THERMODYNAMIC PROPERTIES FROM CUBIC EQUATIONS OF STATE by PATRICK CHUNG-NIN MAK B.A.Sc, The University of British Columbia, 1985 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1988 © PATRICK CHUNG-NIN MAK, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Cti&AiCAL- W SrZ-£l"J6 The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date /W - Z~ / f ABSTRACT The Lielmezs-Merriman equation of state has been modified in such a wa3' that it can be applied over the entire PVT surface except along the critical isotherm. The dimensionless T coordinate has been defined according to the two regions on the PVT surface as : T /T - 1 T = — for T < T Tc/Tb - 1 and * T/T - 1 T = for T > T VTb " 1 where Tc is the critical temperature and is the normal boiling point. The a-function is now given by the following : q a = 1 + p(T*) for T < Tc and a = 1 - p(T )q for T > T c The two substance-dependent constants p and q are generated from the vapor pressure data. The applicability of the proposed modification has been tested by comparing its predictions of various pure compound physical and thermodynamic properties with known experimental data and with predictions from the Soave-Redlich-Kwong and Peng-Robinson equations of state. The proposed equation ii is the most accurate equation of state for calculating vapor pressure, and saturated vapor and liquid volumes. The Peng-Robinson equation is the best for enthalp}' and entropy of vaporization estimations. The Soave-Redlich-Kwong equation is the least accurate equation for pressure and volume predictions in the single phase regions. For temperature prediction, all three equations of state give similar results in the subcritical and supercritical regions. None of the three equations is capable of representing all departure functions accurately. The Peng-Robinson equation and the proposed equation are very similar in accuracy except in the region where the temperature is near the critical. That is, between 0.95 < Tr ^ 1.05, the proposed equation gives rather poor results. For isobaric heat capacity calculation, both Soave-Redlich-Kwong and Peng-Robinson equations are adequate. The Soave-Redlich-Kwong equation gives the lowest overall average RMS % error for Joule-Thomson coefficient estimation. The Soave-Redlich-Kwong equation also provides the most reliable prediction for the Joule-Thomson inversion curve right up to the maximum inversion pressure. None of the cubic equations of state studied in this work is recommended for second virial coefficient calculation below Tr = 0.8. An a-function specifically designed for the calculation of second virial coefficient has been included in this work. The estimation from the proposed function gives equal, if not better, accuracy than the Tsonopoulos correlation. iii TABLE OF CONTENTS ABSTRACT ii LIST OF FIGURES vi LIST OF TABLES x ACKNOWLEDGEMENT xii Chapter 1. INTRODUCTION 1 Chapter 2. LITERATURE REVIEW 7 Chapter 3. DEVELOPMENT OF AN EQUATION OF STATE 19 3.1. Theoretical Considerations 19 3.2. Lielmezs-Merriman Equation of State 25 3.3. Modified Lielmezs-Merriman Equation of State 32 Chapter 4. APPLICATIONS OF THE CUBIC EQUATION OF STATE 45 4.1. Region I : Saturation 46 4.2. Region II : Subcritical 70 4.3. Region III : Supercritical 89 4.4. Region IV : Compressed Liquid 106 4.5. Inversion Curve 123 Chapter 5. SECOND VIRIAL COEFFICIENT 137 5.1. Introduction 137 5.2. Theoretical Background 139 5.3. A New Correlation 141 5.4. Comparison 145 CONCLUSIONS 150 RECOMMENDATIONS FOR FURTHER STUDY 153 NOMENCLATURE 154 REFERENCES 157 APPENDIX A 164 1. Critical Compressibility Factor 164 1.1. The Soave-Redlich-Kwong Equation 169 1.2. The Peng-Robinson Equation 169 1.3. This Work 170 2. Constants A and B 170 2.1. The Soave-Redlich-Kwong Equation 171 2.2. The Peng-Robinson Equation 171 2.3. This Work 172 3. Compressibility Factor Equation 172 iv 3.1. The Soave-Redlich-Kwong Equation 173 3.2. The Peng-Robinson Equation 173 3.3. This Work 173 4. Fugacity Coefficient 173 4.1. The Soave-Redlich-Kwong Equation 176 4.2. The Peng-Robinson Equation 176 4.3. This Work 177 5. Enthalpy Departure Function 177 5.1. The Soave-Redlich-Kwong Equation 179 5.2. The Peng-Robinson Equation 180 5.3. This Work 181 6. Heat Capacity Departure Function 183 6.1. The Soave-Redlich-Kwong Equation 189 6.2. The Peng-Robinson Equation 190 6.3. This Work 190 7. Joule-Thomson Coefficient 191 7.1. The Soave-Redlich-Kwong Equation 191 7.2. The Peng-Robinson Equation 192 7.3. This Work 192 8. Inversion Curve 192 8.1. The Soave-Redlich-Kwong Equation 195 8.2. The Peng-Robinson Equation 195 8.3. This Work 196 9. Virial Coefficients 197 9.1. The Soave-Redlich-Kwong Equation 199 9.2. The Peng-Robinson Equation 199 9.3. This Work 200 v LIST OF FIGURES Figure 1.1 Division of the Pressure-Volume surface 6 Figure 3.1 Maxwell-rule of equal areas 35 Figure 3.2 Algorithm for determining the substance-dependent constants p and q 36 Figure 3.3 Correlations of p with parameter s and with acentric factor u 37 Figure 3.4 Correlations of acentric factor u with parameter s 38 Figure 4.1 Algorithm for calculating saturation properties 61 Figure 4.2 Region I : Error distribution curves of methane 62 Figure 4.3 Region I : Error distribution curves of n-pentane 63 Figure 4.4 Region I : Error distribution curves of 1-propanol 64 Figure 4.5 Region I : Error distribution curves of argon 65 Figure 4.6 Region I : Enthalpy and entropy of vaporization of methane 66 Figure 4.7 Region I : Enthalpy and entropy of vaporization of n-pentane 67 Figure 4.8 Region I : Enthalpj' and entropy of vaporization of 1-propanol 68 Figure 4.9 Region I : Enthalpy and entropy of vaporization of argon 69 Figure 4.10 Region II : Compressibility factors of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature 82 Figure 4.11 Region II : Compressibility factors of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure 82 Figure 4.12 Region II : Reduced enthalpy departure functions of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature 83 Figure 4.13 Region II : Reduced enthalpy departure functions of i-butane (Tr = 0.98) and water (Tr = 0.96) versus • reduced pressure 83 Figure 4.14 Region II : Reduced entropy departure functions of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature 84 Figure 4.15 Region II : Reduced entropy departure functions of i-butane (Tr=0.98) and water (Tr = 0.96) versus reduced pressure 84 vi Figure 4.16 Region II : Reduced Helmholtz departure functions of i-butane (Pr=0.56) and water (Pr = 0.31) versus reduced temperature 85 Figure 4.17 Region II : Reduced Helmholtz departure functions of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure 85 Figure 4.18 Region II : Reduced Gibbs energy departure functions of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature 86 Figure 4.19 Region II : Reduced Gibbs energy departure functions of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure 86 Figure 4.20 Region II : Reduced internal energy departure functions of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature 87 Figure 4.21 Region II : Reduced internal energy departure functions of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure 87 Figure 4.22 Region II : Fugacity coefficients of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature 88 Figure 4.23 Region II : Fugacit3' coefficients of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure 88 Figure 4.24 Region III : Compressibility factors of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature 99 Figure 4.25 Region III : Compressibility factors of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure 99 Figure 4.26 Region III : Reduced enthalpy departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature 100 Figure 4.27 Region III : Reduced enthalpy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure 100 Figure 4.28 Region III : Reduced entropy departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature 101 Figure 4.29 Region III : Reduced entropy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure 101 Figure 4.30 Region III : Reduced Helmholtz departure functions of n-octane (Pr=4.03) and ethanol (Pr=1.63) versus reduced temperature 102 Figure 4.31 Region III : Reduced Helmholtz departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure 102 Figure 4.32 Region III : Reduced Gibbs energy departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature.