Properties Model for Aqueous Sodium Chloride Solutions near the Critical Point of Water

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Properties Model for Aqueous Sodium Chloride Solutions near the Critical Point of Water Bing Liu Brigham Young University - Provo Follow this and additional works at: Part of the Chemical Engineering Commons BYU ScholarsArchive Citation Liu, Bing, "Properties Model for Aqueous Sodium Chloride Solutions near the Critical Point of Water" (2005). All Theses and Dissertations This Dissertation is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu.

2 PROPERTIES MODEL FOR AQUEOUS SODIUM CHLORIDE SOLUTIONS NEAR THE CRITICAL POINT OF WATER by Bing Liu A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Chemical Engineering Brigham Young University December 2005

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4 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a dissertation submitted by Bing Liu This dissertation has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date John L. Oscarson, Chair Date Larry L. Baxter Date Thomas H. Fletcher Date Randy S. Lewis Date Richard L. Rowley

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6 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the dissertation of Bing Liu in its final form and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date John L. Oscarson Chair, Graduate Committee Accepted for the Department W. Vincent Wilding Department Chair Accepted for the College Alan R. Parkinson Dean, Ira A. Fulton College of Engineering and Technology

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8 ABSTRACT PROPERTIES MODEL FOR AQUEOUS SODIUM CHLORIDE SOLUTIONS NEAR THE CRITICAL POINT OF WATER Bing Liu Department of Chemical Engineering Doctor of Philosophy Traditional excess Gibbs energy models in terms of temperature, pressure, and concentration become progressively less effective in describing the thermodynamics of aqueous solutions at temperatures above 300 C, and are totally inadequate in the critical region of water. This deficiency is due to the strong ion association and the large property fluctuations (such as density) with small variations in pressure, temperature, and solute concentration around the critical point of water. In this work, a speciation-based model has been developed to describe the thermodynamic properties of aqueous sodium chloride solutions in the critical region of water. The anomalous fluctuation problem is avoided by adopting a residual Helmholtz energy approach in terms of temperature, density, and solute concentration. Partial ion dissociation is accounted for by including an isochoric equilibrium constant equation and

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10 a mean spherical approximation in the present model. The present model includes such classical interactions or effects as hard-sphere interactions, dipole-dipole interactions, ion dissociation effects, long-range ion-ion interactions, and a non-classical perturbation term. The related parameters that account for these effects were regressed to fit the measured values in the critical region of water. Densities, compressibility factors, apparent molar volumes, heats of dilution, and apparent isobaric molar heat capacities were used to test the validity of the model. The predicted values in this work agree well with the literature data over a wide range of temperatures (350 to 400 C), pressures (17.5 to 40 MPa), and sodium chloride concentrations (0 to 5 mol/kg). Comparisons with other models are also included in this work. This model can be used to predict speciation, solute dissociation reaction, and many other comprehensive properties in aqueous sodium chloride solutions at near-critical conditions.

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12 ACKNOWLEDGEMENTS First, I would like to acknowledge my family, who have been encouraging and believing in me in these years. I would like to thank Dr. Oscarson, who helped, guided and allowed me to work on this project. I would also like to thank my committee members for the help and support that they gave to this project. Finally, I would like to thank Dr. Izatt, who helped review my dissertation. I thank Craig Peterson and Jaime Cardenas-Garcia, who made many heat of dilution measurements for this project, and Josef Sedlbauer, who sent me a copy of apparent isobaric molar heat capacities of sodium chloride measured by Carter.

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14 TABLE OF CONTENTS TABLE OF CONTENTS... vii LIST OF FIGURES... xi LIST OF TABLES... xix NOMENCLATURE... xxi CHAPTER 1 INTRODUCTION... 1 CHAPTER 2 LITERATURE REVIEW AND BACKGROUND Peculiar Properties of Aqueous Electrolyte Solutions in the Critical Region of Water Ionic Behavior in the Critical Region of Water Anomalous Behavior of Thermodynamic Properties in the Critical Region Modeling Approaches Models Based on Complete Ion Association Equilibrium Constant Models RI Model, a Speciation-based Model Models Involving Electrostatic Interactions Molecular Simulations in the Critical Region of Water Experimental Data in the Critical Region of Water vii

15 2.3.1 Vapor and Liquid Phase Equilibria and Densities in the Critical Region of Water Heats of Dilution and Heat Capacities in the Critical Region of Water Apparent Molar Properties of NaCl Electric Conductances and Equilibrium Constants Empirical Equations and Experimental Consistency Summary CHAPTER 3 RESEARCH OBJECTIVES AND APPROACHES Objectives Analysis and Approaches Analysis of Interactions between the Species in Solutions Approaches CHAPTER 4 RII MODEL DEVELOPMENT AND PROPERTY DERIVATIONS RII Model Development Modification in the AP Model Development of the Isochoric Equilibrium Constant Modification in the MSA Model RII Model Application Fraction of Dissociated NaCl, F disso Solution Compressibility Factor and Volume Solution Internal Energy and Solution Enthalpy viii

16 4.2.4 Heats of Dilution Apparent Molar Properties of NaCl Programming Procedure Summary CHAPTER 5 SOLUTION DENSITY CHAPTER 6 COMPRESSIBILITY FACTOR Hard-sphere Compressibility factor, z hs Dipolar Compressibility Factor, z dip Perturbation Compressibility Factor, z per Solute Dissociation Compressibility Factor, z disso MSA Compressibility Factor, z MSA Total Compressibility Factor CHAPTER 7 APPARENT MOLAR VOLUME OF SODIUM CHLORIDE CHAPTER 8 HEAT OF DILUTION CHAPTER 9 APPARENT ISOBARIC MOLAR HEAT CAPACITY OF SODIUM CHLORIDE CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS Conclusions Model Development Properties in the Critical Region of Water Comparison with Other Models Recommendations BIBLIOGRAPHY ix

17 APPENDICES APPENDIX A THE PARAMETERS AND THE MIXING RULES IN THE PERTURBATION TERM IN THE AP MODEL APPENDIX B MEASURED DENSITIES AND COMPRESSIBILITY FACTORS AND THOSE CALCULATED USING THE RII MODEL APPENDIX C MEASURED APPARENT MOLAR VOLUMES OF NACL AND THOSE CALCULATED USING THE RII MODEL APPENDIX D MEASURED HEATS OF DILUTION AND THOSE CALCULATED USING THE RII MODEL APPENDIX E MEASURED APPARENT ISOBARIC MOLAR HEAT CAPACITIES AND THOSE PREDICTED USING THE RII MODEL x

18 LIST OF FIGURES Figure 1. Smoothed association constants of aqueous NaCl solutions in the critical region of water based on conductance measurements Figure 2. Isochoric heat capacities of water measured at water density of g/cm Figure 3. The isochoric heat capacities of pure water and of aqueous KCl solutions at mol/kg H 2 O as a function of temperature at near-critical isochores Figure 4. Comparison of vapor-liquid equilibria correlated using the AP model (solid lines) with the smoothed data (symbols) reported by Bischoff and Pitzer as a function of solute mass percent Figure 5. Comparison of heats of dilution measured by Fuangswasdi et al. with those predicted using the AP model Figure 6. Comparison of heats of dilution measured by Busey et al. with those predicted using the AP model at extremely low concentrations of NaCl Figure 7. Comparison of heats of dilution measured by Busey et al. at 402 C and MPa as a function of NaCl concentration with those predicted using the AP and RI models xi

19 Figure 8. Comparison of heats of dilution measured by Fuangswasdi et al. at 370 C and 24.7 MPa as a function of NaCl concentration with those predicted using the AP and RI models Figure 9. Comparison of the critical points of aqueous NaCl solutions correlated using different empirical power-law series expressions developed by Abdulagatov et al. and by Knight & Bodnar with the experimental data measured by Abdulagatov et al., by Knight & Bodnar, and by Sourirajan & Kennedy Figure 10. Scatter of the measured heats of dilution at low NaCl concentrations in the critical region of water Figure 11. Comparison of received enthalpy signals between two cases Figure 12. Plot of the NaCl density as a function of temperature at 28 MPa correlated using the AP and RII models Figure 13. Interaction parameters used in the AP model (a) and in the RII model (b) as a function of temperature Figure 14. Equilibrium constants correlated using the Gruszkiewicz and Wood equation Figure 15. Comparison of measured equilibrium constants as a function of density at given temperatures with those correlated using the Gruszkiewicz and Wood equation and the equation developed in the RII model Figure 16. Comparison of measured equilibrium constants as a function of density at 400 C with those correlated using the Gruszkiewicz and Wood equation, the fitted Marshall and Franck equation, and the equation developed in the RII model xii

20 Figure 17. Plot of derivatives of three equilibrium constants with respect to temperature as a function of temperature at 0.6 g/cm Figure 18. Plot of derivatives of three equilibrium constants with respect to water density as a function of water density at 380 C Figure 19. Plot of average diameters used in the Myers et al. and RII models Figure 20. Diagram of a typical solution dilution process Figure 21. Flowchart of the programming procedure Figure 22. Comparison of measured solution densities as a function of NaCl concentration at 350 C and at P = 15.5 ~ 16.2 MPa with those calculated using the AP, RI, and RII models Figure 23. Comparison of measured solution densities as a function of NaCl concentration at 400 C and 38 MPa with those calculated using the AP, RI, and RII models Figure 24. Comparison of densities assuming volumes are additive with those experimental & smoothed experimental data at 400 C and 28 MPa Figure 25. Comparison of measured solution densities as a function of pressure at 400 C and 4.28 mol/kg H 2 O with those calculated using the AP, RI, and RII models.. 91 Figure 26. Comparison of measured solution densities as a function of pressure at 400 C and 1.90 mol/kg H 2 O with those calculated using the AP, RI, and RII models.. 91 Figure 27. Plot of solution critical point as a function of NaCl concentration. Experimental data were measured by Bodnar and Costain Figure 28. Comparison of measured densities as a function of temperature at 28 MPa and mol/kg H 2 O with those calculated using the AP, RI, and RII models xiii

21 Figure 29. Comparison of measured densities as a function of temperature at 28 MPa and 3.1 mol/kg H 2 O with those calculated using the AP, RI, and RII models Figure 30. Comparison of measured densities (Literature) with those calculated using the AP and RII models Figure 31. Variation of the hard-sphere contribution to the residual compressibility factor with NaCl concentration at given temperatures and pressures, calculated using the RII model Figure 32. Plot of the dipolar contribution to the residual compressibility factor as a function of NaCl concentration at given temperatures and pressures, calculated using the RII model Figure 33. Plot of the perturbation contribution to the residual compressibility factor as a function of NaCl concentration at given temperatures and pressures, calculated using the RII model Figure 34. Plot of the solute dissociation contribution to the compressibility factor as a function of NaCl concentration at given temperatures and pressures, calculated using the RII model Figure 35. Plot of the solute dissociation quantities as a function of NaCl concentration at 375 C and 24 MPa, calculated using the RII model Figure 36. Plot of the long-range ion-ion contribution to the residual compressibility factor as a function of NaCl concentration at given temperatures and pressures, calculated using the RII model Figure 37. Plot of the MSA quantities calculated using the RII model as a function of NaCl concentration at 375 C and 24 MPa xiv

22 Figure 38. Plot of different contributions to the total compressibility factor at 375 C, 30 MPa Figure 39. Comparison of the literature compressibility factors (symbols) calculated using the measured densities with those calculated using the AP and RII models. 107 Figure 40. Plot of the number of dissociated ions and volume changes due to solute dissociation and long-range ion-ion interactions at 378 C and 28 MPa, calculated using the RII model Figure 41. Plot of apparent molar volume as a function of pressure at 378 C and at given NaCl concentrations Figure 42. Comparison of measured apparent molar volume of NaCl as a function of NaCl concentration at C and 28 MPa with those calculated using the AP and RII models Figure 43. Plot of the isochoric ion association constant and the fraction of solute dissociation as a function of NaCl concentration at C and 28 MPa, calculated using the RII model Figure 44. Apparent molar volumes plotted as a function of NaCl concentration at 396 C and at 33 MPa Figure 45. Apparent molar volumes plotted as a function of NaCl concentration at 378 C and at 28 MPa Figure 46. Temperature dependence of the apparent molar volumes of NaCl at 28 MPa Figure 47. Temperature dependence of the apparent molar volumes of NaCl at mol/kg H 2 O xv

23 Figure 48. Compressibility plotted as a function of temperature at 28 MPa and at given NaCl concentrations, calculated using the RII model Figure 49. Comparison of the measured apparent molar volumes of NaCl with those calculated using the AP (dotted lines) and RII models (solid lines) Figure 50. Volume change due to solute dissociation and long-range ion-ion interactions as a function of temperature, calculated using the RII model Figure 51. Comparison of measured heats of dilution (symbols) with those correlated using the RII model (lines) at initial NaCl concentration of 0.5 mol/kg H 2 O Figure 52. Comparison of measured heats of dilution (symbols) with those correlated using the RII model (lines) at initial NaCl concentration of 0.5 mol/kg H 2 O Figure 53. Comparison of measured heats of dilution as a function of NaCl concentration at 350 C and 20.5 MPa with those calculated using the AP, RI, and RII models Figure 54. Comparison of measured heats of dilution as a function of NaCl concentration at 350 C and 20.4~21.2 MPa with those calculated using the AP, RI, and RII models Figure 55. Comparison of measured heats of dilution as a function of NaCl concentration at 370 C, 24.7 MPa with those calculated using the AP, RI, and RII models Figure 56. Comparison of measured heats of dilution as a function of NaCl concentration at 380 C, 24.7 MPa with those calculated using the AP, RI, and RII models xvi

24 Figure 57. Comparison of measured heats of dilution as a function of NaCl concentration at 380 C, 24.7 MPa with those calculated using the AP, RI, and RII models Figure 58. Comparison of the accuracy of the heats of dilution calculated using the AP, RI, and RII models Figure 59. Comparison of solute dissociation enthalpy changes calculated using the RI model with those calculated using the RII model at 375 C and 25 MPa Figure 60. Comparison of measured apparent molar heat capacities of NaCl as a function of solute concentration at 350 C and 20 MPa with those predicted using the Archer and RII models Figure 61. Comparison of measured apparent molar heat capacities of NaCl as a function of solute concentration at C and 30.3 MPa with those predicted using the Archer and RII models Figure 62. Plot of the heat capacity profile shift with the addition of NaCl Figure 63. Apparent molar heat capacities plotted as a function of temperature at 32 MPa and mol/kg H 2 O Figure 64. Apparent molar heat capacities plotted as a function of NaCl concentration at 351 C and at given pressures Figure 65. Comparison of measured apparent molar heat capacities of NaCl (symbols) with those predicted using the RII model (solid lines) in the critical region xvii

25 Figure 66. Comparison of measured apparent molar heat capacities of NaCl (symbols) with those predicted using the RII model (solid lines) at 28 MPa and at given temperatures xviii

26 LIST OF TABLES Table 1. Parameters of pure water used in the AP and RII models Table 2. Comparison of pure water densities provided by NIST in the critical region of water with those calculated using the AP and RII models Table 3. Parameters of pure NaCl used in the AP and RII models Table 4. Interaction parameters used in the AP and RII models Table 5. Average diameter parameters used in the Myers et al. and RII models Table 6. Comparison of measured solution densities at 378 C and mol/kg H 2 O with those calculated using the AP, RI, and RII models xix

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28 NOMENCLATURE Abbreviations AP the model developed by Anderko and Pitzer (Reference 69) EOS HNC MC MD MSA PCB equation of state hypernetted chain theory Monte Carlo simulation molecular dynamics simulation mean spherical approximation polychlorinated biphenyl RI the model developed by Oscarson et al. (Reference 54) RII SCWO the model developed in the present work supercritical water oxidation Roman letters A a Helmholtz energy parameter in the perturbation term in the original and the modified AP model A 1 -A 10 parameters in the Archer and Wang dielectric constant equation xxi

29 a 1 -a 6 A 2, A 3 A m aq parameters in the Gruszkiewicz and Wood equation expansion terms used in the dipolar term in the AP model parameter in the Debye-Hückel equation aqueous b (1) van der Waals co-volume (2) constant in the Pitzer ion-interaction model B m C c parameter in the extended Debye-Hückel equation heat capacity parameter in the perturbation term in the original and the modified AP model c 1 -c 6 CaCl 2 Cl - CsBr D d coefficients used in the dipolar term in the AP model calcium chloride chloride ion cesium bromine parameter in the repulsive term in the AP model parameter in the perturbation term in the original and the modified AP model E parameter in the repulsive term in the AP model e (1) elementary charge (2) parameter in the perturbation term in the original and the modified AP model xxii

30 F (1) parameter in the repulsive term in the AP model (2) fraction of solute dissociation with superscript disso f[i m ] modified extended Debye-Hückel limiting law function in the Pitzer ion-interaction model G H H + H 2 O I I 2, I 3 K k KCl LiCl m M MgCl 2 Gibbs energy enthalpy hydrogen ion water ionic strength dipolar functions defined in the dipolar term in the AP model equilibrium constant Boltzmann constant potassium chloride lithium chloride molal concentration molar molecular mass magnesium chloride n (1) number of moles (2) number of components N A Na + NaBr Avogadro s number sodium ion sodium bromine xxiii

31 NaCl OH - P q sodium chloride Hydroxide ion pressure auxiliary variable in the perturbation term in the original or the modified AP model R T U v W X universal gas constant temperature internal energy molar volume solute mass percent ratio of the moles of components to the moles of solute + water before dissociation x mole fraction z (1) compressibility factor (2) ion charge number Greek Letters A Φ Debye-Hückel limiting slope for osmotic coefficients in the Pitzer ion-interaction model α binary mixture parameter in the perturbation term in the original or the modified AP model Γ screening parameter in the MSA model xxiv

32 γ (1) ternary mixture parameter in the perturbation term in the original or the modified AP model (2) Activity coefficient Δ δ molar property change Quadruplet mixture parameter in the perturbation term in the original or the modified AP model ε (1) dielectric constant (2) quintuplet mixture parameter in the perturbation term in the original or the modified AP model ε 0 η θ permittivity of free space reduced density defined in the AP model parameter in the perturbation term in the original or the modified AP model κ κ T λ μ μ ρ σ σ 1, σ 2 Debye screening length isothermal compressibility of water binary interaction parameter in the Pitzer ion-interaction model dipole moment Reduced dipole moment density component diameter parameters in determining the average ion diameter in the MSA model xxv

33 τ auxiliary parameter in the perturbation term in the original or the modified AP model Subscripts added c Cl - dil final H 2 O hyp water added for the dilution process critical chloride ion dilution after dilution water hypothetical properties i, j, k, l, m components i, j, k, l, m (replaceable with digits or letters) ij ijk ijkl ijklm initial binary interaction parameter (replaceable with digits) ternary interaction parameter (replaceable with digits) quaternary interaction parameter (replaceable with digits) quinary interaction parameter (replaceable with digits) before dilution m (1) molality basis (2) constant molal concentration Na + NaCl p sodium ion Sodium chloride constant pressure xxvi

34 r T v reduced quantity constant temperature constant volume X constant number of components based on one mole of solute + water without dissociatoin x φ ρ constant mole fraction of solute apparent molar properties of solute constant density Superscript AP Born dip disso ex hs ig ion MAP mix per prod quantity calculated using the original AP model quantity calculated using the Born model dipolar quantity defined in the AP model dissociation excess thermodynamic quantity hard-sphere quantity ideal gas dissociated ion system quantity calculated using the modified AP model mixing quantity of species perturbation quantity defined in the AP model dissociation products like sodium and chloride ions xxvii

35 ref rep res RI RII reference quantity repulsive quantity defined in the AP model residual thermodynamic quantity quantity calculated using the model developed by Palmer et al. quantity calculated using the model in this work * dimensionless quantity infinite dilution o (1) standard state at infinite dilution (2) per unit quantity (3) interaction parameters without the auxiliary τ term in the perturbation term in the original or the modified AP model Other Annotations # number + cation ion - anion ion ± mean ion ^ quantity based on one kilogram of water ~ Reduced Letters used temporarily in this work to shorten equation lengths are not included. xxviii

36 CHAPTER 1 INTRODUCTION Supercritical water oxidation (SCWO) has become a promising technology in recent years. It can be used to achieve high destruction efficiencies for a wide variety of hazardous organic wastes 1-6 including munitions, 7 polychlorinated biphenyls (PCBs), 8 aqueous organics, 9, 10 sewage, 11 and sludges. 12, 13 This technology involves the oxidation of organic components with oxidizer in water at temperatures (T) and pressures (P) above the critical point of water (374 C, 22.1 MPa). 14 At supercritical conditions, water acts like a dense gas, and is a good solvent for organic substances and is completely miscible with gases like oxygen and carbon dioxide. Many organic compounds can be rapidly and completely oxidized in homogeneous phase reactions producing carbon dioxide, water, nitrogen, and various acids. 4 A typical SCWO process operates from 400 to 650 ºC. 15 Formation of nitrogen oxides is avoided because the process T is much lower than that 10, 16 required in combustion or incineration. However, there are two major engineering challenges 17, 18 that exist in SCWO processes. First, increased corrosion of the reactor and heat exchanger materials, and second, precipitation of insoluble inorganic salts within the reactor or equipment. Precipitation results because inorganic salts have low solubility at high T values. This results in fouling or plugging problems. 19 Fouling or plugging is a major concern if salts are dissolved or produced in the solutions. 20 SCWO processes provide an extremely 1

37 aggressive oxidation environment at high T, high P, and low PH values. Many stainless steel and corrosion-resistant alloy materials that work well at ambient conditions cannot be used at SCWO conditions due to the serious corrosion. The corrosion problem is only serious if the reactants contain acid-forming elements such as chloride, fluoride, phosphorous, and/or sulfur. 21, 22 Corrosion is most marked at T values between 300 and 400 o C 23, 24 where fluid properties change most markedly with variations in T, P, and solute concentration. At lower T values, the corrosion rate is slow due to slow kinetics. At higher T values, anion concentration is low because most ions are associated at high T values. This results in slow corrosion rates. Knowledge of thermodynamic properties is required to design processes that can minimize the corrosion or fouling/plugging problems. This knowledge is also helpful in understanding many other natural or engineering high-temperature processes involving corrosion, salt deposition, ion 25, 26 speciation, etc., such as those found in geochemical and hydrothermal processes, power plants, 27 and industrial chemistry. 19 To gain knowledge of the properties of aqueous solutions in the critical region of water, a large number of measurements have been made. Measurements have included such properties as volumes/densities of liquids and vapors, enthalpies, heat capacities, ion association, critical PvTx (pressure-volume-temperaturecomposition relationship) values of solutions, 48, 49 50, 51 and vapor-liquid phase equilibria. However, collecting experimental data at all temperatures, pressures, and solute concentrations would be costly and, in fact, impossible. Due to the expense and difficulty of the experiments in the critical region of water, several researchers have proposed mathematical models to correlate the properties of simple aqueous solutions. One of the 2

38 most thoroughly modeled systems is aqueous sodium chloride (NaCl) solution because it is simple in composition (1:1 electrolyte) and extensive experimental thermodynamic data are available for this system. There are three main approaches used to model aqueous electrolyte solutions: molecular simulation, empirical modeling, and semiempirical equations of state (EOS). While simulation is helpful in understanding the relationship between the desired properties and the underlying intermolecular forces and fluid structures, it requires large computer resources and becomes less accurate near the critical point due to the long-range nature of density fluctuations in this region. 52 Empirical modeling is based only on experimental data, and thus, it is risky to use results from such equations at conditions where one must extrapolate or interpolate. This is especially true in the critical region of water where properties change dramatically with small variations in T, P, and molal concentration (m). Furthermore, empirical models can only be used to correlate limited kinds of thermodynamic properties that have been measured directly or can be derived directly. Semi-empirical EOS modeling may be a promising approach because it connects classical thermodynamic theories with experiments. Anderko and Pitzer have developed a semi-empirical EOS model correlating phase equilibria and densities of aqueous NaCl solutions, 53 but this model is valid only in concentrated or gas-like solutions where solute dissociation can be considered negligible. Knowledge of speciation is unavailable in this model. Large errors occur when it is used to predict the properties (e.g. heats of dilution) in dilute solutions. 54 Other models, such as the Pitzer ion-interaction model and the Oelkers & Helgeson models, can be used to calculate speciation or long-range ion-ion interactions at subor supercritical conditions, but these models are not applicable at near-critical conditions. 3

39 Oscarson et al. 54 developed a speciation-based model (RI model) to predict heats of dilution in dilute solutions. The accuracy of this model is poor at medium or high m values. None of these models can predict speciation correctly over a wide range of m values, or predict extensive thermodynamic properties in the critical region of water. In this work, a comprehensive EOS model (present RII model) has been developed that correlates many properties of aqueous NaCl solutions near and above the critical point of water and over a wide range of m values. This model is an improved version of the RII model reported earlier. 58 The main shortcoming of the previous RII model is that it implicitly included solute dissociation effects in the perturbation term. The present RII model overcomes this problem. It is speciation-based and includes various species interactions that exist in real solutions. This model can be used to reproduce experimental densities, compressibility factors, apparent molar volumes, heats of dilution, and apparent isobaric molar heat capacities within acceptable uncertainties. It can also be used to explore other properties such as speciation and the thermodynamic quantities of the ion association or solute dissociation reaction. In the following chapters, the name of RII model refers to the latest RII model developed in the present work. In Chapter 2, the previous modeling and experimental work in the critical region of water are reviewed. In Chapter 3, the research objectives and the modeling approach are listed. In Chapter 4, the development of the RII model is outlined in detail. Various property derivations are also included in this chapter for further comparisons, analyses and discussion in the following chapters. In Chapter 5, the solution densities correlated using the RII model are compared with the experimental data and those calculated using other models. In Chapter 6, the various compressibility factors are classified according to 4

40 the various interactions. Necessary comparisons and discussion are also included. In Chapters 7, 8, and 9, the validity and ability of the RII model are tested using measured apparent molar volumes of NaCl, heats of dilution, and apparent isobaric molar heat capacities of NaCl, respectively. Some interesting phenomena and behavior of the apparent molar properties are discussed and explained using the RII model. Comparisons with other available models are also included. Final conclusions and recommendations for future work can be found in Chapter 10. 5

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42 CHAPTER 2 LITERATURE REVIEW AND BACKGROUND In this chapter, the literature on aqueous solutions in the critical region of water is reviewed. First, a brief background is presented to describe the peculiar properties of aqueous electrolyte solutions in the critical region of water. Two major problems in modeling near-critical solutions are discussed. Next, a general modeling overview is provided, where advantages and disadvantages among different models are compared. Some typical EOS models are also discussed in detail according to the different molecular/ionic interactions that exist in the solutions. Last, a literature data review is presented including such properties as densities, apparent molar volumes, heats of dilution, and apparent isobaric molar heat capacities, etc. 2.1 Peculiar Properties of Aqueous Electrolyte Solutions in the Critical Region of Water Two major problems increase the difficulty in modeling aqueous electrolyte solutions at near-critical conditions as compared to modeling aqueous electrolyte solutions at low temperatures: strong ion association and anomalous behavior of the thermodynamic properties. 7

43 2.1.1 Ionic Behavior in the Critical Region of Water Many electrolytes can be considered completely dissociated in aqueous solutions at ambient T values. As T increases, all electrolytes, even alkali halides, become more associated. The T value where ion association becomes significant differs with each electrolyte. Models based on the assumption of complete solute dissociation become increasingly unreliable as T increases. This association behavior has been demonstrated by both simulations and conductance measurements. 44, 46, 62 Figure 1 illustrates the changes of the association constant as a function of T at constant densities (ρ). The logarithm of the association constant, log K m, increases markedly as T increases at low or medium water density ( constant ( 1 K m ρ H2O ) values. For example, at 0.2 g/cm 3, the association ) increases dramatically from 706 at 350 C to at 400 C. At high T values, the fraction of the dissociated electrolyte approaches unity only at extremely low m values. Ion pairs are dominant at high m values or in vapor-like solutions, though some ions are dissociated. The properties of aqueous electrolyte solutions at high T values are significantly different from those at low T values because the ionic interactions change from long-range to short-range as ion association proceeds. This ionic behavior must be accounted for in determining speciation and the interactions between species, and thus, increases the complexity in modeling thermodynamic properties at high T values. 8

44 -logk m ρ H2O = 0.7 g/cm3 -logk m ρ H2O = 0.45 g/cm3 -logk m 4 ρ H2O = 0.2 g/cm T ( o C) Figure 1. Smoothed association constants of aqueous NaCl solutions in the critical region of water based on conductance measurements. 44 The value of log K m increases as temperature increases and water density decreases. Lower density values indicate solutions are more vapor like Anomalous Behavior of Thermodynamic Properties in the Critical Region The second modeling problem arises from the anomalous behavior of many thermodynamic properties in the critical region. As conditions approach the critical point, many properties of all fluids have large fluctuations due to the long-range nature of density fluctuation correlations. One example of such behavior is isochoric heat capacity, 9

45 C v. In Figure 2, the C v value of water is plotted as a function of T at a given ρ H2O value. The C v value varies smoothly at conditions far away from the critical point, but changes dramatically as the critical point of water is approached. Cv (kj/kg K) T ( C) Figure 2. Isochoric heat capacities 63 of water measured at water density of g/cm 3. This density value is very close to the critical density of water (0.322 g/cm 3 ). Solution thermodynamic properties are often modeled in terms of Gibbs energy with T, P, and m as the independent variables. A typical Gibbs energy model is usually expressed in an equation form shown in Eq. (1), ( ) o ex GTPm,, = G( TP, ) + G ( TPm,, ) (1) where G o and G ex are the standard Gibbs energy and the excess Gibbs energy, respectively. The standard state in the asymmetric convention is the solution referenced 10

46 to infinite dilution at unit solute concentration (usually molal concentration) and at the T and P values of interest. This formulation is convenient, partly because it avoids using a hypothetical liquid solute as a standard state, and partly because it uses experimentally accessible parameters as its independent variables. In the critical region, however, Gibbs energy models in terms of T, P, and m are impractical for correlation purposes. At near-critical conditions, water is highly compressible. The thermodynamic properties at the standard state derived from the G o term are extremely sensitive to small P changes. 52, 64 Some properties (e.g. standard molar volume and heat capacity) may even diverge at the critical point of water. 65 The G ex term in Eq. (1) is also affected by this anomalous behavior because it accounts for the difference between the infinitely dilute solution and the solution at finite concentrations. Small variations in P may cause large changes in other related excess thermodynamic properties. Addition of solute changes the critical point of solution 48 resulting in large shifts of many other property profiles with respect to T, P or m. An example is the variation of C v profile with respect to T at near-critical conditions. In Figure 3, the pure water C v values at g/cm 3 are compared with those of aqueous KCl solutions at g/cm 3 and at mol/kg H 2 O as a function of T. It is seen that the C v values of water change dramatically around the critical T value of water (374 C). The C v values of solution in Figure 3, however, change more smoothly and slowly, and its maximum at 376 C is much smaller than that of pure water at 374 C. In other words, T c increases and the C v profile around T c becomes more smooth with the addition of salt at temperatures near the T c value of water. 11

47 H 2 O, g/cm 3 KCl + H 2 O, g/cm Cv (kj/kg K) T ( C) Figure 3. The isochoric heat capacities of pure water and of aqueous KCl solutions at mol/kg H 2 O as a function of temperature at near-critical isochores. 42 This inherent anomalous behavior indicates that Gibbs energy correlations in terms of T, P, and m are inappropriate in modeling thermodynamic properties accurately at high T values, especially at near-critical conditions where fluid compressibility becomes extremely large. 2.2 Modeling Approaches A successful model should be able to solve or avoid the two major problems that exist in G ex models in near-critical solutions. Different methods have been developed to model aqueous electrolyte solutions in the critical region of water. Some methods are aimed at solving the parameter sensitivity problem in the critical region, and some methods are used to solve the ionic behavior problem at high T values. 12

48 It has been pointed out 65 that the anomalous behavior of the thermodynamic properties in the critical region may be avoided using correlations in terms of T, ρ or volume (v), and m rather than T, P, and m. The appropriate thermodynamic quantity in terms of T, ρ, and m is Helmholtz energy, A, and thus, this quantity is conventionally applied in modeling aqueous solutions at high T values. This approach removes the parameter-sensitive problem that exists in Gibbs energy models in terms of T, P, and m. The available electrolyte models for the treatment of ions in solutions can be grouped into three types, i.e. 1. models based on the assumption of complete ion association, 2. models based on the assumption of complete solute dissociation, and 3. speciation-based models. Models of the first type are analogous to non-electrolyte mixture models. They are particularly suitable for supercritical or high-temperature solutions, where most ions are associated with each other. However, this kind of model fails to predict reliable results in dilute or dense solutions, where solute dissociation is appreciable. Models of the second type are usually applicable in the sub-critical region of water, and are invalid at higher T values where ion association becomes dominant. Speciation-based models reflect the real nature of aqueous electrolyte solutions at near-critical conditions. They are more accurate but more computationally intensive. One reason is that the algorithm must include the computation of the species in equilibrium. Another reason is that large amounts of data are needed for the properties of various species. Speciation is important 13

49 in modeling aqueous electrolyte solutions in the critical region of water, because this knowledge is required to understand the corrosion phenomena that exist in many hightemperature processes like SCWO. The goal of this research is to develop a speciationbased model. The following subsections will discuss some typical models Models Based on Complete Ion Association During the last decade, several equations of state were developed to model the properties of aqueous electrolyte solutions at high T values based on the assumption of complete ion association. Tanger and Pitzer developed an EOS model for aqueous NaCl solutions based on an expansion around the critical point of water. 66 Sengers and Gallagher 67 proposed an equation on the basis of the corresponding-state principle. These models reproduced vapor-liquid equilibrium data for the NaCl-H 2 O system, but failed to predict densities correctly in liquid-like solutions. Lvov and Wood 68 developed a model to incorporate the volumes of liquid-like solutions over a wide range of T, but their model did not include chemical potentials resulting in limited ability in predicting other thermodynamic properties. Anderko and Pitzer developed a more comprehensive EOS model (AP model) to correlate both phase equilibria and volumetric properties of aqueous NaCl solutions. 69 In this model, they assumed that all ions were completely associated, and the model successfully correlated both liquid-vapor equilibria and solution densities at T values from 300 to 927 C and at P values up to 5 kbar. The AP model is one of the most successful ion association models, and is introduced in the following subsections. 14

50 Overview of the AP Model The AP model was developed using the Helmholtz energy approach for concentrated or vapor-like NaCl solutions near the critical point of water. 69 It was assumed that there are three kinds of forces that exist in solutions: water molecule-water molecule, ion pair-ion pair, and water molecule-ion pair interactions. Traditionally, the residual Helmholtz energy, A res, is often used in A(T, ρ, m) models. This is defined as the difference between the actual Helmholtz energy, A, and the ideal gas Helmholtz energy, A ig, at the same T, ρ, and m values, that is: res ig A ( T, ρ, m) = A( T, ρ, m) A ( T, ρ, m) (2) In Eq. (2), the A ig (T, ρ, m) term can be easily calculated using thermodynamic identities if ideal gas heat capacities are known. The AP model includes a reference term that describes the properties of a system consisting of dipolar ion pairs and solvent molecules. A perturbation term is used in the AP model to account for the difference between the properties of the real solutions and of the reference solutions. Eq. (3) gives the expression used to calculate the residual Helmholtz energy in the AP model. res ref per A = A + A (3) The A res, A ref and A per terms in Eq. (3) refer to the residual Helmholtz energy, the reference Helmholtz energy, and the perturbation Helmholtz energy, respectively. In the AP model, A ref accounts for the properties of a mixture of hard spheres with dipole or quadrapole moments. The related interactions include those between ion pairs and water molecules or other ion pairs and between water molecules themselves. 15

51 Therefore, the reference Helmholtz energy can be divided into two terms: the repulsive term, A rep, which is based on a system consisting of simple hard spheres and has no temperature dependence, and the dipolar term, A dip, which accounts for the dipolar interactions. The correlation is shown in Eq.(4), ref rep dip A ( T, v, x) = A ( v, x) + A ( T, v, x) (4) where v is the molar volume of solutions, and x is solute mole fraction. Combining Eq. (3) with Eq. (4) gives: res rep dip per A ( T, v, x) = A ( v, x) + A ( T, v, x) + A ( T, v, x) (5) The subsequent subsections briefly describe the formulations of these three terms The Repulsive Term A rep Boublik 70 and Mansoori et al. 71 developed a set of equations for the hard-sphere system. These equations were used in the AP model to calculate the molar Helmholtz energy of the repulsive term: rep A RT 3 3 3DE E E η E = F F + F + ( 1)ln(1 η) η (1 η) F (6) In Eq. (6), R is the universal gas constant, D, E and F are parameters calculated using each component s diameter σ i as shown in Eqs. (7), (8) and (9), respectively. n D= xσ (7) i= 1 i i n E = xσ (8) i= 1 i 2 i 16

52 n F = xσ (9) i= 1 i 3 i The η term in Eq. (6) is the reduced density and is defined in Eq. (10): η = b/4v (10) In Eq. (10), b is the van der Waals co-volume parameter and can be calculated using Eq. (11): 2πN 2πN b= x = F 3 3 n A 3 iσ i A (11) i= 1 where N A is Avogadro s number, n is the number of components with a value of two for the NaCl-H 2 O system, and x i is the mole fraction of each component The Dipolar Term A dip Stell et al. 72, 73 and Rushbrooke et al. 74 developed a perturbation theory for dipolar hard spheres. They expessed the dipolar Helmholtz energy using a formula shown in Eq. (12). dip A A RT 1 A / A 2 = (12) 3 2 where the values of A 2 and A 3 can be calculated from the formulations developed by Twu et al., 75 Flytzani-Stephanopolous et al., 76 and Gubbins and Twu. 77 The specific equations for the A 2 and A 3 terms are shown in Eqs. (13) and (14). 4 bb A = xx I ( ) b η % μμ % η (13) n n i j i j 2 ij i j 2 3 i= 1 j= 1 ij 10 bb b A = x x x % % % I ( η ) (14) n n n i j k i j k ijk i j k 3 9 i 1 j 1 k 1 bb ij jkb η μμμ = = = ik 17

53 The b i term in Eqs. (13) and (14) is the co-volume parameter of species i. Other parameters in Eqs. (13) and (14) can be calculated using Eqs. (15) through (21): 1/3 1/3 ( ) b = b + b ij i j /2 3 (15) η = b /4v (16) ij ij ( ) 1/3 η = b /4 v= b b b /4v (17) ijk ijk ij jk ik 1/2 1/2 2 2 μ i μ i % μi = 3 = (18) σ 3b i kt i kt 2π N A I ( η) = 1+ cη+ cη + cη (19) I ( η) I 1 cη cη cη ( η ) = (20) n n b 1 η = = xixb j ij (21) 4v 4v i= 1 j= 1 In Eq. (18), k is the Boltzmann constant, μ is the dipole moment, and μ is the reduced dipole moment. In Eqs. (19) and (20), c 1 = , c 2 = , c 3 = , c 4 = , c 5 = , and c 6 = Anderko and Pitzer fitted the polynomial parameters of the I 2 (η) and I 3 (η) functions using the approach as shown in Rushbrooke et al. s work The Perturbation Term A per The perturbation term accounts for the departure of the real solution from the reference solution. A generalized van der Waals attraction expression is used to calculate the truncated virial expansion: 78 18

54 per A 1 a acb adb aeb = RT RT v 4v 16v 64v (22) In Eq. (22), the quantities of a, acb, adb 2, and aeb 3 are the terms related to the second, third, fourth, and fifth virial coefficients, and can be calculated using pure component correlations according to the mixing rules. 53 Anderko and Pitzer developed some empirical equations to account for the non-classical behavior included in the A per term (Appendix A). Many parameters in these equations were regressed to fit solution densities and phase equilibria Algorithm for Determining Properties According to the Gibbs equation, the value of P can be obtained using Eq. (24). da = SdT Pdv (23) A = P v T (24) Use of the AP model requires an iterative algorithm to find the solution v value at given T, P, and x values. Once v is found, other solution properties can be calculated using classical thermodynamic identities Evaluation on the AP Model The AP model is mainly used to correlate phase equilibria and solution densities. Figure 4 shows the ability of the AP model in fitting the vapor-liquid phase equilibria of aqueous NaCl solutions in the critical region of water. Though this model is based on the 19

55 assumption of complete ion association, it still fits the smoothed phase equilibrium data 69 well over a wide range of T, P, and m values in the critical region of water (Figure 4). P (MPa) C 360 C C C 390 C C %W NaCl Figure 4. Comparison of vapor-liquid equilibria correlated using the AP model (solid lines) with the smoothed data (symbols) reported by Bischoff and Pitzer 50 as a function of solute mass percent. The main shortcoming of the AP model is that all sodium and chloride ions are assumed to be paired in solutions. It cannot be used to predict the speciation in solutions, which is especially important in understanding the corrosion problem involving many high-temperature processes. This model fits the experimental data well in the concentrated or vapor-like solutions where most ions are associated. However, in dilute solutions or in solutions with liquid-like densities, solute dissociation is appreciable and its effect cannot be considered negligible. This dissociation introduces long-range ion-ion interactions that are not accounted for in the AP model (water molecule-water molecule, 20