  i  -  l 4  2J  Jf¥  Once again, the fluxes are defined as shown in Figure 4.5. From Fick's first law,  59  "  1 6 ]  J~Znl -Q1L.) ,tot<il 2  -^ZnLj-HL  '  ll  +  ^ZnLj-CHL^  [4-20]  ^ZoLj-HL [cZn^-HL^ZnLj.HL.] +fcznLj.flfl.^(^ZaLj-CHLfe ZnI,.<HL)J  =  —C  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 Z n l ^ H L a n d ZnL (HL) are equal to zero, an expression 2  2  for/^LJ-CHL),u*ai  solely  in terms of C ^ - H L may be developed, i.e. <  •^ZnL-(HL).»o«j; 2  l  4*. \ . ^ . H L + ^.(HL) p p ^ ''ZnLj-HL J  i3/  [4-23]  i4  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^  #  - r — — <-Zn (HL)j  [4-24]  -  K  - =  C  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 p H 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 p H 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) in the organic phase, but rather 2  the diffusion of Z n  2+  in the aqueous phase. At high zinc concentrations, moderate to high  D2EHPA concentrations, and at all values of p H , 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) , Znl^-HL, and ZnI^-(HL) . In particular, the model is most 2  2  sensitive to the diffusion coefficient of (HL) since, under conditions of organic rate control, 2  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) . The greatest error 2  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) at the interface, resulting in a 2  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) diffusion profile. 2  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 H L , and ZnI^(HL) ) were altered by up to ± 20%. As 2  2  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 K  a/  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  Zinc Flux vs. Zinc Concentration w = 100 rpm  7 o  CD  <0 6  o E CO  5  Experimental Data Mathematical Model  O ^  3  x  2  LL  O C  N  1 0  0  0.05  0.10  0.15  0.20  [Zn] (kmol/m ) 3  Figure 4.9  Effect of changing bulk zinc concentration on zinc flux : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  Zinc Flux vs. Zinc Concentration w = 300 rpm  8 o  CD CO  7 6  o E  5  co  4  O  Experimental Data Mathematical Model  3 X  _g LL.  O C  Kl  2 1 0  0  0.05  0.10  0.15  0.20  [Zn] (kmol/m ) 3  Figure 4.10  Effect of changing bulk zinc concentration on zinc flux: Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25°C, filter = 0.45um, co = 300 rpm  64  Zinc Flux vs. D2EHPA Concentration 12 o C D w  10  w = 100 rpm  Experimental Data Mathematical Model  o E CO O  X ID o c N  0.10 Formal [D2EHPA] (kmol/m ) 3  Figure 4.11  Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  Zinc Flux vs. D2EHPA Concentration 14 o CO 12 o E  w = 300 rpm  Experimental Data Mathematical Model  10  GO o X J2  Ll_ O c Kl  0.10 Formal [D2EHPA] (kmol/m ) 3  Figure 4.12  Effect of changing D2EHPA concentration on zinc flux: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 300 rpm  65  Zinc Flux vs. pH w = 100 rpm  7 o CD co  o E CO o  6 5  Experimental Data Mathematical Model  4 3  X  3  1  O  c "N  0 3.0  3.5  4.0  4.5  5.0  5.5  PH Figure 4.13  Effect of changing bulk p H on zinc flux : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45pm, co = 100 rpm  Zinc Flux vs. pH w = 300 rpm  8 o  CD CO  76-  O  E co  5  Experimental Data Mathematical Model  4-\  O  3 X  _3  2  LL  1  O  c N  0 3.0  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  Sensitivity Analysis - Zinc Flux vs. [Zn] Vary Organic Diffusion Coefficients  8 o CD  76-  o E  5-  00  4-  o  3 2  LL  O  c kl  Experimental Data  f  +20 % +10%  Model Value  -10% -20 %  I  1 H 0 0  0.05  0.10 [Zn]  Figure 4.15  0.15  0.20  (kmol/m ) 3  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  • Experimental Data  o  CD CO  10  +20% +10%  8  -10% —- -20%  O  E CO  Model Value  6  O X  o  4H 2  c N 0.02  0.04  0.06  0.08  0.10  Formal [D2EHPA] (kmol/m ) 3  Figure 4.16  Zinc flux vs. D2EHPA concentration for changes in the organic species diffusion coefficients : [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  67  Sensitivity Analysis - Zinc Flux vs. [Zn] Vary Equilibrium Constant  8  CD o  SO  6 o  E  5  CO  4  O  »-I  — 'if  Experimental Data x4 x2 Model Value x0.5 x0.25  3  X  _z> 2 | LL O c N  1-  o  0  0.05  0.10 [Zn]  0.15  0.20  (kmol/m ) 3  Figure 4.17 Zinc flux vs. zinc concentration for changes in the equiUbrium constant  : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 °C,  filter = 0.45um, co = 100 rpm  Sensitivity Analysis - Zinc Flux vs. [D2EHPA] Vary Equilibrium Constant  12 o  CD CO  O  E co  O  X  _3 LL O c  10  8H 6  • Experimental Data x4 Model Value x0.5 x0.25  ......  x  2  4 2 0  0.02  0.04  0.06  0.08  0.10  Formal [D2EHPA] (kmol/m ) 3  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  Sensitivity Analysis - Zinc Flux vs. [Zn] Vary L/or  8 o  CD W  1  1  7H 6  O  E  5  GO o  4  =  Experimental Data  r  +20 % +10%  3  Model Value  22 "  -10% -20 %  UL  o c  kl  1 H 0  0  0.05  0.10  0.15  0.20  [Zn] (kmol/m ) 3  Figure 4.19  Zinc flux vs. zinc concentration for changes in the filter equivalent thickness L/a: 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 Ua  12  • Experimental Data  o  CD CO  10 H  o  8  00  6  E  — --  +20% +10%  y  Model Value  — -10% —- -20%  y..-\'  O X  4  •'.yyyy.  J2 LL-  CS  2  c  kl  0  0.02  0.04  0.06  0.08  0.10  Formal [D2EHPA] (kmol/m ) 3  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 p H , 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 D2EHPA.  2+  as the rate controlling step to organic mass transfer of  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 (from Figure 4.23), the interfacial zinc concentration 3  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 , the interfacial p H , and the interfacial zinc and D2EHPA avg  concentrations. The average association factor, n , is defined as the computed total flux of avg  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  avg  .  The decrease in interfacial p H 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 The decrease in interfacial Z n  2+  2+  ions.  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 p H has on the association factor n , the interfacial p H , and the interfacial zinc and D2EHPA concentrations. In the p H region avg  examined, D2EHPA transport is the rate controlling step.  71  Figure 4.29 shows the effect of changes in p H on the value of n . For low values of avg  pH, the interfacial p H will decrease, resulting in a larger interfacial D2EHPA concentration, which will result in a slight increase in n . avg  The interfacial p H increases slightly as the bulk p H 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 p H 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 p H 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 (n ) vs. Zinc Cone. avg  1.75  >  CCD  1.70-  o o  1.65-  c g  1.60-  03 LL  co  "o o to w  <  1.55-  1.50  0.05  0.20  0.10  [Zn] (kmol/m ) 3  Figure 4.21  Predicted change in association factor (n ) with bulk zinc avg  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/m ) 3  Figure 4.22  Predicted change in interfacial p H 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/m ) 3  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 LU CM Q  "5  "8 CP  0.05  0.10  0.20  [Zn] (kmol/m ) 3  Figure 4.24  Predicted change in interfacial D2EHPA concentration with bulk zinc concentration : Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 "C, filter = 0.45pm, co = 100 rpm  74  Association Factor (n ) vs. D2EHPA Cone. avg  1.65-j _  « 1.60-  Fad  o  Associat  c o 1.55-  i  1.50  1  0  1 0.02  1  1 0.04  1  1 0.06  1  1 0.08  1 0.10  Formal [D2EHPA] ( kmol/m ) 3  Figure 4.25  Predicted change in association factor ( «  avg  ) with bulk D2EHPA  concentration: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45pm, co = 100 rpm  Interfacial pH vs. D2EHPA Concentration  2.2  ~\  0  1  1 0.02  1  1 0.04  1  1  0.06  1  1 0.08  1  0.10  Formal [D2EHPA] (kmol/m ) 3  Figure 4.26  Predicted change in interfacial p H with bulk D2EHPA concentration: [Zn] = 0.05M, p H = 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, p H = 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  K^tota, , interface / /  yy // yy  X CM 0.003-  Q «  o  0.002  CO  t= 0.001 CD  0.02  0.04  0.06  0.08  0.10  Formal [D2EHPA] ( kmol/m ) 3  Figure 4.28 Predicted change in interfacial D2EHPA concentration with bulk D2EHPA concentration: [Zn] = 0.05M, p H = 4.5, T = 25 °C, filter = 0.45um, co = 100 rpm  76  Association Factor (n ) vs. pH avg  1.70i  1.65O CO  u_ o  1.60  CO  o to  1.55-  CO  < 1.50 -I 3.0  1  1 3.5  1  1 4.0  1  1 4.5  1  1 5.0  1 5.5  PH Figure 4.29  Predicted change in association factor ( « ) with bulk p H : avg  [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , T = 25 °C, filter = 0.45um, co = 100 rpm  Interfacial pH vs. Bulk pH  X Q_ "CO  "o co CD  3.0  4.0  4.5  5.5  Bulk pH Figure 4.30  Predicted change in interfacial p H 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  *|  o E  0.049  0.048 N.  0.047 co  o  co t £ 0.0460.045 3.0  1— 4.0  -i  3.5  P  —i 5.0  4.5  r  5.5  H  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 = 0.45pm, co = 100 rpm  Interfacial D2EHPA Concentration vs. pH 0.0030 co  E  ^  E  [(HL'2'total. ).] Intarfaca  0.0025  j*:  ~ <  0.0020  X  0.0015  LL!  I(HL2).'interlace  CM  ^ co  0.0010  ^  0.0005  'o ' CD -*—>  C  3.0  [HL]Interlace 3.5  1 4.0  ~i  1  PH  1— 4.5  -i  1— 5.0  5.5  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 = 0.45um, 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: H L , (HL)^ Znl^, Z n L H L , 2  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 Z n L ^ H L ) ^  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  Zn  +  + 3HL  ZnLj • HL + 2FT  [4-26]  79  Zn  2+  + 4HL <—> ZnLj • (HL) + 2H+  p  2  14  =  r ^ C  r  [4-27]  2  4  2J> HL C  where P ^ Pi , and p are the formation constants for the species ZnL?, ZnL^-HL, and ZnLj-dTL^ 3  :4  respectively. A similar expression can be developed for the dimerization equihbrium between the monomelic and dimeric form of D2EHPA: 2HL  <—> (HL)  cc™* K  2  D  = C  where K is the dimerization constant for D2EHPA. D  [4-28]  2  HL  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. A n 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]:  Zn  „  3 _  +  + -(HL^  _  C^.HLC^  K,  <—> ZnLj • HL + 2fT  [4-29]  ,1.5  Equations can be developed from equations [4-25], [4-26], and [4-27] which interrelate the speciation of Znl^, Z n l ^ H L , and Z n L ^ H L ^ a s follows:  ZnLj • HL  P /p =  <—> ZnLj + HL  12  C2aLj CHL ZnLj-HL  13  2  p /p =  <—> ZnLj • HL + HL  13  [4-31]  ZnLj-HL HL  C  ZnLj • (HL)  [4-30]  C  C  14  The quotient Pi /pi( D is equal to the stepwise formation constant K m  m+  m+V  At any point in the organic, the zinc species ZnLj, Z n L j H L , and Z n L ( H L ) are related by 2  2  the following equations: Zn,total  C  —  ZnLj  C  +  ZnLj-HL + ZnLj-(HL)j  C  [4 32]  C  and  80  [4-33]  -ZnLj-HL Pu/Pis CHL  [4-34]  ZnLj-HL HL  C  t-ZnLj-CHLjj  C  PVPu  Combining equations [4-32], [4-33], and [4-34], C  £zn,tt>tal  ZnLj-HL  Pl2/Pl3  C  +  C  ZDLJ-HL  ZnLj-HL +  C  [4-35]  H L  PIS/PU  [4-36]  '•Zn.total ^ZnLj-HL M>14  Performing a mass balance over the entire organic/interface region for zinc and "L", for zinc, •^Zn,total ^Zn.toul  +  [4-37]  ^ZnLj + ^ZnLj-HL + ^ZaLj-CHL^  —  ^ZIILJ  [4-38]  + ^ZnLj-HL + ^ZnLj-CMJj  and for "L", •^HL + 2/(HL)j +  + 3/2^. HL + 4 /  ZnLj  .  (HL  )  j  [4-39]  = 0  Expanding equation [4-38], [4-40]  " ^ZnLj-f.HL)^' Z Q L J - C . H L J J  •'zn.total  —  ^ZnLjCzaLj + ^ZnLj-HL^ZnLj-HL —  +  '-ZnLj-rHLJjl [4-41]  ^^(HL^ZoLj-rHLJj ^^ZnLjCznLj + ^ZnL -HL ZnL -HL + ^^(HL^ZnLj-fHLjJ C  2  2  let KCZnbulk — ^ZriL^ZriL, + ^ZnLj-HL^ZnLj-HL + ^ZnLj-CHL^^Lj-CHL^  'Za.total  —  - KCZnbulk ^ZnLj ZnLj + ^ZnL,-HL ZnL,-HL + ^ZnL,-(HL>, ; Lj-HL^ZnLj-l nLj-CHL^ZaLj-CHLJj  Substitute in values of  C  C  C  and c ^ - m L ) , from equations [4-33] and [4-34],  81  [4-42]  'Zn .total  [4-43]  13  ^ZnL  + ^ZnLj-HL ZnLj-HL C  CHL  ZnLj-HL HL  C  + ^ZnLj-CHLJj  C  KCZnbulk  Pl3/PM [4-44]  /za.to.ai + KCZnbulk ZnLj-HL ^  e'  + ^ZnLj.CHLJj ( J ^  Note that in this equation C ^ L J - H L is a function of J^uu KCZnbulk, and c^.  Expanding equation [4-39], *HL( HL-4L) C  +  2/:  L) ( (HL) -C( HL) ) C  ( H  + 3/^  I  2  2  n L 2  .  H L  ^c  2  .  Z n L j  H L  ^HLHL 2^(HL)ij''(HL) + 2&ZnL C C  +  2  —  J  -c  +  ZNLJ  +  Z N L 2  .HL] + 4£ ^. z  ( H L ) 2  (c  Z n L 2  .  ( H L ) 2  - c^.pn.^j  = 0  ^^hLj-HL^Lj-HL + ^^hLj.fHL^ZnLj-CHL^  ^"HL^HL 2^(HL)^'(HL) 2&2nL2CzL 2  [4-45]  2^^^-C^j  2  n  2  [4-46]  3/*2i^.j|LCzL.HL ^^Zol^.n^L^ZnLj'CHL^ = 0 n  n  i  let KCbulk  =  £HL HL 2£ C  +  ( H L  ^C  ( I  ^HLHL + 2A^ j C( L) + 2 £  ^ + 2/^^2^+3/^^  C  HL I  H  i  ZNLJ  C L Zll  z  + SfcznL^.jjLCznLj.HL  ^  From equations [4-28], [4-33], and [4-34], equation [4-47] can be rewritten as:  ^HLHL + 2£HLv[^D HL ] C  C  (  +  2&ZnL  [4-48]  HL (VP 13 CHL  ZnLj-HL HL  C  C  Pl /Pl4  - KCbulk = 0  3  From equation [4-44], an expression has been developed for c ^ v m .  a sa  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 K (equation [4-49]): ex  [4-49] Ok  C  (HL)j  1.5  -  K  m  /(•^Zn,total)  where/^zntota,) should equal 0, and the correct value for the flux Jz^totai can be determined by iterating /zn,totai t i l /(/zn,totai) converges. A general flowchart showing the basic program structure of the u n  extended mathematical model is shown in Figure 4.33.  Input Data Compute Bulk Speciation Iterate Flux  Figure 4.33  Guess Zinc Flux Compute Interfacial Concentrations Compute Equilbrium Constant  Basic program structure of the extended mathematical model  83  4.5 Preload Results A n 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 Z n L j H L species, p\ /p\3, as a suitable value could not 2  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 10 , which gave a fairly good fit to s  the experimental results in the high preload range. For any value of p\ /Pi3 selected, the model fit is rather poor over the preload range. 2  A small value of p /Pi3 provides a fairly good fit at low preload values, but prevents 12  significant amounts of the Znl^ complex from forming at high preload values: A larger value of P12/P13 ts better in the intermediate to high preload concentration range, but n  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 H L or ZnL2-(HL) , 2  2  and requires no additional H L molecules for complexation. Thus, in the case where the system is rate controlled by the diffusion of H L , 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  6  Experimental Data  5  ft //?i3=6x10-  #2/013 = 1 X 1 0 "  4  5  Z  #2/63 = 3x10"  5  ( W l 3 = 1 x10"  5  4  A //?13 2  =1 x l O *  00  O  3  X  ZD  2  O  1  c "N  0%  20%  40%  60%  80%  100%  Preload (% Loading) Figure 4.34  Comparison of experimental data and extended model predictions for flux vs. preload for selected values of p\ /Pi3: 2  [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 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 K . a  85  Zinc Flux vs. Preload w = 100 rpm  Preload (% Loading) Figure 4.35  Effect of preload on zinc flux : [Zn] = 0.05M, Formal [D2EHPA]  = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm,  total  co = 100 rpm, $ /p\ = 6 x 10 12  s  3  Zinc Flux vs. Preload  0%  Figure 4.36  20%  40% 60% Preload (% Loading)  80%  100%  Effect of preload on zinc flux: [Zn] = 0.05M, Formal [D2EHPA]  totaI  co = 300 rpm,  = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm,  pVp^  = 6 x 10  86  s  Sensitivity Analysis - Zinc Flux vs. Preload vary Organic Diffusion Coefficients  o  \  6-  CD CO  W  i  5-  o E  \ \ \ \ \  4  " - ? \ W \  oo  O  W  3  X X  2  o c  1  • Experimental Data — +20% +10% — Model Value —- -10% —- -20%  \  W  A  N  0%  20%  40%  60%  80%  100%  Preload (% Loading) Figure 4.37 Zinc flux vs. preload for changes in the organic species diffusion coefficients : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25°C, filter = 0.45pm, co = 100 rpm, pVpa3 = 6 x 10 total  s  Sensitivity Analysis - Zinc Flux vs. Preload Vary "Equilibrium Constant  o  CD CO  o E  6 A W \ N 'W  Experimental Data x4 x2 Model Value x0.5 xO.25  W -NSN  5 4  co  O  X  3  o c  w  H  3  w - w  2  \ \ \ \ \  1  N  0%  20%  40%  60%  80%  100%  Preload (% Loading) Figure 4.38 Zinc flux vs. preload for changes in the equilibrium constant : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25 'C, filter = 0.45um, co = 100 rpm, p / p = 6 x 10" total  5  12  87  13  Sensitivity Analysis - Zinc Flux vs. Preload 7 . .  •  V  a  r  V  U  o  • Experimental Data  CD 6 o E  +20%  S > ^ X  5  \ \ \  4  < - X \  sX>-  00  O  ------  +10%  —-  -10%  —-  -20%  Model Value  3 j  X  2  o cr N  1  13  a  0 0%  Figiure 4.39  20%  40% 60% Preload (% Loading)  80%  100%  Zinc flux vs. preload for changes in the filter equivalent thickness L / a : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M ,  p H = 4.5,T = 25°C, filter = 0.45pm, co = 100 rpm, p / p = 6 x 10 12  s  13  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 p H both increase as expected for increasing preload CFigures 4.40 and 4.41). As zinc flux decreases, less Z n required and fewer H  +  2+  is  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) ) to 2  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 -HL. Finally, for high preload values the amount of free D2EHPA becomes so low 2  (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) and then Z n L j H L become greater than their bulk counterparts. 2  Thus, the  direction of zinc transport reverses for both Znl^-CHL^ and Z n L j H L , and these two species diffuse to the interface where their additional complexed H L extracts Z n  and forms Z n L  2+  2  according to the equilibria: Zn  2+  Zn  2+  + Znlv(HL)  2  <—> 2ZnL2 + 2fT  [4-50]  <—> 3 2 ^ + 2H+  t" ]  and + 2ZIUVHL  4  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) and Z n L H L ; at loadings 2  greater than ~70%, free D2EHPA is no longer the major extractant.  90  2  Aqueous Interfacial Zinc Cone. vs. Preload 0.050 i  :  1  0.049-  E  "o E  -*  0.048-  8 | 0.047-  5, 0.0461  0.045 -I 0%  1 20%  1 40%  1  1  1 60%  1  1 80%  1  100%  Preload (% Loading) Figure 4.40  Predicted change in interfacial zinc concentration (aqueous) with preload: [Zn] = 0.05M, Formal [D2EHPA]  = 0.05 M , p H = 4.5,  total  T = 25°C, filter = 0.45pm, co = 100 rpm, p / p = 6 x 10 12  s  13  Interfacial pH vs. Preload  2.4 -I 0%  1  1 20%  1  1 40%  1 60%  1  1  1 80%  1  100%  Preload (% Loading) Figure 4.41 Predicted change in interfacial p H with preload: [Zn] = 0.05M, Formal [D2EHPA]  tota]  = 0.05 M , p H = 4.5, T = 25°C,  filter = 0.45pm, co = 100 rpm, p / p \ = 6 x 10 12  91  3  s  Bulk D2EHPA Cone. vs. Preload  100%  Preload (% Loading) Figure 4.42 Predicted change in bulk D2EHPA concentration with preload : [Zn] = 0.05M, Formal [D2EHPA] filter = 0.45pm, co = 100 rpm,  total  pVPis  = 0.05 M , p H = 4.5, T = 25°C, = 6 x 10  s  Interfacial D2EHPA Cone. vs. Preload 0.0022 0.0020  o 0.0018-1 E  [(HL) j 2  totaJ  inigrface  0.0016  <  0.0014  Q_ 0.0012  I UJ 0.0010 CM  Q  0.0008 -\  .4 0.0006 TO 0.0004  I...,  'o  0.0002  o%  [HL]  Interface  \  20%  40%  80%  60%  100%  Preload (% Loading) Figure 4.43 Predicted change in interfacial D2EHPA concentration with preload : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 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%  100%  Preload (% Loading) Figure 4.44  Predicted association factors vs. preload: [Zn] = 0.05M, Formal [D2EHPA]  total  co = 100 rpm,  = 0.05 M , p H = 4.5, T = 25 °C, filter = 0.45pm, pVPo = 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]  total  = 0.05 M , p H = 4.5, T = 25 °C,  filter = 0.45pm, co = 100 rpm, p / p 12  93  = 6 x 10"  5  13  Organic Bulk Zinc Cone. vs. Preload  ZnLpHl  20%  40%  60%  80%  100%  Preload (% Loading) Figure 4.46 Predicted change in bulk zinc concentrations (organic) with preload: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25'C, filter = 0.45pm, co = 100 rpm, p /p\ = 6 x 10 total  s  12  _  3  Organic Interfacial Zinc Cone. vs. Preload  20%  40%  60%  100%  Preload (% Loading) Figure 4.47 Predicted change in interfacial zinc concentrations (organic) with preload: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, T = 25'C, filter = 0.45pm, co = 100 rpm, p V P i a = 6 x 10" total  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  2,bulk  — _ ZnLHL, 2  ni  o 0.010 g 0.008  ZnLg-HL  bulk  £ 0.006  0%  Figure 4.48  20%  40% 60% Preload (% Loading)  80%  100%  Predicted change in organic zinc concentrations with preload : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 4.5, T = 25 °C,  filter = 0.45pm, co = 100 rpm, p\ /p\ = 6 x 10"  5  2  3  Species Flux vs. Preload 10 "o cu  CO  o E CO O  A  8  ('""-^.effective  ZnLg C—- ZnL-HL D—- ZnL-(HL) B  6  2  4  E  Z  2  total  n  2H  2  c__  .  0 -2  -B'  X  3  -4 -6 0%  20%  40% 60% Preload (% Loading)  80%  100%  Figure 4.49 Predicted flux of organic species with preload : [Zn] = 0.05M, Formal [D2EHPA]  total  = 0.05 M , p H = 4.5, T = 25 °C,  filter = 0.45um, co = 100 rpm,  95  pVPis  = 6 x 10'  5  4.6 V a r i a b l e 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.  A n 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. Data  1201  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 V M R 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  /sec)  Zinc Flux vs. Temperature  CM  E "o E  9  -i  8-  • •  7•  65;  CO  o X X 33 LL O  c N  43-  • Experimental Data VMR:(HL)-n=1.5 VMR : (HL) - n=2  2-  2  1o15  2  i  i  i  i  i  i  20  25  30  35  40  45  50  Temperature (°C) Figure 4.50  V M R predictions and experimental data for changes in temperature : [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 4.5, filter = 0.45pm, co = 100 rpm  -16.2  Arrhenius Plot - Ln flux vs 1/T  -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 10 (1/K) 3  Figure 4.51  Arrhenius Plot for experimental temperature data with linear regression fit: [Zn] = 0.05M, Formal [D2EHPA] = 0.05 M , p H = 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 Size  Filter Length,  (pm)  (pm)  L  Porosity,  L/a  oc  (pm)  0.05 0.22  150  0.72  208  150  0.75  200  0.45  150  190  0.80  150  0.79 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) and ZnLj molecules was estimated to be approximately 2  0.0014pm.  (3)  Thus, for the 0.05pm filter the (HL) and Z n l ^ molecules are only 35 times smaller 2  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. 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) and Znl^ molecules. [16)  2  98  Zinc Flux vs. Filter Pore Size w = 100 rpm  o  CD CO  6-  ^  5  o  E  Experimental Data Mathematical Model  A  CO O X  LL O  c  1-  Ki 0.20  0.40  0.60  0.80  Filter Pore Size (/iim) 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 o  CD CO  w = 300 rpm  7H 6  O  E CO  O  5  Experimental Data Mathematical Model  4H  3 X 3  2  O  1  c  KJ  0  0  0.20  0.40  0.60  0.80  Filter Pore Size (/im) 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 Juang  1651  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 S L M system will be very similar to that encountered in this study. Under most conditions, it is likely that aqueous mass transport of Z n to the interface 2+  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 will 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) to the interface on the feed side, 2  and perhaps the diffusion of ZnL -(HL) to the interface on the strip side. This hypothesis must 2  x  still be subjected to experimental verification.  100  CHAPTER 5 - Conclusions The extraction of zinc is controlled by the mass transfer of reactants ( Z n and (HL) ) to the 2  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 p H 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 p H 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 Z n L j d T L ^ 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, E , equal to 12.4 kj/mole. This value is consistent with a t  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 S L M  applications, because it shows that there is a minimumfilterpore 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. A n 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. Since the ligand exchange rates for these metals are slower, it is likely that the system would operate in either a mixed or chemical reaction regime. The mathematical model would have to be extended to accommodate a slow chemical reaction step. Finally, by using a solvent impregnated filter membrane in the RDC, an aqueous feed solution in the outer compartment, and an aqueous strip solution in the inner compartment, a SLM-equivalent system could be studied.  102  REFERENCES  1. Developments in Solvent Extraction, S. Alegret, ed., Ellis Horwood Ltd., Chichester, England, 1988. 2. Tavlarides, L.L., Bae, J.H., and Lee, C.K. Solvent extraction, membranes, and ion exchange in hydrometallurgical dilute metals separation. Sep. Sci. Technol., 22 (2&3), pp. 581-617,1987.  3. Ritcey, C M . and Ashbrook, A.W. Solvent Extraction: Principles and Applications to Proce Metallurgy, Part II, Elsevier Scientific Publishing Co., The Netherlands, 1979. 4. Brooks, C S . 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Simulation of interfacial two-step consecutive reactions by diffusion in the mass-transfer kinetics of liquid-liquid extraction of metal cations. /. Phys. Chem., 84, pp. 3582-3587,1980. 25. Roddy, J.W., Coleman, C.F, and Arai, S. Mechanism of the slow extraction of iron(LTI) from acid perchlorate solutions by di(2-ethylhexyl) phosphoric acid in n-octane. /. Inorg. Nucl. Chem., 33, pp. 1099-1118,1971.  104  26. Komasawa, I. and Otake, T. Kinetic studies of the extraction of divalent metals from nitrate media with bis(2-ethylhexyl) phosphoric acid. Ind. Eng. Chem. Fundam., 22 (4), pp. 367-371,1983. 27. Albery, W.J., Choudhery, R. A. and Fisk, P.R. Kinetics and mechanism of interfacial reactions in the solvent extraction of copper. Faraday Discuss. Chem. Soc, 77, pp. 53-65, 1984. 28. Danesi, P.R and Vandegrift, G.F. Kinetics and mechanism of the interfacial mass transfer of Eu * and Am * in the system bis(2-ethylhexyl) 3  3  phosphate-n-dodecane-NaCl-HCl-Water. /. Phys. 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Percutaneous absorption: interfacial transfer kinetics. /. Pharmacy Pharmacology, 31, pp. 65-68,1979. 48. Albery, W.J. and Fisk, P.R. The kinetics of extraction of copper with Acorga P50 studied by a diffusion cell. Proceedings of Hydrometallurgy '81, IMM, Manchester, 1981. 49. Dreisinger, D.B., Cooper, W.C., and Distin, P. A. The kinetics of divalent metal extraction with HDEHP, HEHEHP and HDTMPP using the rotating diffusion cell. Proceedings 2nd Int'l. Conf. Sep. Sci. Technol, CS.Ch.E., Hamilton, Ontario, pp. 151-157,1989. 50. Huang, T.C. and Juang, RS. Extraction equilibrium of zinc from sulfate media with bis(2-ethylhexyl) phosphoric acid. Ind. Eng. Chem. Fundam., 25 (4), pp. 752-757,1986.  106  51. Li, Z.C., Fiirst, W., and Renon, H . Extraction of zinc(LT) from chloride and perchlorate aqueous solutions by di(2-ethylhexyl) phosphoric acid in Escaid 100: experimental equiUbrium study. Hydrometallurgy, 16, pp. 231-241,1986. 52. Sato, T., Kawamura, M . , Nakamura, T. and Ueda, M . The extraction of divalent manganese, iron, cobalt, nickel, copper and zinc from hydrochloric acid solutuions by di(2-ethylhexyl) phosphoric acid. /. appl. Chem. Biotechnol., 28, pp. 85-94,1978. 53. Smelov, V.S., Lanin, V . R , Smyk, Z.A., and Chubkov, V.V. Radiokhimya, 14 (3), p.352, 1972. 54. Teramoto, M . , Matsuyama, H . , Takaya, H . , and Asano, S. Development of spiral-type supported Uquid membrane module for separation and concentration of metal ions. Sep. Sci. Technol., 22 (11), pp. 2175-2201,1987. 55. Grimm, R. and Kolarik, Z. Acidic organophosphorous extractants - XLX: extraction of Cu(H), Co(II), Ni(H), Zn(LT), and Cd(II) by di(2-ethylhexyl) phosphoric acid. /. Inorg. Nucl. Chem., 36, pp. 189-192,1974. 56. Smelov, V.S., Lanin, V.P., Smyk, Z.A., and Chubkov, V.V. Sov. Radiochem., 14, p.364, 1972. 57. Sastre, A . M . and Muhammed, M . The extraction of zinc(LI) from sulphate and perchlorate solutions by di(2-ethylhexyl) phosphoric acid dissolved in Isopar-H. Hydrometallurgy, 12, pp. 177-193,1984. 58. Forrest, C. and Hughes, M.A. The separation of zinc from copper by di(2-ethylhexyl) phosphoric acid - an equiUbrium study. Hydrometallurgy, 3, pp. 327-342,1978. 59. Cianetti, C. and Danesi, P.R Kinetics and mechanism of the interfacial mass transfer of Z n , C o , and N i 2+  2+  2+  in the system: bis(2-ethylhexyl) phosphoric acid, n-dodecane - K N 0 , 3  water. Solv. Ex. Ion Ex., 1 (1), pp. 9-26,1983. 60. Ajawin, L.A., Demetrion, J., Perez de Ortiz, E.S., and Sawistowski, H . Effect of mass transfer parameters on zinc extraction by HDEHP. Inst. Chem. Eng. Symp. Ser., 88, pp. 183-191,1985. 61. Huang, T.C. and Juang, R.S. Kinetics and mechanism of zinc extraction from sulfate medium with di(2-ethylhexyl) phosphoric acid. /. Chem. Eng. Japan, 19 (5), pp. 379-396, 1986.  107  62. Aparicio, J. and Muhamrned, M . Extraction kinetics of zinc from aqueous perchlorate solution by D2EHPA dissolved in Isopar-H. Hydrometallurgy, 21, pp. 385-399,1989. 63. Danesi, P.R. Separation of metal species by supported liquid membranes. Sep. Sci. Technol., 19 (11&12), pp. 857-894,1984-85. 64. Fernandez, L., Aparicio, J., and Muhamrned, M . The role of feed metal concentration in the coupled transport of zinc through a bis(2-ethylhexyl) phosphoric acid solid supported liquid membrane from aqueous perchlorate media. Sep. Sci. Technol, 22 (6), pp. 1577-1595,1987. 65. Huang, T.C. and Juang, R.S. Rate and mechanism of divalent metal transport through supported liquid membrane containing di(2-ethylhexyl) phosphoric acid as a mobile carrier. /. Chem. Tech. Biotechnol, 42, pp. 3-17,1988. 66. Partridge, J.A. and Jensen, R C . Purification of di(2-ethylhexyl) phosphoric acid by precipitation of copper(LI) di(2-ethylhexyl) phosphate. /. Inorg. Nucl. Chem., 31, pp. 2587-2589,1969. 67. Rossotti, F.J.C. and Rossotti, H . Potentiometric titrations using Gran plots. /. Chem. 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).  A n 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.  for Belt I  j  Figure A.1 - The RDC and the Optical R P M Sensor A frequency counter (Advance mstruments TC9A Timer/Counter) with adjustable sensitivity input (lOmV, lOOmV, or I V - the I V 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 R P M could be directly determined according to the following equation:  109  Figure A.2 - The Tachometer Circle  6 counts x revolution  , ,. . 10 second sampling time x  1 minute 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  —  oR1 5V Regulated D3 i  JT  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 wn idows 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 — Repeat until keypressed Reset real-time clock I End 1 1  Figure B.2 Major elements of the data acquisition program  Table B.l: Pascal Listing of Data Acquisition Computer Program Program A D _ D a t a _ A c q u i s i t i o n ; const CountMAX = 12001; type timestring Filestring timearray countarray Maxstring Regpack =  { Max. number o f a r r a y elements - l i m i t  = string[8]; = string[14]; = array[0..countmax] o f b y t e ; = array[0..countmax] o f i n t e g e r ; = string[255]; record AX,BX,CX,DX,BP,SI,DI,DS,ES,Flags: end;  i s imposed  by memory r e s t r i c t i o n s }  integer;  i : integer; miscellaneous counter index } n : integer; index f o r a r r a y s A, B, C, and t c o u n t } A, B, C : t i m e a r r a y ; a r r a y s f o r hour, minute, and second v a l u e s } tcount : countarray; a r r a y f o r count v a l u e s } Datafile : text; Output t e x t f i l e } dummy : c h a r ; Base_Address, D a t a _ R e g l s t e r : I n t e g e r ;  Function Keyboardlnput : char; { Waits u n t i l var  a key i s p r e s s e d , and t h e n r e p o r t s t h e v a l u e o f t h e key. }  Key : c h a r ;  begin { Keyboardlnput } Repeat D e l a y ( 5 0 ) u n t i l  KeyPressed;  113  Read(Kbd.Key); K e y b o a r d l n p u t := Key; end; { Keyboardlnput }  Function  Question  (prompt:MaxString)  :  boolean;  { Asks t h e Yes o r No q u e s t i o n c o n t a i n e d i n t h e s t r i n g prompt and then w a i t s u n t i l a Y o r N ) ( answer has been r e c e i v e d . The v a l u e o f t h e answer i s then r e p o r t e d by the v a l u e o f t h i s ) { b o o l e a n f u n c t i o n - True = Yes and F a l s e = No. } v a r query  : char;  begin ( Question } Repeat W r i t e ( prompt + ' <Y/N> query := K e y b o a r d l n p u t ; writeln(query); Until If end;  (query  (query  i n ['Y','y','N','n']);  i n ['Y','y']) then Q u e s t i o n  { Question  Procedure  ' + chr(8)  );  { Keep a s k i n g u n t i l := t r u e e l s e Q u e s t i o n  :=  a Y or N answer i s r e c e i v e d }  false;  }  Beep;  ( Beeps the s p e a k e r - used when u s e r has made a d a t a e n t r y e r r o r ) { o r t o a l e r t the u s e r t o a p o s s i b l e problem. } b e g i n ( Beep } sound(880); Delay(125); Nosound; end; { Beep }  Procedure  { Beep at 880 Hz ) { f o r 125 m i l l i s e c o n d s . ) { Turn o f f sound. }  MakeScreen;  { T h i s p r o c e d u r e draws the t e x t s c r e e n which c o n t a i n s a s t a t u s window, a window f o r } ( u s e r i n p u t , and a window f o r d i s p l a y i n g a c q u i r e d d a t a . The windows a r e drawn u s i n g ) { IBM e x t e n d e d g r a p h i c s c h a r a c t e r s . ) var begin  i : Integer;  loop v a r i a b l e  { MakeScreen }  ClrScr; G o t o X Y d , 1); W r i t e (chr (201) ) ,f o r i := 2 t o 78 do Write(chr(187)); GotoXYd , 2); GotoXY(79, 2 ) ;  upper l e f t - h a n d c o r n e r } draw upper l e f t c o r n e r c h a r a c t e r ) draw l i n e ) draw upper r i g h t c o r n e r c h a r a c t e r }  Write(chr(205));  W r i t e (chr (186) ) ; Write(chr(186));  GotoXYd, 3); Write(chr(204)); f o r i := 2 t o 78 do Write(chr(185));  left right  2 } 2 }  row 3 ) l e f t side l i n e join character } draw l i n e ) right side line join character )  Write(chr(205));  f o r i := 4 t o 10 do begin GotoXYd , i ) ; Write(chr(186)); GotoXY(79, i ) ; W r i t e (chr (186) ) ; end; G o t o X Y d , 11); Write(chr(204)); f o r i := 2 t o 78 do Write(chr(185));  margin, row margin, row  rows 4 t h r o u g h 10 } draw l e f t draw r i g h t  margin b o r d e r margin b o r d e r  row 11 } left side l i n e draw l i n e ) right side l i n e  Write(chr(205));  f o r i := 12 t o 24 do begin G o t o X Y d , i ) ; W r i t e (chr (186) ) ; GotoXY(79, i ) ; W r i t e (chr (186) ) ; end;  rows 12  114  join  character }  join  character }  t h r o u g h 24  draw l e f t draw r i g h t  ) }  }  margin b o r d e r margin b o r d e r  } }  GotoXYd, 25); Write(chr(200)); f o r i := 2 t o 78 do W r i t e ( c h r ( 2 0 5 ) ) ; Write(chr(188));  ( 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  LowVideo; end;  ( select  corner character )  low i n t e n s i t y  video )  {MakeScreen)  Procedure  ResetWindow;  ( R e s e t s t h e t e x t s c r e e n by s e t t i n g the window back t o f u l l s c r e e n . } { A l s o moves t h e c u r s o r t o t h e bottom l e f t - h a n d c o r n e r o f the s c r e e n . ) b e g i n { ResetWindow ) W i n d o w ( l , l , 80,25); GotoXYd,24) ; end; ( ResetWindow }  Procedure  Time(var  h o u r , m i n , s e c , f r a c : b y t e ;  v a r time  :  t i m e s t r i n g ) ;  { Reads t h e r e a l - t i m e c l o c k and s e t s t h e v a l u e s hour, min, sec, frac (hundredths o f seconds) ) ( u s i n g DOS I n t e r r u p t 21h. The v a r i a b l e s t r i n g time i s a l s o s e t e q u a l t o t h e t i m e , e x p r e s s e d ) ( i n t h e form HH:MM:SS. ) Regs : Regpack; AH, AL : b y t e ; tempi, temp2, temp3 b e g i n { Time } AL := $00; AH := $2C; Regs.AX := AH s h l Intr($21,Regs) hour Regs.CX min = Regs.CX sec = Regs.DX frac = Regs.DX  end;  not r e q u i r e d , j u s t be neat s e l e c t "get system t i m e " f u n c t i o n c o n v e r t AH and AL t o 16 b i t AX v a l u e )  + AL;  shr mod shr mod  str(hour,tempi); s t r ( m i n , temp2); s t r ( s e c , temp3); i f hour < 10 t h e n i f min < 10 t h e n i f s e c < 10 t h e n time := tempi +  string[2];  { r e g i s t e r v a r i a b l e passed t o i n t e r r u p t handler ) { 8 b i t registers } ( temporary v a r i a b l e s used f o r h o l d i n g time d i g i t s  { ( { ( {  8; 256; 8; 256;  tempi '0' temp2 '0' temp3 '0' + temp2 +  c a l l i n t e r r u p t 21 h e x a d e c i m a l g e t v a l u e f o r hour ) g e t v a l u e f o r min ) get value f o r sec ) get value f o r f r a c ) ( { { { ( { {  + tempi; + temp2; + temp3; ' :' + temp3;  }  c o n v e r t numeric hour v a l u e i n t o s t r i n g } c o n v e r t numeric min v a l u e i n t o s t r i n g } c o n v e r t numeric s e c v a l u e i n t o s t r i n g } add a l e a d i n g z e r o i f hour < 10 } add a l e a d i n g z e r o i f min < 10 ) add a l e a d i n g z e r o i f s e c < 10 } c r e a t e time s t r i n g i n t h e form HH:MM:SS }  ( Time }  Procedure  SetTime(hour,min,sec:byte);  { S e t s t h e time t o t h e v a l u e s s p e c i f i e d by t h e arguments hour, min, sec u s i n g DOS I n t e r r u p t { 21h. T h i s p a r t i c u l a r r o u t i n e i s n o t p a r t i c u l a r l y e f f i c i e n t , b u t i t i s n o t i n a t i m e - c r i t i c a l ( s e c t i o n , and i s much c l e a r e r t h e way i t i s s t r u c t u r e d h e r e . } Regs : Regpack; AH, AL, CH, CL, DH, DL begin AL AH CH CL DH DL  : byte;  ( r e g i s t e r v a r i a b l e passed ( 8 bit registers )  to interrupt  { SetTime } = = = = = =  $00; $2D; hour; min; sec; 0;  { { { { { {  not r e q u i r e d , j u s t be neat s e l e c t " s e t system t i m e " f u n c t i o n s e t hour s e t minutes s e t seconds s e t h u n d r e d t h s o f seconds e q u a l t o  115  handler  Regs.AX Regs.CX Regs.DX  := AH s h l 8 + AL; := CH s h l 8 + CL; := DH s h l 8 + DL;  ( c o n v e r t AH and AL t o 16 b i t AX v a l u e } { c o n v e r t CH and CL t o 16 b i t CX v a l u e ) { c o n v e r t DH and DL t o 16 b i t DX v a l u e )  Intr($21,Regs); end;  { call  interrupt  21 h e x a d e c i m a l  )  { SetTime )  Procedure  Error;  { Catastrophic e r r o r handler. T h i s p r o c e d u r e i s i n v o k e d i f t h e A/D c o n v e r t e r r e p o r t s } ( an e r r o r , s o m e t h i n g which s h o u l d not happen under normal c i r c u m s t a n c e s . When r u n , i t } ( r e s e t s t h e t e x t window and i m m e d i a t e l y h a l t s program e x e c u t i o n . } begin ( E r r o r } Writeln; W r i t e l n ( ' A / D CONVERSION Writeln;  ERROR');  Re setWindow;  Halt; end; { Error }  Procedure ( { ( (  ResetTime(oldhour,oldmin,oldsec,A,B,C:byte)  ;  T h i s p r o c e d u r e r e s e t s t h e r e a l - t i m e c l o c k a t t h e end o f t h e a c q u i s i t i o n r u n . The time } a t t h e s t a r t o f a c q u i s i t i o n was s t o r e d i n oldhour, oldmin, £ oldsec, and A, B, & C h o l d ) t h e e l a p s e d time o f t h e r u n (hours, minutes, and seconds, r e s p e c t i v e l y . The c l o c k time ) i s o b t a i n e d by summing t h e i n i t i a l time and t h e e l a p s e d time, w i t h c a r r y . )  v a r hour, min, s e c : r e a l ; begin { sec min hour  ResetTime ) := o l d s e c + C; := o l d m i n + B; := o l d h o u r + A;  if  ( s e c >= 60) t h e n begin sec := s e c - 60; min := min + 1; end; i f (min >= 60) t h e n begin min := min - 60; hour := hour + 1; end; i f (hour >= 24) t h e n hour SetTime(hour,min,sec); end; { ResetTime }  Procedure  := hour - 24;  AcquireData(Data var var  R e g i s t e r : i n t e g e r ; v a r f c o u n t : D a t a f i l e  c o u n t a r r a y ; : t e x t ) ;  A , B , C : t i m e a r r a y ; v a r  n  :  i n t e g e r ;  ( A c q u i r e D a t a i s t h e m a s t e r d a t a 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 s e 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 s e c t i o n s a r e w r i t 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 p r e p a r e s t h e b o a r d f o r command i n p u t , and t h e second s e c t i o n ( d u p l i c a t e d t w i c e i n t h i s p r o c e d u r e ) a c q u i r e s t h e A/D r e a d i n g c h a n n e l 1. )  from  { The program c o n s t a n t l y c h e c k s f o r a v o l t a g e t r a n s i t i o n i n t h e c h a r t f e e d o u t p u t (read i n t o t h e A/D c o n v e r t e r t h r o u g h a n a l o g c h a n n e l #1. I f a t r a n s i t i o n i s d e t e c t e d , t h e n countinc is i n c r e m e n t e d and count flag i s set. The d e t e c t i o n l o o p then c o n t i n u e s u n t i l ( a f t e r r e a d i n g t h e c l o c k ) t h e new h u n d r e d t h s second v a l u e i s l e s s t h a n t h e o l d h u n d r e d t h s second v a l u e ( i . e . a new second). A t t h i s p o i n t t h e a r r a y tcount[n] i s s e t e q u a l t o tcount[n-l] p l u s t h e number o f t r a n s i t i o n s ( c o u n t i n c ) d e t e c t e d i n t h e p r e v i o u s second. The a r r a y s A[n], B[n], & C[n] a r e s e t e q u a l t o t h e e l a p s e d time (hours, minutes, and s e c o n d s ) . The time and t h e count v a l u e s a r e output t o t h e s c r e e n so t h a t program o p e r a t i o n may be v e r i f i e d ; t h e v a l u e s o f n, A[n], B[n], C[n], and tcount are a l s o outputted to the f i l e . Thus, a t t h e end o f a c q u i s i t i o n , t h e a r r a y s A, B, C, and tcount c o n t a i n t h e c o u n t / t i m e p r o f i l e , w i t h a new a r r a y element whenever a t l e a s t one v o l t a g e t r a n s i t i o n o c c u r r e d 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 H i g h , Low : countflag : countinc :  : boolean; byte; boolean; integer;  timex : t i m e s t r i n g ; D : byte; oldhour,oldmin,oldsec Dold : byte; dummy : keyflag volume : begin  byte;  char; : boolean;  { { { {  a p a i r o f 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 High and Low b y t e s o f A / D c o n v e r t e d v a l u e } i n d i c a t e s i f a t r a n s i t i o n has o c c u r r e d } number o f t r a n s i t i o n s i n t h e c u r r e n t s e c o n d }  occurrs }  { { { {  s t r i n g i n d i c a t i n g the c u r r e n t time i n HH:MM:SS format } h u n d r e d t h s o f a second d i g i t from r e a l - t i m e c l o c k } O l d time r e a d from c l o c k p r i o r t o a c q u i s i t i o n } o l d h u n d r e d t h s o f a second v a l u e - f o r second d e t e c t i o n }  { dummy c h a r a c t e r f o r k e y p r e s s e d 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 p r e s s e d }  real;  { t o t a l volume which has been d i s p e n s e d }  { AcquireData }  { I n l i n e machine code which i n i t i a l i z e s inline  <$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);  Time(oldhour,oldmin,oldsec,D,timex), n := 0; c o u n t i n c := 0; k e y f l a g := f a l s e ;  the A / D c o n v e r t e r .  { { ( { { { ( { { { ( ( { {  }  Port[Command_Register] := Command_Stop TEMP := P o r t [ D a t a _ R e g i s t e r ] Repeat u n t i l ((Port[Status_Register] and $02) = 0) Repeat u n t i l ((Port[Status_Register] and $04) = $4) Port[Command_Register] := Command_Reset Repeat u n t i l ((Port[Status_Register] and $1) = $1) TEMP := P o r t [ D a t a _ R e g i s t e r ] Repeat u n t i l ((Port[Status_Register] and $02) = 0) Repeat u n t i l ((Port[Status_Register] and $04) = $4) { get  time  from c l o c k p r i o r  to  start  of a c q u i s i t i o n }  { s e t the count i n d e x = 0 } { s e t the c u r r e n t count i n c r e m e n t = 0 } { no key has been p r e s s e d t o t e r m i n a t e e x e c u t i o n )  { Read A / D c h a n n e l t o get i n i t i a l s t a t e o f the c h a r t f e e d } { LOW c o n t a i n s t h e low b y t e , and HIGH c o n t a i n s t h e h i g h b y t e ) inline  ($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 := P o r t [ D a t a _ R e g i s t e r ] ; inline ($BA/$ED/$02/$EC/$24/$01/ $3C/$01/$75/$F9); HIGH := P o r t [ D a t a _ R e g i s t e r ] ; i n l i n e ($BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $EC/$24/$04/ $3C/$04/$75/$F9); IF ( ( P o r t [ D a t a _ R e g i s t e r ] AND $80) SetTime(0,0,0); T i m e (A[0] , B [ 0 ] , C [ 0 ] , D, timex) ; t c o u n t [ 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); GotoXYU.l) ; Write('Initialization Window( 5 , 4 , 7 5 , 1 0 ) ; Clrscr; Volume  :=  Port[Command_Register] Command ADIN Repeat u n t i l ((Port[Status_Register] and $02) = 0) Port[Data_Register] := A D g a i n Repeat u n t i l ((Port[Status_Register] and $02) = 0) Port[Data_Register] := ADchannel Repeat until  tcount[0]/400;  ',timex);  and $01)  =  $1)  ((Port[Status_Register]  and $01)  =  $1)  Repeat until  Repeat u n t i l ((Port[Status_Register] and $02) = 0) Repeat u n t i l ((Port[Status_Register] and $04) = $4) $80) t h e n E r r o r ; Set c l o c k t o 0 , 0 , 0 - z e r o e l a p s e d time } Read back the time from the c l o c k ) T o t a l c o u n t s at time z e r o = 0 ) No t r a n s i t i o n s have o c c u r r e d ) Set Aflag i f t h r e s h o l d o u t p u t i s exceeded ("HIGH") Set Bflag so i t i s the same as A F l a g ) Select  :  ((Port[Status_Register]  s t a t u s window }  Display " I n i t i a l i z a t i o n " and time S e l e c t d a t a o u t p u t window )  1;  to  show a c t i v a t i o n  { Compute the d i s p e n s e d volume )  { W r i t e t h e i n i t i a l i z a t i o n i n f o r m a t i o n t o the f i l e } Writeln(Datafile,A[0]:2,' ',B[0]:2,' ',C[0]:2,' ',tcount[0]:5,' n +  )  Increment  the  117  array  index  ',Volume:7:4);  )  { Master a c q u i s i t i o n repeat loop. T h i s l o o p r e p e a t s u n t i l an o v e r f l o w { i n t h e number o f c o u n t s o r a key i s p r e s s e d t o end a c q u i s i t i o n . ) Repeat if  Keypressed then  { Read A/D  channel  Keyflag  occurs  := t r u e ;  to get s t a t e o f the chart  inline  ($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 := P o r t [ D a t a _ R e g i s t e r ] ; i n l i n e ($BA/$ED/$02/$EC/$24/$01/ $3C/$01/$75/$F9); HIGH := P o r t [ D a t a _ R e g i s t e r ] ; i n l i n e ($BA/$ED/$02/$EC/ $24/$02/$75/$FB/ $EC/$24/$04/ $3C/$04/$75/$F9); IF ( ( P o r t [ D a t a _ R e g i s t e r ] AND $80) Dold := d; T i m e (A [n] ,B[n],C[n] ,D,timex) ;  feed. )  Port[Command_Register] : = Command_ADIN Repeat u n t i l ( ( P o r t [ S t a t u s _ R e g i s t e r ] and $02) = 0) P o r t [ D a t a _ R e g i s t e r ] := ADgain Repeat u n t i l ( ( P o r t [ S t a t u s _ R e g i s t e r ] and $02) = 0) P o r t [ D a t a _ R e g i s t e r ] := ADchannel Repeat until  ( ( P o r t [ S t a t u s _ R e g i s t e r ] and $01) = $1)  Repeat until  ( ( P o r t [ S t a t u s _ R e g i s t e r ] and $01) = $1)  Repeat u n t i l ( ( P o r t [ S t a t u s _ R e g i s t e r ] and $02) = 0) Repeat u n t i l ( ( P o r t [ S t a t u s _ R e g i s t e r ] and $04) = $4) $80) Keep then the o El r rdo hundredths r; second v a l u e f o r comparison } Read the e l a p s e d time }  ( i f a count has been measured and t h e r e i s a new second t h e n compute ( t h e t o t a l number o f c o u n t s , c l e a r count flag, w r i t e the d a t a t o t h e ( s c r e e n and t o t h e f i l e , increment n, and r e s e t countinc. } i f c o u n t f l a g and (D < Dold) t h e n begin t c o u n t [ n ] := t c o u n t [ n - 1 ] + c o u n t i n c ; c o 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 Volume := t c o u n t [ n ] / 4 0 0 ; w r i t e l n ( D a t a f i l e , A [ n ] : 2 , ' ' , B [ n ] : 2 , ' ',C[n]:2,' ' , t c o u n t [ n ] : 5 , ' n := n + 1; c o u n t i n c := 0; end; Aflag  :=  (HIGH > 9 ) ;  ( i f HIGH i s g r e a t a e r than  } }  counts'); ',Volume:7:4),  v o l t a g e t h r e s h o l d , s e t Aflag  { i f A f l a g i s not e q u a l t o B f l a g , then a v o l t a g e t r a n s i t i o n has o c c u r r e d . } { Increment countinc, s e t countflag, and r e s e t Bflag so i t i s e q u a l t o Aflag. I f not ( A f l a g = B f l a g ) t h e n begin c o u n t i n c := c o u n t i n c + 1; c o u n t f l a g := t r u e ; b f l a g := a f l a g ; end; { I f a r r a y space l i m i t i s h i t o r a key has been p r e s s e d u n t i l ((n = CountMAX) OR K e y f l a g ) ;  then  terminate  }  acquisition }  w h i l e k e y p r e s s e d 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) ; ( r e a d t h e time } Writeln; Window(5, 14, 75, 2 4 ) ; { S e l e c t t h e output window ) Clrscr; W r i t e l n ( ' E x e c u t i o n terminated: '.timex); { D i s p l a y s t a t u s , t e r m i n a t i o n time (elapsed) ResetTime(oldhour,oldmin,oldsec,A[n],B[n],C[n]); { Reset t h e r e a l - t i m e c l o c k } writeln; w r i t e l n ( ' T o t a l volume d i s p e n s e d : end;  { AcquireData  Procedure ( ( { {i {  ',volume:7:4,' m l ' ) ;  }  OpenDataFile(var  D a t a f i l e  :  t e x t ) ;  T h i s p r o c e d u r e prompts t h e u s e r f o r t h e d e s i r e d d a t a f i l e f i l e n a m e , f i l l s i n t h e e x t e n s i o n " d a t " i f i t i s not s p e c i f i e d , and t h e n c h e c k s t o see i f t h i s name i s a l r e a d y p r e s e n t on t h e d i s k . I f i t i s , the user i s given the choice of o v e r w r i t i n g t. I f t h e u s e r d e c l i n e s , t h e n t h e y a r e prompted t o e n t e r t h e f i l e n a m e a g a i n . The f i l e v a r i a b l e Datafile, i s used i n other 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 .  118  )  }  var  Filename : F i l e s t r i n g ; Exist : boolean; OKflag : boolean;  begin  { OpenDataFile \  { name o f d a t a f i l e } { t r u e i f s p e c i f i e d f i l e n a m e a l r e a d y 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 l e n a m e has been s u c c e s s f u l l y s e l e c t e d )  Window(5,12,75,24); O K f l a g := f a l s e ;  ( Set and s e l e c t the u s e r i n p u t window ) ( f i l e n a m e has not been s u c c e s s f u l l y s e l e c t e d y e t  repeat Clrscr;  { r e p e a t u n t i l OKFlag } { c l e a r the s c r e e n i n s i d e the window }  { Read f i l e n a m e s u n t i l repeat Write('Save filename: '); Readln(Filename); u n t i l (length(filename) > 0);  at  least  one c h a r a c t e r i s  }  entered  ( i f t h e f i l e n a m e i s l o n g e r t h a n 4 c h a r a c t e r s , check t o see i f the e x t e n s i o n ) ( is ".dat". I f s o , s t r i p i t o f f ). i f (length(filename) > 4) t h e n i f (copy(filename,length(filename)-3,length(filename))=' .dat' ) t h e n f i l e n a m e := 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 t h e f i l e n a m e i s l o n g e r than 8 c h a r a c t e r s , f i l e n a m e i s e q u a l t o i f (length(filename) > 8) then f i l e n a m e := c o p y ( f i l e n a m e , 1 , 8 ) ; Filename  := F i l e n a m e +  '.dat';  first  8 characters  { Add the e x t e n s i o n " . d a t "  Assign(Datafile,Filename) ;  { D e f i n e the  file  }  variable }  { T u r n o f f e r r o r c h e c k i n g . Reset d a t a f i l e , and then r e - e n a b l e e r r o r c h e c k i n g . If } { D a t a f i l e e x i s t s , t h e n I O r e s u l t w i l l be e q u a l t o 0. } ($1-} Reset(Datafile) ($1+) ; E x i s t := ( I O r e s u l t = 0 ) ; {If the f i l e a l r e a d y e x i s t s on t h e d i s k . E x i s t = t r u e O K f l a g := not E x i s t ; { The f i l e n a m e i s OK t o use i f i t d o e s n ' t E x i s t ) { I f the f i l e n a m e e x i s t s , i f Exist then begin  then warn the u s e r and ask whether o r not t o  overwrite  }  Beep;  Beep; Writeln; Writeln('WARNING:  File  " ' , F i l e n a m e , ' " already e x i s t s ! ' ) ;  ( I f t h e y answer y e s t o the q u e s t i o n , then the f i l e n a m e i s OK, so s e t { OKflag=true. C l o s e the D a t a f i l e and E r a s e i t t o s t a r t w i t h a c l e a n s l a t e . I f q u e s t i o n ( ' D o you want t o e r a s e i t ? ' ) then begin O K f l a g := t r u e ; Close (Datafile) ; Erase(Datafile) ; end;  } }  end until  OKflag;  { Repeat a s k i n g f o r  Clrscr;  {  datafile  :  '.Filename),  (  Identify  the o u t p u t  filename  selected )  { OpenDataFile }  Main  Begin  one i s OK  ( C l e a r the window )  writeln; writeln('Output end;  filenames u n t i l  -  Main program code  }  { Main }  Base_Address Data_Register  := :=  $2EC; Base_Address;  Makescreen;  { Memory A d d r e s s o f the Data A c q u i s i t i o n b o a r d } { Data R e g i s t e r A d d r e s s i s t h e { Draw the  same as t h e Base A d d r e s s }  t e x t windows )  OpenDataFile(Datafile); Rewrite(Datafile) ;  ( Get t h e f i l e F i l e n a m e and p r e p a r e f o r f i l e w r i t e ) ( Open and empty the f i l e - s e t p o i n t e r t o the b e g i n n i n g } AcquireData(Data_Register,A,B,C,tcount,n,DataFile); { Master A c q u i s i t i o n Routine ) Close(Datafile);  { C l o s e the  file  119  }  for  i  := 1 t o  20000 d o ;  Writeln; W r i t e ( ' P r e s s any key t o dummy := K e y b o a r d l n p u t ; ResetWindow; Clrscr; end. { Main )  { d o - n o t h i n g w a i t l o o p t o pause program - i t i s h e l p f u l { see f i n a l count v a l u e s b e f o r e t e r m i n a t i n g program. continue...'); ( Wait f o r a k e y p r e s s b e f o r e t e r m i n a t i n g ( Reset t e x t s c r e e n } ( that's  all  folks }  120  execution }  to  ) )  APPENDIX C - Raw Experimental Data  NOTE: A n asterisk (*) signifies that the marked parameter is being varied from the baseline.  STUDYA01 STUDYA02 STUDYA03 STUDYA04 STUDYA05 STODYA06 STUDYA07 STUDYA08 STUDYA09 STUDYA10 STUDYA11 STUDYA12 STUDYA13 STODYA14 STUDYA15 STUDYA16 STUDYA17 STUDYA18 STUDYA19 STUDYA20 STUDYA21 STUDYA22 STUDYA25 STUDYA26 STUDYA27 STUDYA28 STUDYA30 STUDYA31 STUDYA32 STUDYA33 STUDYA34 STUDYA35 STUDYA36 STUDYA37 STUDYA38 STUDYA39 STUDYA40 STUDYA41 STUDYA42 STUDYA43 STUDYA44 STUDYA45 STUDYA4 8 STUDYA50 STUDYA51 STUDYA52  [Zn] ( M )  [D2EHPA] ( F )  pH  Temp. ( 'C )  0 .05 0 .05 0 .05 0 .05 0. 001* 0. 03* 0. 01* 0. 005* 0. 003* 0. 10* 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0. 005* 0. 10* 0. 005* 0 .05 0 .05 0 .05 0. 005* 0. 10* 0. 20* 0. 20* 0. 02* 0. 04* 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05  0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0. 10* 0. 025* 0. 010* 0. 005* 0.0025* 0. 005* 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0 .05 0. 10* 0. 075* 0. 035* 0 .05 0 .05 0 .05 0 .05 0 .05  5. 50* 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 5. 00* 3. 25* 4 .00* 3. 75* 3. 50* 5. 50* 4 . 50 4 .50 4 .50 4 . 50 4 . 50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50 4 .50  25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 30* 15* 20* 35* 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 35* 50* 50* 40* 45*  Intercept ( x 10' ) 1..20225 1..27801 1..60495 1..44678 0..46758 1..56962 1..62316 1..87731 1..44404 1..56957 0..87816 2..78002 5..63237 0..93892 14 .39475 , 12 .06940 . 1..38820 1..88770 1..56045 1..33529 1..41360 1..40901 1..15114 1..62001 1..63941 1..07196 1..27783 1..25067 1..31220 1..21560 1..44421 1..35095 1..30828 1..01799 1..23857 1..33987 1..34332 1..40675 0..71916 0..92083 1..87262 1..09105 0..92457 0..88095 1,,03295 0,.95741  121  Slope ( x 10' ) 0,.48573 0..64266 0..52152 0..53018 5..80803 0..58608 0..81708 1..31886 1..59222 0..51383 0..33828 1..22596 1..98924 5..15378 11..31931 4 .45671 . 0..58543 0.. 12249 0..53104 0..68410 0..77253 0,.63507 0..49328 0..63318 0,.28731 0..44648 1..03012 0 .56321 , 1,.09904 0..59272 0..57371 0.. 61098 1,. 12605 0..84444 0..49103 0,.43416 0.. 64144 0..54474 0.,30998 0..36531 0,. 68720 0..41156 0..47012 0..43449 0,.38181 0,.39778  J(100rpm) kmol/mVs  J(300rpm) kmol/mVs  6 .34E-08 5 .63E-08 4 .98E-08 5 .38E-08 2 .01E-08 4 . 94E-08 4 .43E-08 3 . 45E-08 3 .74E-08 5 .08E-08 8 .77E-08 2 .68E-08 1 . 39E-08 2 .03E-08 4 .32E-09 6 .44E-09 5 .43E-08 5 .04E-08 5 .07E-08 5 .36E-08 4 .97E-08 5 .26E-08 6 . 52E-08 4 . 74E-08 5 .37E-08 7 .05E-08 4 .82E-08 5 .93E-08 4 .62E-08 5 . 97E-08 5 .29E-08 5 .48E-08 4 .59E-08 5 .98E-08 6 .18E-08 5 .97E-08 5 .43E-08 5 .47E-08 1 .04E-07 8 .31E-08 4 .16E-08 7 .09E-08 7 .76E-08 8 .21E-08 7 .53E-08 7 .90E-08  7.. 04E-08 6..39E-08 5..44E-08 5.. 94E-08 3..26E-08 5.. 46E-08 5..03E-08 4 . 05E-08 4 . 64E-08 5.. 56E-08 9..71E-08 3..00E-08 1.. 53E-08 3..08E-08 5..14E-09 7..11E-09 6..06E-08 5..15E-08 5..56E-08 6..09E-08 5.. 68E-08 5..91E-08 7 .29E-08 . 5..25E-08 5.. 66E-08 7.. 86E-08 5..75E-08 6.. 66E-08 5.. 54E-08 6..75E-08 5..88E-08 6.. 16E-08 5.. 52E-08 7..17E-08 6..86E-08 6..52E-08 6.. 13E-08 6.,06E-08 1.. 17E-07 9..22E-08 4 .59E-08 . 7..84E-08 8..81E-08 9.•30E-08 8.•31E-08 8..81E-08  [D2EHPA] ( F )  % Loading  0..05 0..05 0..05 0..05 0..05 0..05 0..05 0,.05 0..05 0..05 0,.05  0..05 0..05 0..05 0..05 0..05 0..05 0.,05 0..05 0..05 0..05 0,.05  8.2 24 .3 0 18.7 40.0 49.2 28.5 57 .7 82.2 67.6 88.1  [Zn] ( M)  [D2EHPA] (F )  filter  0..05 0..05 0..05 0..05 0..05 0.,05 0..05  0.,05 0.,05 0.,05 0.,05 0..05 0.,05 0..05  [Zn] ( M ) STUDYB01 STUDYB02 STUDYB03 STUDYB04 STUDYB05 STUDYB06 STUDYB07 STUDYB08 STUDYB09 STUDYB10 STUDYB11  STUDYD01 STUDYD02 STUDYD03 STUDYD04 STUDYD05 STUDYD06 STUDYD07  Intercept ( x 10 )  Slope ( x 10 )  J(100rpm) kmol/mVs  1.,44008 1..87071 1,,29804 1..77354 2,,45371 3..29994 2..02465 4..03017 12..01515 6,.45097 28 .52638  0,,62974 0..75522 0,.58299 0..87059 1.,32527 1..49803 0..95048 2..09985 3..40072 2.,35808 14 .30439  5., 19E-08 4..07E-08 5., 72E-08 4 .09E-08 . 2,,87E-08 2..24E-08 3.,62E-08 1.,77E-08 6,.83E-09 1..21E-08 2.. 52E-09  7  7  Intercept Slope ( x 10 ) ( x 10 ) 7  0,. 22|lm 0..80|im' 0,. 05u.m 0,. 80u.m 0.,22u.m 0,. 05u.m 0,. 22U-m  1,.34584 1,.27965 2,.86257 1..25882 1..27255 2,.87390 1,.38042  122  7  0.,61641 0.,47415 0.,56463 0..54710 0.,54201 0..50528 0..54523  J(300rpm) kmol/mVs  .  5..81E-08 4,.53E-08 6..42E-08 4 .. 62E-08 3.•28E-08 2.. 52E-08 4 .08E-08 . 2..01E-08 7..39E-09 1.•33E-08 2..86E-09  J(100rpm) kmol/mVs  J(300rpm) kmol/m /s  5,. 48E-08 6.. 07E-08 3..03E-08 5..94E-08 5..91E-08 3..06E-08 5..55E-08  6., 17E-08 6., 70E-08 3 .21E-08 . 6..65E-08 6.,60E-08 3..23E-08 6.. 16E-08  2  APPENDIX D - Basic Mathematical Model The mathematical derivation and programflowchartfor 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 , . Hp s _ H \ , . HL = HL , HL2 = (HL) „, ZnL2HLl = Z n L - H L , and ZnL2HL2 s ZnL -(HL) , „ „ } o f b i n d i c a t e s a b u l k 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 a q )  q l  ( o t g ) 2  ( o r  2  2  { a suffix const  betal3_over_betal4  =  le-3;  Kd = 5 . 0 1 2 e 4 ; K e x = 3.236e-2; c _ Z L _ b u l k = 0; w = 100; type  ( o t g )  Filestring MaxString Extension  2  ( r a t i o of ZnL -HL and ZnL -(HL) f o r m a t i o n c o n s t a n t s , = 10 " -3 } ( D i m e r i z a t i o n c o n s t a n t f o r D2EHPA, = 10 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 organic 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 } 2  2  string[14]; string[255]; string[4]; mass t r a n s f e r c o e f f i c i e n t s } species bulk concentrations } species i n t e r f a c i a l concentrations species i n t e r f a c i a l concentrations zinc flux i n kmol/m /sec } association factor ) i n i t i a l , f i n a l , i n c r e m e n t , and c u r r e n t parameter v a l u e s }  b e g i n { Beep } sound(880); Delay(125); Nosound; end; ( Beep }  )  { Beep a t 880 Hz } ( f o r 125 m i l l i s e c o n d s . } { Turn o f f sound. }  Keyboardlnput  { Waits u n t i l v a r Key :  2  Beep;  { Beeps t h e s p e a k e r - u s e d when u s e r has made a d a t a e n t r y e r r o r { or t o a l e r t t h e u s e r t o a p o s s i b l e p r o b l e m . }  Function  2  A  aq_Zn, k_aq_Hp, k _ o r _ H L 2 , k _ o r _ Z L l , k _ o r _ Z L 2 , HL e f f _ b u l k , c _ Z n _ b u l k , pH, c _ H p _ b u l k , c_ZLtot_i, c_Zn_i, c_HL2tot_i, c_Hp_i, c _ H L _ i , c HL2 i , c ZnL2HLl i , c ZnL2HL2 i . JZn, n. V a r y _ i , Vary_f, Vary_inc, Vary_value : r e a l ;  Procedure  species  :  char;  a key i s p r e s s e d , and t h e n r e p o r t s  the v a l u e o f the k e y .  char;  begin ( Keyboardlnput ) Repeat D e l a y ( 5 0 ) u n t i l K e y P r e s s e d ; Read(Kbd,Key); K e y b o a r d l n p u t := K e y ; end; ( Keyboardlnput }  123  }  } }  Function  Question  (prompt:MaxString)  :  boolean;  { Asks t h e Yes o r No q u e s t i o n c o n t a i n e d i n the s t r i n g prompt and then w a i t s u n t i l a Y o r N } { answer has been r e c e i v e d . The v a l u e o f the answer i s then r e p o r t e d by t h e v a l u e o f t h i s } { b o o l e a n f u n c t i o n - True s Yes and F a l s e s No. ) v a r query  : char;  begin { Question ) Repeat W r i t e ( prompt + ' <Y/N> query := Keyboardlnput; writeln(query); Until If end;  (query  (query  + chr(8)  );  ['Y','y','N','n']);  ['Y','y'])  ( Keep a s k i n g u n t i l  then Q u e s t i o n := t r u e e l s e Q u e s t i o n :=  a Y o r N answer i s r e c e i v e d )  false;  { Question )  Procedure { ( { { (  in  in  '  OpenDataFile(var  D a t a f i l e  :  text;  filename_extension  :  extension);  T h i s p r o c e d u r e prompts the u s e r f o r the d e s i r e d d a t a f i l e f i l e n a m e , adds t h e e x t e n s i o n s p e c i f i e d , and t h e n c h e c k s t o see i f t h i s name i s a l r e a d y p r e s e n t on t h e d i s k . If i t i s , t h e u s e r i s g i v e n the c h o i c e o f o v e r w r i t i n g i t . I f the u s e r d e c l i n e s , then they are prompted t o e n t e r the f i l e n a m e 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 t h e r 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  Filename : F i l e s t r i n g ; Exist : boolean; OKflag : boolean;  } ) } }  { name o f d a t a f i l e ) { t r u e i f s p e c i f i e d f i l e n a m e a l r e a d y 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 l e n a m e has been s u c c e s s f u l l y s e l e c t e d )  begin { OpenDataFile ) O K f l a g := f a l s e ;  ( f i l e n a m e has not been s u c c e s s f u l l y s e l e c t e d y e t  repeat { repeat u n t i l OKFlag ) repeat ( Read f i l e n a m e s u n t i l at Write('Save filename: '); Readln(Filename); u n t i l (length(filename) > 0) ;  least  }  one c h a r a c t e r i s e n t e r e d )  { i f t h e f i l e n a m e i s l o n g e r than 4 c h a r a c t e r s , check t o see i f the e x t e n s i o n ) { matches filename_extension. If so, s t r i p i t o f f . ) i f (length(filename) > 4) then if (copy(filename,length(filename)-3,length(filename))=filename_extension) then f i l e n a m e := copy(filename,1,length(filename)-4); { i f t h e f i l e n a m e i s l o n g e r than 8 c h a r a c t e r s , f i l e n a m e i s e q u a l t o If (length(filename) > 8) then f i l e n a m e := c o p y ( f i l e n a m e , 1 , 8 ) ; Filename  := F i l e n a m e + f i l e n a m e _ e x t e n s i o n ;  { D e f i n e the  { Turn o f f e r r o r c h e c k i n g . Reset { D a t a f i l e e x i s t s , then I O r e s u l t {$1-} Reset(Datafile) ($1+) ; E x i s t := ( I O r e s u l t = 0 ); O K f l a g := not E x i s t ;  8 characters }  ( Add the e x t e n s i o n )  Assign(Datafile,Filename);  { I f the f i l e n a m e e x i s t s , i f Exist then begin  first  d a t a f i l e , and then r e - e n a b l e e r r o r w i l l be e q u a l t o 0. }  file  checking.  variable } If  }  (If the f i l e a l r e a d y e x i s t s on the d i s k . E x i s t = t r u e ( The f i l e n a m e i s OK t o use i f i t d o e s n ' t E x i s t }  then warn the u s e r and ask whether  or not t o o v e r w r i t e  }  Beep; Beep;  Writeln; Writeln('WARNING:  File  "',Filename, "• already  exists!');  { I f t h e y answer yes t o the q u e s t i o n , then the f i l e n a m e i s OK, so set ( OKflag=true. C l o s e the D a t a f i l e and E r a s e i t t o s t a r t w i t h a c l e a n s l a t e . I f QuestionCDo you want t o e r a s e i t ? ' ) then begin O K f l a g := t r u e ; Close(Datafile); Erase(Datafile);  124  ) )  )  end; end u n t i l OKflag;  { Repeat a s k i n g  Writeln('Output Writeln;  datafile  : '.Filename);  Rewrite(Datafile);  { Identify  ( Open and  empty the  { 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 [ H L ] e f f , b [ZnL]b Writeln(Datafile,'[(HL)2]i [HL2]ieff [ZnLl]i end;  { OpenDataFile  Function  for filenames  file  pH,b JZn [ZnL2]i  until  the o u t p u t  one  filename  { Computes t h e v a l u e  n  :  n [ZnL]i,t  [Zn]i [H+]i');  r e a l ) : r e a l ;  o f x" and  r e t u r n s the value  as Power. }  b e g i n { Power ) power := e x p ( n * l n ( x ) ) ; end; ( Power }  Procedure  Quadratic(a,b,c  { S o l v e s the q u a d r a t i c e q u a t i o n  : and  r e a l ;  var  r o o t l ,  r e t u r n s the two  root2  r o o t s as  :  rootl  r e a l ) ; and  root2.  )  begin { Quadratic } r o o t l := (-b + s q r t ( b * b - 4 * a * c ) ) / ( 2 * a ) ; r o o t 2 := (-b - s q r t < b * b - 4 * a * c ) ) / ( 2 * a ) ; end; ( Quadratic )  Procedure  Calculate_Constants;  { Computes t h e 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 } ( e q u i v a l e n 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 t h e n 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 a p p r o p r i a t e f o r m u l a . ) const  D_HL2 D_ZnL2HLl D_ZnL2HL2 D_HC104 D_Zn nu H20 nu_heptane = L_over_alpha  var  zD_aq_Zn,  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 k_aq_Zn k_aq_Hp k_or_HL2 k_or_ZLl k o r ZL2 end;  1.003e-9 0.781e-9 0.659e-9 2.854e-9 1.038e-9 8.93e-7; 5.64e-7; = 1.90e-4;  zD_aq_Hp,  zD_or_ZLl,  zD_or_ZL2  : real;  }  0.643/sqrt(w/60)*Power(nu_H2O, (1/6) ) * P o w e r (D_Zn, (1/3) ) ; 0 . 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u _ H 2 0 , ( 1 / 6 ) ) * Power(D_HC104, ( 1 / 3 ) ) ; 0.643/sqrt(w/60)*Power(nu_heptane,(1/6)) * P o w e r ( D _ H L 2 , ( 1 / 3 ) ) ; 0.643/sqrt(w/60)*Power(nuheptane,(1/6)) * Power(D_ZnL2HLl,(1/3)); 0 . 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u h e p t a n e , (1/6) ) * P o w e r f D ZnL2HL2, (1/3)), D_Zn D_HC104 D_HL2 D_ZnL2HLl D ZnL2HL2  { Calculate  Procedure  zD_or_HL2,  zD_aq_Zn; zD_aq_Hp; (zD_or_HL2 (zD_or_ZLl (zD o r ZL2  L_over_alpha); L_over_alpha); L_over_alpha);  Constants  Vary_Setup;  { T h i s procedure j u s t gets the i n i t i a l , f i n a l , { a range o f c o n c e n t r a t i o n s i s examined. }  and  125  increment  }  selected }  - set p o i n t e r to the beginning  }  Power(x,  i s OK  v a l u e when }  [HL]i  )  ');  begin { Vary_Setup } Write (' I n i t i a l Value Readln ( V a r y i ) ; Write (' Final Value Readln ( V a r y _ f ) ; Write (' Increment Readln (Vary_inc); i f Vary_i > Vary_f then  r e c u r s e t o get  correct  values i f  initial  i s > than  final  Vary_Setup  else if  { r e c u r s e t o get  V a r y _ i n c <= 0 then Vary_Setup; { Vary_Setup }  end;  Procedure I n p u t _ P a r a m e t e r s ( v a r var  { t ( ( { {  correct values i f  c _ Z n _ b u l k , c_HL e f f _ b u l k , code : integer")";  increment  pH  :  is < 0 }  r e a l ;  T h i s p r o c e d u r e prompts the u s e r f o r v a l u e s f o r the b u l k z i n c c o n c e n t r a t i o n , b u l k D2EHPA c o n c e n t r a t i o n ( e x p r e s s e d as d i m e r ) , and b u l k pH. The u s e r can examine a range o f v a l u e s by e n t e r i n g a v a l u e o f - 1 . In t h i s c a s e , the s u b r o u t i n e V a r y _ S e t u p i s c a l l e d which prompts the u s e r f o r the i n i t i a l , f i n a l , and i n c r e m e n t a l 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 o r n o t ; a v a l u e of zero i n d i c a t e s no r a n g e , w h i l e 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  :  { temporary  real;  begin ( Input_Parameters } code := 0; Writeln; W r i t e l n (' E n t e r model p a r a m e t e r s : Write (' [Zn]bulk = ' ) ; Readln (Input); i f (Input = -1) then begin code := 1;  ( no range y e t (to  :=  -1  at  the  prompt)'); )  input;  (  end else c_HL_eff_bulk  :=  select  a range o f v a l u e s }  ( range i s f o r HL ) ( get the range )  Vary_setup;  input;  Write (' pH = ' ) ; Readln (Input); i f (Input = -1) then begin code := 3;  (  select  a range o f v a l u e s  )  ( range i s f o r H* ) { get the range )  Vary_eetup;  end;  enter  ( range i s f o r Zn ) ( get the range )  Write (' [HL]bulk = ' ) ; Readln (Input); i f (Input = -1) then begin code := 2;  end else pH :=  vary,  )  }  { s e l e c t a range o f v a l u e s  Vary_setup;  end e l se c_Zn_bulk  storage v a r i a b l e  input;  ( Input_Parameters  }  Procedure 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 ( J Z n : R e a l ) ; { T h i s p r o c e d u r e computes t h e 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 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 b o t h ZnL -HL and ZnL -(HL) c o m p l i c a t e s the p r o c e d u r e somewhat, so t h a t 2  2  126  2  an i t e r a t i v e a p p r o a c h i s needed t o f i n d the c o r r e c t v a l u e 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 t h a t h e e r r o r c r i t e r i a eps. } eps  const  = le-4;  v a r J H L 2 e f f , JHp, JZL : R e a l ; a, b, c, r o o t l , r o o t 2 : r e a l ; n _ o l d , n_new : r e a l ; begin  { Compute_Interface_Concentrations  ]  c _ Z n _ i := - JZn / k_aq_Zn + c_Zn_bulk; i f c _ z n _ i < 0 t h e n c _ z n _ i := l e - 1 0 ;  ( Compute the i n t e r f a c i a l [Zn *] ) { i f i t ' s n e g a t i v e , make i t v e r y s m a l l }  JHp := -2 c _ H p _ i := i f c_hp_i  { from the mass b a l a n c e } { Compute t h e i n t e r f a c i a l [H ] ) { i f i t ' s n e g a t i v e , make i t v e r y  n_new  :=  2  * JZn; - JHp / k_aq_Hp + c_Hp_bulk; < 0 t h e n c _ h p _ i := l e - 1 0 ;  +  { s e t up  n;  initial  { iterate  repeat n _ o l d := n; n := n_new;  n's  value  until  }  w i t h i n e r r o r eps  { from the mass b a l a n c e } J H L 2 e f f := JZn * n; c _ H L 2 t o t _ i := - J H L 2 e f f / k_or_HL2 + 0.5 * c _ H L _ e f f _ b u l k ; i f c _ H L 2 t o t _ i < 0 then c _ H L 2 t o t _ i := l e - 1 0 ; { i f i t ' s negative, { s o l v e f o r p a r t i t i o n between HL a := Kd; b := 1/2; c := - c _ H L 2 t o t _ i ; Q u a d r a t i c ( a , b, c , r o o t l , root2) ; c _ H L _ i := r o o t l ; c HL2 i := c . H L 2 t o t  i-l/2*c  and  (HL)  u s i n g the q u a d r a t i c e q u a t i o n  2  small }  )  make i t v e r y  small  }  { i n t e r f a c i a l HL i s e q u a l t o the f i r s t r o o t } ( compute i n t e r f a c i a l (HL) from mass b a l a n c e  HL i ;  2  JZL := JZn; { from the mass b a l a n c e } { compute 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 o f ZnL -HL and ZnL -(HL) } c _ Z n L 2 H L l _ i := (JZL) / ( k _ o r _ Z L l + k_or_ZL2 * c _ H L _ i / b e t a l 3 _ o v e r _ b e t a l 4 ) ; c_ZnL2HL2_i := (JZL) / ( k _ o r _ Z L l * b e t a l 3 _ o v e r _ b e t a l 4 / c _ H L _ i + k _ o r _ Z L 2 ) ; c_ZLtot_i := c _ Z n L 2 H L l _ i + c_ZnL2HL2_i; 2  n_new := Until n end;  :=  (1.5  * c_ZnL2HLl_i  + 2 * c_ZnL2HL2_i)  ( (abs(n_new-n_old)/n_new)  < eps  /  2  2  c_ZLtot_i;  );  n_new;  { Compute_Interface_Concentrations  Function  F ( x : r e a l )  :  }  r e a l ;  { 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 computed f o r t h e g i v e n f l u x X by t h e f u n c t i o n Compute_Interface_Concentrations. The e q u i l i b r i u m c o n s t a n t f o r the g i v e n 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 i s t h e n computed and compared w i t h the a c t u a l e q u i l i b r i u m c o n s t a n t . Note t h a t 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 a r e u s e d i n t h e Kex e x p r e s s i o n r a t h e r t h a n 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 a r e assumed t o be e q u a l t o 1. ) begin  { F }  Compute_Interface_Concentrations(X);  F end  := | F  c_ZnL2HLl_i  * Power(c_Hp_i,2)  / c_Zn_i  / Power(c_HL2_i,1.5) -  K_ex;  );  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 ;  var  X3  :  r e a l ) ;  { t h i s p r o c e d u r e u s e s an i n c r e m e n t a l s e a r c h f o l l o w e d by a b i s e c t i o n r o u t i n e t o f i n d the r o o t o f the f u n c t i o n F ( X ) . F ( X ) i s d e f i n e d as t h e d i f f e r e n c e between the e q u i l i b r i u m c o n s t a n t computed  127  from t h e 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 a c t u a l v a l u e o f 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 , t h e 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 t h e n the d i f f e r e n c e between t h e two e q u i l i b r i u m c o n s t a n t s i s computed. } var XI,  YI,  X2,  Y2,  X30LD,  Y3  :  REAL;  LABEL LOOP; begin XI YI IF  { Solve } := X i n i t i a l ; := F(X1) ; (YI = 0) then begin X3 := X I ; Exit; end;  repeat X2 := XI + dX; i f (X2 > X f i n a l ) then begin Writeln('Solution Exit; end; Y2 := F ( X 2 ) ; if  until  not  found on s e a r c h  (Y1*Y2) > 0 then begin XI := X2; YI := Y2; end else i f (Y1*Y2) = 0 then begin X3 := X2; Exit; end; (Y1*Y2) < 0;  { Start Bisection u n t i l  accuracy in X f a l l s  LOOP: X3 := (XI + X2) / 2; i f (abs((X3 - XI) / XI) < eps) Y3 := F ( X 3 ) ; i f (YI * Y3) = 0 then E x i t ; i f (YI * Y3) < 0 then begin ( yl*y3 < 0 } X2 := X3; Y2 := Y 3 ; end else begin { yl*y3 > 0 } XI := X 3 ; YI := Y 3 ; end; Goto LOOP; end;  interval');  then  within error  eps }  Exit;  { Solve }  Procedure  Write_Console_Data;  ( This 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 , the bulk c o n c e n t r a t i o n s , ) ( and t h e computed 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 t o t h e s c r e e n . } begin { Write_Console_Data } W r i t e l n ( ' F l u x = ',JZn:8,' kmol/m 2/sec n = ',n:6:4); Write ('[Zn],b = ' , c_Zn_bulk: 6: 4,' ..- ' ) ; Write (' [ H L ] e f f , b = ' , c_HL_ef f _ b u l k : 6: 4, ' '); W r i t e l n ( ' [ Z n L 2 ] , b = ',c_ZL_bulk:6,'. pH = ',pH:4:2); W r i t e l n (' [ Z n ] , i = ' , c _ Z n _ i : 8 : 6 ) ; Write ( ' [ H L ] 2 e f f , i = ' , c _ H L 2 t o t _ i : 8: 6) ; WritelnC [HL] , i = ' , c _ H L _ i : 8 : 6, ' [ H L ] 2 , i = ' , c_HL2_i : 8 : 6) ; Write ( ' [ Z n L ] t o t , i = ' , c _ Z L t o t _ i : 8 : 6) ; WritelnC [ Z n L 2 H L ] , i = ' , c _ Z n L 2 H L l _ i : 8: 6,' [ZnL2HL2],i = ', c_ZnL2HL2_i: 8: 6) ; Writeln; end; { Write_Console_Data } A  128  Procedure  Write_File_Data(  v a r  D a t a f i l e  :  text  );  { T h i s p r o c e d u r e w r i t e s t h e v a l u e s f o r the b u l k c o n c e n t r a t i o n s , the z i n c f l u x , { 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 c o n c e n t r a t i o n s t o the D a t a f i l e . } begin { Write_File_Data } Write (Datafile,c_Zn_bulk:9,' ',c_HL_eff_bulk:9,' ',c_ZL_bulk: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_ZnL2HLl_i:9,' '); Writeln(Datafile,c_ZnL2HL2_i:9,' ' , c _ Z L t o t _ i : 9 , ' ',c_Hp_i:9); end; { W r i t e F i l e Data }  {  Main -  Main program  code  the  }  ');  }  ( T h i s i s the main p r o g r a m . P r o c e d u r e s 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 l e n a m e and s e t up the f i l e f o r w r i t i n g , and o b t a i n the b u l k 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 l o o p i s then e x e c u t e d which f i n d s the z i n c f l u x f o r the c o n d i t i o n s s p e c i f i e d , and w r i t e s v a l u e s t o the s c r e e n and t h e d a t a f i l e . This loop c o n t i n u e s u n t i l t h e f l u x f o r a l l c o n c e n t r a t i o n v a l u e s s p e c i f i e d has been computed. The d a t a f i l e i s t h e n c l o s e d and e x e c u t i o n i s t e r m i n a t e d . } ( The v a r i a b l e vary i n i t s v a r i o u s p e r m u t a t i o n s i s u s e d to accomodate the c i r c u m s t a n c e where the z i n c f l u x i s computed f o r a range o f v a l u e s . In t h i s c a s e , code i s s e t t o some v a l u e o t h e r than z e r o , i t s p a r t i c u l a r v a l u e s p e c i f y i n g the parameter t o be a l t e r e d . Vary_i, vary_f, vary_inc, and vary_value a r e t h e n u s e d t o c y c l e the parameter t h r o u g h the r a n g e . }  begin  Xi = l e - 9 ; Xf = 5e-7; dX = 5 e - 9 ; eps = l e - 4 ;  { { ( (  Datafile : text; code : i n t e g e r ;  ( Output D a t a f i l e ) { Parameter code }  Initial flux value for i t e r a t i v e search Final flux value for i t e r a t i v e search Incremental f l u x value f o r i t e r a t i v e search 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  routine routine routine }  ) } )  { Main )  Clrscr; Calculate_Constants;  OpenDataFile(Datafile,'.out') ;  { 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 F i l e n a m e and p r e p a r e f o r  file  write  )  { E x e c u t e I/O r o u t i n e t o get v a l u e s f o r the aqueous z i n c c o n c e n t r a t i o n , D2EHPA c o n c e n t r a t i o n ) { ( e x p r e s s e d as monomer), and aqueous pH. The r o u t i n e r e t u r n s t h e s e v a l u e s , as w e 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 o v e r a r a n g e . } I n p u t _ P a r a m e t e r s ( c _ Z n _ b u l k , c _ H L _ e f f _ b u l k , pH, c o d e ) ; Vary_value  := V a r y _ i ;  { s e t the  range v a r i a b l e e q u a l t o  { i f code i s n o n z e r o , s e t the a p p r o p r i a t e parameter i f code = 1 t h e n c _ Z n _ b u l k := V a r y _ v a l u e ; i f code = 2 t h e n c _ H L _ e f f _ b u l k := V a r y _ v a l u e ; i f code = 3 t h e n pH := V a r y _ v a l u e ; Repeat  { Loop o n c e ,  n :=  1.5;  c_Hp_bulk  := e x p ( - p H *  2.303);  e q u a l t o the  or u n t i l  the  range i n i t i a l  range v a r i a b l e  value }  }  range maximum i s exceeded )  { An i n i t i a l v a l u e f o r n ) { c o n v e r t pH i n t o H* c o n c e n t r a t i o n }  Solve(Xi,Xf,dX,eps,JZn);  ( Conduct an i t e r a t i v e s e a r c h f o r JZn o v e r 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  Write_Console_Data; W r i t e F i l e Data(DataFile) ,  ( D i s p l a y the s o l u t i o n on the s c r e e n } ( Output v a l u e s t o s p e c i f i e d D a t a f i l e }  { I f code i s not e q u a l t o z e r o , i . e . run f o r a range o f v a l u e s , i n c r e m e n t the range ) { v a r i a b l e and t h e n s e t the a p p r o p r i a t e parameter e q u a l t o t h e new range v a l u e . } i f not (code = 0) t h e n begin V a r y _ v a l u e := V a r y _ v a l u e + V a r y _ i n c ; i f code = 1 t h e n c_Zn_bu'lk := V a r y _ v a l u e ; i f code = 2 t h e n c _ H L _ e f f _ b u l k := V a r y _ v a l u e ; i f code = 3 t h e n pH := V a r y _ v a l u e ; end; until  ((code = 0)  or  (Vary_value  > Vary_f)) ,  Close(Datafile); end.  ( C l o s e the  { Main }  129  file  }  eps.  APPENDIX E - Extended Mathematical Model The mathematical derivation and programflowchartfor 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 : Program  Pascal Listing of Extended Mathematical Model Preload;  { guide to v a r i a b l e  notation:  Zn = Z n * , , Hp 3 _ H * , HL = HL , HL2 = (HL) , „ „ , ZL = t o t a l o r g a n i c z i n c , ZnL2HL0 = Z n L , ZnL2HLl = Z n L - H L , and ZnL2HL2 H ZnL -(HL) , } a b u l k 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 2  (  q l  ( a q l  ( o r g | 2  2 ( o r g l  2  ( o r g ] 2  ( a suffix const  of b i n d i c a t e s  bl2_over_bl3 bl3_over_bl4  = =  6e-5; le-3;  2  ( o r g  species }  { r a t i o o f ZnL and ZnL -HL f o r m a t i o n c o n s t a n t s ) { r a t i o of ZnL -HL and Z n L - ( H L ) f o r m a t i o n c o n s t a n t s , = 10 -3 } { D i m e r i z a t i o n c o n s t a n t f o r D2EHPA, = 10 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 c o n c e n t r a t i o n } { B u l k pH } { T o t a l D2EHPA (as monomer) i n o r g a n i c b u l k ) { RDC r o t a t i o n a l s p e e d , i n rpm } 2  2  2  2  2  2  A  Kd = 5 . 0 1 2 e 4 ; K_ex = 3 . 2 3 6 e - 2 ; c_Zn_bulk = 0.05; pH = 4 . 5 ; c_HL_total_bulk = w = 100; type Filestring = MaxString = Extension =  A  0.05;  string[14]; string[255]; string[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 _ H L _ e f f _ b u l k , c _ H L _ b u l k , c_HL2_bulk, c _ Z L _ b u l k , c_ZnL2HL0_b, c _ Z n L 2 H L l _ 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_ c ZnL2HLl i , c ZnL2HL2 i . JZn, JZL, n, KCZnbulk, KCbulk, V a r y _ i , V a r y _ f , Vary i n c Vary_value : r e a l ;  Procedure  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 } species bulk concentrations ) species bulk concentrations ) species bulk concentrations ) species i n t e r f a c i a l concentrations ) species i n t e r f a c i a l concentrations ) zinc flux i n kmol/m /sec ) association factor } i n t e r m e d i a t e mass t r a n s f e r v a l u e s } i n i t i a l , f i n a l , i n c r e m e n t , and c u r r e n t parameter v a l u e s } 2  Beep;  { Beeps the s p e a k e r - u s e d when u s e r has made a d a t a e n t r y e r r o r { or t o a l e r t t h e u s e r t o a p o s s i b l e p r o b l e m . ) b e g i n { Beep } sound(880); Delay(125); Nosound; end; { Beep }  { Beep at 880 Hz } { f o r 125 m i l l i s e c o n d s . } { Turn o f f sound. }  130  )  Function  Keyboardlnput  { Waits u n t i l v a r Key :  :  char;  a key i s p r e s s e d ,  and t h e n r e p o r t s  the v a l u e o f t h e k e y .  )  char;  begin { Keyboardlnput ) Repeat Delay(50) u n t i l Read(Kbd,Key); K e y b o a r d l n p u t := K e y ; end; ( Keyboardlnput )  Function  Question  KeyPressed;  (prompt:MaxString)  :  boolean;  ( Asks the Yes o r No q u e s t i o n c o n t a i n e d i n the s t r i n g prompt and t h e n w a i t s u n t i l a Y o r N ) ( answer has been r e c e i v e d . The v a l u e o f the answer i s t h e n r e p o r t e d by t h e v a l u e o f t h i s ) { b o o l e a n f u n c t i o n - True s Yes and F a l s e = No. ) v a r query  :  char;  begin { Question } Repeat W r i t e ( prompt + ' <Y/N> q u e r y := K e y b o a r d l n p u t ; writeln(query); Until If end;  (query  (query  + chr(8)  );  ['Y','y','N','n']);  ['Y','y'])  ( Keep a s k i n g u n t i l  then Q u e s t i o n := t r u e e l s e Q u e s t i o n :=  a Y or N answer i s  received )  false;  { Question )  Procedure  ( { ( { {  in  in  '  OpenDataFile(var  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 p r o c e d u r e prompts the u s e r f o r the d e s i r e d d a t a f i l e f i l e n a m e , adds t h e e x t e n s i o n s p e c i f i e d , and then c h e c k s t o see i f t h i s name i s a l r e a d y p r e s e n t on the d i s k . If i t i s , t h e u s e r i s g i v e n the c h o i c e o f o v e r w r i t i n g i t . I f the u s e r d e c l i n e s , t h e n t h e y a r e prompted t o e n t e r the f i l e n a m e 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 t h e r 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  Filename : F i l e s t r i n g ; Exist : boolean; OKflag : boolean;  begin  { OpenDataFile  OKflag  :=  false;  } } } }  ( name o f d a t a f i l e } ( t r u e i f s p e c i f i e d f i l e n a m e a l r e a d y 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 l e n a m e has been s u c c e s s f u l l y s e l e c t e d )  ) { f i l e n a m e has not been s u c c e s s f u l l y s e l e c t e d y e t  repeat { repeat u n t i l OKFlag ) repeat { Read f i l e n a m e s u n t i l at Write('Save filename: '); Readln(Filename); u n t i l (length(filename) > 0);  least  one c h a r a c t e r  }  i s entered }  { i f t h e f i l e n a m e i s l o n g e r t h a n 4 c h a r a c t e r s , check t o see i f the e x t e n s i o n ) { matches filename_extension. If so, s t r i p i t o f f . ) i f (length(filename) > 4) t h e n if (copy(filename,length(filename)-3,length(filename))=filename_extension) t h e n f i l e n a m e := copy(filename,1,length(filename)-4); { i f the f i l e n a m e i s l o n g e r than 8 c h a r a c t e r s , f i l e n a m e i s e q u a l t o If (length(filename) > 8) then f i l e n a m e := c o p y ( f i l e n a m e , 1 , 8 ) ; Filename  := F i l e n a m e + f i l e n a m e _ e x t e n s i o n ;  A s s i g n ( D a t a f i l e , Filename);  first  8 characters }  ( Add the e x t e n s i o n ) { Define  the  file  variable  { Turn o f f e r r o r c h e c k i n g . Reset d a t a f i l e , and t h e n r e - e n a b l e e r r o r c h e c k i n g . If ) { D a t a f i l e e x i s t s , t h e n I O r e s u l t w i l l be e q u a l t o 0. } ($1-) R e s e t ( D a t a f i l e ) ($1+) ; E x i s t := ( I O r e s u l t = 0 ); {If t h e f i l e a l r e a d y e x i s t s on the d i s k . E x i s t = t r u e O K f l a g := not E x i s t ; { The f i l e n a m e i s OK t o use i f i t d o e s n ' t E x i s t )  131  )  )  { I f the filename i f Exist then begin  exists,  t h e n warn t h e u s e r and ask whether  o r not t o o v e r w r i t e  )  Beep; Beep;  Writeln; Writeln('WARNING:  File  "'.Filename,'"  already  exists!');  { I f t h e y answer y e s t o t h e q u e s t i o n , t h e n t h e f i l e n a m e i s OK, so s e t { OKflag=true. C l o s e t h e D a t a f i l e and E r a s e i t t o s t a r t w i t h a c l e a n s l a t e . I f Q u e s t i o n ( ' D o you want t o e r a s e i t ? ' ) then begin O K f l a g := t r u e ; Close(Datafile); Erase(Datafile); end; end u n t i l OKflag;  { Repeat  Writeln('Output Writeln;  datafile  asking f o r filenames  : ',Filename);  Rewrite(Datafile);  { Identify  ( Open and empty t h e f i l e  until  the output  - set pointer  one i s OK }  filename  { OpenDataFile }  Function  Power(x,  f Computes t h e v a l u e  n  :  r e a l ) : r e a l ;  o f x" and r e t u r n s  the value  as Power. }  b e g i n { Power } power := e x p ( n * l n ( x ) ) ; end; { Power }  Procedure  Quadratic(a,b,c  ( Solves the q u a d r a t i c begin ( Quadratic r o o t l := (-b + r o o t 2 := <-b end; { Quadratic  equation  :  r e a l ;  v a r r o o t l ,  and r e t u r n s  root2  t h e two r o o t s  :  as rootl  r e a l ) ; and root2.  )  } sqrt(b*b-4*a*c))/(2*a); s q r t ( b * b - 4 * a * c ) ) / (2*a) ,}  Procedure C a l c u l a t e  Constants;  ( Computes t h e 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 } { e q u i v a l e n 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 t h e ) ( 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 t h e a p p r o p r i a t e f o r m u l a . ) const  var  D_HL D_HL2 D_ZnL2HL0 D_ZnL2HLl D_ZnL2HL2 D_HC104 D_Zn nu H20 nu_heptane = L_over_alpha zD_aq_Zn,  begin  1.520e 1.003e 0.993e 0.781e 0.659e 2.854e 1.038e 8.93e-- 7 ; 5.64e-7; = 1.90e-4;  zD_aq_Hp,  zD_or_HL,  zD_or_HL2,  zD_or_ZL0,  z D _ o r _ Z L l , zD_or_ZL2  { Calculate_Constants )  zD_aq_Zn  := 0 . 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u _ H 2 0 , ( 1 / 6 ) )  132  * Power(D_Zn, (1/3));  selected }  to the beginning )  ( W r i t e f i l e header } Writeln(Datafile,'w = ',w:4); Write (Datafile, ' loading [Znjbulk [ H L ] e f f , b [ZnL]b '),Write (Datafile,'[HL]bulk [HL2]bulk [ZnL0]b [ZnLl]b [ZnL2]b'); Write (Datafile,' pH,b JZn n'); Write (Datafile,' [Zn]i [HL] i [(HL)2]i [HL2]ieff); W r i t e l n ( D a t a f i l e , ' [ZnL0]i [ZnLl]i [ZnL2]i [ZnL]i,t [H+]i'); end;  } )  :  real;  zD_aq_Hp zD_or_HL zD_or_HL2 zD_or_ZLO zD_or_ZLl zD o r ZL2 k_aq_Zn k_aq_Hp k_or_HL k_or_HL2 k_or_ZL0 k_or_ZLl k o r ZL2 end;  0. 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u _ H 2 0 , (1/6)) * Power(D_HC104, ( 1 / 3 ) ) ; Power(D_HL , ( 1 / 3 ) ) ; 0.643/sqrt(w/60)*Power(nu_heptane,(1/6)] Power(D_HL2 , (1/3) ) ; 0 . 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u _ h e p t a n e , (1/6)) Power(D_ZnL2HL0, ( 1 / 3 ) ) , 0 . 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u _ h e p t a n e , (1/6)) Power(D_ZnL2HLl, (1/3)), 0 . 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u _ h e p t a n e , (1/6)) P o w e r ( D ZnL2HL2, ( 1 / 3 ) ) ; 0 . 6 4 3 / s q r t ( w / 6 0 ) * P o w e r ( n u _ h e p t a n e , (1/6)) 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 o r ZL2  L_over_alpha) ; L_over_alpha) ; L_over_alpha) ; L_over_alpha) ; L_over a l p h a ) ;  { Calculate_Constants }  Procedure  Vary_Setup;  { T h i s procedure just gets the i n i t i a l , f i n a l , ( a range o f c o n c e n t r a t i o n s i s examined. } begin { Vary_Setup } Write (' I n i t i a l Value: Readln (Vary_i); Write (' Final Value: Readln (Vary_f); Write (' Increment : Readln (Vary_inc); i f Vary i > Vary_f then  and i n c r e m e n t v a l u e when )  '); '); '); { r e c u r s e t o get  correct values i f  initial  is  { r e c u r s e t o get  correct values i f  increment  > than  final  Vary_Setup  else if end;  V a r y _ i n c <= 0 then V a r y _ S e t u p ; ( Vary_Setup }  Procedure { ( ( { (  Input_Parameters(var  l o a d i n g  v a r y c o d e  < 0 }  i n t e g e r ) ;  :  T h i s p r o c e d u r e prompts the u s e r f o r a v a l u e f o r the p e r c e n t a g e z i n c l o a d i n g i n the o r g a n i c } phase. The u s e r can examine a range o f v a l u e s by e n t e r i n g a v a l u e o f - 1 . In t h i s c a s e , ) the s u b r o u t i n e V a r y _ S e t u p i s c a l l e d which prompts the u s e r f o r t h e i n i t i a l , f i n a l , and ) incremental values. varycode t e l l s t h e program whether a range has been s e l e c t e d or n o t ; ) a v a l u e o f z e r o i n d i c a t e s no r a n g e , w h i l e 1 i n d i c a t e s a l o a d i n g r a n g e . }  var  input  : real;  { temporary  begin { Input_Parameters ) v a r y c o d e := 0; Writeln; W r i t e l n C E n t e r model p a r a m e t e r s : Write (' loading = ' ) ; Readln (Input); i f (Input = -1) then begin v a r y c o d e := 1;  (to v a r y , (  Vary_setup;  end;  storage v a r i a b l e  enter  -1  at  the  }  prompt)');  s e l e c t a range o f v a l u e s }  { get  the  range }  :  r e a l ;  v a r  end else l o a d i n g := i n p u t ; { Input_Parameters }  Procedure  Solve(  X i n i t i a l , X f i n a l , d X , e p s f u n c t i o n c o d e  { ( { ( { I  v a r  r e a l ;  :  is  :  i n t e g e r ) ;  X3  :  r e a l ; f o r w a r d ;  Master root f i n d i n g r o u t i n e . S i n c e S o l v e r e f e r e n c e s S o l v e f u n c t i o n , which r e f e r e n c e s } 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 , which r e f e r e n c e s S o l v e , a f o r w a r d d e c l a r a t i o n i s ) required. functioncode i n d i c a t e s which f u n c t i o n i s t o be s o l v e d f o r ; i t a l s o ) s e l e c t s the i n c r e m e n t method; i f functioncode i s e q u a l t o 1 t h e n dx i s added t o the ) c u r r e n t v a l u e o f X f o r e a c h i n t e r v a l , but i f functioncode i s not e q u a l t o 1, t h e n X ) i s m u l t i p l i e d by dX f o r e a c h i n t e r v a l . )  133  )  Procedure  Compute_Bulk_Concentrations;  ( T h i s p r o c e d u r e computes t h e c o n c e n t r a t i o n o f HL, ( H L ) , Z n L , ZnL -HL, and Z n L - ( H L ) { b a s e d on t h e c o n d i t i o n s i n the b u l k ( t o t a l D2EHPA c o n c e n t r a t i o n and l o a d i n g ) . ) 2  const  2  2  2  }  Xi= l e - 1 0 ; dX = 2; eps = l e - 4 ;  var Xf, begin  2  temp  :  real;  { Compute_Bulk_Concentrations ) c_Hp_bulk  := e x p ( - p H  * 2.303);  ( compute  [H*]  from pH )  c _ Z L _ b u l k := c _ H L _ t o t a l _ b u l k 12* loading; { 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 b u l k z i n c c o n c e n t r a t i o n i s e q u a l t o z e r o , t h e n t h e r e i s no s p e c i a t i o n o f ) { o r g a n i c z i n c i n the b u l k . T h e r e f o r e , the e f f e c t i v e c o n c e n t r a t i o n o f HL i s e q u a l ) { t o the t o t a l c o n c e n t r a t i o n o f HL i n the b u l k . The p a r t i t i o n between HL and } { (HL) can t h e r e f o r e be c a l c u l a t e d . ) i f ( c _ Z L _ b u l k = 0) then begin c _ H L _ e f f _ b u l k := c _ H L _ t o t a l _ b u l k ; Q u a d r a t i o ( K d , 0 . 5 , - c _ H L _ e f f bulk/2,c_HL_bulk,temp); c_HL2_bulk := (c_HL_eff_buTk-c_HL_bulk)/2; 2  c_ZnL2HL0_b := c _ Z n L 2 H L l _ b := c_ZnL2HL2_b :=  0; 0; 0;  end else begin X f := 2 * c _ H L _ t o t a l _ b u l k ; Solve(Xi,Xf,dX,eps,c_HL_bulk,3);  { S o l v e o v e r the  interval,  SOLVEFUNCTION #3  )  c_HL2_bulk := Kd * c_HL_bulk * c _ H L _ b u l k ; c _ H L _ e f f _ b u l k := c_HL_bulk + 2 * c _ H L 2 _ b u l k ; c _ Z n L 2 H L l _ b := c _ Z L _ b u l k / ( b l 2 _ o v e r _ b l 3 / c HL_bulk+l+c_HL_bulk/bl3_over_bl4); c_ZnL2HL0_b := c _ Z n L 2 H L l _ b * b l 2 _ o v e r _ b l 3 7 c _ H L _ b u l k ; c_ZnL2HL2_b := c _ Z n L 2 H L l _ b * c_HL_bulk / b l 3 _ o v e r _ b l 4 ; end; end;  { Compute_Bulk_Concentrations }  Compute_Interface_Concentrations(  Procedure  x  :  r e a l  );  ( T h i s p r o c e d u r e computes 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 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 u s e d t o compute 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 o f the f i v e o r g a n i c s p e c i e s . The p r o c e d u r e S o l v e i s used t o compute t h e c o n c e n t r a t i o n o f HL at the interface. The upper l i m i t f o r the s e a r c h i n t e r v a l , Yf, i s s e t a t 1.5 t i m e s the t o t a l HL c o n c e n t r a t i o n i n t h e b u l k ; a m u l t i p l i e r o f 1.5 i s used s i n c e dY i s m u l t i p l i c a t i v e and a l s o e q u a l to 1.5. Thus, complete coverage of the e n t i r e i n t e r v a l i s e n s u r e d . 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 t h e n c a l c u l a t e d , and t h e 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 a t the i n t e r f a c e i s found by a d d i n g t h e 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 o r g a n i c z i n c s p e c i e s may be computed. )  const  var  Yi = le-10; dY = 1 . 5 ; eps = l e - 4 ;  J H L 2 e f f , JHp, J Z n : Yf : r e a l ;  begin  { Compute_Interface_Concentrations  JZn ':= x ; c _ Z n _ i := i f c_zn_i JHp := -2 c_Hp_i := i f c_hp_i JZL Yf  real;  < * <  )  J z n / k_aq_Zn + c _ Z n _ b u l k ; 0 t h e n c _ z n _ i := l e - 1 0 ; Jzn; JHp / k_aq_Hp + c _ H p _ b u l k ; 0 t h e n c _ h p _ i := l e - 1 0 ;  := J Z n ; := c _ H L _ e f f _ b u l k  *  1.5;  Solve(Yi,Yf,dY,eps,c_HL_i,2); c_HL2_i  := Kd * c_HL i  *  { Solve over the  c_HL_i;  134  interval,  SOLVEFUNCTION #2 }  c_HL2tot_i  :=  c_ZnL2HLl_i  c_HL_i /  2 + c_HL2_i;  :=  (JZL + KCZnbulk) / (k or_ZLO * b l 2 _ o v e r _ b l 3 + k_or_ZL2 * c _ H L _ i 7 b l 3 _ o v e r _ b l 4 ) ; c_ZnL2HL0_i := c _ Z n L 2 H L l _ i * b l 2 _ o v e r _ b l 3 / c _ H L _ i ; c_ZnL2HL2_i := c _ Z n L 2 H L l _ i * c _ H L _ i / b l 3 _ o v e r _ b l 4 ; c_ZLtot_i := c_ZnL2HL0_i + c _ Z n L 2 H L l _ i + c _ Z n L 2 H L 2 _ i ; { compute t h e n :=  end;  association factor  (c_ZnL2HL0_i + 1.5  at  the  interface  Function SolveFunction(  x  :  r e a l ;  c_HL_i + k_or_ZLl  )  * c _ Z n L 2 H L l _ i + 2 * c_ZnL2HL2_i)  { Compute_Interface_Concentrations  /  /  c_ZLtot_i;  )  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 p r o c e d u r e SOLVE. SOLVE i s u s e d t o f i n d t h e r o o t s o f t h r e e 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 t h e v a l u e o f Functioncode. I f Functioncode i s e q u a l t o 1, then SOLVE has been c a l l e d from M a i n , where i t i s u s e d t o f i n d the Z i n c f l u x , JZn. T h i s 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 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 , 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 o f HL v i a an i t e r a t i v e approach. It t h e r e f o r e c a l l s SOLVE a g a i n , t h i s time w i t h a Functioncode e q u a l to 2. F i n a l l y , SOLVE i s u s e d by the 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 t o f i n d the b u l k c o n c e n t r a t i o n o f HL, and S o l v e f u n c t i o n i s c a l l e d w i t h Functioncode e q u a l t o 3. ) v a r C Z N L l _ i , t e m p i , temp2 : begin i f ( F u n c t i o n C o d e = 1) then begin  real;  Compute_Interface_Concentrations(X);  SolveFunction end;  := c _ Z n L 2 H L l _ i * P o w e r ( c _ H p _ i , 2 )  /  c_Zn_i / P o w e r ( c _ H L 2 _ i , 1 . 5 )  -  K_ex;  if  ( F u n c t i o n C o d e = 2) then begin C Z N L l _ i := (JZL+KCZnbulk) / ( k _ o r _ Z L 0 * b l 2 _ o v e r _ b l 3 / x + k _ o r _ Z L l + 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 _ o r _ H L 2 * K d * x * x + 2 * k _ o r _ Z L 0 * C Z N L l _ i * b l 2 _ o v e r _ b l 3 / x + 3*k"_or_ZLl*CZNLl_i + 4*k_or_ZL2*CZNLl_i*x/bl3_over_bl4 - KCbulk; end;  if  ( F u n c t i o n C o d e = 3) t h e n S o l v e F u n c t i o n := c _ H L _ t o t a l _ b u l k - x - 2 * Kd * x * x + 4*x/bl3_over_bl4 ) * c_ZL_bulk /  - ( 2*bl2_over_bl3/x + 3 (bl2_over_bl3/x + 1 + x/bl3_over_bl4);  end;  Procedure 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 ; f u n c t i o n c o d e : i n t e g e r ) } ;  v a r  X3  :  r e a l ;  { Master root f i n d i n g r o u t i n e . T h i s p r o c e d u r e u s e s an i n c r e m e n t a l s e a r c h f o l l o w e d by a b i s e c t i o n method t o f i n d t h e r o o t o f the f u n c t i o n S O L V E F U N C T I O N ( x , f u n c t i o n c o d e ) . functioncode i n d i c a t e s which f u n c t i o n i s t o be s o l v e d f o r ; i t a l s o s e l e c t s the increment method; i f functioncode i s e q u a l t o 1 t h e n dX i s added t o the c u r r e n t v a l u e o f X f o r each i n t e r v a l , but i f functioncode i s not e q u a l t o 1, t h e n 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 . F o r f u n c t i o n #1, the v a l u e o f the f u n c t i o n changes v e r y r a p i d l y , and t h e r e f o r e s m a l l s t e p s were r e q u i r e d . Conversely, for f u n c t i o n s #2 and 3, t h e e q u i l i b r i u m changes s l o w l y w i t h x and a l a r g e i n t e r v a l must be s e a r c h e d , and t h e r e f o r e i n s t e a d o f a d d i n g dX to X, a m u l t i p l i c a t i v e approach i s u s e d . ) var XI,  Yl,  X2,  Y2,  X30LD, Y3 : REAL;  LABEL LOOP; begin XI Yl IF  { Solve } := X i n i t i a 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 ) ; (Yl = 0) t h e n begin X3 := X I ; Exit; end;  135  Repeat i f ( F u n c t i o n c o d e = 1)  t h e n X2 e l s e X2  := XI := XI  + dx * dx;  { select  (X2 > X f i n a l ) t h e n begin W r i t e l n ( ' S o l u t i o n not found on s e a r c h Exit; 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 t e r a t i o n method }  if  if  until ( Start  interval');  <Y1*Y2) > 0 then begin XI := X2; YI := Y2; end else IF (Y1*Y2) = 0 then begin X3 := X2; Exit; end; (Y1*Y2)  < 0;  Bisection until  LOOP: X3 if Y3 IF IF  accuracy in X f a l l s  within error  eps }  12;  := (XI + X2) (abs((X3 - XI) / XI) < eps) then E x i t ; := 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 ) ; (YI * Y3) = 0 t h e n E x i t ; (YI * Y3) < 0 then begin ( yl*y3 < 0 } X2 := X 3 ; Y2 := Y 3 ; end else begin { yl*y3 > 0 ) XI := X 3 ; Y l := Y 3 ; end; GOTO'LOOP; end; { Solve }  Procedure  Write_Console_Data;  { T h i s p r o c e d u r e 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 b u l k and 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 t o the s c r e e n . }  }  begin { Write_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); Write ('[Zn],b = ',c_Zn_bulk:6:4,' '); Writeln('[ZnL2],b = ' ,c_ZL_bulk:6:4,' pH = ' , p H : 4 : 2 ) ; W r i t e l n (' [Zn] , i = ' , c _ Z n _ i : 8: 6) ; Write ('[HL]eff,b = ',c_HL_eff_bulk:6:4,' '); WritelnC [HL] , b = ' , c _ H L _ b u l k : 8: 6 , ' [ H L ] 2 , b = ' , c _ H L 2 _ b u l k : 8: 6) ; Write ( ' [ H L ] 2 e f f , i = ' , c _ H L 2 t o t _ i : 8: 6) ; WritelnC [HL] , i = ' , c _ H L _ i : 8: 6 , ' [ H L ] 2 , i = ' , c_HL2_i : 8: 6) ; Writeln('[ZnL]tot,i = ',c_ZLtot_i:8:6); WritelnC [ Z n L 2 ] , i = ' , c _ Z n L 2 H L 0 _ i : 8: 6 , ' [ Z n L 2 H L ] , i = ' , c _ Z n L 2 H L l _ i : 8: 6, ' ',c_ZnL2HL2_i:8:6); Writeln; end; { W r i t e C o n s o l e Data } /,  Procedure  Write_File_Data(  v a r  D a t a f i l e  :  text  );  ( T h i s p r o c e d u r e w r i t e s t h e v a l u e s f o r t h e b u l k c o n c e n t r a t i o n s , the z i n c ( 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 c o n c e n t r a t i o n s t o the D a t a f i l e . }  136  [ZnL2HL2],i  flux,  the  }  =  begin { W r i t e _ F i l e _ D a t a } Write (Datafile,loading:6:4,' ',c_Zn_bulk:9,' ' , c _ H L _ e f f _ b u l k : 9 , ' ',c_ZL_bulk:9,' ' ) ; Write ( D a t a f i l e , c _ H L _ b u l k : 9 , ' ',c_HL2_bulk:9,' ' ) ; Write ( D a t a f i l e , c _ Z n L 2 H L 0 _ b : 9 , ' ',c_ZnL2HLl_b:9,' ',c_ZnL2HL2_b:9, ' ' ) ; Write ( D a t a f i l e , p H : 4 : 2 , ' ',JZn:9,' ',n:6:4,' ' , c _ Z n _ i : 9 , ' ' , c _ H L _ i : 9 , ' ' ) ; Write ( D a t a f i l e , c _ H L 2 _ i : 9 , ' ' , c _ H L 2 t o t _ i : 9 , ' ',c_ZnL2HL0_i:9, ' ' , c _ Z n L 2 H L l _ i : 9 , ' ' ) ; Writeln(Datafile,c_ZnL2HL2_i:9,' ' , c _ Z L t o t _ i : 9 , ' ',c_Hp_i:9); end; { Write_File_Data )  {  Main  -  Main program code  }  { T h i s i s t h e main program. P r o c e d u r e s a r e c a l l e d which c a l c u l a t e t h e 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 l e n a m e and s e t up t h e f i l e f o r w r i t i n g , and o b t a i n t h e l o a d i n g v a l u e (or r a n g e ) . A l o o p i s e x e c u t e d which t h e n computes t h e b u l k s p e c i e s c o n c e n t r a t i o n s , f i n d s the z i n c f l u x f o r t h e c o n d i t i o n s s p e c i f i e d , and w r i t e s v a l u e s t o t h e s c r e e n and t h e d a t a f i l e . T h i s l o o p c o n t i n u e s u n t i l t h e z i n c f l u x f o r a l l l o a d i n g v a l u e s s p e c i f i e d has been computed. The d a t a f i l e i s t h e n c l o s e d and e x e c u t i o n i s t e r m i n a t e d . } ( The v a r i a b l e vary i n i t s v a r i o u s p e r m u t a t i o n s i s used t o accomodate t h e c i r c u m s t a n c e where t h e z i n c f l u x i s computed f o r a range o f l o a d i n g v a l u e s . varycode e q u a l t o 1 i n d i c a t e s t h a t t h e f l u x s e a r c h r o u t i n e i s t o be e x e c u t e d f o r l o a d i n g s i n t h e i n t e r v a l Vary_i - vary_f, w i t h s t e p s i z e  vary_inc. )  const  X i = le-11; X f = 5e-7; dX = 5e-9; eps = l e - 4 ;  { ( ( (  var  Datafile varycode  ( Output D a t a f i l e ) ( Parameter code )  begin  : text; : integer;  Initial flux value f o r i t e r a t i v e search r o u t i n e ) Final flux value f o r i t e r a t i v e search routine } 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 routine }  { Main }  Clrscr; Calculate_Constants; OpenDataFile(Datafile,'.out');  { Compute mass t r a n s f e r c o e f f i c i e n t s } ( Get t h e f i l e F i l e n a m e and p r e p a r e f o r f i l e  { E x e c u t e I/O r o u t i n e t o g e t a v a l u e f o r t h e l o a d i n g . { w e l l as varycode which i n d i c a t e s i f t h e u s e r wishes Input_Parameters(loading, varycode); if  varycode = 1 then begin V a r y _ v a l u e := V a r y _ i ; l o a d i n g := V a r y _ v a l u e ; end;  Repeat  write )  The r o u t i n e r e t u r n s t h e l o a d i n g , as } t o v a r y t h e p a r a m e t e r o v e r a range. }  ( I f a range i s s e l e c t e d , { and l o a d i n g )  { Loop once, o r u n t i l  initialize  range  vary_value }  maximum i s exceeded }  Compute_Bulk_Concentrations;  ( Compute c o n s t a n t s which a r e used by 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 and S o l v e f u n c t i o n } KCZnbulk := k_or_ZL0*c_ZnL2HL0_b + k _ o r _ Z L l * c _ Z n L 2 H L l _ 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 _ o r _ Z L l * c _ Z n L 2 H L l _ b + 4*k_or_ZL2*c_ZnL2HL2_b; Solve(Xi,Xf,dX,eps,JZn,1);  { Conduct an i t e r a t i v e range  Write_Console_Data; Write_File_Data(DataFile);  variable  and t h e n  set  ( ( v a r y c o d e = 0) o r ( V a r y _ v a l u e > V a r y _ f ) ) ;  Close(Datafile); end.  dX, and e r r o r c r i t e r i a  ( D i s p l a y t h e s o l u t i o n on t h e s c r e e n } { Output v a l u e s t o s p e c i f i e d D a t a f i l e }  ( I f varycode i s e q u a l t o one, i n c r e m e n t t h e range { l o a d i n g e q u a l t o t h e new range v a l u e . ) i f varycode = 1 then begin V a r y _ v a l u e := V a r y _ v a l u e + v a r y _ i n c ; l o a d i n g := V a r y _ v a l u e ; end; until  s e a r c h f o r JZn over t h e  Xi - Xf, i n t e r v a l  ( Close the f i l e  { Main }  137  )  }  eps. )  

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