RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 1 of 77
Release Stamp DOCUMENT RELEASE AND CHANGE FORM
Prepared For the U.S. Department of Energy, Assistant Secretary for Environmental Management By Washington River Protection Solutions, LLC., PO Box 850, Richland, WA 99352 Contractor For U.S. Department of Energy, Office of River Protection, under Contract DE-AC27-08RV14800 DATE: TRADEMARK DISCLAIMER: Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or any agency thereof or its contractors or subcontractors. Printed in the United States of America. Aug 23, 2016 1. Doc No: RPP-RPT-50703 Rev. 01A
2. Title: Development of a Thermodynamic Model for the Hanford Tank Waste Simulator (HTWOS) 3. Project Number: ☒ N/A 4. Design Verification Required: ☐ Yes ☒ No 5. USQ Number: ☒ N/A 6. PrHA Number Rev. ☒ N/A Clearance Review Restriction Type: public
7. Approvals Title Name Signature Date Checker CREE, LAURA H CREE, LAURA H 08/16/2016 Clearance Review RAYMER, JULIA R RAYMER, JULIA R 08/23/2016 Document Control Approval MANOR, TAMI MANOR, TAMI 08/23/2016 Originator BRITTON, MICHAEL D BRITTON, MICHAEL D 08/16/2016 Quality Assurance DELEON, SOSTEN O DELEON, SOSTEN O 08/16/2016 Responsible Manager CREE, LAURA H CREE, LAURA H 08/16/2016 8. Description of Change and Justification Updated the reduced chemical potential coefficient vlaues for Na2SO4 and Na2SO4·10H2O in Table A.1, as the original values were incorrect. Added mineral names for double salts NaNO2·Na2SO4·H2O, Na3FSO4, and Na7F(PO4)2·19H2O for consistency with other double sales evaluated. Changed the name of the author's company from EnergySolutions to Atkins Global on the cover page.
9. TBDs or Holds ☒ N/A
10. Related Structures, Systems, and Components a. Related Building/Facilities ☒ N/A b. Related Systems ☒ N/A c. Related Equipment ID Nos. (EIN) ☒ N/A
11. Impacted Documents – Engineering ☒ N/A Document Number Rev. Title
12. Impacted Documents (Outside SPF): N/A 13. Related Documents ☐ N/A Document Number Rev. Title RPP-51192 00 Plan for Evaluation of the HTWOS Integrated Solubility Model Predictions RPP-PLAN-46002 00 WASH AND LEACH FACTOR WORK PLAN SVF-2375 00 SVF-2375-Rev0_GEMS.xlsm 14. Distribution Name Organization ARM, STUART T ONE SYS RPP INTEGRTD FLOWSHEET BELSHER, JEREMY D ONE SYS SYSTEM PLNG & MODELING BERGMANN, LINDA M ONE SYS SYSTEM PLNG & MODELING BRITTON, MICHAEL D ONE SYS PROJECT FLOWSHEETS HERTING, DANIEL L PROCESS CHEMISTRY HO, QUYNH-DAO T ONE SYS PROJECT FLOWSHEETS JASPER, RUSSELL T ONE SYS SYSTEM PLNG & MODELING REAKSECKER, SEAN D ONE SYS SYSTEM PLNG & MODELING REYNOLDS, JACOB G TNK WST INVENTORY & CHARACTZTN
1 SPF-001 (Rev.D1) RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 2 of 77
RPP-RPT-50703, Rev. 1A
Development of a Thermodynamic Model for the Hanford Tank Waste Simulator (HTWOS)
R. Carter Atkins Global, LLC 2345 Stevens Drive, Suite 240 Richland, WA 99352 U.S. Department of Energy Contract DE-AC27-08RV14800
EDT/ECN: DRF UC: Cost Center: 2PH00 Charge Code: B&R Code: Total Pages: 77 TM 08/23/16 Key Words: Thermodynamics, Pitzer ion interaction model, water activity, solute activity, molality, ionic strength, solubility, Hanford crystal phases, osmotic coefficient, activity coefficient, HTWOS.
Abstract: This report describes the multicomponent Pitzer ion interaction model and the development of a database of temperature dependent parameter coefficients for ultimate use with the model in the Hanford Tank Waste Simulator (HTWOS). The bulk components included in the termodynamic model include sodium nitrate, sodium nitrite, sodium hydroxide, sodium fluoride, sodium chloride, sodium carbonate, sodium phosphate, sodium oxalate and gibbsite. To ensure the final parameters were self-consistent, they were optimized by fitting the model to experimentally determined solubility data.This optimized model allows predictions of phase speciation to high ionic strengths and temperatures from 0 to 100 °C for Hanford Tank waste.
TRADEMARK DISCLAIMER. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors.
DATE: Aug 23, 2016
By Julia Raymer at 10:32 am, Aug 23, 2016 Release Approval Date Release Stamp
Approved For Public Release
A-6002-767 (REV 3) RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 3 of 77
RPP-RPT-50703 Rev.1A
Development of a Thermodynamic Model for the Hanford Tank Waste Operations Simulator (HTWOS)
R Carter Atkins Global, LLC Richland, WA 99354
U.S. Department of Energy Contract DE-AC27-08RV14800
ABSTRACT
Complex equilibria exist between the aqueous phase and solid phases in Hanford waste. Because these solutions contain components at high concentration, it is necessary to obtain accurate parameterizations of water activity for retrieval, transfer, and for the vitrification plant at relevant temperatures and concentrations where storage, processing, and treatment are to be performed. This information can be used to predict identity and concentrations of solid hydrates known to exist in Hanford waste at the same time as aqueous species concentrations. The components included in the thermodynamic model described in this report are those constituents considered most important in Hanford Tank waste, i.e. NaNO2, NaNO3, NaOH, NaAl(OH)4, NaF, Na2CO3, Na2SO4, Na2C2O4, Na2HPO4, Na3PO4, and water. The solid phase components considered are: Al(OH)3, Na2C2O4, Na2CO3·H2O, Na2CO3·7H2O, Na2CO3·10H2O, Na2SO4, Na2SO4·10H2O, NaF, NaF·Na2SO4, Na2HPO4·12H2O, NaNO3·Na2SO4·H2O, NaNO2, NaNO3, Na3PO4·¼NaOH·12H2O, NaF·2Na3PO4·19H2O, and NaAlCO3(OH)2. The thermodynamic model described here is the well-known Pitzer ion-interaction model for calculation of ion activity coefficients and water activity (via the osmotic coefficient) in aqueous multicomponent electrolyte systems. The parameters required by the Pitzer model for the components considered here have been obtained from the open literature. To ensure the final model is self-consistent, these parameters were optimized by fitting the model to solubility data of simple mixtures available in the open literature. This optimized model allows predictions of phase speciation to high ionic strengths and temperatures from 0 to 100 °C. The ultimate goal is to include this thermodynamic model into the Hanford Tank Waste Operations Simulator (HTWOS) to replace existing simple wash and leach factors for the species listed above.
KEY WORDS
Thermodynamics, Pitzer ion-interaction model, water activity, solute activity, molality, ionic strength, solubility, Hanford crystal phases, osmotic coefficient, activity coefficient, HTWOS. RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 4 of 77
TABLE OF CONTENTS
1 INTRODUCTION ...... 1 2 THERMODYNAMIC MODEL DESCRIPTION...... 3 2.1 EXCESS GIBBS FREE ENERGY...... 3 2.2 MULTICOMPONENT OSMOTIC COEFFICIENT ...... 4 2.3 MULTICOMPONENT ACTIVITY COEFFICIENTS ...... 5 2.4 HIGHER ORDER UNSYMMETRICAL MIXING PARAMETERS...... 6 2.5 TEMPERATURE DEPENDENCE OF THE PITZER PARAMETERS ...... 8 3 GIBBS ENERGY MINIMIZATION ...... 10 3.1 GIBBS ENERGY MINIMIZATION SPREADSHEET (GEMS) ...... 10 3.1.1 GEM Worksheet...... 11 3.1.2 Pitzer Model Worksheet...... 12 3.1.3 Gibbs Energy (Felmy) Worksheet...... 13 3.1.4 Gibbs Energy (Weber) Worksheet ...... 13 3.1.5 Gibbs Energy (HTWOS) Worksheet...... 14 3.1.6 Binary Parameters Worksheet ...... 14 3.1.7 Ternary Parameters Worksheet ...... 14 3.1.8 V & V Worksheet...... 15 4 VERIFICATION OF THE INITIAL PITZER MODEL IMPLEMENTATION...... 16 5 DEVELOPMENT OF THE HTWOS PITZER DATABASE...... 20 5.1 EVALUATION OF SINGLE SOLUTES...... 20
5.1.1 NaNO3 ...... 20
5.1.2 NaNO2 ...... 23 5.1.3 NaOH ...... 25 5.1.4 NaF...... 27 5.1.5 NaCl ...... 27
5.1.6 Na2SO4...... 28
5.1.7 Na2CO3 ...... 30
5.1.8 Na3PO4...... 32
5.1.9 Na2C2O4 ...... 35 5.2 EVALUATION OF SOLUTE MIXTURES...... 36
5.2.1 Na-NO2-NO3-H2O ...... 36
5.2.2 Na-NO3-OH-H2O ...... 37
5.2.3 Na-NO3-F-H2O...... 38
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5.2.4 Na-NO3-C2O4-H2O ...... 38
5.2.5 Na-NO3-CO3-H2O ...... 39
5.2.6 Na-NO2-OH-H2O ...... 39
5.2.7 Na-NO2-CO3-H2O ...... 40
5.2.8 Na-NO2-SO4-H2O...... 40
5.2.9 Na-OH-F-H2O ...... 41
5.2.10 Na-OH-C2O4-H2O ...... 41
5.2.11 Na-OH-CO3-H2O...... 42
5.2.12 Na-OH-SO4-H2O ...... 43
5.2.13 Na-OH-PO4-H2O ...... 43
5.2.14 Na-NO2-PO4-H2O...... 44
5.2.15 Na-SO4-PO4-H2O ...... 45
5.2.16 Na-CO3-PO4-H2O ...... 46
5.2.17 Na-Cl-OH-H2O...... 47
5.2.18 Na-Cl-OH-Al-H2O ...... 47 5.3 DOUBLE SALT SYSTEMS ...... 48
5.3.1 Na-NO3-SO4-H2O...... 49
5.3.2 Na-F-SO4-H2O...... 49
5.3.3 Na-F-PO4-H2O...... 50
5.3.4 Na-CO3-HCO3-H2O...... 50
5.3.5 Na-CO3-SO4-H2O ...... 51 5.4 ADDITIONAL SALT SYSTEMS...... 52
5.4.1 NaAlCO3(OH)2 (Dawsonite) ...... 52 5.4.2 AlOOH (Boehmite) ...... 53 6 CONCLUSIONS ...... 54 7 REFERENCES...... 55 APPENDIX A – COEFFICIENTS FOR GIBBS ENERGY OF FORMATION...... 61 APPENDIX B – BINARY PITZER PARAMETERS...... 63 APPENDIX C – TERNARY PITZER PARAMETERS...... 65 APPENDIX D – VBA CODE LISTING FOR EXCEL FUNCTION ETHETA ...... 68
LIST OF FIGURES
Figure 3-1. GEM Worksheet Layout...... 11 Figure 3-2. Solver Parameters Window...... 12
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Figure 5-1. NaNO3 Osmotic Coefficients at 0 °C...... 21 Figure 5-2. NaNO3 Osmotic Coefficients at 25 °C...... 21 Figure 5-3. NaNO3 Osmotic Coefficients at 50 °C...... 22 Figure 5-4. NaNO3 Osmotic Coefficients at 75 °C...... 22 Figure 5-5. NaNO3 Osmotic Coefficients at 100 °C...... 22 Figure 5-6. Solubility of NaNO₃ in H2O...... 22 Figure 5-7. NaNO2 Osmotic Coefficients at 0 °C...... 24 Figure 5-8. NaNO2 Osmotic Coefficients at 25 °C...... 24 Figure 5-9. NaNO2 Osmotic Coefficients at 50 °C...... 24 Figure 5-10. NaNO2 Osmotic Coefficients at 100 °C...... 24 Figure 5-11. Solubility of NaNO2 in H2O...... 24 Figure 5-12. NaOH Osmotic Coefficients at 0 °C...... 26 Figure 5-13. NaOH Osmotic Coefficients at 25 °C...... 26 Figure 5-14. NaOH Osmotic Coefficients at 40 °C...... 26 Figure 5-15. NaOH Osmotic Coefficients at 70 °C...... 26 Figure 5-16. NaOH Osmotic Coefficients at 100 °C...... 26 Figure 5-17. Solubility of NaF in H2O from 0 to 100 °C...... 27 Figure 5-18. Solubility of NaCl in H2O from 0 to 100 °C...... 28 Figure 5-19. Na2SO4 Osmotic Coefficients at 20 °C...... 29 Figure 5-20. Na2SO4 Osmotic Coefficients at 25 °C...... 29 Figure 5-21. Na2SO4 Osmotic Coefficients at 45 °C...... 29 Figure 5-22. Na2SO4 Osmotic Coefficients at 60 °C...... 29 Figure 5-23. Na2SO4 Osmotic Coefficients at 80 °C...... 29 Figure 5-24. Na2SO4 Osmotic Coefficients at 99.6 °C...... 29 Figure 5-25: Solubility of Na2SO4 in H2O from 0 to 100 °C ...... 30 Figure 5-26: Solubility of Na2CO3 in H2O from 0 to 100 °C...... 32 Figure 5-27. Na2HPO4 Osmotic Coefficients at 0 °C...... 33 Figure 5-28. Na2HPO4 Osmotic Coefficients at 25 °C...... 33 Figure 5-29. Na2HPO4 Osmotic Coefficients at 100 °C...... 33 Figure 5-30. Na2HPO4 Osmotic Coefficients for all the Data...... 33 Figure 5-31: Solubility of Na2HPO4 in H2O from 0 to 100 °C ...... 34 Figure 5-32. Na3PO4 Osmotic Coefficients at 0 °C...... 35 Figure 5-33. Na3PO4 Osmotic Coefficients at 25 °C...... 35 Figure 5-34. Na3PO4 Osmotic Coefficients at 100 °C...... 35 Figure 5-35. Solubility of Na3PO4 in H2O...... 35 Figure 5-36. Solubility of Na2C2O4 in H2O from 0 to 100 °C...... 35 Figure 5-37. Solubilities of NaNO2 and NaNO3 in the NaNO2-NaNO3-H2O System from 0 to 103 °C. .. 37 Figure 5-38. Solubility of NaNO3 in the NaOH-NaNO3-H2O System from 0 to 100 °C...... 37 Figure 5-39. Solubilities of NaF and NaNO3 in the NaF-NaNO3-H2O System at 25 and 50 °C...... 38 Figure 5-40. Solubilities of NaNO3 and Na2C2O4 in the NaNO3-Na2C2O4-H2O System at 20, 50 and 75 °C...... 38 Figure 5-41. Solubilities of NaNO3 and Na2CO3 in the NaNO3-Na2CO3-H2O System at 10, 24.2 and 25 °C...... 39 Figure 5-42. Solubility of NaNO2 in the NaNO2-NaOH-H2O System at 20 and 25 °C...... 40 Figure 5-43. Solubilities of NaNO2 and Na2CO3 in the NaNO2-Na2CO3-H2O System at 20, 23.1 and 25 °C...... 40 Figure 5-44. Solubilities of NaNO2 and Na2SO4 in the NaNO2-Na2SO4-H2O System at 0, 25 and 50 °C...... 41 Figure 5-45. Solubility of NaF in the NaF-NaOH-H2O System from 0 to 94 °C...... 41 Figure 5-46. Solubility of Na2C2O4 in the Na2C2O4-NaOH-H2O System from 0 to 50 °C...... 42
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Figure 5-47. Solubility of Na2CO3 in the Na2CO3-NaOH-H2O System from 0 to 100 °C...... 42 Figure 5-48. Solubility of Na2SO4 in the Na2SO4-NaOH-H2O System from 0 to 100 °C...... 43 Figure 5-49. Solubility of Na3PO4 in the Na3PO4-NaOH-H2O System from 0 to 100 °C...... 44 Figure 5-50 - Solubility of Na3PO4 and NaNO2 in the Na3PO4-NaNO2-H2O System at 25 °C ...... 45 Figure 5-51 - Solubility of Na2HPO4 in Na2SO4 in the Na2HPO4-Na2SO4-H2O System at 25 °C...... 45 Figure 5-52 - Solubility of Na3PO4 in Na2SO4 in the Na3PO4-Na2SO4-H2O System at 25 °C...... 46 Figure 5-53. Solubilities of Na2CO3 and Na3PO4 in the Na2CO3-Na3PO4-H2O System from 0 to 100 °C...... 47 Figure 5-54. Solubility of NaCl in the NaCl-NaOH-H2O System from 0 to 90 °C...... 47 Figure 5-55. Solubility of Al(OH)3 in the NaCl-NaOH-Al(OH)3-H2O System from 6 to 80 °C...... 48 Figure 5-56. Solubility of Al(OH)3 in the NaOH-Al(OH)3-H2O System at 40, 70, and 100 °C...... 48 Figure 5-57. Solubilities of NaNO3 and Na2SO4 in the NaNO3-Na2SO4-H2O System from 0 to 100 °C...... 49 Figure 5-58. Solubilities of NaF and Na2SO4 in the NaF-Na2SO4-H2O System from 0 to 80 °C...... 50 Figure 5-59. Solubilities of NaF and Na3PO4 in the NaF-Na3PO4-H2O System from 25 to 50 °C...... 50 Figure 5-60. Solubilities of Na2CO3 and NaHCO3 in the Na2CO3-NaHCO3-H2O System from 0 to 100 °C...... 51 Figure 5-61 - Solubilities of Na2CO3 and Na2SO4 in the Na2CO3-Na2SO4-H2O System from 15 to 100 °C...... 52
LIST OF TABLES
Table 2-1. Numerical Arrays for Calculating J(x) and J'(x)...... 8 Table 2-2. Debye-Hückel Parameter Coefficients...... 9 Table 4-1. Verification Tests...... 16 Table 4-2. Test 18 Results Comparison for Tank AW-103 at 25 °C...... 17 Table 4-3. Test 19 Results Comparison for Tank AW-103 at 50 °C...... 18 Table 4-4. Comparison of the Solubility Results for the System Na-CO3-PO4-H2O at 25 °C...... 19 Table 5-1. Data Sources for properties of NaNO3...... 20 Table 5-2. Weber Pitzer Parameters and Reduced Chemical Potentials for NaNO3...... 21 Table 5-3. Data Sources for properties of NaNO2...... 23 Table 5-4. Weber Pitzer Parameters and Reduced Chemical Potentials for NaNO2...... 23 Table 5-5. Data Sources for properties of NaOH...... 25 Table 5-6. Weber Pitzer Parameters and Reduced Chemical Potentials for NaOH...... 25 Table 5-7. Data Sources for solubility of NaF in H2O...... 27 Table 5-8. Data Sources for properties of Na2SO4...... 28 Table 5-9. Comparison of Osmotic Coefficients for Na2CO3 from 5 to 45 °C...... 31 Table 5-10. Data Sources for properties of Na2HPO4...... 33 Table 5-11. Data Sources for properties of Na3PO4...... 34 Table 5-12. Solute Mixtures Analyzed and Applicable Range of Temperature...... 36 Table 5-13. Reduced Ideal Chemical Potentials for Dawsonite in the Temperature Range 0 to 100 °C... 53 Table 5-14. Reduced Ideal Chemical Potentials for Boehmite in the Temperature Range 0 to 100 °C. ... 53
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LIST OF TERMS
Abbreviations and Acronyms
DC Dependent Component DSP Disodium Hydrogen Phosphate GEMS Gibbs Energy Minimization Spreadsheet HTWOS Hanford Tank Waste Operations Simulator IAP Ion Activity Product IC Independent Component MSE Mixed Solvent Electrolyte SF Scaling Factor SI Solubility Index TSP Trisodium Phosphate VBA Visual Basic for Applications WTP Hanford Waste Treatment and Immobilization Plant
Units
atm atmosphere oC Celsius J Joules K Kelvin kg kilogram kPa kilopascals m molal mbar millibar mol mole
Definitions
T Absolute Temperature (K) γ Activity coefficient (molal basis) GEX Excess Gibbs free energy I Ionic Strength (molal basis) m Molality Osmotic coefficient (molal basis) Pw Pure water vapor pressure Ω Solubility Index Ps Solution vapor pressure R Universal Gas Constant aw Water activity
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RPP-RPT-50703 Rev. 1A
1 INTRODUCTION This report describes a thermodynamic model for incorporation into the Hanford Tank Waste Operations Simulator (HTWOS) to replace existing simple wash and leach factors. Currently, HTWOS uses these simple wash and leach factors to predict partitioning of chemical species between the liquid and solid phases during retrieval operations in Tank Farms and, in addition, washing and leaching operations in the Hanford Waste Treatment and Immobilization Plant (WTP) (RPP-RPT-17152, Hanford Tank Waste Operations Simulator (HTWOS) Version 6.5 Model Design Document). In order to improve the retrieval, washing and leaching predictions, constituents tracked in HTWOS will be divided into four categories based on relative solubility as described in RPP-PLAN-46002, Wash and Leach Factor Work Plan: Extremely soluble constituents. These constituents are assumed to be in the liquid phase at all times once they are dissolved from saltcake during retrieval. Extremely insoluble constituents. These constituents will continue to use existing simple wash and leach factors, or newer simplified correlations, as so little dissolves into the liquid phase that large errors in relative concentrations will have negligible effects on the accuracy of absolute concentration. Constituents of intermediate solubility. The solubility of these constituents depends highly on waste processing conditions. Their wash and leach factors will be replaced and a more rigorous thermodynamic model will be used to predict their solubility. Kinetic dependent. The solubility of these constituents depends on kinetics, i.e. the amount dissolved or precipitated is a function of time. The thermodynamic model described in this report deals with only those constituents which fall into the intermediate solubility group. These constituents are those in significant concentrations in Hanford waste or deemed important contributors to waste transfer and storage issues, corrosion mitigation, predicted mission length, glass product quality, or all of these. Constituents included are, but not limited to, gibbsite, trisodium phosphate, sodium fluoride, sodium sulfate, sodium oxalate, sodium carbonate. In addition, sodium hydroxide, sodium nitrite and sodium nitrate are included, even though they are highly soluble, as they are the largest contributors to ionic strength. The thermodynamic model chosen is the well-known Pitzer ion-interaction model for mixed electrolytes as found in “Thermodynamics of Electrolytes: IV. Activity and Osmotic Coefficients for Mixed Electrolytes” (Pitzer and Kim 1974) and extended in “The Prediction of Mineral Solubilities in Neutral Waters: The Na-K-Mg-Ca-H-Cl-SO4-OH-HCO3-CO3-CO2-H2O System to High Ionic Strengths at 25 °C” (Harvie et al. 1984); TWRS-PP-94-090, A Chemical Model for the Major Electrolyte Components of the Hanford Waste Tanks. The Binary Electrolytes in the System: Na-NO3-NO2-SO4-CO3-F-PO4-OH- Al(OH)4-H2O; and SAND2009-3115, Implementation of Equilibrium Aqueous Speciation and Solubility (EQ3 type) Calculations into Cantera for Electrolyte Solutions. This model is widely accepted in the scientific community, underpinning many software programs, and has been used to predict Hanford waste speciation to some success in the past. The Pitzer model calculates ion activity coefficients and water activity for a given mixture composition and temperature. To do this requires up to four parameters (known as binary parameters) per solute and requires at least two other parameters (known as mixing parameters) per solute when predicting multicomponent systems. Of the solutes chosen for this application, all have binary and mixing parameters available in the open literature.
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Before the Pitzer model is incorporated into HTWOS, a Microsoft Excel1 based application (hereafter called GEMS) has been developed so that offline testing, verification and validation can be performed. To verify the model was translated into Excel correctly, results from GEMS have been compared to those obtained from GMIN, a Gibbs free energy minimization package developed by Felmy in PNL-7281, GMIN: A Computerized Chemical Equilibrium Model Using a Constrained Minimization of the Gibbs Free Energy, also based on Pitzer’s equations, using the same input compositions and model parameters. Following successful verification, a self-consistent set of Pitzer parameters was generated by optimization against solubility data of simple (binary and ternary) systems available in the open literature and relevant to the constituents of interest for HTWOS. The starting values for these parameters were taken from several sources in the open literature; ORNL/TM-2000/317, Modeling of Sulfate Double-salts in Nuclear Waste; ORNL/TM-2000/348, Waste and Simulant Precipitation Issues; ORNL/TM-2001/102, Thermodynamic Modeling of Savannah River Evaporators; and ORNL/TM-2001/109, Phase Equilibrium Studies of Savannah River Tanks and Feed Streams for the Salt Waste Processing Facility.
1 Excel is a registered trademark of Microsoft Corporation, Redmond, Washington
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2 THERMODYNAMIC MODEL DESCRIPTION
2.1 EXCESS GIBBS FREE ENERGY In order to predict solubility of a salt accurately, knowledge of ion activity coefficients of its constituents is required. For hydrated salts, the activity of water is also required. To calculate ion activity coefficients and water activity, a thermodynamic model is required. The model chosen for implementation in HTWOS is the well-known Pitzer ion-interaction model described first in “Thermodynamics of Electrolytes: I. Theoretical Basis and General Equations” (Pitzer 1973). This model has been extended by several researchers (e.g., Harvie et al. 1984; TWRS-PP-94-090; SAND2009-3115) to include interactions with neutral aqueous species. Pitzer’s model is based on a virial expansion of the excess Gibbs free energy of the solution, which on a ‘per kg of solvent’ basis, is defined as: = (1 − + ln ) (2-1) where GEX is the difference or “excess” in the Gibbs free energy between a real solution and an ideal solution defined on the molality scale, ww is the mass of solvent (usually water) in the solution in th kilograms, mi is the molality of the i ion, is the osmotic coefficient on a molality basis, and γi is the activity coefficient of the ith ion on a molality basis. R is the universal gas constant and T is the absolute temperature. The general expression adopted by Pitzer to represent the excess Gibbs free energy is a combination of the Debye-Hückel theory for long range ionic interactions with a second and third order virial coefficient expansion to account for short range interactions (Pitzer and Kim 1974). Using the notation given by Moffat in SAND2009-3115, the expression is given as: 4 = − ln 1 + √ + 2 + ( ) 3
+ 2Φ + + 2Φ + (2-2) + 2 + 2 + 2 +
+ where is a subscript extending over all anions, extends over all cations, and extends only over all neutral solute molecules. The summations denoted by c 1 2 / / = (2-3) 8 1000 Where is Avogadro’s number, is the density of water, is the electronic charge, = K 0 is the permittivity of water, 0 is the permittivity of free space, and K is the dielectric constant for water. 3 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 12 of 77 RPP-RPT-50703 Rev. 1A The ionic strength of the solution, I, is defined on a molality basis as: 1 = 2 (2-4) th where is the magnitude of charge on the i ion, and i is a subscript extending over all anions and cations. The remaining coefficients in the excess Gibbs energy expression (2-2) are defined as follows: = | | (2-5) ( ) ( ) ( ) = + √ + √ (2-6) where (1 − (1 + ) ) ( ) = 2 (2-7) The terms , , , , , , and in Equation (2-2) are experimentally derived coefficients that can have temperature and pressure dependencies. The exponential coefficient = 2.0 1/2 -1/2 1/2 -1/2 kg mol , except for 2-2 electrolytes, where = 1.4 kg mol is used in combination with = 12 1/2 -1/2 ( ) kg mol . For electrolytes other than 2-2, the term √ is not used (as is the case described herein for the HTWOS thermodynamic model implementation). ( ) ( ) ( ) The , , , and binary coefficients are referred to as ion-interaction or Pitzer parameters. These Pitzer parameters are not ionic strength dependent, but can vary with temperature and pressure. 2.2 MULTICOMPONENT OSMOTIC COEFFICIENT An expression for osmotic coefficient arises from differentiation of the excess Gibbs free energy Equation (2-2) with respect to the mass of water. The following equation is used: 1 − 1 = − (2-8) ∑ Thus, after substitution of GEX/RT given by Equation (2-2), differentiation, rearrangement, and collection of terms, the osmotic coefficient is given by: 2 ⁄ − 1 = − ∑ 1 + √ + + + Φ + (2-9) + Φ + + + 1 + + + 2 where is the osmotic coefficient Debye-Hückel term defined as: = (2-10) 3 The sums over i include all solute species, i.e. all cations, anions and neutrals in solution. The binary Pitzer parameters, , are derived from the , interaction terms as: 4 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 13 of 77 RPP-RPT-50703 Rev. 1A ( ) ( ) ( ) = + = + √ + √ (2-11) / = 2| | (2-12) The unsymmetrical mixing terms, Φ and Φ are defined similarly as: Φ Φ = Φ + (2-13) Φ Φ = Φ + (2-14) These higher order unsymmetrical mixing terms are defined in Section 2.4. The activity of water, aw, is derived from the osmotic coefficient, , by the following definition: 1000 = − ln (2-15) ∑ where Mw is the molecular weight of water (g/mol) and the summation over i includes all cations, anions and neutral species in solution. 2.3 MULTICOMPONENT ACTIVITY COEFFICIENTS The activity coefficients for the ions and neutrals can be derived in a similar way as the osmotic coefficient described in Section 2.2. The excess Gibbs free energy expression (2-2) can be differentiated with respect to each ion concentration to obtain the natural logarithm of its activity coefficient. The relevant equation is: ln = − (2-16) Thus, after substitution of the expression for excess Gibbs free energy (2-2) and differentiation with respect to the ith ion molality (all others are treated as constant), the following equation is obtained for a particular cation, M: ln = + (2 + ) + + 2Φ + (2-17) + + 2 + For anion, X: ln = + (2 + ) + | | + 2Φ + (2-18) + + 2 + where √ 2 Φ = − + ln 1 + √ + + 1 + √ (2-19) Φ + 5 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 14 of 77 RPP-RPT-50703 Rev. 1A The function dBca/dI may be calculated by taking the derivative of (2-6) with respect to ionic strength, as: ( )ℎ √ ( )ℎ √ = + (2-20) The function h(x) is defined as: 2 1 − 1 + + ℎ( ) = ′( ) = − (2-21) 2 The activity coefficient of neutral species is given by: ln = 2 + (2-22) The sum over i refers to all solute species (charged and uncharged), including neutral species. 2.4 HIGHER ORDER UNSYMMETRICAL MIXING PARAMETERS The parameters Φ and Φ , and their derivatives, account for long range electrical forces which appear only for unsymmetrical mixing of same-sign charged ions with different magnitudes (e.g. between - 3- OH and PO4 ). Φ and Φ , where ij is either ′ or ′, are given by: Φ = + ( ) (2-23) From equations (2-13) or (2-14) Φ = + ( ) + ( ) (2-24) The ionic strength derivative of Φ used in Equations (2-13), (2-14) and (2-19) is defined as: Φ = ( ) (2-25) The θij is a small virial coefficient expansion term which is ionic strength independent, and can vary with E temperature and pressure. The θij(I) term accounts for the electrostatic unsymmetrical mixing effects and is dependent only on the charges of the ions i, j, the total ionic strength, I, and on the dielectric constant and density of the solvent (through the Debye-Hückel term, ). It is evaluated using the method described by Pitzer in LBL-23554, A Thermodynamic Model for Aqueous Solutions of Liquid- Like Density: 1 1 ( ) = − ( ) − (2-26) 4 2 2 where = 6 √ (2-27) 1 1 ( ) = 1 + + − (2-28) 2 = − (2-29) The ionic strength derivative function of ( ) is given by: ( ) 1 1 ( ) = − + − ( ) − (2-30) 8 2 2 6 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 15 of 77 RPP-RPT-50703 Rev. 1A An efficient way of solving Equations (2-25) to (2-30), without significant loss of accuracy, is to use the Harvie method described in LBL-23554 which uses two Chebyshev polynomial approximations, one for x ≤ 1 and the other for x ≥ 1. The appropriate equations for these regions are: For region I, where x ≤ 1 = 4 − 2 4 = 5 (2-31) = − + = + − k = <20, 0, -1> For region II, where x ≥ 1 40 22 = − 9 9 40 = − 90 (2-32) = − + = + − k = <20, 0, -1> Using the calculated values for bk and dk, J(x) and J´(x) can be calculated from the following formulas: 1 1 ( ) = − 1 + ( − ) (2-33) 4 2 1 1 ′( ) = + ( − ) (2-34) 4 2 The coefficients and are given in Table 2-1. By definition b21 = b22 = d21 = d22 = 0, therefore, the numbers bk are generated in decreasing sequence from 20 down to 0 denoted by k = <20, 0, -1>. The same is true for calculation of the dk values. Pitzer gave the values of J(1) and J´(1) as 0.116437 and 0.160527 respectively, which can be used for checking any program for such calculation. This method of calculating the unsymmetrical mixing contribution was incorporated into GEMS and can be accessed via a user function as Etheta(z1,z2,I,Aφ,true|false). The code, written in Visual Basic for Applications (VBA), is given in Appendix D. Setting the final argument to false returns the value of E θij(I), the last term in Equation (2-23). Setting it to true returns the value of the derivative multiplied by E the ionic strength, i.e. I θ'ij(I), the last term in Equation (2-24). 7 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 16 of 77 RPP-RPT-50703 Rev. 1A Table 2-1. Numerical Arrays for Calculating J(x) and J'(x). k 0 1.925154014814667 0.628023320520852 1 -0.060076477753119 0.462762985338493 2 -0.029779077456514 0.150044637187895 3 -0.007299499690937 -0.028796057604906 4 0.000388260636404 -0.036552745910311 5 0.000636874599598 -0.001668087945272 6 0.000036583601823 0.006519840398744 7 -0.000045036975204 0.001130378079086 8 -0.000004537895710 -0.000887171310131 9 0.000002937706971 -0.000242107641309 10 0.000000396566462 0.000087294451594 11 -0.000000202099617 0.000034682122751 12 -0.000000025267769 -0.000004583768938 13 0.000000013522610 -0.000003548684306 14 0.000000001229405 -0.000000250453880 15 -0.000000000821969 0.000000216991779 16 -0.000000000050847 0.000000080779570 17 0.000000000046333 0.000000004558555 18 0.000000000001943 -0.000000006944757 19 -0.000000000002563 -0.000000002849257 20 -0.000000000010991 0.000000000237816 2.5 TEMPERATURE DEPENDENCE OF THE PITZER PARAMETERS The form of the expression chosen to represent the temperature dependence of the Pitzer parameters in the HTWOS thermodynamic model was that adopted by several researchers (ORNL/TM-2000/317, ORNL/TM-2000/348, ORNL/TM-2001/102, SAND2009-3115). This form of expression is centric to a reference temperature, Tr, and is shown below: 1 1 ( ) = + ( − ) + − + ln + ( − ) (2-35) In equation (2-35), P(T) represents: i) a species reduced chemical potential (μ°/RT), ii) a binary Pitzer (0) (1) (2) parameter, β , β , β , or Cca, or iii) a mixing parameter, θij, ψijk, λni, or ζnij. The reference temperature, Tr, is chosen as 298.15 K. At 25 °C, the value of parameter P is given by the A coefficient, and the other coefficients determine the variation with temperature. No variation with pressure has been modeled as the upper limit on temperature has been restricted to 100 °C. Although other more complicated expressions have been used in the literature to fit Pitzer parameters, notably “Thermodynamics of NaOH(aq) in Hydrothermal Solutions” (Pabalan and Pitzer 1987); “The Prediction of Mineral Solubilities in Natural Waters: A Chemical Equilibrium Model for the Na-K-Ca-Cl-SO4-H2O System to High Concentration from 0 to 250 °C” (Greenberg and Møller 1989); “An Isopiestic Study of Aqueous Solutions of the Alkali Metal Bromides at Elevated Temperatures” (Holmes and Mesmer 1998); “Thermodynamic Properties of the NaCl + H2O System II. Thermodynamic Properties of NaCl(aq), 8 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 17 of 77 RPP-RPT-50703 Rev. 1A NaCl.2H2O(cr), and Phase Equilibria” (Archer 1992); and “Thermodynamic Properties of the NaNO3 + H2O System” (Archer 2000); expression (2-35) does have a physical basis, from assuming a parabolic fit to the partial molar heat capacity of aqueous electrolytes (ANL-EBS-MD-000045, In-Drift Precipitates/Salts Model). In practice, however, not all of the coefficients in Equation (2-35) are required to adequately model the parameter variations within the chosen temperature range of 0 to 100 °C. For all of the binary parameters, it was found that the E coefficient was not required and was therefore set to zero. Additionally, for the θ mixing parameters, all were adequately modeled independently of temperature; hence B, C, D, and E were all set to zero. For the ψ mixing parameters, all but six were found to be independent of temperature. The remaining six ψ parameters required a value for the C coefficient to capture the temperature variation. Of the four λ parameters required, two were found to be constant, and two required only the use of the A and B coefficients. Of the two ζ parameters required, both were modeled with only the A and B coefficients, the remaining coefficients were set to zero. (See Appendices A, B, and C for a full listing of the final optimized Pitzer parameters developed for the HTWOS thermodynamic model). The only exception to using the above temperature dependence expression (2-35) is the Debye-Hückel term, Aφ, which uses the following formulation from Greenberg and Møller 1989: = + + + ln( ) + + + (2-36) ( − 263) (680 − ) where T is the solution temperature (K). The values of the coefficients are given in Table 2-2. Table 2-2. Debye-Hückel Parameter Coefficients. Coefficient Value a 0.336901532 b -0.00063210043 c 9.14252359 d -0.0135143986 e 0.00226089488 f 0.00000192118597 45.2586464 9 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 18 of 77 RPP-RPT-50703 Rev. 1A 3 GIBBS ENERGY MINIMIZATION The method of Gibbs energy minimization is used to solve for the chemical equilibrium composition at constant temperature and pressure for a given input bulk composition. The problem may be expressed in terms of the following constrained minimization using notation from “A Chemical Equilibrium Algorithm for Highly Non-Ideal Multiphase Systems: Free Energy Minimization” (Harvie et al. 1987) minimize: = (3-1) subject to: = , = 1, (3-2) = 0, for each phase in (3-3) ≥ 0 for all (3-4) Where: = the Gibbs energy of the system = the reduced chemical potential of species j = the number of moles of species j nt = the total number of species in the system = the number of independent components (IC) = the number of moles of IC i in one mole of species j = the charge of species j in electrolyte solution phase s e = the number of electrolyte solution phases = the number of moles of each independent component i Equations (3-2) and (3-3) are the mass and charge balance constraints respectively, and Equations (3-4) are species constraints. The constrained minimization problem can be solved by use of Lagrange multipliers to form the Lagrangian, thus Equation (3-1) becomes: = − − − (3-5) in In Equation (3-5), is a Lagrange multiplier for a mass balance constraint and is a Lagrange multiplier for a charge balance constraint (one in each electrolyte phase e). Excel’s built-in Solver routine has been utilized to solve this minimization problem as described next. 3.1 GIBBS ENERGY MINIMIZATION SPREADSHEET (GEMS) The Pitzer ion-interaction model, described in Section 2, was translated into Microsoft Excel so that a database of parameters could be developed for the components of interest and tested before incorporation into HTWOS. Briefly, Pitzer’s equations are solved in worksheet “Pitzer Model” which also displays the relevant Pitzer parameters in use. Additionally, scaling factors for solids are calculated and can be used to check that the current problem has converged on a composition that is in equilibrium. A scaling factor equal to unity indicates that this solid has been predicted to form. The problem is set-up on worksheet “GEM” by supplying the bulk input solution composition, in terms of neutral species, and the solution temperature. An initial guess is supplied in the “nj mol” column and then Excel’s built-in Solver routine is used to minimize the total system Gibbs energy to produce the ‘true’ composition at equilibrium. The 10 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 19 of 77 RPP-RPT-50703 Rev. 1A following sections describe in detail the layout and storage of information used by GEMS (SVF-2375, HISI id = 3024). 3.1.1 GEM Worksheet This is the input sheet, shown in Figure 3-1, for entering bulk solution composition in cells C2:P2 and the temperature in cell Q2. Figure 3-1. GEM Worksheet Layout. Rows 3:15 are hidden from the user, but transform the input composition to the cells in D16:O16 which are the mole amounts of the ICs listed in cells D17:O17. The dependent components (DC) are listed in cells B18:B59 which comprise 13 ions, 26 pure solids, 2 neutral aqueous species and water. The values listed in cells C18:C59 are the DCs’ reduced ideal chemical potentials (i.e. Gibbs free energy of formation divided by RT, R is the universal gas constant and T is the absolute temperature). These cells are linked to worksheet “Gibbs Energy (Felmy)”, “Gibbs Energy (Weber)” or “Gibbs Energy (HTWOS)” depending on which table name is referenced (see Sections 3.1.3 to 3.1.5). Cells D18:O59 are a matrix of coefficients (Aji, including any ion charge) that describes the number of moles of ICs in one mole of DC. Cells D60:O60 are the Lagrange multipliers used in the Gibbs energy minimization (GEM) method in “Modeling Chemical Mass Transfer in Geochemical Processes: Thermodynamic Relations, Conditions of Equilibria and Numerical Algorithms” (Karpov et al. 1997). Cells P18:P59 are the mole amounts of DCs at the minimum Gibbs energy of the system (displayed in cell Q60) and are calculated by the Solver routine in Excel when minimizing the Lagrangian displayed in cell Q63. Several constraints have to be met in obtaining the solution: i) cells D58:O58 are the mass and charge balance residuals, which should be zero (within tolerance) at convergence, ii) cells V18:V56 are the stability criteria (Karpov et al. 1997) and must be greater than or equal to zero at equilibrium, iii) cells W18:W56 are the orthogonality criteria (Karpov et al. 1997) and zeroes off the molar amounts of unstable species and phases and, hence, should all be zero (within tolerance) at equilibrium, and iv) the molar amounts of all species present at equilibrium (cells P18:P56) must be greater than or equal to zero. All these constraints have been entered into the Solver constraints section as shown in Figure 3-2. 11 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 20 of 77 RPP-RPT-50703 Rev. 1A Figure 3-2. Solver Parameters Window. The mole fraction of a DC is listed in cells R18:R59 and is used to calculate the molality of an aqueous species listed in cells X18:X32. These molalities are linked to the corresponding cells in worksheet “Pitzer Model” so that the activity coefficients (listed in cells T18:T32 in logarithmic form) and water activity (listed in cell T33 as the logarithm) can be calculated. Cells S18:S33 list the logarithm of the DCs’ molality which is used with the activity coefficients and water activity to calculate the actual reduced chemical potentials listed in cells Q18:Q59. (Note, the activity of the pure solids are assumed to be unity, hence cells S34:T59 are all zero). The solubility index (SI), Ω, of a pure solid is listed in cells X34:X59. The SI of a solid is the logarithm of the ratio of the ion activity product to the solubility product. When a solid is present in the system at equilibrium, its SI has a value of zero (within tolerance). Values listed in cells Z18:AB33 are checks of the activity of a given species. Values in cells AA35:AA40 are the ionic strength (I), along with electron activity (pe), redox potential (Eh, volts), and pH of the aqueous phase as defined in “Modeling Chemical Equilibrium Partitioning with the GEMS-PSI Code” (Kulic et al. 2004). (Note: the latter three, pe, Eh and pH, only appear to give sensible values when the OH- ion is present in the system because the hydrogen ion is not included in the model). These latter cells are not shown in Figure 3-1. 3.1.2 Pitzer Model Worksheet This worksheet calculates activity coefficients and water activity using Pitzer’s multi-component ion- interaction model (Pitzer and Kim 1974; Harvie et al. 1984; TWRS-PP-94-090; SAND2009-3115). In this worksheet, cells A3:D16 list the usual binary Pitzer parameters, β(0), β(1), and Cᶲ and are linked to the data in worksheet “Binary Params”. Cells B18:B21 contain terms for neutral-ion interactions and cells C22:C23 for neutral-ion-ion interactions for species NaNO2(aq) and NaNO3(aq) (the only neutral species included at this time) and are linked to the data in worksheet “Ternary Params”. Cells G3:I16 contain interaction terms Bca, B′ca and Cca used by the activity coefficient equations and the osmotic coefficient equation. The second order ion-ion mixing terms, θ, are contained in cells K3:K16, T3:T15, and AC3:AC10 and the corresponding third order terms for ion-ion-ion mixing, ψ, are contained in cells M3:M16, V3:V15, and AE3:AE10. Again, they are linked to the data stored in worksheet “Ternary Params”. The higher order mixing terms, for like-charged ions of different magnitude, Eθ(I) and IEθ′(I), are contained in cells N3:O16, W3:X15, and AF3:AG10. These terms are calculated by a VBA procedure called Etheta (Section 2.4 and Appendix D) which implements the use of Chebyshev 12 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 21 of 77 RPP-RPT-50703 Rev. 1A polynomials as described by Pitzer (LBL-23554). These higher order terms are used to calculate the Φ, Φ′ and Φᶲ terms stored in cells P3:R16, Y3:AA15, and AH3:AJ10. These terms are also required by the activity and osmotic coefficient equations. The Debye-Hückel constant, , and the Pitzer b, α₁ and α₂ terms are contained in cells K19:K22 respectively. The species activity coefficients are evaluated in cells D27:BH39. The complicated Equations (2-17) and (2-18) for the activity coefficient for multi-component electrolytes is broken into sections so that individual terms (and summations) can be performed (and checked) in a simpler way. Molality of a particular DC is contained in cells F27:F41 and is linked to the corresponding cell in worksheet “GEM”. Cells C27:C41 contain the corresponding activity of the DCs. The osmotic coefficient Equation (2-9) is calculated in cells F45:AI45 and, again, has been broken into smaller terms (and summations) for clarity and ease of checking. The water activity is given in cell D45, its mole fraction in cell C45 and its rational activity coefficient in cell B45. The block of cells in range A47:M81 contain the pure solid solubility product (Ksp), aqueous species equilibrium constant (Keq) and ion activity product (IAP) information. This information was used in the original version of the workbook to calculate phase compositions prior to implementation of the GEM method. This information is retained for checking the GEM method as the scaling factor (SF) on this worksheet should agree with the corresponding values on the “GEM” worksheet in cells Y34:Y59. 3.1.3 Gibbs Energy (Felmy) Worksheet This worksheet contains a database of thermodynamic data that allow calculation of the Gibbs free energy of formation at any temperature. Felmy reports that the valid temperature range is between 25 to 100 °C. This information was compiled from three sources; TWRS-PP-94-090; PNWD-3120, Development of an Enhanced Thermodynamic Database for the Pitzer Model in ESP: The Fluoride and Phosphate Components; and an e-mail from A. R. Felmy to R. Carter, “RE: GEM modeling of Hanford Tank Waste with Excel”, (Felmy A.R., 2011-04-15). This database, herein called the “Felmy-Pitzer” database, was used to independently verify that the Pitzer equations had been translated into Excel correctly and the built-in Solver gave the correct equilibrium concentrations for the same input bulk compositions (see Section 4). The data is organized into a table, labeled as “TblGFE_F”, with the species name in cells A5:A49 and the reduced chemical potentials in cells B5:B49. The temperature dependent coefficients are located in cells J7:T45. The form of the parameter temperature dependence equation is given in cell J4. (Note that the Felmy database uses a different form of temperature dependence from that used in the Weber and HTWOS Pitzer databases. Felmy adopted the same temperature expression as Greenberg and Møller 1989). The information given in cells C5:F41 is retained for compatibility between databases. In this way, all database tables can be referenced on sheets “GEM” and “Pitzer Model” using the same method. Not all of the species of interest for HTWOS could be obtained from Felmy; hence, the missing species have zero values for all their coefficients. 3.1.4 Gibbs Energy (Weber) Worksheet This worksheet contains data for reduced chemical potentials and coefficients that allow for temperature dependence that were taken from published papers by Weber and co-workers (ORNL/TM-2000/348; ORNL/TM-2001/102 and 109). Like the Felmy-Pitzer database, the data is organized by tables, one labeled as “TblGFE_W” for the Steele data and one labeled as “TblGFE_W2” for the Weber data, with the species name located in cells A5:A49 and A53:A97. A species reduced chemical potential at 25 °C is found in cells B5:B49 and B53:B97, with the coefficients for the temperature dependency equation given in cells C5:F49 and C53:F97. Not all of the species of interest for HTWOS could be found in the 13 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 22 of 77 RPP-RPT-50703 Rev. 1A references of Steele and Weber; the missing species have, therefore, been given zero values for all of their coefficients. The values listed in cells G5:G49 are the original reference indices given by Steele (ORNL/TM-2000/348) and those listed in cells G53:G97 are the original reference indices from Weber (ORNL/TM-2001/102). This database, herein called the “Weber-Pitzer” database, was tested for the reason that it is available in the open literature, whereas not all of the data obtained from Felmy is available publicly. To test the database, it was used to predict pure component solubility and generate solubility isotherms for simple binary systems. The accuracy of such predictions can be used to assess the self-consistency of the information contained therein, as only an internally self-consistent database should be incorporated into HTWOS (see Section 5). 3.1.5 Gibbs Energy (HTWOS) Worksheet This worksheet contains data for reduced chemical potentials and coefficients that allow for temperature dependence. This is the database that will be used by HTWOS. The initial data was taken from the published paper by Steele and co-workers (ORNL/TM-2000/348) with additions from Toghiani (ORNL/TM-2000/317) and Weber (ORNL/TM-2001/102 and 109) for components not available in Steele. The data is organized in a Table, labeled as “TblGFE_H2”, with the species name located in cells A5:A49. The species reduced chemical potential at 25 °C is found in cells B5:B49, with the coefficients for the temperature dependency equation given in cells C5:F49. The reduced chemical potentials for the ions (cells B5:F20) have been retained from the original sources, but the neutral species and solids have had their values optimized to ensure self-consistency (Section 5). This database is herein referred to as the “HWTOS-Pitzer” database. 3.1.6 Binary Parameters Worksheet (0) (1) The “Binary Params” worksheet contains data for the Pitzer binary parameters (β , β , and Cca) for all 3 databases. The data is stored in separate tables named “TblPitz2_F”, “TblPitz2_W” and “TblPitz2_H2” for the Felmy, Weber and HTWOS Pitzer databases respectively. These labels are used on the “GEMS” and “Pitzer Model” sheets for retrieving data from the required database. For each of the tables, the text stored in column “A” is used as a label so that the table can be searched for the correct parameter values stored in columns “E:H”. Again, the data in the Felmy table is organized differently due to use of a different temperature dependence equation. Felmy coefficients are stored in cells S7:Z46 with the value of a parameter computed in cells AA7:AB46. Computed parameter values are then linked to cells E5:E40. The Na-Cl parameters were not obtained from Felmy because this species was added to the model after the initial verification was complete, so were taken from Greenberg and Møller 1989. They are included for compatibility with the other databases. 3.1.7 Ternary Parameters Worksheet The “Ternary Params” worksheet contains data for the Pitzer mixing parameters (θij, ψijk, λni, and ζnij) for all 3 databases. The data is stored in separate tables named “TblPitz3_F”, “TblPitz3_W” and “TblPitz3_H2” for the Felmy, Weber and HTWOS databases respectively. These labels are used on the “GEMS” and “Pitzer Model” sheets for retrieving data from the required database. For each table, the text stored in column “A” is used as a label so that it can be searched for the correct parameter values stored in columns “F:H”. The chloride interaction parameters in the Felmy database were set equal to those in the Weber database. These parameters were not obtained from Felmy because this species was added into the model after completion of the initial verification, but is included for compatibility with the other databases. 14 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 23 of 77 RPP-RPT-50703 Rev. 1A 3.1.8 V & V Worksheet The “V & V” worksheet contains experimentally derived solubility data from many sources in the open literature for many simple binary and ternary systems in the temperature range 0 to 100 °C. Additionally, predicted results from model simulations using all 3 databases are stored alongside the literature data. This data was used to generate charts of solubility isotherms for comparison of the predicted results against the experimental data. These charts were used to visually assess the ‘accuracy’ of the model for a given set of database parameters. The experimental data was also used to optimize the parameters in the HTWOS-Pitzer database as described in Section 5. 15 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 24 of 77 RPP-RPT-50703 Rev. 1A 4 VERIFICATION OF THE INITIAL PITZER MODEL IMPLEMENTATION A series of simple tests were performed with the model using the Felmy-Pitzer database in order to verify that the Pitzer equations, detailed in Section 2, had been translated into the spreadsheet correctly and confirm that Excel’s built-in Solver routine could find the minimum Gibbs energy. These tests were designed to check specific features of the Pitzer equations in a logical and progressive way. The results of each test were compared to the output from an external and independent, validated model (herein called the Felmy model; PNL-7281) using the same bulk compositions and temperature. A list of the tests performed and their purpose is given in Table 4-1. Table 4-1. Verification Tests. Test # Descriptor¹ Purpose of Test 1 nano3 Test a 1-1 electrolyte (NaNO3) at 25 °C 2 no3ip As #1, but with neutral species NaNO3(aq) 3 nano2 As #2, but with ion-neutral interaction terms (λ & ζ) included 4 naco3 Test a 1-2 electrolyte (Na2CO3) at 25 °C (no hydrolysis reaction) 5 hco3 As #4, but with the hydrolysis reaction included 6 po4 Test a 1-3 electrolyte (Na3PO4) at 25 °C (no hydrolysis reaction) 7 hpo4 As #6, but with the hydrolysis reaction included 8 co3so4 Test high order (2-2) unsymmetrical mixing terms (Eθ and Eθ') at 25 °C 9 no3co3 As #8, but for 1-2 unsymmetrical mixing 10 po4no3 As #8, but for 1-3 unsymmetrical mixing 11 po4so4 As #8, but for 2-3 unsymmetrical mixing 12 no380 As #1, but at 80 °C to test temperature dependency of equations 13 no3ip80 As #2, but at 80 °C 14 nano280 As #3, but at 80 °C 15 po440 As #6, but at 40 °C 16 po4so480 As #11, but at 80 °C 17 co3no380 As #9, but at 80 °C 18 feed Test with all components specified (complex feed) at 25 °C 19 feed50 As #18, but at 50 °C 20 co3xx² Calculate solubility isotherm for the Na-CO3-PO4-H2O system at 25 °C ¹ Descriptor is the filename (excluding “.out”) received from Dr. A. R. Felmy containing his results. ² xx denotes 00, 05, 10, 15, 20, or 25 in the names of the 6 files received to generate the solubility curve. During the verification process, several corrections were made to formulas entered into the “Pitzer Model” spreadsheet, especially to correct the calculation of the NaNO2(aq) activity coefficient. Other minor corrections were made to temperature dependency coefficients to increase accuracy. After these corrections were made, the average relative deviation between the two sets of results was less than 1%. 16 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 25 of 77 RPP-RPT-50703 Rev. 1A For tests 18 and 19, the bulk feed composition was chosen to represent Hanford Tank AW-103. The results of test 18, at a temperature of 25 °C, are listed in Table 4-2 which shows the very close agreement between the two models. Table 4-3 lists the results generated at 50 °C. Table 4-2. Test 18 Results Comparison for Tank AW-103 at 25 °C. Felmy Model GEMS Model Name Moles Molality ln a μ*¹ Moles Molality ln a μ* H2O 5.55E+01 5.55E+01 -1.71E-01 -9.58E+01 5.55E+01 5.55E+01 -1.71E-01 -9.58E+01 Na+ 4.83E+00 4.83E+00 1.11E+00 -1.05E+02 4.83E+00 4.83E+00 1.11E+00 -1.05E+02 - NO2 6.72E-01 6.72E-01 -7.87E-01 -1.38E+01 6.77E-01 6.77E-01 -7.80E-01 -1.38E+01 - NO3 1.43E+00 1.43E+00 -4.76E-01 -4.27E+01 1.43E+00 1.43E+00 -4.77E-01 -4.27E+01 F- 1.58E-01 1.58E-01 -2.18E+00 -1.15E+02 1.58E-01 1.58E-01 -2.18E+00 -1.15E+02 - HCO3 9.94E-06 9.94E-06 -1.25E+01 -2.49E+02 1.01E-05 1.01E-05 -1.25E+01 -2.49E+02 -2 CO3 4.26E-01 4.26E-01 -3.66E+00 -2.17E+02 4.27E-01 4.27E-01 -3.65E+00 -2.17E+02 -2 HPO4 6.40E-06 6.40E-06 -1.63E+01 -4.56E+02 6.40E-06 6.40E-06 -1.63E+01 -4.56E+02 -3 PO4 2.75E-02 2.75E-02 -1.20E+01 -4.23E+02 2.75E-02 2.75E-02 -1.20E+01 -4.23E+02 OH- 1.40E+00 1.40E+00 2.73E-01 -6.32E+01 1.40E+00 1.40E+00 2.73E-01 -6.32E+01 -2 SO4 3.38E-02 3.38E-02 -7.18E+00 -3.08E+02 3.39E-02 3.39E-02 -7.18E+00 -3.08E+02 - Al(OH)4 1.46E-01 1.46E-01 -2.35E+00 -5.29E+02 1.46E-01 1.46E-01 -2.35E+00 -5.29E+02 -2 C2O4 9.08E-03 9.08E-03 -7.84E+00 -2.80E+02 9.01E-03 9.01E-03 -7.84E+00 -2.80E+02 NANO3(aq) 1.70E-01 1.70E-01 -1.77E+00 -1.47E+02 1.71E-01 1.71E-01 -1.77E+00 -1.47E+02 NANO2(aq) 1.36E-01 1.36E-01 -1.92E+00 -1.18E+02 1.31E-01 1.31E-01 -1.91E+00 -1.18E+02 Al(OH)3(s) 2.06E-01 2.06E-01 NaF (s) 1.66E+00 1.66E+00 Na2C2O4(s) 1.73E-02 1.74E-02 ¹ μ* = reduced chemical potential (= μ/RT = μ°/RT + ln a) NAME log SI² log SI Al(OH)3(s) 0 -8.8E-10 NaNO2 (s) -1.539 -1.535 NaNO3 (s) -0.815 -0.814 Na2CO3.10H2O -0.587 -0.577 Na2CO3.7H2O -0.841 -0.831 Na2CO3.1H2O -1.293 -1.283 Na2HPO4.12H2O -5.064 -5.061 Na2SO4(s) -1.9 -1.898 Na2SO4.10H2O -1.68 -1.677 NaF.Na2SO4 -0.596 -0.593 Na3PO4.¼NaOH.12H2O -0.923 -0.918 Na7FPO4.19H2O -0.834 -0.826 NaF (s) 0 3.99E-10 NaNO3.Na2SO4.H2O -2.196 -2.192 Na2C2O4(s) 0 -2.5E-06 ² SI = solubility index = (ion activity product)/(equilibrium constant). For a precipitated solid, SI = 1; for an unsaturated solid, SI < 1. 17 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 26 of 77 RPP-RPT-50703 Rev. 1A Table 4-3. Test 19 Results Comparison for Tank AW-103 at 50 °C. Felmy Model GEMS Model Name Moles Molality ln a μ*¹ Moles Molality ln a μ* H2O 5.55E+01 5.55E+01 -1.71E-01 -9.42E+01 5.55E+01 5.55E+01 -1.71E-01 -9.42E+01 Na+ 4.40E+00 4.40E+00 1.15E+00 -1.05E+02 4.39E+00 4.39E+00 1.15E+00 -1.04E+02 - NO2 6.97E-01 6.97E-01 -7.67E-01 -1.38E+01 6.97E-01 6.97E-01 -7.68E-01 -1.38E+01 - NO3 9.54E-01 9.54E-01 -3.59E-01 -4.26E+01 9.52E-01 9.52E-01 -3.61E-01 -4.26E+01 F- 1.71E-01 1.71E-01 -2.19E+00 -1.15E+02 1.70E-01 1.70E-01 -2.20E+00 -1.15E+02 - HCO3 5.58E-05 5.58E-05 -1.07E+01 -2.47E+02 5.59E-05 5.59E-05 -1.07E+01 -2.47E+02 -2 CO3 4.26E-01 4.26E-01 -3.73E+00 -2.17E+02 4.27E-01 4.27E-01 -3.73E+00 -2.17E+02 -2 HPO4 2.80E-05 2.80E-05 -1.47E+01 -4.54E+02 2.80E-05 2.80E-05 -1.47E+01 -4.54E+02 -3 PO4 2.75E-02 2.75E-02 -1.19E+01 -4.23E+02 2.75E-02 2.75E-02 -1.19E+01 -4.23E+02 OH- 1.25E+00 1.25E+00 7.01E-02 -6.34E+01 1.25E+00 1.25E+00 6.95E-02 -6.34E+01 -2 SO4 3.38E-02 3.38E-02 -7.18E+00 -3.08E+02 3.39E-02 3.39E-02 -7.18E+00 -3.08E+02 - Al(OH)4 2.89E-01 2.89E-01 -1.80E+00 -5.27E+02 2.89E-01 2.89E-01 -1.80E+00 -5.27E+02 -2 C2O4 1.30E-02 1.30E-02 -7.92E+00 -2.80E+02 1.29E-02 1.29E-02 -7.92E+00 -2.80E+02 NANO3(aq) 6.48E-01 6.48E-01 -4.33E-01 -1.47E+02 6.50E-01 6.50E-01 -4.31E-01 -1.47E+02 NANO2(aq) 1.10E-01 1.10E-01 -2.13E+00 -1.18E+02 1.11E-01 1.11E-01 -2.13E+00 -1.18E+02 Al(OH)3(s) 6.21E-02 6.21E-02 NaF (s) 1.65E+00 1.65E+00 Na2C2O4(s) 1.34E-02 1.35E-02 ¹ μ* = reduced chemical potential (= µ/RT = µ°/RT + ln a) NAME log SI² log SI Al(OH)3(s) 0 -1.7E-13 NaNO2 (s) -1.716 -1.714 NaNO3 (s) -1.021 -1.020 Na2CO3.10H2O -1.573 -1.567 Na2CO3.7H2O -1.8 -1.795 Na2CO3.1H2O -1.377 -1.373 Na2HPO4.12H2O -16.646 -16.641 Na2SO4(s) -1.744 -1.741 Na2SO4.10H2O -3.59 -3.585 NaF.Na2SO4 -0.598 -0.595 Na3PO4.¼NaOH.12H2O -1.589 -1.583 Na7FPO4.19H2O -2.218 -2.206 NaF (s) 0 3.17E-10 NaNO3.Na2SO4.H2O -1.373 -1.369 Na2C2O4(s) 0 -7.4E-14 ² SI = solubility index = (ion activity product)/(equilibrium constant). For a precipitated solid, SI = 1; for an unsaturated solid, SI < 1. 18 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 27 of 77 RPP-RPT-50703 Rev. 1A Results generated for test 20 are shown in Table 4-4 which compares the solubility predictions of a simple Na-CO3-PO4-H2O system at 25 °C from the Felmy and GEMS models. Table 4-4 shows that the predictions are very consistent, the relative error values being < 1% for all the tests performed. The very small differences could be due to different methods for calculating the higher order unsymmetrical mixing terms Eθ and Eθ' between the two models and/or due to differences in the number of significant figures between the parameter values as stored in the two different databases. Table 4-4. Comparison of the Solubility Results for the System Na-CO3-PO4-H2O at 25 °C. Felmy Model GEMS Model (Felmy-Pitzer database) % Relative Error¹ m Na2CO3² m Na3PO4 Solid³ m Na2CO3 m Na3PO4 Solid Na2CO3 Na3PO4 0.0 0.73086 TSP 0.0 0.73078 TSP 0.0 0.011 0.5050 0.66392 TSP 0.5050 0.66389 TSP 0.0 0.005 1.0244 0.60762 TSP 1.0244 0.60760 TSP 0.0 0.003 1.5528 0.56579 TSP 1.5528 0.56577 TSP 0.0 0.004 2.0889 0.53029 TSP 2.0889 0.53027 TSP 0.0 0.004 2.5542 0.50940 TSP + C.10 2.5412 0.50959 TSP + C.10 0.509 -0.037 ¹ % relative error = (m-Felmy – m-GEMS)/m-Felmy×100% where m-Felmy is the molality from the Felmy model and m- GEMS is the molality from the GEMS model. ² m x = aqueous concentration of solute x where x = Na2CO3 is total carbonate molality, x = Na3PO4 is total phosphate molality. ³ TSP = trisodium phosphate (Na3PO4.¼NaOH.12H2O), C.10 = sodium carbonate decahydrate (Na2CO3.10H2O). Verification of the correctness of the Pitzer equations, as translated in Excel, was successfully completed at the conclusion of these tests. The next step in the verification process was to check the accuracy of the solubility predictions for simple binary and ternary mixtures. This eventually led to the development of a new database of parameters for the HTWOS implementation, known herein as the HTWOS-Pitzer database. This process is described next, in Section 5. 19 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 28 of 77 RPP-RPT-50703 Rev. 1A 5 DEVELOPMENT OF THE HTWOS PITZER DATABASE In order to ensure that the Pitzer parameters and chemical potentials of species to be included in the HTWOS-Pitzer database are thermodynamically self-consistent, the parameters of Toghiani, Steele and Weber (ORNL/TM-2000/317, ORNL/TM-2000/348, ORNL/TM-2001/102 and 109) were obtained. These parameters are freely available in the open literature for all the species required by the HTWOS thermodynamic model, except boehmite and dawsonite. These parameters are referred to, herein, as the Weber-Pitzer database and their values are stored in the GEMS workbook. Assessments were performed to test the consistency of these starting Pitzer parameters and to derive new values as required based on the accuracy of model predictions. Section 5.1 describes details of the assessment of binary electrolytes NaNO3, NaNO2, NaOH, NaF, NaCl, Na2CO3, Na2SO4, Na2C2O4 and Na3PO4. Although sodium chloride is not one of the solutes listed in RPP-PLAN-46002 it was necessary to include it so that other systems could be evaluated, such as the Na- Al-OH-Cl-H2O system reported in “Aluminum Speciation and Equilibria in Aqueous Solutions: I. The Solubility of Gibbsite in the System Na-K-Cl-OH-Al(OH)4 from 0 to 100 °C” (Wesolowski 1992). The assessment of sodium carbonate (Na2CO3) required the inclusion of sodium bicarbonate (NaHCO3) as the - 2- two ions HCO3 and CO3 are in equilibrium in aqueous solutions. In addition, the assessment of trisodium phosphate (Na3PO4) required the inclusion of sodium hydrogen phosphate (Na2HPO4) as the 2- 3- two ions HPO4 and PO4 are in equilibrium in aqueous solutions also. Section 5.2 describes the evaluation of solute mixtures, where the quality of the mixing parameters θ and ψ was assessed. Section 5.3 extends the mixture evaluations to include systems where double salts were known to exist. These assessments allowed the chemical potentials of the double salts to be evaluated. Finally, Section 5.4 details evaluations of species not covered by the previous sections and not available in the original Weber-Pitzer database. This section includes the solids dawsonite (NaAlCO3(OH)2) and boehmite (AlOOH). 5.1 EVALUATION OF SINGLE SOLUTES 5.1.1 NaNO3 Sources of data used to evaluate the Pitzer parameters for sodium nitrate solutions are given in Table 5-1. The data types include osmotic coefficients ( ) from isopiestic measurements, solution vapor pressure (Ps) measurements, freezing point temperature depression (∆Tf) measurements and solubility data (msat). Table 5-1. Data Sources for properties of NaNO3. Temperature Concentration Reference (Legend Entry) Number of Points Data Type Range (°C) Range (mol/kg) International Critical Tables, 1928 (28ICT) 0 – 100 0.6 – 21 66 Ps Pearce and Hopson, 1937 (37PEA/HOP) 25 0.1 – 10.83 18 Ps Kangro and Groeneveld, 1962 (62KAN/GRO) 20 – 25 1 – 10 19 Ps Shpigel and Mishchenko, 1967 (67SHP/MIS) 1 – 75 0.3 – 17 101 Ps Robinson, 1935 (35ROB) 25 0.1 – 6.0 49 Voigt et al., 1990 (90VOI/DIT) 100 0.9 – 16 18 Felmy, 1994 (94FEL) 50 – 100 0.2 – 19 39 De Coppet, 1904 (04DEC) Tfus 1.4 – 8.2 7 ∆Tf Seidell, 1958 (58SEI) 0 – 100 8.5 – 20.7 10 msat The vapor pressure data was converted to water activities, using Equation (5-1) and then to osmotic coefficients using Equation (2-15). Water activity is defined as: 20 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 29 of 77 RPP-RPT-50703 Rev. 1A = (5-1) where aw is water activity, Ps is the solution vapor pressure, and Pw is the vapor pressure of pure water at the same temperature. This equation is valid in the temperature range (0 to 100 °C) and total pressure (1 bar) of interest to the HTWOS model. For the freezing point temperature depression data, the equation from Electrolyte Solutions (Robinson and Stokes 2002), Equation (5-2), was used to convert temperature depression (θ) to osmotic coefficient ( ). = (1 + 4.9 × 10 ) (5-2) where v is the number of moles of ions per mole of solute (assuming complete dissociation), m is the solute molality, λ is the molal lowering of the freezing point and has a value of 1.86 for water, and θ is the lowering of the freezing point given by (To – Tf) where To is the freezing point of pure water and Tf is the freezing point of the solution. The temperatures chosen to evaluate the Pitzer parameters were 0, 25, 50, 75 and 100 °C. Using the data from the Weber-Pitzer database, the following values are obtained at each temperature: Table 5-2. Weber Pitzer Parameters and Reduced Chemical Potentials for NaNO3. Parameter 0 °C 25 °C 50 °C 75 °C 100 °C β(0) -0.03167 0.00204 0.02378 0.03661 0.04271 β(1) 0.12443 0.2368 0.3239 0.39174 0.44465 C 0.00182 0.00008 -0.0009 -0.00131 -0.00131 μ°Na+/RT -114.505 -105.642 -98.158 -91. 7552 -86.216 μ°NO3-/RT -52.2791 -44.707 -38.208 -32.5579 -27.5918 μ°NaNO3(s)/RT -165.02 -147.822 -133.409 -121.172 -110.667 Ksp* 5.83308 12.5159 19.2429 23.1327 23.1304 * Ksp = EXP(μ°NaNO3(s)/RT – μ°Na+/RT – μ°NO3-/RT) is the solubility product for the dissociation reaction NaNO3(s) ⇄ Na+ + NO3- Using the information given in Table 5-2, plots of the experimental osmotic coefficients and model predictions can be made at each temperature as can predictions of the NaNO3 saturation curve. These plots are given in Figures 5-1 to 5-6 where the Weber predictions are dashed lines (- - - -) and symbols are the experimental data. Figure 5-1. NaNO3 Osmotic Coefficients at 0 °C. Figure 5-2. NaNO3 Osmotic Coefficients at 25 °C. 0.95 1 04DEC 28ICT 28ICT 67SHP/MIS 0.9 0.95 67SHP/MIS 37PEA/HOP t t n 0.85 n 35ROB e e i Weber i 0.9 c c i i 62KAN/GRO f f f HTWOS f e 0.8 e Weber o o C C 0.85 HTWOS c c i i t 0.75 t o o m m 0.8 s 0.7 s O O 0.65 0.75 0.6 0.7 0 2 4 6 8 10 0 2 4 6 8 10 12 NaNO₃ molality NaNO₃ molality 21 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 30 of 77 RPP-RPT-50703 Rev. 1A Figure 5-3. NaNO3 Osmotic Coefficients at 50 °C. Figure 5-5. NaNO3 Osmotic Coefficients at 100 °C. 1 94FEL 1 90VOI/DIT 0.95 28ICT 28ICT 0.95 67SHP/MIS 94FEL t t 0.9 n n e e Weber Weber i i 0.9 c c i i f f 0.85 f f HTWOS HTWOS e e o o C C 0.8 0.85 c c i i t t o o 0.75 m m 0.8 s s O O 0.7 0.75 0.65 0.6 0.7 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 NaNO₃ molality NaNO₃ molality Figure 5-4. NaNO3 Osmotic Coefficients at 75 °C. Figure 5-6. Solubility of NaNO₃ in H2O. 1 22 67SHP/MIS 28ICT 0.95 20 Weber t n 18 y e HTWOS i 0.9 t 58SEI i c l i f a l f Weber-1 o e 16 o m C 0.85 ₃ Weber-2 c i O t 14 N o HTWOS a m 0.8 N s 12 O 0.75 10 0.7 8 0 2 4 6 8 10 12 14 16 18 0 20 40 60 80 100 120 NaNO₃ molality Temperature °C Using the Weber-Pitzer parameters and chemical potential coefficients with the precision given in the original papers (ORNL/TM-2001/102 and 109), the fit to the osmotic coefficient is generally poor, as shown in Figures 5-1 to 5-5, except at 25 °C and up to about 8 molal at 50 °C. The solubility curve generated in Figure 5-6 shows that dual solutions exists at temperatures greater than about 40 °C with a discontinuity centered at about 70 °C. The line designated as “Weber-1” is the solubility curve obtained below 70 °C and the line designated as “Weber-2” is the curve obtained above 70 °C, both solutions satisfy the solubility equation, = × × × = ± (5-3) As an example at 50 °C, two solutions exist: at ms = 13.2405 molal and at ms = 20.0721 molal, where Ksp = 19.2428; m²γ±² = (13.2405×0.33131)² = 19.2428; m²γ±² = (20.0721×0.21854)² = 19.2428. For these reasons, it was decided to recalculate the Pitzer parameters and chemical potential coefficients for NaNO3 using the experimental data referenced in Table 5-1. Initially, the Pitzer parameter (0) (1) coefficients for β , β , and C for the Na-NO3 ion pairing, along with the chemical potential for the neutral species NaNO3(aq), were regressed simultaneously with the osmotic coefficients utilizing the original Steele coefficients for the sodium ion and the nitrate ion chemical potentials (ORNL/TM- 2000/348). It was found that the best fit to the experimental data was obtained only when the neutral NaNO3(aq) was included (TWRS-PP-94-090). Then, using these new parameter values and the solubility data of Seidell 1958, the chemical potentials of the solid NaNO3 were regressed. The new coefficients thus obtained for the chemical potential coefficients are listed in Appendix A and the Pitzer parameters coefficients are listed in Appendix B. New predictions for the osmotic coefficients and the solubility isotherm are shown in Figures 5-1 to 5-6 for comparison with the experimental data and the original 22 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 31 of 77 RPP-RPT-50703 Rev. 1A Weber predictions. It can be seen that the new HWTOS coefficients give a superior fit to the experimental osmotic data over the entire concentration range than the original Weber coefficients and that the duality and discontinuity in the Weber solubility predictions are corrected by these new coefficients. The standard deviation2 of fit to the solubility data is 0.041. 5.1.2 NaNO2 For the assessment of the consistency of the Weber sodium nitrite parameters, the following experimental data, listed in Table 5-3, was obtained. Table 5-3. Data Sources for properties of NaNO2. Temperature Concentration Reference (Legend Entry) Number of Points Data Type Range (°C) Range (mol/kg) Staples, 1981 (81STA) 25 0.1 – 12.25 16 Ps Ray and Ogg, 1956 (56RAY/OGG) 25 0.3 – 12.34 7 Ps International Critical Tables, 1928 (28ICT) 100 1 – 16 8 Ps Zaytsev and Aseyev, CRC, 1992 (92CRC) 25, 100 0.76 – 14.49 19 Ps Felmy et al, 1994 (94FEL) 50, 100 0.58 – 17.74 32 Helberg, 1925 (25HEL)¹ Tfus 1.5 – 8.9 7 ∆Tf Bureau, 1937 (37BUR) ¹ Tfus 0.9 – 5.7 6 ∆Tf Protsenko and Savenkov, 1972 (72PRO/SAV) ¹ Tfus 1.6 – 9.3 12 ∆Tf Bureau, 1937 (37BUR) ¹ 0 - 103 10.3 – 24.3 8 msat Erdos and Simkova, 1957 (57ERD/SIM) ¹ 11.9 - 52 11 – 15.3 7 msat Söhnel and Novotny, 1985 (85SOH/NOV) 0 – 100 10.4 – 23.4 10 msat Apelblat and Korin, 1998 (98APE/KOR) 5 - 50 10.7 - 15 10 msat ¹ Taken from the MSE Validation set (OLI Systems, Inc.) The source data in Table 5-3 was processed as described in section 5.1.1. Again, the temperatures chosen to evaluate the Pitzer parameters from the Weber-Pitzer database were 0, 25, 50, 75 and 100 °C. Using the Weber parameters (ORNL/TM-2001/102 and 109), the following values are obtained at each temperature: Table 5-4. Weber Pitzer Parameters and Reduced Chemical Potentials for NaNO2. Parameter 0 °C 25 °C 50 °C 75 °C 100 °C β(0) 0.04869 0.00204 0.02378 0.03661 0.04271 β(1) 0.0137 0.2368 0.3239 0.39174 0.44465 C 0.0013 0.00008 -0.0009 -0.00131 -0.00131 μ°Na+/RT -114.505 -105.642 -98.158 -91. 7552 -86.216 μ°NO2-/RT -16.7237 -12.931 -9.61741 -6.68748 -4.0703 μ°NaNO2(s)/RT -127.849 -114.658 -103.508 -93.9597 -85.6906 Ksp* 29.3677 50.1491 71.3196 88.5018 99.0582 * Ksp = EXP(μ°NaNO2(s)/RT – μ°Na+/RT – μ°NO2-/RT) is the solubility product for the dissociation reaction NaNO2(s) ⇄ Na+ + NO2- 2 Standard deviation = , where mexpt = experimental molality, mcalc = predicted molality, and N = number of data points. 23 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 32 of 77 RPP-RPT-50703 Rev. 1A Using the information listed in Table 5-4, plots of the experimental osmotic coefficient versus model predictions can be made at each temperature as can predictions of the NaNO2 saturation curve. These plots are given in Figures 5-7 to 5-11. Figure 5-7. NaNO2 Osmotic Coefficients at 0 °C. Figure 5-10. NaNO2 Osmotic Coefficients at 100 °C. 1.1 1.05 94FEL 1 1 92CRC t t n 0.9 n Weber e e i i c c i i HTWOS f f f f 0.95 e 0.8 e o o C C c c i i t 0.7 25HEL t 0.9 o o m 37BUR m s 0.6 s O 72PRO/SAV O 0.85 0.5 Weber HTWOS 0.4 0.8 0 2 4 6 8 10 12 0 5 10 15 20 25 NaNO₂ molality NaNO₂ molality Figure 5-8. NaNO2 Osmotic Coefficients at 25 °C. Figure 5-11. Solubility of NaNO2 in H2O. 26 24 22 y t i l a 20 l o 85SOH/NOV m 18 ₂ 37BUR O N 57ERD/SIM a 16 N 98APE/KOR 14 Weber 12 HTWOS 10 0 20 40 60 80 100 120 Temperature °C Figure 5-9. NaNO2 Osmotic Coefficients at 50 °C. 1.02 1 t n e i 0.98 c i f f e o C 0.96 c 94FEL i t o 98APE/KOR m 0.94 s O Weber 0.92 HTWOS 0.9 0 2 4 6 8 10 12 14 16 NaNO₂ molality In Figures 5-7 to 5-11, the Weber predictions are the dashed lines (- - - -) whereas the symbols are the experimental data. It can be seen that the Weber curves do not fit the experimental data very well. At the higher temperatures, the Weber predictions show a rapid decrease in predicted osmotic coefficient with increasing concentration of NaNO2 which results in non-convergence of the solubility prediction at temperatures greater than 50 °C (Figure 5-11). As an example, at 75 °C, the solubility product is 88.5108 (from Table 5-1), but the ion activity product, (ms×γ±)², only achieves a maximum value of 65.2873 at a concentration of 16.332 molal, hence no solid is predicted to form and NaNO2 is predicted to be infinitely soluble by the Weber model. 24 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 33 of 77 RPP-RPT-50703 Rev. 1A To better fit the experimental osmotic data, and to better predict the solubility over the whole temperature range, new Pitzer parameters were regressed. Again, like with NaNO3, the best fit to the osmotic data was achieved when a neutral aqueous NaNO2 molecule is included, as Felmy found (TWRS-PP-94-090). The resulting Pitzer parameter coefficients are listed in Appendix B. The chemical potential coefficients for the solid NaNO2 were obtained from regression with the solubility data taken from the references given in Table 5-3. The coefficients thus obtained are listed in Appendix A. Again, the reduced chemical potential coefficients for the sodium ion were taken from Steele (ORNL/TM-2000/348); however the nitrite ion coefficients were taken from ORNL/TM-2001/102 as the original coefficients in Steele appear to calculate the actual Gibbs free energy, not the reduced chemical potentials. Predictions of the osmotic coefficients and solubility curve are shown in Figures 5-7 to 5-11, where it can be seen that the HTWOS coefficients give predictions that are a great improvement over Weber’s, especially as the complete solubility curve can be predicted over the entire temperature range, from 0 to 100 °C. The standard deviation of fit to the solubility data is 0.039. 5.1.3 NaOH For the assessment of the consistency of the Weber sodium hydroxide parameters, the experimental data listed in Table 5-5 was obtained. Table 5-5. Data Sources for properties of NaOH. Temperature Concentration Reference (Legend Entry) Number of Points Data Type Range (°C) Range (mol/kg) Smithsonian Physical Tables, 1916 (16SMI) 100 0.52 – 10.43 9 Ps Paranjpe, 1918 (18PAR) 0 – 40 3.75 – 22.5 82 Ps Hayward and Perman, 1927 (27HAY/PER) 30 – 80 2.4 – 39.0 51 Ps International Critical Tables, 1928 (28ICT) 20 – 100 1.25 – 75 73 Ps Kangro and Groeneveld, 1962 (62KAN/GRO) 20, 25 1 – 27 19 Ps Dibrov et al, 1964 (64DIB) 25 – 97 1.7 – 20.6 50 Ps Akerlof and Kegeles, 1940 (40AKE/KEG) 0 – 70 0.25 – 16.9 192 Stokes, 1945 (45STO) 25 1.98 – 28.8 33 Zhou et al, 2003 (03ZHO) 25, 40 0.09 – 3.0 6 Smithsonian Physical Tables, 1908 (08SMI) Tfus 0.025 – 0.125 5 ∆Tf International Critical Tables, 1928 (28ICT) Tfus 0.01 – 6.11 10 ∆Tf The temperatures chosen to assess the Weber model were 0, 25, 40, 70 and 100 °C. These temperatures were chosen because multiple datasets were available at each temperature. The Weber model parameters are listed in Table 5-6. (Note: no solid phase was modeled as sodium hydroxide is extremely soluble in water; saturation is approximately 29 molal at 25 °C). Table 5-6. Weber Pitzer Parameters and Reduced Chemical Potentials for NaOH. Parameter 0 °C 25 °C 40 °C 70 °C 100 °C β(0) 0.091478 0.0869 0.091017 0.091478 0.084255 β(1) 0.341649 0.2481 0.279541 0.341649 0.402266 C 0.000685 0.0039 0.002535 0.000685 -0.00028 o μ Na+/RT -114.505 -105.642 -101.007 -92.9606 -86.216 o μ OH-/RT -71.8581 -63.446 -58.9815 -51.1057 -44.3624 25 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 34 of 77 RPP-RPT-50703 Rev. 1A The Pitzer parameters listed in Table 5-6 were plotted with the experimental data to evaluate the accuracy of the Weber model. The results are shown in Figures 5-12 to 5-16. Figure 5-12. NaOH Osmotic Coefficients at 0 °C. Figure 5-15. NaOH Osmotic Coefficients at 70 °C. 4 3 3.5 2.5 t 3 t n n e e i i 2 c c i 2.5 i f f f f e e o o C 2 C 1.5 40AKE/KEG c c i i t 28ICT t o 1.5 o 40AKE/KEG m 08SMI m 1 s s 27HAY/PER O 1 18PAR O Weber 0.5 Weber 0.5 HTWOS HTWOS 0 0 0 5 10 15 20 0 5 10 15 20 NaOH molality NaOH molality Figure 5-13. NaOH Osmotic Coefficients at 25 °C. Figure 5-16. NaOH Osmotic Coefficients at 100 °C. 3 2.5 2.5 2 t t n n e e i 2 i c c i i f f f f 1.5 e 45STO e o o 28ICT C 1.5 C c 64DIB c i i t 64DIB t o 62KAN/GRO 1 o m 1 m 08SMI s 18PAR s O 03ZHO O Weber 0.5 0.5 Weber HTWOS HTWOS 0 0 0 5 10 15 20 0 5 10 15 20 25 NaOH molality NaOH molality Figure 5-14. NaOH Osmotic Coefficients at 40 °C. 3 2.5 t n e i 2 c i f f e 40AKE/KEG o C 1.5 c 03ZHO i t o 28ICT m 1 s 18PAR O 0.5 Weber HTWOS 0 0 5 10 15 20 NaOH molality It appears from Figures 5-12 to 5-16, that the Weber model was fit to approximately 6 molal NaOH only. To extend the Weber model, new Pitzer parameters were regressed with the experimental data up to 10 molal NaOH. The new Pitzer parameters obtained are listed in Appendix B. The chemical potential coefficients of Steele (ORNL/TM-2000/348) for the ions (Na+ and OH-) were retained in the HTWOS- Pitzer database. Predictions of the osmotic coefficient using these new Pitzer parameters are also shown in Figures 5-12 to 5-16 for comparison. The HTWOS model is shown to fit the data accurately to about 10 molal at temperatures at or below 70 °C and up to about 22 molal at 100 °C. 26 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 35 of 77 RPP-RPT-50703 Rev. 1A 5.1.4 NaF The majority of the data for sodium fluoride is reported at 25 °C, with sparse data at other temperatures. For this reason, the Pitzer parameters of Weber will be used (ORNL/TM-2001/102 and 109). A list of the experimental solubility data used to check these Pitzer parameters is given in Table 5-7 and is shown in Figure 5-17. The data was taken from the OLI Systems Mixed Solvent Electrolyte (MSE) validation set (OLI Systems, Inc.), a series of Excel workbooks that can be downloaded from their support website. Table 5-7. Data Sources for solubility of NaF in H2O. Temperature Concentration Reference (Legend Entry) Number of Points Data Type Range (°C) Range (mol/kg) Seidell, 1958 (58SEI) 0 – 100 0.87 – 1.21 10 msat (smoothed) Tananeev, 1941 (41TAN) 0 – 94 0.98 – 1.18 5 msat Foote and Schairer, 1930 (30FOO/SCH) 10, 25, 35 0.97 – 1.00 8 msat Payne, 1937 (37PAY) 0, 25, 35 0.87 – 0.99 3 msat Lopatkina, 1959 (59LOP) 25, 50 0.93 – 0.98 2 msat As can be seen from Figure 5-17, there is a lot of scatter in the original experimental solubility data, and the Weber model tends to greatly over predict the solubility across the entire temperature range. For this reason, the assessment was repeated starting with the Pitzer parameters and reduced chemical potentials for the ions and solid NaF from Steele (ORNL/TM-2000/348). The results from Steele’s model are also shown in Figure 5-17 labeled as “HTWOS” because the Steele model parameters were retained as the coefficients for the HTWOS model as the fit to the solubility curve was acceptable. These coefficients are listed in Appendix A and B. Figure 5-17. Solubility of NaF in H2O from 0 to 100 °C. 2.4 58SEI 2.2 41TAN 30FOO/SCH 2 37PAY 59LOP y t i 1.8 l Weber a l o HTWOS 1.6 m F a 1.4 N 1.2 1 0.8 0 20 40 60 80 100 120 140 Temperature (°C) 5.1.5 NaCl Sodium chloride was not required by the original Wash and Leach Factor Plan (RPP-RPT-46002), but is required to evaluate parameters for other species, for example gibbsite from Wesolowski’s data (Wesolowski 1992). For the HTWOS model, the reduced chemical potential coefficients for the sodium and chloride ion were taken from Steele (ORNL/TM-2000/348). The Pitzer parameters for binary interactions were taken from “Thermodynamics of Electrolytes. 8. High Temperature Properties, Including Enthalpy and Heat Capacity, with Application to Sodium Chloride” (Silvester and Pitzer 1977), but refit to the temperature 27 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 36 of 77 RPP-RPT-50703 Rev. 1A expression of Equation (2-35) over the range 0 to 100 °C. The solubility data from Seidell 1958 was regressed for the coefficients of the reduced chemical potentials of solid NaCl. The resulting parameters are listed in Appendix A and B and stored in the HTWOS-Pitzer database. The accuracy of the solubility predictions, shown in Figure 5-18, has a standard deviation of 0.004 over the temperature range. Figure 5-18. Solubility of NaCl in H2O from 0 to 100 °C. Experimental data taken from Seidell, (1958) 6.8 6.7 6.6 y t i l 6.5 a l o 6.4 m l 58SEI C a 6.3 N HTWOS 6.2 6.1 6 0 20 40 60 80 100 120 Temperature (°C) 5.1.6 Na2SO4 To test Weber’s Pitzer model coefficients for sodium sulfate (ORNL/TM-2001/102 and 109), the experimental data referenced in Table 5-8 was compared to his model predictions of osmotic coefficient. These predictions are shown in Figures 5-19 to 5-24. In Figures 5-19 to 5-24 the Weber model does fit the experimental osmotic coefficient at 20 and 25 °C, but at higher temperatures is not as good. Therefore, the experimental data was regressed to provide new coefficients for the temperature dependence of the Pitzer parameters for the HTWOS model. The results of predictions with the new HTWOS coefficients are plotted in Figures 5-19 to 5-24 also. The fit to the experimental data is much tighter than Weber’s fit, with a standard deviation of 0.003 for all the data over the temperature range. Table 5-8. Data Sources for properties of Na2SO4. Temperature Concentration Number of Reference (Legend Entry) Data Type Range (°C) Range (mol/kg) Points Kangro and Groeneveld, 1962 (62KAN/GRO) 20, 25 0.5 – 3.0 12 Robinson, Wilson and Stokes, 1941 (41ROB/WIL) 25 0.09 – 4.20 23 Hellams et al, 1965 (65HEL/PAT) 45 0.7 – 3.5 13 Humphries, Kohrt and Patterson, 1968 (68HUM/KOH) 60 0.9 – 3.5 13 Moore, Humphries, and Patterson, 1972 (72MOO/HUM) 80 0.93 – 2.98 30 Patterson, Gilpatrick and Soldano, 1960 (60PAT/GIL) 99.6 0.89 – 3.18 5 28 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 37 of 77 RPP-RPT-50703 Rev. 1A Figure 5-19. Na2SO4 Osmotic Coefficients at 20 °C. Figure 5-22. Na2SO4 Osmotic Coefficients at 60 °C. 0.75 0.75 0.7 0.7 t t n n e e i i c c i i f f f 0.65 f 0.65 e e o o C C c c i i t 0.6 t 68HUM/KOH o 0.6 62KAN/GRO o m m s s Weber O Weber O 0.55 0.55 HTWOS HTWOS 0.5 0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Na₂SO₄ molality Na₂SO₄ molality Figure 5-20. Na2SO4 Osmotic Coefficients at 25 °C. Figure 5-23. Na2SO4 Osmotic Coefficients at 80 °C. 0.85 0.75 0.8 0.7 t t n 0.75 n e e i i c c i i f f f f 0.65 e 0.7 e o o C C c c i 62KAN/GRO i t 0.65 t o 0.6 o 72MOO/HUM m 41ROB/WIL m s 0.6 s O O Weber Weber 0.55 0.55 HTWOS HTWOS 0.5 0.5 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 Na₂SO₄ molality Na₂SO₄ molality Figure 5-21. Na2SO4 Osmotic Coefficients at 45 °C. Figure 5-24. Na2SO4 Osmotic Coefficients at 99.6 °C. 0.75 0.75 0.7 0.7 t t n n e e i i c c i i f f f 0.65 f 0.65 e e o o C 65HEL/PAT C c c i i t 0.6 t o Weber 0.6 o m 60PAT/GIL m s s O HTWOS O Weber 0.55 0.55 HTWOS 0.5 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 Na₂SO₄ molality Na₂SO₄ molality Next, predictions of Na2SO4 solubility were made using the HTWOS aqueous ion reduced chemical potentials, the new HTWOS Pitzer parameters, and the solid Na2SO4 reduced chemical potentials of Weber (ORNL/TM-2001/102 and 109). These predictions are compared to the experimental solubility data from Seidell 1958 in Figure 5-25. 29 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 38 of 77 RPP-RPT-50703 Rev. 1A Figure 5-25: Solubility of Na2SO4 in H2O from 0 to 100 °C Experimental data taken from Seidell, (1958) 58SEI 4 Weber 3.5 HTWOS 3 y t i l a l 2.5 o m 2 ₄ O S ₂ 1.5 a N 1 0.5 0 0 20 40 60 80 100 120 Temperature (°C) Figure 5-25 shows that Weber’s reduced chemical potential coefficients are not consistent with the HTWOS’ parameters, so they were refit to the experimental data and the new coefficients were placed into the HTWOS-Pitzer database (Appendix A). Prediction of the solubility from the complete HTWOS model is shown in Figure 5-25 also, where the standard deviation of fit is 0.007 over the temperature range from 0 to 100 °C. 5.1.7 Na2CO3 The thermodynamics of sodium carbonate cannot be studied as a simple binary system due to the hydrolysis of carbonate ions leading to bicarbonate and hydroxide ions. Therefore, the system has to be treated as a mixture of sodium carbonate, sodium bicarbonate and sodium hydroxide. The hydrolysis of the carbonate ion can be described by: + ⇄ + (5-4) The equilibrium constant for this reaction is given by: × = = (5-5) × or ln = ln − ln (5-6) Where Kh is the hydrolysis constant, K2 is the second ionization constant of carbonic acid, and Kw is the self-ionization constant of water. In terms of chemical potentials of the ions, this equilibrium can be written as: ln = + − − (5-7) The temperature dependent expressions of Steele and co-workers (ORNL/TM-2000/348) for the reduced chemical potentials of the ions were used together with their expressions for the Pitzer parameters, except for NaOH which were taken from the HTWOS-Pitzer database. To test the consistency and accuracy of Steele’s model, the predictions of osmotic coefficient from the model developed in “Thermodynamics of Aqueous Carbonate Solutions Including Mixtures of Sodium Carbonate, Bicarbonate, and Chloride” (Peiper and Pitzer 1982) in the temperature range of 5 to 45 °C were employed because experimentally determined osmotic coefficients at temperatures other than 25 °C could not be found in the literature. The results of this analysis, which included the hydrolysis reaction, are given in Table 5-9. 30 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 39 of 77 RPP-RPT-50703 Rev. 1A Table 5-9. Comparison of Osmotic Coefficients for Na2CO3 from 5 to 45 °C. t (°C) 5.0 15.0 25.0 35.0 45.0 m P&P1 Steele2 δ3 P&P Steele δ P&P Steele δ P&P Steele δ P&P Steele δ 0.001 0.968 0.968 0.000 0.970 0.970 0.000 0.971 0.971 0.000 0.972 0.973 -0.001 0.974 0.974 0.000 0.002 0.955 0.955 0.000 0.956 0.957 -0.001 0.957 0.958 -0.001 0.959 0.959 0.000 0.960 0.961 -0.001 0.005 0.931 0.932 -0.001 0.932 0.933 -0.001 0.933 0.934 -0.001 0.934 0.935 -0.001 0.936 0.936 0.000 0.010 0.909 0.909 0.000 0.910 0.910 0.000 0.910 0.911 -0.001 0.911 0.912 -0.001 0.912 0.912 0.000 0.020 0.883 0.883 0.000 0.884 0.884 0.000 0.884 0.885 -0.001 0.884 0.885 -0.001 0.884 0.885 -0.001 0.050 0.846 0.846 0.000 0.847 0.847 0.000 0.847 0.848 -0.001 0.846 0.847 -0.001 0.845 0.845 0.000 0.100 0.814 0.815 -0.001 0.817 0.817 0.000 0.817 0.818 -0.001 0.817 0.818 -0.001 0.815 0.815 0.000 0.200 0.777 0.777 0.000 0.782 0.782 0.000 0.784 0.785 -0.001 0.785 0.786 -0.001 0.783 0.784 -0.001 0.400 0.725 0.726 -0.001 0.736 0.737 -0.001 0.743 0.744 -0.001 0.746 0.747 -0.001 0.746 0.747 -0.001 0.600 0.687 0.687 0.000 0.703 0.704 -0.001 0.715 0.716 -0.001 0.722 0.723 -0.001 0.724 0.725 -0.001 0.800 0.657 0.657 0.000 0.679 0.680 -0.001 0.696 0.696 0.000 0.706 0.707 -0.001 0.711 0.711 0.000 1.000 0.634 0.634 0.000 0.662 0.662 0.000 0.683 0.683 0.000 0.697 0.698 -0.001 0.704 0.705 -0.001 S.D.4 0.00044 S.D. 0.00053 S.D. 0.00087 S.D. 0.00085 S.D. 0.00065 1 P&P = osmotic coefficient from the model of Peiper and Pitzer 1982 2 Steele = Osmotic coefficient from the model in ORNL/TM-2000/348 3 δ = φ(P&P) – φ(Steele) 4 S.D. = √{( ²)/( – 1)} where S.D. = standard deviation and N = number of data points This analysis shows that the Steele model parameters lead to predictions of osmotic coefficient that compare very favorably with Peiper’s model predictions. During this analysis, the B coefficient for β(¹) of Na-CO3 was adjusted from Steele’s original value of 0.09989 to 0.09784 so that the first temperature derivative (∂β(¹)/∂T) value at 25 °C matched that given by Peiper. With this change included, the remaining parameter coefficients from Steele were placed into the HTWOS-Pitzer database as listed in Appendices A and B. As Steele does not have solid sodium carbonate reduced chemical potential coefficients, those of Weber were chosen for the solids, Na2CO3·10H2O, Na2CO3·7H2O, and Na2CO3·H2O. The solubility data listed in Seidell 1958 was employed to test the consistency of using Steele’s model with Weber’s coefficients. Therefore, using Weber’s reduced chemical potentials for the solid carbonates, Steele’s reduced chemical potentials for all the ions and modified Pitzer parameters for Na-CO3 and Na-HCO3 interactions, and the new HTWOS Pitzer parameters for Na-OH interactions derived previously, the predicted solubility of Na2CO3 solutions versus temperature was calculated. The prediction is shown in Figure 5-26 showing that Weber’s coefficients lead to a poor fit of the experimental data. Also, the heptahydrate was not predicted to form at temperatures between 32 and 35.37°C, so the solid reduced chemical potentials were refit using the value of the ion activity product at the saturated concentration as the target Ksp for each temperature. The ions and water reduced chemical potentials were calculated as before from Steele’s coefficients. The expressions for the solid carbonates chemical potentials were calculated at each temperature from: = 2 + + 10 . for Na2CO3·10H2O: (5-8) + ln , . for Na2CO3·7H2O: = 2 + + 7 + ln , . (5-9) . for Na2CO3·H2O: = 2 + + + ln , . (5-10) . For temperatures less than 32 °C, the solid phase is the decahydrate, between 32 ° and 35.37 °C the solid phase is the heptahydrate, and above 35.37 °C the solid is the monohydrate. The reduced chemical potential values were then regressed to calculate the coefficients of the temperature dependency of 31 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 40 of 77 RPP-RPT-50703 Rev. 1A Equation (2-35). The predictions using these revised coefficients are also shown in Figure 5-26, where the standard deviation of fit is 0.033 over the temperature range from 0 to 100 °C. These new coefficients were placed into the HTWOS-Pitzer database listed in Appendix A. Figure 5-26: Solubility of Na2CO3 in H2O from 0 to 100 °C 6 Experimental data taken from Seidell (1958) 58SEI Weber 5 HTWOS y t i 4 l a l o m 3 ₃ O C ₂ a 2 N 1 0 0 20 40 60 80 100 120 Temperature (°C) 5.1.8 Na3PO4 Like the sodium carbonate system described above, trisodium phosphate (TSP) cannot be treated as a single solute when dissolved in water as some fraction of phosphate ions will hydrolyze to form hydrogen phosphate ions and hydroxide ions. Therefore, the system must be treated as a mixture of TSP, disodium hydrogen phosphate (DSP) and sodium hydroxide. The hydrolysis of the phosphate ion is described by: + ⇄ + (5-11) The degree to which the phosphate ions are hydrolyzed is determined by the hydrolysis constant, Kh, which in terms of chemical potentials of the ions is given by: ln = + − − (5-12) The sources of experimental data used to test the self-consistency of Weber’s Pitzer parameters for DSP are given in Table 5-10. The amount of data for this system is limited to only 3 temperatures, 0, 25 and 100 °C. After conversion of the experimental data to osmotic coefficients, the data was compared to Weber’s model predictions. The comparison can be seen in Figures 5-27 to 5-29. In general, Weber’s parameters fit the data well at 0 and 25 °C, but at 100 °C the osmotic coefficients at concentrations greater than 5 molal DSP are under predicted. To improve the fit to the experimental data for the HTWOS model, the whole dataset was plotted versus solute concentration with the resulting graph shown in Figure 5-30. It can be seen that the data forms a single trend irrespective of the temperature the data was measured at. For this reason, it was assumed that the whole data set reflected the osmotic coefficient at any temperature and, therefore, Pitzer parameters were fit at 0, 25, 50, 75 and 100 °C to obtain the temperature dependent coefficients. These coefficients were placed into the HTWOS-Pitzer database (Appendix B). The HTWOS model predictions are included in Figures 5-27 to 5-29. 32 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 41 of 77 RPP-RPT-50703 Rev. 1A Table 5-10. Data Sources for properties of Na2HPO4. Temperature Concentration Number of Reference (Legend Entry) Data Type Range (°C) Range (mol/kg) Points th Lide, CRC Handbook, 90 Ed., 2009 (09CRC) 0 0.03 – 0.11 3 ∆Tf International Critical Tables, 1928 (28ICT) 0 0.01 – 0.104 5 ∆Tf Scatchard and Breckenridge, 1954 (54SCA/BRE) 25 0.09 – 1.0 25 Platford, 1974 (74PLA) 25 0.23 – 2.12 10 th Lide, CRC Handbook, 90 Ed., 2009 (09CRC) 100 0.5 – 7.5 7 Ps International Critical Tables, 1928 (28ICT) 100 0.5 – 5.5 8 Ps Figure 5-27. Na2HPO4 Osmotic Coefficients at 0 °C. Figure 5-29. Na2HPO4 Osmotic Coefficients at 100 °C. 0.92 0.7 28ICT 0.9 09CRC 0.65 09CRC 28ICT t 0.88 t 0.6 Weber n n e Weber e i i c c i 0.86 i HTWOS f 0.55 f f HTWOS f e e o o C 0.84 C 0.5 c c i i t t o 0.82 o 0.45 m m s s O 0.8 O 0.4 0.78 0.35 0.76 0.3 0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5 6 7 8 Na₂HPO₄ molality Na₂HPO₄ molality Figure 5-28. Na2HPO4 Osmotic Coefficients at 25 °C. Figure 5-30. Na2HPO4 Osmotic Coefficients for all the Data 0.85 1 54SCA/BRE 0.8 74PLA 0.9 Weber t t n n 0.75 0.8 e e i i HTWOS c c i i f f f f 0.7 e 0.7 e o o C C c c i i t 0.6 t 0.65 o o m m s s 0.5 O O 0.6 0.55 0.4 0.5 0.3 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 Na₂HPO₄ molality Na₂HPO₄ molality There are three solid species formed when DSP is dissolved in water in the temperature range from 0 to 100 °C; the decahydrate, the heptahydrate and the dihydrate. To evaluate the temperature coefficients of the reduced chemical potentials for these solid species, the solubility data presented in Seidell 1958 was used together with the new HTWOS Pitzer parameters and reduced chemical potentials for the ions. The resulting solubility isotherm, shown in Figure 5-31, has a standard deviation of 0.099 over the fitted temperature range. The coefficients obtained for these three solids were placed into the HTWOS-Pitzer database (Appendix A). 33 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 42 of 77 RPP-RPT-50703 Rev. 1A Figure 5-31: Solubility of Na2HPO4 in H2O from 0 to 100 °C 8 Experimental data taken from Seidell (1958) 7 6 y t i l a l 5 o m ₄ 4 O 58SEI P H ₂ 3 a HTWOS N 2 1 0 0 20 40 60 80 100 120 Temperature (°C) Once the DSP parameters had been obtained, the TSP parameters could be evaluated to obtain the temperature dependent Pitzer coefficients. The experimental data for TSP is scarce and comes from similar sources to the DSP data and is given below in Table 5-11. Table 5-11. Data Sources for properties of Na3PO4. Temperature Concentration Number of Reference (Legend Entry) Data Type Range (°C) Range (mol/kg) Points th Lide, CRC Handbook, 90 Ed., 2009 (09CRC) 0 0.03 – 0.16 5 ∆Tf International Critical Tables, 1928 (28ICT) 0 0.01 – 0.1 4 ∆Tf Scatchard and Breckenridge, 1954 (54SCA/BRE)* 25 0.039 – 0.42 20 th Lide, CRC Handbook, 90 Ed., 2009 (09CRC) 100 0.52 – 2.13 3 Ps International Critical Tables, 1928 (28ICT) 100 0.6 – 2.8 5 Ps * Equimolal mixtures of DSP and TSP were examined to reduce the effects of the hydrolysis reaction. Each system listed in Table 5-11 was analyzed with the hydrolysis reaction included and due to the asymmetry of the anion charges (i.e. -1, -2, and -3) the asymmetric mixing terms were also included (described in Section 2.4). The temperature dependent Pitzer coefficients were regressed to obtain the best fit to the experimental data and, simultaneously, values for the mixing terms θ-PO4-OH and ψ-Na- PO4-OH were obtained. Results of the regression are shown in Figures 5-32 to 5-34. There are 3 solids that exist between 0 and 100 °C when TSP is dissolved in water; the dodecahydrate (Na3PO4·¼NaOH·12H2O) which exists below 50 °C, the octahydrate between 50 and 70 °C, and the hexahydrate from 70 to 100 °C. The reduced chemical potentials for these solids were regressed against the solubility data presented in Seidell 1958 from experiments performed by Wendrow and Kobe, Kobe and Leipper, and Apfel. This data is plotted in Figure 5-35 together with the predictions from the model which resulted in a standard deviation of 0.012 over the fitted temperature range. The reduced chemical potential coefficients for the solids, Pitzer coefficients for the ions, and the mixing parameters (θ and ψ) were placed into the HTWOS-Pitzer database (Appendices A, B and C respectively). Note that after this modeling was complete but before this report was issued, new osmotic coefficient data was published by El Guenouzi and Benbiyi (2014) at 25 C. Inspection of this data indicated that the data did not cover a larger composition range than the data that was used in the modeling from Scatchard and Breckenridge (1954). Thus, it was decided not to update the model for this data at this time. 34 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 43 of 77 RPP-RPT-50703 Rev. 1A Figure 5-32. Na3PO4 Osmotic Coefficients at 0 °C. Figure 5-34. Na3PO4 Osmotic Coefficients at 100 °C. 1 1 0.95 28ICT 0.9 09CRC t 0.9 t HTWOS n n e e i i 0.8 c c i 0.85 i f f f f e o o e C 09CRC 0.8 C 0.7 c c i i t t o 0.75 28ICT o m m 0.6 s s O HTWOS 0.7 O 0.5 0.65 0.6 0.4 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 3 Na₃PO₄ molality Na₃PO₄ molality Figure 5-33. Na3PO4 Osmotic Coefficients at 25 °C. Figure 5-35. Solubility of Na3PO4 in H2O. 0.8 7 0.75 6 t n 5 y e i 0.7 t i c l i f a f l e o 4 Wen/Kob o m C 0.65 ₄ c i 54SCA/BRE O Kob/Lei t 3 P o ₃ a m 0.6 HTWOS s Apfel N O 2 HTWOS 0.55 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 Na₃PO₄ molality Temperature (°C) 5.1.9 Na2C2O4 The data available against which to test Steele’s (ORNL/TM-2000/348) sodium oxalate model is extremely sparse. In fact, Weber (ORNL/TM-2001/102) had to rely on solubility data to obtain his parameters. For this reason, Steele’s Pitzer parameters were assumed to be reasonable and were placed into the HTWOS-Pitzer database unaltered (Appendix B). Figure 5-36. Solubility of Na2C2O4 in H2O from 0 to 100 °C. 0.5 0.45 y 0.4 t i l a l o 0.35 m 51NOR ₄ O ₂ 0.3 C 04MEN/APE ₂ a N 0.25 HTWOS 0.2 0.15 0 20 40 60 80 100 Temperature °C 35 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 44 of 77 RPP-RPT-50703 Rev. 1A To test the self-consistency of Steele’s reduced chemical potential coefficients for the solid sodium oxalate with the HTWOS ion parameters, the solubility data from “The System Oxalic Acid – Sodium Hydroxide – Water” (Norris 1951) and “The Molar Enthalpies of Solution and Solubilities of Ammonium, Sodium, and Potassium Oxalate in Water” (Menczel, Apelblat and Korin 2004) was employed. The predictions using the coefficients of Steele ranged from infinitely soluble at 0 °C to extremely insoluble at 50 °C and above. Similar behavior was observed when the coefficients of Weber (ORNL/TM-2001/102 and 109) were utilized. For these reasons, the experimental solubility data was regressed and new coefficients for the reduced chemical potentials of solid sodium oxalate were obtained. The solubility isotherm predicted from this analysis is shown in Figure 5-36, which has a standard deviation of 0.002 over the fitted temperature range. The coefficients obtained were placed in the HTWOS-Pitzer database (Appendix A). 5.2 EVALUATION OF SOLUTE MIXTURES Once the single solute analysis was completed, attention could be turned towards the predictions of solute mixture solubility. Table 5-12 lists the solute mixtures that were analyzed during the development of the HTWOS Pitzer database and the temperature range of the experimental data over which a particular system was evaluated. The experimental data was collected from the open literature, with the majority taken from the compilations of Seidell 1958 and OLI Systems Mixed Solvent Electrolyte (MSE) validation set (in the form of Excel workbooks downloaded from OLI Systems, Inc. support website). The word “model” refers to utilizing the HTWOS-Pitzer database with GEMS to make the solubility predictions described in this section and includes the legend entries in all the figures presented. Table 5-12. Solute Mixtures Analyzed and Applicable Range of Temperature. Chemical Temperature Chemical Temperature Chemical Temperature System Range (°C) System Range (°C) System Range (°C) Na-NO3-NO2-H2O 0 – 103 Na-NO2-CO3-H2O 20 – 25 Na-CO3-SO4-H2O 15 – 100 Na-NO3-OH-H2O 0 – 100 Na-NO2-SO4-H2O 0 – 50 Na-CO3-PO4-H2O 0 – 100 Na-NO3-F-H2O 25 – 50 Na-OH-F-H2O 0 – 94 Na-F-SO4-H2O 0 – 80 Na-NO3-C2O4-H2O 20 – 75 Na-OH-C2O4-H2O 0 – 70 Na-F-PO4-H2O 25 – 50 Na-NO3-CO3-H2O 10 – 25 Na-OH-CO3-H2O 0 – 100 Na-Cl-OH-H2O 0 – 90 Na-NO3-SO4-H2O 0 – 100 Na-OH-SO4-H2O 0 – 100 Na-Cl-OH-Al-H2O 6 – 80 Na-NO2-OH-H2O 20 – 25 Na-OH-PO4-H2O 0-100 Na-CO3-HCO3-H2O 0 – 100 Na-NO2-PO4-H2O 25 Na-SO4-PO4-H2O 25 - - 5.2.1 Na-NO2-NO3-H2O The predicted solubilities of NaNO2 and NaNO3 in water from 0 to 103 °C are shown in Figure 5-37. The experimental data was taken from the extensive compilation of Seidell 1958 from original data measured by Oswald. As can be seen in Figure 5-37, the model predicts the experimental data very well; the largest deviations are in the predictions for the solid NaNO2 at the highest temperature of 103 °C, which is just above the maximum temperature limit of the original Pitzer parameter fitting (Section 5.1.2). No further model parameters were adjusted during the calculations for this system. 36 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 45 of 77 RPP-RPT-50703 Rev. 1A Figure 5-37. Solubilities of NaNO2 and NaNO3 in the NaNO2-NaNO3-H2O System from 0 to 103 °C. Experimental Data taken from Seidell (1958) Exptl - 0 °C Exptl - 21 °C 25 Exptl - 52 °C Exptl - 103 °C Model - 0 °C 20 Model - 21 °C y Model - 52 °C t i l Model - 100 °C a l 15 o m ₃ O 10 N a N 5 0 0 5 10 15 20 25 NaNO₂ molality 5.2.2 Na-NO3-OH-H2O The experimental data for to this system was taken from the compilation of Seidell 1958 (original data from Engel [91ENG]) at 0 °C and from the OLI Systems Mixed Solvent Electrolyte validation set (OLI Systems, Inc.) at 25, 65 and 100 °C (original data from Sadokhina, Sbitneva and Zimina [85SAD], Plekhotkin and Bobrovskaya [70PLE/BOB], Kurnakow and Nikolajew [27KUR/NIK] and Jänecke [30JAN]). Figure 5-38 shows the model predictions at the four separate temperatures together with the experimental data. The additional mixing parameters, θ OH-NO3, ψ Na-OH-NO3, λ OH-NaNO3, and ζ Na-OH-NaNO3 were adjusted so that the predictions matched better the experimental data. This was achieved by minimizing the square errors between the predicted and the experimental NaNO3 concentration at the same NaOH concentration. The final optimized values for the mixing parameters were -0.0921, 0.0036 for θ OH-NO3 and ψ Na-OH-NO3 respectively. The neutral species mixing terms, λ OH-NaNO3 and ζ Na-OH-NaNO3, were found to linearly depend on the temperature (requiring A and B coefficients in Equation (2-35), see Appendix C). Also, Figure 5-38 shows that the model can make accurate solubility predictions in this system at very high NaOH concentrations, 12 molal and above, even though the original fitting range was restricted to 10 molal NaOH (Section 5.1.3). Figure 5-38. Solubility of NaNO3 in the NaOH-NaNO3-H2O System from 0 to 100 °C. 30 91ENG 0 °C 85SAD 25 °C 27KUR/NIK 25 °C 25 70PLE/BOB 25 °C 27KUR/NIK 65 °C y t i 20 30JAN 100 °C l a l Model 0 °C o Model 25 °C m 15 ₃ Model 65 °C O N Model 100 °C a 10 N 5 0 0 2 4 6 8 10 12 14 16 18 20 22 NaOH molality 37 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 46 of 77 RPP-RPT-50703 Rev. 1A 5.2.3 Na-NO3-F-H2O The NaNO3-NaF-H2O system has been studied experimentally in “Solubility in the Na+F+NO3 and Na+PO4+NO3 Systems in Water and in Sodium Hydroxide Solutions” (Selvaraj et al. 2008) at 25 and 50 °C. Their data was used to compare model predictions for this system as shown in Figure 5-39. Again no new model parameters were adjusted during the calculations. Figure 5-39 shows that the solubility of NaF is relatively insensitive to temperature in the range 25 to 50 °C, when in solution with NaNO3, as the predicted curves lay on top of each other. The exception to this is at the invariant points due to the difference in solubility of NaNO3; 10.8 m at 25 and 13.4 m at 50 °C. Figure 5-39. Solubilities of NaF and NaNO3 in the NaF-NaNO3-H2O System at 25 and 50 °C. 1.2 Experimental Data taken from Salvaraj et al. (2008) 1 25 °C - Exptl 50 °C - Exptl y 0.8 t i l 25 °C - Model a l o 0.6 50 °C - Model m F a N 0.4 0.2 0 0 2 4 6 8 10 12 14 NaNO₃ molality 5.2.4 Na-NO3-C2O4-H2O Two sources of experimental data were utilized to analyze the NaNO3-Na2C2O4-H2O system; “The Na2C2O4-NaNO3-H2O System at 20 °C” (Zhikharev et al. 1979) and “The Na2C2O4-NaNO3-H2O System at 50 ° and 75 °C” (Kol’ba et al. 1980). The experimental data spans the temperature range from 20 to 75 °C and is shown in Figure 5-40 together with model predictions at each temperature. Figure 5-40. Solubilities of NaNO3 and Na2C2O4 in the NaNO3-Na2C2O4-H2O System at 20, 50 and 75 °C. 0.45 Experimental Data taken from Zhikharev et al. (1979) and Kol'ba et al. (1980) 0.4 20 °C - Exptl 0.35 50 °C - Exptl ₄ O 75 °C - Exptl ₂ 0.3 C ₂ a 0.25 20 °C - Model N y 50 °C - Model t i 0.2 l a l 75 °C - Model o 0.15 m 0.1 0.05 0 0 5 10 15 20 molality NaNO₃ 38 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 47 of 77 RPP-RPT-50703 Rev. 1A During the calculations, the mixing terms, θ NO3-C2O4 and ψ Na-NO3-C2O4 had to be adjusted so that the predictions better matched the experimental data. The final optimized values were 0.02369 and 0.04069 respectively, which were placed into the HTWOS-Pitzer database (Appendix C). 5.2.5 Na-NO3-CO3-H2O The NaNO3-Na2CO3-H2O model includes the additional species NaOH and NaHCO3 due to the hydrolysis of the carbonate ion to form bicarbonate and hydroxide ions (Section 5.1.7). Experimental data at 10 and 24.2 °C was obtained from the compilation of Seidell 1958 and at 25 °C from OLI Systems MSE validation set (OLI Systems, Inc.) and is shown in Figure 5-41. Model predictions indicate that four solids precipitate at 24.2 and 25 °C: Na2CO3·10H2O, Na2CO3·7H2O, Na2CO3·H2O, and NaNO3. Initially, Na2CO3·10H2O is precipitated with increasing NaNO3 additions from 0 molal until about 4 molal. Solid Na2CO3·7H2O is predicted to precipitate from 4 molal until around 6 molal NaNO3. Na2CO3·H2O is predicted to precipitate from 6 molal until around 7 molal NaNO3, although the experimental data indicates that this solid was not found. Beyond 7 molal NaNO3, further addition of NaNO3 results in the predicted precipitation of solid NaNO3 up to the saturation point of 10.8 molal. At 10 °C, the model predicts that Na2CO3·10H2O is precipitated up to about 7.5 molal NaNO3, the predicted invariant point. Upon further additions of NaNO3, up to the saturation concentration of 9.5 molal, solid NaNO3 is predicted to precipitate. During the calculations, the mixing parameters, θ CO3-NO3 and ψ Na- CO3-NO3, were adjusted to better fit the experimental data. The final values obtained were -0.0891 and 0.0648 respectively (see Appendix C). Figure 5-41. Solubilities of NaNO3 and Na2CO3 in the NaNO3-Na2CO3-H2O System at 10, 24.2 and 25 °C. 4.5 Experimental Data taken from Seidell (1958) and OLI Systems Inc. (2011) 4 Exptl - 25 °C 3.5 Exptl - 24.2 °C y t Exptl - 10 °C i 3 l a l Model - 25 °C o 2.5 m Model - 24.2 °C ₃ O 2 Model - 10 °C C ₂ a 1.5 N 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 NaNO₃ molality 5.2.6 Na-NO2-OH-H2O The experimental data for the NaNO2-NaOH-H2O system was available at 20 and 25 °C only and came from the OLI Systems MSE validation set (OLI Systems, Inc.) where they were taken originally from Plekhotkin and Bobrovskaya. The data is displayed graphically in Figure 5-42 together with model predictions at both temperatures. The only mixing parameter required so that the predictions matched the experimental data was λ OH-NaNO2 with a value of 0.02 (TWRS-PP-94-090). The experimental data indicates that solid NaOH precipitates between 14 to 15 molal NaNO2 at both temperatures, but since this solid is not part of the HTWOS inventory, the model continues to predict NaNO2 as the solid phase and, hence, a deviation away from the experimental data can be seen. 39 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 48 of 77 RPP-RPT-50703 Rev. 1A Figure 5-42. Solubility of NaNO2 in the NaNO2-NaOH-H2O System at 20 and 25 °C. Experimental data: Plekhotkin and Bobrovskaya, (from OLI Systems, Inc.) 14 Exptl - 20 °C 12 Exptl - 25 °C y Model - 20 °C t i 10 l a l Model - 25 °C o 8 m ₂ O 6 N a 4 N 2 0 0 5 10 15 20 NaOH molality 5.2.7 Na-NO2-CO3-H2O For the system NaNO2-Na2CO3-H2O, experimental data exist at three temperatures; 20, and 25 °C data taken from the OLI Systems MSE validation set (OLI Systems, Inc.) and 23.1 °C data taken from the compilation of Seidell 1958. This data is plotted in Figure 5-43 together with model predictions for the solubilities at each temperature. At all three temperatures, the initial solid predicted was Na2CO3·10H2O, transitioning to Na2CO3·7H2O at NaNO2 concentrations in the range of 3 to 6 molal and Na2CO3·H2O at still higher concentrations of NaNO2 between 6 and 11 molal. At around 11 m NaNO2, the saturation concentration of NaNO2 was reached and that solid was precipitated. Figure 5-43. Solubilities of NaNO2 and Na2CO3 in the NaNO2-Na2CO3-H2O System at 20, 23.1 and 25 °C. Experimental Data from Seidell (1958) and OLI Systems Inc. (2008) 3.5 Exptl - 20 °C Exptl - 23.1 °C 3 Exptl - 25 °C Model - 20 °C y 2.5 t i l Model - 23.1 °C a l o 2 Model - 25 °C m ₃ O 1.5 C ₂ a N 1 0.5 0 0 2 4 6 8 10 12 14 NaNO₂ molality 5.2.8 Na-NO2-SO4-H2O Two sources of experimental data were obtained for the NaNO2-Na2SO4-H2O system; the compilation from OLI Systems MSE validation set (OLI Systems, Inc.) and that of Toghiani and colleagues (ORNL/TM-2000/317). The experimental data at three temperatures are plotted in Figure 5-44 together with model predictions of the solubility at each temperature. During calculations, the mixing parameters, θ NO2-SO4 and ψ Na-NO2-SO4 were adjusted to better fit the experimental data. The ternary mixing parameter, ψ Na-NO2-SO4, was found to be dependent on the inverse of temperature (i.e. the A and C 40 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 49 of 77 RPP-RPT-50703 Rev. 1A coefficients of Equation (2-35) were required). Toghiani’s experimental data was not used in the fitting of the mixing parameters. The final optimized values for the mixing parameters are listed in Appendix C. Figure 5-44. Solubilities of NaNO2 and Na2SO4 in the NaNO2-Na2SO4-H2O System at 0, 25 and 50 °C. 3.5 Experimental Data taken from Toghiani et al. (2008) and OLI Systems Inc. (2008) 3 08OLI - 0 °C Model - 0 °C 2.5 08OLI- 25 °C y t i l 08TOG - 25 °C a l o 2 Model - 25 °C M ₄ 08OLI - 50 °C O 1.5 S ₂ 08TOG - 50 °C a N 1 Model - 50 °C 0.5 0 0 2 4 6 8 10 12 14 16 NaNO₂ Molality 5.2.9 Na-OH-F-H2O Two sources of experimental data were obtained for the NaF-NaOH-H2O system; one taken from the compilation of Seidell at 0, 20, 40, 80, and 94 °C (Seidell 1958) and the second at 25 and 50 °C was taken from “Solubility in the Na+F+SO4 in Water and in Sodium Hydroxide Solutions” (Toghiani et al. 2005). Only the data from Seidell was used to evaluate the mixing parameters θ OH-F and ψ Na-OH-F as the data from Toghiani was more scattered. All the experimental data is plotted in Figure 5-45 together with solubility predictions from the model, at the same temperatures as Seidell’s data, using the optimized mixing parameter values (Appendix C). Figure 5-45. Solubility of NaF in the NaF-NaOH-H2O System from 0 to 94 °C. Experimental Data taken from Seidell, (1958) 0 °C and Toghiani et al. (2005) 20 °C 25 °C 1.4 40 °C 50 °C 1.2 80 °C 94 °C F a 1 Model - 0 °C N f Model - 20 °C o 0.8 Model - 40 °C y t i Model - 80 °C l 0.6 a l Model - 94 °C o m 0.4 0.2 0 0 1 2 3 4 5 6 7 molality of NaOH 5.2.10 Na-OH-C2O4-H2O The experimental data for the Na2C2O4-NaOH-H2O system was taken from Norris at 0, 30, 40 and 50 °C (Norris 1951) and is plotted in Figure 5-46. This data was used to optimize the mixing parameters θ OH- C2O4 and ψ Na-OH-C2O4. The resulting model predictions are shown in Figure 5-46. It can be seen that as the hydroxide concentration in solution increases, the oxalate concentration decreases, initially very 41 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 50 of 77 RPP-RPT-50703 Rev. 1A quickly, but then plateaus at very low levels between 2 to 10 molal hydroxide before slowly rising again at greater than 10 molal hydroxide concentration. Figure 5-46. Solubility of Na2C2O4 in the Na2C2O4-NaOH-H2O System from 0 to 50 °C. Experimental data from Norris, (1951) 0.4 Exptl - 0 °C Exptl - 30 °C 0.35 Exptl - 40 °C Exptl - 50 °C y 0.3 t i l Model 0 °C a l 0.25 o Model 30 °C m Model 40 °C ₄ 0.2 O Model 50 °C ₂ C 0.15 ₂ a N 0.1 0.05 0 0 2 4 6 8 10 12 NaOH molality 5.2.11 Na-OH-CO3-H2O For the NaOH-Na2CO3-H2O system, the extensive experimental data was taken from compilation of Seidell from 0 to 100 °C (Seidell 1958). The original data was listed at 5 °C intervals in the range 0 to 50 °C and then at 10 °C intervals up to 100 °C. For clarity, only the data at 20 °C intervals has been plotted in Figure 5-47, together with solubility predictions from the model. During the calculations, the mixing parameters θ OH-CO3 and ψ Na-OH-CO3 were adjusted to better fit the experimental data. The ternary mixing parameter, ψ Na-OH-CO3, was found to be dependent on the inverse of temperature. The optimized values for the coefficients of the mixing parameters are listed in Appendix C. At temperatures up to 30 °C, the initial solid predicted was Na2CO3·10H2O transitioning to the monohydrate, Na2CO3·H2O, at higher hydroxide concentrations. At temperatures above 35 °C, the only solid predicted was the monohydrate. Figure 5-47. Solubility of Na2CO3 in the Na2CO3-NaOH-H2O System from 0 to 100 °C. 5 Experimental Data taken from Seidell (1958) Exptl - 0 °C 4.5 Exptl - 20 °C 4 Exptl - 40 °C Exptl - 60 °C y 3.5 t i l Exptl - 80 °C a l 3 o Exptl - 100 °C m 2.5 Model - 0 °C ₃ O Model - 20 °C C ₂ 2 a Model - 40 °C N 1.5 Model - 60 °C 1 Model - 80 °C 0.5 Model - 100 °C 0 0 5 10 15 20 NaOH molality 42 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 51 of 77 RPP-RPT-50703 Rev. 1A 5.2.12 Na-OH-SO4-H2O Experimental data for the Na2SO4-NaOH-H2O system was taken from the compilation of Seidell at temperatures in the range from 0 to 100 °C (Seidell 1958). During calculations, the mixing parameters, θ OH-SO4 and ψ Na-OH-SO4, were adjusted to better fit the experimental data. The ternary mixing parameter, ψ Na-OH-SO4, was found to be dependent on the inverse of temperature. Data at 90 °C was used in the fitting of the mixing parameters also, but is not shown in Figure 5-48 for clarity. Figure 5-48 displays the experimental data (as symbols) plotted together with model predictions (as solid curves) and shows that the model predicts the experimental solubility data very well. Figure 5-48. Solubility of Na2SO4 in the Na2SO4-NaOH-H2O System from 0 to 100 °C. 3.5 Experimental Data taken from Seidell (1958) Exptl - 0 °C 3 Exptl - 25 °C Exptl - 50 °C 2.5 y Exptl - 70 °C t i l a Exptl - 100 °C l o 2 m Model - 0 °C ₄ O 1.5 Model - 25 °C S ₂ a Model - 50 °C N 1 Model - 70 °C 0.5 Model - 100 °C 0 0 2 4 6 8 10 12 14 NaOH molality At 0 and 25 °C, the solid predicted to form initially is Na2SO4·10H2O, but transitions to Na2SO4 upon further additions of sodium hydroxide. At 40 °C and up to 100 °C, the only solid predicted was Na2SO4, which is shown to be practically insoluble at very high hydroxide concentrations (greater than 11 molal). 5.2.13 Na-OH-PO4-H2O The experimental data for the Na3PO4-NaOH-H2O system was taken from the compilation of Seidell from 0 to 100 °C (Seidell 1958). The published data was in the form of Na2O and P2O5 weight percent, so was converted to equivalent NaOH, Na2HPO4 and Na3PO4 molal concentrations for comparison with model predictions. The experimental data exhibited many different solids, but for purposes of fitting the HTWOS model parameters, only data pertaining to solids Na3PO4. ¼NaOH.12H2O, Na3PO4.8H2O, Na3PO4·6H2O, Na2HPO4·12H2O, Na2HPO4·7H2O, and Na2HPO4·2H2O was included. During the calculations, the mixing parameters, θ OH-PO4 and ψ Na-OH-PO4, and θ HPO4-PO4 and ψ Na-HPO4-PO4 were adjusted to better fit the experimental data. The final values obtained were 0.06705 and -0.00682, and 0.06796 and -0.01928 respectively at 25 °C. Both the ψ -parameters where found to be temperature dependent, with ψ Na-OH-PO4 requiring a ‘C’ coefficient of 20.0725 whereas, ψ Na-HPO4-PO4 required ‘B’ and ‘C’ coefficients of 0.00398 and -405.128 respectively (Appendix C). The resulting model predictions, together with the experimental data, from 0 °C to 100 °C are shown in Figure 5-49. Generally, the model predicts the trend in the experimental data, but as the temperature increases, the accuracy of the HTWOS model decreases. The invariant points, which lie between the sodium to phosphate ratios of 2:1 and 3:1 are not predicted by the model with great accuracy (the experimental invariant point at 80 °C was not given in Seidell’s compilation which complicated the model fitting process). In addition, at 60 °C, the experimental data suggests that the solid Na3PO4·¼NaOH·12H2O precipitates between 0.5 and 2 molal phosphate, however the HTWOS model predicts the solids Na3PO4·6H2O and Na3PO4·8H2O precipitate in the same phosphate solution concentration range. 43 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 52 of 77 RPP-RPT-50703 Rev. 1A Although Weber and coworkers (1999) were able to fit the 60 °C solubility data at low phosphate concentrations (< 2 molal) with their model, it does appear that they did not account for the presence of other hydrated trisodium phosphate solids (namely the hexa- and octa-hydrates that are accounted for in the HTWOS model) . The HTWOS model does not predict the experimental data for the hydrogen phosphate solids either at temperatures greater than 40 °C. The model predictions are highly curvilinear from the Na/P = 2:1 line to the invariant points. This is probably caused by the magnitudes of the mixing parameters, θ HPO4-PO4 and ψ Na-HPO4-PO4, at the higher temperatures. Figure 5-49. Solubility of Na3PO4 in the Na3PO4-NaOH-H2O System from 0 to 100 °C. 5.2.14 Na-NO2-PO4-H2O The system Na3PO4-NaNO2- H2O has to include both the neutral species [NaNO2(aq)] and the hydrolysis of the phosphate ion. Experimental data at 25 °C was taken from the IUPAC compilation Solubility Data Series, Volume 31, Alkali Metal Phosphates, edited by Kertes (1988). During the model fitting analysis, the mixing parameters θ NO2-PO4 and Ψ Na- NO2-PO4 were adjusted to give the best fit, with final values of -0.03784 and 0.07283 respectively (see Appendix C). The results of the model predictions are shown in Figure 5-50, where the solid phase between 0 and 9 molal nitrite is predicted to be Na3PO4·¼NaOH.12H2O, between 9 and ~11.5 molal is predicted to be Na3PO4·6H2O, and finally NaNO2 is predicted to form between ~11.5 and 12.3 molal nitrite. As can be seen from the figure, addition of nitrite to the solution causes phosphate to precipitate due to the salting-out effect. 44 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 53 of 77 RPP-RPT-50703 Rev. 1A Figure 5-50 - Solubility of Na3PO4 and NaNO2 in the Na3PO4-NaNO2-H2O System at 25 °C 0.9 Experimental Data taken from IUPAC Solubility 0.8 Data Series (1988) 0.7 y t i 0.6 l a l o 0.5 m 4 O 0.4 Exptl - 25 °C P 3 a 0.3 Model - 25 °C N 0.2 0.1 0 0 2 4 6 8 10 12 14 NaNO2 molality 5.2.15 Na-SO4-PO4-H2O Experimental data for the Na2SO4-Na3PO4-H2O system was taken from the compilation of Kertes 1988. Only the data at 25 °C was analyzed as the remainder of the data was above the 100 °C upper temperature limit allowed by the HTWOS model. This system also requires the phosphate ion hydrolysis reaction to properly predict solubility behavior, so the data for the Na2SO4-Na2HPO4-H2O system at 25 °C was analyzed first to independently evaluate the mixing parameters θ SO4-HPO4 and Ψ Na-SO4- HPO4. During the model fitting process, the mixing parameters θ SO4-HPO4 and Ψ Na-SO4-HPO4 were adjusted to give the optimum fit to the experimental data (see Figure 5-51) with resulting values of -1.18119 and 0.49405 respectively (see Appendix C). Figure 5-51 shows that the model does not predict well the region from 0 to about 1.5 molal Na2SO4 where DSP is predicted to form. The prediction is concave whereas the experimental data is convex to the invariant point. An alternative fitting approach was tested whereby only the DSP data was used, but that model was unable to fit the invariant point or the region where Na2SO4·10H2O forms, from about 1.5 to 2 molal Na2SO4. Figure 5-51 - Solubility of Na2HPO4 in Na2SO4 in the Na2HPO4-Na2SO4-H2O System at 25 °C 1 Experimental Data taken from IUPAC Solubility 0.9 Data Series (1988) 0.8 y t 0.7 i l a l o 0.6 m 4 0.5 O Exptl - 25 °C P 0.4 H 2 a Model - 25 °C N 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Na2SO4 molality Using the new values of the mixing parameters for θ SO4-HPO4 and Ψ Na-SO4-HPO4, the data for the Na2SO4-Na3PO4-H2O system was evaluated. During the model fitting process, the mixing parameters θ SO4-PO4 and Ψ Na- SO4-PO4 were adjusted to give the optimum fit to the experimental data with 45 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 54 of 77 RPP-RPT-50703 Rev. 1A resulting values of 1.09208 and -0.30557 respectively (see Appendix C). The model prediction can be seen in Figure 5-52 where it can be seen that the solubility of TSP is reduced by increasing the concentration of sulfate ions in solution, up to the invariant point predicted by the model to be 1.89 molal Na2SO4. At this point, the model predicts the solids Na2SO4·10H2O and Na3PO4·¼NaOH·12H2O co- precipitate. Figure 5-52 - Solubility of Na3PO4 in Na2SO4 in the Na3PO4-Na2SO4-H2O System at 25 °C 0.9 Experimental Data taken from IUPAC Solubility 0.8 Data Series (1988) 0.7 y t i 0.6 l a l o 0.5 m 4 O 0.4 Exptl - 25 °C P 3 a 0.3 Model - 25 °C N 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Na2SO4 molality 5.2.16 Na-CO3-PO4-H2O The Na2CO3-Na3PO4-H2O system has to include bicarbonate, hydrogen phosphate and hydroxide ions as well as the obvious sodium, carbonate and phosphate ions due to the two hydrolysis reactions involved. The experimental data was taken from the compilation of Seidell 1958 for temperatures ranging from 0 to 100 °C. The mixing parameters, θ CO3-PO4 and ψ Na- CO3-PO4, were adjusted during the model fitting process to minimize the error between experimental and predicted saturation concentrations. The mixing parameter values that gave the best fit are 0.15952 and -0.06168 respectively at 25 °C. The ψ Na- CO3- PO4 parameter was found to be temperature dependent, with a C coefficient of 60.488 (see Appendix C). The results are shown in Figure 5-53 for both the experimental data and the model predictions. At 0 and 25 °C, the solids predicted to exist were Na3PO4·¼NaOH·12H2O and Na2CO3·10H2O; at 40 °C the solids predicted were Na3PO4·¼NaOH·12H2O and Na2CO3·H2O; at 60 °C the solids predicted were Na3PO4·8H2O and Na2CO3·H2O; at 80 and 100 °C the solids predicted were Na3PO4·6H2O and Na2CO3·H2O. 46 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 55 of 77 RPP-RPT-50703 Rev. 1A Figure 5-53. Solubilities of Na2CO3 and Na3PO4 in the Na2CO3-Na3PO4-H2O System from 0 to 100 °C. 6 Experimental Data taken from Seidell (1958) Exptl - 0 °C Exptl - 25 °C 5 Exptl - 40 °C Exptl - 60 °C Exptl - 80 °C y t i 4 l Exptl - 100 °C a l o Model - 0 °C m 3 Model - 25 °C ₄ O Model - 40 °C P ₃ a 2 Model - 60 °C N Model - 80 °C Model - 100 °C 1 0 0 1 2 3 4 5 Na₂CO₃ molality 5.2.17 Na-Cl-OH-H2O The NaCl-NaOH-H2O system was included because the evaluation of mixing between chloride and hydroxide ions was required before the aluminate system (Section 5.2.18) could be analyzed. Again, the experimental data was taken from the compilation of Seidell 1958 which ranged from 0 to 90 °C. The data is plotted in Figure 5-51 at 0, 60 and 90 °C together with the model predictions of solubility. The fit to the experimental data at the other temperatures was equally as good, but is not shown in Figure 5-54 for clarity. During the calculations, the mixing parameters, θ OH-Cl and ψ Na-OH-Cl, were adjusted to better fit the experimental data. The ternary mixing parameter, ψ Na-OH-Cl, was found to vary inversely with temperature. The final value obtained for θ was 0.02096, and the two coefficients, A and C, for ψ were found to be -0.01071 and 12.3654 respectively (see Appendix C). Figure 5-54. Solubility of NaCl in the NaCl-NaOH-H2O System from 0 to 90 °C. 6 Experimental Data taken from Seidell (1958) Exptl - 0 °C 5 Exptl - 60 °C Exptl - 90 °C y 4 t Model - 0 °C i l a l Model - 60 °C o 3 m Model - 90 °C l C a N 2 1 0 0 2 4 6 8 10 12 14 16 NaOH molality 5.2.18 Na-Cl-OH-Al-H2O The experimental data for the NaCl-NaOH-Al(OH)3-H2O system was taken from Wesolowski 1992 and ranged from 6 to 80 °C. Wesolowski used NaCl as an ionic strength adjuster so that the mixtures studied ranged from 0.01 to 5 molal ionic strengths. The Pitzer parameters, mixing parameters and solubility product constant expression were chosen using the recommendations of Wesolowski. These 47 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 56 of 77 RPP-RPT-50703 Rev. 1A recommendations were then used in the model to predict the solubilities of aluminate in sodium chloride- hydroxide mixtures at the same temperatures as the experimental data and are shown in Figure 5-55. Figure 5-55. Solubility of Al(OH)3 in the NaCl-NaOH-Al(OH)3-H2O System from 6 to 80 °C. 1.6 Experimental Data taken from Weslowski (1992) ] ) l A 1.4 - Exptl - 6 °C m / 1.2 Exptl - 25 °C H O Exptl - 50 °C - m 1 ( Exptl - 70 °C ₀ ₁ g Exptl - 80 °C o l 0.8 [ Model - 6 °C Q f 0.6 Model - 25 °C o Model - 50 °C m h 0.4 t i Model - 70 °C r a g 0.2 Model - 80 °C o l 0 0 1 2 3 4 5 6 Ionic Strength (molal) To independently check these parameters, experimental data was obtained from “Solubility and Density of Hydrated Aluminas in NaOH solutions” (Russell et al 1955). The model was used to predict the solubility of Al(OH)3 in NaOH solutions in the absence of NaCl at 40, 70, and 100 °C and is shown to fit the data very well in Figure 5-56. Figure 5-56. Solubility of Al(OH)3 in the NaOH-Al(OH)3-H2O System at 40, 70, and 100 °C. 10 Experimental Data taken from Russell et al., (1955) 9 Exptl - 40 °C 8 Exptl - 70 °C Exptl - 100 °C y t i 7 l Model - 40 °C a l o 6 Model - 70 °C m Model - 100 °C ₄ ) 5 H O ( 4 l A a 3 N 2 1 0 0 1 2 3 4 5 6 7 8 NaOH molality 5.3 DOUBLE SALT SYSTEMS The following systems are more complex than the previous systems analyzed as they contain solids that are a combination of the solutes mixed. These solids are known as double salts and four systems were analyzed. For these new solids, the reduced chemical potentials listed in ORNL/TM-2001/102 were used as initial values, which were then optimized so that model predictions matched the experimental data as necessary. At the same time, any additional mixing parameters (θ, ψ) were obtained simultaneously during this optimization exercise. 48 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 57 of 77 RPP-RPT-50703 Rev. 1A 5.3.1 Na-NO3-SO4-H2O The experimental data for the NaNO3-Na2SO4-H2O system was obtained from the compilation of Seidell 1958 in the temperature range from 0 to 100 °C. This data is plotted in Figure 5-54 and shows that the double salt Darapskite (NaNO3·Na2SO4·H2O) is predicted to form between 20 and 50 °C, but not at 0, 75 or 100 °C. The experimental data at 75 °C is not shown in Figure 5-57 for clarity, but was used with the rest of the data to determine the reduced chemical potential coefficients of the double salt. At the same time, the mixing parameters, θ NO3-SO4 and ψ Na-NO3-SO4, were adjusted to better fit the experimental data. Again, the ternary mixing parameter, ψ Na-NO3-SO4, was found to be inversely proportional to the temperature. The optimized coefficients for the double salt and the mixing parameters are listed in Appendix A and C respectively. Figure 5-57. Solubilities of NaNO3 and Na2SO4 in the NaNO3-Na2SO4-H2O System from 0 to 100 °C. 4 Experimental Data taken from Seidell (1958) Exptl - 0 °C 3.5 Exptl - 20 °C Exptl - 25 °C 3 Exptl - 35 °C y t Exptl - 50 °C i l a 2.5 l Exptl - 100 °C o Model - 0 °C m 2 ₄ Model - 20 °C O S ₂ 1.5 Model - 25 °C a N Model - 35 °C 1 Model - 50 °C Model - 100 °C 0.5 0 0 2 4 6 8 10 12 14 16 18 20 22 NaNO₃ molality 5.3.2 Na-F-SO4-H2O The experimental data for the NaF-Na2SO4-H2O system was taken from three sources; the compilation of Seidell 1958 at temperatures of 10, 15, 25, and 35 °C, Toghiani and co-workers at 25 and 50 °C (Toghiani et al. 2005) and the MSE validation set from OLI Systems at 0 and 80 °C (OLI Systems, Inc.). This data is plotted in Figure 5-58 as symbols together with model predictions as solid curves. The double salt, Kogarkoite (Na3FSO4), was predicted to exist at 15 °C and above at intermediate concentrations of fluoride in solution. As can be seen in the plot, the double salt solubility is fairly insensitive to temperature as all the curves lay very close to each other. During the calculations, the coefficients for the reduced chemical potential of the double salt were adjusted to optimize the fit to the experimental data. The final values of the coefficients are listed in Appendix A. 49 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 58 of 77 RPP-RPT-50703 Rev. 1A Figure 5-58. Solubilities of NaF and Na2SO4 in the NaF-Na2SO4-H2O System from 0 to 80 °C. 4 Experimental Data taken from Seidell (1958) Exptl - 0 °C 3.5 and Toghiani et al. (2005) Exptl - 10 °C Exptl - 15 °C 3 Exptl - 25 °C y Exptl - 25 °C t i l Exptl - 35 °C a 2.5 l Exptl - 50 °C o m Exptl - 80 °C 2 ₄ Model - 0 °C O S Model - 10 °C ₂ 1.5 a Model - 15 °C N 1 Model - 25 °C Model - 35 °C 0.5 Model - 50 °C Model - 80 °C 0 0 0.2 0.4 0.6 0.8 1 1.2 NaF molality 5.3.3 Na-F-PO4-H2O The experimental data for the NaF-Na3PO4-H2O system was taken from the OLI Systems MSE validation set (OLI Systems, Inc.). The data was recorded at temperatures of 25, 35 and 50 °C only. The data is displayed in Figure 5-59 as symbols together with model predictions as the solid curves. The double salt, Natrophosphate (Na7F(PO4)2·19H2O), was predicted to exist at all three temperatures from extremely low concentrations of fluoride in solution with the effect of rapidly lowering the phosphate concentration in solution initially. Figure 5-59. Solubilities of NaF and Na3PO4 in the NaF-Na3PO4-H2O System from 25 to 50 °C. 1.2 Experimental Data taken from OLI Systems Inc (2008) 1 y t i l 0.8 a Exptl - 25 °C l o Exptl - 35 °C m 0.6 ₄ Exptl - 50 °C O P ₃ Model - 25 °C a 0.4 N Model - 35 °C Model - 50 °C 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 NaF molality During calculations, the mixing parameter θ F-PO4 was adjusted to better fit the experimental data with a final value of 0.9 obtained (see Appendix C). The chemical potential coefficients for the double salt were the original values of Weber (ORNL/TM-2001/102 and 109) and are listed in Appendix A. 5.3.4 Na-CO3-HCO3-H2O The experimental data for this system was taken from the compilation of Seidell and ranges from 0 °C to 100 °C (Seidell 1958). The experimental data is displayed in Figure 5-60 as symbols and the model predictions as the solid curves. This system required the addition of the bicarbonate and hydroxide ions 50 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 59 of 77 RPP-RPT-50703 Rev. 1A due to the hydrolysis reaction of the carbonate ion. The mixing parameters for θ OH-CO3 and Ψ Na-OH- HCO3 were obtained in section 5.2.11, whereas the mixing parameters θ HCO3-CO3 and Ψ Na-HCO3- HCO3 were obtained during the model fitting process. A value of -0.07045 was obtained for θ HCO3- CO3, whereas Ψ Na-HCO3-HCO3 varied inversely with temperature leading to values of coefficient ‘A’ of 0.01994 and ‘C’ of -30.6057 (Appendix C). At temperatures above 25 °C, the double salt Trona (Na2CO3·NaHCO3·2H2O) is predicted to form at Na2CO3 concentrations greater than 2 molal. At 100 °C, the additional double salt Wegscheiderite (Na2CO3·2NaHCO3) precipitates from about 1.5 molal carbonate to about 3 molal carbonate, before changing to Trona. The near-vertical lines are the single salts of sodium bicarbonate solids. Figure 5-60. Solubilities of Na2CO3 and NaHCO3 in the Na2CO3-NaHCO3-H2O System from 0 to 100 °C 3 Experimental Data taken from Seidell (1958) Exptl 0 °C Exptl 25 °C 2.5 Exptl 50 °C Exptl 75 °C y Exptl 100 °C t i l 2 Model 0 °C a l o Model 25 °C m 3 1.5 Model 50 °C O C Model 75 °C H a 1 Model 100 °C N 0.5 0 0 1 2 3 4 5 Na2CO3 molality 5.3.5 Na-CO3-SO4-H2O The experimental data for this system was taken from the compilation of Seidell at temperatures that range from 15 to 100 °C (Seidell 1958). In this system, the double salt Burkeite (Na2CO3·2Na2SO4) is known to form above 25 °C and, although this double salt was not required for the HTWOS model, it is included so that the mixing parameters (θ and Ψ) could be evaluated up to 100 °C. The data extracted from Seidell is shown in Figure 5-56 as symbols together with model predictions as the solid curves. Only results at 15, 25, 70 and 100 °C are shown for clarity, with similar fits at the other temperatures. This system required the addition of the bicarbonate and hydroxide ions because of the hydrolysis reaction of the carbonate ion. The mixing parameter values for θ CO3-SO4, θ HCO3-SO4 and ψ Na-CO3- SO4, ψ Na-HCO3-SO4 were taken from Weber (ORNL/TM-2001/102 and 109) whilst those for the HCO3-CO3, OH-CO3 and OH-SO4 interactions were those calculated earlier in Sections 5.2.11 and 5.2.12 respectively. The coefficients for all these mixing parameters are listed in Appendix C. In Figure 5-61, the horizontal sections of the curves, at 15 and 25 °C, are the predictions for solid Na2SO4·10H2O solubility whilst the vertical sections of the curves are the predictions for solid Na2CO3·10H2O solubility. At temperature above 25 °C, the double salt Burkeite is predicted to precipitate from about 3 to 4 molal carbonate at 35 °C to about 0.5 to 4 molal carbonate at 100 °C. Although Burkeite is known to be a solid solution, the model predictions fit the experimental data very well, with an overall standard error of fit of 0.124, by assuming it is a pure phase. 51 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 60 of 77 RPP-RPT-50703 Rev. 1A Figure 5-61 - Solubilities of Na2CO3 and Na2SO4 in the Na2CO3-Na2SO4-H2O System from 15 to 100 °C. 5.4 ADDITIONAL SALT SYSTEMS This section details the modeling of solids not covered by the models of Weber, Steele and co-workers, or Toghiani and colleagues used as the starting point for the HTWOS model generation. Two solids of great importance for Hanford waste that have not been mentioned so far are dawsonite (NaAlCO3(OH)2) and boehmite (AlOOH), both of which contain aluminum. These solids have been included in the HTWOS model database for completeness, but only dawsonite is currently enabled in the Gibbs energy minimization routine in GEMS. At present, boehmite cannot be predicted to form. The reason for excluding boehmite is that it is more stable than gibbsite, thermodynamically, and the model would predict it to form instead of gibbsite. However, gibbsite is known to be the dominant aluminum solid in the majority of Hanford waste, so including boehmite in the HTWOS thermodynamic model would give incorrect predictions in the absence of any kinetic mechanism to control its formation. 5.4.1 NaAlCO3(OH)2 (Dawsonite) The solubility product constant for dawsonite as a function of temperature was taken from “Dawsonite Synthesis and Reevaluation of its Thermodynamic Properties from Solubility Measurements: Implications for Mineral Trapping of CO2” (Benezeth et al. 2007). Their expression is valid in the temperature range from 0 to 300 °C and represents the following chemical equilibrium: ( ) ( ) + 2 ⇄ ( ) + + + (5-13) From Benezeth, the expression for the solubility product is given by: 6510.1 log = 8.797 − − 0.01625 (5-14) From the thermodynamic definition of the equilibrium constant: −ΔG + 2 − ( ) − − − ln = = (5-15) the reduced ideal chemical potential (μº/RT) for dawsonite can be found. The coefficients for the temperature dependence of the reduced chemical potential can be found by regressing against the temperature functions of Equation (2-35) at different temperatures in the range 0 to 100 °C. The relevant information for this calculation is given in Table 5-13. 52 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 61 of 77 RPP-RPT-50703 Rev. 1A Table 5-13. Reduced Ideal Chemical Potentials for Dawsonite in the Temperature Range 0 to 100 °C. Reduced Chemical Potentials T (°C) T(K) T - To 1/To - 1/T ln T/To T² - To² log Ks4 ln Ks4 Al(OH)4- HCO3- Na+ H2O Dawsonite 0 273.15 -25 -0.000307 -0.0875756 -14282.5 -19.475112 -44.843103 -574.3536 -261.84937 -114.413196 -106.23287 -782.993482 5 278.15 -20 -0.0002412 -0.0694362 -11526 -19.127935 -44.043698 -563.0933 -256.62201 -112.58823 -103.96784 -768.411515 10 283.15 -15 -0.0001777 -0.0516199 -8719.5 -18.795888 -43.279132 -552.1027 -251.51819 -110.807427 -101.782111 -754.143185 15 288.15 -10 -0.0001164 -0.0341155 -5863 -18.478184 -42.547592 -541.3818 -246.53791 -109.070788 -99.6720171 -740.19401 20 293.15 -5 -5.721E-05 -0.0169123 -2956.5 -18.17409 -41.847388 -530.9305 -241.68118 -107.378312 -97.6340803 -726.569255 25 298.15 0 0 0 0 -17.88292 -41.176945 -520.749 -236.948 -105.73 -95.665 -713.273945 30 303.15 5 5.5319E-05 0.01663102 3006.5 -17.604035 -40.534788 -510.8372 -232.33836 -104.125851 -93.7616401 -700.31288 35 308.15 10 0.00010884 0.03298996 6063 -17.336837 -39.919542 -501.195 -227.85227 -102.565866 -91.9210185 -687.690647 40 313.15 15 0.00016066 0.04908559 9169.5 -17.080766 -39.329918 -491.8225 -223.48972 -101.050044 -90.140297 -675.411635 45 318.15 20 0.00021084 0.06492625 12326 -16.835298 -38.764706 -482.7198 -219.25071 -99.578386 -88.4167721 -663.48004 50 323.15 25 0.00025948 0.0805199 15532.5 -16.59994 -38.222775 -473.8867 -215.13525 -98.1508913 -86.7478673 -651.899884 55 328.15 30 0.00030663 0.09587411 18789 -16.374231 -37.70306 -465.3233 -211.14334 -96.76756 -85.1311242 -640.67502 60 333.15 35 0.00035237 0.11099612 22095.5 -16.157735 -37.20456 -457.0296 -207.27497 -95.4283923 -83.5641963 -629.809138 65 338.15 40 0.00039675 0.12589287 25452 -15.950045 -36.726335 -449.0056 -203.53015 -94.133388 -82.0448419 -619.305783 70 343.15 45 0.00043984 0.14057095 28858.5 -15.750774 -36.267498 -441.2513 -199.90887 -92.8825473 -80.5709181 -609.168354 75 348.15 50 0.00048169 0.15503671 32315 -15.559561 -35.827214 -433.7666 -196.41113 -91.6758701 -79.1403746 -599.400115 80 353.15 55 0.00052236 0.16929618 35821.5 -15.376064 -35.404695 -426.5517 -193.03694 -90.5133563 -77.7512494 -590.004204 85 358.15 60 0.00056189 0.18335518 39378 -15.199958 -34.999197 -419.6065 -189.7863 -89.3950061 -76.4016628 -580.983634 90 363.15 65 0.00060033 0.19721926 42984.5 -15.03094 -34.610017 -412.9309 -186.6592 -88.3208193 -75.0898136 -572.341307 95 368.15 70 0.00063773 0.21089375 46641 -14.868719 -34.23649 -406.525 -183.65565 -87.2907961 -73.8139741 -564.080011 100 373.15 75 0.00067413 0.22438377 50347.5 -14.713023 -33.877987 -400.3888 -180.77564 -86.3049363 -72.572487 -556.202432 Regression of the data given in Table 5-13 lead to the following equation for the reduced ideal chemical potential for dawsonite: = −713.2739 + 10.4537( − ) + 14990.1(1 − 1 ) − 648.08 ln − 0.0097( − ) (5-16) Where T is the temperature in Kelvin, R is the universal gas constant and T0 is the reference temperature of 298.15 K. These coefficient values are listed in Appendix A. As yet, the model has not been used to predict the formation of dawsonite using GEMS. 5.4.2 AlOOH (Boehmite) Using a similar procedure to that described for dawsonite in Section 5.4.1, the coefficients for boehmite were evaluated using the solubility product constant values of Apps and Neil in the temperature range of 0 to 100 °C from “Solubilities of Aluminum Hydroxides and Oxyhydroxides in Alkaline Solutions” (Apps and Neil 1990). The relevant information for regression is given in Table 5-14. Table 5-14. Reduced Ideal Chemical Potentials for Boehmite in the Temperature Range 0 to 100 °C. Reduced Chemical Potentials T (°C) T (K) T - To 1/To - 1/T ln(T/To) T² - To² log Ks4 ln Ks4 Al(OH)4- OH- H2O AlOOH 0 273.15 -25 -0.000307 -0.0875756 -14282.5 -1.462 -3.3663794 -574.35355 -71.768186 -106.23287 -399.71887 25 298.15 0 0 0 0 -1.24 -2.8552055 -520.749 -63.534 -95.665 -364.40521 50 323.15 25 0.00025948 0.0805199 15532.5 -1.022 -2.353242 -473.8867 -56.233414 -86.747867 -333.25866 60 333.15 35 0.00035237 0.11099612 22095.5 -0.937 -2.1575222 -457.02961 -53.574587 -83.564196 -322.04835 100 373.15 75 0.00067413 0.22438377 50347.5 -0.623 -1.4345105 -400.38885 -44.433041 -72.572487 -284.81783 Regression of the data given in Table 5-14 lead to the following equation for the reduced ideal chemical potential for boehmite: = −364.4052 + 6.0996( − ) + 16635.9(1 − 1 ) − 497.63 ln − 0.0055( − ) (5-17) Where, as before, T is the temperature in Kelvin, R is the universal gas constant and T0 is the reference temperature of 298.15 K. These coefficient values are listed in Appendix A. Again, the model has not been used to predict the formation of boehmite using GEMS. 53 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 62 of 77 RPP-RPT-50703 Rev. 1A 6 CONCLUSIONS A thermodynamic model for the Na-NO3-NO2-OH-F-CO3-HCO3-SO4-PO4-HPO4-C2O4-Al(OH)4-H2O system from 0 to 100 °C has been developed, based on Pitzer’s multi-component ion interaction model, which can be used to predict the solubility of important Hanford waste solids to high ionic strength within experimental accuracy. The model has been parameterized for all the constituent binary solutes and the majority of ternary systems and thus, in principle, can be applied to more complex combinations of the constituent solutes. However, because at higher temperatures, there is a lack of data for some ternary mixtures, predictions from the model should be used cautiously. Binary systems where the data is limited for temperatures other than 25 °C include NaF-H2O, Na2CO3- H2O, Na2C2O4-H2O, Na2HPO4-H2O and Na3PO4-H2O. Ternary systems where the data is limited in terms of temperature range include NaNO3-NaF-H2O (25 – 50 °C), NaNO3-Na2C2O4-H2O (20 – 75 °C), NaNO3-Na2CO3-H2O (10 – 25 °C), NaNO2-NaOH-H2O (20 – 25 °C), NaNO2-Na2CO3-H2O (20 – 25 °C), NaNO2-Na2SO4-H2O (0 – 50 °C), NaOH-Na2C2O4-H2O (0 – 70 °C), NaNO2-Na3PO4-H2O (25 °C) , Na2SO4-Na3PO4-H2O (25 °C) , and NaF-Na3PO4-H2O (25 – 50 °C). For the ternary systems, NaNO3- NaAl(OH)4-H2O and NaNO2-NaAl(OH)4-H2O , no solubility data was found in the open literature even though mixing parameters were found in Weber (ORNL/TM-2001/102). A Microsoft Excel workbook, known as GEMS, comprising the Pitzer thermodynamic model and a Gibbs energy minimization routine has been developed to aid testing and verification before incorporation into HTWOS. The workbook contains all of the model parameter coefficients obtained during development that is described in detail in Section 5. Complementary to the work reported here, work to translate this model into HTWOS is progressing. Simultaneously, a version of GEMS has been developed in PTC Mathcad3 to develop a method to solve the Gibbs energy minimization problem since HTWOS does not have an Excel Solver routine available. As part of the validation process, once the Pitzer equations, HTWOS-Pitzer database and Gibbs energy minimization solver have been installed in HTWOS, speciation predictions will be compared to analytical results of the complex waste stored in several Hanford Tanks as discussed in “Plan for Evaluation of the HTWOS Integrated Solubility Model Predictions” (RPP-51192). 3 Mathcad is a registered trademark of PTC, Needham, Massachusetts. 54 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 63 of 77 RPP-RPT-50703 Rev. 1A 7 REFERENCES Åkerlöf, G., and Kegeles, G., 1940, “Thermodynamics of Concentrated Aqueous Solutions of Sodium Hydroxide”, Journal of the American Chemical Society, Vol. 62, Issue 3. 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Zhou, J., Qi-Yuan, C., Zhou, Y, and Yin, Z-L., 2003, “A New Isopiestic Apparatus for the Determination of Osmotic Coefficients”, The Journal of Chemical Thermodynamics, Vol. 35. 59 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 68 of 77 RPP-RPT-50703 Rev. 1A APPENDIX A COEFFICIENTS FOR GIBBS ENERGY OF FORMATION 60 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 69 of 77 RPP-RPT-50703 Rev. 1A APPENDIX A – COEFFICIENTS FOR GIBBS ENERGY OF FORMATION Table A.1. Coefficients for Gibbs Energy of Formation μ₀/RT = A + B (T - T₀) + C (1/T₀ - 1/T) + D ln(T/T₀) + E (T² - T₀²) Species A B C D E Ar 0 0 0 0 0 H₂O -95.665 -1.0029 0 324.04 0.000508 Na+ -105.73 0.85194 0 0 -0.000883 H+ 0 0 0 0 0 - NO3 -43.984 0.68002 0 0 -0.000675 OH- -63.534 0.75606 0 0 -0.000747 Cl- -52.928 0.367999 0 0 -0.000358 F- -112.59 1.1322 0 0 -0.001143 -3 PO4 -411.192 4.33069 0 0 -0.004362 -2 HPO4 -439.592 4.74018 0 0 -0.004998 - NO2 -12.931 0 7624.179 16.5833 0 - Al(OH)4 -520.749 5.22566 0 0 -0.005394 -2 CO3 -213.14 2.28402 0 0 -0.002298 - HCO3 -236.948 2.40768 0 0 -0.002471 -2 SO4 -300.531 3.11291 0 0 -0.003155 -2 C2O4 -272.165 2.782581 0 0 -0.002792 Al(OH)₃ -459.8459 4.437403 -5468.36 37.02324 -0.004647 NaNO₃(aq) -148.5809 1.546306 421.4583 -4.69261 -0.001558 NaNO₃ -147.1552 -0.32363 -43090.9 515.8652 -0.000487 NaNO₂(aq) -116.4318 0.622791 0 91.20799 -0.000756 NaNO₂ -114.7734 -0.633181 -22669.2 410.5574 0 Na₂CO₃·H₂O -519.2349 2.707537 0 443.1349 -0.003707 Na₂CO₃·7H₂O -1095.284 -0.305245 0 1862.046 -0.002667 Na₂CO₃·10H₂O -1383.078 -3.021157 0 2800.334 -0.001406 NaCl -155.0132 1.019301 0 39.13024 -0.001117 NaF -219.391 2.022907 0 0 -0.002087 Na₃FSO₄ -734.908 6.876225 4528.925 19.03678 -0.00723 Na₃PO₄·0.25NaOH·12H₂O -1926.967 97.86802 3111025 -27143.2 -0.057377 Na₂HPO₄·12H₂O -1803.293 -4.33176 148507.1 3047.789 -0.000662 Na₇F(PO₄)₂·19H₂O -3512.445 36.15256 0 0 -0.03774 Na₂C₂O₄ -489.4015 5.041464 37072.55 -247.939 -0.004768 Na₂SO₄ -512.67937 5.05620714 -0.861425 -32.036813 -0.0051488 - Na₂SO₄·10H₂O -1471.4337 -2.63592218 29.81119632 2907.249632 0.002107565 Na₃SO₄NO₃·H₂O -756.32266 24.5571748 678946.664 -6002.3363 -0.0153931 Na₃SO₄NO₃·2H₂O -852.8336 7.083058 111461.4 -301.954 -0.006786 Na₃PO₄·6H₂O -1308.091 76.46452 2095321 -19860 -0.04745 Na₃PO₄·8H₂O -1499.161 35.48944 0 -3324.01 -0.031149 NaAlCO₃(OH)₂ -713.2739 10.45366 14990.06 -648.08 -0.009765 Na₂HPO₄·2H₂O -844.1315 26.25363 795747.9 -6578.85 -0.016687 Na₂HPO₄·7H₂O -1323.937 -0.57624 3440.016 2268.28 -0.003205 NaHCO₃ -343.561 4.737928 28198.59 -347.771 -0.004361 AlOOH -364.4052 6.099599 16635.87 -497.626 -0.00551 Na₂CO₃·NaHCO₃·2H₂O -960.82817 5.78781702 0 648.08 -0.0072018 Na₂CO₃·3NaHCO₃ -1306.6421 56.0926586 -2328162.2 99.376742 -0.0493835 Na₂CO₃·2Na₂SO₄ -1449.9997 14.6377275 0 -0.0003172 -0.0155139 61 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 70 of 77 RPP-RPT-50703 Rev. 1A APPENDIX B BINARY PITZER PARAMETERS 62 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 71 of 77 RPP-RPT-50703 Rev. 1A APPENDIX B – BINARY PITZER PARAMETERS Table B.1. Binary Pitzer Parameters P(T) = A + B (T - T₀) + C (1/T₀- 1/T) + D ln(T/T₀) Label Cation Anion Param A B C D Temp. Na-NO3-B0 Na+ NO3- β⁰ 0.028327 -0.01831 -1406.73 10.51503 0-100 Na-NO3-B1 Na+ NO3- β¹ 0.330682 0.004124 0 0 Na-NO3-C Na+ NO3- C 0 0 0 0 Na-OH-B0 Na+ OH- β⁰ 0.091763 -0.07308 -7118.75 45.78472 0-100 Na-OH-B1 Na+ OH- β¹ 0.212694 0.414149 46598 -279.171 Na-OH-C Na+ OH- C 0.001748 0.002725 267.9999 -1.7259 Na-Cl-B0 Na+ Cl- β⁰ 0.075318 0.006943 1042.211 -5.34011 0-100 Na-Cl-B1 Na+ Cl- β¹ 0.276964 0.016939 1843.033 -10.9493 Na-Cl-C Na+ Cl- C 0.000703 -0.00052 -72.0357 0.379806 Na-F-B0 Na+ F- β⁰ 0.033 0 246.8 -0.6728 0-100 Na-F-B1 Na+ F- β¹ 0.2456 0 2833 -9.451 Na-F-C Na+ F- C 0.00281 0 12.25 -0.0436 Na-PO4-B0 Na+ PO4-3 β⁰ 0.146652 0 5839.131 -17.765 0-100 Na-PO4-B1 Na+ PO4-3 β¹ 4.614956 0 -27717.8 88.28649 Na-PO4-C Na+ PO4-3 C 0 0 0 0 Na-HPO4-B0 Na+ HPO4-2 β⁰ -0.002305 0 -217.774 0.778228 0-100 Na-HPO4-B1 Na+ HPO4-2 β¹ 0.964385 0 -3325.85 14.23582 Na-HPO4-C Na+ HPO4-2 C 0.000101 0 6.474899 -0.0223 Na-NO2-B0 Na+ NO2- β⁰ 0.072382 0.073633 8977.806 -51.5162 0-100 Na-NO2-B1 Na+ NO2- β¹ 0.16036 -0.25287 -34241.4 187.8694 Na-NO2-C Na+ NO2- C -0.001711 -0.00465 -560.099 3.246429 Na-Al(OH)4-B0 Na+ Al(OH)4- β⁰ 0.057093 -0.07308 -7118.75 45.78472 0-100 Na-Al(OH)4-B1 Na+ Al(OH)4- β¹ 0.212694 0.414149 46598 -279.171 Na-Al(OH)4-C Na+ Al(OH)4- C -0.000856 0.002725 267.9999 -1.7259 Na-CO3-B0 Na+ CO3-2 β⁰ 0.03623 -0.0233 -1108.38 11.19856 0-90 Na-CO3-B1 Na+ CO3-2 β¹ 1.50975 -0.09784 -4412.51 44.58207 Na-CO3-C Na+ CO3-2 C 0.00184 0 0 0 Na-HCO3-B0 Na+ HCO3- β⁰ 0.028 -0.01446 -682.886 6.899586 0-90 Na-HCO3-B1 Na+ HCO3- β¹ 0.044 -0.02447 -1129.39 11.41086 Na-HCO3-C Na+ HCO3- C 0 0 0 0 Na-SO4-B0 Na+ SO4-2 β⁰ 0.017271 0 758.5359 -2.00966 0-100 Na-SO4-B1 Na+ SO4-2 β¹ 1.147943 0 -6174.03 22.67425 Na-SO4-C Na+ SO4-2 C 0.001957 0 -34.5971 0.085673 Na-C2O4-B0 Na+ C2O4-2 β⁰ 0.3249 0 17.36 0-110 Na-C2O4-B1 Na+ C2O4-2 β¹ -0.0288 0 0.1478 Na-C2O4-C Na+ C2O4-2 C -0.0483 0 4.58 63 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 72 of 77 RPP-RPT-50703 Rev. 1A APPENDIX C TERNARY PITZER PARAMETERS 64 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 73 of 77 RPP-RPT-50703 Rev. 1A APPENDIX C – TERNARY PITZER PARAMETERS Table C.1. Ternary Pitzer Parameters (2 sheets) P(T) = A + B (T - T₀) + C (1/T₀ - 1/T) Label Ion-i Ion-j Ion-k Parameter A B C Temp. NO3-OH NO3- OH- θ -0.092138 NO3-Cl NO3- Cl- θ 0.016 NO3-F NO3- F- θ 0.081 NO3-Al(OH)4 NO3- Al(OH)4- θ -0.020963 NO3-SO4 NO3- SO4-2 θ 0.083466 NO3-C2O4 NO3- C2O4-2 θ 0.016578 20-75 NO3-CO3 NO3- CO3-2 θ -0.089054 OH-Cl OH- Cl- θ -0.020963 OH-F OH- F- θ 0.074501 OH-PO4 OH- PO4-3 θ 0.067052 OH-CO3 OH- CO3-2 θ 0.021381 OH-SO4 OH- SO4-2 θ -0.06578 OH-Al(OH)4 OH- Al(OH)4- θ 0.013833 OH-C2O4 OH- C2O4-2 θ -0.003262 0-70 NO2-Al(OH)4 NO2- Al(OH)4- θ 0.00197 NO2-CO3 NO2- CO3-2 θ 0.10961 NO2-SO4 NO2- SO4-2 θ 0.010879 0-50 Cl-Al(OH)4 Cl- Al(OH)4- θ -0.020963 Cl-F Cl- F- θ -0.01 Cl-PO4 Cl- PO4-3 θ 0.2559 Cl-CO3 Cl- CO3-2 θ -0.053 Cl-HCO3 Cl- HCO3-2 θ 0.036 Cl-SO4 Cl- SO4-2 θ 0.03 F-CO3 F- CO3-2 θ 0 F-HPO4 F- HPO4-2 θ 0 F-PO4 F- PO4-3 θ 0.9 PO4-HPO4 PO4-3 HPO4-2 θ 0.06796 PO4-NO2 PO4-3 NO2- θ -0.037844 PO4-SO4 PO4-3 SO4-2 θ 1.092079 CO3-HCO3 CO3-2 HCO3- θ -0.070453 CO3-SO4 CO3-2 SO4-2 θ -0.008119 CO3-PO4 CO3-2 PO4-3 θ 0.097406 HCO3-SO4 HCO3- SO4-2 θ -0.027841 HPO4-SO4 HPO4- SO4-2 θ -1.18119 Na-NO3-OH Na+ NO3- OH- ψ 0.003629 Na-NO3-Cl Na+ NO3- Cl- ψ -0.006 Na-NO3-F Na+ NO3- F- ψ 0 Na-NO3-Al(OH)4 Na+ NO3- Al(OH)4- ψ -0.010714 12.36544 Na-NO3-SO4 Na+ NO3- SO4-2 ψ 0.012309 -26.3572 Na-NO3-C2O4 Na+ NO3- C2O4-2 ψ 0.042081 20-75 Na-NO3-CO3 Na+ NO3- CO3-2 ψ 0.064766 Na-OH-Cl Na+ OH- Cl- ψ -0.010714 12.36544 0-90 65 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 74 of 77 RPP-RPT-50703 Rev. 1A Label Ion-i Ion-j Ion-k Parameter A B C Temp. Na-OH-F Na+ OH- F- ψ -0.041229 Na-OH-PO4 Na+ OH- PO4-3 ψ -0.006823 20.07246 Na-OH-CO3 Na+ OH- CO3-2 ψ -0.005314 12.93982 Na-OH-SO4 Na+ OH- SO4-2 ψ 0.000798 19.89553 Na-OH-Al(OH)4 Na+ OH- Al(OH)4- ψ -0.004669 Na-OH-C2O4 Na+ OH- C2O4-2 ψ 0.017838 0-70 Na-Cl-Al(OH)4 Na+ Cl- Al(OH)4- ψ -0.010714 12.36544 Na-Cl-F Na+ Cl- F- ψ -0.00218 Na-Cl-PO4 Na+ Cl- PO4-3 ψ 0 Na-Cl-CO3 Na+ Cl- CO3-2 ψ 0.0085 Na-Cl-HCO3 Na+ Cl- HCO3- ψ -0.015 Na-Cl-SO4 Na+ Cl- SO4-2 ψ 0 Na-F-PO4 Na+ F- PO4-3 ψ 0 Na-F-HPO4 Na+ F- HPO4-2 ψ 0 Na-PO4-HPO4 Na+ PO4-3 HPO4-2 ψ -0.019277 0.003975 -405.128 Na-PO4-NO2 Na+ PO4-3 NO2- ψ 0.072827 Na-PO4-SO4 Na+ PO4-3 SO4-2 ψ -0.305574 Na-CO3-HCO3 Na+ CO3-2 HCO3- ψ 0.019942 -30.6057 0-100 Na-CO3-SO4 Na+ CO3-2 SO4-2 ψ -3.48E-05 Na-CO3-PO4 Na+ CO3-2 PO4-3 ψ -0.016412 -0.15727 Na-HCO3-SO4 Na+ HCO3- SO4-2 ψ 0.010681 Na-HPO4-SO4 Na+ HPO4-2 SO4-2 ψ 0.494049 Na-NO2-Al(OH)4 Na+ NO2- Al(OH)4- ψ 0.0054 Na-NO2-C2O4 Na+ NO2- C2O4-2 ψ 0.23 Na-NO2-CO3 Na+ NO2- CO3-2 ψ -0.001604 Na-NO2-SO4 Na+ NO2- SO4-2 ψ 0.019572 -32.6211 0-50 OH-NaNO2 OH- NaNO2 λ 0.02 20-25 OH-NaNO3 OH- NaNO3 λ 0.114519 -0.0011 10-25 CO3-NaNO2 CO3-2 NaNO2 λ 0.033166 NO2-NaNO2 NO2- NaNO2 λ 0.012787 -0.00162 20-25 Na-NO2-NaNO2 Na+ NO2- NaNO2 ζ 0.00151 0.000168 Na-OH-NaNO3 Na+ OH- NaNO3 ζ -0.00818 8.53E-05 10-25 66 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 75 of 77 RPP-RPT-50703 Rev. 1A APPENDIX D VBA CODE LISTING FOR EXCEL FUNCTION ETHETA 67 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 76 of 77 RPP-RPT-50703 Rev. 1A APPENDIX D – VBA CODE LISTING FOR EXCEL FUNCTION ETHETA VBA code listing for calculating higher order unsymmetrical mixing parameters: Eθ and I×Eθ′. Dim akI As Variant Dim akII As Variant Public Function Etheta(zi, zj, Im, Aphi, Deriv) ' ' K S Pitzer "A Thermodynamic Model for Aqueous Solutions of Liquid-Like Density", LBL--23554 ' ' Calculates higher order unsymmetrical mixing coefficients ' local variables Dim E As Double Dim b(22) As Double Dim d(22) As Double Dim J(2) As Double Dim Jp(2) As Double Dim x(2) As Double Dim dzdx As Double Dim z As Double ' set up data x(0) = 6 * zi * zj * Aphi * Sqr(Im) x(1) = 6 * zi * zi * Aphi * Sqr(Im) x(2) = 6 * zj * zj * Aphi * Sqr(Im) ' set up aK arrays Call Coeffs ' calculate bk and dk b(22) = 0: b(21) = 0: d(22) = 0: d(21) = 0 For i = 0 To 2 If x(i) < 1 Then ' Region I: x <= 1 z = 4 * x(i) ^ (0.2) - 2 dzdx = 0.8 * x(i) ^ (-0.8) For k = 20 To 0 Step -1 b(k) = z * b(k + 1) - b(k + 2) + akI(k) d(k) = b(k + 1) + z * d(k + 1) - d(k + 2) Next k Else ' Region II: x >= 1 z = 40 / 9 * x(i) ^ (-0.1) - 22 / 9 dzdx = -4 / 9 * x(i) ^ (-1.1) For k = 20 To 0 Step -1 b(k) = z * b(k + 1) - b(k + 2) + akII(k) d(k) = b(k + 1) + z * d(k + 1) - d(k + 2) Next k End If J(i) = 0.25 * x(i) - 1 + 0.5 * (b(0) - b(2)) Jp(i) = 0.25 + 0.5 * dzdx * (d(0) - d(2)) Next i ' calculate the E-theta-ij terms E = (zi * zj / 4 / Im) * (J(0) - 0.5 * J(1) - 0.5 * J(2)) If Deriv = False Then Etheta = E Else Etheta = -E + (zi * zj / 8 / Im) _ * (x(0) * Jp(0) - 0.5 * x(1) * Jp(1) - 0.5 * x(2) * Jp(2)) End If End Function Sub Coeffs() 68 RPP-RPT-50703 Rev.01A 8/23/2016 - 10:10 AM 77 of 77 RPP-RPT-50703 Rev. 1A ' ' Table A1 data ' ' Region I: for x <= 1 akI = Array( _ 1.925154014814670, -0.060076477753119, -0.029779077456514, -0.007299499690937, _ 0.000388260636404, 0.000636874599598, 0.000036583601823, -0.000045036975204, _ -0.000004537895710, 0.000002937706971, 0.000000396566462, -0.000000202099617, _ -0.000000025267769, 0.000000013522610, 0.000000001229405, -0.000000000821969, _ -0.000000000050847, 0.000000000046333, 0.000000000001943, -0.000000000002563, _ -0.000000000010991) ' Region II: for x >= 1 akII = Array( _ 0.628023320520852, 0.462762985338493, 0.150044637187895, -0.028796057604906, _ -0.036552745910311, -0.001668087945272, 0.006519840398744, 0.001130378079086, _ -0.000887171310131, -0.000242107641309, 0.000087294451594, 0.000034682122751, _ -0.000004583768938, -0.000003548684306, -0.000000250453880, 0.000000216991779, _ 0.000000080779570, 0.000000004558555, -0.000000006944757, -0.000000002849257, _ 0.000000000237816) End Sub 69