Equation of State for Natural Gas Systems. m A thesis submitted to the University of London for the Degree of Doctor of Philosophy and for the Diploma of Imperial College by Jorge Francisco Estela-Uribe. Department of Chemical Engineering and Chemical Technology Imperial College of Science, Technology and Medicine. Prince Consort Road, SW7 2BY London, United Kingdom. March 1999. Acknowledgements. I acknowledge and thank the sponsorship I received from the following institutions: Fundacion para el Futuro de Colombia, Colfiituro; Pontificia Universidad Javeriana, Seccional Cali; the Committee of Vice-Chancellors and Principal of the Universities of the United Kingdom and Ruhrgas AG. Without their economic support my stay in London and the completion of this work would have been impossible. My greatest gratitude goes to my Supervisor, Dr Martin Trusler. His expert guidance and advice were always fundamental for the success of this research. He helped me with abundant patience and undying commitment and his optimism regarding the possibilities of the project was always inspirational for me. He also worked shoulder by shoulder with me on solving a good number of experimental problems, some of them were just cases of bad luck and for some prominent ones, I was the only one to blame. Others helped me willingly as well. My labmate, Dr Andres Estrada-Alexanders, was quite helpful in the beginning of my research, and has carried on being so. He not only did do well in his role as the senior student in the laboratory, but also became a good fnend of mine. Another good friend of mine, Dr Abdel Fenghour, has always been an endless source of help and information. My other colleagues were also important for me. Dr Margarida Costa-Gomes was an endless source of enlightening conversation. With Dr Maria Joao Cebola I spent countless hours of light-hearted chat which made my long hours in the lab much more fun. My other labmate, Mr Peter H Stanley, was also very helpful, particularly in computer-related matters. My all-time best friends, Dr Andres Lema-Hincapie, Dr Mario A Llano-Restrepo and Doctor-to-be Manuel Salazar, always cared about my well-being during these years. And Irene, of whom I have become fond, has encouraged me to bring this thesis to its completion, and helped me release much stress these months, I am thankful to her. Last but not least, my family back home have my given to me their unrestricted and loving support since I started this project. They have longed patiently for my return home all the time during these very long years. To all the persons I have mentioned above, I wish them well. Abstract. The development of an accurate equation of state for natural gas systems and an experimental programme of measurements of the speed of sound in binary gaseous mixtures is presented. Accounts are given of the statistical mechanical foundations of the extended principle of corresponding states, the theory of spherical acoustic resonators, the relationship between the speed of sound and thermodynamic properties and the theory of virial coefficients and intermolecular potential energy fiinctions. Detailed information is presented on the development of two separate extended corresponding states models based on correlations of molecular shape factors over a wide range of reduced temperature and density, with methane as the reference fluid. One of the models constitute a generalised functionality in terms of the critical constants and the acentric factors; the other is a substance-dependent formulation whose parameters are reported for hydrocarbons from methane to w-pentane, nitrogen, carbon monoxide, carbon dioxide and normal hydrogen. Predicted volumetric, vapour-liquid equilibria and caloric properties are presented and compared to up-to-date experimental data. The method by which these models may be extended to multicomponent systems is also presented along with values of parameters in correlations of temperature- and density- dependent binary interaction parameters for eighteen binary systems. Calculated compression factors and speeds of sound are presented for binary and natural gas systems and compared to predictions from other models. Compression factors and speeds of sound in natural gas mixtures are calculated on average within ± 0.05 per cent for temperatures between 270 K and 330 K and pressures up to 12 MPa. The technique of the measurement of the speed of sound in gases from the resonance frequencies of an spherical cavity is described. Results of measurements on two binary mixtures of methane with nitrogen and one of methane plus carbon dioxide are reported. These results were utilised to validate the models mentioned above. These speeds of sound are precise within about 15 parts per million. Second and third acoustic virial coefficients were also determined for these mixtures. Parameters of isotropic and site-site Maitland-Smith intermolecular potentials and Axilrod-Teller triple-dipole terms were obtained, from which interaction second and third ordinary virial coefficients are calculated. List of Contents. Page Acknowledgements 2 Abstract 3 List of Contents 4 List of Figures 9 List of Tables 16 Chapter 1. Introduction 20 References 25 Chapter 2. Corresponding States Theory 26 2.1. Introduction 26 2.2. The Simple Corresponding States Principle 26 2.2.1. Classical Theory of Corresponding States 26 2.2.2. Conformal Fluid Theory 28 2.2.2. Molecidar Theory of Corresponditig States for Simple Fluids 30 2.3. The Extended Corresponding States Principle 35 2.2.1. Deviations from the Simple Corresponding States Principle 35 2.2.2. Molecular Shape Factors 41 2.2.2. Scaling of Configurational Properties 45 2.4. Extended Corresponding States Models for Mixtures 47 2.4.1. Conformal Solution Theory 47 2.4.2. Scaling of Mixture Properties 50 2.5. Applications of Extended Corresponding States Models 53 References 59 Chapter 3. Thermodynamic Properties and the Speed of Sound 61 3.1. Introduction 61 3.2. Relationships between the Speed of Sound and Thermodynamic Properties. 61 2.2.1. Thermodynamic Speed of Sound 61 3.2.2. Perfect-gas Limit 64 3.2.3. Acoustic Virial Coefficients 65 3.3. Measurement of the Speed of Sound 66 3.3. L Acoustic Model 66 3.3.2. Dissipation Processes 68 3.3.3. Relaxation Phenomena 71 References 75 Chapter 4. Virial Coefficients 76 4.1. Introduction 76 4.2. The Virial Equation of State 76 4.3. Intermolecular Potential Functions 77 4.3. L Intermolecular Forces. 77 4.3.2. Total Intermolecular Potential Energy. 79 4.3.3. Spherical Pair-Potential Models for Nonpolar Molecules. 79 4.3.4. Intermolecular Potential of Non-Spherical Molecules. 81 4.4. Virial Coefficients 83 4.4.1. Second and Third Virial Coefficients in the Classical Limit. 83 4.4.2. Quantum Corrections. 85 4.4.3. Virial Coefficients for Mixtures 88 References 90 Chapter 5. Natural Gas Systems 91 5.1. Introduction 91 5.2. Natural Gas Characterisation 91 5.3. Equations of State for Natural Gas Systems 94 5.3.1. The GERG-88 Virial Equations of State 95 5.3.2. The AGA8-DC92 Equation of State 99 5.3.3. Extended Corresponding States Model 100 5.3.4. Multiparameter Corresponding States Model. 101 5.3.5. The NISTModel, NGAS 101 References 103 Chapter 6. Extended Corresponding States Equation of State for Natural Gas Systems 105 6.1. Introduction 105 6.2. Experimental Information 105 6.2.1. Ethane 106 6.2.2. Propane 108 6.2.3. Isobutane 108 6.2.4. Normal Butane 110 6.2.5. Nitrogen Ill 6.2.6. Carbon Dioxide 112 6.2.7. Carbon Monoxide 113 6.2.8. Normal Hydrogen 114 6.2.9. Normal Pentane 114 6.3. Reference Fluid and Equation of State 116 6.4. Calculation of Empirical Shape Factors from Volumetric Data 117 6.5. Thermodynamically-Consistent Extended Corresponding States Model 131 6.6. Independent Shape Factors Extended Corresponding States Model 137 6.7. Extension to Mixtures 139 6.7.1. Extension to Mixtures with Model 1 140 6.7.2. Extension to Mixtures with Model 2 142 6.8. Results for Pure Components 144 6.8.1. Results from Model 1 144 6.8.2. Result from Model 2 151 6.8.3. Generalised Ancillaiy Equations for Vapour-Liquid Equilibria Properties. .... 157 6.9. Results for Mixtures and Natural Gases 159 6.9. L Results for Mixtures 160 6.9.2. Results for Natural Gases. 165 6.10. Results for Vapour-Liquid Equilibria in Mixtures 172 6.11. Discussion 174 References 181 Chapter 7. Experimental Programme and Results 186 7.1. Introduction 186 7.2. Description of the Apparatus 186 7.2.1. Spherical Resonator. 186 7.2.2. Thermometers. 190 7.2.3. Pressure Transducers. 191 7.2.4. Electroacoustic Transducers. 193 7.3. Experimental Procedures 195 7.4. Tests on the Spherical Resonator 198 7.5. Calibration of the Resonator Zero-Pressure Radius 201 7.5.1. Results using Nitrogen 201 7.5.2. Results using Argon 202 7.6. Low-Density Measurements on Nitrogen 209 7.6.1. Speeds of Sound. 210 7.6.2. Acoustic Virial Coefficients. 214 7.6.3. Intermolecular Potential Models and Virial Coefficients. 215 7.7. Measurements on Mixtures Methane + Nitrogen 221 7.7.1. Speeds of Sound. 221 7.7.2. Mixture Compositions. 232 7.7.3. Acoustic Virial Coefficients. 233 7.7.4. Intermolecular Potential Models and Virial Coefficients. 235 7.8. Measurements on the Mixture (jcCH^ + (1 - a:)C02), with x « 0.8 244 7.8.1. Speeds of Sound. 245 7.8.2. Composition of the Mixture 249 7.8.3. Acoustic Virial Coefficients. 250 7.8.4. Intermolecular Potential Model and Virial Coefficients. 251 7.10. Comparison between Experimental and Computational Results 254 References 259 8 Chapter 8.